randomized quantile residual for assessing generalized ...longhai/doc/talks/slides_glmm_rqr.pdf ·...

Page 1

Randomized Quantile Residual for Assessing GeneralizedLinear Mixed Models with Application to Zero-inflated

Microbiome Data

Longhai Li

Department of Mathematics and StatisticsUniversity of SaskatchewanSaskatoon, SK, CANADA

5 June 2018Annual Meeting of Statistical Society of Canada

McGill University, Montreal, Canada

Page 2

Acknowledgements

This talk is based on the results of the M.Sc thesis project undertakenby Wei Bai, co-supervised with Cindy X. Feng (U of S).

Thank Prof. Wei Xu (U of T) for providing the microbiome data forthis research.

Thank NSERC and CFI for providing grants for my research.

Thank the ICSA Canada Chapter, particularly Prof. Changbao Wu(U of Waterloo), for organizing and sponsoring this session.

Page 3

Outline

1 Introduction

2 Zero-inflated/modified Generalized Linear Mixed Models

3 Randomized Quantile Residual

4 Simulation StudiesDescription of Data Generating ProcessAssessing Models for Datasets Simulated from ZMP ModelAssessing Models for Datasets Simulated from ZMNB Model

5 Application to a Twin Study OTU Dataset

6 Conclusions and Discussions

Page 4

Section 1

Introduction

Page 5

Introduction

The operational taxonomic unit (OTU) counts in microbiome datasetshave characteristics of zero-inflation and over-dispersion. Variousgeneralized mixed models have been proposed to to fit the data.

Correctness in model specification plays extremely important role instatistical inference, for example in calculating p-values/q-values forselecting OTUs that are related to a phenotype.

Pearson and deviance residuals are often used in practice withoutjustification. However, when applied to count data, the distributionsof these residuals are far from the normal distribution.

Randomized quantile residual (RQR) was originally proposed by Dunnand Smyth (1996) as an alternative for Pearson and devianceresiduals. However, it has NOT been used much by statisticians.

We investigate the performance of RQR in checkingzero-inflated/modified generalized linear mixed effect (GLMM)models using simulated and real datasets.

1. Introduction/ 1/39

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Section 2

Zero-inflated/modified Generalized Linear MixedModels

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Generalized Linear Mixed Model

As an example of GLMM, NB mixed model (NB) is described as follows:

A probability distribution for the response (yi ) given a mean functionµi and other parameters, eg.

yi |µi ∼ Negative-Binomial(µi , k)

A link function for linking the mean µi to a linear function of fixedfactor (Xi ) and random factors (Zi ), eg.

log(µi ) = Xiβ + Ziu

Certain penalization (often normal) is imposed to u.

2. Zero-inflated/modified Generalized Linear Mixed Models/ 2/39

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Zero-inflated Poisson (ZIP) Model: I

The zero-inflated Poisson with parameters λi and pi , denoted byZIP(λi , pi ), is defined as:

yi ∼

{0, with probability pi

Poisson(µi ), with probability 1− pi .(1)

The following link functions are often used:

log(µi ) = offseti + Xiβ + Ziu (2)

log

(pi

1− pi

)= offseti + Xi β + Zi u, (3)

2. Zero-inflated/modified Generalized Linear Mixed Models/ 3/39

Page 9

Zero-inflated Poisson (ZIP) Model: II

The PMF and CDF of the ZIP distribution:

dzip(yi = 0) = pi + (1− pi )× e−µi (4)

dzip(yi = j) = (1− pi )e−µiµji

j!, for j > 0 (5)

pzip(yi = J;µi , pi ) = pi + (1− pi )ppois(J, µi ). (6)

The mean and variance of a ZIP random variable can be calculated by

E (yi ) = (1− pi )× µi (7)

V (yi ) = (1− pi )×(µi + pi × µi 2

). (8)

2. Zero-inflated/modified Generalized Linear Mixed Models/ 4/39

Page 10

Zero-inflated Negative-Binomial (ZINB) Model: I

Zero-inflated NB(ZINB) can be defined similarly as ZIP, with Poissonreplaced by NB.

The PMF and CDF of the ZINB distribution:

dzinb(yi = 0) = pi + (1− pi )×(

k

k + µi

)k

(9)

dzinb(yi = j) = (1− pi )× dnb(j , µi , k), for j > 0 (10)

pzinb(yi ;µi , k , pi ) = pi + (1− pi )pnb(yi , µi , k) (11)

The mean and variance of a ZIP random variable can be calculated by

E (yi ) = (1− pi )× µi (12)

V (yi ) = (1− pi )×(µi +

µi2

k

)+ µi

2 ×(pi

2 + pi)

(13)

2. Zero-inflated/modified Generalized Linear Mixed Models/ 5/39

Page 11

Zero-modified Poisson (ZMP): I

Zero-Modified Model: Zero-modified models are also called hurdle models.A logistic regression for the zero indicator (Zi ):

Pr(Zi = z) =

{πi , z = 0

1− πi , z = 1.(14)

Given Zi , the conditional probability mass function for Yi is:Pr(Yi = yi |Zi = 0) = I (yi = 0)

Pr(Yi = yi |Zi = 1) = dpois(yi )

1−dpois(0)I (yi > 0).

(15)

The unconditional probability mass function for Yi is

Pr(Yi = yi ) =

πi , if yi = 0

(1− πi ) dpois(yi )

1−dpois(0), if yi > 0.

(16)

2. Zero-inflated/modified Generalized Linear Mixed Models/ 6/39

Page 12

Zero-modified Poisson (ZMP): II

We often used the log link functions for non-zero count mean µi and logisticlink for πi :

log(µi ) = offseti + Xiβ + Ziu (17)

log

(πi

1− πi

)= offseti + Xi β + Zi u, (18)

The PMF and CDF of ZMP distribution:

dzmp(yi = 0) = πi (19)

dzmp(yi = j) = (1− πi )dpois(j)

1− ppois(0), for j > 0 (20)

pzmp(yi ;µi , πi ) = πi + (1− πi )ppois(yi ;µi , πi )− ppois(0)

1− ppois(0), (21)

2. Zero-inflated/modified Generalized Linear Mixed Models/ 7/39

Page 13

Zero-modified NB (ZMNB)

ZMNB model can be defined analogously. The PMF and CDF of ZMNBdistribution:

dzmnb(yi = j) = πi I (j = 0) + (1− πi )dnb(yi )

1− pnb(0)I (j > 0) (22)

pzmnb(yi ;µi , k, πi ) = πi + (1− πi )pnb(yi )− pnb(0)

1− pnb(0). (23)

The same link functions as in ZMP are used for ZMNB.

The mean and variance of ZMNB:

E (yi ) =1− πi1− p0

× µi (24)

V (yi ) =1− πi1− p0

×(µi + µ2

i +µi

2

k

)−(

1− πi1− p0

× µi

)2

. (25)

2. Zero-inflated/modified Generalized Linear Mixed Models/ 8/39

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Section 3

Randomized Quantile Residual

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Problems with Pearson and Deviance Residuals

In regression models for discrete outcomes, the residuals are far fromnormality, with residuals clustering on lines according to distinctresponse values, which poses great challenges for visual inspection.Therefore, residual plots for the diagnosis of models for discreteoutcome variables give very limited meaningful information for modeldiagnosis.

The Pearson χ2 statistic is written as, X 2 =∑n

i=1 r2i , and the

deviance (χ2 statistic) is written as, D =∑n

i=1 d2i . The asymptotic

distribution of D and X 2 under the true model is often assumed to beχ2n−p. However, the use of this asymptotic distribution for both X 2

and D is lack of theoretical underpinning.

3. Randomized Quantile Residual/ 9/39

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A First Look at Three Residuals

Pearson

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−1.5 −0.5 0.0 0.5 1.0 1.5

−1

01

23

45

x

Pea

rson

Deviance

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−1.5 −0.5 0.0 0.5 1.0 1.5−

2−

10

12

3

x

Dev

ianc

e

Randomized Quantile

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−1.5 −0.5 0.0 0.5 1.0 1.5

−3

−2

−1

01

23

x

Ran

dom

ized

Qua

ntile

A simulated dataset is checked against the true generating model.However, Pearson and deviance residuals exhibit trend and cluster inlines.

In addition, the often used χ2 tests are not well-calibrated.

3. Randomized Quantile Residual/ 10/39

Page 17

Definition of Randomized Quantile Residual

Predictive p-value for continuous yi :

F (yi ; µi , φ) = P(Yi ≤ yi | µi , φ)

Randomized predictive p-valueIf F is discrete, the estimated lower tail probability is randomized intoa uniform random number.

F ∗(yi ; µi , φ, ui ) = F (yi−; µi , φ) + ui P(yi ; µi , φ), (26)

where ui from uniform distribution on (0, 1], F (yi−; µi , φ) is the lowerlimit of F at yi , i.e., supy<yi F (y ; µi , φ), the lower limit in the “gap”

of F (·, µi , φ) at yi .Randomized quantile residual

qi = Φ−1(F ∗(yi ; µi , φ, ui )) (27)

where Φ−1 is the quantile function of a standard normal distribution

3. Randomized Quantile Residual/ 11/39

Page 18

An Illustrative Example for RQR: I

The true model:We simulate a response variable of size n = 1000 from a Poissonmodel with

log(µi ) = −1 + 2sin(2xi ),

where µi is the expected mean count for the ith subject andxi ∼ Uniform(0, 2π), i = 1, · · · , nA wrong model:Poisson model with mean structure

log(µi ) = β0 + β1xi

with xi as a predictor with linear effect.

3. Randomized Quantile Residual/ 12/39

Page 19

An Illustrative Example for RQR: II

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

1.0

x

F*

true model

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F*

wrong model

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● ● ● ● ● ● ● ●0 1 2 3 4 5 6 7

3. Randomized Quantile Residual/ 13/39

Page 20

Normality of Randomized Quantile Residual (RQR)

Theorem

Suppose a continuous random variable Y has the CDF F (y), then F (Y ) isuniformly distributed on (0,1].

Theorem

Suppose the true distribution of Yi given Xi has the CDF F (yi ;µi , φ) andPMF P(yI ;µi , φ), where µi is a function of Xi involving the modelparameters. The randomized lower tail probability F ∗(yi ;µi , φ, ui ) isdefined as F (yi−;µi , φ) + ui P(yI ;µi , φ) (26). Suppose Ui is uniformlydistributed on (0,1]. Then, we have

F ∗(Yi ;µi , φ,Ui ) ∼ Uniform((0, 1]), (28)

andqi = φ−1(F ∗(Yi ;µi , φ,Ui )) ∼ N(0, 1). (29)

3. Randomized Quantile Residual/ 14/39

Page 21

Proof of Normality of RQR

For any interval B ⊆ (0, 1],

P(F ∗(Yi ;µi , φ,Ui ) ∈ B|Yi = k(j)) =length(F (j) ∩ B)

p(j),

where length(·) is the length of interval. By the law of total probability,

P(F ∗(Yi ;µi , φ,Ui ) ∈ B) (30)

=∞∑j=1

P(F ∗(Yi ;µi , φ,Ui ) ∈ B|Yi = k(j))× P(Yi = k(j)) (31)

=∞∑j=1

length(F (j) ∩ B)

p(j)× p(j) (32)

=∞∑j=1

length(F (j) ∩ B) (33)

= length(∪∞j=1F(j) ∩ B) = length(B) (34)

3. Randomized Quantile Residual/ 15/39

Page 22

Section 4

Simulation Studies

Page 23

Subsection 1

Description of Data Generating Process

Page 24

General Form of Microbiome Dataset

OTU1 ... OTUm Total Reads Host Factors Sample VariablesY1 Ym Offset Fixed Factors Random Factors

sample 1 Y11 ... Y1m T1 X11 ... X1s Z11 ... Z1t

. . ... . . ... ...

. . ... . . ... ...

. . ... . . ... ...sample n Yn1 ... Ynm Tn Xn1 ... Xns Zn1 ... Znt

4. Simulation Studies/Description of Data Generating Process 16/39

Page 25

Link Functions and Parameters in Data Generation

Link Functions

log(µi ) = log(Ti ) + β0 + β(1)Xi1

+ ...+ β(s)Xis

+ u(1)Zi1

+ ...+ u(t)Zit,

log(πi

1− πi) = β0 + β

(1)Xi1

+ ...+ β(s)Xis

+ u(1)Zi1

+ ...+ u(t)Zit

Parameters:

Parameter Generator

β0 -0.2

β, β N(0, 0.12)u, u N(0, 22)k (ZMNB) Unif(1,2)Ti Poisson(3× 105)

Other Settings: m = 3000, s = 3, t = 3; each fixed factor has 5 levelsand each random factor has 10 levels.

4. Simulation Studies/Description of Data Generating Process 17/39

Page 26

Steps to Generate OTUs with ZMP/ZMNB Model

Step 1: Generate matrix of fixed and random factors, and total reads Ti

randomly (used for all Yj).For each response Yj :Step 2: Compute πij and µij using link functions with randomly generatedparameters.Step 3: We generate a count indicator Zij as a binary Bernoulli randomvariable:

Zij =

{0, with probability πij

1 with probability 1− πij .(35)

Step 4: If the indicator Zij = 0, then Yij = 0. If the indicator Zij = 1,then Yij follows a truncated Poisson or NB model, e.g.,

Yij ∼ Truncated-Poisson(µij).

4. Simulation Studies/Description of Data Generating Process 18/39

Page 27

Subsection 2

Assessing Models for Datasets Simulated from ZMP Model

Page 28

Checking One Yj : RQR plot vs Fitted Values

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400000 600000 800000 1000000 1400000

−3−2

−10

12

Fitted values

Rand

omize

d Qu

antile

ZMP

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400000 600000 800000 1000000 1400000

−2−1

01

2

Fitted values

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ZIP

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400000 600000 800000 1000000 1400000

−3−2

−10

12

Fitted values

Rand

omize

d Qu

antile

ZMNB

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2e+05 5e+05 1e+06 2e+06

−6−4

−20

24

6

Fitted values

Rand

omize

d Qu

antile

Poisson

4. Simulation Studies/Assessing Models for Datasets Simulated from ZMP Model 19/39

Page 29

Checking One Yj : QQ-plot of RQR

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−3 −2 −1 0 1 2 3

−3−2

−10

12

Randomized Quantile

Theoretical Quantiles

Sam

ple Q

uant

iles

ZMP

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−3 −2 −1 0 1 2 3

−2−1

01

2

Randomized Quantile

Theoretical Quantiles

Sam

ple Q

uant

iles

ZIP

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−3 −2 −1 0 1 2 3

−3−2

−10

12

Randomized Quantile

Theoretical Quantiles

Sam

ple Q

uant

iles

ZMNB

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−3 −2 −1 0 1 2 3

−6−4

−20

24

6

Randomized Quantile

Theoretical Quantiles

Sam

ple Q

uant

iles

Poisson

4. Simulation Studies/Assessing Models for Datasets Simulated from ZMP Model 20/39