Robust Methods for Moderation Analysis with a Two-Level Regression Model

ABSTRACT

Moderation analysis has many applications in social sciences. Most widely used estimation methods for moderation analysis assume that errors are normally distributed and homoscedastic. When these assumptions are not met, the results from a classical moderation analysis can be misleading. For more reliable moderation analysis, this article proposes two robust methods with a two-level regression model when the predictors do not contain measurement error. One method is based on maximum likelihood with Student's t distribution and the other is based on M-estimators with Huber-type weights. An algorithm for obtaining the robust estimators is developed. Consistent estimates of standard errors of the robust estimators are provided. The robust approaches are compared against normal-distribution-based maximum likelihood (NML) with respect to power and accuracy of parameter estimates through a simulation study. Results show that the robust approaches outperform NML under various distributional conditions. Application of the robust methods is illustrated through a real data example. An R program is developed and documented to facilitate the application of the robust methods.

KEYWORDS:

• Complete moderation
• partial moderation
• robust methods
• sandwich-type standard errors

Article information

Conflict of interest disclosures: Each author signed a form for disclosure of potential conflicts of interest. No authors reported any financial or other conflicts of interest in relation to the work described.

Ethical principles: The authors affirm having followed professional ethical guidelines in preparing this work. These guidelines include obtaining informed consent from human participants, maintaining ethical treatment and respect for the rights of human or animal participants, and ensuring the privacy of participants and their data, such as ensuring that individual participants cannot be identified in reported results or from publicly available original or archival data.

Funding: This work was supported by Grant SES-1461355 from the National Science Foundation.

Role of the Funders/Sponsors: None of the funders or sponsors of this research had any role in the design and conduct of the study; collection, management, analysis, and interpretation of data; preparation, review, or approval of the manuscript; or decision to submit the manuscript for publication.

Acknowledgments: The authors thank Dr. Stephen West and two anonymous reviewers for their comments on prior versions of this manuscript. The ideas and opinions expressed herein are those of the authors alone, and endorsement by the authors’ institution or the National Science Foundation is not intended and should not be inferred.

Notes

1 Even for the two-level model that is to be studied in the current article, the  Level 1 error e and the Level 2 error ϵ₀ are not separable.

2 Another assumption in the two-level regression model is that x_i and z_i do not contain measurement errors. We will discuss measurement errors in predictors in the concluding section.

3 A scale parameter determines the statistical dispersion of a probability distribution. In general, the larger the scale parameter, the more spread out the distribution. In a normal distribution, scale typically means standard deviation.

4 With the LS estimates of $\mathbf{\gamma }$ and the population values of $\mathbf{\sigma }$ as starting values, the criterion for convergence is defined as the total absolute difference for $\widehat{\mathbf{\theta }}$ between two consecutive iterations being smaller than .0001 within 500 iterations.

5 For TMLE of $\widehat{\mathbf{\sigma }}$, h_i is replaced by [(m − 2)/m]h_i in (B2(B2) $$\begin{array}{c}\hfill \widehat{\mathbf{\sigma }}={\left(\sum _{i=1}^n\frac{1}{{\tau }_i^4}{\mathbf{h}}_i{\mathbf{h}}_i^{\text{'}}\right)}^{-1}\left(\sum _{i=1}^n\frac{1}{{\tau }_i^2}{w}_{i2}{r}_i^2{\mathbf{h}}_i\right).\end{array}$$(B2) ). This is because the connection between τ_i and ${\mathbf{h}}_i^{\text{'}}\mathbf{\sigma }$ is different under different distributional assumption of δ_i.

6 For sandwich-type covariance matrix of TMLEs, h_i is replaced by [(m − 2)/m]h_i when obtaining the derivatives in (C1(C1) $$\frac{\partial {\mathbf{g}}_{i1}\left(\mathbf{\gamma }\right)}{\partial {\mathbf{\gamma }}^{\text{'}}}=-\frac{r_i}{{\tau }_i^2}{\mathbf{c}}_i{\mathbf{c}}_i^{\text{'}}\frac{\partial w_{i1}}{\partial r_i}-\frac{1}{{\tau }_i^2}{w}_{i1}{\mathbf{c}}_i{\mathbf{c}}_i^{\text{'}},\left[4pt\right]$$(C1) ) to (C4(C4) $$\frac{\partial {\mathbf{g}}_{i2}\left(\mathbf{\sigma }\right)}{\partial {\mathbf{\sigma }}^{\text{'}}}=-\frac{r_i^3}{4{\tau }_i^4}{\mathbf{h}}_i{\mathbf{h}}_i^{\text{'}}\frac{\partial w_{i2}}{\partial r_i}-\frac{1}{2{\tau }_i^4}(2w_{i2}{r}_i^2-1){\mathbf{h}}_i{\mathbf{h}}_i^{\text{'}}.$$(C4) ).