### Example 43.18 Weighted Multilevel Model for Survey Data

Multilevel models are a useful tool for analyzing survey data from multistage sampling designs. In multistage designs, the first-stage clusters (primary sampling units) are selected by using a probability sample from a list of first-stage units. In the second stage, second-stage clusters are selected by using a probability sample from a list of second-stage clusters for every first-stage cluster in the sample. The third and subsequent stages of clusters are selected similarly. You can use multilevel models to analyze data from multistage designs in which each stage of sampling corresponds to one level of random effects in the model. The characteristics of the units at each stage become the explanatory variables at that level. If units are drawn with unequal selection probabilities at each stage, then the unweighted estimators from standard multilevel models can be biased. Pfeffermann et al. (1998) and Rabe-Hesketh and Skrondal (2006) discuss using sampling weights to reduce this bias.

To learn how to fit a two-level model, consider a two-stage design simulation that is similar to the simulation discussed in Rabe-Hesketh and Skrondal (2006).

First, you generate a finite population of 500 level-2 units and 50 level-1 units from a two-level random intercept probit model with dichotomous responses and linear predictor,

where , , and . The between-cluster covariate x_1i and within-cluster covariate x_2ij are generated from a Bernoulli distribution with probability 0.5.

Then, you sample subsets of level-2 units and level-1 units from a two-stage sampling design similar to the one discussed in Rabe-Hesketh and Skrondal (2006). The generated sample contains a total of 7,530 observations. The inverse selection probabilities are used as weights. To reduce the bias in the variance parameter estimator, you scale the level-1 weights by using what Pfeffermann et al. (1998) and Rabe-Hesketh and Skrondal (2006) refer to as Method 2.

The first 30 observations of the data set are shown in Output 43.18.1. In this data set, y is the dichotomous response variable, and x1 and x2 are the between-cluster and within-cluster covariates, respectively. The variables id and w2 identify the first-stage clusters (level-2 units) and their weights, respectively. The observation units are the second-stage clusters (level-1 units), and their weights are scaled using Method 2 in Pfeffermann et al. (1998) and Rabe-Hesketh and Skrondal (2006) and stored in sw1. The first-stage sampling weights, w2, range from 1.3359 to 4.0513, with an average of 1.6965. The second-stage sampling weights, w1, range from 1.8571 to 2.000, with an average of 1.9588. The scaled weights, sw1, range from 0.9657 to 1.0126, with an average of 1.000.

Output 43.18.1: Simulated Two-Stage Sampling Survey Data (First 30 Observations)

Obs x1 x2 y w2 id sw1
1 1 0 1 1.33594 1 1.00286
2 1 0 1 1.33594 1 1.00286
3 1 1 1 1.33594 1 1.00286
4 1 1 1 1.33594 1 1.00286
5 1 0 1 1.33594 1 1.00286
6 1 1 1 1.33594 1 1.00286
7 1 1 1 1.33594 1 1.00286
8 1 1 1 1.33594 1 1.00286
9 1 0 1 1.33594 1 1.00286
10 1 0 1 1.33594 1 1.00286
11 1 1 1 1.33594 1 1.00286
12 1 0 1 1.33594 1 1.00286
13 1 1 1 1.33594 1 1.00286
14 1 0 1 1.33594 1 1.00286
15 1 0 1 1.33594 1 0.99667
16 1 1 1 1.33594 1 0.99667
17 1 0 1 1.33594 1 0.99667
18 1 0 1 1.33594 1 0.99667
19 1 0 1 1.33594 1 0.99667
20 1 1 1 1.33594 1 0.99667
21 1 0 1 1.33594 1 0.99667
22 1 0 1 1.33594 1 0.99667
23 1 0 1 1.33594 1 0.99667
24 1 0 1 1.33594 1 0.99667
25 1 1 1 1.33594 1 0.99667
26 1 1 1 1.33594 1 0.99667
27 1 0 1 1.33594 2 0.99667
28 1 0 1 1.33594 2 0.99667
29 1 1 1 1.33594 2 0.99667
30 1 1 1 1.33594 2 0.99667

The following statements fit a two-level model without using weights at any level:


class id;
model y = x1 x2 / dist=binomial link=probit solution;
random int  / subject=id;
run;


The parameter estimates are shown in Output 43.18.2. The estimated fixed-effects parameters are all close to their corresponding true values, but the estimated covariance parameter is almost half of its true value.

Output 43.18.2: Analysis of Two-Stage Sampling Survey Data

The GLIMMIX Procedure

Covariance Parameter Estimates
Cov Parm Subject Estimate Standard Error
Intercept id 0.5612 0.08777

Solutions for Fixed Effects
Effect Estimate Standard Error DF t Value Pr > |t|
Intercept 1.0300 0.07608 293 13.54 <.0001
x1 1.0809 0.1186 7234 9.11 <.0001
x2 0.9934 0.06344 7234 15.66 <.0001

You can incorporate the unequal weights at both levels into the model by using the OBSWEIGHT= and WEIGHT= options as follows:


class id;
model y = x1 x2 / dist=binomial link=probit obsweight=sw1 solution;
random int  / subject=id weight=w2;
run;


To fit a weighted multilevel model, you should use METHOD=QUAD. The EMPIRICAL=CLASSICAL option in the PROC GLIMMIX statement instructs PROC GLIMMIX to compute the empirical (sandwich) variance estimators for the fixed effect and the variance. The empirical variance estimators are recommended for the inference about fixed effects and variance estimated by pseudo-likelihood.

The OBSWEIGHT= option in the MODEL statement specifies the weight variable for the observation level. The WEIGHT= option in the RANDOM statement specifies the weight variable for the level that is specified by the SUBJECT= option.

The parameter estimates are shown in Output 43.18.3.

Output 43.18.3: Analysis of Two-Stage Sampling Survey Data Using Weights

The GLIMMIX Procedure

Covariance Parameter Estimates
Cov Parm Subject Estimate Standard Error
Intercept id 1.0106 0.2503

Solutions for Fixed Effects
Effect Estimate Standard Error DF t Value Pr > |t|
Intercept 1.0322 0.1263 497.3 8.17 <.0001
x1 1.1182 0.2142 12275 5.22 <.0001
x2 1.0074 0.07887 12275 12.77 <.0001

Note that the estimated variance component for the random id effect is much closer to the true value of 1 in the weighted analysis. When you use the raw level-2 weights and the scaled level-1 weights, the multilevel model reduces the bias in the variance parameter estimate. However, the standard errors for the weighted analysis are larger than those for the unweighted analysis.