1. Generating Data from Mixed Models and Estimating parameters using PROC MIXED
Consider a situation of Mixed Models where one gets n observations from k groups. Assume that response variable Y follows the mixed model
Yij = μ + βi Χij + εij i = 1,...k; j = 1,...,n,
where μ is a fixed common effect, βi is a random effect (common for a given group and different for different groups) and εij are error terms. Assume that the all random effects are normally distributed as
βi ~ N(0, σb2 ) and that εij ~ N(0, σe2 )
(a). Χij = 1 for all i and j as in a simple random effects model, k=5 and n=10, using SAS rand('normal') function generate a set of Yij when μ =10, σb = .5 and σe = 1.
(b). Using the above data estimate parameters μ and βi as well as σb = .5 and σe parameters by PROC MIXED and compare estimates with actual values (note: they can be very different due to small sample size, but we will study average performance below).
(c) Repeat above two steps when the covariate vector for each group is a vector from 1 to 10; i.e. Χi. = c(1,2,....n).
2. Simulation: Testing Accuracy of SAS Estimates
Convert the data generation in Exercise 1 above into a macro with σb and σe as parameters. Under the above two scenarios on covariate X, call your macro 1,000 times and report
(a) average of βi estimates when σb = .5 and σe = 1,
(b) average of σe estimates when σb = .1 and σe = 1,
(c) average of σb estimates when σb = .1 and σe = 1.