1 STA 617 – Chp12 Generalized Linear Mixed Models STA 617 – Chp12 Generalized Linear Mixed Models 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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1 STA 617 – Chp12 Generalized Linear Mixed Models 12.3.4 Modeling Heterogeneity among Multicenter Clinical Trials compare two groups on a response for.
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
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1STA 617 – Chp12 Generalized Linear Mixed ModelsSTA 617 – Chp12 Generalized Linear Mixed Models
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.
2STA 617 – Chp12 Generalized Linear Mixed ModelsSTA 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
3STA 617 – Chp12 Generalized Linear Mixed ModelsSTA 617 – Chp12 Generalized Linear Mixed Models
A logistic-normal model permitting treatment-by-center interaction is
The log odds ratio equals +bi in center i.
The model parameters
4STA 617 – Chp12 Generalized Linear Mixed ModelsSTA 617 – Chp12 Generalized Linear Mixed Models
(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
17STA 617 – Chp12 Generalized Linear Mixed ModelsSTA 617 – Chp12 Generalized Linear Mixed Models
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.
18STA 617 – Chp12 Generalized Linear Mixed ModelsSTA 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.
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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;
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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.
25STA 617 – Chp12 Generalized Linear Mixed ModelsSTA 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), 11-36.