1. Lecture 11 (Chapter 9)

2. Generalized Linear Mixed Models with Random Effects
The logistic regression model with random intercept
Example: 2x2 crossover trial
Example: Indonesian Children’s Health Study
The Poisson regression model with random intercept
Example: seizure data

3. Basic idea: There is natural heterogeneity among subjects.
Random Effects GLM
Basic idea: There is natural heterogeneity among subjects.
Systematic part
Random part

4. Example: 2x2 crossover trial
In random effects models, the regression coefficients measure the more direct influence of explanatory variables on the responses for heterogeneous individuals.
For example:

5. Example: 2x2 crossover trial
(Recall) Example:
This model states that:
1. each person has his/her own probability of positive response under a placebo (B)

6. Example: 2x2 crossover trial
This model also states that:
2. a person’s odds of a normal response are multiplied by when taking drug A, regardless of the initial risk
Individual odds of normal response when taking drug (x=1)
Individual odds of normal response when on placebo (x=0) (“initial risk”)

7. In Logistic Models
In other words:
Marginal model estimates are smaller than random effects model estimates
Tests of hypotheses approximately the same in random effects as in marginal

8. Let β be the vector of regression coefficients under a marginal model
Correspondence between regression parameters in random effects and marginal models
Let β be the vector of regression coefficients under a marginal model
Let β* be the vector of regression coefficients under a random effects model
G is the variance of the random effects Ui ~ N(0,G)
Random effects model
β* > β
G=0 => β* = β
Marginal model

9. Estimation of Generalized Linear Mixed Models
Setting:
f(Yij|Ui) in the exponential family
Yi1,…,Yini | Ui are independent
Ui ~ f(Ui,G)
Maximum Likelihood Estimation
Ui is a set of unobserved variables which we integrate out of the likelihood

10. Maximum Likelihood Estimation of G and β
Assume: Ui ~ N(0, G)
We can learn about one individual’s coefficients by understanding the variability in coefficients across the population (G)

11. Maximum Likelihood Estimation of G and β (cont’d)
If Gi is small, then rely on population average coefficients to estimate those for an individuals
We weight the cross-sectional information more heavily and we borrow strength across subjects
If Gi is large, then rely heavily on the data from each individual to estimate their own coefficients
We weight the longitudinal information more heavily since comparisons within a subject are likely to be more precise than comparisons among subjects

12. Maximum Likelihood Estimation
“Average-away” the random effects (Ui)
Use all the data
Use an EM algorithm
Use numerical integration

13. Example: 2x2 crossover trial

14. Max. Likelihood. Estimation - Example: 2x2 crossover trial

15. Max. Likelihood. Estimation - Example: 2x2 crossover trial
95% of the subjects would fall between ±(2 x 4.9) logit units of the overall mean
This range on the logit scale translates into probabilities between 0 and 1, i.e. some people have little change and others have very high hance of a normal reading in the placebo and the treatment groups

16. Max. Likelihood. Estimation - Example: 2x2 crossover trial
Assuming a constant treatment effect for all persons, the odds of a normal response for a subject are estimated to be
= exp(1.9) = 6.7
times higher on the active drug than on the placebo

17. *robust SE
**model-based SE

18. Assume: constant treatment effect for everybody
exp(1.9)=6.7 => odds of a normal response for a subject are 6.7 times higher on the active drug than on the placebo
exp(0.57)=1.67 => population average odds of a normal response are 1.67 times higher on the active drug than on the placebo

19. Example: 2x2 crossover trial Summary
Marginal model
Random effects model (maximum likelihood)
The smaller value from the marginal analysis is consistent with the theoretical inequality:

20. Example: Indonesian study
Logistic regression with random intercept
Relative odds of infection associated with Xerophthalmia are:
exp(0.57)=1.7,
but this is not stat sig different from 1
β* more similar to β because G is smaller

21. Indonesian Study: Maximum Likelihood Approach
With a random effects model, we can address the question of how an individual child’s risk for respiratory infection will change if his/her vitamin A status were to change.
We assume that each child has a distinct intercept which represents his/her propensity to infection.
We account for correlation by including random intercepts, Ui ~ N(0,G).

22. Indonesian Study: Maximum Likelihood Approach (cont’d)
considerable heterogeneity among children
Among children with linear predictor equal to the intercept (-2.2), i.e. children of average age and height, who are female, and who are vitamin A sufficient, about 95% would have a probability of infection between (0.025) and (0.31):
Relative odds of infection associated with vitamin A deficiency are exp(0.54)=1.7

23. Indonesian Study: Maximum Likelihood Approach (cont’d)
The longitudinal age effect on the risk of respiratory infection in Model 2 can be explained by the seasonal trend in Model 3
Because of the small heterogeneity (summarized by G), the estimates of the coefficients obtained under a random effects model are similar to those obtained under a marginal model
Ratio of the random effects coefficients and marginal coefficients are close to

24. Poisson model with random intercept: Epileptic seizure example

25. Poisson model with random intercept: Epileptic seizure example
2 weeks each
8 weeks
Randomly assigned to either placebo or progabide

26. Example: Seizure

27. Example: Seizure (cont’d)

28. Example: Seizure (cont’d)

29. Example: Seizure (cont’d)

30. Example: Seizure (cont’d)
The model does not fit well =>
Extend the model by including a random slope Ui2

31. Model 1: random intercept only (similar result to conditional likelihood)
Model 2: random intercept + random slope

32. Poisson-Gaussian Random Effects Models: Epileptic Seizure
Here, we assume there might be heterogeneity among subjects in the ratio of the expected seizure counts before and after randomization.
This degree of heterogeneity can be measured by G22.

33. Poisson-Gaussian Random Effects Models: Epileptic Seizure
We use maximum likelihood estimation.

34. Poisson-Gaussian Random Effects Models: Epileptic Seizure
The ratio of seizure counts in the placebo post-to-pre treatment is subject specific:
The ratio of seizure counts in the progabide post-to-pre treatment is subject specific:

35. Poisson-Gaussian Random Effects Models: Epileptic Seizure
Results
The estimate of G22 is statistically significant, therefore the data give support for between-subject variability in the ratio of the expected seizure counts before and after randomization.
subjects in the placebo group with Ui2=0 have expected seizure rates after treatment, which are estimated to be roughly similar to those before treatment

36. Poisson-Gaussian Random Effects Models: Epileptic Seizure
Results (cont’d)
Among subjects in the progabide group with Ui2=0, the seizure rates are reduced after treatment by about 27%,
The treatment seems to have a modest effect:
Without patient 207, the evidence for progabide is stronger: