Effect Partitioning in Cross-Sectionally Clustered Data Without Multilevel Models

Abstract

Effect partitioning is almost exclusively performed with multilevel models (MLMs) – so much so that some have considered the two to be synonymous. MLMs are able to provide estimates with desirable statistical properties when data come from a hierarchical structure; but the random effects included in MLMs are not always integral to the analysis. As a result, other methods with relaxed assumptions are viable options in many cases. Through empirical examples and simulations, we show how generalized estimating equations (GEEs) can be used to effectively partition effects without random effects. We show that more onerous steps of MLMs such as determining the number of random effects and the structure for their covariance can be bypassed with GEEs while still obtaining identical or near-identical results. Additionally, violations of distributional assumptions adversely affect estimates with MLMs but have no effect on GEEs because no such assumptions are made. This makes GEEs a flexible alternative to MLMs with minimal assumptions that may warrant consideration. Limitations of GEEs for partitioning effects are also discussed.

Keywords:

• Multilevel modeling
• contextual effect
• compositional effect
• hierarchical linear modeling

Notes

1 Note that GEEs are technically an estimation method and not a model. In the context of this paper, when we say “GEEs” by itself, this typically is shorthand for “regression model accounting for clustering estimated with generalized estimating equations”.

2 The most general way to express the generalized estimating equation is ${\sum }_{j=1}^J\frac{\partial {\mathbf{\mu }}_j}{\partial \mathbf{\gamma }}{\mathbf{V}}_j^{-1}\left({\mathbf{y}}_j-{\mathbf{X}}_j\mathbf{\gamma }\right)=0.$ Note the difference in the first term after the summation than what is presented in Equation 3. The general form allows GEEs to easily extend to discrete outcomes with a link function where $${\mathbf{\mu }}_j=g^{-1}({\mathbf{X}}_j\mathbf{\gamma }).$$ However, with continuous outcomes where g is the identity link, $\frac{\partial {\mathbf{\mu }}_j}{\partial \mathbf{\gamma }}$ simplifies simply to $${\mathbf{X}}_j$$ because $${\mathbf{\mu }}_j={\mathbf{X}}_j\mathbf{\gamma }$$ so $\frac{\partial {\mathbf{\mu }}_j}{\partial \mathbf{\gamma }}=\frac{\partial {\mathbf{X}}_j\mathbf{\gamma }}{\partial \mathbf{\gamma }}={\mathbf{X}}_j.$ Readers may note that this results in Equation 3 taking the same form of the least squares estimating equations, though the estimates can still be obtained with quasi-likelihood in the spirit of GEEs.

3 Consistent in statistical terminology means that the probability that the estimator produces the population value becomes arbitrary close to 1 as the sample size increases indefinitely.

4 Effect partitioning is much more widespread in longitudinal data than this review would seem to indicate. The low value here is likely attributable to the fact that Raudenbush and Bryk (2002) is a seminal reference for cross-sectionally clustered data while other sources are typically more comprehensive and more frequently cited for longitudinal applications (e.g., Curran & Bauer, 2011).

5 A 50:50 mixture χ² test (Stram & Lee, 1994) for the intercept variance (${}_{50:50}{\chi }^2(1,2)=270.2,p<.001$) and slope variance (${}_{50:50}{\chi }^2(1,2)=9.6,p=.003$) were both significant, indicating that the model with both random effects fits better. The covariance between the intercepts and slopes was not significant (${\chi }^2(1)=1.19,p=.274$).