Permutation Tests for Random Effects in Linear Mixed Models

The best linear unbiased estimator (BLUE) and the best linear unbiased predictor (BLUP) are used to estimate, respectively, the parameter vectors of fixed and random effects in linear mixed models. But, when multicollinearity problem is arised, alternative estimators and predictors to BLUE and BLUP are preferred because of bad variance property of BLUE. Commonly used prediction approaches are the ridge and Liu prediction under multicollinearityin linear mixed models and in this article, we suggest a new prediction approachto combat multicollinearity problem by expanding the Kibria–Lukman (KL) prediction approach in linear regression models to linear mixed models. We do comparisons between the KL estimator/predictor and several other estimators/predictors, namely BLUE/BLUP, ridge and Liu estimators/predictors by using the matrix mean square error criterion. We give the selection of the ridge biasing parameter. Lastly, to demonstrate the performance of our new suggested prediction approach, we make greenhouse gases data analysis and a Monte-Carlo simulation study.

BackgroundIn many medical studies the likelihood ratio test (LRT) has been widely applied to examine whether the random effects variance component is zero within the mixed effects models framework; whereas little work about likelihood-ratio based variance component test has been done in the generalized linear mixed models (GLMM), where the response is discrete and the log-likelihood cannot be computed exactly. Before applying the LRT for variance component in GLMM, several difficulties need to be overcome, including the computation of the log-likelihood, the parameter estimation and the derivation of the null distribution for the LRT statistic.MethodsTo overcome these problems, in this paper we make use of the penalized quasi-likelihood algorithm and calculate the LRT statistic based on the resulting working response and the quasi-likelihood. The permutation procedure is used to obtain the null distribution of the LRT statistic. We evaluate the permutation-based LRT via simulations and compare it with the score-based variance component test and the tests based on the mixture of chi-square distributions. Finally we apply the permutation-based LRT to multilocus association analysis in the case–control study, where the problem can be investigated under the framework of logistic mixed effects model.ResultsThe simulations show that the permutation-based LRT can effectively control the type I error rate, while the score test is sometimes slightly conservative and the tests based on mixtures cannot maintain the type I error rate. Our studies also show that the permutation-based LRT has higher power than these existing tests and still maintains a reasonably high power even when the random effects do not follow a normal distribution. The application to GAW17 data also demonstrates that the proposed LRT has a higher probability to identify the association signals than the score test and the tests based on mixtures.ConclusionsIn the present paper the permutation-based LRT was developed for variance component in GLMM. The LRT outperforms existing tests and has a reasonably higher power under various scenarios; additionally, it is conceptually simple and easy to implement.

Pharmacokinetic-Pharmacodynamic Modeling and Simulation

Previous chapters discussed linear mixed effects models and autoregressive linear mixed effects models for analysis of longitudinal data. This chapter discusses multivariate extensions of these models. In longitudinal clinical studies, multivariate responses are often collected at each measurement time point from each subject. When two response variables, such as an efficacy measurement and a safety measurement are obviously correlated, there are advantages in analyzing the bivariate responses jointly. Parathyroid hormone (PTH) and serum calcium (Ca) measurements in the treatment of secondary hyperparathyroidism in chronic hemodialysis patients provide an example in which joint bivariate responses are of interest. We introduce multivariate longitudinal data and explain bivariate autoregressive linear mixed effects models in which the current responses are regressed on the previous responses of both variables, fixed effects, and random effects. The dependent bivariate responses approach equilibria, and the equilibria are modeled using fixed and random effects. These type of profiles are observed in long-term clinical studies. We also explain bivariate linear mixed effects models.

Ciprian M. Crainiceanu 'Likelihood ratio testing for zero variance components in linear mixed effects models.' - Daowen Zhang & Xihong Lin 'Variance component testing in generalized linear mixed models for longitudinal/clustered data and other related topics.' -Satkartar Kinney & David Dunson 'Bayesian model uncertainty in mixed effects models.' -Bo Cai & David Dunson 'Bayesian variable selection in generalized linear mixed models.' - Peter M. Bentler & Jiajuan Liang 'A unified approach to two-level structural equation models and linear mixed effects models.' - Sik-Yum Lee & Xin-Yuan Song 'Bayesian model comparison of structural equation models.' - Joyee Ghosh & David Dunson 'Bayesian model selection in factor analytic models.'

A central point of disagreement, in certain long-standing discussions about a particular whaling dataset in the Scientific Committee of the International Whaling Commission, has directly involved model selection issues for linear mixed effect models. The biological question under discussion is associated with a clearly defined parameter of primary interest, a focus parameter, which makes model selection with the Focused Information Criterion (FIC) more appropriate than other selection methods. Since the existing FIC methodology has not covered the case of linear mixed effects models, this article sets up the required framework and develops the necessary formulae for the relevant FIC. Our new criterion requires the asymptotic distribution of estimators derived for a given candidate linear mixed model but with behaviour examined under a wider linear mixed model. These results, needed here to build our FIC, also have independent interest.

The number of parameters mushrooms in a linear mixed effects (LME) model in the case of multivariate repeated measures data. Computation of these parameters is a real problem with the increase in the number of response variables or with the increase in the number of time points. The problem becomes more intricate and involved with the addition of additional random effects. A multivariate analysis is not possible in a small sample setting. We propose a method to estimate these many parameters in bits and pieces from baby models, by taking a subset of response variables at a time, and finally using these bits and pieces at the end to get the parameter estimates for the mother model, with all variables taken together. Applying this method one can calculate the fixed effects, the best linear unbiased predictions (BLUPs) for the random effects in the model, and also the BLUPs at each time of observation for each response variable, to monitor the effectiveness of the treatment for each subject. The proposed method is illustrated with an example of multiple response variables measured over multiple time points arising from a clinical trial in osteoporosis.

Diagnostic tests are of vital importance in a number of veterinary fields and are essential for decision-making by veterinary practioners, researchers and the veterinary authorities. The development and evaluation of laboratory diagnostic tests as well as the interpretation of test results engage a number of different veterinary disciplines including virologists, parasitologists, bacteriologists, laboratory experts, clinicians as well as epidemiologists and biostatisticians. This thesis was devised shortly after the section of epidemiology came into existence at the Vetsuisse Faculty end of 2009. The aim is to interrelate veterinary epidemiology - being closely linked to virology, parasitology, bacteriology and clinical medicine - with biostatistics. Thus allowing innovative concepts and recent software developments to provide solutions and answers in our veterinary field. After a brief introduction into diagnostic tests, the thesis is structured into three main parts. First, from a laboratory perspective, aspects of analytical sensitivity and specificity are illustrated by three peer-reviewed papers presenting practical applications of veterinary diagnostic tests for feline calicivirus and porcine parvovirus. Second, in the context of assessing agreement and method comparison studies, categorical test results are compared by Cohen’s k and continuous measurements by linear mixed effects models extending the classical use of Bland-Altman plots with limits of agreement. The former is illustrated in a publication on the diagnostic test Interferon-Gamma to classify bovines being infected with Mycobacterium bovis when analysing samples from the same animals in five different laboratories and from different anatomical locations. Two peer-reviewed papers aim to assess the agreement of serumspecific lipase, as a diagnostic test for feline pancreatitis with a commercial test kit and ultrasonographic findings. In the context of comparing continuous measurements, i.e. cardiac output and blood pressure measured directly and indirectly, two peer-reviewed paper illustrate the application of linear mixed effects models to simultaneously assess bias, precision and covariate information. Typical pitfalls arising in method comparison studies are summarised in a review paper. Building upon the previous parts, the third part adds to the existing knowledge by presenting novel and innovative approaches when assessing the performance of diagnostic tests in the absence of a perfect gold standard. So-called no gold standard models (NGS) are applied to assess the diagnostic test accuracies in the context of zoonotic diseases such as bovine tuberculosis, porcine toxoplasmosis as well as canine and vulpine echinococcosis. Additionally, these models are applied to assess diagnostic test accuracies to diagnose Brachyspira hyodysenteriae, an economically important disease in pigs. Furthermore, they are also utilised to assess diagnostic accuracies of diagnostic tools for subclinical mastitis in dairy cattle. In this part, Bayesian approaches are presented, together with technical aspects including conditional dependencies, Markov Chain Monte Carlo simulations and aspects of model selection. The innovative aspects and main novelty of the work presented in this thesis pertain to the incorporation of covariate information and random effects. Whereas covariate information in the form of risk factors has been included in statistical analyses since decades, considering random effects to account for clustering of animals at herd level just became possible with increased computational power. Nowadays dealing appropriately with the hierarchical structure of animal data is considered as good statistical practice and warranted in epidemiological studies to avoid biased results. However, diagnostic test studies including covariate information and random effects are scarce. The aim of this thesis is to fill this gap by showcasing a number of Bayesian latent class analyses for various infectious diseases in animals.

BackgroundSystolic blood pressure (SBP) after endovascular thrombectomy (EVT) for large artery occlusive stroke is dynamic, requiring adaptable early prediction tools for improving outcomes. We investigated if post-EVT SBP course was...

BackgroundPreventive strategies for musculoskeletal disorders require a better understanding of age-related changes in muscle strength, physical function and body composition (including bone). Many publications use simple change measures from observations...

The forest dynamics are usually explained by the precipitation and temperature through fixed effects models using ordinary least squares and geographically weighted regression methods. However, forest dynamics were found insufficiently explained by meteorological factors as the fixed effects models were not designed to account for random effects. In this study, we utilized three types of forests located in the Gulf of Mexico Coast region, including softwood, hardwood, and mixed forests to investigate the underlying forest dynamics to meteorological variations by incorporating random effects into fixed effects models. Four types of linear mixed effects models (LMMs) were developed for regressing the normalized difference of vegetation index (NDVI) against two explanatory variables: precipitation and temperature. By assuming that the intercept and slope parameters estimated from LMMs would vary randomly, we intended to explore if the amount of variation in the NDVI variables could be reduced by the use of random effects variables. The results suggested that the random intercept and random slope model fitted the data better than the random intercept model with higher R 2, lower Akaike information criterion, and Bayesian information criterion values. The R 2 value indicated that the explanatory power of the LMM varies between forest types. Moreover, this study revealed that a linear mixed effects model could significantly reduce the unexplained variance by introducing random effects variables, and forest dynamics is a synthetic result of the mixed effects of temperature and fixed effects of precipitation.

We consider a general class of varying coefficient mixed models where random effects are introduced to account for between-subject variation. To address the question of whether a varying coefficient mixed model can be reduced to a simpler varying coefficient model, we develop one-sided tests for the null hypothesis that all the variance components are zero. In addition to the purely null-based standard quasi-score test (SQT), we propose an extended quasi-score test (EQT) by constructing estimators that are consistent under both the null and alternative hypotheses. No assumptions are required for the distributions of random effects and random errors. Both SQT and EQT are consistent for global alternatives and local alternatives distinct at certain rates from the null. Furthermore, the asymptotic null distributions are simple and easy to use in practice. For comparison, we also adapt the one-sided score test (SST) in Silvapulle and Silvapulle (1995) and the likelihood ratio test (LRT) in Fan, Zhang, and Zhang (2001). Extensive simulations indicate that all proposed tests perform well and the EQT is more powerful than SQT, SST, and LRT. A data example is analyzed for illustration.

SUMMARY The effect of non-normality in estimating variance components in a random effects or mixed effects linear model is studied. It is shown that for a wide variety of balanced situations the interval estimation of the upper variance component is likely to be relatively little affected by non-normality in the error component. A method is suggested for constructing a confidence interval for a ratio of variance components in a balanced model, without assuming normality.

In the previous chapters, we discussed autoregressive linear mixed effects models. In this section, we discuss the relationships between the autoregressive linear mixed effects models and nonlinear mixed effects models, growth curves, and differential equations. The autoregressive model shows a profile approaching an asymptote, where the change is proportional to the distance remaining to the asymptote. Autoregressive models in discrete time correspond to monomolecular curves in continuous time. Autoregressive linear mixed effects models correspond to monomolecular curves with random effects in the baseline and asymptote, and special error terms. The autoregressive coefficient is a nonlinear parameter, but all random effects parameters in the model are linear. Therefore, autoregressive linear mixed effects models are nonlinear mixed effects models without nonlinear random effects and have a closed form of likelihood. When there are time-dependent covariates, autoregressive linear mixed effects models are represented by a differential equation and random effects. The monomolecular curve is one of the popular growth curves. We introduce other growth curves, such as the logistic curves and von Bertalanffy curves, and generalizations of growth curves. Re-parameterization is often performed in nonlinear models, and various representations of re-parameterization in monomolecular and other curves are provided herein.

Linear mixed effects (LME) models are important statistical tools for analysis of clustered and correlated data. High breakdown estimators are currently the robust methods of choice for multivariate linear regression, but extensions of such estimators have been developed only for completely balanced LME models. In this work, we propose a generalized S-estimator for a general unbalanced LME model. Our GS-estimator reduces to the classic high breakdown S-estimator when the LME model reduces to a multivariate normal location and scale model or mul- tivariate regression model. The asymptotic properties are established, and we show that the estimator may be viewed as a redescending M-estimator. A small simu- lation study is conducted to compare performance of the GS-estimates, monotone M-estimates, and restricted maximum likelihood (REML) estimates under various contamination patterns. The proposed estimator is used for analysis of age-related changes in hemoglobin levels of sickle cell disease patients.