Testing random effects in linear mixed‐effects models with serially correlated errors

Abstract

In linear mixed‐effects models, random effects are used to capture the heterogeneity and variability between individuals due to unmeasured covariates or unknown biological differences. Testing for the need of random effects is a nonstandard problem because it requires testing on the boundary of parameter space where the asymptotic chi‐squared distribution of the classical tests such as likelihood ratio and score tests is incorrect. In the literature several tests have been proposed to overcome this difficulty, however all of these tests rely on the restrictive assumption of i.i.d. measurement errors. The presence of correlated errors, which often happens in practice, makes testing random effects much more difficult. In this paper, we propose a permutation test for random effects in the presence of serially correlated errors. The proposed test not only avoids issues with the boundary of parameter space, but also can be used for testing multiple random effects and any subset of them. Our permutation procedure includes the permutation procedure in Drikvandi, Verbeke, Khodadadi, and Partovi Nia (2013) as a special case when errors are i.i.d., though the test statistics are different. We use simulations and a real data analysis to evaluate the performance of the proposed permutation test. We have found that random slopes for linear and quadratic time effects may not be significant when measurement errors are serially correlated.

A framework for analysing longitudinal data involving time-varying covariates

“…As a consequence, there is no open set containing the true variance components under the null hypothesis. Therefore, the classical asymptotic chi-squared distribution of the likelihood ratio (LR) or restricted LR test statistic is not valid (see, for example, Stram and Lee, 1994;Drikvandi et al, 2012Drikvandi et al, , 2013Drikvandi and Noorian, 2019). For testing zero variance components, it is shown that the correct asymptotic distribution of the LR or restricted LR statistic is a mixture of chi-squared distributions, provided that the response variable can be partitioned into independent subvectors and the number of subvectors tends to infinity (e.g., Stram and Lee, 1994).…”

Sparse Principal Component Analysis for Natural Language Processing

“…Again, care ought to be taken when calculating the caught variance with associated or correlated loadings. Note that assessing and testing a significant variance in correlated models is a nonstandard testing problem [11][12][13][14].…”

Optimal tests for random effects in linear mixed models

“…More recently, this approach has been considered by several authors in conjunction with empirical Bayesian and permutation test (e.g. [24]) while Drikvandi and Noorian [7] have considered the permutation test for a more broad class of linear mixed models with correlated errors. The results were shown that both tests to perform well, albeit the permutation test with the likelihood ratio statistic tends to provide a relatively higher power when testing multiple random effects.…”