politecnico di milano · abstract lameness detection and quantification in horses is an important,...
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Politecnico di Milano
SCUOLA DI INGEGNERIA INDUSTRIALE E DELL’INFORMAZIONE
Corso di Laurea Magistrale in Ingegneria Matematica
Tesi di Laurea Magistrale
Permutational inference for functional-on-scalarlinear mixed-effect models applied to equine
accelerometric data
Relatore
Prof. Simone Vantini
Correlatore
Ing. Alessia Pini
Candidato
Alessia SerafiniMatr. 819016
Anno Accademico 2014–2015
“When you have eliminated the impossible,whatever remains, however improbable,
must be the truth.”
— Sir Arthur Conan Doyle, “The Sign of the Four”
Alessia Serafini: Permutational inference for functional-on-scalar linear mixed-effect models applied to equine accelerometric data | Tesi di Laurea Magistrale inIngegneria Matematica, Politecnico di Milano.c© Copyright Aprile 2016.
Politecnico di Milano:www.polimi.it
Scuola di Ingegneria Industriale e dell’Informazione:www.ingindinf.polimi.it
Abstract
Lameness detection and quantification in horses is an important, but oftendifficult, task for veterinarians. For this reason, many instrumented methodshave been developed to offer objective measurements as a support to clinicalvisual examinations. In this Thesis, kinematic data consisting of three-dimensionalacceleration signals of eight horses are studied. Each horse was measured ninetimes: before induction of lameness (sound) and after induction of two degrees oflameness in each of the four legs by means of a modified horseshoe with a screweliciting pressure on the sole of the hoof. The aim of this work is to investigateif and how the location of the lameness influences the shape of the observedcurves. To achieve this goal, a functional-on-scalar linear mixed-effect model forthe acceleration signals has been proposed. Techniques coming from FunctionalData Analysis and non-parametric inference have been exploited and combined,giving birth to a new approach to test for significance of fixed and random effectsbased on permutation procedures. Furthermore, identification of portions of thedomain imputable of rejecting the null hypothesis is made possible through theapplication of the Interval-Wise Testing method, which relies on the definition ofan adjusted p-value function able to control the interval-wise error rate. Resultsshow that different lame limbs have different effects on vertical acceleration curves.Instead, transversal and longitudinal accelerations of horses with lame hindlimbscannot be easily distinguished from signals of a sound horse.
KEYWORDS: horses; lameness; 3D accelerations; functional data analysis; non-parametric inference; permutation test; functional linear mixed model; functionalmixed-effect ANOVA.
v
Sommario
Rilevare e quantificare il livello di zoppia nei cavalli è un compito molto importan-te, ma spesso difficile, per i veterinari. Per questa ragione, molti metodi sono statisviluppati per offrire delle misurazioni oggettive come supporto all’esame clinico,che consiste in una valutazione visiva dei movimenti dell’animale. In questa tesivengono studiati dei dati cinematici, nello specifico i segnali tridimensionali relativialle misurazioni delle accelerazioni di otto cavalli. Ogni cavallo è stato misuratonove volte: prima dell’induzione della zoppia (cavallo sano) e dopo l’induzione didue diversi gradi di zoppia in ognuna delle quattro zampe, tramite un apposito ferrodi cavallo munito di una vite che provoca una pressione sulla suola dello zoccolo.L’obiettivo di questo lavoro è quello di esaminare se e come la posizione della zoppiainfluenza la forma delle curve di accelerazione osservate. Per raggiungere questoobiettivo, un modello lineare a effetti misti per risposte funzionali e covariate scalariè stato proposto. Varie tecniche provenienti dall’ambito della Functional DataAnalysis e da quello dell’inferenza non parametrica sono state sfruttate e combinate,dando vita a un nuovo metodo per testare la significatività di effetti fissi e casualitramite procedure permutazionali. Inoltre, la selezione degli intervalli del dominioresponsabili del rifiuto dell’ipotesi nulla è resa possibile attraverso l’utilizzo delmetodo Interval-Wise Testing, che si basa sulla definizione di una funzione p-valueaggiustata in grado di controllare l’interval-wise error rate. I risultati mostranoche ogni zampa infortunata provoca un diverso effetto nelle curve di accelerazioneverticale. Invece, le accelerazioni trasversali e longitudinali di cavalli con una zoppianelle zampe posteriori non sono distinguibili da quelle di un cavallo sano.
PAROLE CHIAVE: cavalli; zoppia; accelerazioni 3D; analisi di dati funzionali;inferenza non-parametrica; test permutazionali; modelli lineari funzionali a effettimisti; ANOVA funzionale a effetti misti.
vii
Contents
Introduction 1
1 State of the Art 31.1 Equine Lameness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 Lameness diagnosis . . . . . . . . . . . . . . . . . . . . . . . 51.1.3 Quantitative methods for lameness identification . . . . . . . 6
1.2 Permutational inference for functional data . . . . . . . . . . . . . . 91.2.1 Inference for functional-on-scalar linear models . . . . . . . . 11
2 Methodology 172.1 Interval-Wise Testing for functional data . . . . . . . . . . . . . . . 17
2.1.1 Functional IWT procedure . . . . . . . . . . . . . . . . . . . 182.2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.1 Functional-on-scalar linear model . . . . . . . . . . . . . . . 212.2.2 Functional-on-scalar linear mixed model . . . . . . . . . . . 222.2.3 A particular case: one-way FANOVA . . . . . . . . . . . . . 24
2.3 Testing for fixed effects . . . . . . . . . . . . . . . . . . . . . . . . . 252.3.1 Application of IWT procedure to test fixed effects . . . . . . 26
2.4 Testing for random effects . . . . . . . . . . . . . . . . . . . . . . . 282.4.1 Application of IWT procedure to test a random effect . . . . 29
2.5 Freedman and Lane permutation scheme . . . . . . . . . . . . . . . 30
3 Dataset 333.1 Data acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.1 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2 Preprocessing of data . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3 Different representations of data . . . . . . . . . . . . . . . . . . . . 35
3.3.1 Two-peaks representations . . . . . . . . . . . . . . . . . . . 353.3.2 Difference between peaks representations . . . . . . . . . . . 37
3.4 Covariates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.4.1 lameLeg and lameForeHind . . . . . . . . . . . . . . . . . . 413.4.2 lameDegree . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.4.3 horse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
ix
x CONTENTS
4 Results of test for the fixed effect 474.1 Results of the Original Datasets with fixed effect lameLeg . . . . . . 49
4.1.1 VERTICAL Original Dataset . . . . . . . . . . . . . . . . . 494.1.2 TRANSVERSAL Original Dataset . . . . . . . . . . . . . . 534.1.3 LONGITUDINAL Original Dataset . . . . . . . . . . . . . . 57
4.2 Results of the Modified Datasets with fixed effect lameForeHind . . 614.2.1 VERTICAL Modified Dataset . . . . . . . . . . . . . . . . . 614.2.2 TRANSVERSAL Modified Dataset . . . . . . . . . . . . . . 654.2.3 LONGITUDINAL Modified Dataset . . . . . . . . . . . . . 68
4.3 Results of the Modified Datasets with fixed effect lameForeHind_Degree 724.3.1 VERTICAL Modified Dataset . . . . . . . . . . . . . . . . . 724.3.2 TRANSVERSAL Modified Dataset . . . . . . . . . . . . . . 764.3.3 LONGITUDINAL Modified Dataset . . . . . . . . . . . . . 80
5 Results of test for the random effect 855.1 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.2 Results of the Original Datasets with fixed effect lameLeg . . . . . . 86
5.2.1 VERTICAL Original Dataset . . . . . . . . . . . . . . . . . 865.2.2 TRANSVERSAL Original Dataset . . . . . . . . . . . . . . 885.2.3 LONGITUDINAL Original Dataset . . . . . . . . . . . . . . 90
5.3 Results of the Modified Datasets with fixed effect lameForeHind . . 925.3.1 VERTICAL Modified Dataset . . . . . . . . . . . . . . . . . 925.3.2 TRANSVERSAL Modified Dataset . . . . . . . . . . . . . . 945.3.3 LONGITUDINAL Modified Dataset . . . . . . . . . . . . . 96
5.4 Results of the Modified Datasets with fixed effect lameForeHind_Degree 985.4.1 VERTICAL Modified Dataset . . . . . . . . . . . . . . . . . 985.4.2 TRANSVERSAL Modified Dataset . . . . . . . . . . . . . . 1005.4.3 LONGITUDINAL Modified Dataset . . . . . . . . . . . . . 102
6 Conclusions 105
A Graphical results of test for the fixed effect of Peak1 - Peak2Datasets 109A.1 Results of the Original Peak1 - Peak2 Datasets with fixed effect
lameLeg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110A.2 Results of the Modified Peak1 - Peak2 Datasets with fixed effect
lameForeHind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117A.3 Results of the Modified Peak1 - Peak2 Datasets with fixed effect
lameForeHind_Degree . . . . . . . . . . . . . . . . . . . . . . . . . 124
B Graphical results of test for the random effect of Peak1 - Peak2Datasets 131B.1 Results of the Original Peak1 - Peak2 Datasets with fixed effect
lameLeg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132B.2 Results of the Modified Peak1 - Peak2 Datasets with fixed effect
lameForeHind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
CONTENTS xi
B.3 Results of the Modified Peak1 - Peak2 Datasets with fixed effectlameForeHind_Degree . . . . . . . . . . . . . . . . . . . . . . . . . 140
Bibliography 145
List of Figures
1.1 Trotting horse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.1 Comparison between Original and Modified Datasets . . . . . . . . 373.2 Comparison between Original and Modified Peak1-Peak2 Datasets . 393.3 lameLeg groups in the Original Datasets . . . . . . . . . . . . . . . 423.4 lameForeHind groups in the Modified Datasets . . . . . . . . . . . . 433.5 Vertical Original accelerations of the eight horses involved in the
experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1 Results of the four different tests for significance of the fixed effectlameLeg in the Vertical Original Dataset . . . . . . . . . . . . . . . 51
4.2 Results of the restricted - LM pairwise test for significance of thefixed effect lameLeg in the Vertical Original Dataset . . . . . . . . 52
4.3 Results of the four different tests for significance of the fixed effectlameLeg in the Transversal Original Dataset . . . . . . . . . . . . . 55
4.4 Results of the restricted - LM pairwise test for significance of thefixed effect lameLeg in the Transversal Original Dataset . . . . . . 56
4.5 Results of the four different tests for significance of the fixed effectlameLeg in the Longitudinal Original Dataset . . . . . . . . . . . . 59
4.6 Results of the restricted - LM pairwise test for significance of thefixed effect lameLeg in the Longitudinal Original Dataset . . . . . . 60
4.7 Results of the four different tests for significance of the fixed effectlameForeHind in the Vertical Modified Dataset . . . . . . . . . . . 63
4.8 Results of the restricted - LM pairwise test for significance of thefixed effect lameForeHind in the Vertical Modified Dataset . . . . 64
4.9 Results of the four different tests for significance of the fixed effectlameForeHind in the Transversal Modified Dataset . . . . . . . . . 66
4.10 Results of the restricted - LM pairwise test for significance of thefixed effect lameForeHind in the Transversal Modified Dataset . . 67
4.11 Results of the four different tests for significance of the fixed effectlameForeHind in the Longitudinal Modified Dataset . . . . . . . . 70
4.12 Results of the restricted - LM pairwise test for significance of thefixed effect lameForeHind in the Longitudinal Modified Dataset . . 71
4.13 Results of the four different tests for significance of the fixed effectlameForeHind_Degree in the Vertical Modified Dataset . . . . . . 74
4.14 Results of the restricted - LM pairwise test for significance of thefixed effect lameForeHind_Degree in the Vertical Modified Dataset 75
xiii
xiv LIST OF FIGURES
4.15 Results of the four different tests for significance of the fixed effectlameForeHind_Degree in the Transversal Modified Dataset . . . . 78
4.16 Results of the restricted - LM pairwise test for significance of the fixedeffect lameForeHind_Degree in the Transversal Modified Dataset 79
4.17 Results of the four different tests for significance of the fixed effectlameForeHind in the Longitudinal Modified Dataset . . . . . . . . 82
4.18 Results of the restricted - LM pairwise test for significance of thefixed effect lameForeHind_Degree in the Longitudinal ModifiedDataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.1 Results of the test for significance of the random effect horse in theVertical Original Dataset . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2 Results of the test for significance of the random effect horse in theTransversal Original Dataset . . . . . . . . . . . . . . . . . . . . . . 89
5.3 Results of the test for significance of the random effect horse in theLongitudinal Original Dataset . . . . . . . . . . . . . . . . . . . . . 91
5.4 Results of the test for significance of the random effect horse in theVertical Modified Dataset . . . . . . . . . . . . . . . . . . . . . . . 93
5.5 Results of the test for significance of the random effect horse in theTransversal Modified Dataset . . . . . . . . . . . . . . . . . . . . . 95
5.6 Results of the test for significance of the random effect horse in theLongitudinal Modified Dataset . . . . . . . . . . . . . . . . . . . . . 97
5.7 Results of the test for significance of the random effect horse in theVertical Modified Dataset . . . . . . . . . . . . . . . . . . . . . . . 99
5.8 Results of the test for significance of the random effect horse in theTransversal Modified Dataset . . . . . . . . . . . . . . . . . . . . . 101
5.9 Results of the test for significance of the random effect horse in theLongitudinal Modified Dataset . . . . . . . . . . . . . . . . . . . . . 103
A.1 Results of the four different tests for significance of the fixed effectlameLeg in the Vertical Original Peak1-Peak2 Dataset . . . . . . . 111
A.2 Results of the restricted - LM pairwise test for significance of thefixed effect lameLeg in the Vertical Original Peak1-Peak2 Dataset . 112
A.3 Results of the four different tests for significance of the fixed effectlameLeg in the Vertical Original Peak1-Peak2 Dataset . . . . . . . 113
A.4 Results of the restricted - LM pairwise test for significance of thefixed effect lameLeg in the Transversal Original Peak1-Peak2 Dataset114
A.5 Results of the four different tests for significance of the fixed effectlameLeg in the Longitudinal Original Peak1-Peak2 Dataset . . . . . 115
A.6 Results of the restricted - LM pairwise test for significance of thefixed effect lameLeg in the Longitudinal Original Peak1-Peak2 Dataset116
A.7 Results of the four different tests for significance of the fixed effectlameForeHind in the Vertical Modified Peak1-Peak2 Dataset . . . 118
A.8 Results of the restricted - LM pairwise test for significance of thefixed effect lameForeHind in the Vertical Modified Peak1-Peak2Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
LIST OF FIGURES xv
A.9 Results of the four different tests for significance of the fixed effectlameForeHind in the Vertical Modified Peak1-Peak2 Dataset . . . 120
A.10 Results of the restricted - LM pairwise test for significance of thefixed effect lameForeHind in the Transversal Modified Peak1-Peak2Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
A.11 Results of the four different tests for significance of the fixed effectlameForeHind in the Longitudinal Modified Peak1-Peak2 Dataset 122
A.12 Results of the restricted - LM pairwise test for significance of the fixedeffect lameForeHind in the Longitudinal Modified Peak1-Peak2Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
A.13 Results of the four different tests for significance of the fixed effectlameForeHind_Degree in the Vertical Modified Peak1-Peak2 Dataset125
A.14 Results of the restricted - LM pairwise test for significance of the fixedeffect lameForeHind_Degree in the Vertical Modified Peak1-Peak2Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
A.15 Results of the four different tests for significance of the fixed effectlameForeHind_Degree in the Vertical Modified Peak1-Peak2 Dataset127
A.16 Results of the restricted - LM pairwise test for significance of thefixed effect lameForeHind_Degree in the Transversal ModifiedPeak1-Peak2 Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . 128
A.17 Results of the four different tests for significance of the fixed effectlameForeHind_Degree in the Longitudinal Modified Peak1-Peak2Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
A.18 Results of the restricted - LM pairwise test for significance of thefixed effect lameForeHind_Degree in the Longitudinal ModifiedPeak1-Peak2 Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . 130
B.1 Results of the test for significance of the random effect horse in theVertical Original Peak1-Peak2 Dataset . . . . . . . . . . . . . . . . 133
B.2 Results of the test for significance of the random effect horse in theTransversal Original Peak1-Peak2 Dataset . . . . . . . . . . . . . . 134
B.3 Results of the test for significance of the random effect horse in theLongitudinal Original Peak1-Peak2 Dataset . . . . . . . . . . . . . 135
B.4 Results of the test for significance of the random effect horse in theVertical Modified Peak1-Peak2 Dataset . . . . . . . . . . . . . . . . 137
B.5 Results of the test for significance of the random effect horse in theTransversal Modified Peak1-Peak2 Dataset . . . . . . . . . . . . . . 138
B.6 Results of the test for significance of the random effect horse in theLongitudinal Modified Peak1-Peak2 Dataset . . . . . . . . . . . . . 139
B.7 Results of the test for significance of the random effect horse in theVertical Modified Peak1-Peak2 Dataset . . . . . . . . . . . . . . . . 141
B.8 Results of the test for significance of the random effect horse in theTransversal Modified Peak1-Peak2 Dataset . . . . . . . . . . . . . . 142
B.9 Results of the test for significance of the random effect horse in theLongitudinal Modified Peak1-Peak2 Dataset . . . . . . . . . . . . . 143
List of Tables
2.1 Methods to test fixed and random effects. . . . . . . . . . . . . . . 31
3.1 Categorical variables relative to horse B1 . . . . . . . . . . . . . . . 40
4.1 Methods to test fixed effects. . . . . . . . . . . . . . . . . . . . . . . 47
xvii
List of Acronyms
AAEP American Association of Equine Practitioners
ANOVA ANalysis Of VAriance
BLUP Best Linear Unbiased Predictors
FDA Functional Data Analysis
FWER Family Wise Error Rate
GLMM Generalized Linear Mixed Model
GRF Ground Reaction Force
ITP Interval Testing Procedure
IWER Interval Wise Error Rate
IWT Interval Wise Testing
LF Left Fore
LH Left Hind
LM Linear Model
LMM Linear Mixed Model
LR Likelihood Ratio
ML Maximum Likelihood
OLS Ordinary Least Squares
PVF Peak Vertical Force
REML REstricted Maximum Likelihood
RF Right Fore
RH Right Hind
xix
Introduction
In recent years, the growing interest in horses for racing and riding activities has
stimulated scientific research in equine locomotion. The most frequently reported
health issue affecting horses is lameness, an alteration of the gait pattern which can
be caused by either a structural or a functional disorder of the locomotor system.
This physical problem can be devastating to the athletic performance of the horse,
since it may force the withdrawal of the animal from sports competitions. Attempts
to avoid a great economical loss due to lameness and difficulties in establishing a
diagnosis, have led to the development of a great number of instrumented methods
for equine gait analysis and lameness quantification.
Motivated by all of these facts, data collected at the University of Copenhagen
and consisting of 3D acceleration signals of trotting horses are the object of the
analyses presented in this Thesis. Eight horses were led by hand in trot over a
distance of 24 m at constant velocity and were equipped with an accelerometer
placed at the lowest point of the back in the midline. The accelerometer measured
the three-dimensional accelerations of trunk movements of each horse nine times:
before induction of lameness (sound) and after mechanical induction of two degrees
(low and moderate) of lameness in each of the four legs.
The aim of this work is to analyse these data to investigate how lameness
influences the vertical, transversal and longitudinal component of the 3D records.
Each of the signal components consists of a series of discrete observations from
a continuous-time signal. As a consequence, they can be regarded as functions
of time and can be studied with the tools provided by Functional Data Analysis
(FDA), a lively research area in statistics. As a first step, a functional-on-scalar
linear mixed-effect model for the data is developed, i.e. a linear model characterised
by a functional response, in this case represented by one of the three components
of the acceleration vector, and scalar fixed and random effects. In our model, the
1
2 INTRODUCTION
label identifying the lame limb is treated as a fixed effect, while the horse’s ID is
introduced in the model as a random intercept in order to take into account the
variation induced in the acceleration pattern due to each horse’s particular gait.
After that, we are interested in seeing if these effects are statistically significant in
our model and most of all, we want to select the intervals of the domain in which
they are significant.
In the field of FDA, the development of suitable inferential techniques is still a
recent and challenging topic of high importance for practitioners. This problem
is currently addressed from a parametric or a non-parametric perspective. The
former relies on distributional assumptions (e.g. homoscedasticity, normality,
regular exponential family, random sampling, etc.) and/or asymptotic results
which could be unrealistic or assumed for mere convenience. For example, in the
case of functional data, normality is a very demanding assumption that proves to
be practically impossible to verify. For this reason, in this work we opt for the
non-parametric approach, generally based on computational intensive permutation
or bootstrap techniques.
Testing for significance of fixed and random effects by means of permutational
approaches is an element of novelty when dealing with functional linear mixed
models. In fact this approach has not been explored yet in other works.
To detect the portions of the domain imputable of the rejection of the null
hypothesis, we make use of a purely non-parametric inferential procedure called
Interval-Wise Testing (IWT), that introduces an adjusted p-value function able
to control the interval-wise error rate, in detail given any interval of the domain
where the null hypothesis is not violated, the probability of wrongly selecting it is
controlled.
In this Thesis, all these methods are explained in detail and then applied to
our dataset. In particular this work is structured as follows: Chapter 1 presents a
review of literature regarding both equine lameness identification and inferential
methods based on permutations. Chapter 2 describes the different methodological
aspects used to make inference on our functional data. Chapter 3 contains the
description of the dataset and finally, in Chapter 4 and Chapter 5 the results of our
analyses as applications of the methodologies described in Chapter 2 are presented.
Conclusions are reported in Chapter 6.
Chapter 1
State of the Art
1.1 Equine Lameness
During the traditional clinical lameness examination, the veterinarian evaluates
the locomotion pattern of the horse at trot to score subjectively the severity of
the lameness, to identify the lame limb and to hypothesise the anatomical location
and the severity of the locomotor injury. Recognition of lameness is a key skill to
successful diagnosis, but even for an expert orthopaedic, the correct identification
of the lame limb is not easily achievable, especially when the lameness degree is not
severe and the clinical symptoms are not displayed. Besides, due to the subjectivity
of the evaluation, different clinicians may give different diagnosis. Consequently the
use of mathematical and statistical tools is an important aspect to allow lameness
quantification and support the analysis of the clinicians.
1.1.1 Terminology
Some technical terms related to the horses’ movements are defined (Barrey,
1999) in this Section, in order to allow a better comprehension of the following
analysis and results.
• A gait is a complex and strictly coordinated rhythmic and automatic move-
ment of the limbs and the entire body of the animal which result in the
production of progressive movements. Two types of gait can be distinguished
by the symmetry or asymmetry of the limb movement sequence with respect
to time and the median plane of the horse:
3
4 CHAPTER 1. STATE OF THE ART
Figure 1.1: Trotting horse: stance phase of the right diagonal, suspension phase, stancephase of the left diagonal, suspension phase.
– symmetric gaits: walk, tölt, pace, trot;
– asymmetric gaits: canter, gallop.
• The stride is a full cycle of limb motion. Since the pattern is repeated, the
beginning of the stride can be at any point in the pattern and the end of that
stride at the same place in the beginning of the next pattern.
• A complete gait cycle includes a stance phase when the limb is in contact
with the ground and swing phase when the limb is not in contact with the
ground. During the suspension phase at trot, pace, canter or gallop, there
is no hoof contact with the ground.
Among the large variety of gaits a horse can perform, lameness detection is
usually made at trot.
The trot is a two-beat diagonal gait of the horse where the diagonal pairs of legs
move forward at the same time with a moment of suspension between each beat.
As shown in Figure 1.1, there is a stance phase of the right-fore and left-hindlimb
and then a stance phase of the left-fore and right-hindlimb, each one followed by a
suspension phase. In the following right-fore and left-hindlimb will be referred to
as right diagonal or RF-LH diagonal, while the left-fore and right-hindlimb as
left diagonal or LF-RH.
The fact that trot is the fastest symmetrical gait (on average about 8 miles per hour
- 13 km/h) makes it the most suitable one for lameness examination and assessment.
1.1. EQUINE LAMENESS 5
1.1.2 Lameness diagnosis
Lameness is an alteration of the horse’s gait due to pain and can be manifested
as a change in attitude or performance.
The veterinarians usually watch the horse walking and trotting in a straight line
on a flat, hard surface (this provides an opportunity to listen to the footfall and to
consider this information along with the visual appraisal). The horse is observed
from the front, back and both side views, so that any deviations in gait (such as
winging or paddling), failure to land squarely on all four feet and the unnatural
shifting of weight from one limb to another can be noted. The detection of the
compensatory movements performed by a horse to adapt to lameness is an integral
part of the diagnosis. The most consistent sign of a unilateral forelimb lameness is
the head nod. The head and neck of the horse rise when the lame forelimb strikes
the ground and is weight bearing, and fall when the sound limb strikes the ground.
The sacral (pelvic) rise is the most easily observed sign of hindlimb lameness. The
entire pelvis and sacrum rise when the lame limb strikes the ground and is weight
bearing, and fall when the sound limb strikes the ground. Both head nod and sacral
rise serve to reduce concussion on the lame limb (Adams and Stashak’s, 2011).
The horse also walks and trots in circles, on a lunge, in a round pen and under
saddle. The veterinarians look for signs, such as shortening of the stride, irregular
foot placement, head bobbing, stiffness, weight shifting, etc.. Both forelimb and
hindlimb lameness may become worse when the horse is circled; most of the time,
the lameness is accentuated when the affected limb is on the inside of the circle.
Evaluating the physical condition can be challenging since more subtle lameness
causes fewer compensatory changes, making lameness diagnosis more difficult, and
problems within the axial skeleton (back) may also alter the movement of the limbs.
Moreover lameness may appear as a slight shortening of the stride, or the condition
may be so severe that the horse will not bear weight on the affected limb. With such
extremes of lameness possible, a lameness grading system has been developed by the
American Association of Equine Practitioners (AAEP) to aid both communication
and record-keeping.
Description of the six levels of the AAEP scoring system, as reported on their
website (http://www.aaep.org/):
• 0: Lameness is not perceptible under any circumstances.
6 CHAPTER 1. STATE OF THE ART
• 1: Lameness is difficult to observe and is not consistently apparent, regardless
of circumstances (e.g. under saddle, circling, inclines, hard surface, etc.).
• 2: Lameness is difficult to observe at a walk or when trotting in a straight line
but consistently apparent under certain circumstances (e.g. weight-carrying,
circling, inclines, hard surface, etc.).
• 3: Lameness is consistently observable at a trot under all circumstances.
• 4: Lameness is obvious at a walk.
• 5: Lameness produces minimal weight bearing in motion and/or at rest or a
complete inability to move.
Even if using this scale helps the clinicians to express their diagnosis, the visual
identification of a lame limb depends also on some characteristics of the observing
subject (his experience and training, his personal judgment parameters, etc.) and
these may lead to disagreeing diagnosis given by different examiners. This fact
is confirmed by the analysis carried out by Thomsen et al., 2010a, in which two
experienced equine practitioners independently scored lameness based on video
recordings of trotting horses using the AAEP scale: the percentage of agreements
was 70%, while the percentages of one- and 2-point disagreements were 23.3 and
6.7%, respectively.
Since the visual examination of the physical condition of a horse presents so
many difficulties, it would be helpful for veterinarians to support their clinical
judgment with the application of objective instrumented methods. Some of these
procedures will be presented in the following Section (1.1.3).
1.1.3 Quantitative methods for lameness identification
The first experimental measurements on horse’s locomotion were undertaken
independently by Marey and Muybridge at the end of 19th century (Barrey, 1999).
Since then, many techniques of lameness identification and quantification have been
developed to provide an objective analysis of horses’ gait by means of statistical
and mathematical tools. In literature several approaches have been explored in
order to examine how the gait pattern of horses is affected by the presence of a
lame limb: some of them are reported below.
1.1. EQUINE LAMENESS 7
Abnormalities in gait can be recorded with kinetic equipment: external forces
can be measured through electronic force sensors that record the ground reaction
forces when the hooves are in contact with the ground. The sensors can be installed
either on the ground in a force plate or in a force shoe device, attached under
the hoof. When a body stands on or moves across a force plate, this measuring
instrument can provide the force amplitude and orientation (vector coordinates in
three dimensions), the coordinates of the application point of the force and the
moment value at this point. In Back et al., 2007 a floor-mounted force plate was
used to determine peak vertical force (PVF) expressed in percent body weight (N
force/N body weight); a lameness score was visually assigned to each horse using
the AAEP lameness scale described above. The relationship between the PVF and
the AAEP score was explored with a regression analysis: the results pointed out
that horses with a front lame limb showed a decrease in front-limb loadings. The
paper also demonstrate that interbreed difference should be taken into account
when comparing specific grades of lameness of two groups of horses. While the
force platform has a fixed location, the force shoe, directly fixed on the horse hoof,
allows the evaluation of the ground reaction in various types of exercise; the main
disadvantage of this device is that it adds weight and thickness to the horse’s regular
hoof. Furthermore, force plates recently have been combined with pressure plates,
which allows new possibilities to study not only balance in conformation and gait,
but also hoof balance.
Evaluation of the asymmetry in head and trunk movements has a major role in
lameness examination. Because of Newton’s second law and the fact that trunk
contains most body mass, asymmetries in head and trunk movements allow lame
horses to reduce the vertical ground reaction force in their painful limb: these are
called compensatory movements. Therefore several methods have been developed to
quantify objectively the severity of equine lameness using head and trunk kinematic
data: Fourier analysis of kinematic variables has proved to be an effective tool
for the study of cyclic live motion pattern since it separates the symmetrical and
the asymmetrical parts. Usually these data are collected using skin markers and
cameras to record their motion, such as in Peham et al., 1996 and in Audigié et al.,
2002. In the former one marker is placed on the head and the other one on the
lateral hoof wall of one forelimb (hindlimb lameness is not taken into account in this
8 CHAPTER 1. STATE OF THE ART
work). In the latter the analysis of the symmetry of movements of the cranial and
caudal parts of the trunk are combined: to distinguish fore- and hindlimb lameness,
it could be hypothesized that the symmetry of trunk displacements is more altered
in the cranial part of the trunk in forelimb lamenesses and in the caudal part of
the trunk in hindlimb lamenesses. In order to detect also a lame hindlimb, four
retroreflective skin markers were placed on the dorsal midline of the trunk of each
horse and one additional marker was glued to the dorsolateral wall of each fore
hoof.
Not only the peak vertical displacement, but also velocity and acceleration
of head, withers, tuber sacrale and both tuber coxae were quantified at different
phases of the stride in Buchner et al., 1996. Changes in these variables due to
lameness and symmetry indices calculated as quotients of the values during the
lame and nonlame stance phase were analysed using a two-way analysis of variance.
This work confirmed that during both fore- and hindlimb lameness at the trot, the
vertical velocity of the trunk at impact of the lame limb decreased, during the lame
stance phase the trunk was kept higher above the ground, maximal acceleration
decreased and displacement amplitude was smaller than without lameness.
Compared to other approaches, measuring accelerations offers great advantages
such as low cost and simple instrumentation which allows the collection of measure-
ment both in laboratory or in field conditions, because all equipment is mounted on
the horse. Furthermore, since reaction forces take place on the level of second-order
derivatives, acceleration measurements are expected to be more informative about
the locomotive apparatus. This type of analysis can be found in papers like Sørensen
et al., 2012; Thomsen et al., 2010a,b, in which vertical accelerations are smoothed
to a Fourier series expansion and then registered. The obtained signals are used to
compute different symmetry scores which take into account different aspects, for
example the overall symmetry of the acceleration, the symmetry of loading of the
left and right diagonals and the difference in phase of the two diagonals. Symmetry
scores from PCA are also explored in Sørensen et al., 2012. Another example of
analysis of accelerometric data with symmetry indices can be found in Uhlir et al.,
1997, where the head acceleration asymmetry (HAAS) and the sacrum acceleration
asymmetry (SAAS) were used for quantification of fore- and hindlimb lameness
respectively.
1.2. PERMUTATIONAL INFERENCE FOR FUNCTIONAL DATA 9
The data analysed in this Thesis consist in all the three components of the
acceleration vector (as functions of time), hence they set in this last contest. For a
detailed description of the dataset see Chapter 3.
1.2 Permutational inference for functional data
In this Section mathematical and statistical tools regarding the analysis of
functional data are introduced in order to provide a background before approaching
the next chapters.
In the recent decades, the development of more and more precise acquisition
devices in many research areas has led to the recording of high resolution, and so
high dimensional, data (“large p, small n” problems). Functional Data Analysis
(FDA) fits in this framework: data are no longer p-dimensional vectors, but functions
belonging to an infinite dimensional separable Hilbert space, for instance the L2
space, which is the natural extension of the Euclidean geometry to the FDA
(Ramsay and Silverman, 2002, 2005). Although several techniques to explore this
kind of data are available, how to perform inference is still a debated issue in this
field. The major problem for the analysis of these data is represented by the fact
that many classical multivariate inferential tools (e.g., Hotelling’s theorem) are
not straightforwardly generalizable in the FDA framework, since they require the
number of sample units to be greater than the dimension of the space in which data
are observed. Consequently, the growing interest for the analysis of functional data
is urging the development of inferential techniques suited for any value of n and p.
One way to deal with these issues is the use of methods based on paramet-
ric inference, which relies on distributional assumptions (e.g. homoscedasticity,
normality, regular exponential family, random sampling, etc.) and/or asymptotic
results (Cuevas et al., 2004; Horváth and Kokoszka, 2012). This approach, however,
implies assumptions that could be unrealistic, unclear or practically impossible to
verify. In contrast, non-parametric approaches commonly rely on computational
intensive permutation or bootstrap techniques and try to keep assumptions at a
lower workable level, avoiding those that are difficult to justify or interpret, possibly
without excessive loss of inferential efficiency.
Although the first description of permutation tests can be traced back to the first
10 CHAPTER 1. STATE OF THE ART
half of the 20th century (Fisher, 1935), it is only in recent years, with the increase
in computer performances, that they have begun to be exploited in applications.
A permutation test is essentially based on a family of transformations which
preserve the likelihood under the null hypothesis H0 (i.e., it is satisfied the so called
exchangeability condition under the null hypothesis) and on a suitable test statistic
T which is stochastically larger under the alternative hypothesis H1 than under
H0 (Pesarin and Salmaso, 2010). Under the null hypothesis H0, all the possible
rearrangements or permutations of the data which satisfy the exchangeability
condition are equally likely. Hence, if the null hypothesis H0 is true, the discrete
distribution of the test statistic applied on the permuted data T ∗ can be computed
by exploring all the possible permutations of the observed data. The p-value of the
test is given by the proportion of permuted scenarios in which the test statistic
evaluated on the permutations T ∗ is greater than or equal to the value of the test
statistic applied on the observed data T0. As the amount of data increases, the exact
enumeration of all permutations becomes computationally unfeasible. Therefore
only a subset of the permutations is explored through a Conditional Monte Carlo
algorithm, generating an approximated distribution of the test statistic applied
on the permuted data T ∗. While the steps are straightforward to implement, the
challenge lies in selecting an appropriate test statistic and determining how to
permute the data correctly.
In general, permutation tests are known to be conditional methods of inference,
where the conditioning is done with respect to a set of sufficient statistics in H0 for
the underlying population distribution. The conditioning may be done on different
sets of sufficient statistics, but it seems reasonable to condition on a minimal set
in order to make the best use of the available information: in the non-parametric
framework this requirement is generally met by referring to the observed data,
as no further reduction of dimensionality is possible without loss of information.
This kind of conditioning and the assumed exchangeability with respect to groups
under H0 make permutation tests independent of the underlying likelihood model
related to the population distribution. Moreover they imply the exactness of the
permutation tests. When the exchangeability property of data is not satisfied or
cannot be assumed in the null hypothesis, permutation inferences are generally not
exact.
1.2. PERMUTATIONAL INFERENCE FOR FUNCTIONAL DATA 11
Inference related to linear models and linear mixed models is one of the most
interesting and widely studied topic in statistics. In literature many strategies
to test for significance of fixed and random effects have been proposed, both in
the scalar and in the functional framework: among all the possible approaches,
permutation tests are powerful tools in this context.
Some of these methods are reported in Subsection 1.2.1.
1.2.1 Inference for functional-on-scalar linear models
Over the last few years, functional and in particular functional-on-scalar (func-
tional response and parameters, scalar covariates) linear models and linear mixed
models have been extensively studied, as the development of modern techniques,
which enable the collection of high-resolution time-continuous data, has allowed
their application in many fields of research (Ramsay and Silverman, 2002, 2005).
In this context, many of the empirically relevant questions do not only address
the effect of covariates and variance components on a functional response, but also
require identification of time intervals characterized by effects of specific covariates
or variance components significantly different from zero (domain selection). To
do that, global tests for the parameters of the model are not suitable, since they
cannot state which parts of the domain are imputable of the rejection of the null
hypothesis.
To overcome this issue, different approaches can be considered. One solution is
to provide point-wise confidence bands for the functional parameters: the results
indicate in which parts of the domain the covariates have an effect, but the control
of the probability of type I error is just point-wise, because point-wise limits are not
equivalent to confidence regions for the entire estimated curves. As a consequence
conclusion are only to be drawn at specific points.
Another method is the Interval Testing Procedure (ITP), introduced by Pini
and Vantini, 2013 and applied to the functional-on-scalar linear models in Pini,
2014: data are represented on a suitable high-dimensional ordered functional basis,
and a family of tests is performed (either with a parametric or non-parametric
approach) on the coefficients of the basis expansion, by controlling the Family Wise
Error Rate (FWER) on intervals (i.e. the probability that a part of the domain is
wrongly detected as significant is always controlled). A drawback with this type of
12 CHAPTER 1. STATE OF THE ART
procedure is that it strongly relies on the choice of the basis expansion.
A new procedure, called Interval-Wise Testing (IWT) has been recently proposed
by Pini and Vantini, 2015 to test functional data and select intervals of the domain
where the null hypothesis is rejected. With respect to ITP it does not require the
projection of data on a finite dimensional functional basis. This method relies on
the definition of adjusted p-value function provided with a control of the Interval
Wise Error Rate (IWER), i.e., the control of the probability of wrongly rejecting any
interval of the domain. In Abramowicz et al., 2015, IWT is applied to a functional-
on-scalar linear model to test various hypotheses on the functional parameters. In
this Thesis this approach will be extended to test for significance of both fixed and
random effects in functional-on-scalar linear mixed models (see Chapter 2).
Permutation tests for fixed effects
Various permutational strategies have been suggested for a test of significance
for a single regression variable in a linear model. Such tests may be influenced by
the presence of one or more other variables in addition to the one tested, indeed
there’s not a general agreement concerning how to perform exact partial tests in
linear multiple regression (i.e. tests in the presence of concomitant variables). When
taking into account the potential relationship (collinearity) between the predictor
variables, the most well-known techniques are permutation of raw data (Manly,
1991, 1997), permutation of residual under the reduced model (Freedman and Lane,
1983; Kennedy, 1995) and permutation of residuals under the full model (ter Braak
1990, 1992).
Results of simulations designed to compare empirically the behaviour of these
methods are presented in Anderson and Legendre, 1999; Anderson and Robinson,
2001. In these works the properties of the different methods with respect to type
I error and power have been investigated and their relationships theoretically
examined showing that permutation of raw data, permutation under the reduced
model in the manner of Freedman and Lane and permutation under the full model
all gave asymptotically equivalent results in most situations, with no significant
difference in power among any of them. In presence of normal or exponential errors,
all these three methods matched the normal theory t-test and had type I error
rate close to the level α. However in situations of extremely non-normal error
1.2. PERMUTATIONAL INFERENCE FOR FUNCTIONAL DATA 13
distributions, they converge asymptotically to appropriate type I error much more
quickly than the normal-theory t-test. Lastly, Kennedy method resulted in inflated
type I error for data of virtually all kinds, especially at small sample sizes.
The results found for the scalar case can be equally applied in multivariate
situations, where the data generally do not fulfill assumption required by traditional
parametric testing procedures. Likewise, these results are valid for individual
terms in ANOVA designs, since ANOVA with fixed factors is simply a special case
of multiple regression, but with categorical predictor variables. Anyway, when
dealing with complex ANOVA designs (such as two-way ANOVA, nested hierarchies,
presence of interaction terms, etc.) two additional strategies can be considered:
restricted permutations (i.e., allowing permutations to occur only within levels of
other factors) and permutation of units other than the observations (e.g., permuting
the units induced by a nested factor in the test of a higher ranked factor). The
second strategy may be pertinent in designs with random factors: to construct an
exact test for the fixed effect, permutations should be restricted to occur within
levels of the random factor (Anderson and Ter Braak, 2003).
Permutation tests for random effects
Linear mixed models (LMMs) are a rich class of models characterized by the
presence of both fixed and random effects. LMMs are often used to fit longitudinal
or repeated measures data, where outcomes for a limited number of subjects are
collected repeatedly over time, or with multilevel or clustered data, where random
effects are used to account for the within-level or within-cluster correlations.
Inference regarding the inclusion or exclusion of random effects is one of the most
challenging problems in this context due to the fact that the variance components
are equal to 0 under the null hypothesis, a value that is located on the boundary
of the parameter space. Consequently, the asymptotic distribution of the Wald,
score and likelihood ratio (LR) statistics under H0 will no longer be the typical χ2.
Instead, the correct null distribution for these statistics has been proved to be a
mixture of χ2 distributions.
Even in this framework, permutation tests are a useful alternative to the
parametric methods, as they are known to have nominal size in finite samples while
requiring only a few weak assumptions. A permutation approach to test for random
14 CHAPTER 1. STATE OF THE ART
effects was presented by Fitzmaurice et al., 2007. The test was specifically designed
for multilevel studies where inclusion of a single random effect to quantify the
heterogeneity among the different levels may be required. Under H0, the cluster
indices are simply random labels and any permutation of them is equally likely.
Specifically, the cluster indices are randomly permuted while holding fixed the
number of units within a cluster. This strategy provides a one-sided p-value for
the LR (or score) test statistic that has the correct type I error rate under the
null, regardless of the number of levels in the data structure, the number of units
at any given level and the choice of link function. However, this test is limited to
the setting at hand and cannot be generalized to longitudinal studies and other
correlated data sources if there are multiple random effects or a single continuous
random effect, such as time.
The work of Lee and Braun, 2012 is a generalization to the approach of Fitz-
maurice et al. and leads to a pair of permutation tests that allow for inference
with any number and type of random effects in a LMM. The first test statistic is
given by the sample variance of the Best Linear Unbiased Predictors (BLUPs) for
the random effect and the second statistic is the restricted likelihood ratio. The
null permutation distributions of these statistics are computed by permuting the
residuals both within and among subjects and are valid both asymptotically and in
small samples. While the test based on the BLUPs can test one random effect at a
time, one of the advantages of the LR based permutation test is that it can address
simultaneous inference on multiple random effects. Detailed theoretical properties
and simulation studies on these methods (both in the case of LMM and GLMM)
can be found in Lee, 2012.
Also the strategy proposed by Drikvandi et al., 2013 handles testing any subset
of variance components: the distribution of a suitable statistic is approximated
through randomly permuting the individual indices of the correct exchangeable
units, while the number of repeated measurements for each individual are kept
fixed. The simulation results suggest that this test has the correct type I error rate,
and its efficiency is reasonably well in comparison to the F-test, and the LR test
based on Fitzmaurice et al., 2007 (in case of non-normal random effects this latter
test appears to be the most powerful).
Regarding functional linear mixed models, although much work has been done on
1.2. PERMUTATIONAL INFERENCE FOR FUNCTIONAL DATA 15
the estimation, there are only few papers which deal with inference in these or more
complex models. One of the first to take into account inference in this framework
was Guo, 2002: it suggested a likelihood ratio test for testing the fixed-effects
using the connection between cubic smoothing splines (at the design points) and
LMMs, and the non-standard asymptotic theory for LR tests. However, test for the
random effects was not considered. In Antoniadis and Sapatinas, 2007 a wavelet
decomposition approach was used to model both fixed effects and random effects
in the same functional space, implying that the population-average curve and the
subject-specific curves have the same smoothness property. Unlike the previous
one, this work provides formal frequentist functional testing procedures for both
fixed effects and random effects, adapting methodologies in linear mixed effects and
non-parametric regression models.
From a review of literature it has been found that, although methods involving
permutation tests have been exploited in the field of LMMs (as described above),
they have not been applied yet in the context of functional linear mixed models.
Motivated by this gap in literature and by the many benefits provided by permuta-
tion techniques, in the next Chapter a strategy based on IWT and permutation
tests will be developed in order to investigate the significance of both fixed and
random effects in a functional LMM.
Chapter 2
Methodology
In this Chapter all the methodologies used to develop our analysis of the
accelerometric data are introduced. In particular in Section 2.1 the Interval-Wise
Testing procedure on which all our tests are based is described. Afterwards, we will
see how this method is applied to testing for significance of fixed and random effects
in LM and LMM (2.3 and 2.4). In Section 2.5 the Freedman and Lane technique to
perform permutation tests is explained.
2.1 Interval-Wise Testing for functional data
The overwhelming majority of non-parametric and parametric available methods
for hypothesis testing in Functional Data Analysis are global, in the sense that they
provide a unique result over the whole domain of the curves. Hence, if there is
enough evidence to reject the null hypothesis, they are not able to identify which
parts of the domain has led to rejection. This fact could make them not fully
satisfactory, since the selection of the intervals where the null hypothesis is false
(domain selection) is a desired property, especially in some applications.
The Interval-Wise Test (IWT) introduced by Pini and Vantini, 2015 is a a purely
non-parametric inferential procedure able to detect the portions of the domain
imputable of rejecting the null hypothesis for functional data. This approach has
the great advantage that it neither rely on a discretisation of the data by means
of a basis expansion nor on the choice of the initial partition of the domain in
sub-intervals: therefore, the IWT is a totally data-driven inferential method.
The procedure will be presented in a general inferential framework in FDA
17
18 CHAPTER 2. METHODOLOGY
(Subsection 2.1.1), while two specific applications of the IWT will be reported in
Sections 2.3 and 2.4.
2.1.1 Functional IWT procedure
Consider a set of functional data belonging to the Hilbert space L2 and defined
over the domain T = (a, b) ⊂ R: the aim is to test a functional null hypothesis
H0 against an alternative H1, defining an R-valued functional test statistic that
could be used to test H0 globally. In case of rejection of the null hypothesis H0,
the purpose is the selection of the intervals of the variable t ∈ T where significant
differences are detected. This problem can be virtually addressed by performing an
infinite family of tests on all the points t of the domain (a, b). Then the challenge
is to control the IWER arising from the continuous infinity of multiple (dependent)
hypotheses tests. To achieve this goal, the IWT procedure is applied, following
these three steps:
Interval-wise testing
Let I ⊆ T be an interval or a complementary interval of the form I = (t1, t2)
or I = T \ (t1, t2) with a ≤ t1 < t2 ≤ b, respectively. Furthermore we define as HI0and HI1 the restriction of the null and alternative hypotheses on I.
For every open interval, and complementary interval I of the domain, we perform
the functional test:
HI0 vs HI1 (2.1)
Testing these hypotheses can be done using different strategies depending on
the assumptions on data and on the test at hand. If the assumption of functional
normality seems not realistic, one can rely on non-parametric permutation methods.
In the case of testing difference between two or more functional populations, an exact
permutation test for (2.1) can be achieved by evaluating a suitable test statistic
T I over all possible permutations of the data over the sample units. In a more
general framework an exact (asymptotically exact) permutation test can be achieved
by choosing a family of (asymptotically) likelihood invariant transformations of
the data set, and evaluating T I over all possible transformations. The p-value of
the corresponding test, denoted by pI , is the proportion of the corresponding test
2.1. INTERVAL-WISE TESTING FOR FUNCTIONAL DATA 19
statistics evaluated on permuted data exceeding the statistics evaluated on the
original data set. As usual, low values of pI indicate statistical evidence to reject
the null hypothesis in I.
Definition of the p-value functions
The family of interval-wise tests defined in the first phase, is used to define the
unadjusted and adjusted functional p-values. A point-wise p-value function is not
trivially defined in L2, since each element of this functional space is an equivalence
class defined on the equivalence relation infinitely often with respect to the Lebesgue
measure.
Definition 2.1. Let H0 and H1 denote, respectively, the null and alternative
functional hypotheses, and HI0 and HI1 their restriction on I. Finally, let pI denote
the p-value of the functional test of HI0 against HI1 . Then, the unadjusted p-value
function p(t) and the adjusted p-value function p(t) are defined ∀t ∈ T as:
p(t) = lim supI→t
pI ; p(t) = supI3t
pI(t),
where with the notation I → t, we indicate that both the extremes of the interval
I converge to t.
Note that, even though in the L2 framework it is meaningless to talk about point-
wise evaluations of the functions, the boundedness of the p-values pI guarantees that
both the unadjusted and the adjusted p-values are instead well defined ∀t ∈ T . Theunadjusted and the adjusted p-value functions p(t) and p(t) have some remarkable
properties:
• The unadjusted p-value p(t) is provided with a control of the point-wise error
rate, that is,
∀t ∈ T s.t. ∃I 3 t : HI0 is true⇒ P[p(t) ≤ α] ≤ α
• The adjusted p-value p(t) is provided with a control of the interval-wise error
rate, that is,
∀I ⊆ T : HI0 is true⇒ P[∀t ∈ I, p(t) ≤ α] ≤ α
20 CHAPTER 2. METHODOLOGY
• The unadjusted p-value function p(t) is point-wise consistent, i.e. the proba-
bility of truly detecting any point t for which the null hypothesis is not true
converge to one as the sample size increases. In details, ∀α ∈ (0, 1):
∀t ∈ T s.t. @I 3 t : HI0 is true⇒ P[p(t) ≤ α] −−−→n→∞
1
• The adjusted p-value function p(t) is interval-wise consistent, i.e the probability
of truly detecting any interval I for which the null hypothesis is not true
∀t ∈ I converge to one as the sample size increases. In details, ∀α ∈ (0, 1):
∀I ⊆ T s.t. @J ⊆ I : HJ0 is true⇒ P[∀t ∈ I, p(t) ≤ α] −−−→n→∞
1
For further details and the proofs of the properties see Pini and Vantini, 2015.
Domain selection
Once a significance level α ∈ (0, 1) for the test is chosen, intervals of the domain
presenting a rejection of the null hypothesis are obtained by thresholding the p-value
functions evaluated in the previous step. In detail, if we are only interested in
controlling the point-wise error rate at level α (i.e., given any point of the domain
where the null hypothesis is not violated, the probability of wrongly selecting it as
significant is controlled), we select the points t ∈ T such that p(t) ≤ α. If instead
we are interested in controlling the interval-wise error rate at level α (i.e., given any
interval of the domain where the null hypothesis is not violated, the probability
of wrongly selecting it as significant is controlled), we select the points t ∈ T such
that p(t) ≤ α.
2.2 Models
Suppose we have observed a sample of n continuous L2 random functions over
time t s.t. t ∈ (a, b). The continuity of the functional data yi(t) is a natural
assumption in the analysis of accelerations. However the procedure can still be
applied even if this assumption is relaxed, as shown in Pini and Vantini, 2015.
In the following we introduce two models for yi(t):
2.2. MODELS 21
• functional-on-scalar Linear Model (Subsection 2.2.1)
• functional-on-scalar Linear Mixed Model (Subsection 2.2.2)
2.2.1 Functional-on-scalar linear model
Model
yi(t) = β0(t) +G∑g=1
xigβg(t) + εi(t), ∀i = 1, ..., n (2.2)
Where:
• yi(t), t ∈ (a, b) is the i-th functional response;
• xi1, ..., xiG ∈ R are known scalar covariates related to the i-th observation;
• βg(t), g = 0, ..., G are the unknown functional parameters;
• errors εi(t), t ∈ (a, b) are i.i.d. (with respect to units) zero-mean random
functions (not necessarily Gaussian) with finite total variance, i.e.
∫ b
a
E[εi(t)]2dt <∞, ∀i = 1, ..., n.
Using the matrix notation this model can be expressed as:
y = Xβ + ε (2.3)
Where y is the functional vector containing the n observed functions, β is the
(G+ 1)-vector of parameter functions, ε is a vector of n residual functions and X is
the n× (G+ 1) design matrix.
The only way in which (2.3) differs from the corresponding equations on the general
linear model is that the parameter β, and hence the predicted observations Xβ,
are vectors of functions rather than vectors of numbers.
Model Estimation
The ordinary least squares (OLS) estimators of the functional parameters
βg(t), g = 0, ..., G, can be found by minimizing the residual sum of squares (with
22 CHAPTER 2. METHODOLOGY
respect to βg(t), g = 0, ..., G), which in this specific case consists of the sum over
units of the squared L2 distances between the functional data yi(t) and the quantity
β0(t) +∑G
g=1 xigβg(t).
n∑i=1
∫ b
a
(yi(t)− β0(t)−G∑g=1
xigβg(t))2dt (2.4)
If there are no particular restrictions on the way in which the parameters vary
as functions of t, the minimization of (2.4) can be done individually for each t, as
in vector-on-scalar linear models. That is, the OLS estimates can be calculated
for a suitable grid of values of t using ordinary regression analysis (Ramsay and
Silverman, 2005):
(β0(t), ..., βG(t)) = arg min(β0(t),...,βG(t))
n∑i=1
(yi(t)− β0(t)−G∑g=1
xigβg(t))2 (2.5)
(In matrix notation: β = (XTX)−1XTy)
For each t, βg(t) are thus the OLS estimators of the corresponding scalar-on-scalar
linear model at point t. Note that the fact that the same design matrix is involved
for each t makes for considerable economy of numerical effort.
2.2.2 Functional-on-scalar linear mixed model
Model
yij(t) = β0(t) +G∑g=1
xijgβg(t) +K∑k=1
zijkbjk(t) + εij(t), ∀i = 1, ..., nj; j = 1, ..., J
(2.6)
• yij(t), t ∈ (a, b) is the i-th functional response of the j-th subject, i =
1, ..., nj; j = 1, ..., J with nj number of observations of the j-th subject, J
number of subjects;
• xij1, ..., xijG ∈ R are known fixed effect (scalar) covariates;
• zij1, ..., zijK ∈ R are known random effect (scalar) covariates;
2.2. MODELS 23
• βg(t), g = 0, ..., G are the unknown population-level fixed effect functional
coefficients;
• bjk(t), k = 1, ..., K are the unknown functional random effects; for any fixed
t ∈ (a, b), the vector b(t)j = (bj1(t), ..., bjK(t)) is assumed to have a multivariate
distribution with mean 0 and covariance matrix Σ(t), in which the respective
variances for bj1(t), ..., bjK(t) are denoted as σ2bj1
(t), ..., σ2bjK
(t);
• errors εij(t) are i.i.d. (with respect to units) zero-mean random functions
with variance σ2ε(t), t ∈ (a, b). A further assumption is that for each j, b(t)j
and εij(t) are independent.
Equivalently, the LMM for subject j can be written using matrix notation as:
yj = Xjβ + Zjbj + εj
Where yj is the functional vector containing the nj observed functions relative to
subject j, β is the (G+ 1)-vector of fixed effect parameters functions, bj is the K
dimensional vector of functional random effects, ε = (ε1j, ..., εnjj) is a vector of njresidual functions and Xj and Zj are the subject-specific design matrices for the
G+ 1 fixed effect and K random effect covariates, respectively.
The general model expression can be obtained combining data from all subjects.
Let y = (y1, ...,yJ) be the n-dimensional vector of functional observations, b =
(b1, ..., bJ) the vector of functional random effects of all subjects and ε = (ε1, ..., εJ)
the n-dimensional vector of functional errors, where n =∑J
j=1 nj . Furthermore, let
X and Z be the respective design matrices for the fixed and random effect covariates
formed by successively placing each subject’s design matrices under each other.
Then the model will be:
y = Xβ + Zb+ ε (2.7)
In addition to that, for any fixed t ∈ (a, b):
V ar
b(t)ε(t)
=
Q(t) 0
0 R(t)
Where Q(t) = Σ(t) ⊗ IQ and R(t) = σ2
ε(t)IR, in which ⊗ denotes the Kronecker
product, and IQ(t) and IR are J × J and n× n identity matrices.
24 CHAPTER 2. METHODOLOGY
Model Estimation
Estimation of the elements of β, Q(t), and R(t) at any fixed t is typically done
through maximum likelihood (ML) or restricted maximum likelihood (REML).
Asymptotically, the ML and REML estimators are equivalent, but for small sample
sizes, the REML estimator is expected to be less biased than the maximum likelihood
estimator. Subject specific random effects b = (b1, ..., bJ) can be predicted using
best linear unbiased prediction (BLUP). The quantities of interest can be obtained
solving the following equations given by Henderson in 1950 (Lee and Braun, 2012),
where for simplicity of notation the time index is omitted:
XTR−1Xβ + XTR−1Zb = XTR−1y
ZTR−1Xβ + (ZTR−1Z +Q−1)b = ZTR−1y
The solutions are:
β = (XT V−1
X)−1XT V−1
y
b = QZV−1e
Where e = y −Xβ and V = ZT QZ + R is the estimated covariance matrix for y.
In general, b can be interpreted as realized values of the random vector b.
For the purpose of this Thesis, only LMM with a single random intercept are
considered. Consequently, in the following we will refer to linear mixed models
meaning the particular case of K = 1 and with zij1 constant and equal to 1, i.e.:
yij(t) = β0(t) +G∑g=1
xijgβg(t) + bj1(t) + εij(t), ∀i = 1, ..., nj; j = 1, ..., J (2.8)
2.2.3 A particular case: one-way FANOVA
Here we present the Functional Analysis of Variance (FANOVA) model as a
particular case of a linear model. Anyway it can be generalized to the case of a
linear mixed model, where besides the fixed factor, one or more random effects are
present.
yig(t) = µ(t) + τg(t) + εig(t), ∀i = 1, ..., ng g = 1, ..., G (2.9)
2.3. TESTING FOR FIXED EFFECTS 25
Where:
• yig(t), t ∈ (a, b) is the i-th observation of the g-th treatment, i = 1, ..., ng g =
1, ..., G with ng number of observations belonging to the g-th treatment, G
number of treatment levels;
• µ(t), t ∈ (a, b) is the grand mean function;
• τg(t), t ∈ (a, b) is the g-th treatment effect; to be able to identify them
uniquely, we require that they satisfy the constraint:
G∑g=1
τg(t) = 0, ∀t ∈ (a, b);
• errors εig(t), t ∈ (a, b) are i.i.d. (with respect to units) zero-mean random
functions (not necessarily Gaussian) with finite total variance, i.e.
∫ b
a
E[εi(t)]2dt <∞, ∀i = 1, ..., n.
The model (2.9) has an equivalent formulation of the form (2.2). In particular,
we can define the matrix X with n =∑G
g=1 ng rows (one for each observation) and
(G+ 1) columns. The label (ig) can be used to indicate the row corresponding to
observation i in group g; this row has a one in the first column and in column g+ 1,
and zeroes in the rest. The element x(ig)k indicates the value in row (ig) and column
k of X. We define also a corresponding set of (G+ 1) regression functions βk such
that β = (β1, β2, ..., βG+1) = (µ, τ1, ..., τG). Consequently the model becomes:
yig(t) =G+1∑k=1
x(ig)kβk(t) + εig(t), ∀i = 1, ..., ng; g = 1, ..., G (2.10)
Or, more compactly:
y = Xβ + ε.
2.3 Testing for fixed effects
In this Section, we will introduce the problem of testing for significance of
fixed effects both in the case of a functional-on-scalar linear model (LM) and a
26 CHAPTER 2. METHODOLOGY
functional-on-scalar linear mixed model (LMM).
Referring to the models (2.2) and (2.6) we are interested in testing if none of
the covariates associated to the fixed effects significantly affects the response, i.e.,
the functional version of classical F -test:H0 : βg(t) = 0 ∀g ∈ {1, ..., G}, ∀t ∈ (a, b)
H1 : βg(t) 6= 0 for some g ∈ {1, ..., G} and some t ∈ (a, b)(2.11)
In case of rejection of the null hypothesis H0, the purpose is the selection of
the intervals of the variable t where significant differences are detected. To do
that, we apply the procedure previously described in a more general framework in
Subsection 2.1.1.
2.3.1 Application of IWT procedure to test fixed effects
Interval-wise testing
A functional test consistent with (2.11) is performed on every interval of the
domain. In particular, given any I ⊆ (a, b), we test the restriction of H0 on interval
I: HI0 : βg(t) = 0 ∀g ∈ {1, ..., G}, ∀t ∈ I
HI1 : βg(t) 6= 0 for some g ∈ {1, ..., G} and some t ∈ I(2.12)
For both LM and LMM we use the following test statistic:
T Ifixed =
∫I
SSE(t)/dfexplSSR(t)/dfres
dt (2.13)
Where SSE(t)/dfexpl is the explained sum of squares divided by its degrees of
freedom and SSR(t)/dfres is the residual sum of squares divided by its degrees of
freedom.
In the particular case of the linear model, the statistic defined above is described
by the formula:
T Ifixed =
∫I
(n−G)∑n
i=1(yi(t)− y(t))2
(G− 1)∑n
i=1(yi(t)− yi(t))2dt (2.14)
Where yi(t) = β0(t) +∑G
g=1 xigβg(t) is the fitted part of the model and y(t) =
2.3. TESTING FOR FIXED EFFECTS 27
∑ni=1 yi(t)/n is the sample mean of the response functions.
Instead, in LMM, the denominator degrees of freedom are upper bounds for the
real df. This is because the F -test is always an approximation for these models.
The reason why these degrees of freedom are not more accurately approximated at
present is because it is difficult to decide exactly how this should be done for this
kind of models (Bates, 2005).
In this work, hypothesis testing is done using functional permutation tests
based on the Freedman and Lane permutation scheme and on the test statistic
mentioned above (the steps of this procedure are reported in 2.5). The Freedman
and Lane permutation strategy relies on the permutation of the estimated residuals
under the reduced model (i.e., the model under the null hypothesis of the test),
which it is the most commonly used scheme for linear models. As pointed out
in Anderson and Legendre, 1999 and Anderson and Robinson, 2001, this method
showed better properties in terms of type I error and power compared to other
permutation techniques.
We also considered two different strategies to permute the residuals:
• Unrestricted permutations
This is the original approach proposed by Freedman and Lane: the residuals
of the reduced models are permuted randomly, without any constraint;
• Restricted permutations
Permutations of the residuals are allowed to occur only within levels of the
(unique) random factor, as suggested by Anderson and Ter Braak, 2003: in
this way, one can take into account the existence of a random effect, even
if not present in the model. Note that using this method, the procedure is
still valid because it only restricts the set of all possible permutations and it
applies likelihood-invariant transformations also if the random effect is not
included in the model.
28 CHAPTER 2. METHODOLOGY
Computation of the adjusted p-value function
The adjusted p-value function is defined as the maximum p-value of all interval-
wise tests on intervals containing t:
p(t) = supI3t
pI (2.15)
where pI is the p-value of the test (2.12).
Theoretically, the definition of adjusted p-value functions implies the evaluation
of the p-values of the infinite family of all possible interval-wise tests. In practice
this can be avoided, thanks to the continuity of the test statistic with respect to
the extremes of integration: the p-values of tests on intervals I can be discretised
on a fine grid, and the sup in the definition of adjusted p-value functions can be
approximated with its discrete counterpart.
Domain selection
Intervals of the domain presenting a rejection of the null hypothesis are obtained
selecting the points t where the adjusted p-value function is smaller than the (chosen)
significance level α. Since the test of hypothesis (2.11) based on the statistic Tfixedis provided with a control of the IWER, by rejecting H0, we have that, given any
interval in which the response is not influenced by any covariate, the probability
that this interval is wrongly selected as significant is lower than α.
2.4 Testing for random effects
Let us consider the model (2.8). We are interested in investigating if the presence
of the random intercept b1 is significant for the model. This is equivalent to testing:H0 : σ2bj1
(t) = 0 ∀j ∈ {1, ..., J}, ∀t ∈ (a, b)
H1 : σ2bj1
(t) > 0 for some j ∈ {1, ..., J} and some t ∈ (a, b)(2.16)
The challenge in testing for random effects lies in the fact that the value of the
variance component under the null hypothesis is on the boundary of the parameter
space. As a result, the usual χ2 asymptotic distributions of the Wald, score, and
2.4. TESTING FOR RANDOM EFFECTS 29
likelihood ratio test statistics do not hold. Instead, the correct null distribution for
the likelihood ratio statistic has been proved to be a mixture of χ2 distributions.
The approach described in the next Subsection relies on the likelihood ratio
based permutation test suggested in Lee and Braun, 2012 and Lee, 2012, extended
to the functional data framework.
2.4.1 Application of IWT procedure to test a random effect
Interval-wise testing
A functional test consistent with (2.16) is performed on every interval of the
domain. In particular, given any I ⊆ (a, b), we test the restriction of H0 on interval
I: HI0 : σ2
bj1(t) = 0 ∀j ∈ {1, ..., J}, ∀t ∈ I
HI1 : σ2bj1
(t) > 0 for some j ∈ {1, ..., J} and some t ∈ I(2.17)
This test is based on the test statistic:
T Irandom = e∫I (−2 log (L
Ht0−L
Ht1))dt (2.18)
Where λt = −2 log (LHt0− LHt
1) is the likelihood ratio test statistic: LHt
0and LHt
1
are the likelihoods under the null and alternative hypotheses restricted on the point
t, respectively. The expression exp(∫I log(f(t))dt
)is called geometric integral of
the function f(·) and can be thought of as a “continuous” version of a “discrete”
product.
Under the null hypothesis of no random effects, the residuals ε = y − Xβ
are exchangeable, and more specifically, independent and identically normally
distributed with mean 0 and variance σ2ε . Hence, following the Freedman and Lane
permutation scheme, the estimated residuals ε can be permuted both within and
between subjects.
Using the same notation as in Section 2.2.2 (omitting the index t), we have
E[y] = Xβ and V ar[y] = V . We want to use the same statistic of the parametric
case, that is built under the assumption y ∼ N (Xβ,V ). Note that normality is
assumed to built the test statistic but it is not needed to perform the permutation
test. At this point, replacing all parameters with their estimates under the null and
30 CHAPTER 2. METHODOLOGY
alternative hypotheses (as denoted by their subscripts), λ becomes:
λ = log
(|V 0||V 1|
)+ eT1 (V
−10 − V
−11 )e1 + log
(|XT V
−10 X|
|XT V−11 X|
)
V 0 and V 1 are re-estimated for each permutation of the residuals. By doing so, we
allow the empirical distribution to “mix” as the rank of Σ varies, thereby generating
a distribution similar to the mixture χ2 asymptotic distribution.
Note that this test, as pointed out in Lee and Braun, 2012 and Lee, 2012, can
also address simultaneous inference on multiple random effects.
Computation of the adjusted p-value function and domain selection
This two steps of the IWT procedure are carried out in the same way as in
Subsection 2.3.1.
2.5 Freedman and Lane permutation scheme
In this Section, we give some details on the implementation of the Freedman
and Lane permutation scheme for testing hypotheses (2.11) and (2.16) for each
interval I ⊆ (a, b).
Freedman and Lane permutational procedure
1. Evaluate the reference test statistic T0 on the observed data;
2. Estimate the residuals of the reduced model;
3. Repeat the following B times:
i. Permute randomly the residuals of the reduced model;
ii. Compute the new permuted data;
iii. Evaluate the statistic T ∗b on the permuted data;
4. Calculate the p-value of the test as the proportion of permuted scenarios in
which the test statistic T ∗b which is greater or equal to the reference value T0.
In Table 2.1 the reduced model associated to the null hypothesis is specified for
each of the tests described in the previous sections.
2.5. FREEDMAN AND LANE PERMUTATION SCHEME 31
Table 2.1: Methods to test fixed and random effects.
Model H0 Reduced Model Permutations
LM βg(t) = 0, ∀g ∀t yi(t) = β0(t) + εi(t)UnrestrictedRestricted
LMM βg(t) = 0, ∀g ∀t yij(t) = β0(t) + bj1(t) + εij(t)UnrestrictedRestricted
LMM σ2bj1
(t) = 0,∀j ∀t yij(t) = β0(t) +∑G
g=1 xijgβg(t) + εij(t) Unrestricted
Chapter 3
Dataset
In this Chapter a description of the dataset provided by the University of
Copenhagen is presented (see also Sørensen et al., 2012; Thomsen et al., 2010b). All
experimental procedures were pre-approved by the Danish Animal Experimentation
Board and the study was performed according to the Danish Animal Testing Act.
3.1 Data acquisition
A 10G, three-axis piezo-resistant accelerometer was held firmly in place by an
elastic girth at the lowest point of the dorsal back region in the midline (approx-
imately above T13) and connected to a data-logger mounted on another girth.
The horses were led by hand in trot over a distance of 24 m at constant velocity
and were videotaped during the measurement. The accelerometer measured the
three-dimensional accelerations of trunk movements at a frequency of 240 Hz, and
the raw data signal was filtered to remove micro-structure noise using a Butterworth
filter with a cut-off frequency of 50 Hz.
Each record is represented by a 3D acceleration signal, which consists of a
number of observations varying from 1100 to 1400 for each component and it is
acquired in a time interval of about 6 s. Each of these signals is originally composed
of M = 8 parts of equal length, referred to as M cycles, and since trot is a two-beat
gait, each cycle is composed of two halves corresponding to the two diagonal pairs
of limbs. In other words, a complete data signal consists of M replications of a
bi-phased signal.
33
34 CHAPTER 3. DATASET
3.1.1 Experiment
The experiment studied in this Thesis involved eight horses. Each horse was
measured nine times: before induction of lameness (sound) and after induction
of two degrees of lameness in each of the four legs. The lameness was induced
mechanically by equipping the horses with a modified horseshoe with a screw
eliciting pressure on the sole of the hoof. The shoe makes stance on the limb painful.
The two degrees, in the following denoted low-degree lameness and moderate-degree
lameness, correspond to different levels of this pressure. Two data signals are
unavailable, resulting in a total of 70 signals.
3.2 Preprocessing of data
Based on the 3D signal, the ground reaction force (GRF) was computed as the
length of the 3D vector after subtracting gravity. The obtained signal was rescaled
to unit interval in order to make signals comparable across time arguments and
treated as a discrete equidistantly observed function. The GRF record was then
smoothed to a Fourier basis with 501 basis functions by minimizing a standard least
squares criterion and after that, to a period Fourier basis with 81 basis functions
and a period of 1/8 to create a periodic target function.
At this point, a warping function was computed to warp the smoothed GRF
record towards the periodic target obtained from the GRF record itself. The
warping function was estimated from the register.fd() function of the R-package
fda. The so called “function parameter object”-argument was chosen to penalize
the integral of the squared second order derivative of the warping function and the
penalty parameter was set to 0.01.
The warping function obtained from the GRF record was used to warp all three di-
rections (vertical, transversal and longitudinal) using the function register.newfd()
of the fda-package. The warped signals were evaluated at 8001 grids points and
averaged over eight gait cycles to give a signal representing an average gait cycle at
1001 grid points. Note that up to this point the preprocessing was done separately
on each record, obtaining a sort of “average cycle”. In the final step, on the sample
of all the “average cycles”, a shift registration toward the cross sectional mean was
performed in order to minimize the L2-norm between each signal and the cross
3.3. DIFFERENT REPRESENTATIONS OF DATA 35
sectional mean. This procedure is required to account for differences in what is
defined as a starting point of a gait cycle for different records and, even if it allows
for non-linear warping functions, for most records the estimated warp turns out to
be more or less linear.
At that point the sample of shift registered “average cycles” were evaluated
at 1001 grid points. Extracting every tenth observation, the version of the data
analysed in this thesis is finally obtained.
3.3 Different representations of data
Lameness may result in non-repetitious movements (single events) that increase
the difference between gait cycles and reduce the overall impression of a repeated
two-beat gait pattern. In order to diminish the effect of such single events, all our
analysis are based on an average signal over eight gait cycles or transformations of
it. For the purpose of our work, we considered four different representations of the
data, which will be introduced in Subsections 3.3.1 and 3.3.2.
3.3.1 Two-peaks representations
These data consist of the average smoothed gait cycles, hence the acceleration
curves are characterised by the presence of two peaks corresponding to the two
diagonals. All the three components of the acceleration vector (i.e. vertical,
transversal and longitudinal) are considered. Positive accelerations for the three
axes are represented by dorsal, left and cranial directions, respectively.
Original Datasets
In these datasets the acceleration curves are given by the functions of time
{aori (t)}i=1,...,n, t ∈ [0, 1], which are cut out such that the stride starts in the
suspension phase before the stance phase of the right diagonal.
The three Original Datasets are:
• VERTICAL_Original Dataset: vertical component of the 3D accelera-
tions. The first positive peak is associated with the ground reaction force of
the RF-LH diagonal, whereas the second positive peak corresponds to the
LF-RH diagonal.
36 CHAPTER 3. DATASET
• TRANSVERSAL_Original Dataset: transversal component of the 3D
accelerations. The first peak is associated with the stance phase of the RF-LH
diagonal, whereas the second peak corresponds to the LF-RH diagonal.
• LONGITUDINAL_Original Dataset: longitudinal component of the 3D
accelerations. The first peak is associated with the stance phase of the RF-LH
diagonal, whereas the second peak corresponds to the LF-RH diagonal.
Modified Datasets
The acceleration functions of the Original Datasets are modified such that the
curves {amodi (t)}i=1,...,n, t ∈ [0, 1] start in the suspension phase before the stance
phase of the sound diagonal. In particular, when the lame limb is the right-fore or
the left-hind, the two halves of the original acceleration curves are swapped.
The three Modified Datasets are:
• VERTICAL_Modified Dataset: vertical component of the modified 3D
accelerations. The first positive peak is associated with the ground reaction
force with the sound diagonal, whereas the second positive peak corresponds
to the lame diagonal.
• TRANSVERSAL_Modified Dataset: transversal component of the
modified 3D accelerations. The first peak is associated with the stance
phase of the sound diagonal, whereas the second peak corresponds to the
lame diagonal.
• LONGITUDINAL_Modified Dataset: longitudinal component of the
modified 3D accelerations. The first peak is associated with the stance phase
of the sound diagonal, whereas the second peak corresponds to the lame
diagonal.
A graphical comparison of the Original and Modified datasets is showed in
Figure 3.1. Transversal accelerations (second row) show a particular feature which
is not present in vertical and longitudinal accelerations: a trotting horse lateral
movements are symmetrical in the two diagonals with respect to the craniocaudal
axis. This translates into a symmetry of the two peaks of the acceleration functions
with respect to the 0-acceleration axis. This fact has to be taken into account in the
3.3. DIFFERENT REPRESENTATIONS OF DATA 37
construction of the TRANSVERSAL Modified Dataset: in addition to the swapping
operation, the exchanged peaks have to be reflected across the 0-acceleration axis.
Another important aspect to point out is that these data are clearly non
Gaussian: this strongly limits the application parametric methods for inference, in
favour of non-parametric techniques.
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Original Vertical Accelerations
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Modified Vertical Accelerations
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Original Transversal Accelerations
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2Modified Transversal Accelerations
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Original Longitudinal Accelerations
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Modified Longitudinal Accelerations
Figure 3.1: Comparison between Original (left column) and Modified (right column)Datasets. The curves are colored according to the lame limb: No, RH, RF, LH, LF. Lowdegrees of lameness are indicated with lighter tones, moderate degrees with darker ones.
3.3.2 Difference between peaks representations
These data are obtained from the datasets introduced in Subsection 3.3.1 by
subtracting the two parts of the curves representing the two diagonals. The results
38 CHAPTER 3. DATASET
of the analysis of these datasets will be presented in Appendices A and B.
Original Peak1-Peak2 Datasets
The data are given by the functions of time {∆aori (t)}i=1,...,n, t ∈ [0, 1] such that:
∆aori (t) = aori (t)− aori(t+
1
2
)t ∈[0,
1
2
]Note that for a sound horse the difference between the two peaks of acceleration
curves is almost null, due to the symmetry of the movements he performs.
The three Original Peak1-Peak2 Datasets are:
• VERTICAL_Original Peak1-Peak2 Dataset: difference between the
first peak (RF-LH diagonal) and the second peak (LF-RH diagonal) of the
vertical component of the 3D accelerations.
• TRANSVERSAL_Original Peak1-Peak2 Dataset: difference between
the first peak (RF-LH diagonal) and the second peak (LF-RH diagonal) of
the transversal component of the 3D accelerations.
• LONGITUDINAL_Original Peak1-Peak2 Dataset: difference between
the first peak (RF-LH diagonal) and the second peak (LF-RH diagonal) of
the longitudinal component of the 3D accelerations.
Modified Peak1-Peak2 Datasets
The data are given by the functions of time {∆amodi (t)}i=1,...,n, t ∈ [0, 1] such that:
∆amodi (t) = amodi (t)− amodi
(t+
1
2
)t ∈[0,
1
2
]The three Modified Peak1-Peak2 Datasets are:
• VERTICAL_Modified Peak1-Peak2 Dataset: difference between the
first peak (sound diagonal) and the second peak (lame diagonal) of the vertical
component of the 3D accelerations.
• TRANSVERSAL_Modified Peak1-Peak2 Dataset: difference between
the first peak (sound diagonal) and the second peak (lame diagonal) of the
transversal component of the 3D accelerations.
3.3. DIFFERENT REPRESENTATIONS OF DATA 39
• LONGITUDINAL_Modified Peak1-Peak2 Dataset: difference between
the first peak (sound diagonal) and the second peak (lame diagonal) of the
longitudinal component of the 3D accelerations.
A graphical comparison of the Original and Modified Peak1-Peak2 datasets is
showed in Figure 3.2.
0.0 0.1 0.2 0.3 0.4 0.5
−1.
5−
0.5
0.5
1.0
1.5
Original Vertical Peak1−Peak2
0.0 0.1 0.2 0.3 0.4 0.5−
1.5
−0.
50.
51.
01.
5
Modified Vertical Peak1−Peak2
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
23
Original Transversal Peak1−Peak2
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
23
Modified Transversal Peak1−Peak2
0.0 0.1 0.2 0.3 0.4 0.5
−6
−4
−2
02
46
Original Longitudinal Peak1−Peak2
0.0 0.1 0.2 0.3 0.4 0.5
−6
−4
−2
02
46
Modified Longitudinal Peak1−Peak2
Figure 3.2: Comparison between Original (left column) and Modified (right column)Peak1-Peak2 Datasets. The curves are colored according to the lame limb: No, RH, RF,LH, LF. Low degrees of lameness are indicated with lighter tones, moderate degrees withdarker ones.
40 CHAPTER 3. DATASET
3.4 Covariates
Independently by the chosen representation of the data, some categorical vari-
ables are associated to each curve:
• horse: B1, B2, B3, B4, B5, B6, B7, B8
Horse ID: each horse is measured 9 times, except for horse B3 which has 2
missing observations (relative to lameLeg RH).
• lameLeg: No, RH, RF, LH, LF
Lame limb: “No” indicates no lameness (i.e. sound horse), “RH” right-hind
limb, “RF” right-fore limb, “LH” left-hind limb, “LF” left-fore limb.
• lameSide: No, Right, Left
• lameForeHind: No, Hind, Fore
• lameDegree: No, Low, Mod
Degrees of the mechanically induced lameness: “No” indicates that no lameness
is induced, “Low” stands for low degree and “Mod” for moderate degree of
lameness.
In Table 3.1, variables related to all the nine measurements of the horse B1 are
presented. This is valid also for all the other horses participating to the experiment,
except for horse B3, whose data relative to the right-hind limb could not be recorded.
Table 3.1: Categorical variables relative to horse B1
horse lameLeg lameSide lameForeHind lameDegree
B1 No No No NoB1 RH Right Hind LowB1 RH Right Hind ModB1 RF Right Fore LowB1 RF Right Fore ModB1 LH Left Hind LowB1 LH Left Hind ModB1 LF Left Fore LowB1 LF Left Fore Mod
3.4. COVARIATES 41
3.4.1 lameLeg and lameForeHind
As confirmed in literature, a sound horse moves symmetrically (up to random
variation) during trot in the sense that the accelerations of the trunk are the same
for the two parts of the gait cycle (black curves of Figures 3.3 and 3.4). On the
contrary, lameness is believed to disturb the gait pattern, leading to an asymmetry
between the two parts of the stride. Specifically, the horse attempts to “unload”
the lame limb during the weight-bearing phase of the stride through compensatory
movements of head and trunk. This causes maximal acceleration and acceleration
amplitude to decrease during the stance phase of the lame limb and to increase
during the stance phase of the contralateral nonlame limb. This is very evident in
our data. If we consider the Original Datasets (Figure 3.3) one can note that when
the lameness is induced in the right-fore or the left-hind limb (violet and orange
curves), the first peak resulting from the stance phase of the right diagonal is lower
than the second peak. The opposite situation arises during the stance phase of the
left diagonal (blue and green curves).
The definition of the Modified Datasets implies that the first peak is always
referred to the sound diagonal and as a result it is always higher or equal to the second
peak. It should be noticed that in the modified acceleration curves the information of
the lameSide is lost due to the peak-swapping procedure. Consequently a distinction
can be made only among levels of the factor lameForeHind (see Figure 3.4).
The two categorical variables lameLeg and lameForeHind are introduced as fixed
effects in our models for the original and modified accelerations, respectively. In
fact, we are interested in studying the effects of the induced lameness on the shape
of acceleration functions. This problem sets in the Functional Analysis of Variance
(FANOVA) framework presented in Subsection 2.2.3.
3.4.2 lameDegree
The degree of lameness induced in a limb is an important aspect to take into
account because the more the lameness is mild, the more the visual diagnosis
becomes difficult. The distinction between low and moderate degrees of lameness is
represented in Figures 3.3 and 3.4 by using lighter or darker tones of the same color.
Anyway, this variable cannot be included in our models as a second fixed effect,
since the factors lameDegree - lameLeg and lameDegree - lameForeHind are not
42 CHAPTER 3. DATASET
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−2
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−2
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−6
−2
26
0.0 0.2 0.4 0.6 0.8 1.0
−6
−2
26
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−2
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−2
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−6
−2
26
0.0 0.2 0.4 0.6 0.8 1.0
−6
−2
26
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−2
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−2
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−6
−2
26
0.0 0.2 0.4 0.6 0.8 1.0
−6
−2
26
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−2
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−2
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−6
−2
26
0.0 0.2 0.4 0.6 0.8 1.0
−6
−2
26
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−2
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−2
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−6
−2
26
0.0 0.2 0.4 0.6 0.8 1.0
−6
−2
26
Figure 3.3: Representation of the groups induced by the factor lameLeg on the VerticalOriginal accelerations (left column), on the Transversal Original accelerations (centralcolumn) and on the Longitudinal Original accelerations (right column). In each row,acceleration curves relative to one of the levels of the factor lameLeg are colored, while allthe other curves are represented in gray. Colors associated to lameLeg levels are: No, RH,RF, LH, LF. Darker colors (of the same tone) are used to indicate a moderate lamenessdegree, while lighter colors indicate a low lameness degree.
3.4. COVARIATES 43
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Figure 3.4: Representation of the groups induced by the factor lameForeHind on theVertical Modified accelerations (left column), on the Transversal Modified accelerations(central column) and on the Longitudinal Modified accelerations (right column). In eachrow, acceleration curves relative to one of the levels of the factor lameForeHind are colored,while all the other curves are represented in gray. Colors associated to lameForeHind levelsare: No, Hind, Fore. Darker colors (of the same tone) are used to indicate a moderatelameness degree, while lighter colors indicate a low lameness degree.
crossed (i.e. there is not at least one observation in every combination of categories
for the two factors).
What can be done is to construct new variables containing the information com-
ing from both the factors of the above mentioned couples. The factors lameDegree
- lameForeHind give birth to the new factor:
• lameForeHind_Degree: No, Hind_low, Hind_mod, Fore_low, Fore_mod
In practice, the couple lameDegree - lameLeg is not considered, since the presence of
nine groups would not allow the application of the permutation procedures described
in the previous Chapter, in particular the ones based on restricted permutations.
3.4.3 horse
A noteworthy feature of our data is that there are multiple observations from
the same horse. Every horse has a slightly different gait, and this is going to be
44 CHAPTER 3. DATASET
an idiosyncratic factor that affects all signals from the same horse, thus rendering
these different signals inter-dependent rather than independent. This fact is clearly
visible in Figure 3.5, where curves relative to a single horse are represented in the
each panel.
To deal with this problem the variable horse is treated as a random effect,
associated with a sample randomly drawn from a population. This approach allows
to model correlations between observations over the same “subject” which is typical,
for example, for longitudinal and repeated measurements data. In particular, by
introducing a random intercept, we are able to characterise the variation induced
in the response by the different levels of the covariate (Abramovich and Angelini,
2006).
3.4. COVARIATES 45
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Horse B1
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Horse B2
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Horse B3
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Horse B4
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Horse B5
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Horse B6
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Horse B7
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Horse B8
Figure 3.5: Vertical Original accelerations of the eight horses involved in the experiment.In each panel all the curves relative to one horse are colored, while all the others arerepresented in gray. Acceleration curves are colored according to the levels of the factorlameLeg: No, RH, RF, LH, LF. Darker colors (of the same tone) are used to indicate amoderate lameness degree, while lighter colors indicate a low lameness degree. Note thatfor horse B3 two signals are missing (blue curves).
Chapter 4
Results of test for the fixed effect
In this Chapter, the results relative to tests for significance of the fixed effect
are presented. All the tests have been performed as described in Section 2.3,
considering both linear and linear mixed models and both unrestricted and restricted
permutation of residuals. In Table 4.1 there is a summary of the four different
approaches used to test fixed effects.
Table 4.1: Methods to test fixed effects.
Model Permutations
Linear Model (LM) UnrestrictedRestricted
Linear Mixed Model (LMM) UnrestrictedRestricted
In addition to that we also investigate the differences between the accelerations
curves associated to different levels of the fixed effect, in a pairwise perspective. This
test detects significant differences between all pairs of lameLeg or lameForeHind
or lameForeHind_Degree levels. Furthermore, it is able to identify the regions of
the domain which contribute to the rejection of the null hypothesis thanks to the
application of the same IWT procedure as above.
From the analysis carried out with the aforementioned four types of test, we
will see that not taking into account the presence of the random effect either in
the model or in the permutations will translate into lower sensitivity in detecting
parts of the domain where significant differences between groups are present. As a
consequence the Unrestricted - LM test will prove to be less powerful than the other
47
48 CHAPTER 4. RESULTS OF TEST FOR THE FIXED EFFECT
tests. Since the three tests that do consider the presence of the random effect behave
similarly, only the graphical results of the Restricted - LM test will be provided for
the pairwise comparisons. This test is much more computationally efficient than
the tests based on the Linear Mixed Model, whose fitting for each permutation
requires a long time. All the tests are performed choosing two significance levels:
1% (dark gray bands) and 5% (light gray bands). For simplicity of exposition the
comments are based on the results obtained with a significance level of α = 5%,
which is the most commonly used level in statistical analysis.
The results of the Original and Modified Datasets (two peaks representations)
are discussed in the following Sections. As concerns the Original and Modified
Peak1-Peak2 Datasets, their analysis is reported in Appendix A.
4.1 Results of the Original Datasets with fixed ef-
fect lameLeg
4.1.1 VERTICAL Original Dataset
Fixed Effect: lameLeg (No, RH, RF, LH, LF)
Random Effect: horse (B1, B2, B3, B4, B5, B6, B7, B8)
Linear Model
aV ERTorig (t) = µ(t) + lameLegg(t) + εig(t) (4.1)
Linear Mixed Model
aV ERTorijg (t) = µ(t) + lameLegg(t) + horsej(t) + εijg(t) (4.2)
Where:
• index i indicates the i-th observation, i = 1, ..., 70;
• index j indicates the j-th level of horse, j = 1, ..., 8;
• index g indicates the g-th level of lameLeg, g = 1, ..., 5;
• horsej(t) are i.i.d. zero-mean random functions with variance σ2j (t), indepen-
dent from εijg(t), j = 1, ..., 8 t ∈ [0, 1]
TestH0 : lameLegg(t) = 0 ∀g ∈ {1, ..., 5}, ∀t ∈ [0, 1]
H1 : lameLegg(t) 6= 0 for some g ∈ {1, ..., 5} and some t ∈ [0, 1](4.3)
Test statistic
T Ifixed =
∫I
SSE(t)/dfexplSSR(t)/dfres
dt
49
50 CHAPTER 4. RESULTS OF TEST FOR THE FIXED EFFECT
Results
From Figure 4.1 it can be noted that each of the four types of test detects a
significant effect in the positive peaks of the vertical acceleration functions and also
in parts of the domain before the positive peaks. This corresponds approximately
to the phase of landing, followed by the hoof contact with the ground of the right
fore and left hindlimb for the first positive peak, and of the left fore and right
hindlimb for the second one.
Figure 4.2 gives further details on the reasons that have led to the rejection of
the null hypothesis, comparing two levels of the fixed effect lameLeg at a time. In
the first row the level “No” (sound horse) is compared to all the other levels: in all
the panels the IWT procedure finds differences in the peak relative to the stance
phase of the lame diagonal (first peak for RF and LH, second peak for LF and
RH) and in a time interval before the maximum acceleration of the sound diagonal
(second peak for RF and LH, first peak for LF and RH). This means that, with
respect to a sound horse, lameness influences the vertical acceleration of the trunk
not only when the lame limb strikes the ground, but also during the landing of the
sound diagonal. This is due to compensatory movements of the horse, which aims
at the reduction of the loading on the painful leg through abnormal movement of a
body part (head nod or pelvic hike), weight shifting (to the contralateral or diagonal
limb or torso), change in joint angles (lack of fetlock extension), and alterations in
foot flight (Adams and Stashak’s, 2011).
When comparing the levels of lameLeg belonging to the same diagonal, i.e.
RF-LH and LF-RH, we can see that most of the differences are detected in the part
of the stride relative to the sound diagonal (second part for RF-LH and first part
for LF-RH). Great differences are also found when comparing lame limb belonging
to different diagonals, both in the case of lame limbs on the same side (fore-hind
comparisons) and in the case of lame forelimbs or hindlimbs (right-left comparisons).
4.1. RESULTS OF THE ORIGINAL DATASETS 51
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Test fixed effect: unrestricted − LM
Ver
tical
Acc
eler
atio
ns
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Test fixed effect: restricted − LM
Ver
tical
Acc
eler
atio
ns
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Test fixed effect: unrestricted − LMM
Ver
tical
Acc
eler
atio
ns
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Test fixed effect: restricted − LMM
Ver
tical
Acc
eler
atio
ns
(a)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Test fixed effect
Adj
uste
d p−
valu
e
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
unrestricted − LMrestricted − LMunrestricted − LMMrestricted − LMM
(b)
Figure 4.1: (a) Results of the four different tests for significance of the fixed effectlameLeg in the Vertical Original Dataset: regions presenting p-value ≤ 1% are representedin dark gray, p-value ≤ 5% in light gray. Acceleration curves are colored depending onthe levels of the factor lameLeg: No, RH, RF, LH, LF.(b) Adjusted p-value function of the four different tests for significance of the fixed effectlameLeg in the Vertical Original Dataset. The two red lines represent the thresholds 0.01and 0.05.
52 CHAPTER 4. RESULTS OF TEST FOR THE FIXED EFFECT
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
No
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
RH
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
RF
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
LH
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
LF
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
No−RH
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
No−RH
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
No−RF
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
No−RF
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
No−LH
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
No−LH
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
No−LF
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
No−LF
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
RH−RF
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
RH−RF
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
RH−LH
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
RH−LH
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
RH−LF
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
RH−LF
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
RF−LH
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
RF−LH
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
RF−LF
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
RF−LF
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
LH−LF
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
LH−LF
X
p−va
lue
Figure 4.2: Results of the restricted - LM pairwise test for significance of the fixedeffect lameLeg in the Vertical Original Dataset. Diagonal: plots of the functional data ofeach group, all the other data are represented in gray. Superior diagonal: means of eachpairwise comparison, and regions presenting a significant difference controlling the IWERat 1% (dark gray areas) and 5% (light gray areas). Inferior diagonal: p-value (dashed line)and adjusted p-value (solid line) functions associated to the pairwise differences betweenthe vertical accelerations of the five group representing the lame limb.
4.1. RESULTS OF THE ORIGINAL DATASETS 53
4.1.2 TRANSVERSAL Original Dataset
Fixed Effect: lameLeg (No, RH, RF, LH, LF)
Random Effect: horse (B1, B2, B3, B4, B5, B6, B7, B8)
Linear Model
aTRANSorig (t) = µ(t) + lameLegg(t) + εig(t) (4.4)
Linear Mixed Model
aTRANSorijg (t) = µ(t) + lameLegg(t) + horsej(t) + εijg(t) (4.5)
Where:
• index i indicates the i-th observation, i = 1, ..., 70;
• index j indicates the j-th level of horse, j = 1, ..., 8;
• index g indicates the g-th level of lameLeg, g = 1, ..., 5;
• horsej(t) are i.i.d. zero-mean random functions with variance σ2j (t), indepen-
dent from εijg(t), j = 1, ..., 8 t ∈ [0, 1]
Test H0 : lameLegg(t) = 0 ∀g ∈ {1, ..., 5}, ∀t ∈ [0, 1]
H1 : lameLegg(t) 6= 0 for some g ∈ {1, ..., 5} and some t ∈ [0, 1]
Test statistic
T Ifixed =
∫I
SSE(t)/dfexplSSR(t)/dfres
dt
Results
Looking at Figure 4.3 we note that the factor lameLeg is considered significant
for the model of transversal accelerations in almost the whole the domain, excluding
the regions where the highest (in absolute value) peaks are present.
54 CHAPTER 4. RESULTS OF TEST FOR THE FIXED EFFECT
Observing in detail Figure 4.4, we note that the results are very noisy, in fact
several intervals are highlighted in many comparisons. This is probably due to the
high variability of the data themselves, and has to be considered when discussing
the results. The adjusted p-value in the IWT procedure is defined in such a way
that given any interval of the domain where the null hypothesis is not violated, the
probability of wrongly selecting it as significant is controlled. Consequently, when
a great number of intervals is found, there is a real possibility of having selected
some of them incorrectly.
One of the first things to notice is that, when lameness affects one of the
hindlimbs, it appears to have no effect on the transversal acceleration signals: in
panels No - RH, No - LH and RH - LH no gray bands are present. In all the other
panels differences in both shift and amplitude of the peaks are detected.
4.1. RESULTS OF THE ORIGINAL DATASETS 55
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Test fixed effect: unrestricted − LM
Tran
sver
sal A
ccel
erat
ions
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Test fixed effect: restricted − LM
Tran
sver
sal A
ccel
erat
ions
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Test fixed effect: unrestricted − LMM
Tran
sver
sal A
ccel
erat
ions
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Test fixed effect: restricted − LMM
Tran
sver
sal A
ccel
erat
ions
(a)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Test fixed effect
Adj
uste
d p−
valu
e
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
unrestricted − LMrestricted − LMunrestricted − LMMrestricted − LMM
(b)
Figure 4.3: (a) Results of the four different tests for significance of the fixed effectlameLeg in the Transversal Original Dataset: regions presenting p-value ≤ 1% arerepresented in dark gray, p-value ≤ 5% in light gray. Acceleration curves are coloreddepending on the levels of the factor lameLeg: No, RH, RF, LH, LF.(b) Adjusted p-value function of the four different tests for significance of the fixed effectlameLeg in the Transversal Original Dataset. The two red lines represent the thresholds0.01 and 0.05.
56 CHAPTER 4. RESULTS OF TEST FOR THE FIXED EFFECT
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
No
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
RH
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
RF
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
LH
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
LF
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
No−RH
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
No−RH
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
No−RF
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
No−RF
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
No−LH
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
No−LH
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
No−LF
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
No−LF
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
RH−RF
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
RH−RF
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
RH−LH
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
RH−LH
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
RH−LF
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
RH−LF
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
RF−LH
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
RF−LH
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
RF−LF
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
RF−LF
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
LH−LF
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
LH−LF
X
p−va
lue
Figure 4.4: Results of the restricted - LM pairwise test for significance of the fixed effectlameLeg in the Transversal Original Dataset. Diagonal: plots of the functional data ofeach group, all the other data are represented in gray. Superior diagonal: means of eachpairwise comparison, and regions presenting a significant difference controlling the IWERat 1% (dark gray areas) and 5% (light gray areas). Inferior diagonal: p-value (dashed line)and adjusted p-value (solid line) functions associated to the pairwise differences betweenthe vertical accelerations of the five group representing the lame limb.
4.1. RESULTS OF THE ORIGINAL DATASETS 57
4.1.3 LONGITUDINAL Original Dataset
Fixed Effect: lameLeg (No, RH, RF, LH, LF)
Random Effect: horse (B1, B2, B3, B4, B5, B6, B7, B8)
Linear Model
aLONGorig (t) = µ(t) + lameLegg(t) + εig(t) (4.6)
Linear Mixed Model
aLONGorijg (t) = µ(t) + lameLegg(t) + horsej(t) + εijg(t) (4.7)
Where:
• index i indicates the i-th observation, i = 1, ..., 70;
• index j indicates the j-th level of horse, j = 1, ..., 8;
• index g indicates the g-th level of lameLeg, g = 1, ..., 5;
• horsej(t) are i.i.d. zero-mean random functions with variance σ2j (t), indepen-
dent from εijg(t), j = 1, ..., 8 t ∈ [0, 1]
Test H0 : lameLegg(t) = 0 ∀g ∈ {1, ..., 5}, ∀t ∈ [0, 1]
H1 : lameLegg(t) 6= 0 for some g ∈ {1, ..., 5} and some t ∈ [0, 1]
Test statistic
T Ifixed =
∫I
SSE(t)/dfexplSSR(t)/dfres
dt
Results
Similarly to the case of transversal accelerations, effects of hindlimb lamenesses
are not detected in longitudinal signals: see panels No - RH, No - LH and RH - LH
of Figure 4.6. On the contrary, the presence of a lame limb in one of the forelegs
affects the shape of longitudinal acceleration curves, modifying mostly the peak
58 CHAPTER 4. RESULTS OF TEST FOR THE FIXED EFFECT
relative to the stance phase of the lame diagonal. According to that, in panels No -
RF and No - LF the presence of gray bands is concentrated in the first half and in
the second half of the stride, respectively. Also an interval preceding the negative
peak of the sound diagonal is selected: in both cases the colored curves are slightly
shifted to the left, which is interpretable as an attempt to anticipate the stance
phase of the sound limbs pair in order to unload the painful leg. Comparing the
two groups corresponding to the forelimbs (RF - LF) many intervals are selected
and here it is easy to note that swapping the two halves of one of the two curves,
the resulting would be very similar to the other one: this is what happens in the
construction of the Modified Datasets, in which the information on the lame side is
lost during the swapping process. When considering the comparison between a fore
and a hindlimb we note that the longitudinal accelerations relative to hindlimbs
lameness (blue and orange curves) are almost null (constant velocity) after the
positive peak of the lame diagonal and before the negative peak of the healty
diagonal. On the contrary in violet and green curves we can see that there is a
more gradual decrease of the acceleration between the two peaks.
4.1. RESULTS OF THE ORIGINAL DATASETS 59
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Test fixed effect: unrestricted − LM
Long
itudi
nal A
ccel
erat
ions
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Test fixed effect: restricted − LM
Long
itudi
nal A
ccel
erat
ions
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Test fixed effect: unrestricted − LMM
Long
itudi
nal A
ccel
erat
ions
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Test fixed effect: restricted − LMM
Long
itudi
nal A
ccel
erat
ions
(a)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Test fixed effect
Adj
uste
d p−
valu
e
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
unrestricted − LMrestricted − LMunrestricted − LMMrestricted − LMM
(b)
Figure 4.5: (a) Results of the four different tests for significance of the fixed effectlameLeg in the Longitudinal Original Dataset: regions presenting p-value ≤ 1% arerepresented in dark gray, p-value ≤ 5% in light gray. Acceleration curves are coloreddepending on the levels of the factor lameLeg: No, RH, RF, LH, LF.(b) Adjusted p-value function of the four different tests for significance of the fixed effectlameLeg in the Longitudinal Original Dataset. The two red lines represent the thresholds0.01 and 0.05.
60 CHAPTER 4. RESULTS OF TEST FOR THE FIXED EFFECT
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
No
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
RH
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
RF
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
LH
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
LF
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
No−RH
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
No−RH
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
No−RF
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
No−RF
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
No−LH
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
No−LH
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
No−LF
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
No−LF
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
RH−RF
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
RH−RF
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
RH−LH
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
RH−LH
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
RH−LF
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
RH−LF
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
RF−LH
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
RF−LH
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
RF−LF
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
RF−LF
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
LH−LF
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
LH−LF
X
p−va
lue
Figure 4.6: Results of the restricted - LM pairwise test for significance of the fixed effectlameLeg in the Longitudinal Original Dataset. Diagonal: plots of the functional data ofeach group, all the other data are represented in gray. Superior diagonal: means of eachpairwise comparison, and regions presenting a significant difference controlling the IWERat 1% (dark gray areas) and 5% (light gray areas). Inferior diagonal: p-value (dashed line)and adjusted p-value (solid line) functions associated to the pairwise differences betweenthe vertical accelerations of the five group representing the lame limb.
4.2 Results of the Modified Datasets with fixed ef-
fect lameForeHind
As anticipated in Subsection 3.4.1, the information of the lame side is lost due to
the swapping operation performed during the construction of the Modified Datasets.
This fact was confirmed by pairwise tests to compare levels RF-LF and RH-LH,
which detected no significant differences between the acceleration curves belonging
to these groups at a significance level of 5%. Consequently, the analyses presented
in this Section consider as fixed effect the factor lameForeHind.
4.2.1 VERTICAL Modified Dataset
Fixed Effect: lameForeHind (No, Hind, Fore)
Random Effect: horse (B1, B2, B3, B4, B5, B6, B7, B8)
Linear Model
aV ERTmodig (t) = µ(t) + lameForeHindg(t) + εig(t) (4.8)
Linear Mixed Model
aV ERTmodijg (t) = µ(t) + lameForeHindg(t) + horsej(t) + εijg(t) (4.9)
Where:
• index i indicates the i-th observation, i = 1, ..., 70;
• index j indicates the j-th level of horse, j = 1, ..., 8;
• index g indicates the g-th level of lameForeHind, g = 1, 2, 3;
• horsej(t) are i.i.d. zero-mean random functions with variance σ2j (t), indepen-
dent from εijg(t), j = 1, ..., 8 t ∈ [0, 1]
TestH0 : lameForeHindg(t) = 0 ∀g ∈ {1, 2, 3}, ∀t ∈ [0, 1]
H1 : lameForeHindg(t) 6= 0 for some g ∈ {1, 2, 3} and some t ∈ [0, 1]
(4.10)
61
62 CHAPTER 4. RESULTS OF TEST FOR THE FIXED EFFECT
Test statistic
T Ifixed =
∫I
SSE(t)/dfexplSSR(t)/dfres
dt
Results
Figure 4.7 shows how the effect of the factor lameForeHind is significant in most
part of the domain, except for the two negative peaks.
In Figure 4.8 pairwise comparison are presented. As in the case of the Vertical
Original Dataset, differences with curves of sound horses (first row) are detected
mostly in the stance phase of the lame diagonal (second peak), where the vertical
accelerations of lame horses are lessened with respect to the ones associated to
healty horses. As regards the level “Fore”, we can note that the vertical accelerations
curves are (on average) lower than the accelerations of the level “Hind” in the first
peak (sound diagonal), while the second peak (lame diagonal) seems to be slightly
shifted to the right with respect to the corresponding peak of both level “Hind” and
“No”. This shift of the second peak with respect to curves of a sound horse is also
present, even if less emphasised, when the lameness is induced in a hindlimb. The
rationale is that the horse seeks to minimize the pressure on the painful limb by a
faster change of weight to the healthy diagonal. This corresponds to a delay in the
part of the signal associated with the lame diagonal (i.e. the second peak) and it is
consistent with results found by Sørensen et al., 2012.
4.2. RESULTS OF THE MODIFIED DATASETS 63
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Test fixed effect: unrestricted − LM
Ver
tical
Acc
eler
atio
ns
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Test fixed effect: restricted − LM
Ver
tical
Acc
eler
atio
ns
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Test fixed effect: unrestricted − LMM
Ver
tical
Acc
eler
atio
ns
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Test fixed effect: restricted − LMM
Ver
tical
Acc
eler
atio
ns
(a)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Test fixed effect
Adj
uste
d p−
valu
e
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
unrestricted − LMrestricted − LMunrestricted − LMMrestricted − LMM
(b)
Figure 4.7: (a) Results of the four different tests for significance of the fixed effectlameForeHind in the Vertical Modified Dataset: regions presenting p-value ≤ 1% arerepresented in dark gray, p-value ≤ 5% in light gray. Acceleration curves are coloreddepending on the levels of the factor lameForeHind: No, Hind, Fore.(b) Adjusted p-value function of the four different tests for significance of the fixedeffect lameForeHind in the Vertical Modified Dataset. The two red lines represent thethresholds 0.01 and 0.05.
64 CHAPTER 4. RESULTS OF TEST FOR THE FIXED EFFECT
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
No
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Hind
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Fore
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
No−Hind
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
No−Hind
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
No−Fore
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
No−Fore
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Hind−Fore
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Hind−Fore
X
p−va
lue
Figure 4.8: Results of the restricted - LM pairwise test for significance of the fixed effectlameForeHind in the Vertical Modified Dataset. Diagonal: plots of the functional dataof each group, all the other data are represented in gray. Superior diagonal: means of eachpairwise comparison, and regions presenting a significant difference controlling the IWERat 1% (dark gray areas) and 5% (light gray areas). Inferior diagonal: p-value (dashed line)and adjusted p-value (solid line) functions associated to the pairwise differences betweenthe vertical accelerations of the five group representing the lame limb.
4.2. RESULTS OF THE MODIFIED DATASETS 65
4.2.2 TRANSVERSAL Modified Dataset
Fixed Effect: lameForeHind (No, Hind, Fore)
Random Effect: horse (B1, B2, B3, B4, B5, B6, B7, B8)
Linear Model
aTRANSmodig (t) = µ(t) + lameForeHindg(t) + εig(t) (4.11)
Linear Mixed Model
aTRANSmodijg (t) = µ(t) + lameForeHindg(t) + horsej(t) + εijg(t) (4.12)
Where:
• index i indicates the i-th observation, i = 1, ..., 70;
• index j indicates the j-th level of horse, j = 1, ..., 8;
• index g indicates the g-th level of lameForeHind, g = 1, 2, 3;
• horsej(t) are i.i.d. zero-mean random functions with variance σ2j (t), indepen-
dent from εijg(t), j = 1, ..., 8 t ∈ [0, 1]
TestH0 : lameForeHindg(t) = 0 ∀g ∈ {1, 2, 3}, ∀t ∈ [0, 1]
H1 : lameForeHindg(t) 6= 0 for some g ∈ {1, 2, 3} and some t ∈ [0, 1]
Test statistic
T Ifixed =
∫I
SSE(t)/dfexplSSR(t)/dfres
dt
Results
Results shown in Figure 4.9 and Figure 4.10 are consistent with what has
previously been found in the case of Transversal Original Dataset.
66 CHAPTER 4. RESULTS OF TEST FOR THE FIXED EFFECT
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2Test fixed effect: unrestricted − LM
Tran
sver
sal A
ccel
erat
ions
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Test fixed effect: restricted − LM
Tran
sver
sal A
ccel
erat
ions
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Test fixed effect: unrestricted − LMM
Tran
sver
sal A
ccel
erat
ions
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Test fixed effect: restricted − LMM
Tran
sver
sal A
ccel
erat
ions
(a)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Test fixed effect
Adj
uste
d p−
valu
e
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
unrestricted − LMrestricted − LMunrestricted − LMMrestricted − LMM
(b)
Figure 4.9: (a) Results of the four different tests for significance of the fixed effectlameForeHind in the Transversal Modified Dataset: regions presenting p-value ≤ 1%are represented in dark gray, p-value ≤ 5% in light gray. Acceleration curves are coloreddepending on the levels of the factor lameForeHind: No, Hind, Fore.(b) Adjusted p-value function of the four different tests for significance of the fixed effectlameForeHind in the Transversal Modified Dataset. The two red lines represent thethresholds 0.01 and 0.05.
4.2. RESULTS OF THE MODIFIED DATASETS 67
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
No
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Hind
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Fore
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
No−Hind
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
No−Hind
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
No−Fore
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
No−Fore
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Hind−Fore
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Hind−Fore
X
p−va
lue
Figure 4.10: Results of the restricted - LM pairwise test for significance of the fixed effectlameForeHind in the Transversal Modified Dataset. Diagonal: plots of the functionaldata of each group, all the other data are represented in gray. Superior diagonal: meansof each pairwise comparison, and regions presenting a significant difference controllingthe IWER at 1% (dark gray areas) and 5% (light gray areas). Inferior diagonal: p-value (dashed line) and adjusted p-value (solid line) functions associated to the pairwisedifferences between the vertical accelerations of the five group representing the lame limb.
68 CHAPTER 4. RESULTS OF TEST FOR THE FIXED EFFECT
4.2.3 LONGITUDINAL Modified Dataset
Fixed Effect: lameForeHind (No, Hind, Fore)
Random Effect: horse (B1, B2, B3, B4, B5, B6, B7, B8)
Linear Model
aLONGmodig (t) = µ(t) + lameForeHindg(t) + εig(t) (4.13)
Linear Mixed Model
aLONGmodijg (t) = µ(t) + lameForeHindg(t) + horsej(t) + εijg(t) (4.14)
Where:
• index i indicates the i-th observation, i = 1, ..., 70;
• index j indicates the j-th level of horse, j = 1, ..., 8;
• index g indicates the g-th level of lameForeHind, g = 1, 2, 3;
• horsej(t) are i.i.d. zero-mean random functions with variance σ2j (t), indepen-
dent from εijg(t), j = 1, ..., 8 t ∈ [0, 1]
TestH0 : lameForeHindg(t) = 0 ∀g ∈ {1, 2, 3}, ∀t ∈ [0, 1]
H1 : lameForeHindg(t) 6= 0 for some g ∈ {1, 2, 3} and some t ∈ [0, 1]
Test statistic
T Ifixed =
∫I
SSE(t)/dfexplSSR(t)/dfres
dt
Results
The null hypothesis of no lameForeHind effect on longitudinal accelerations is
rejected in the first part of the stance phase of the sound diagonal and in all the
second half of the gait cycle (Figure 4.11).
4.2. RESULTS OF THE MODIFIED DATASETS 69
As in the vertical case, even in the longitudinal component it can be noted
that when the horse is affected by forelimb or hindlimb lameness, the positive
and negative peaks of the second half of the stride changes with respect to the
nonlame mean curve (first row of Figure 4.12). In particular the amplitude of the
peaks decreases and their position is shifted to the right. This is more noticeable
in the case of forelimb lameness, where also the first couple of peaks tends to be
anticipated. In general the presence of a lame forelimb seems to have a larger
impact on the acceleration signals and our tests are able to highlight the intervals
where these effects are more evident.
70 CHAPTER 4. RESULTS OF TEST FOR THE FIXED EFFECT
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Test fixed effect: unrestricted − LMLo
ngitu
dina
l Acc
eler
atio
ns
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Test fixed effect: restricted − LM
Long
itudi
nal A
ccel
erat
ions
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Test fixed effect: unrestricted − LMM
Long
itudi
nal A
ccel
erat
ions
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Test fixed effect: restricted − LMM
Long
itudi
nal A
ccel
erat
ions
(a)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Test fixed effect
Adj
uste
d p−
valu
e
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
unrestricted − LMrestricted − LMunrestricted − LMMrestricted − LMM
(b)
Figure 4.11: (a) Results of the four different tests for significance of the fixed effectlameForeHind in the Longitudinal Modified Dataset: regions presenting p-value ≤ 1%are represented in dark gray, p-value ≤ 5% in light gray. Acceleration curves are coloreddepending on the levels of the factor lameForeHind: No, Hind, Fore.(b) Adjusted p-value function of the four different tests for significance of the fixed effectlameForeHind in the Longitudinal Modified Dataset. The two red lines represent thethresholds 0.01 and 0.05.
4.2. RESULTS OF THE MODIFIED DATASETS 71
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
No
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Hind
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Fore
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
No−Hind
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
No−Hind
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
No−Fore
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
No−Fore
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Hind−Fore
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Hind−Fore
X
p−va
lue
Figure 4.12: Results of the restricted - LM pairwise test for significance of the fixed effectlameForeHind in the Longitudinal Modified Dataset. Diagonal: plots of the functionaldata of each group, all the other data are represented in gray. Superior diagonal: meansof each pairwise comparison, and regions presenting a significant difference controllingthe IWER at 1% (dark gray areas) and 5% (light gray areas). Inferior diagonal: p-value (dashed line) and adjusted p-value (solid line) functions associated to the pairwisedifferences between the vertical accelerations of the five group representing the lame limb.
72 CHAPTER 4. RESULTS OF TEST FOR THE FIXED EFFECT
4.3 Results of the Modified Datasets with fixed ef-
fect lameForeHind_Degree
In this Section we examine the significance of the factor lameForeHind_Degree,
which introduces the distinction between two degrees of induced lameness. This
further information carried by the factor lameDegree can be considered only in
the analysis of Modified Datasets. In fact in the Original Datasets this additional
information would make the permutational procedure impossible to apply because
of the reduction of the number of permutable observations.
4.3.1 VERTICAL Modified Dataset
Fixed Effect: lameForeHind_Degree (No, Hind_low, Hind_mod, Fore_low, Fore_mod)
Random Effect: horse (B1, B2, B3, B4, B5, B6, B7, B8)
Linear Model
aV ERTmodig (t) = µ(t) + lameForeHind_Degreeg(t) + εig(t) (4.15)
Linear Mixed Model
aV ERTmodijg (t) = µ(t) + lameForeHind_Degreeg(t) + horsej(t) + εijg(t) (4.16)
Where:
• index i indicates the i-th observation, i = 1, ..., 70;
• index j indicates the j-th level of horse, j = 1, ..., 8;
• index g indicates the g-th level of lameForeHind_Degree, g = 1, ..., 5;
• horsej(t) are i.i.d. zero-mean random functions with variance σ2j (t), indepen-
dent from εijg(t), j = 1, ..., 8 t ∈ [0, 1]
TestH0 : lameForeHind_Degreeg(t) = 0 ∀g ∈ {1, ..., 5}, ∀t ∈ [0, 1]
H1 : lameForeHind_Degreeg(t) 6= 0 for some g ∈ {1, ..., 5} and some t ∈ [0, 1]
(4.17)
4.3. RESULTS OF THE MODIFIED DATASETS 73
Test statistic
T Ifixed =
∫I
SSE(t)/dfexplSSR(t)/dfres
dt
Results
The fixed effect lameForeHind_Degree appears to be significant in the initial
part of the first half of the stride (sound diagonal) and in all the second half (lame
diagonal).
During trotting the horse is expected to modify the pressure on the lame limb
pair, in particular to diminish it with respect to the pressure the healthy limb pair,
in a way that depends on the degree of lameness. The extreme case happens when
lameness is so severe that the affected limb is completely unable to bear weight.
Analysing Figure 4.14 we expect to see a confirmation of this fact in a reduction
of the peak height and in a more evident shifting to the right of the second peak
at increasing lameness degree. Actually, this is what happens in the comparison
Fore_low - Fore_mod, while no difference is detected between the groups Hind_low
- Hind_mod. The inability to distinguish the low from the moderate lameness
degree in the hindlimbs might be caused by the position of the accelerometer, which
is placed at the lowest point of the back in the midline (above T13) near the head.
For this reason, compensatory movements of the caudal part of the trunk (which
are more consistent in hindlimb lameness) are not fully captured by the sensor.
Considering the fifth column of Figure 4.14, we can see that the higher the
lameness degree is, the lower the second peak of the signal and the more right
shifted becomes with respect to the baseline signal coming from sound horses.
Another fact to point out is that the first peak of the acceleration curves tends to
be slightly left shifted, in a more manifest way for moderate lameness degrees. This
means that with growing level of pain the horse tries both to retard the stance
phase of the injured leg and to anticipate the stance phase of the sound limb pair.
As we expected, differences in all the comparison are more easily detectable when
lameness is more severe and affects one of the forelimbs.
74 CHAPTER 4. RESULTS OF TEST FOR THE FIXED EFFECT
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Test fixed effect: unrestricted − LMV
ertic
al A
ccel
erat
ions
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Test fixed effect: restricted − LM
Ver
tical
Acc
eler
atio
ns
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Test fixed effect: unrestricted − LMM
Ver
tical
Acc
eler
atio
ns
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Test fixed effect: restricted − LMM
Ver
tical
Acc
eler
atio
ns
(a)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Test fixed effect
Adj
uste
d p−
valu
e
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
unrestricted − LMrestricted − LMunrestricted − LMMrestricted − LMM
(b)
Figure 4.13: (a) Results of the four different tests for significance of the fixed effectlameForeHind_Degree in the Vertical Modified Dataset: regions presenting p-value≤ 1% are represented in dark gray, p-value ≤ 5% in light gray. Acceleration curves arecolored depending on the levels of the factor lameForeHind_Degree: No, Fore_low,Fore_mod, Hind_low, Hind_mod.(b) Adjusted p-value function of the four different tests for significance of the fixed effectlameForeHind_Degree in the Vertical Modified Dataset. The two red lines representthe thresholds 0.01 and 0.05.
4.3. RESULTS OF THE MODIFIED DATASETS 75
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Fore_low
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Fore_mod
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Hind_low
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Hind_mod
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
No
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Fore_low−Fore_mod
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Fore_low−Fore_mod
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Fore_low−Hind_low
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Fore_low−Hind_low
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Fore_low−Hind_mod
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Fore_low−Hind_mod
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Fore_low−No
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Fore_low−No
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Fore_mod−Hind_low
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Fore_mod−Hind_low
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Fore_mod−Hind_mod
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Fore_mod−Hind_mod
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Fore_mod−No
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Fore_mod−No
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Hind_low−Hind_mod
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Hind_low−Hind_mod
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Hind_low−No
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Hind_low−No
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Hind_mod−No
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Hind_mod−No
X
p−va
lue
Figure 4.14: Results of the restricted - LM pairwise test for significance of the fixedeffect lameForeHind_Degree in the Vertical Modified Dataset. Diagonal: plots ofthe functional data of each group, all the other data are represented in gray. Superiordiagonal: means of each pairwise comparison, and regions presenting a significant differencecontrolling the IWER at 1% (dark gray areas) and 5% (light gray areas). Inferior diagonal:p-value (dashed line) and adjusted p-value (solid line) functions associated to the pairwisedifferences between the vertical accelerations of the five group representing the lame limb.
76 CHAPTER 4. RESULTS OF TEST FOR THE FIXED EFFECT
4.3.2 TRANSVERSAL Modified Dataset
Fixed Effect: lameForeHind_Degree (No, Hind_low, Hind_mod, Fore_low, Fore_mod)
Random Effect: horse (B1, B2, B3, B4, B5, B6, B7, B8)
Linear Model
aTRANSmodig (t) = µ(t) + lameForeHind_Degreeg(t) + εig(t) (4.18)
Linear Mixed Model
aTRANSmodijg (t) = µ(t) + lameForeHind_Degreeg(t) + horsej(t) + εijg(t) (4.19)
Where:
• index i indicates the i-th observation, i = 1, ..., 70;
• index j indicates the j-th level of horse, j = 1, ..., 8;
• index g indicates the g-th level of lameForeHind_Degree, g = 1, ..., 5;
• horsej(t) are i.i.d. zero-mean random functions with variance σ2j (t), indepen-
dent from εijg(t), j = 1, ..., 8 t ∈ [0, 1]
TestH0 : lameForeHind_Degreeg(t) = 0 ∀g ∈ {1, ..., 5}, ∀t ∈ [0, 1]
H1 : lameForeHind_Degreeg(t) 6= 0 for some g ∈ {1, ..., 5} and some t ∈ [0, 1]
Test statistic
T Ifixed =
∫I
SSE(t)/dfexplSSR(t)/dfres
dt
Results
In Figure 4.16 we can see that the obtained results are similar to the cases of
the factors lameLeg and lameForeHind. In particular the null hypothesis is not
rejected when comparing the groups of signals relative to lame hindlimbs with the
group of sound horses. Moreover, we note that the distinction between the low and
4.3. RESULTS OF THE MODIFIED DATASETS 77
the moderate lameness degree in the forelimbs is not as marked as in the vertical
accelerations.
78 CHAPTER 4. RESULTS OF TEST FOR THE FIXED EFFECT
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Test fixed effect: unrestricted − LMTr
ansv
ersa
l Acc
eler
atio
ns
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Test fixed effect: restricted − LM
Tran
sver
sal A
ccel
erat
ions
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Test fixed effect: unrestricted − LMM
Tran
sver
sal A
ccel
erat
ions
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Test fixed effect: restricted − LMM
Tran
sver
sal A
ccel
erat
ions
(a)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Test fixed effect
Adj
uste
d p−
valu
e
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
unrestricted − LMrestricted − LMunrestricted − LMMrestricted − LMM
(b)
Figure 4.15: (a) Results of the four different tests for significance of the fixed effectlameForeHind_Degree in the Transversal Modified Dataset: regions presenting p-value≤ 1% are represented in dark gray, p-value ≤ 5% in light gray. Acceleration curves arecolored depending on the levels of the factor lameForeHind_Degree: No, Fore_low,Fore_mod, Hind_low, Hind_mod.(b) Adjusted p-value function of the four different tests for significance of the fixed effectlameForeHind_Degree in the Transversal Modified Dataset. The two red lines representthe thresholds 0.01 and 0.05.
4.3. RESULTS OF THE MODIFIED DATASETS 79
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Fore_low
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Fore_mod
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Hind_low
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Hind_mod
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
No
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Fore_low−Fore_mod
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Fore_low−Fore_mod
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Fore_low−Hind_low
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Fore_low−Hind_low
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Fore_low−Hind_mod
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Fore_low−Hind_mod
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Fore_low−No
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Fore_low−No
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Fore_mod−Hind_low
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Fore_mod−Hind_low
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Fore_mod−Hind_mod
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Fore_mod−Hind_mod
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Fore_mod−No
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Fore_mod−No
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Hind_low−Hind_mod
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Hind_low−Hind_mod
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Hind_low−No
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Hind_low−No
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Hind_mod−No
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Hind_mod−No
X
p−va
lue
Figure 4.16: Results of the restricted - LM pairwise test for significance of the fixedeffect lameForeHind_Degree in the Transversal Modified Dataset. Diagonal: plots ofthe functional data of each group, all the other data are represented in gray. Superiordiagonal: means of each pairwise comparison, and regions presenting a significant differencecontrolling the IWER at 1% (dark gray areas) and 5% (light gray areas). Inferior diagonal:p-value (dashed line) and adjusted p-value (solid line) functions associated to the pairwisedifferences between the vertical accelerations of the five group representing the lame limb.
80 CHAPTER 4. RESULTS OF TEST FOR THE FIXED EFFECT
4.3.3 LONGITUDINAL Modified Dataset
Fixed Effect: lameForeHind_Degree (No, Hind_low, Hind_mod, Fore_low, Fore_mod)
Random Effect: horse (B1, B2, B3, B4, B5, B6, B7, B8)
Linear Model
aLONGmodig (t) = µ(t) + lameForeHind_Degreeg(t) + εig(t) (4.20)
Linear Mixed Model
aLONGmodijg (t) = µ(t) + lameForeHind_Degreeg(t) + horsej(t) + εijg(t) (4.21)
Where:
• index i indicates the i-th observation, i = 1, ..., 70;
• index j indicates the j-th level of horse, j = 1, ..., 8;
• index g indicates the g-th level of lameForeHind_Degree, g = 1, ..., 5;
• horsej(t) are i.i.d. zero-mean random functions with variance σ2j (t), indepen-
dent from εijg(t), j = 1, ..., 8 t ∈ [0, 1]
TestH0 : lameForeHind_Degreeg(t) = 0 ∀g ∈ {1, ..., 5}, ∀t ∈ [0, 1]
H1 : lameForeHind_Degreeg(t) 6= 0 for some g ∈ {1, ..., 5} and some t ∈ [0, 1]
Test statistic
T Ifixed =
∫I
SSE(t)/dfexplSSR(t)/dfres
dt
Results
As we can see from Figure 4.17, the fixed effect lameForeHind_Degree is
significant mostly in the regions preceding the first negative peak, between the
second negative peak and the second positive one and following this last peak.
4.3. RESULTS OF THE MODIFIED DATASETS 81
In Figure 4.18 we can observe that there are no differences between the two
lameness degrees induced in the hindlimbs. Confronting these two groups with the
sound one, at 5% level only a slight shift of the lame diagonal peak is detected
as significantly different with respect to the black curves. Instead the impact of
forelimbs lameness on the accelerations pattern is much more marked, especially in
all the part of the stride corresponding to the stance phase of the painful leg and
preceding the landing of the sound diagonal (see all the panels containing green
curves).
82 CHAPTER 4. RESULTS OF TEST FOR THE FIXED EFFECT
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Test fixed effect: unrestricted − LMLo
ngitu
dina
l Acc
eler
atio
ns
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Test fixed effect: restricted − LM
Long
itudi
nal A
ccel
erat
ions
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Test fixed effect: unrestricted − LMM
Long
itudi
nal A
ccel
erat
ions
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Test fixed effect: restricted − LMM
Long
itudi
nal A
ccel
erat
ions
(a)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Test fixed effect
Adj
uste
d p−
valu
e
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
unrestricted − LMrestricted − LMunrestricted − LMMrestricted − LMM
(b)
Figure 4.17: (a) Results of the four different tests for significance of the fixed effectlameForeHind in the Longitudinal Modified Dataset: regions presenting p-value ≤ 1%are represented in dark gray, p-value ≤ 5% in light gray. Acceleration curves are coloreddepending on the levels of the factor lameForeHind_Degree: No, Fore_low, Fore_mod,Hind_low, Hind_mod.(b) Adjusted p-value function of the four different tests for significance of the fixedeffect lameForeHind_Degree in the Longitudinal Modified Dataset. The two red linesrepresent the thresholds 0.01 and 0.05.
4.3. RESULTS OF THE MODIFIED DATASETS 83
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Fore_low
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Fore_mod
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Hind_low
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Hind_mod
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
No
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Fore_low−Fore_mod
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Fore_low−Fore_mod
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Fore_low−Hind_low
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Fore_low−Hind_low
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Fore_low−Hind_mod
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Fore_low−Hind_mod
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Fore_low−No
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Fore_low−No
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Fore_mod−Hind_low
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Fore_mod−Hind_low
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Fore_mod−Hind_mod
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Fore_mod−Hind_mod
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Fore_mod−No
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Fore_mod−No
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Hind_low−Hind_mod
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Hind_low−Hind_mod
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Hind_low−No
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Hind_low−No
X
p−va
lue
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Hind_mod−No
X
Y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Hind_mod−No
X
p−va
lue
Figure 4.18: Results of the restricted - LM pairwise test for significance of the fixedeffect lameForeHind_Degree in the Longitudinal Modified Dataset. Diagonal: plots ofthe functional data of each group, all the other data are represented in gray. Superiordiagonal: means of each pairwise comparison, and regions presenting a significant differencecontrolling the IWER at 1% (dark gray areas) and 5% (light gray areas). Inferior diagonal:p-value (dashed line) and adjusted p-value (solid line) functions associated to the pairwisedifferences between the vertical accelerations of the five group representing the lame limb.
Chapter 5
Results of test for the random effect
In this Chapter, the results of tests for significance of the random effect are
presented. All the tests have been performed according to the procedure described
in Section 2.4.
Unlike the case of testing for fixed effect, here testing the difference between all
the pairs of horses would not provide interesting information. In fact, the obtained
results would not be generalizable because they would be referred to the particular
pair of horses taken into account. For this reason no such test has been done in
this case.
The results regarding the Original and Modified Peak1-Peak2 Datasets are
reported in Appendix B, while the analysis concerning the Original and Modified
Datasets (two peaks representations) is presented in the following Sections.
5.1 Comments
Test for significance of the random effect is an important step in the analysis
presented in this Thesis, since it validates the introduction of a variance component
depending on the horse ID in our LMM and the restriction of the set of all possible
permutations in the test for fixed effect procedures.
Initially, the choice of considering the horse as a random intercept was suggested
by the structure of the data and was supported by some graphical inspection of the
acceleration signals (e.g. Figure 3.5 on page 45). Now a formal confirmation that
our initial choice was correct is given by the results of our tests. In fact, in all our
LMMs the random effect is significant in almost the whole domain.
85
86 CHAPTER 5. RESULTS OF THE ORIGINAL DATASETS
5.2 Results of the Original Datasets with fixed ef-
fect lameLeg
5.2.1 VERTICAL Original Dataset
Fixed Effect: lameLeg (No, RH, RF, LH, LF)
Random Effect: horse (B1, B2, B3, B4, B5, B6, B7, B8)
Linear Mixed Model
aV ERTorijg (t) = µ(t) + lameLegg(t) + horsej(t) + εijg(t)
Where:
• index i indicates the i-th observation, i = 1, ..., 70;
• index j indicates the j-th level of horse, j = 1, ..., 8;
• index g indicates the g-th level of lameLeg, g = 1, ..., 5;
• horsej(t) are i.i.d. zero-mean random functions with variance σ2j (t), indepen-
dent from εijg(t), j = 1, ..., 8 t ∈ [0, 1]
Test H0 : σ2j (t) = 0 ∀j ∈ {1, ..., 8}, ∀t ∈ [0, 1]
H1 : σ2j (t) 6= 0 for some j ∈ {1, ..., 8} and some t ∈ [0, 1]
(5.1)
Test statistic
T Irandom = e∫I (−2 log (L
Ht0−L
Ht1))dt
87
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Test random effect
Ver
tical
Acc
eler
atio
ns
(a)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Test Random Effect
Adj
uste
d p−
valu
e
(b)
Figure 5.1: (a) Results of the test for significance of the random effect horse in theVertical Original Dataset: regions presenting p-value ≤ 1% are represented in dark gray,p-value ≤ 5% in light gray. Acceleration curves are colored depending on the levels of thefactor lameLeg: No, RH, RF, LH, LF.(b) Adjusted p-value function of the test for significance of the random effect horse in theVertical Original Dataset. The two red lines represent the thresholds 0.01 and 0.05.
88 CHAPTER 5. RESULTS OF THE ORIGINAL DATASETS
5.2.2 TRANSVERSAL Original Dataset
Fixed Effect: lameLeg (No, RH, RF, LH, LF)
Random Effect: horse (B1, B2, B3, B4, B5, B6, B7, B8)
Linear Mixed Model
aTRANSorijg (t) = µ(t) + lameLegg(t) + horsej(t) + εijg(t)
Where:
• index i indicates the i-th observation, i = 1, ..., 70;
• index j indicates the j-th level of horse, j = 1, ..., 8;
• index g indicates the g-th level of lameLeg, g = 1, ..., 5;
• horsej(t) are i.i.d. zero-mean random functions with variance σ2j (t), indepen-
dent from εijg(t), j = 1, ..., 8 t ∈ [0, 1]
Test H0 : σ2j (t) = 0 ∀j ∈ {1, ..., 8}, ∀t ∈ [0, 1]
H1 : σ2j (t) 6= 0 for some j ∈ {1, ..., 8} and some t ∈ [0, 1]
Test statistic
T Irandom = e∫I (−2 log (L
Ht0−L
Ht1))dt
89
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Test random effect
Tran
sver
sal A
ccel
erat
ions
(a)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Test random effect
Adj
uste
d p−
valu
e
(b)
Figure 5.2: (a) Results of the test for significance of the random effect horse in theTransversal Original Dataset: regions presenting p-value ≤ 1% are represented in darkgray, p-value ≤ 5% in light gray. Acceleration curves are colored depending on the levelsof the factor lameLeg: No, RH, RF, LH, LF.(b) Adjusted p-value function of the test for significance of the random effect horse in theTransversal Original Dataset. The two red lines represent the thresholds 0.01 and 0.05.
90 CHAPTER 5. RESULTS OF THE ORIGINAL DATASETS
5.2.3 LONGITUDINAL Original Dataset
Fixed Effect: lameLeg (No, RH, RF, LH, LF)
Random Effect: horse (B1, B2, B3, B4, B5, B6, B7, B8)
Linear Mixed Model
aLONGorijg (t) = µ(t) + lameLegg(t) + horsej(t) + εijg(t)
Where:
• index i indicates the i-th observation, i = 1, ..., 70;
• index j indicates the j-th level of horse, j = 1, ..., 8;
• index g indicates the g-th level of lameLeg, g = 1, ..., 5;
• horsej(t) are i.i.d. zero-mean random functions with variance σ2j (t), indepen-
dent from εijg(t), j = 1, ..., 8 t ∈ [0, 1]
Test H0 : σ2j (t) = 0 ∀j ∈ {1, ..., 8}, ∀t ∈ [0, 1]
H1 : σ2j (t) 6= 0 for some j ∈ {1, ..., 8} and some t ∈ [0, 1]
Test statistic
T Irandom = e∫I (−2 log (L
Ht0−L
Ht1))dt
91
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Test random effect
Long
itudi
nal A
ccel
erat
ions
(a)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Test random effect
Adj
uste
d p−
valu
e
(b)
Figure 5.3: (a) Results of the test for significance of the random effect horse in theLongitudinal Original Dataset: regions presenting p-value ≤ 1% are represented in darkgray, p-value ≤ 5% in light gray. Acceleration curves are colored depending on the levelsof the factor lameLeg: No, RH, RF, LH, LF.(b) Adjusted p-value function of the test for significance of the random effect horse in theLongitudinal Original Dataset. The two red lines represent the thresholds 0.01 and 0.05.
92 CHAPTER 5. RESULTS OF THE MODIFIED DATASETS
5.3 Results of the Modified Datasets with fixed ef-
fect lameForeHind
5.3.1 VERTICAL Modified Dataset
Fixed Effect: lameForeHind (No, Hind, Fore)
Random Effect: horse (B1, B2, B3, B4, B5, B6, B7, B8)
Linear Mixed Model
aV ERTmodijg (t) = µ(t) + lameForeHindg(t) + horsej(t) + εijg(t)
Where:
• index i indicates the i-th observation, i = 1, ..., 70;
• index j indicates the j-th level of horse, j = 1, ..., 8;
• index g indicates the g-th level of lameForeHind, g = 1, 2, 3;
• horsej(t) are i.i.d. zero-mean random functions with variance σ2j (t), indepen-
dent from εijg(t), j = 1, ..., 8 t ∈ [0, 1]
Test H0 : σ2j (t) = 0 ∀j ∈ {1, ..., 8}, ∀t ∈ [0, 1]
H1 : σ2j (t) 6= 0 for some j ∈ {1, ..., 8} and some t ∈ [0, 1]
Test statistic
T Irandom = e∫I (−2 log (L
Ht0−L
Ht1))dt
93
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Test random effect
Ver
tical
Acc
eler
atio
ns
(a)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Test random effect
Adj
uste
d p−
valu
e
(b)
Figure 5.4: (a) Results of the test for significance of the random effect horse in theVertical Modified Dataset: regions presenting p-value ≤ 1% are represented in dark gray,p-value ≤ 5% in light gray. Acceleration curves are colored depending on the levels of thefactor lameForeHind: No, Hind, Fore.(b) Adjusted p-value function of the test for significance of the random effect horse in theVertical Modified Dataset. The two red lines represent the thresholds 0.01 and 0.05.
94 CHAPTER 5. RESULTS OF THE MODIFIED DATASETS
5.3.2 TRANSVERSAL Modified Dataset
Fixed Effect: lameForeHind (No, Hind, Fore)
Random Effect: horse (B1, B2, B3, B4, B5, B6, B7, B8)
Linear Mixed Model
aTRANSmodijg (t) = µ(t) + lameForeHindg(t) + horsej(t) + εijg(t)
Where:
• index i indicates the i-th observation, i = 1, ..., 70;
• index j indicates the j-th level of horse, j = 1, ..., 8;
• index g indicates the g-th level of lameForeHind, g = 1, 2, 3;
• horsej(t) are i.i.d. zero-mean random functions with variance σ2j (t), indepen-
dent from εijg(t), j = 1, ..., 8 t ∈ [0, 1]
Test H0 : σ2j (t) = 0 ∀j ∈ {1, ..., 8}, ∀t ∈ [0, 1]
H1 : σ2j (t) 6= 0 for some j ∈ {1, ..., 8} and some t ∈ [0, 1]
Test statistic
T Irandom = e∫I (−2 log (L
Ht0−L
Ht1))dt
95
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Test random effect
Tran
sver
sal A
ccel
erat
ions
(a)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Test random effect
Adj
uste
d p−
valu
e
(b)
Figure 5.5: (a) Results of the test for significance of the random effect horse in theTransversal Modified Dataset: regions presenting p-value ≤ 1% are represented in darkgray, p-value ≤ 5% in light gray. Acceleration curves are colored depending on the levelsof the factor lameForeHind: No, Hind, Fore.(b) Adjusted p-value function of the test for significance of the random effect horse in theTransversal Modified Dataset. The two red lines represent the thresholds 0.01 and 0.05.
96 CHAPTER 5. RESULTS OF THE MODIFIED DATASETS
5.3.3 LONGITUDINAL Modified Dataset
Fixed Effect: lameForeHind (No, Hind, Fore)
Random Effect: horse (B1, B2, B3, B4, B5, B6, B7, B8)
Linear Mixed Model
aLONGmodijg (t) = µ(t) + lameForeHindg(t) + horsej(t) + εijg(t)
Where:
• index i indicates the i-th observation, i = 1, ..., 70;
• index j indicates the j-th level of horse, j = 1, ..., 8;
• index g indicates the g-th level of lameForeHind, g = 1, 2, 3;
• horsej(t) are i.i.d. zero-mean random functions with variance σ2j (t), indepen-
dent from εijg(t), j = 1, ..., 8 t ∈ [0, 1]
Test H0 : σ2j (t) = 0 ∀j ∈ {1, ..., 8}, ∀t ∈ [0, 1]
H1 : σ2j (t) 6= 0 for some j ∈ {1, ..., 8} and some t ∈ [0, 1]
Test statistic
T Irandom = e∫I (−2 log (L
Ht0−L
Ht1))dt
97
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Test random effect
Long
itudi
nal A
ccel
erat
ions
(a)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Test random effect
Adj
uste
d p−
valu
e
(b)
Figure 5.6: (a) Results of the test for significance of the random effect horse in theLongitudinal Modified Dataset: regions presenting p-value ≤ 1% are represented in darkgray, p-value ≤ 5% in light gray. Acceleration curves are colored depending on the levelsof the factor lameForeHind: No, Hind, Fore.(b) Adjusted p-value function of the test for significance of the random effect horse in theLongitudinal Modified Dataset. The two red lines represent the thresholds 0.01 and 0.05.
98 CHAPTER 5. RESULTS OF THE MODIFIED DATASETS
5.4 Results of the Modified Datasets with fixed ef-
fect lameForeHind_Degree
5.4.1 VERTICAL Modified Dataset
Fixed Effect: lameForeHind_Degree (No, Hind_low, Hind_mod, Fore_low, Fore_mod)
Random Effect: horse (B1, B2, B3, B4, B5, B6, B7, B8)
Linear Mixed Model
aV ERTmodijg (t) = µ(t) + lameForeHind_Degreeg(t) + horsej(t) + εijg(t)
Where:
• index i indicates the i-th observation, i = 1, ..., 70;
• index j indicates the j-th level of horse, j = 1, ..., 8;
• index g indicates the g-th level of lameForeHind_Degree, g = 1, ..., 5;
• horsej(t) are i.i.d. zero-mean random functions with variance σ2j (t), indepen-
dent from εijg(t), j = 1, ..., 8 t ∈ [0, 1]
Test H0 : σ2j (t) = 0 ∀j ∈ {1, ..., 8}, ∀t ∈ [0, 1]
H1 : σ2j (t) 6= 0 for some j ∈ {1, ..., 8} and some t ∈ [0, 1]
Test statistic
T Irandom = e∫I (−2 log (L
Ht0−L
Ht1))dt
99
0.0 0.2 0.4 0.6 0.8 1.0
−1
01
2
Test random effect
Ver
tical
Acc
eler
atio
ns
(a)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Test random effect
Adj
uste
d p−
valu
e
(b)
Figure 5.7: (a) Results of the test for significance of the random effect horse in theVertical Modified Dataset: regions presenting p-value ≤ 1% are represented in dark gray,p-value ≤ 5% in light gray. Acceleration curves are colored depending on the levels of thefactor lameForeHind_Degree: No, Fore_low, Fore_mod, Hind_low, Hind_mod.(b) Adjusted p-value function of the test for significance of the random effect horse in theVertical Modified Dataset. The two red lines represent the thresholds 0.01 and 0.05.
100 CHAPTER 5. RESULTS OF THE MODIFIED DATASETS
5.4.2 TRANSVERSAL Modified Dataset
Fixed Effect: lameForeHind_Degree (No, Hind_low, Hind_mod, Fore_low, Fore_mod)
Random Effect: horse (B1, B2, B3, B4, B5, B6, B7, B8)
Linear Mixed Model
aTRANSmodijg (t) = µ(t) + lameForeHind_Degreeg(t) + horsej(t) + εijg(t)
Where:
• index i indicates the i-th observation, i = 1, ..., 70;
• index j indicates the j-th level of horse, j = 1, ..., 8;
• index g indicates the g-th level of lameForeHind_Degree, g = 1, ..., 5;
• horsej(t) are i.i.d. zero-mean random functions with variance σ2j (t), indepen-
dent from εijg(t), j = 1, ..., 8 t ∈ [0, 1]
Test H0 : σ2j (t) = 0 ∀j ∈ {1, ..., 8}, ∀t ∈ [0, 1]
H1 : σ2j (t) 6= 0 for some j ∈ {1, ..., 8} and some t ∈ [0, 1]
Test statistic
T Irandom = e∫I (−2 log (L
Ht0−L
Ht1))dt
101
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
2
Test random effect
Tran
sver
sal A
ccel
erat
ions
(a)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Test random effect
Adj
uste
d p−
valu
e
(b)
Figure 5.8: (a) Results of the test for significance of the random effect horse in theTransversal Modified Dataset: regions presenting p-value ≤ 1% are represented in darkgray, p-value ≤ 5% in light gray. Acceleration curves are colored depending on the levelsof the factor lameForeHind_Degree: No, Fore_low, Fore_mod, Hind_low, Hind_mod.(b) Adjusted p-value function of the test for significance of the random effect horse in theTransversal Modified Dataset. The two red lines represent the thresholds 0.01 and 0.05.
102 CHAPTER 5. RESULTS OF THE MODIFIED DATASETS
5.4.3 LONGITUDINAL Modified Dataset
Fixed Effect: lameForeHind_Degree (No, Hind_low, Hind_mod, Fore_low, Fore_mod)
Random Effect: horse (B1, B2, B3, B4, B5, B6, B7, B8)
Linear Mixed Model
aLONGmodijg (t) = µ(t) + lameForeHind_Degreeg(t) + horsej(t) + εijg(t)
Where:
• index i indicates the i-th observation, i = 1, ..., 70;
• index j indicates the j-th level of horse, j = 1, ..., 8;
• index g indicates the g-th level of lameForeHind_Degree, g = 1, ..., 5;
• horsej(t) are i.i.d. zero-mean random functions with variance σ2j (t), indepen-
dent from εijg(t), j = 1, ..., 8 t ∈ [0, 1]
Test H0 : σ2j (t) = 0 ∀j ∈ {1, ..., 8}, ∀t ∈ [0, 1]
H1 : σ2j (t) 6= 0 for some j ∈ {1, ..., 8} and some t ∈ [0, 1]
Test statistic
T Irandom = e∫I (−2 log (L
Ht0−L
Ht1))dt
103
0.0 0.2 0.4 0.6 0.8 1.0
−6
−4
−2
02
46
Test random effect
Long
itudi
nal A
ccel
erat
ions
(a)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Test random effect
Adj
uste
d p−
valu
e
(b)
Figure 5.9: (a) Results of the test for significance of the random effect horse in theLongitudinal Modified Dataset: regions presenting p-value ≤ 1% are represented in darkgray, p-value ≤ 5% in light gray. Acceleration curves are colored depending on the levelsof the factor lameForeHind_Degree: No, Fore_low, Fore_mod, Hind_low, Hind_mod.(b) Adjusted p-value function of the test for significance of the random effect horse in theLongitudinal Modified Dataset. The two red lines represent the thresholds 0.01 and 0.05.
Chapter 6
Conclusions
In this Thesis, we focused on inference for functional parameters of both
functional-on-scalar linear models (LMs) and functional-on-scalar linear mixed-effect
models (LMMs). The former are linear models where the response variable is a
function and the covariates are fixed scalar variables multiplied by fixed functional
parameters. Moreover, smooth and potentially correlated error curves can be
included adding random effects to model dependence between functional observa-
tions related to the same individual: this is the case of functional-on-scalar linear
mixed-effect models.
After providing estimates (Ordinary Least Squares for LM and REstricted
Maximum Likelihood for LMM) for the functional regression parameters and
predictions for subject specific random effects (Best Linear Unbiased Predictor),
we proposed two tests aimed at verifying if fixed and random effects actually have
to be included in the model. In particular the classical F -test and the Likelihood
Ratio Based Permutation Test proposed by Lee and Braun, 2012 in the scalar
case are extended to the functional setting to investigate the significance of the
functional parameters and variance components, respectively. All tests are based
on permutations of residuals under the reduced model (Freedman and Lane, 1985,
Section 2.5) and on the Interval-Wise Testing (IWT) procedure (Pini and Vantini,
2015, Section 2.1), a non-parametric method for testing functional data which is
able to select the intervals of the domain where the null hypothesis is rejected.
Testing for significance of fixed and random effects by means of permutational
approaches is an element of novelty when dealing with functional linear mixed
models. In fact this approach has not been explored yet in other works.
105
106 CHAPTER 6. CONCLUSIONS
The motivation of this work was the analysis of equine accelerometric data
provided by the University of Copenhagen. Vertical, transversal and longitudinal
acceleration signals of eight horses trotting in a straight line, before and after
the induction of lameness, were studied. Specifically, groups of curves identified
by the lame limb were compared in order to determine how lameness affects the
gait of the horse. Such problems are considered within the functional analysis
of variance (FANOVA) framework, which is a particular case of functional linear
model. The design of the experiment, which consists of multiple measurements per
horse, suggested adding a random intercept to our model. In this way, we were able
to characterise the variation induced in the response by the different “subjects”.
Tests for the fixed effect were performed initially in a one-way FANOVA model
(LM) and then considering the same model with the addition of the random effect
(LMM). For each of these two models, permutations of the residuals were made
following two approaches: the first was the original method proposed by Freedman
and Lane and the second was a variant of the former, where permutations were
restricted to occur only within levels of the random factor. These four different
methods gave all equivalent results, except for the one relying on the linear model
and unrestricted permutations. In this latter case, the fact that the presence of the
horse effect is not taken into account has led to a decrease in power of the test.
The signals we actually analysed consisted in two-peaked curves, representing
average gait cycles. The first half of a curve was relative to the stance phase of the
right diagonal and the second half to the left diagonal. A new approach to examine
our accelerometric data was the introduction of the “modified” representation. The
original signals were modified in such a way that the first peak of the curves
was associated with the stance phase of the sound diagonal and the second peak
with the lame diagonal: this caused the loss of information of the lame side
(right/left). This representation allowed us to study the effect of the degree of
lameness (low/moderate) on the accelerations, which could not be done with the
original data due to the insufficient number of permutable units.
The following conclusions can be drawn from the results of our tests for the fixed
effect both on the original and on the modified datasets. In these tests, domain
selection was able to provide detailed information on the parts of the stride which
are more influenced by lameness. In particular it can be observed that:
107
• Lameness has a more evident impact on the vertical component of the accel-
eration signals, especially in the landing and in the stance phase of the lame
diagonal;
• Hindlimbs lameness does not induce any detectable alteration in transversal
and longitudinal accelerations;
• A moderate lameness degree causes more evident effects than a low lameness
degree;
• The horse tends to reduce the norm of the acceleration vector during the
stance phase of the lame limb and to anticipate the landing of the sound limb
pair.
As regards the random intercept, it was found to be significant in almost the whole
domain, as we expected. This result validates the introduction of the horse effect
in the model and/or in the permutations.
The analyses presented in this work can be used as a preliminary study for the
construction of a classification method for functional data. In particular it would
be of sure interest to exploit the information provided by the domain selection
focusing only on the intervals where the groups of curves are actually different. This
objective method, combined with the clinical examination of the horse physical
condition, would be very helpful to provide a correct and precise diagnosis.
Another future development could be a combined analysis of accelerations of
horse trotting both in a straight line and in a circle. In the latter case lameness is
accentuated when the affected limb is on the inside of the circle. Consequently the
information of the lame side (right/left) could be retrieved by these data, while the
distinction between lame forelimbs and hindlimbs could be provided by the analysis
of the modified signals of horses trotting in a straight line. Besides also the severity
of the lameness could be determined by the modified representation.
Appendix A
Graphical results of test for the fixedeffect of Peak1 - Peak2 Datasets
In this Appendix, the results of tests for significance of the fixed effect of Peak1- Peak2 Datasets are presented. In particular:
• Section A.1: results of test for the fixed effect lameLeg of the Original Peak1- Peak2 Datasets;
• Section A.2: results of test for the fixed effect lameForeHind of the ModifiedPeak1 - Peak2 Datasets;
• Section A.3: results of test for the fixed effect lameForeHind_Degree of theModified Peak1 - Peak2 Datasets.
All the tests have been performed according to the procedure described in Section2.3.
109
110 APPENDIX A. GRAPHICAL RESULTS FIXED EFFECT
A.1 Results of the Original Peak1 - Peak2 Datasetswith fixed effect lameLeg
Fixed Effect: lameLeg (No, RH, RF, LH, LF)Random Effect: horse (B1, B2, B3, B4, B5, B6, B7, B8)
Models
• VERTICAL Original Peak1-Peak2{LM: ∆aV ERTorig (t) = µ(t) + lameLegg(t) + εig(t)
LMM: ∆aV ERTorijg (t) = µ(t) + lameLegg(t) + horsej(t) + εijg(t)
• TRANSVERSAL Original Peak1-Peak2{LM: ∆aTRANSorig (t) = µ(t) + lameLegg(t) + εig(t)
LMM: ∆aTRANSorijg (t) = µ(t) + lameLegg(t) + horsej(t) + εijg(t)
• LONGITUDINAL Original Peak1-Peak2{LM: ∆aLONGorig (t) = µ(t) + lameLegg(t) + εig(t)
LMM: ∆aLONGorijg (t) = µ(t) + lameLegg(t) + horsej(t) + εijg(t)
Test{H0 : lameLegg(t) = 0 ∀g ∈ {1, ..., 5}, ∀t ∈ [0, 1/2]
H1 : lameLegg(t) 6= 0 for some g ∈ {1, ..., 5} and some t ∈ [0, 1/2]
Test statistic
T Ifixed =
∫I
SSE(t)/dfexplSSR(t)/dfres
dt
Results
• VERTICAL Original Peak1-Peak2: Figure A.1 on the next page, Figure A.2on page 112
• TRANSVERSAL Original Peak1-Peak2: Figure A.3 on page 113, Figure A.4on page 114
• LONGITUDINAL Original Peak1-Peak2: Figure A.5 on page 115, Figure A.6on page 116
A.1. ORIGINAL PEAK1 - PEAK2 DATASETS 111
0.0 0.1 0.2 0.3 0.4 0.5
−1.
5−
1.0
−0.
50.
00.
51.
01.
5
Test fixed effect: unrestricted − LM
Ver
tical
Pea
k1 −
Pea
k2
0.0 0.1 0.2 0.3 0.4 0.5
−1.
5−
1.0
−0.
50.
00.
51.
01.
5
Test fixed effect: restricted − LM
Ver
tical
Pea
k1 −
Pea
k2
0.0 0.1 0.2 0.3 0.4 0.5
−1.
5−
1.0
−0.
50.
00.
51.
01.
5
Test fixed effect: unrestricted − LMM
Ver
tical
Pea
k1 −
Pea
k2
0.0 0.1 0.2 0.3 0.4 0.5
−1.
5−
1.0
−0.
50.
00.
51.
01.
5
Test fixed effect: restricted − LMM
Ver
tical
Pea
k1 −
Pea
k2
(a)
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Test fixed effect
Adj
uste
d p−
valu
e
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
unrestricted − LMrestricted − LMunrestricted − LMMrestricted − LMM
(b)
Figure A.1: (a) Results of the four different tests for significance of the fixed effectlameLeg in the Vertical Original Peak1-Peak2 Dataset: regions presenting p-value ≤ 1%are represented in dark gray, p-value ≤ 5% in light gray. Acceleration curves are coloreddepending on the levels of the factor lameLeg: No, RH, RF, LH, LF.(b) Adjusted p-value function of the four different tests for significance of the fixed effectlameLeg in the Vertical Original Peak1-Peak2 Dataset. The two red lines represent thethresholds 0.01 and 0.05.
112 APPENDIX A. GRAPHICAL RESULTS FIXED EFFECT
0.0 0.1 0.2 0.3 0.4 0.5
−1.
5−
0.5
0.5
1.0
1.5
No
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−1.
5−
0.5
0.5
1.0
1.5
RH
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−1.
5−
0.5
0.5
1.0
1.5
RF
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−1.
5−
0.5
0.5
1.0
1.5
LH
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−1.
5−
0.5
0.5
1.0
1.5
LF
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−1.
5−
0.5
0.0
0.5
1.0
1.5
No−RH
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
No−RH
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−1.
5−
0.5
0.0
0.5
1.0
1.5
No−RF
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
No−RF
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−1.
5−
0.5
0.0
0.5
1.0
1.5
No−LH
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
No−LH
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−1.
5−
0.5
0.0
0.5
1.0
1.5
No−LF
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
No−LF
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−1.
5−
0.5
0.0
0.5
1.0
1.5
RH−RF
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
RH−RF
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−1.
5−
0.5
0.0
0.5
1.0
1.5
RH−LH
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
RH−LH
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−1.
5−
0.5
0.0
0.5
1.0
1.5
RH−LF
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
RH−LF
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−1.
5−
0.5
0.0
0.5
1.0
1.5
RF−LH
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
RF−LH
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−1.
5−
0.5
0.0
0.5
1.0
1.5
RF−LF
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
RF−LF
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−1.
5−
0.5
0.0
0.5
1.0
1.5
LH−LF
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
LH−LF
X
p−va
lue
Figure A.2: Results of the restricted - LM pairwise test for significance of the fixed effectlameLeg in the Vertical Original Peak1-Peak2 Dataset. Diagonal: plots of the functionaldata of each group, all the other data are represented in gray. Superior diagonal: meansof each pairwise comparison, and regions presenting a significant difference controllingthe IWER at 1% (dark gray areas) and 5% (light gray areas). Inferior diagonal: p-value (dashed line) and adjusted p-value (solid line) functions associated to the pairwisedifferences between the curves of the five group representing the lame limb.
A.1. ORIGINAL PEAK1 - PEAK2 DATASETS 113
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
23
Test fixed effect: unrestricted − LM
Tran
sver
sal P
eak1
− P
eak2
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
23
Test fixed effect: restricted − LM
Tran
sver
sal P
eak1
− P
eak2
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
23
Test fixed effect: unrestricted − LMM
Tran
sver
sal P
eak1
− P
eak2
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
23
Test fixed effect: restricted − LMM
Tran
sver
sal P
eak1
− P
eak2
(a)
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Test fixed effect
Adj
uste
d p−
valu
e
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
unrestricted − LMrestricted − LMunrestricted − LMMrestricted − LMM
(b)
Figure A.3: (a) Results of the four different tests for significance of the fixed effectlameLeg in the Vertical Original Peak1-Peak2 Dataset: regions presenting p-value ≤ 1%are represented in dark gray, p-value ≤ 5% in light gray. Acceleration curves are coloreddepending on the levels of the factor lameLeg: No, RH, RF, LH, LF.(b) Adjusted p-value function of the four different tests for significance of the fixed effectlameLeg in the Transversal Original Peak1-Peak2 Dataset. The two red lines representthe thresholds 0.01 and 0.05.
114 APPENDIX A. GRAPHICAL RESULTS FIXED EFFECT
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
23
No
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
23
RH
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
23
RF
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
23
LH
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
23
LF
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
2
No−RH
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
No−RH
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
2
No−RF
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
No−RF
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
2
No−LH
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
No−LH
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
2
No−LF
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
No−LF
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
2
RH−RF
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
RH−RF
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
2
RH−LH
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
RH−LH
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
2
RH−LF
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
RH−LF
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
2
RF−LH
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
RF−LH
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
2
RF−LF
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
RF−LF
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
2
LH−LF
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
LH−LF
X
p−va
lue
Figure A.4: Results of the restricted - LM pairwise test for significance of the fixedeffect lameLeg in the Transversal Original Peak1-Peak2 Dataset. Diagonal: plots ofthe functional data of each group, all the other data are represented in gray. Superiordiagonal: means of each pairwise comparison, and regions presenting a significant differencecontrolling the IWER at 1% (dark gray areas) and 5% (light gray areas). Inferior diagonal:p-value (dashed line) and adjusted p-value (solid line) functions associated to the pairwisedifferences between the curves of the five group representing the lame limb.
A.1. ORIGINAL PEAK1 - PEAK2 DATASETS 115
0.0 0.1 0.2 0.3 0.4 0.5
−6
−4
−2
02
4
Test fixed effect: unrestricted − LM
Long
itudi
nal P
eak1
− P
eak2
0.0 0.1 0.2 0.3 0.4 0.5
−6
−4
−2
02
4
Test fixed effect: restricted − LM
Long
itudi
nal P
eak1
− P
eak2
0.0 0.1 0.2 0.3 0.4 0.5
−6
−4
−2
02
4
Test fixed effect: unrestricted − LMM
Long
itudi
nal P
eak1
− P
eak2
0.0 0.1 0.2 0.3 0.4 0.5
−6
−4
−2
02
4
Test fixed effect: restricted − LMM
Long
itudi
nal P
eak1
− P
eak2
(a)
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Test fixed effect
Adj
uste
d p−
valu
e
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
unrestricted − LMrestricted − LMunrestricted − LMMrestricted − LMM
(b)
Figure A.5: (a) Results of the four different tests for significance of the fixed effectlameLeg in the Longitudinal Original Peak1-Peak2 Dataset: regions presenting p-value≤ 1% are represented in dark gray, p-value ≤ 5% in light gray. Acceleration curves arecolored depending on the levels of the factor lameLeg: No, RH, RF, LH, LF.(b) Adjusted p-value function of the four different tests for significance of the fixed effectlameLeg in the Longitudinal Original Peak1-Peak2 Dataset. The two red lines representthe thresholds 0.01 and 0.05.
116 APPENDIX A. GRAPHICAL RESULTS FIXED EFFECT
0 10 20 30 40 50
−6
−4
−2
02
4
No
X
Y
0 10 20 30 40 50
−6
−4
−2
02
4
RH
X
Y
0 10 20 30 40 50
−6
−4
−2
02
4
RF
X
Y
0 10 20 30 40 50
−6
−4
−2
02
4
LH
X
Y
0 10 20 30 40 50
−6
−4
−2
02
4
LF
X
Y
0 10 20 30 40 50
−4
−2
02
4
No−RH
X
Y
0 10 20 30 40 50
0.0
0.2
0.4
0.6
0.8
1.0
No−RH
X
p−va
lue
0 10 20 30 40 50
−4
−2
02
4
No−RF
X
Y
0 10 20 30 40 50
0.0
0.2
0.4
0.6
0.8
1.0
No−RF
X
p−va
lue
0 10 20 30 40 50
−4
−2
02
4
No−LH
X
Y
0 10 20 30 40 50
0.0
0.2
0.4
0.6
0.8
1.0
No−LH
X
p−va
lue
0 10 20 30 40 50
−4
−2
02
4
No−LF
X
Y
0 10 20 30 40 50
0.0
0.2
0.4
0.6
0.8
1.0
No−LF
X
p−va
lue
0 10 20 30 40 50
−4
−2
02
4
RH−RF
X
Y
0 10 20 30 40 50
0.0
0.2
0.4
0.6
0.8
1.0
RH−RF
X
p−va
lue
0 10 20 30 40 50
−4
−2
02
4
RH−LH
X
Y
0 10 20 30 40 50
0.0
0.2
0.4
0.6
0.8
1.0
RH−LH
X
p−va
lue
0 10 20 30 40 50
−4
−2
02
4
RH−LF
X
Y
0 10 20 30 40 50
0.0
0.2
0.4
0.6
0.8
1.0
RH−LF
X
p−va
lue
0 10 20 30 40 50
−4
−2
02
4
RF−LH
X
Y
0 10 20 30 40 50
0.0
0.2
0.4
0.6
0.8
1.0
RF−LH
X
p−va
lue
0 10 20 30 40 50
−4
−2
02
4
RF−LF
X
Y
0 10 20 30 40 50
0.0
0.2
0.4
0.6
0.8
1.0
RF−LF
X
p−va
lue
0 10 20 30 40 50
−4
−2
02
4
LH−LF
X
Y
0 10 20 30 40 50
0.0
0.2
0.4
0.6
0.8
1.0
LH−LF
X
p−va
lue
Figure A.6: Results of the restricted - LM pairwise test for significance of the fixedeffect lameLeg in the Longitudinal Original Peak1-Peak2 Dataset. Diagonal: plots ofthe functional data of each group, all the other data are represented in gray. Superiordiagonal: means of each pairwise comparison, and regions presenting a significant differencecontrolling the IWER at 1% (dark gray areas) and 5% (light gray areas). Inferior diagonal:p-value (dashed line) and adjusted p-value (solid line) functions associated to the pairwisedifferences between the curves of the five group representing the lame limb.
A.2 Results of the Modified Peak1 - Peak2 Datasetswith fixed effect lameForeHind
Fixed Effect: lameForeHind (No, Hind, Fore)Random Effect: horse (B1, B2, B3, B4, B5, B6, B7, B8)
Models
• VERTICAL Modified Peak1-Peak2{LM: ∆aV ERTmodig (t) = µ(t) + lameForeHindg(t) + εig(t)
LMM: ∆aV ERTmodijg (t) = µ(t) + lameForeHindg(t) + horsej(t) + εijg(t)
• TRANSVERSAL Modified Peak1-Peak2{LM: ∆aTRANSmodig (t) = µ(t) + lameForeHindg(t) + εig(t)
LMM: ∆aTRANSmodijg (t) = µ(t) + lameForeHindg(t) + horsej(t) + εijg(t)
• LONGITUDINAL Modified Peak1-Peak2{LM: ∆aLONGmodig (t) = µ(t) + lameForeHindg(t) + εig(t)
LMM: ∆aLONGmodijg (t) = µ(t) + lameForeHindg(t) + horsej(t) + εijg(t)
Test{H0 : lameForeHindg(t) = 0 ∀g ∈ {1, ..., 5}, ∀t ∈ [0, 1/2]
H1 : lameForeHindg(t) 6= 0 for some g ∈ {1, ..., 5} and some t ∈ [0, 1/2]
Test statistic
T Ifixed =
∫I
SSE(t)/dfexplSSR(t)/dfres
dt
Results
• VERTICAL Modified Peak1-Peak2: Figure A.7 on the next page, Figure A.8on page 119
• TRANSVERSAL Modified Peak1-Peak2: Figure A.9 on page 120, Figure A.10on page 121
• LONGITUDINAL Modified Peak1-Peak2: Figure A.11 on page 122, Fig-ure A.12 on page 123
117
118 APPENDIX A. GRAPHICAL RESULTS FIXED EFFECT
0.0 0.1 0.2 0.3 0.4 0.5
−0.
50.
00.
51.
01.
5Test fixed effect: unrestricted − LM
Ver
tical
Pea
k1 −
Pea
k2
0.0 0.1 0.2 0.3 0.4 0.5
−0.
50.
00.
51.
01.
5
Test fixed effect: restricted − LM
Ver
tical
Pea
k1 −
Pea
k2
0.0 0.1 0.2 0.3 0.4 0.5
−0.
50.
00.
51.
01.
5
Test fixed effect: unrestricted − LMM
Ver
tical
Pea
k1 −
Pea
k2
0.0 0.1 0.2 0.3 0.4 0.5
−0.
50.
00.
51.
01.
5
Test fixed effect: restricted − LMM
Ver
tical
Pea
k1 −
Pea
k2
(a)
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Test fixed effect
Adj
uste
d p−
valu
e
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
unrestricted − LMrestricted − LMunrestricted − LMMrestricted − LMM
(b)
Figure A.7: (a) Results of the four different tests for significance of the fixed effectlameForeHind in the Vertical Modified Peak1-Peak2 Dataset: regions presenting p-value≤ 1% are represented in dark gray, p-value ≤ 5% in light gray. Acceleration curves arecolored depending on the levels of the factor lameForeHind: No, Hind, Fore.(b) Adjusted p-value function of the four different tests for significance of the fixed effectlameForeHind in the Vertical Modified Peak1-Peak2 Dataset. The two red lines representthe thresholds 0.01 and 0.05.
A.2. MODIFIED PEAK1 - PEAK2 DATASETS 119
0.0 0.1 0.2 0.3 0.4 0.5
−0.
50.
00.
51.
01.
5
No
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−0.
50.
00.
51.
01.
5
Hind
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−0.
50.
00.
51.
01.
5
Fore
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−0.
50.
00.
51.
01.
5
No−Hind
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
No−Hind
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−0.
50.
00.
51.
01.
5
No−Fore
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
No−Fore
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−0.
50.
00.
51.
01.
5
Hind−Fore
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Hind−Fore
X
p−va
lue
Figure A.8: Results of the restricted - LM pairwise test for significance of the fixedeffect lameForeHind in the Vertical Modified Peak1-Peak2 Dataset. Diagonal: plots ofthe functional data of each group, all the other data are represented in gray. Superiordiagonal: means of each pairwise comparison, and regions presenting a significant differencecontrolling the IWER at 1% (dark gray areas) and 5% (light gray areas). Inferior diagonal:p-value (dashed line) and adjusted p-value (solid line) functions associated to the pairwisedifferences between the curves of the five group representing the lame limb.
120 APPENDIX A. GRAPHICAL RESULTS FIXED EFFECT
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
23
Test fixed effect: unrestricted − LMTr
ansv
ersa
l Pea
k1 −
Pea
k2
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
23
Test fixed effect: restricted − LM
Tran
sver
sal P
eak1
− P
eak2
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
23
Test fixed effect: unrestricted − LMM
Tran
sver
sal P
eak1
− P
eak2
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
23
Test fixed effect: restricted − LMM
Tran
sver
sal P
eak1
− P
eak2
(a)
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Test fixed effect
Adj
uste
d p−
valu
e
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
unrestricted − LMrestricted − LMunrestricted − LMMrestricted − LMM
(b)
Figure A.9: (a) Results of the four different tests for significance of the fixed effectlameForeHind in the Vertical Modified Peak1-Peak2 Dataset: regions presenting p-value≤ 1% are represented in dark gray, p-value ≤ 5% in light gray. Acceleration curves arecolored depending on the levels of the factor lameForeHind: No, Hind, Fore.(b) Adjusted p-value function of the four different tests for significance of the fixed effectlameForeHind in the Transversal Modified Peak1-Peak2 Dataset. The two red linesrepresent the thresholds 0.01 and 0.05.
A.2. MODIFIED PEAK1 - PEAK2 DATASETS 121
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
23
No
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
23
Hind
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
23
Fore
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
2
No−Hind
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
No−Hind
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
2
No−Fore
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
No−Fore
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
2
Hind−Fore
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Hind−Fore
X
p−va
lue
Figure A.10: Results of the restricted - LM pairwise test for significance of the fixedeffect lameForeHind in the Transversal Modified Peak1-Peak2 Dataset. Diagonal: plotsof the functional data of each group, all the other data are represented in gray. Superiordiagonal: means of each pairwise comparison, and regions presenting a significant differencecontrolling the IWER at 1% (dark gray areas) and 5% (light gray areas). Inferior diagonal:p-value (dashed line) and adjusted p-value (solid line) functions associated to the pairwisedifferences between the curves of the five group representing the lame limb.
122 APPENDIX A. GRAPHICAL RESULTS FIXED EFFECT
0.0 0.1 0.2 0.3 0.4 0.5
−4
−2
02
46
Test fixed effect: unrestricted − LMLo
ngitu
dina
l Pea
k1 −
Pea
k2
0.0 0.1 0.2 0.3 0.4 0.5
−4
−2
02
46
Test fixed effect: restricted − LM
Long
itudi
nal P
eak1
− P
eak2
0.0 0.1 0.2 0.3 0.4 0.5
−4
−2
02
46
Test fixed effect: unrestricted − LMM
Long
itudi
nal P
eak1
− P
eak2
0.0 0.1 0.2 0.3 0.4 0.5
−4
−2
02
46
Test fixed effect: restricted − LMM
Long
itudi
nal P
eak1
− P
eak2
(a)
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Test fixed effect
Adj
uste
d p−
valu
e
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
unrestricted − LMrestricted − LMunrestricted − LMMrestricted − LMM
(b)
Figure A.11: (a) Results of the four different tests for significance of the fixed effectlameForeHind in the Longitudinal Modified Peak1-Peak2 Dataset: regions presentingp-value ≤ 1% are represented in dark gray, p-value ≤ 5% in light gray. Acceleration curvesare colored depending on the levels of the factor lameForeHind: No, Hind, Fore.(b) Adjusted p-value function of the four different tests for significance of the fixed effectlameForeHind in the Longitudinal Modified Peak1-Peak2 Dataset. The two red linesrepresent the thresholds 0.01 and 0.05.
A.2. MODIFIED PEAK1 - PEAK2 DATASETS 123
0.0 0.1 0.2 0.3 0.4 0.5
−4
−2
02
46
No
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−4
−2
02
46
Hind
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−4
−2
02
46
Fore
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−4
−2
02
46
No−Hind
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
No−Hind
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−4
−2
02
46
No−Fore
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
No−Fore
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−4
−2
02
46
Hind−Fore
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Hind−Fore
X
p−va
lue
Figure A.12: Results of the restricted - LM pairwise test for significance of the fixedeffect lameForeHind in the Longitudinal Modified Peak1-Peak2 Dataset. Diagonal: plotsof the functional data of each group, all the other data are represented in gray. Superiordiagonal: means of each pairwise comparison, and regions presenting a significant differencecontrolling the IWER at 1% (dark gray areas) and 5% (light gray areas). Inferior diagonal:p-value (dashed line) and adjusted p-value (solid line) functions associated to the pairwisedifferences between the curves of the five group representing the lame limb.
124 APPENDIX A. GRAPHICAL RESULTS FIXED EFFECT
A.3 Results of the Modified Peak1 - Peak2 Datasetswith fixed effect lameForeHind_Degree
Fixed Effect: lameForeHind_Degree (No, Hind_low, Hind_mod, Fore_low, Fore_mod)Random Effect: horse (B1, B2, B3, B4, B5, B6, B7, B8)
Models
• VERTICAL Modified Peak1-Peak2{LM: ∆aV ERTmodig (t) = µ(t) + lameFH_Degreeg(t) + εig(t)
LMM: ∆aV ERTmodijg (t) = µ(t) + lameFH_Degreeg(t) + horsej(t) + εijg(t)
• TRANSVERSAL Modified Peak1-Peak2{LM: ∆aTRANSmodig (t) = µ(t) + lameFH_Degreeg(t) + εig(t)
LMM: ∆aTRANSmodijg (t) = µ(t) + lameFH_Degreeg(t) + horsej(t) + εijg(t)
• LONGITUDINAL Modified Peak1-Peak2{LM: ∆aLONGmodig (t) = µ(t) + lameFH_Degreeg(t) + εig(t)
LMM: ∆aLONGmodijg (t) = µ(t) + lameFH_Degreeg(t) + horsej(t) + εijg(t)
Test{H0 : lameForeHind_Degreeg(t) = 0 ∀g ∈ {1, ..., 5}, ∀t ∈ [0, 1/2]
H1 : lameForeHind_Degreeg(t) 6= 0 for some g ∈ {1, ..., 5} and some t ∈ [0, 1/2]
Test statistic
T Ifixed =
∫I
SSE(t)/dfexplSSR(t)/dfres
dt
Results
• VERTICAL Modified Peak1-Peak2: Figure A.13 on the next page, Figure A.14on page 126
• TRANSVERSAL Modified Peak1-Peak2: Figure A.15 on page 127, Fig-ure A.16 on page 128
• LONGITUDINAL Modified Peak1-Peak2: Figure A.17 on page 129, Fig-ure A.18 on page 130
A.3. MODIFIED PEAK1 - PEAK2 DATASETS 125
0.0 0.1 0.2 0.3 0.4 0.5
−0.
50.
00.
51.
01.
5
Test fixed effect: unrestricted − LM
Ver
tical
Pea
k1 −
Pea
k2
0.0 0.1 0.2 0.3 0.4 0.5
−0.
50.
00.
51.
01.
5
Test fixed effect: restricted − LM
Ver
tical
Pea
k1 −
Pea
k2
0.0 0.1 0.2 0.3 0.4 0.5
−0.
50.
00.
51.
01.
5
Test fixed effect: unrestricted − LMM
Ver
tical
Pea
k1 −
Pea
k2
0.0 0.1 0.2 0.3 0.4 0.5
−0.
50.
00.
51.
01.
5
Test fixed effect: restricted − LMM
Ver
tical
Pea
k1 −
Pea
k2
(a)
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Test fixed effect
Adj
uste
d p−
valu
e
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
unrestricted − LMrestricted − LMunrestricted − LMMrestricted − LMM
(b)
Figure A.13: (a) Results of the four different tests for significance of the fixed effectlameForeHind_Degree in the Vertical Modified Peak1-Peak2 Dataset: regions presentingp-value ≤ 1% are represented in dark gray, p-value ≤ 5% in light gray. Acceleration curvesare colored depending on the levels of the factor lameForeHind_Degree: No, Fore_low,Fore_mod, Hind_low, Hind_mod.(b) Adjusted p-value function of the four different tests for significance of the fixed effectlameForeHind_Degree in the Vertical Modified Peak1-Peak2 Dataset. The two redlines represent the thresholds 0.01 and 0.05.
126 APPENDIX A. GRAPHICAL RESULTS FIXED EFFECT
0.0 0.1 0.2 0.3 0.4 0.5
−0.
50.
00.
51.
01.
5
Fore_low
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−0.
50.
00.
51.
01.
5
Fore_mod
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−0.
50.
00.
51.
01.
5
Hind_low
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−0.
50.
00.
51.
01.
5
Hind_mod
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−0.
50.
00.
51.
01.
5
No
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−0.
50.
00.
51.
01.
5
Fore_low−Fore_mod
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Fore_low−Fore_mod
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−0.
50.
00.
51.
01.
5
Fore_low−Hind_low
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Fore_low−Hind_low
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−0.
50.
00.
51.
01.
5
Fore_low−Hind_mod
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Fore_low−Hind_mod
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−0.
50.
00.
51.
01.
5
Fore_low−No
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Fore_low−No
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−0.
50.
00.
51.
01.
5
Fore_mod−Hind_low
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Fore_mod−Hind_low
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−0.
50.
00.
51.
01.
5
Fore_mod−Hind_mod
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Fore_mod−Hind_mod
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−0.
50.
00.
51.
01.
5
Fore_mod−No
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Fore_mod−No
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−0.
50.
00.
51.
01.
5
Hind_low−Hind_mod
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Hind_low−Hind_mod
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−0.
50.
00.
51.
01.
5
Hind_low−No
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Hind_low−No
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−0.
50.
00.
51.
01.
5
Hind_mod−No
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Hind_mod−No
X
p−va
lue
Figure A.14: Results of the restricted - LM pairwise test for significance of the fixedeffect lameForeHind_Degree in the Vertical Modified Peak1-Peak2 Dataset. Diagonal:plots of the functional data of each group, all the other data are represented in gray.Superior diagonal: means of each pairwise comparison, and regions presenting a significantdifference controlling the IWER at 1% (dark gray areas) and 5% (light gray areas). Inferiordiagonal: p-value (dashed line) and adjusted p-value (solid line) functions associated tothe pairwise differences between the curves of the five group representing the lame limb.
A.3. MODIFIED PEAK1 - PEAK2 DATASETS 127
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
23
Test fixed effect: unrestricted − LM
Tran
sver
sal P
eak1
− P
eak2
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
23
Test fixed effect: restricted − LM
Tran
sver
sal P
eak1
− P
eak2
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
23
Test fixed effect: unrestricted − LMM
Tran
sver
sal P
eak1
− P
eak2
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
23
Test fixed effect: restricted − LMM
Tran
sver
sal P
eak1
− P
eak2
(a)
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Test fixed effect
Adj
uste
d p−
valu
e
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
unrestricted − LMrestricted − LMunrestricted − LMMrestricted − LMM
(b)
Figure A.15: (a) Results of the four different tests for significance of the fixed effectlameForeHind_Degree in the Vertical Modified Peak1-Peak2 Dataset: regions presentingp-value ≤ 1% are represented in dark gray, p-value ≤ 5% in light gray. Acceleration curvesare colored depending on the levels of the factor lameForeHind_Degree: No, Fore_low,Fore_mod, Hind_low, Hind_mod.(b) Adjusted p-value function of the four different tests for significance of the fixed effectlameForeHind_Degree in the Transversal Modified Peak1-Peak2 Dataset. The two redlines represent the thresholds 0.01 and 0.05.
128 APPENDIX A. GRAPHICAL RESULTS FIXED EFFECT
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
23
Fore_low
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
23
Fore_mod
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
23
Hind_low
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
23
Hind_mod
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
23
No
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
2
Fore_low−Fore_mod
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Fore_low−Fore_mod
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
2
Fore_low−Hind_low
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Fore_low−Hind_low
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
2
Fore_low−Hind_mod
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Fore_low−Hind_mod
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
2
Fore_low−No
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Fore_low−No
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
2
Fore_mod−Hind_low
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Fore_mod−Hind_low
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
2
Fore_mod−Hind_mod
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Fore_mod−Hind_mod
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
2
Fore_mod−No
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Fore_mod−No
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
2
Hind_low−Hind_mod
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Hind_low−Hind_mod
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
2
Hind_low−No
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Hind_low−No
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
2
Hind_mod−No
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Hind_mod−No
X
p−va
lue
Figure A.16: Results of the restricted - LM pairwise test for significance of the fixed effectlameForeHind_Degree in the Transversal Modified Peak1-Peak2 Dataset. Diagonal:plots of the functional data of each group, all the other data are represented in gray.Superior diagonal: means of each pairwise comparison, and regions presenting a significantdifference controlling the IWER at 1% (dark gray areas) and 5% (light gray areas). Inferiordiagonal: p-value (dashed line) and adjusted p-value (solid line) functions associated tothe pairwise differences between the curves of the five group representing the lame limb.
A.3. MODIFIED PEAK1 - PEAK2 DATASETS 129
0.0 0.1 0.2 0.3 0.4 0.5
−4
−2
02
46
Test fixed effect: unrestricted − LM
Long
itudi
nal P
eak1
− P
eak2
0.0 0.1 0.2 0.3 0.4 0.5
−4
−2
02
46
Test fixed effect: restricted − LM
Long
itudi
nal P
eak1
− P
eak2
0.0 0.1 0.2 0.3 0.4 0.5
−4
−2
02
46
Test fixed effect: unrestricted − LMM
Long
itudi
nal P
eak1
− P
eak2
0.0 0.1 0.2 0.3 0.4 0.5
−4
−2
02
46
Test fixed effect: restricted − LMM
Long
itudi
nal P
eak1
− P
eak2
(a)
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Test fixed effect
Adj
uste
d p−
valu
e
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
unrestricted − LMrestricted − LMunrestricted − LMMrestricted − LMM
(b)
Figure A.17: (a) Results of the four different tests for significance of the fixed effectlameForeHind_Degree in the Longitudinal Modified Peak1-Peak2 Dataset: regions pre-senting p-value ≤ 1% are represented in dark gray, p-value ≤ 5% in light gray. Accelerationcurves are colored depending on the levels of the factor lameForeHind_Degree: No,Fore_low, Fore_mod, Hind_low, Hind_mod.(b) Adjusted p-value function of the four different tests for significance of the fixed effectlameForeHind_Degree in the Longitudinal Modified Peak1-Peak2 Dataset. The twored lines represent the thresholds 0.01 and 0.05.
130 APPENDIX A. GRAPHICAL RESULTS FIXED EFFECT
0.0 0.1 0.2 0.3 0.4 0.5
−4
−2
02
46
Fore_low
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−4
−2
02
46
Fore_mod
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−4
−2
02
46
Hind_low
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−4
−2
02
46
Hind_mod
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−4
−2
02
46
No
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
−4
−2
02
46
Fore_low−Fore_mod
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Fore_low−Fore_mod
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−4
−2
02
46
Fore_low−Hind_low
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Fore_low−Hind_low
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−4
−2
02
46
Fore_low−Hind_mod
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Fore_low−Hind_mod
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−4
−2
02
46
Fore_low−No
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Fore_low−No
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−4
−2
02
46
Fore_mod−Hind_low
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Fore_mod−Hind_low
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−4
−2
02
46
Fore_mod−Hind_mod
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Fore_mod−Hind_mod
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−4
−2
02
46
Fore_mod−No
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Fore_mod−No
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−4
−2
02
46
Hind_low−Hind_mod
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Hind_low−Hind_mod
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−4
−2
02
46
Hind_low−No
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Hind_low−No
X
p−va
lue
0.0 0.1 0.2 0.3 0.4 0.5
−4
−2
02
46
Hind_mod−No
X
Y
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Hind_mod−No
X
p−va
lue
Figure A.18: Results of the restricted - LM pairwise test for significance of the fixed effectlameForeHind_Degree in the Longitudinal Modified Peak1-Peak2 Dataset. Diagonal:plots of the functional data of each group, all the other data are represented in gray.Superior diagonal: means of each pairwise comparison, and regions presenting a significantdifference controlling the IWER at 1% (dark gray areas) and 5% (light gray areas). Inferiordiagonal: p-value (dashed line) and adjusted p-value (solid line) functions associated tothe pairwise differences between the curves of the five group representing the lame limb.
Appendix B
Graphical results of test for therandom effect of Peak1 - Peak2Datasets
In this Appendix, the results of tests for significance of the random effect ofPeak1 - Peak2 Datasets are presented. In particular:
• Section B.1: results of test for the random effect of the Original Peak1 - Peak2Datasets, where the fixed effect of the LMM is represented by the factorlameLeg;
• Section B.2: results of test for the random effect of the Modified Peak1 -Peak2 Datasets, where the fixed effect of the LMM is represented by thefactor lameForeHind;
• Section B.3: results of test for the random effect of the Modified Peak1 -Peak2 Datasets, where the fixed effect of the LMM is represented by thefactor lameForeHind_Degree.
All the tests have been performed according to the procedure described in Section2.4.
131
132 APPENDIX B. GRAPHICAL RESULTS RANDOM EFFECT
B.1 Results of the Original Peak1 - Peak2 Datasetswith fixed effect lameLeg
Fixed Effect: lameLeg (No, RH, RF, LH, LF)Random Effect: horse (B1, B2, B3, B4, B5, B6, B7, B8)
Models
• VERTICAL Original Peak1-Peak2
∆aV ERTorijg (t) = µ(t) + lameLegg(t) + horsej(t) + εijg(t)
• TRANSVERSAL Original Peak1-Peak2
∆aTRANSorijg (t) = µ(t) + lameLegg(t) + horsej(t) + εijg(t)
• LONGITUDINAL Original Peak1-Peak2
∆aLONGorijg (t) = µ(t) + lameLegg(t) + horsej(t) + εijg(t)
Test {H0 : σ2
j (t) = 0 ∀j ∈ {1, ..., 8}, ∀t ∈ [0, 1/2]
H1 : σ2j (t) 6= 0 for some j ∈ {1, ..., 8} and some t ∈ [0, 1/2]
Test statistic
T Irandom = e∫I (−2 log (L
Ht0−L
Ht1))dt
Results
• VERTICAL Original Peak1-Peak2: Figure B.1 on the next page
• TRANSVERSAL Original Peak1-Peak2: Figure B.2 on page 134
• LONGITUDINAL Original Peak1-Peak2: Figure B.3 on page 135
B.1. ORIGINAL PEAK1 - PEAK2 DATASETS 133
0.0 0.1 0.2 0.3 0.4 0.5
−1.
5−
1.0
−0.
50.
00.
51.
01.
5
Test random effect
Ver
tical
Pea
k1 −
Pea
k2
(a)
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Test random effect
Adj
uste
d p−
valu
e
(b)
Figure B.1: (a) Results of the test for significance of the random effect horse in theVertical Original Peak1-Peak2 Dataset: regions presenting p-value ≤ 1% are representedin dark gray, p-value ≤ 5% in light gray. Acceleration curves are colored depending onthe levels of the factor lameLeg: No, RH, RF, LH, LF.(b) Adjusted p-value function of the test for significance of the random effect horse in theVertical Original Peak1-Peak2 Dataset. The two red lines represent the thresholds 0.01and 0.05.
134 APPENDIX B. GRAPHICAL RESULTS RANDOM EFFECT
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
23
Test random effect
Tran
sver
sal P
eak1
− P
eak2
(a)
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Test random effect
Adj
uste
d p−
valu
e
(b)
Figure B.2: (a) Results of the test for significance of the random effect horse inthe Transversal Original Peak1-Peak2 Dataset: regions presenting p-value ≤ 1% arerepresented in dark gray, p-value ≤ 5% in light gray. Acceleration curves are coloreddepending on the levels of the factor lameLeg: No, RH, RF, LH, LF.(b) Adjusted p-value function of the test for significance of the random effect horse in theTransversal Original Peak1-Peak2 Dataset. The two red lines represent the thresholds0.01 and 0.05.
B.1. ORIGINAL PEAK1 - PEAK2 DATASETS 135
0.0 0.1 0.2 0.3 0.4 0.5
−6
−4
−2
02
4
Test random effect
Long
itudi
nal P
eak1
− P
eak2
(a)
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Test random effect
Adj
uste
d p−
valu
e
(b)
Figure B.3: (a) Results of the test for significance of the random effect horse inthe Longitudinal Original Peak1-Peak2 Dataset: regions presenting p-value ≤ 1% arerepresented in dark gray, p-value ≤ 5% in light gray. Acceleration curves are coloreddepending on the levels of the factor lameLeg: No, RH, RF, LH, LF.(b) Adjusted p-value function of the test for significance of the random effect horse in theLongitudinal Original Peak1-Peak2 Dataset. The two red lines represent the thresholds0.01 and 0.05.
136 APPENDIX B. GRAPHICAL RESULTS RANDOM EFFECT
B.2 Results of the Modified Peak1 - Peak2 Datasetswith fixed effect lameForeHind
Fixed Effect: lameForeHind (No, Hind, Fore)Random Effect: horse (B1, B2, B3, B4, B5, B6, B7, B8)
Models
• VERTICAL Modified Peak1-Peak2
∆aV ERTmodijg (t) = µ(t) + lameForeHindg(t) + horsej(t) + εijg(t)
• TRANSVERSAL Modified Peak1-Peak2
∆aTRANSmodijg (t) = µ(t) + lameForeHindg(t) + horsej(t) + εijg(t)
• LONGITUDINAL Modified Peak1-Peak2
∆aLONGmodijg (t) = µ(t) + lameForeHindg(t) + horsej(t) + εijg(t)
Test {H0 : σ2
j (t) = 0 ∀j ∈ {1, ..., 8}, ∀t ∈ [0, 1/2]
H1 : σ2j (t) 6= 0 for some j ∈ {1, ..., 8} and some t ∈ [0, 1/2]
Test statistic
T Irandom = e∫I (−2 log (L
Ht0−L
Ht1))dt
Results
• VERTICAL Modified Peak1-Peak2: Figure B.4 on the facing page
• TRANSVERSAL Modified Peak1-Peak2: Figure B.5 on page 138
• LONGITUDINAL Modified Peak1-Peak2: Figure B.6 on page 139
B.2. MODIFIED PEAK1 - PEAK2 DATASETS 137
0.0 0.1 0.2 0.3 0.4 0.5
−0.
50.
00.
51.
01.
5
Test random effect
Ver
tical
Pea
k1 −
Pea
k2
(a)
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Test random effect
Adj
uste
d p−
valu
e
(b)
Figure B.4: (a) Results of the test for significance of the random effect horse in theVertical Modified Peak1-Peak2 Dataset: regions presenting p-value ≤ 1% are representedin dark gray, p-value ≤ 5% in light gray. Acceleration curves are colored depending onthe levels of the factor lameForeHind: No, Hind, Fore.(b) Adjusted p-value function of the test for significance of the random effect horse in theVertical Modified Peak1-Peak2 Dataset. The two red lines represent the thresholds 0.01and 0.05.
138 APPENDIX B. GRAPHICAL RESULTS RANDOM EFFECT
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
23
Test random effect
Tran
sver
sal P
eak1
− P
eak2
(a)
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Test random effect
Adj
uste
d p−
valu
e
(b)
Figure B.5: (a) Results of the test for significance of the random effect horse inthe Transversal Modified Peak1-Peak2 Dataset: regions presenting p-value ≤ 1% arerepresented in dark gray, p-value ≤ 5% in light gray. Acceleration curves are coloreddepending on the levels of the factor lameForeHind: No, Hind, Fore.(b) Adjusted p-value function of the test for significance of the random effect horse in theTransversal Modified Peak1-Peak2 Dataset. The two red lines represent the thresholds0.01 and 0.05.
B.2. MODIFIED PEAK1 - PEAK2 DATASETS 139
0.0 0.1 0.2 0.3 0.4 0.5
−4
−2
02
46
Test random effect
Long
itudi
nal P
eak1
− P
eak2
(a)
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Test random effect
Adj
uste
d p−
valu
e
(b)
Figure B.6: (a) Results of the test for significance of the random effect horse inthe Longitudinal Modified Peak1-Peak2 Dataset: regions presenting p-value ≤ 1% arerepresented in dark gray, p-value ≤ 5% in light gray. Acceleration curves are coloreddepending on the levels of the factor lameForeHind: No, Hind, Fore.(b) Adjusted p-value function of the test for significance of the random effect horse in theLongitudinal Modified Peak1-Peak2 Dataset. The two red lines represent the thresholds0.01 and 0.05.
140 APPENDIX B. GRAPHICAL RESULTS RANDOM EFFECT
B.3 Results of the Modified Peak1 - Peak2 Datasetswith fixed effect lameForeHind_Degree
Fixed Effect: lameForeHind_Degree (No, Hind_low, Hind_mod, Fore_low, Fore_mod)Random Effect: horse (B1, B2, B3, B4, B5, B6, B7, B8)
Models
• VERTICAL Modified Peak1-Peak2
∆aV ERTmodijg (t) = µ(t) + lameForeHind_Degreeg(t) + horsej(t) + εijg(t)
• TRANSVERSAL Modified Peak1-Peak2
∆aTRANSmodijg (t) = µ(t) + lameForeHind_Degreeg(t) + horsej(t) + εijg(t)
• LONGITUDINAL Modified Peak1-Peak2
∆aLONGmodijg (t) = µ(t) + lameForeHind_Degreeg(t) + horsej(t) + εijg(t)
Test {H0 : σ2
j (t) = 0 ∀j ∈ {1, ..., 8}, ∀t ∈ [0, 1/2]
H1 : σ2j (t) 6= 0 for some j ∈ {1, ..., 8} and some t ∈ [0, 1/2]
Test statistic
T Irandom = e∫I (−2 log (L
Ht0−L
Ht1))dt
Results
• VERTICAL Modified Peak1-Peak2: Figure B.7 on the next page
• TRANSVERSAL Modified Peak1-Peak2: Figure B.8 on page 142
• LONGITUDINAL Modified Peak1-Peak2: Figure B.9 on page 143
B.3. MODIFIED PEAK1 - PEAK2 DATASETS 141
0.0 0.1 0.2 0.3 0.4 0.5
−0.
50.
00.
51.
01.
5
Test random effect
Ver
tical
Pea
k1 −
Pea
k2
(a)
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Test random effect
Adj
uste
d p−
valu
e
(b)
Figure B.7: (a) Results of the test for significance of the random effect horse in theVertical Modified Peak1-Peak2 Dataset: regions presenting p-value ≤ 1% are representedin dark gray, p-value ≤ 5% in light gray. Acceleration curves are colored depending onthe levels of the factor lameForeHind_Degree: No, Fore_low, Fore_mod, Hind_low,Hind_mod.(b) Adjusted p-value function of the test for significance of the random effect horse in theVertical Modified Peak1-Peak2 Dataset. The two red lines represent the thresholds 0.01and 0.05.
142 APPENDIX B. GRAPHICAL RESULTS RANDOM EFFECT
0.0 0.1 0.2 0.3 0.4 0.5
−3
−2
−1
01
23
Test random effect
Tran
sver
sal P
eak1
− P
eak2
(a)
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Test random effect
Adj
uste
d p−
valu
e
(b)
Figure B.8: (a) Results of the test for significance of the random effect horse inthe Transversal Modified Peak1-Peak2 Dataset: regions presenting p-value ≤ 1% arerepresented in dark gray, p-value ≤ 5% in light gray. Acceleration curves are coloreddepending on the levels of the factor lameForeHind_Degree: No, Fore_low, Fore_mod,Hind_low, Hind_mod.(b) Adjusted p-value function of the test for significance of the random effect horse in theTransversal Modified Peak1-Peak2 Dataset. The two red lines represent the thresholds0.01 and 0.05.
B.3. MODIFIED PEAK1 - PEAK2 DATASETS 143
0.0 0.1 0.2 0.3 0.4 0.5
−4
−2
02
46
Test random effect
Long
itudi
nal P
eak1
− P
eak2
(a)
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Test random effect
Adj
uste
d p−
valu
e
(b)
Figure B.9: (a) Results of the test for significance of the random effect horse inthe Longitudinal Modified Peak1-Peak2 Dataset: regions presenting p-value ≤ 1% arerepresented in dark gray, p-value ≤ 5% in light gray. Acceleration curves are coloreddepending on the levels of the factor lameForeHind_Degree: No, Fore_low, Fore_mod,Hind_low, Hind_mod.(b) Adjusted p-value function of the test for significance of the random effect horse in theLongitudinal Modified Peak1-Peak2 Dataset. The two red lines represent the thresholds0.01 and 0.05.
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