Testing Variance Components in Linear Mixed Modeling Using Permutation

Abstract

Inference of variance components in linear mixed modeling (LMM) provides evidence of heterogeneity between individuals or clusters. When only nonnegative variances are allowed, there is a boundary (i.e., 0) in the variances’ parameter space, and regular inference statistical procedures for such a parameter could be problematic. The goal of this article is to introduce a practically feasible permutation method to make inferences about variance components while considering the boundary issue in LMM. The permutation tests with different settings (i.e., constrained vs. unconstrained estimation, specific vs. generalized test, different ways of calculating p values, and different ways of permutation) were examined with both normal data and non-normal data. In addition, the permutation tests were compared to likelihood ratio (LR) tests with a mixture of chi-squared distributions as the reference distribution. We found that the unconstrained permutation test with the one-sided p-value approach performed better than the other permutation tests and is a useful alternative when the LR tests are not applicable. An R function is provided to facilitate the implementation of the permutation tests, and a real data example is used to illustrate the application. We hope our results will help researchers choose appropriate tests when testing variance components in LMM.

Keywords:

• Hierarchical linear modeling
• nonparametric statistics
• variance testing

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: Lijuan Wang was supported by NIH grants 1R01HD087319 and 1R01HD091235 during the study period.

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 ideas and opinions expressed herein are those of the authors alone, and endorsement by the authors’ institutions is not intended and should not be inferred.

Notes

1 Some researchers distinguish specific versus generalized tests by whether setting the covariances at 0 in alternative hypotheses (e.g., Drikvandi et al., 2013). That is, if all covariances are freely estimated, it is a generalized test; otherwise, it is a specific test.

2 The Wald test is asymptotically equivalent to the LR test, thus it is not elaborated here.

3 Drikvandi et al., (2013) denoted the test statistics as T.

4 We examined three more generalized variance test statistics. In the first one, $\widehat{\mathit{D}}$ and ${\widehat{\mathit{D}}}_{\mathit{11}}$ instead of ${\widehat{\mathit{D}}}^{\mathit{*}}$ and ${\widehat{\mathit{D}}}_{\mathit{11}}^{\mathit{*}}$ were used in Equation (5). In the second one, ${\widehat{\mathit{D}}}_{\mathit{11}}$ was replaced by ${\widehat{\mathit{D}}}_{\mathit{12}}{\widehat{\mathit{D}}}_{\mathit{22}}^{\mathit{-1}}{\widehat{\mathit{D}}}_{\mathit{12}},$ and the updated whole matrix $\widehat{\mathit{D}}$ was used in Equation (2). In the third one, we took the absolute values of the off-diagonal elements of $\widehat{\mathit{D}},$ and applied Equation (5). But none of them controlled the Type I error rates well in the simulation.

5 We also evaluated a constrained estimation of D which is calculated in two steps, as a layman’s method. We first allowed negative variances, and if the estimated variances are negative, we constrained the negative variances to be 0 and re-estimated D. In the simulation, we found that the two-step constrained estimation method generally led to either too liberal or too conservative test results.

6 We also evaluated unconstrained specific and generalized LR tests with two steps, which may be adopted by researchers for convenience in real data analysis. That is, when the slope variance is estimated to be negative, the decision is to fail to reject the null hypothesis $H_0:D=\left[\begin{array}{cc}{\sigma }_0^2& 0\\ 0& 0\end{array}\right];$ otherwise the specific and generalized LR test statistics are compared to the critical values of ${\chi }_1^2$ and ${\chi }_2^2$ at $\alpha =0.05,$ respectively. Thus, there are two-step unconstrained normal-theory-based specific and generalized LR tests for normal data, and two-step unconstrained robust specific and generalized LR tests for non-normal data. Our simulation results showed that the two-step unconstrained LR tests provided too conservative Type I error rates. The main reason is that only a large positive variance leads to the rejection of the null hypothesis at the second step, thus we lose power to allow for negative variances and fail to reject the null hypothesis at the first step.