proc QLIM, proc LOGISTIC, PROC GENMOD, PROC GLIMMIX to fit Population-averaged model vs. Subject-specific model

The GLM consists of three elements:

1. A probability distribution from the exponential family.

2. A linear predictor η = Xβ .

3. A link function g such that E(Y) = μ = g^-1(η).

For GLM, there are two different models: population-averaged model without mix effects, subject-specific model with mix effects.

The population-averaged approach is focused on modeling the mean response across the population of units at each time point as a function of time. Thus, the model described how the averages across the population of responses at different time points are related over time.

The subject-specific approach is focused on modeling the individual unit trajectories rather than the mean across all units. We did this by the introduction of random effects, e.g., the random coefficient model that says each unit has its own intercept and slope.

The MLE of beta is the solution of the normal equation, (sum_i x_i inv(sigma) (y_i-x_i*beta)=0). It is consistent and asymptotically normal with estimated asymptotic variance matrix H. There are three types of variances can be computed.
First, Hessian matrix H=-der^2.(l(beta))/der(beta)der(beta')
Second, asymptotic variance matrix of outer product B=der.(l(beta))/der(beta). der.(l(beta))/der(beta').
Third, asymptotic robust-sandwich estimate H^(-1)BH^(-1).

proc QLIM compute all three variances.
proc Logistic compute Hessian variance.
proc GENMOD and proc GLIMMIX compute Hessian variance and robust-sandwich variances.