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1、Correlated Binary and Survival Data Analysis1Correlated Binary and Survival Correlated binary and survival response data can occur in similar condition as correlated continuous response. Cardiovscular disease in for individuals coming from several families Survival time for patients coming from seve
2、ral institutions Methods for Correlated Binary Responses Generalized mixed effects model (GLMM) Generalized estimating equations (GEE)Generalized Linear Mixed Effects ModelsGeneralized mixed effects models (GLMMs) are generalized linear models with fixed and random effects as covariates. These model
3、s are useful for correlated non-normal response. The model notations are as follows: Conditional on fixed xyplot(yweek|trt,group=ID,data=bacteria,type=“o“)table(y, as.numeric(y) y 1 2n 43 0y 0 177 prop.y 2)+(1|ID),family=binomial,data=bacteria)Generalized linear mixed model fit by maximum likelihood
4、 (Laplace Approximation) glmerModFamily: binomial ( logit ) Formula: y trt + I(week 2) + (1 | ID)Data: bacteriaAIC BIC logLik deviance df.resid 202.3 219.2 -96.1 192.3 215 Scaled residuals: Min 1Q Median 3Q Max -4.5615 0.1359 0.3022 0.4217 1.1276 Random effects:Groups Name Variance Std.Dev.ID (Inter
5、cept) 1.543 1.242 Number of obs: 220, groups: ID, 50Fixed effects:Estimate Std. Error z value Pr(|z|) (Intercept) 3.5479 0.6958 5.099 3.41e-07 * trtdrug -1.3667 0.6770 -2.019 0.043517 * trtdrug+ -0.7826 0.6831 -1.146 0.251926 I(week 2)TRUE -1.5985 0.4759 -3.359 0.000783 * - Signif. codes: 0 * 0.001
6、* 0.01 * 0.05 . 0.1 112Other Models Consideredb=6)+(1|ID),family=binomial,data=bacteria)c2)+(1|ID),family=binomial,data=bacteria)d2)+(1|ID),family=binomial,data=bacteria)e=2*order of polynomials) Model DiagnosisTest of dispersion parameter: ratio of observed standard residual variance vs expected Be
7、low is an function from http:/ overdisp_fun expected (i.e. n*p(1-p). The inference uses expected variance therefore can be inflated.Thats why overdispersion is a more serious problem than underdispersion that results in a conservative result (less likely to be significant or lower power). Correction
8、 for over-dispersion includes including more covariates, interactions, non-linear effect quasi-binomial model (not available in lme4) individual level random effectsGeneralized Estimating EquationsGeneralized estimating equations belongs to marginal models in that the correlation among observations
9、are modelled in the random errors, in contrast to mixed effects models in additional random effects Assuming yi=( yi1, , yim) is a vector of observations taken at the ith unit; Here the covariance of Y is estimated separately from the coefficients. 16GEE Parameter EstimationsBased on Quasi-likelihoo
10、d models, can be estimated by solving the following generalized estimating equations (GEE): (Liang and Zeger 1986) Vi is unknown and can be provided by user called working correlation matrix estimate is the same as the GLM-logistic or LSQ estimates without accounting for correlation because the firs
11、t GEE equation is coincide with the score function from likelihood when the link is logistic and identity.17Parameter Variance Estimate in GEEwhere This variance estimate can be biased if Vi is not correctly specified. In order to avoid the bias, instead of using 2Vi as the covariance of Yi , the co
12、variance of Yi is estimated using fitted residuals. That is,18Robust Variance Estimator in GEEwhere are fitted residualsThis is called “sandwich variance estimator”, “empirical variance estimator”, “Robust variance estimator”19Fit GEE models in R1.gee() in package gee 2.geeglm() in package in geepac
13、k() The two functions yields identical results in most cases. geepack was developed more recently with more modeling options and more reliable than gee for extremely unbalanced design. For example, one may also specify working correlation as a function of covariates in geeglm()but cannot do the same
14、 in gee().library(gee) summary(gee(unclass(y)-1)trt+I(week2), family=binomial, id=ID, data=bacteria)20Beginning Cgee S-function, (#) geeformula.q 4.13 98/01/27 running glm to get initial regression estimate(Intercept) trtdrug trtdrug+ I(week 2)TRUE 2.8332459 -1.1186848 -0.6372256 -1.2948525 GEE: GEN
15、ERALIZED LINEAR MODELS FOR DEPENDENT DATAgee S-function, version 4.13 modified 98/01/27 (1998) Model:Link: Logit Variance to Mean Relation: Binomial Correlation Structure: Independent Call: gee(formula = (unclass(y) - 1) trt + I(week 2), id = ID, data = bacteria, family = binomial)Summary of Residuals:Min 1Q Median 3Q Max -0.94444615 0.05555385 0.15257307 0.17676895 0.39658649 21Coefficients:Estimate
