STAT 526 Advanced Statistical Methodology

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1 STAT 526 Advanced Statistical Methodology Fall 2017 Lecture Note 10 Analyzing Clustered/Repeated Categorical Data 0-0

2 Outline Clustered/Repeated Categorical Data Generalized Linear Mixed Models Generalized Estimating Equations Dabao Zhang Page 1

3 Clustered/Repeated Categorical Data Example: Effects of Surface and Vision on Balance An experiment was conducted to study the effects of surface and vision on balance. Forty subjects were studied, twenty males and twenty females. Each subject was tested twice in each of the surface (foam or a normal surface) and eye combinations (with eyes closed or open or with a dome placed over the head) for a total of 12 measures per subject. > data(ctsib, package="faraway"); > str(ctsib) data.frame : 480 obs. of 8 variables: $ Subject: int $ Sex : Factor w/ 2 levels "female","male": $ Age : int $ Height : num $ Weight : num $ Surface: Factor w/ 2 levels "foam","norm": $ Vision : Factor w/ 3 levels "closed","dome",..: $ CTSIB : int > ctsib$stable <- ifelse(ctsib$ctsib==1,1,0); > sum(ctsib$stable); [1] 114 Dabao Zhang Page 2

4 > csglm1 <- glm(stable~sex+age+height+weight+surface+vision,binomial,data=ctsib); > csglm2 <- glm(stable~sex+age+height+weight+surface+vision+factor(subject), binomial,data=ctsib); Warning message: fitted probabilities numerically 0 or 1 occurred in: glm.fit(x = X, y = Y, weights = weights, start = start, etastart = etastart, > anova(csglm1,csglm2,test="chi"); Analysis of Deviance Table... Resid. Df Resid. Dev Df Deviance P(> Chi ) e-20 The small p-value implies a significant subject effect. Note that the fixed subject effects confound these effects of subject-specific variables. When studying y ij = stable ij, i.e., whether the j-th measure of subject i is completely stable, we would rather consider unobserved, along with observed, subject-specific variables which affect y ij. Let x ij be the observed predictors for the j-th measure of subject i. Then the generalized linear model for subject i can be g(e[y ij γ i,x ij ]) = η ij = γ i + x ij β γ i accounts for the effect of unobserved subject-specific variables. x ij includes those observed subject-specific varaibles. Dabao Zhang Page 3

5 Example: Seizure Rates of Epileptics under Treatment In a clinical trial of 59 epileptics, patients were observed for 8 weeks and the number of seizures recorded for a baseline. Then the patients were randomized to treatment by the drug Progabide (31 patients) or the the placebo group (28 patients), and were observed for four 2-week periods with the number of seizures recorded. > data(epilepsy, package="faraway"); > str(epilepsy) data.frame : 295 obs. of 6 variables: $ seizures: num $ id : int $ treat : num $ expind : num $ timeadj : num $ age : num expind: indicates the baseline period by 0 and the treatment period by 1 timeadj: length of the time period Dabao Zhang Page 4

6 ># use matplot (x,y,...) to plot the columns of x against the columns of y > y <- matrix(epilepsy$seizures,nrow=5); > matplot(0:4,sqrt(y),type="l",lty=epilepsy$treat[5*(1:59)]+1, xlab="period",ylab="sqrt(seizures)"); > sepil <- epilepsy[(epilepsy$id<=5) (epilepsy$id<=33&epilepsy$id>=29),]; > sy <- matrix(sepil$seizures,nrow=5); > matplot(0:4,sqrt(sy),type="l",lty=sepil$treat[5*(1:20)]+1, xlab="period",ylab="sqrt(seizures)"); Sqrt(Seizures) Sqrt(Seizures) Period Q: How to model the correlated within-group responses for longitudinal data? Period Dabao Zhang Page 5

7 Generalized Linear Mixed Models For j-th observation of i-th subject, y ij γ i EFD(θ ij,φ), γ i iid N(0,Σ γ ) j = 1,,n i ; i = 1,,m θ ij is the canonical parameter φ is the (known) dispersion parameter E[y ij γ i ] = µ ij = b (θ ij ) var(y ij γ i ) = φb (θ ij ) = φv(µ ij ), assuming a ij (φ) = φ Generalized Linear Mixed Model: (Conditional Models) g(µ ij ) = η ij x ij β + z ij γ i, γ i iid N(0,Σ γ ) subject-wise: g(µ i ) = η i x i β + z i γ i, where g(µ i ) = (g(µ i1 ),,g(µ ini )) T sample-wise: g(µ) = η Xβ + Zγ, where γ = (γ 1,,γ m ) T For Y = (y 11 y 1n1 y m1 y mn m )T E[Y] = E{E[Y γ]} = E[µ ij ] = E[g 1 (Xβ + Zγ)] var(y) = var(e[y γ]) + E[var(Y γ)] = var(g 1 (Xβ + Zγ)) + φe[v(g 1 (Xβ + Zγ))] Dabao Zhang Page 6

8 Fitting GLMMs with Maximum Likelihood Approach The likelihood of the GLMM is L Σ γ m/2 m i=1 R q exp Σ γ m/2 R mq exp { ni { m i=1 j=1 n i j=1 Y ij θ ij b(θ ij ) a(φ) Y ij θ ij b(θ ij ) a(φ) Each γ i is assumed to be a q-dimensional column vector; D = diag{σ γ Σ γ } is mq mq; How to calculate the (possibly high-dimensional) integrals? 1 2 γt i Σ 1 γ γ i } dγ i, 1 2 γt Σ 1 γ γ }dγ, Laplace s method can be applied to approximate the integrals (Wolfinger, 1993); Alternative numerical methods are available to calculate the integrals, see Sinha (2004). Function lmer of package lme4 in R applies Laplace s method. Dabao Zhang Page 7

9 > library(lme4); > csglmm2 <- lmer(stable~sex+age+height+weight+surface+vision+(1 Subject),family=binomial,data=ctsib) > summary(csglmm2) Generalized linear mixed model fit using Laplace Formula: stable ~ Sex + Age + Height + Weight + Surface + Vision + (1 Subject) Data: ctsib Family: binomial(logit link) AIC BIC loglik deviance Random effects: Groups Name Variance Std.Dev. Subject (Intercept) number of obs: 480, groups: Subject, 40 Estimated scale (compare to 1 ) Fixed effects: Estimate Std. Error z value Pr(> z ) (Intercept) Sexmale Age Height Weight Surfacenorm < 2e-16 *** Visiondome Visionopen e-14 ***... > #try: lmer(stable~sex+age+height+weight+surface+vision+(1 Subject),method="PQL",family=binomial,data=ctsib) Dabao Zhang Page 8

10 Fitting GLMMs with Penalized Quasi-Likelihood Approach Since E[y ij γ i ] = µ ij, var(y ij γ i ) = φv(µ ij ), the log quasi-likelihood for y i is (assuming γ i is known) Q i = n i j=1 µij y ij y ij t φv(t) dt The integrated quasi-likelihood for the GLMM is L Σ γ m/2 m i=1 R q exp { } Q i 1 2 γt i Σ 1 γ γ i dγ i { m } = Σ γ m/2 R mq exp Q i 1 2 γt D 1 γ dγ i=1 Each γ i is assumed to be a q-dimensional column vector D = diag{σ γ Σ γ } is mq mq 1 2 γt D 1 γ can be considered as a penalty to the quasi-likelihood = penalized quasi-likelihood (PQL; Breslow and Clayton, 1993) Dabao Zhang Page 9

11 PQL is an approximate method of inference in GLMMs On the basis of current estimates (ˆβ,ˆγ), generate the working observation ỹ ij = ˆη ij + g (ˆµ ij )(y ij ˆµ ij ) update the estimates of (β,γ) from the following linear mixed models ỹ ij = x ij β + z ij γ i + ǫ ij γ i N(0,Σ γ ), ǫ ij N(0,φ[g (ˆµ ij )] 2 V(ˆµ ij )) PQL is implemented in package MASS of R > library(mass); help(glmmpql); glmmpql(fixed, random, family, data, correlation, weights, niter = 10,...) fixed: a two-sided linear formula giving fixed-effects part of the model. random: a formula or list of formulae describing the random effects. family: a GLM family. correlation: an optional correlation structure. weights: optional case weights as in glm. niter: maximum number of iterations....: Further arguments for lme. Dabao Zhang Page 10

12 PQL may sometimes yield badly biased estimates of variance components, especially in binary outcomes. An alternative algorithm is to apply Laplace s method to the integrals of the likelihood function for the GLMM, e.g., function lmer of package lme4 in R Example: Effects of Surface and Vision on Balance (Continued) > csglmm1 <- glmmpql(stable~sex+age+height+weight+surface+vision,random=~1 Subject,family=binomial,data=ctsib) > summary(csglmm1);... Random effects: Formula: ~1 Subject (Intercept) Residual StdDev: Variance function: Structure: fixed weights Formula: ~invwt Fixed effects: stable ~ Sex + Age + Height + Weight + Surface + Vision Value Std.Error DF t-value p-value (Intercept) Sexmale Age Height Weight Surfacenorm Visiondome Visionopen Dabao Zhang Page 11

13 Generalized Estimating Equations Instead of explicitly specifying random effects as in GLMM, we can directly model the correlation between within-group observations (marginal models) = Generalized Estimation Equations (GEE; Liang and Zeger, 1986) models Need to specify a link function, i.e., g(µ ij ) = x ij β where µ ij = E[y ij ] Need to specify a variance function V(µ ij ) Need to specify the correlation matrix R(α) = corr(y i ) GEE provides consistent estimates of the fixed effects in β even if the proposed correlation structure R(α) is incorrect! GEE is used to iteratively get an estimate of β by solving the following equations m ( ) T µi V 1 i (y i µ i ) = 0 β µ i = (µ i1 µ ini ) T y i = (y i1 y ini ) T i=1 V i = φa 1/2 i R(α)A 1/2 i where A i = diag{v(µ i1 ) V(µ ini )} GEE does not estimate the dispersion parameter φ, which can be estimated through Pearson s residuals. Dabao Zhang Page 12

14 The correlation matrix R(α) is generally unknown, so a working correlation matrix is specified and used instead. Examples of the working correlation matrix AR(1), i.e., corr(y ij,y ik ) = α j k Exchangle (compound symmetry), i.e., corr(y ij,y ik ) equals to 1 if j = k, α otherwise Unstructured, i.e., corr(y ij,y ik ) = α jk Independent, i.e., corr(y ij,y ik ) equals to 1 if j = k, 0 otherwise The sandwich estimator of the covariance of ˆβ Ĉov(ˆβ) = I 1 0 I 1I 1 ( µi I 0 = β i ( µi I 1 = β i ) T ( ) µi V i 0, β, ) T ( V 1 i (Y i ˆµ i )(Y i ˆµ i ) T µi β I 1 0 is a naive estimate of Cov(ˆβ) and may be very misleading. The sandwich estimator (sometimes called the robust or empirical estimator) provide a consistent estimate of Cov(ˆβ) even if the working correlation is misspecified. Dabao Zhang Page 13 ).

15 GEE in R Package gee ( > library(gee); help(gee); gee(formula, id, data, subset, family = gaussian, corstr = "independence", Mv = 1, contrasts = NULL, scale.fix = FALSE, scale.value = 1) id: a vector which identifies the clusters. Physically contiguous records possessing the same value are assumed to be in the same cluster. The following vector specifies 4 clusters of size 4: c(0,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1). corstr: specifying the correlation structure with "independence", "fixed", "stat M dep", "non stat M dep", "exchangeable", "AR-M" or "unstructured" Mv: must be specified for corstr = "stat M dep", "non stat M dep", or "AR-M". contrasts: a list giving contrasts for some or all of the factors appearing in the model formula. scale.fix: a logical variable; if true, the scale parameter is fixed at the value of scale.value. scale.value: numeric variable giving the value to which the scale parameter should be fixed; used only if scale.fix == TRUE. Dabao Zhang Page 13

16 Example: Effects of Surface and Vision on Balance (Continued) > csgee <- gee(stable~sex+age+height+weight+surface+vision,id=subject,corstr="exchangeable",family=binomial,data=ctsib); > summary(csgee); GEE: GENERALIZED LINEAR MODELS FOR DEPENDENT DATA gee S-function, version 4.13 modified 98/01/27 (1998) Model: Link: Logit Variance to Mean Relation: Binomial Correlation Structure: Exchangeable... Coefficients: Estimate Naive S.E. Naive z Robust S.E. Robust z (Intercept) Sexmale Age Height Weight Surfacenorm Visiondome Visionopen Estimated Scale Parameter: Working Correlation [,1] [,2] [,3] [,4] [,5] [,6] [,7] [1,] [2,] [3,] Dabao Zhang Page 14

17 Example: Seizure Rates of Epileptics under Treatment (Continued) > segee <- gee(seizures~offset(log(timeadj))+expind+treat+i(expind*treat),id=id,corstr="ar-m",mv=1, family=poisson,data=sepil,subset=(id!=49)); > summary(segee);... Model: Link: Logarithm Variance to Mean Relation: Poisson Correlation Structure: AR-M, M = 1... Coefficients: Estimate Naive S.E. Naive z Robust S.E. Robust z (Intercept) expind treat I(expind * treat) Estimated Scale Parameter: Number of Iterations: 4 Working Correlation [,1] [,2] [,3] [,4] [,5] [1,] [2,] [3,] [4,] [5,] Dabao Zhang Page 15

18 GLMM or GEE? The parameter β may have different interpretations In a GLMM, β measures the effect of the covariates for an individual In a GEE, β measures the effect of the covariates for a population, usually smaller. If a GLMM is a good model, then often so is a GEE model but for substantially different parameter estimates (Zeger et al., 1988) For log-linear models, only the intercepts will differ For logistic models, the slopes will be attenuated Zeger et al. (1988) pointed out that GLMM models can also be fitted by GEE methods, provided the latter are extended to estimate the variance components of the random effects. Dabao Zhang Page 16