Samples were selected from the generated finite population with unequal probability. First, level two observations (individuals) were selected as a function of the random slopes where selection is proportional to exp (δβi). This selection mechanism results in
overselection of observations with larger slope values. It also affects the intercept values since these are correlated with the slopes. Five hundred replicates with three different
level two sample sizes were selected. The level two sample sizes are 50, 200, and 500. Next, from each of the replicate level two samples, level one observations (time points) were selected as a function of the random error terms. Only the first 5 time points are included in the level one selection of samples to be more comparable with many panel studies and the empirical example provided in this paper. Errors at each time point are stratified into two strata where for stratum oneεit >0 and for stratum twoεit 50. Level
one observations (time points) are selected using stratified sampling in the proportions presented in Table 4.2.
This selection mechanism results in overselection of level one observations that have values greater than the average value as given by the population intercept and slope. The selection results in nonignorable (MNAR) missing values at level one since observations are dependent on the missing values ofyit. The majority (two-thirds) of cases have at least
four time points and around one-third have three or fewer time points. The particular selection design used in this simulation is designed to affect the average intercept and slope values, E(αi) andE(βi) where the average slope value should be inflated for both
models and the average intercept value should be deflated in the unconditional model and inflated in the conditional model due to the negative and positive covariances between the random effects. Random intercepts and slopes may also be affected by such a design, but are not necessarily affected. The random variation around the newly biased fixed estimates may remain the same as the random variation in the population for the true parameters. This is indeed the case for most of this simulation data as we shall observe in the results. The covariate effects should not be affected by the selection design because these effects are homogenous across the different levels of the slope and level one error values.
Unequal selection of observations was done in two stages in the SAS procedure SUR- VEYSELECT. Weights for the individuals selected into the samples and the conditional weights for the repeated measures selected into the samples are calculated automatically
by the SURVEYSELECT procedure as the inverse of the probability of selection of each observation. Time invariant weights are equal to the inverse of the probability of selection for level two observations, wi = π1
i where πi is proportional to exp (δβi). These weights
correct for unequal inclusion of individuals into the sample. The conditional weights for the selection of repeated measures is the inverse of the conditional probability of selection of level one observations given selection of the level two observation, λit = π1
it|πi where
πit|πi is the conditional probability of level one observation at time t given level two
observation iis selected into the sample and is proportional to values presented in Table 4.2.
Time varying weights were calculated as the inverse of the product of the probability of selection of level two observations and the probability of selection of level one obser- vations. Specifically, the probability of inclusion into the sample is multiplied by the conditional probability of inclusion for each time point separately. Time varying weights are therefore: ωit = 1 πi · 1 πit|πi
where wit is the inverse of the unconditional probability of selection of level one ob-
servation at time t. The wit are the traditional single level weights for cross-sectional
analysis at each time point separately. Multilevel weights are equal to wi for level two
and wt = πit1|πi for level one. Panel weights,w
p
i, are derived as the inverse of the product
of the level two probability of inclusion and the conditional probability of inclusion for every level one time point because selection probabilities are independent at each time point, i.e., wpi = 1 πi · 1 πi0|πi · 1 πi1|πi · 1 πi2|πi · 1 πi3|πi · 1 πi4|πi
The degree of unequal selection may be evaluated using the unequal weighting effect (UWE), which measures the amount of noise added to estimates. The UWE for a mean estimate is equal to 1 +cv2
w = 1 + var(w)
by a factor of √UWE. Table 4.3 presents the √UWE for each of the weights used by generating model and level two sample size. From the table it can be seen that the panel weights have the most variation relative to the other weights. The time varying weights have the second greatest variability as these combine unequal selection from both levels one and two. Finally, the selection of level two observations as indicated by √UWE for the time invariant weights is more variable than the selection of level one observations conditional on the level two selection.