1 STA 617 – Chp12 Generalized Linear Mixed Models Modeling Heterogeneity among Multicenter Clinical Trials  compare two groups on a response for data stratified on a third variable.  The main focus relates to studying the association in the 2X2 tables and whether and how it varies among the strata.  such as schools or medical clinics.  The fit of the random effects model provides a simple summary such as an estimated mean and standard deviation of log odds ratios for the population of strata.  In each stratum it also provides a predicted log odds ratio that shrinks the sample value toward the mean.  This is especially useful when the sample size in a stratum is small and the ordinary sample odds ratio has large standard error.

2 STA 617 – Chp12 Generalized Linear Mixed Models Clinical Trial Relating Treatment to Response for Eight Centers  The purpose: compare an active drug and a control, for curing an infection.  t : =1 active drug; =2 control  logistic-normal no interaction

3 STA 617 – Chp12 Generalized Linear Mixed Models  A logistic-normal model permitting treatment-by-center interaction is  The log odds ratio equals +b i in center i.  The model parameters

4 STA 617 – Chp12 Generalized Linear Mixed Models SAS proc nlmixed data=table12_5; /*model page 508*/ eta = alpha + beta*t + u; p = exp(eta) / (1 + exp(eta)); model success ~ binomial(n,p); random u ~ normal(0,sigma*sigma) subject=center; proc nlmixed data=table12_5; /*model page 509*/ eta = alpha + (beta + b) * t + u; p = exp(eta) / (1 + exp(eta)); model success ~ binomial(n,p); random u b ~ normal([0, 0], [sigma_a*sigma_a, 0, sigma_b*sigma_b]) subject=center; run;

5 STA 617 – Chp12 Generalized Linear Mixed Models  Considerable evidence of a drug effect occurs.

6 STA 617 – Chp12 Generalized Linear Mixed Models

RANDOM EFFECTS MODELS FOR MULTINOMIAL DATA  Random effects models for binary responses extend to multicategory responses.  models for nominal and ordinal responses.

8 STA 617 – Chp12 Generalized Linear Mixed Models Cumulative Logit Model with Random Intercept  A GLMM for the cumulative logits  Marginal effects tend to be smaller, increasingly so as sigma increases.

9 STA 617 – Chp12 Generalized Linear Mixed Models Insomnia Study Revisited

10 STA 617 – Chp12 Generalized Linear Mixed Models  Marginal model  random-intercept model

11 STA 617 – Chp12 Generalized Linear Mixed Models  Results are substantively similar to the marginal model, but estimates and standard errors are about 50% larger.  This reflects the relatively large heterogeneity sigma=1.90 and the resultant strong association between the responses at the two occasions.

12 STA 617 – Chp12 Generalized Linear Mixed Models SAS

13 STA 617 – Chp12 Generalized Linear Mixed Models Marginal model proc genmod data=francom; class case; model outcome = treat time treat*time /dist=multinomial link=clogit; repeated subject=case / type=indep corrw; run; /*GEE*/

14 STA 617 – Chp12 Generalized Linear Mixed Models proc nlmixed qpoints=40; bounds i2 > 0; bounds i3 > 0; eta1 = i1 + treat*beta1 + time*beta2 + treat*time*beta3 + u; eta2 = i1 + i2 + treat*beta1 + time*beta2 + treat*time*beta3 + u; eta3 = i1 + i2 + i3 + treat*beta1 + time*beta2 + treat*time*beta3 + u; p1 = exp(eta1)/(1 + exp(eta1)); p2 = exp(eta2)/(1 + exp(eta2)) - exp(eta1)/(1 + exp(eta1)); p3 = exp(eta3)/(1 + exp(eta3)) - exp(eta2)/(1 + exp(eta2)); p4 = 1 - exp(eta3)/(1 + exp(eta3)); ll=(outcome=1)*log(p1)+(outcome=2)*log(p2) +(outcome=3)*log(p3)+(outcome=4)*log(p4); model outcome ~ general(ll); estimate 'interc2' i1+i2; * this is alpha_2 in model, and i1 is alpha_1; estimate 'interc3' i1+i2+i3; * this is alpha_3 in model; random u ~ normal(0, sigma*sigma) subject=case; run;

15 STA 617 – Chp12 Generalized Linear Mixed Models

16 STA 617 – Chp12 Generalized Linear Mixed Models 12.5 MULTIVARIATE RANDOM EFFECTS MODELS FOR BINARY DATA  random effects are often univariate, taking the form of random intercepts.  bivariate random effects are helpful for describing heterogeneity in multicenter clinical trials.  multivariate random effects are natural

17 STA 617 – Chp12 Generalized Linear Mixed Models Matched Pairs with a Bivariate Binary Response  A sample of schoolboys were interviewed twice, several months apart, and asked about their self-perceived membership in the ‘‘leading crowd’’ and about whether they sometimes needed to go against their principles to belong to that group.  two binary response variables, which we refer to as membership and attitude, measured at two interview times for each subject.

18 STA 617 – Chp12 Generalized Linear Mixed Models  categories for attitude as (positive, negative)  ‘‘positive’’ refers to disagreeing with the statement that one must go against his principles.

19 STA 617 – Chp12 Generalized Linear Mixed Models  For subject i, let y itv be the response at interview time t on variable v, where v=M for membership and v=A for attitude.  The logit model  is a bivariate random effect that describes subject heterogeneity for (membership, attitude).

20 STA 617 – Chp12 Generalized Linear Mixed Models SAS data new; set crowd; case=_n_; x1m=1; x1a=0; x2m=0; x2a=0; var=1; resp=mem1; output; x1m=0; x1a=1; x2m=0; x2a=0; var=0; resp=att1; output; x1m=0; x1a=0; x2m=1; x2a=0; var=1; resp=mem2; output; x1m=0; x1a=0; x2m=0; x2a=1; var=0; resp=att2; output; drop mem1 att1 mem2 att2;

21 STA 617 – Chp12 Generalized Linear Mixed Models proc nlmixed data=new qpoints=50; eta=beta1m*x1m + beta1a*x1a + beta2m*x2m + beta2a*x2a + um*var + ua*(1-var); p=exp(eta)/(1+exp(eta)); model resp ~ binary(p); random um ua ~ normal([0,0],[s1*s1, cov12, s2*s2]) subject=case; replicate count; estimate 'mem change' beta2m-beta1m; estimate 'att change' beta2a-beta1a; run;

22 STA 617 – Chp12 Generalized Linear Mixed Models

23 STA 617 – Chp12 Generalized Linear Mixed Models  For both variables, the probability of the first outcome category is higher at the second interview.  For instance, for a given subject the odds of self- perceived membership in the leading crowd at interview 2 are estimated to be exp(0.379)=1.46 times the odds at interview 1.

24 STA 617 – Chp12 Generalized Linear Mixed Models Hierarchical (Multilevel) Modeling  Hierarchical data structures, with units grouped at different levels, are common in education.  A statewide study of factors that affect student performance might measure students’ scores on a battery of exams but use a model that takes into account the student, the school or school district, and the county.  Just as two observations on the same student might tend to be more alike than observations on different students, so might two students in the same school tend to be more alike than students from different schools.  Student, school, and county terms might be treated as random effects, with different ones referring to different levels of the model. For instance, a model might have students at level 1, schools at level 2, and counties at level 3.  GLMMs for data having a hierarchical grouping of this sort are called multi-level models.  Random effects enter the model at each level of the hierarchy.

25 STA 617 – Chp12 Generalized Linear Mixed Models For normal data  Tekwe, C. D., Carter, R. L., Ma, C., Algina, J., Lucas, M., Roth, J., Ariet, M., Fisher, T., & Resnick, M. B.(2002). An empirical comparison of statistical models for value-added assessment of school performance. Journal of Educational and Behavioral Statistics. 29(1), An empirical comparison of statistical models for value-added assessment of school performance.  atistical_assessment.pdf atistical_assessment.pdf

26 STA 617 – Chp12 Generalized Linear Mixed Models Summary:  Match pairs data  Marginal modeling  Conditional modeling (random effects)