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12.3.4 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.

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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

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A logistic-normal model permitting treatment-by-center interaction is

The log odds ratio equals +bi in center i.

The model parameters

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SAS

proc nlmixed data=table12_5; /*model 12.11 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 12.12 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;

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Considerable evidence of a drug effect occurs.

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12.4 RANDOM EFFECTS MODELS FOR MULTINOMIAL DATA Random effects models for binary responses extend to

multicategory responses. models for nominal and ordinal responses.

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12.4.1 Cumulative Logit Model with Random Intercept A GLMM for the cumulative logits

Marginal effects tend to be smaller, increasingly so as sigma increases.

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12.4.2 Insomnia Study Revisited

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Marginal model

random-intercept model

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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.

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SAS

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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*/

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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;

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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

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12.5.1 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.

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categories for attitude as (positive, negative) ‘‘positive’’ refers to disagreeing with the statement that

one must go against his principles.

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For subject i, let yitv 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).

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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;

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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;

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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.

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12.5.3 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.

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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), 11-36.

http://www.acsu.buffalo.edu/~cxma/STA617/tekwe_statistical_assessment.pdf

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Summary:

Match pairs data Marginal modeling Conditional modeling (random effects)