mixed - Stata

40 mixed — Multilevel mixed-effects linear regression
magnitudes of these weights need to be normalized so that they are “consistent” from group to group.
This is in stark contrast to a standard analysis, where the scale of sampling weights does not factor
into estimation, instead only affecting the estimate of the total population size.
mixed offers three methods for standardizing weights in a two-level model, and you can specify
which method you want via the pwscale() option. If you specify pwscale(size), then the w i|j (or
w ij , it does not matter) are scaled to sum to the cluster size n j . Method pwscale(effective) adds
in a dependence on the sum of the squared weights so that level-one weights sum to the “effective”
sample size. Just like pwscale(size), pwscale(effective) also behaves the same whether you
have w i|j or w ij , and so it can be used with either.
Although both pwscale(size) and pwscale(effective) leave w j untouched, the pwscale(gk)
method is a little different in that 1) it changes the weights at both levels and 2) it does assume
you have w i|j for level-one weights and not w ij (if you have the latter, then first divide by w j ).
Using the method of Graubard and Korn (1996), it sets the weights at the group level (level two) to
the cluster averages of the products of both level weights (this product being w ij ). It then sets the
individual weights to 1 everywhere; see Methods and formulas for the computational details of all
three methods.
Determining which method is “best” is a tough call and depends on cluster size (the smaller
the clusters, the greater the sensitivity to scale), whether the sampling is informative (that is, the
sampling weights are correlated with the residuals), whether you are interested primarily in regression
coefficients or in variance components, whether you have a simple random-intercept model or a
more complex random-coefficients model, and other factors; see Rabe-Hesketh and Skrondal (2006),
Carle (2009), and Pfeffermann et al. (1998) for some detailed advice. At the very least, you want
to compare estimates across all three scaling methods (four, if you add no scaling) and perform a
sensitivity analysis.
If you choose to rescale level-one weights, it does not matter whether you have w i|j or w ij . For
the pwscale(size) and pwscale(effective) methods, you get identical results, and even though
pwscale(gk) assumes w i|j , you can obtain this as w i|j = w ij /w j before proceeding.
If you do not specify pwscale(), then no scaling takes place, and thus at a minimum, you need
to make sure you have w i|j in your data and not w ij .
Example 12
Rabe-Hesketh and Skrondal (2006) analyzed data from the 2000 Programme for International
Student Assessment (PISA) study on reading proficiency among 15-year-old American students, as
performed by the Organisation for Economic Co-operation and Development (OECD). The original
study was a three-stage cluster sample, where geographic areas were sampled at the first stage, schools
at the second, and students at the third. Our version of the data does not contain the geographic-areas
variable, so we treat this as a two-stage sample where schools are sampled at the first stage and
students at the second.