randomized quantile residuals: an omnibus model diagnostic...

Randomized Quantile Residuals: an Omnibus ModelDiagnostic Tool with Unified Reference Distribution

Longhai Li

Department of Mathematics and StatisticsUniversity of SaskatchewanSaskatoon, SK, CANADA

Presented on 19 June 2017Xiamen University, China

Acknowledgements

Joint work with Alireza Sadeghpour and Cindy X. Feng.

The work was supported by grants from Natural Sciences andEngineering Research Council of Canada (NSERC) and CanadaFoundation for Innovation (CFI).

Outline

1 Introduction

2 Non-normal Regression Models

3 Pearson and Deviance Residuals

4 Randomized Quantile Residual

5 Simulation StudiesDetection of Non-linearityDetection of Zero-Inflation

6 Application to a Big Health Dataset

7 Conclusion and Discussion

Section 1

Introduction

Introduction

Examining residuals, such as Pearson and deviance residuals, is aprimary method to identify the discrepancies between models anddata and to assess the overall goodness-of-fit of a model. In normallinear regression, both of these residuals coincide and are normallydistributed;

However, in non-normal regression models, the residuals are far fromnormality, with residuals aligning nearly parallel curves according todistinct response values, which imposes great challenges for visualinspection.

Randomized quantile residual was proposed by Dunn and Smyth(1996) to circumvent the above-mentioned problems in traditionalresiduals.

Randomized quantile residual is still lack of applications in practice.We will demonstrate how good it is.

1. Introduction/ 1/36

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

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Pea

rson

Deviance

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−1.5 −0.5 0.0 0.5 1.0 1.5−

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3

x

Dev

ianc

e

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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 corrected model. However,Pearson and deviance residuals exhibit trend and cluster in lines.

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

Randomized quantile residuals can be checked as traditional residualsfor normal regression.

1. Introduction/ 2/36

Section 2

Non-normal Regression Models

Generalized Linear Model

Generalized Linear Model (GLM): The GLM assumes that a responsevariable yi given xi follows a non-normal distribution, such as Poissonand Gamma, etc. A unified form for the PDF is:

f (yi ; θi , φ) = exp

{yiθi − b(θi )

a(φ)+ c(yi , φ)

}. (1)

A link function is used to connect the conditional expected value ofthe response variable, µi = E (yi |xi ), to a linear combination of thecovariates and regression parameters as,

g(µi ) = ηi = xiβ. (2)

2. Non-normal Regression Models/ 3/36

Zero-Inflated Models: I

Zero-Inflated Model: In practice, very often, we have excessive zerosin count data, which might not be captured by a conventional GLMmodel. One popular approach to model such data is to use a mixtureof point mass at zero modelling the non-risk group (structural zeros)and a GLM modelling the at-risk group.Example:The zero-inflated Poisson response variable with parameters λi andpi , denoted by ZIP(λi , pi ), is defined as:

yi ∼

{δ0 with probability pi

Poisson(λi ) with probability 1− pi ,(3)


Zero-Inflated Models: II

The PMF of the ZIP distribution:

dzip(yi = 0) = pi + (1− pi )e−λi (4)

dzip(yi = j) = (1− pi )e−λiλji

j!, j = 1, 2, · · · . (5)

The CDF of the ZIP distribution

pzip(yi = J;λi , pi ) =J∑

j=0

dzip(yi = j) = pi +(1−pi )ppois(J, λi ), (6)

Links of pi and λi to covariates

logit(pi ) = ziγ and log(λi ) = xiβ, (7)

where zi and xi are vectors of explanatory variables for pi and λi withγ and β corresponding to their parameter vectors, respectively.


Section 3

Pearson and Deviance Residuals

Pearson and Deviance Residuals

Pearson residual is defined as

ri =yi − µ̂i√V̂ (yi )

. (8)

The deviance residual for the ith observation is defined as signedsquare root of the corresponding component of D (y ; µ̂), i.e.

di = sign(yi − µ̂i )√

2{ωi

[yi (θ̃i − θ̂i )− b(θ̃i ) + b(θ̂i )

]}, (9)

where θ̃i and θ̂i denote the parameters in the saturated and fittedmodels, respectively.

3. Pearson and Deviance Residuals/ 6/36

Problems with Traditional Residuals

In regression models for discrete outcomes, the residuals are far fromnormality, with residuals aligning nearly parallel curves according todistinct response values, which poses great challenges for visualinspection. Therefore, residual plots for the diagnosis of models fordiscrete outcome variables give very limited meaningful informationfor model diagnosis, which renders it of no practical use.

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, where n is the sample size and p is the number of parameters.

However, the use of this asymptotic distribution for both X 2 and Dappears lack of theoretical underpinning.

3. Pearson and Deviance Residuals/ 7/36

Section 4

Randomized Quantile Residual

Definition of Randomized Quantile Residual

Predictive p-value for continuous yi :

F (yi ; µ̂i , φ̂) = P(Yi ≤ yi | µ̂i , φ̂)

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

F ∗(yi ; µ̂i , φ̂, ui ) = F (yi−; µ̂i , φ̂) + ui d(yi ; µ̂i , φ̂), (10)

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

of F (·, µ̂i , φ̂) at yi .

Randomized quantile residual

qi = Φ−1(F ∗(yi ; µ̂i , φ̂, ui )) (11)

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

4. Randomized Quantile Residual/ 8/36

Illustrative Example #1: I

The true model for yi has the PMF:

yi 0 1 2

d0(yi ) 0.25 0.5 0.25(12)

We compare with a wrong model with PMF d1:

yi 0 1 2

d1(yi ) 0.1 0.8 0.1(13)


Illustrative Example #1: II

Figure 2: The randomized predictive p-value for the true model and the wrongmodel.

(a) F ∗ for true model (b) F̃ ∗ for wrong model


Illustrative Example #2: 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.


Illustrative Example #2: II

CDF linesThe CDF of the response variable Yi given xi (under a consideredmodel with parameters estimated with sample) is denoted byF (k|xi ) = P(Yi ≤ k |xi ), for k = 0, 1, · · · .An illustrative picture


Illustrative Example #2: III


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 d(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 d(yi ;µi , φ) (10). Suppose Ui is uniformlydistributed on (0,1]. Then, we have

F ∗(Yi ;µi , φ,Ui ) ∼ Uniform(0, 1], (14)

andqi = φ−1(F ∗(Yi ;µi , φ,Ui )) ∼ N(0, 1). (15)


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) (16)

=∞∑j=1

P(F ∗(Yi ;µi , φ,Ui ) ∈ B|Yi = k(j))× P(Yi = k(j)) (17)

=∞∑j=1

length(F (j) ∩ B)

p(j)× p(j) (18)

=∞∑j=1

length(F (j) ∩ B) (19)

= length(∪∞j=1F(j) ∩ B) = length(B) (20)


Section 5

Simulation Studies

Subsection 1

Detection of Non-linearity

Simulation Setup

We simulate a covariate x ∼ Uniform(−1.5, 1.5) of size n = 1000.

The response variable is simulated from a negative binomialregression model

log(µi ) = β0 + β1x2i ,

where µi is the expected count for the ith subject.

Then, we consider fitting a wrong model assuming

log(µi ) = β0 + β1xi .

We set β0 = 0, β1 = 1.

The reciprocal for the dispersion parameter associated with thenegative binomial distribution is set as k = 2.

5. Simulation Studies/Detection of Non-linearity 16/36

Scatterplots of Residuals for a Single DatasetPearson

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01

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−1.5 −0.5 0.0 0.5 1.0 1.5−

3−

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10

12

3x

Ran

dom

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ntile


QQ-plots of Residuals for a Single Dataset

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01

23

45

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01

23

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−3 −2 −1 0 1 2 3

−3

−2

−1

01

23

Randomized Quantile

Theoretical QuantilesS

ampl

e Q

uant

iles


Replicated Overall GOF Checking with Normality TestPearson

Pearson

p−value

Fre

quen

cy

0.0 0.2 0.4 0.6 0.8 1.0

020

0060

0010

000

DevianceDeviance

p−value

Fre

quen

cy0.0 0.2 0.4 0.6 0.8 1.00

2000

6000

1000

0

Randomized QuantileRandomized Quantile

p−value

Fre

quen

cy

0.0 0.2 0.4 0.6 0.8 1.0

010

030

050

0

Pearson

p−value

Fre

quen

cy

0.0 0.2 0.4 0.6 0.8 1.0

020

0060

0010

000

Deviance

p−value

Fre

quen

cy

0.0 0.2 0.4 0.6 0.8 1.0

020

0060

0010

000

Randomized Quantile

p−valueF

requ

ency

0.0 0.2 0.4 0.6 0.8 1.00

2000

6000


Replicated Overall GOF Checking with χ2 Test

Pearson

p−value

Fre

quen

cy

0.0 0.2 0.4 0.6 0.8 1.0

020

040

060

080

0Deviance

p−valueF

requ

ency

0.0 0.2 0.4 0.6 0.8 1.00

2000

6000

1000

0 Randomized Quantile

p−value

Fre

quen

cy

0.0 0.2 0.4 0.6 0.8 1.0

020

040

060

080

0

Pearson

p−value

Fre

quen

cy

0.0 0.2 0.4 0.6 0.8 1.0

020

0060

0010

000 Deviance

p−value

Fre

quen

cy

0.0 0.2 0.4 0.6 0.8 1.0

020

0060

0010

000 Randomized Quantile

p−valueF

requ

ency

0.0 0.2 0.4 0.6 0.8 1.0

020

040

060

080

010

00


Subsection 2

Detection of Zero-Inflation

Simulation Setup

We simulate a data set of size 1000 with a covariate x from a uniformdistribution over (−1, 2) and the response variable from a ZIP model,where the expected mean of the Poisson component is

log(λi ) = β0 + β1xi

A Poisson model with the same expected mean λi is used as anwrong model to compare the power of various residuals.

5. Simulation Studies/Detection of Zero-Inflation 21/36

Scatterplots of Residuals for a Single Dataset

Pearson

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02

46

x

Pea

rson

Deviance

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3−

2−

10

12

3x

Dev

ianc

e

Randomized Quantile

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−1

01

23

x

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dom

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ntile

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QQ-plots of Residuals for a Single Dataset

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−3 −2 −1 0 1 2 3

−15

−10

−5

05

Randomized Quantile


Sam

ple

Qua

ntile

s


Replicated Overall GOF Checking with Normality Test

PearsonPearson

p−value

Fre

quen

cy

0.0 0.2 0.4 0.6 0.8 1.0

020

0060

0010

000

DevianceDeviance

p−value

Fre

quen

cy

0.0 0.2 0.4 0.6 0.8 1.0

020

0060

0010

000

Randomized QuantileRandomized Quantile

p−value

Fre

quen

cy

0.0 0.2 0.4 0.6 0.8 1.0

010

020

030

040

050

0

Pearson

p−value

Fre

quen

cy

0.0 0.2 0.4 0.6 0.8 1.0

020

0060

0010

000

Deviance

p−value

Fre

quen

cy

0.0 0.2 0.4 0.6 0.8 1.0

020

0060

0010

000

Randomized Quantile

p−valueF

requ

ency

0.0 0.2 0.4 0.6 0.8 1.0

020

0060

0010

000


Replicated Overall GOF Checking with χ2 Test

Pearson

p−value

Fre

quen

cy

0.0 0.2 0.4 0.6 0.8 1.0

010

030

050

070

0

Deviance

p−valueF

requ

ency

0.0 0.2 0.4 0.6 0.8 1.0

020

0060

0010

000

Randomized Quantile

p−value

Fre

quen

cy

0.0 0.2 0.4 0.6 0.8 1.0

010

030

050

0

Pearson

p−value

Fre

quen

cy

0.0 0.2 0.4 0.6 0.8 1.0

020

0060

0010

000

Deviance

p−value

Fre

quen

cy

0.0 0.2 0.4 0.6 0.8 1.0

020

0060

0010

000

Randomized Quantile

p−valueF

requ

ency

0.0 0.2 0.4 0.6 0.8 1.0

020

0060

0010

000


Section 6

Application to a Big Health Dataset

Data Description

We apply the randomized quantile residual to examine thegoodness-of-fits of the above mentioned more flexible models in alarge survey dataset on 4406 individuals, which was collected by theNational Medical Expenditure Survey (NMES) for studying thedemand of health care of the elderly in the United States.

The response considered is the number of emergency departmentvisits.

The covariates include health measures: self-perceived health, thenumber of chronic conditions, and a measure of disability status;demographic variables: age, race, sex, marital status, education andregion; economic variables: family income, employment status,supplementary private insurance status, and public insurance status.

6. Application to a Big Health Dataset/ 26/36

List of Variables

Variable Definition

Emer number of emergency department visitsExclhlth =1 if self-perceived health is excellentAvghlth =1 if self-perceived health is averagePoorhlth =1 if self-perceived health is poorNumchron number of chronic conditions (cancer, heart attack, gall bladder problems,

emphysema, arthritis, diabetes, other heart disease)Adldiff =1 if the person has a condition that limits activities of daily livingNoreast =1 if the person lives in the northeastern USMidwest =1 if the person lives in the midwestern USWest =1 if the person lives in the western USAge age in years (divided by 10)Black =1 if the person is African AmericanMale =1 if the person is maleMarried =1 if the person is marriedSchool number of years of educationFaminc family income in 10,000Employed =1 if the person is employedPrivins =1 if the person is covered by private health insuranceMedicaid =1 if the person is covered by Medicaid


Histogram of Numbers of Emergency Visits

Figure 12: Frequency distribution for the number of emergency department visits.

Number of emergency room visits

Fre

quen

cy

0 2 4 6 8 10 12

010

0020

0030

00


Four Fitted Models for Further Diagnosis

We may suspect that there are over-dispersion and/or excessive zerocounts; therefore, we consider fitting four models:

PoissonNegative Binomial (NB)Zero-inflated Poisson (ZIP) andZero-inflated Negative Binomial (ZINB)

Comparing models with Akaike Information CriterionPoisson=5648, NB=5352, ZIP=5418 and ZINB=5354

AIC can be used to compare the goodness-of-fit of competing models,but it cannot tell

whether a model fits the data adequately,whether the distribution assumption in the response variable is valid.


Scatterplots of Residuals for Poisson and ZIPPearson

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0.1 0.2 0.5 1.0 2.0

05

1015

Poisson

Fitted values

Pea

rson

Res

idua

ls

Deviance

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0.1 0.2 0.5 1.0 2.0−

20

24

6

Poisson

Fitted values

Dev

ianc

e R

esid

uals

Randomized Quantile

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0.1 0.2 0.5 1.0 2.0

−6

−4

−2

02

46

Poisson

Fitted values

Ran

dom

ized

Qua

ntile

Res

idua

ls

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0.1 0.2 0.5 1.0 2.0

02

46

810

12

ZIP

Fitted values

Pea

rson

Res

idua

ls

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0.1 0.2 0.5 1.0 2.0

02

46

ZIP

Fitted values

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ianc

e R

esid

uals

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0.1 0.2 0.5 1.0 2.0

−6

−4

−2

02

46

ZIP

Fitted valuesR

ando

miz

ed Q

uant

ile R

esid

uals


Scatterplots of Residuals for NB and ZINB

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0.1 0.2 0.5 1.0 2.0

02

46

810

12

NB

Fitted values

Pea

rson

Res

idua

ls

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0.1 0.2 0.5 1.0 2.0

−1

01

23

4

NB

Fitted valuesD

evia

nce

Res

idua

ls

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0.1 0.2 0.5 1.0 2.0

−6

−4

−2

02

46

NB

Fitted values

Ran

dom

ized

Qua

ntile

Res

idua

ls

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0.1 0.2 0.5 1.0 2.0

02

46

810

12

ZINB

Fitted values

Pea

rson

Res

idua

ls

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0.1 0.2 0.5 1.0 2.0

−1

01

23

4

ZINB

Fitted values

Dev

ianc

e R

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uals

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0.1 0.2 0.5 1.0 2.0

−6

−4

−2

02

46

ZINB

Fitted valuesR

ando

miz

ed Q

uant

ile R

esid

uals


QQ-plots of Residuals for Poisson and ZIPPearson

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02

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−2 0 2

−2

02

46

ZIP


ampl

e Q

uant

iles


QQ-plots of Residuals for NB and ZINB

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−2 0 2

−2

02

4

ZINB


ampl

e Q

uant

iles


Replicated Normality Test p-values of RQR

Figure 17: Frequency of the p-values of the Shaprio-Wilk normality test for the 1000replicated randomized quantile residuals.

Poisson

Shaprio−Wilk p−value

Freq

uenc

y

0.0 0.2 0.4 0.6 0.8 1.0

010

020

030

040

0NB


Freq

uenc

y

0.0 0.2 0.4 0.6 0.8 1.0

010

2030

4050

6070

ZIP


Freq

uenc

y

0.0 0.2 0.4 0.6 0.8 1.0

010

020

030

040

050

0

ZINB


Freq

uenc

y

0.0 0.2 0.4 0.6 0.8 1.0

010

2030

4050

60


Section 7

Conclusion and Discussion

Conclusion

Model diagnosis in non-normal regression is very important but difficult.This paper empirically demonstrates that randomized quantile residual isan excellent diagnostic tool: We have empirically demonstrated that

under any true model, randomized residual quantiles are normaldistributed.

the overall GOF tests by applying normality test to randomizedquantile residuals are well-calibrated

randomized quantile residual has great power in detecting many kindsof model inadequacy

7. Conclusion and Discussion/ 35/36

Future work

Randomized quantile residual can be applied to mixed-effect modelsfor data with complex structures, such as clustered, temporal, andlongitudinal data. We are working to develop residual-based modeldiagnostic tools for widely used R packages such as lme4, and mgcv

packages in R.

Randomized quantile residual can be applied to check hierarchicalBayesian models for datasets with complex structure, as analternative of the widely used posterior predictive checking.

Utilize randomized quantile residual in many contemporaryapplications, such as in diagnosing mixed-effect models forhigh-throughput data.

7. Conclusion and Discussion/ 36/36

randomized quantile residuals: an omnibus model diagnostic...

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