Nonlinear mixed-effects models with misspecified random-effects distribution.

Nonlinear mixed-effects models are being widely used for the analysis of longitudinal data, especially from pharmaceutical research. They use random effects which are latent and unobservable variables so the random-effects distribution is subject to misspecification in practice. In this paper, we first study the consequences of misspecifying the random-effects distribution in nonlinear mixed-effects models. Our study is focused on Gauss-Hermite quadrature, which is now the routine method for calculation of the marginal likelihood in mixed models. We then present a formal diagnostic test to check the appropriateness of the assumed random-effects distribution in nonlinear mixed-effects models, which is very useful for real data analysis. Our findings show that the estimates of fixed-effects parameters in nonlinear mixed-effects models are generally robust to deviations from normality of the random-effects distribution, but the estimates of variance components are very sensitive to the distributional assumption of random effects. Furthermore, a misspecified random-effects distribution will either overestimate or underestimate the predictions of random effects. We illustrate the results using a real data application from an intensive pharmacokinetic study.

Permutation and Bayesian tests for testing random effects in linear mixed-effects models.

In many applications of linear mixed-effects models to longitudinal and multilevel data especially from medical studies, it is of interest to test for the need of random effects in the model. It is known that classical tests such as the likelihood ratio, Wald, and score tests are not suitable for testing random effects because they suffer from testing on the boundary of the parameter space. Instead, permutation and bootstrap tests as well as Bayesian tests, which do not rely on the asymptotic distributions, avoid issues with the boundary of the parameter space. In this paper, we first develop a permutation test based on the likelihood ratio test statistic, which can be easily used for testing multiple random effects and any subset of them in linear mixed-effects models. The proposed permutation test would be an extension to two existing permutation tests. We then aim to compare permutation tests and Bayesian tests for random effects to find out which test is more powerful under which situation. Nothing is known about this in the literature, although this is an important practical problem due to the usefulness of both methods in tackling the challenges with testing random effects. For this, we consider a Bayesian test developed using Bayes factors, where we also propose a new alternative computation for this Bayesian test to avoid some computational issue it encounters in testing multiple random effects. Extensive simulations and a real data analysis are used for evaluation of the proposed permutation test and its comparison with the Bayesian test. We find that both tests perform well, albeit the permutation test with the likelihood ratio statistic tends to provide a relatively higher power when testing multiple random effects.

Testing random effects in linear mixed-effects models with serially correlated errors.

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.

Diagnosing misspecification of the random-effects distribution in mixed models.

It is traditionally assumed that the random effects in mixed models follow a multivariate normal distribution, making likelihood-based inferences more feasible theoretically and computationally. However, this assumption does not necessarily hold in practice which may lead to biased and unreliable results. We introduce a novel diagnostic test based on the so-called gradient function proposed by Verbeke and Molenberghs (2013) to assess the random-effects distribution. We establish asymptotic properties of our test and show that, under a correctly specified model, the proposed test statistic converges to a weighted sum of independent chi-squared random variables each with one degree of freedom. The weights, which are eigenvalues of a square matrix, can be easily calculated. We also develop a parametric bootstrap algorithm for small samples. Our strategy can be used to check the adequacy of any distribution for random effects in a wide class of mixed models, including linear mixed models, generalized linear mixed models, and non-linear mixed models, with univariate as well as multivariate random effects. Both asymptotic and bootstrap proposals are evaluated via simulations and a real data analysis of a randomized multicenter study on toenail dermatophyte onychomycosis.

Nonlinear mixed-effects models for pharmacokinetic data analysis: assessment of the random-effects distribution.

Nonlinear mixed-effects models are frequently used for pharmacokinetic data analysis, and they account for inter-subject variability in pharmacokinetic parameters by incorporating subject-specific random effects into the model. The random effects are often assumed to follow a (multivariate) normal distribution. However, many articles have shown that misspecifying the random-effects distribution can introduce bias in the estimates of parameters and affect inferences about the random effects themselves, such as estimation of the inter-subject variability. Because random effects are unobservable latent variables, it is difficult to assess their distribution. In a recent paper we developed a diagnostic tool based on the so-called gradient function to assess the random-effects distribution in mixed models. There we evaluated the gradient function for generalized liner mixed models and in the presence of a single random effect. However, assessing the random-effects distribution in nonlinear mixed-effects models is more challenging, especially when multiple random effects are present, and therefore the results from linear and generalized linear mixed models may not be valid for such nonlinear models. In this paper, we further investigate the gradient function and evaluate its performance for such nonlinear mixed-effects models which are common in pharmacokinetics and pharmacodynamics. We use simulations as well as real data from an intensive pharmacokinetic study to illustrate the proposed diagnostic tool.

Testing multiple variance components in linear mixed-effects models.

Testing zero variance components is one of the most challenging problems in the context of linear mixed-effects (LME) models. The usual asymptotic chi-square distribution of the likelihood ratio and score statistics under this null hypothesis is incorrect because the null is on the boundary of the parameter space. During the last two decades many tests have been proposed to overcome this difficulty, but these tests cannot be easily applied for testing multiple variance components, especially for testing a subset of them. We instead introduce a simple test statistic based on the variance least square estimator of variance components. With this comes a permutation procedure to approximate its finite sample distribution. The proposed test covers testing multiple variance components and any subset of them in LME models. Interestingly, our method does not depend on the distribution of the random effects and errors except for their mean and variance. We show, via simulations, that the proposed test has good operating characteristics with respect to Type I error and power. We conclude with an application of our process using real data from a study of the association of hyperglycemia and relative hyperinsulinemia.

Exploring the relationship between government stringency and preventative social behaviours during the COVID-19 pandemic in the United Kingdom.

We constructed a preventive social behaviours (PSB) Index using survey questions that were aligned with WHO recommendations, and used linear regression to assess the impact of reported COVID-19 deaths (RCD), people's confidence of government handling of the pandemic (CGH) and government stringency (GS) in the United Kingdom (UK) over time on the PSB index. We used repeated, nationally representative, cross-sectional surveys in the UK over the course of 41 weeks from 1st April 2020 to January 28th, 2021, including a total of 38,092 participants. The PSB index was positively correlated with the logarithm of RCD (R: 0.881, p < .001), CGH (R: 0.592, p < .001) and GS (R: 0.785, p < .001), but was not correlated with time (R: -0.118, p = .485). A multivariate linear regression analysis suggests that the log of RCD (coefficient: 0.125, p < .001), GS (coefficient: 0.010, p = .019), and CGH (coefficient: 0.0.009, p < .001) had a positive and significant impact on the PSB Index, while time did not affect it significantly. These findings suggest that people's behaviours could have been affected by multiple factors during the pandemic, with the number of COVID-19 deaths being the largest contributor towards an increase in protective behaviours in our model.

MEGH: A parametric class of general hazard models for clustered survival data.

In many applications of survival data analysis, the individuals are treated in different medical centres or belong to different clusters defined by geographical or administrative regions. The analysis of such data requires accounting for between-cluster variability. Ignoring such variability would impose unrealistic assumptions in the analysis and could affect the inference on the statistical models. We develop a novel parametric mixed-effects general hazard (MEGH) model that is particularly suitable for the analysis of clustered survival data. The proposed structure generalises the mixed-effects proportional hazards and mixed-effects accelerated failure time structures, among other structures, which are obtained as special cases of the MEGH structure. We develop a likelihood-based algorithm for parameter estimation in general subclasses of the MEGH model, which is implemented in our R package MEGH. We propose diagnostic tools for assessing the random effects and their distributional assumption in the proposed MEGH model. We investigate the performance of the MEGH model using theoretical and simulation studies, as well as a real data application on leukaemia.

A goodness-of-fit test for the random-effects distribution in mixed models.

In this paper, we develop a simple diagnostic test for the random-effects distribution in mixed models. The test is based on the gradient function, a graphical tool proposed by Verbeke and Molenberghs to check the impact of assumptions about the random-effects distribution in mixed models on inferences. Inference is conducted through the bootstrap. The proposed test is easy to implement and applicable in a general class of mixed models. The operating characteristics of the test are evaluated in a simulation study, and the method is further illustrated using two real data analyses.