Choice of link and variance function for generalized linear mixed models: a case study with binomial response in proteomics

Abstract

Non-normality is a common phenomenon in data from agricultural and biological research, especially in molecular data (for example; -omics, RNAseq, flow cytometric data, etc.). For over half a century, the leading paradigm called for using analysis of variance (ANOVA) after applying a data transformation. The introduction of generalized linear mixed models (GLMM) provides a new way of analyzing non-normal data. Selecting an apt link function in GLMM can be quite influential, however, and is as critical as selecting an appropriate transformation for ANOVA. In this paper, we assess the performance of different parametric link families available in literature. Then, we propose a new estimation method for selecting an appropriate link function with a suitable variance function in a quasi-likelihood framework. We apply these methods to a proteomics data set, showing that GLMMs provide a very flexible framework for analyzing these kinds of data.     

Confidence intervals and tests on the variance ratio in random models with two variance components

The LM test is modified to test any value of the ratio of two variance components in a mixed effects linear model with two variance components. The test is exact, so it can be used to construct exact confidence intervals on this ratio.Exact Neyman-Pearson (NP) tests on the variance ratio are described.Their powers provide attainable upper bounds on powers of tests on the variance ratio.Efficiencies of LM tests, which include ANOVA tests, and NP tests are compared for unbalanced, random, one-way ANOVA models.Confidence intervals corresponding to LM tests and NP tests are described.     

Chisquared and related inducing pivot variables: an application to orthogonal mixed models

Abstract

We use chi-squared and related pivot variables to induce probability measures for model parameters, obtaining some results that will be useful on the induced densities. As illustration we considered mixed models with balanced cross nesting and used the algebraic structure to derive confidence intervals for the variance components. A numerical application is presented.     

Variance Components Testing in ANOVA‐Type Mixed Models

The purpose of this article is threefold. First, variance components testing for ANOVA ‐type mixed models is considered, in which response may not be divided into independent sub‐vectors, whereas most of existing methods are for models where response can be divided into independent sub‐vectors. Second, testing that a certain subset of variance components is zero. Third, as normality is often violated in practice, it is desirable to construct tests under very mild assumptions. To achieve these goals, an adaptive difference‐based test and an adaptive trace‐based test are constructed. The test statistics are asymptotically normal under the null hypothesis, are consistent against all global alternatives and can detect local alternatives distinct from the null at a rate as close to n ^{? 1 ∕ 2} as possible with n being the sample size. Moreover, when the dimensions of variance components in different sets are bounded, we develop a test with chi‐square as its limiting null distribution. The finite sample performance of the tests is examined via simulations, and a real data set is analysed for illustration.     

A semi-parametric additive model for variance heterogeneity

This paper presents a flexible model for variance heterogeneity in a normal error model. Specifically, both the mean and variance are modelled using semi-parametric additive models. We call this model a Mean And Dispersion Additive Model (MADAM). A successive relaxation algorithm for fitting the model is described and justified as maximizing a penalized likelihood function with penalties for lack of smoothness in the additive non-parametric functions in both mean and variance models. The algorithm is implemented in GLIM4, allowing flexible and interactive modelling of variance heterogeneity. Two data sets are used for demonstration.     

A note on fiducial generalized pivots for in one-way heteroscedastic ANOVA with random effects

The paper deals with generalized confidence intervals for the between-group variance in one-way heteroscedastic (unbalanced) ANOVA with random effects. The approach used mimics the standard one applied in mixed linear models with two variance components, where interval estimators are based on a minimal sufficient statistic derived after an initial reduction by the principle of invariance. A minimal sufficient statistic under heteroscedasticity is found to resemble its homoscedastic counterpart and further analogies between heteroscedastic and homoscedastic cases lead us to two classes of fiducial generalized pivots for the between-group variance. The procedures suggested formerly by Wimmer and Witkovský [Between group variance component interval estimation for the unbalanced heteroscedastic one-way random effects model, J. Stat. Comput. Simul. 73 (2003), pp. 333–346] and Li [Comparison of confidence intervals on between group variance in unbalanced heteroscedastic one-way random models, Comm. Statist. Simulation Comput. 36 (2007), pp. 381–390] are found to belong to these two classes. We comment briefly on some of their properties that were not mentioned in the original papers. In addition, properties of another particular generalized pivot are considered.     

Interval estimation of parameters in an unbalanced mixed two-fold nested classification-the last-stage uniformity case

A procedure for the construction of exact simultaneous confidence intervals on functions of the fixed-effects parameters and on functions of variance components in an unbalanced, mixed, two-fold nested classification is introduced. The type of model considered in this paper enables the construction of such intervals to be based on the corresponding ANOVA table using its mean square ratios.     

Extended tests for non-zero between-study variance

The ANOVA F-test, James tests and generalized F-test are extended to test hypotheses on the between-study variance for values greater than zero. Using simulations, we compare the performance of extended test procedures with respect to the actual attained type I error rate. Examples are provided to demonstrate the application of the procedures in ANOVA models and meta-analysis.     

On the coefficient of determination for mixed regression models

For mixed regression models, we define a variance decomposition including three terms, explained individual variance, unexplained individual variance and noise variance. In contrast to traditional variance decomposition, it is thus the unexplained  , not the explained, variance that is split. It gives rise to a coefficient of individual determination (CID) defined as the estimated fraction of explained individual variance. We argue that in many applications CID is a valuable complement to R²$R^2$, since it excludes noise variance (which can never be explained) and thus has one as a natural upper bound.     

The effects of missing serial effects and/or heteroscedastic errors on mixed models using repeated growth data

When a two-level multilevel model (MLM) is used for repeated growth data, the individuals constitute level 2 and the successive measurements constitute level 1, which is nested within the individuals that make up level 2. The heterogeneity among individuals is represented by either the random-intercept or random-coefficient (slope) model. The variance components at level 1 involve serial effects and measurement errors under constant variance or heteroscedasticity. This study hypothesizes that missing serial effects or/and heteroscedasticity may bias the results obtained from two-level models. To illustrate this effect, we conducted two simulation studies, where the simulated data were based on the characteristics of an empirical mouse tumour data set. The results suggest that for repeated growth data with constant variance (measurement error) and misspecified serial effects (ρ > 0.3), the proportion of level-2 variation (intra-class correlation coefficient) increases with ρ and the two-level random-coefficient model is the minimum AIC (or AIC_c) model when compared with the fixed model, heteroscedasticity model, and random-intercept model. In addition, the serial effect (ρ > 0.1) and heteroscedasticity are both misspecified, implying that the two-level random-coefficient model is the minimum AIC (or AIC_c) model when compared with the fixed model and random-intercept model. This study demonstrates that missing serial effects and/or heteroscedasticity may indicate heterogeneity among individuals in repeated growth data (mixed or two-level MLM). This issue is critical in biomedical research.     

Improved estimation of variance components in mixed models

Taking Albert's (1976) formulation of a mixed model ANOVA, we consider improved estimation of the variance components for balanced designs under squared error loss. Two approaches are presented. One extends the ideas of Stein (1964), The other is developed from the fact that variance components can be expressed as linear combinations of chi-square scale parameters. Encouraging simulation results are presented.     

Asymptotic cumulants of the minimum phi-divergence estimator for categorical data under possible model misspecification

Abstract

The asymptotic cumulants of the minimum phi-divergence estimators of the parameters in a model for categorical data are obtained up to the fourth order with the higher-order asymptotic variance under possible model misspecification. The corresponding asymptotic cumulants up to the third order for the studentized minimum phi-divergence estimator are also derived. These asymptotic cumulants, when a model is misspecified, depend on the form of the phi-divergence. Numerical illustrations with simulations are given for typical cases of the phi-divergence, where the maximum likelihood estimator does not necessarily give best results. Real data examples are shown using log-linear models for contingency tables.     

Umvu estimators for the population mean and variance based on random effects models for lognormal data

Cross-classified data are often obtained in controlled experimental situations and in epidemiologic studies. As an example of the latter, occupational health studies sometimes require personal exposure measurements on a random sample of workers from one or more job groups, in one or more plant locations, on several different sampling dates. Because the marginal distributions of exposure data from such studies are generally right-skewed and well-approximated as lognormal, researchers in this area often consider the use of ANOVA models after a logarithmic transformation. While it is then of interest to estimate original-scale population parameters (e.g., the overall mean and variance), standard candidates such as maximum likelihood estimators (MLEs) can be unstable and highly biased. Uniformly minimum variance unbiased (UMVU) cstiniators offer a viable alternative, and are adaptable to sampling schemes that are typiral of experimental or epidemiologic studies. In this paper, we provide UMVU estimators for the mean and variance under two random effects ANOVA models for logtransformed data. We illustrate substantial mean squared error gains relative to the MLE when estimating the mean under a one-way classification. We illustrate that the results can readily be extended to encompass a useful class of purely random effects models, provided that the study data are balanced.     

Bootstrap tests for variance components in generalized linear mixed models

In many applications of generalized linear mixed models to clustered correlated or longitudinal data, often we are interested in testing whether a random effects variance component is zero. The usual asymptotic mixture of chi‐square distributions of the score statistic for testing constrained variance components does not necessarily hold. In this article, the author proposes and explores a parametric bootstrap test that appears to be valid based on its estimated level of significance under the null hypothesis. Results from a simulation study indicate that the bootstrap test has a level much closer to the nominal one while the asymptotic test is conservative, and is more powerful than the usual asymptotic score test based on a mixture of chi‐squares. The proposed bootstrap test is illustrated using two sets of real‐life data obtained from clinical trials. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

Estimating the parameters of mixed linear models with modal estimators

This paper presents a procedure to estimate the variance components and fixed effects of mixed linear models. The mode of the joint posterior distribution of all the parameters is obtained by an iterative technique.

The proposed method is illustrated with one-way and two-fold nested random models. Two numerical examples demonstrate the iterative solution.     

Discrepancy in regression estimates between log-normal and gamma: some case studies

In regression models with multiplicative error, estimation is often based on either the log-normal or the gamma model. It is well known that the gamma model with constant coefficient of variation and the log-normal model with constant variance give almost the same analysis. This article focuses on the discrepancies of the regression estimates between the two models based on real examples. It identifies that even though the variance or the coefficient of variation remains constant, but regression estimates may be different between the two models. It also identifies that for the same positive data set, the variance is constant under the log-normal model but non-constant under the gamma model. For this data set, the regression estimates are completely different between the two models. In the process, it explains the causes of discrepancies between the two models.     

The effect of interaction and rounding error in two-way ANOVA: example of impact on testing for normality

A key issue in various applications of analysis of variance (ANOVA) is testing for the interaction and the interpretation of resulting ANOVA tables. In this note it is demonstrated that for a two-way ANOVA, whether interactions are incorporated or not may have a dramatic influence when considering the usual statistical tests for normality of residuals. The effect of numerical rounding is also discussed.     

Model selection for infinite variance time series

In this we consider the problem of model selection for infinite variance time series. We introduce a group of model selection critera based on a general loss function Ψ. This family includes various generalizations of predictive least square and AIC Parameter estimation is carried out using Ψ. We use two loss functions commonly used in robust estimation and show that certain criteria out perform the conventional approach based on least squares or Yule-Walker estima­tion for heavy tailed innovations. Our conclusions are based on a comprehensive study of the performance of competing criteria for a wide selection of AR(2) models. We also consider the performance of these techniques when the ‘true’ model is not contained in the family of candidate models.     

VARIANCE ESTIMATION IN THE ERROR COMPONENTS REGRESSION MODEL

ABSTRACT

In a regression model with a random individual and a random time effect explicit representations of the nonnegative quadratic minimum biased estimators of the corresponding variances are deduced. These estimators always exist and are unique. Moreover, under normality assumption of the dependent variable unbiased estimators of the mean squared errors of the variance estimates are derived. Finally, confidence intervals on the variance components are considered.     

A new REML (parameter expanded) EM algorithm for linear mixed models

Linear mixed models are regularly applied to animal and plant breeding data to evaluate genetic potential. Residual maximum likelihood (REML) is the preferred method for estimating variance parameters associated with this type of model. Typically an iterative algorithm is required for the estimation of variance parameters. Two algorithms which can be used for this purpose are the expectation‐maximisation (EM) algorithm and the parameter expanded EM (PX‐EM) algorithm. Both, particularly the EM algorithm, can be slow to converge when compared to a Newton‐Raphson type scheme such as the average information (AI) algorithm. The EM and PX‐EM algorithms require specification of the complete data, including the incomplete and missing data. We consider a new incomplete data specification based on a conditional derivation of REML. We illustrate the use of the resulting new algorithm through two examples: a sire model for lamb weight data and a balanced incomplete block soybean variety trial. In the cases where the AI algorithm failed, a REML PX‐EM based on the new incomplete data specification converged in 28% to 30% fewer iterations than the alternative REML PX‐EM specification. For the soybean example a REML EM algorithm using the new specification converged in fewer iterations than the current standard specification of a REML PX‐EM algorithm. The new specification integrates linear mixed models, Henderson's mixed model equations, REML and the REML EM algorithm into a cohesive framework.