The use of score tests for inference on variance components

A kernel of truth: statistical advances in polygenic variance component models for complex human pedigrees

Statistical genetic analysis of quantitative traits in large pedigrees is a formidable computational task due to the necessity of taking the nonindependence among relatives into account. With the growing awareness that rare sequence variants may be important in human quantitative variation, heritability and association study designs involving large pedigrees will increase in frequency due to the greater chance of observing multiple copies of rare variants among related individuals. Therefore, it is important to have statistical genetic test procedures that utilize all available information for extracting evidence regarding genetic association. Optimal testing for marker/phenotype association involves the exact calculation of the likelihood ratio statistic which requires the repeated inversion of potentially large matrices. In a whole genome sequence association context, such computation may be prohibitive. Toward this end, we have developed a rapid and efficient eigen simplification of the likelihood that makes analysis of family data commensurate with the analysis of a comparable sample of unrelated individuals. Our theoretical results which are based on a spectral representation of the likelihood yield simple exact expressions for the expected likelihood ratio test statistic (ELRT) for pedigrees of arbitrary size and complexity. For heritability, the ELRT is where h2 and λgi are, respectively, the heritability and eigenvalues of the pedigree-derived genetic relationship kernel (GRK). For association analysis of sequence variants, the ELRT is given by where ht2, hq2, and hr2 are the total, quantitative trait nucleotide, and residual heritabilities, respectively. Using these results, fast and accurate analytical power analyses are possible, eliminating the need for computer simulation. Additional benefits of eigen simplification include a simple method for calculation of the exact distribution of the ELRT under the null hypothesis which turns out to differ from that expected under the usual asymptotic theory. Further, when combined with the use of empirical GRKs-estimated over a large number of genetic markers-our theory reveals potential problems associated with nonpositive semidefinite kernels. These procedures are being added to our general statistical genetic computer package, SOLAR.

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.

Ancestral haplotype-based association mapping with generalized linear mixed models accounting for stratification

MOTIVATION

In many situations, genome-wide association studies are performed in populations presenting stratification. Mixed models including a kinship matrix accounting for genetic relatedness among individuals have been shown to correct for population and/or family structure. Here we extend this methodology to generalized linear mixed models which properly model data under various distributions. In addition we perform association with ancestral haplotypes inferred using a hidden Markov model.

RESULTS

The method was shown to properly account for stratification under various simulated scenari presenting population and/or family structure. Use of ancestral haplotypes resulted in higher power than SNPs on simulated datasets. Application to real data demonstrates the usefulness of the developed model. Full analysis of a dataset with 4600 individuals and 500 000 SNPs was performed in 2 h 36 min and required 2.28 Gb of RAM.

AVAILABILITY

The software GLASCOW can be freely downloaded from www.giga.ulg.ac.be/jcms/prod_381171/software.

CONTACT

francois.guillaume@jouy.inra.fr

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

A note on a hierarchical interpretation for negative variance components

A lot has been said about the relationship between hierarchical models, such as linear mixed-effects models, and the marginal models they imply. Generally, there is a many-to-one map of hierarchical models onto a given marginal model. Additionally, in some cases, no obvious hierarchical model leads to a given marginal model. For example, it is commonly known that the random-intercepts model produces, marginally, a compound-symmetry model with non-negative intraclass correlation, whereas, on the other hand, a compound-symmetry model with negative intraclass correlation is not induced by a conventional random-intercepts model. We show here that it is still possible, and even intuitively appealing, to formulate hierarchical models inducing structure such as negative compound-symmetry correlation. Thus, the aim of this note is to further clarify the relationship between hierarchical and marginal models, enhancing appeal and establishing symmetry of the concepts. Consequences for interpretation and sensitivity analysis are discussed. The ideas are illustrated in three sets of data.

Permutation tests for random effects in linear mixed models

Inference regarding the inclusion or exclusion of random effects in linear mixed models is challenging because the variance components are located on the boundary of their parameter space under the usual null hypothesis. As a result, the asymptotic null distribution of the Wald, score, and likelihood ratio tests will not have the typical χ(2) distribution. Although it has been proved that the correct asymptotic distribution is a mixture of χ(2) distributions, the appropriate mixture distribution is rather cumbersome and nonintuitive when the null and alternative hypotheses differ by more than one random effect. As alternatives, we present two permutation tests, one that is based on the best linear unbiased predictors and one that is based on the restricted likelihood ratio test statistic. Both methods involve weighted residuals, with the weights determined by the among- and within-subject variance components. The null permutation distributions of our statistics are computed by permuting the residuals both within and among subjects and are valid both asymptotically and in small samples. We examine the size and power of our tests via simulation under a variety of settings and apply our test to a published data set of chronic myelogenous leukemia patients.

Model Selection with the Linear Mixed Model for Longitudinal Data

Model building or model selection with linear mixed models (LMMs) is complicated by the presence of both fixed effects and random effects. The fixed effects structure and random effects structure are codependent, so selection of one influences the other. Most presentations of LMM in psychology and education are based on a multilevel or hierarchical approach in which the variance-covariance matrix of the random effects is assumed to be positive definite with nonzero values for the variances. When the number of fixed effects and random effects is unknown, the predominant approach to model building is a step-up method in which one starts with a limited model (e.g., few fixed and random intercepts) and then additional fixed effects and random effects are added based on statistical tests. A model building approach that has received less attention in psychology and education is a top-down method. In the top-down method, the initial model has a single random intercept but is loaded with fixed effects (also known as an "overelaborate" model). Based on the overelaborate fixed effects model, the need for additional random effects is determined. There has been little if any examination of the ability of these methods to identify a true population model (i.e., identifying the model that generated the data). The purpose of this article is to examine the performance of the step-up and top-down model building approaches for exploratory longitudinal data analysis. Student achievement data sets from the Chicago longitudinal study serve as the populations in the simulations.

CINRG pilot trial of coenzyme Q10 in steroid-treated Duchenne muscular dystrophy

INTRODUCTION

Corticosteroid treatment slows disease progression and is the standard of care for Duchenne muscular dystrophy (DMD). Coenzyme Q10 (CoQ10) is a potent antioxidant that may improve function in dystrophin-deficient muscle.

METHODS

We performed an open-label, "add-on" pilot study of CoQ10 in thirteen 5-10-year-old DMD patients on steroids. The primary outcome measure was the total quantitative muscle testing (QMT) score.

RESULTS

Twelve of 16 children (mean age 8.03 ± 1.64 years) completed the trial. Target serum levels of CoQ10 (≥2.5 μg/ml) were shown to be subject- and administration-dependent. Nine of 12 subjects showed an increase in total QMT score. Overall, CoQ10 treatment resulted in an 8.5% increase in muscle strength (P = 0.03).

CONCLUSIONS

Addition of CoQ10 to prednisone therapy in DMD patients resulted in an increase in muscle strength. These results warrant a larger, controlled trial of CoQ10 in DMD.

Comparison of three tests of homogeneity of odds ratios in multicenter trials with unequal sample sizes within and among centers

Background

Mixed effects logistic models have become a popular method for analyzing multicenter clinical trials with binomial data. However, the statistical properties of these models for testing homogeneity of odds ratios under various conditions, such as within-center and among-centers inequality, are still unknown and not yet compared with those of commonly used tests of homogeneity.

Methods

We evaluated the effect of within-center and among-centers inequality on the empirical power and type I error rate of the three homogeneity tests of odds ratios including likelihood ratio (LR) test of a mixed logistic model, DerSimonian-Laird (DL) statistic and Breslow-Day (BD) test by simulation study. Moreover, the impacts of number of centers (K), number of observations in each center and amount of heterogeneity were investigated by simulation.

Results

As compared with the equal sample size design, the power of the three tests of homogeneity will decrease if the same total sample size, which can be allocated equally within one center or among centers, is allocated unequally. The average reduction in the power of these tests was up to 11% and 16% for within-center and among-centers inequality, respectively. Moreover, in this situation, the ranking of the power of the homogeneity tests was BD≥DL≥LR and the power of these tests increased with increasing K.

Conclusions

This study shows that the adverse effect of among-centers inequality on the power of the homogeneity tests was stronger than that of within-center inequality. However, the financial limitations make the use of unequal sample size designs inevitable in multicenter trials. Moreover, although the power of the BD is higher than that of the LR when K≤6, the proposed mixed logistic model is recommended when K≥8 due to its practical advantages.

Arbitrariness of models for augmented and coarse data, with emphasis on incomplete data and random effects models

Statistical models often extend beyond the data available. First, in coarse data, what is actually observed is less detailed than what might be, owing to incompleteness, censoring, grouping, or a combination thereof. Second, in augmented data, the observed data are hypothetically supplemented with random effects, latent variables/classes, or component membership in mixture distributions. The two settings together will be referred to as enriched data. Reasons for modelling enriched data encompass mathematical and computational convenience, advantages in interpretation, and substantive plausibility. Models for enriched data combine evidence coming from empirical data with unverifiable model components, resting entirely on assumptions.

This has acute consequences for enriched data, but knowledge about this issue is somewhat scattered. We provide a unified framework for enriched data and show, generally and with focus on incomplete-data models and random-effects models on the other hand, that to any given model an entire class of models can be assigned, with all of its members producing the same fit to the observed data but arbitrary regarding the unobservable parts of the enriched data. The concepts developed are illustrated using a clinical trial in toenail dermatophyte onychomycosis and a developmental toxicity study conducted in mice.

Bayesian semiparametric frailty selection in multivariate event time data

Biomedical studies often collect multivariate event time data from multiple clusters (either subjects or groups) within each of which event times for individuals are correlated and the correlation may vary in different classes. In such survival analyses, heterogeneity among clusters for shared and specific classes can be accommodated by incorporating parametric frailty terms into the model. In this article, we propose a Bayesian approach to relax the parametric distribution assumption for shared and specific-class frailties by using a Dirichlet process prior while also allowing for the uncertainty of heterogeneity for different classes. Multiple cluster-specific frailty selections rely on variable selection-type mixture priors by applying mixtures of point masses at zero and inverse gamma distributions to the variance of log frailties. This selection allows frailties with zero variance to effectively drop out of the model. A reparameterization of log-frailty terms is performed to reduce the potential bias of fixed effects due to variation of the random distribution and dependence among the parameters resulting in easy interpretation and faster Markov chain Monte Carlo convergence. Simulated data examples and an application to a lung cancer clinical trial are used for illustration.

Fitting ACE structural equation models to case-control family data

Investigators interested in whether a disease aggregates in families often collect case-control family data, which consist of disease status and covariate information for members of families selected via case or control probands. Here, we focus on the use of case-control family data to investigate the relative contributions to the disease of additive genetic effects (A), shared family environment (C), and unique environment (E). We describe an ACE model for binary family data; this structural equation model, which has been described previously, combines a general-family extension of the classic ACE twin model with a (possibly covariate-specific) liability-threshold model for binary outcomes. We then introduce our contribution, a likelihood-based approach to fitting the model to singly ascertained case-control family data. The approach, which involves conditioning on the proband's disease status and also setting prevalence equal to a prespecified value that can be estimated from the data, makes it possible to obtain valid estimates of the A, C, and E variance components from case-control (rather than only from population-based) family data. In fact, simulation experiments suggest that our approach to fitting yields approximately unbiased estimates of the A, C, and E variance components, provided that certain commonly made assumptions hold. Further, when our approach is used to fit the ACE model to Austrian case-control family data on depression, the resulting estimate of heritability is very similar to those from previous analyses of twin data.

Multivariate association analysis of the components of metabolic syndrome from the Framingham Heart Study

Metabolic syndrome, by definition, is the manifestation of multiple, correlated metabolic impairments. It is known to have both strong environmental and genetic contributions. However, isolating genetic variants predisposing to such a complex trait has limitations. Using pedigree data, when available, may well lead to increased ability to detect variants associated with such complex traits. The ability to incorporate multiple correlated traits into a joint analysis may also allow increased detection of associated genes. Therefore, to demonstrate the utility of both univariate and multivariate family-based association analysis and to identify possible genetic variants associated with metabolic syndrome, we performed a scan of the Affymetrix 50 k Human Gene Panel data using 1) each of the traits comprising metabolic syndrome: triglycerides, high-density lipoprotein, systolic blood pressure, diastolic blood pressure, blood glucose, and body mass index, and 2) a composite trait including all of the above, jointly. Two single-nucleotide polymorphisms within the cholesterol ester transfer protein (CETP) gene remained significant even after correcting for multiple testing in both the univariate (p < 5 x 10-7) and multivariate (p < 5 x 10-9) association analysis. Three genes met significance for multiple traits after correction for multiple testing in the univariate analysis, while five genes remained significant in the multivariate association. We conclude that while both univariate and multivariate family-based association analysis can identify genes of interest, our multivariate approach is less affected by multiple testing correction and yields more significant results.

Comparison of univariate and multivariate linkage analysis of traits related to hypertension

Complex traits are often manifested by multiple correlated traits. One example of this is hypertension (HTN), which is measured on a continuous scale by systolic blood pressure (SBP). Predisposition to HTN is predicted by hyperlipidemia, characterized by elevated triglycerides (TG), low-density lipids (LDL), and high-density lipids (HDL). We hypothesized that the multivariate analysis of TG, LDL, and HDL would be more powerful for detecting HTN genes via linkage analysis compared with univariate analysis of SBP. We conducted linkage analysis of four chromosomal regions known to contain genes associated with HTN using SBP as a measure of HTN in univariate Haseman-Elston regression and using the correlated traits TG, LDL, and HDL in multivariate Haseman-Elston regression. All analyses were conducted using the Framingham Heart Study data. We found that multivariate linkage analysis was better able to detect chromosomal regions in which the angiotensinogen, angiotensin receptor, guanine nucleotide-binding protein 3, and prostaglandin I2 synthase genes reside. Univariate linkage analysis only detected the AGT gene. We conclude that multivariate analysis is appropriate for the analysis of multiple correlated phenotypes, and our findings suggest that it may yield new linkage signals undetected by univariate analysis.

Maternal corticosterone but not testosterone level is associated with the ratio of second-to-fourth digit length (2D:4D) in field vole offspring (Microtus agrestis)

The steroid environment encountered by a foetus can strongly affect its post-natal physiology and behaviour. It has been proposed that steroid concentrations experienced in utero could be estimated from adults by measuring their second-to-fourth digit length ratio (2D:4D). However, there is still little direct evidence that intra-uterine steroid levels affect individual 2D:4D. We examined whether maternal pre-pregnancy testosterone and corticosterone levels (as estimates of intra-uterine testosterone and corticosterone exposure) affected the 2D:4D of pups in non-domesticated field voles (Microtus agrestis), measured by X-rays at the age of weaning (21 days). Furthermore, for the first time in a non-human species, we studied whether testosterone and corticosterone levels correlated with 2D:4D in adult females. We found that the maternal pre-pregnancy level of testosterone was not associated with offspring 2D:4D in either the left or the right paw. Instead, maternal pre-pregnancy corticosterone level was positively correlated with offspring 2D:4D in the right paw, but unrelated to 2D:4D in the left paw. In addition, the 2D:4D of adult females was not associated with either their circulating testosterone or corticosterone levels. Our results suggest that in field voles maternally administered testosterone is not a major determinant of offspring 2D:4D, whereas maternal stress appears to account for some of the variation in the 2D:4D of their offspring.

Non-linear Models for Longitudinal Data

While marginal models, random-effects models, and conditional models are routinely considered to be the three main modeling families for continuous and discrete repeated measures with linear and generalized linear mean structures, respectively, it is less common to consider non-linear models, let alone frame them within the above taxonomy. In the latter situation, indeed, when considered at all, the focus is often exclusively on random-effects models. In this paper, we consider all three families, exemplify their great flexibility and relative ease of use, and apply them to a simple but illustrative set of data on tree circumference growth of orange trees.

Calculating asymptotic significance levels of the constrained likelihood ratio test with application to multivariate genetic linkage analysis

The asymptotic distribution of the multivariate variance component linkage analysis likelihood ratio test has provoked some contradictory accounts in the literature. In this paper we confirm that some previous results are not correct by deriving the asymptotic distribution in one special case. It is shown that this special case is a good approximation to the distribution in many situations. We also introduce a new approach to simulating from the asymptotic distribution of the likelihood ratio test statistic in constrained testing problems. It is shown that this method is very efficient for small p-values, and is applicable even when the constraints are not convex. The method is related to a multivariate integration problem. We illustrate how the approach can be applied to multivariate linkage analysis in a simulation study. Some more philosophical issues relating to one-sided tests in variance components linkage analysis are discussed.

Testing random effects in the linear mixed model using approximate bayes factors

SUMMARY

Deciding which predictor effects may vary across subjects is a difficult issue. Standard model selection criteria and test procedures are often inappropriate for comparing models with different numbers of random effects due to constraints on the parameter space of the variance components. Testing on the boundary of the parameter space changes the asymptotic distribution of some classical test statistics and causes problems in approximating Bayes factors. We propose a simple approach for testing random effects in the linear mixed model using Bayes factors. We scale each random effect to the residual variance and introduce a parameter that controls the relative contribution of each random effect free of the scale of the data. We integrate out the random effects and the variance components using closed-form solutions. The resulting integrals needed to calculate the Bayes factor are low-dimensional integrals lacking variance components and can be efficiently approximated with Laplace's method. We propose a default prior distribution on the parameter controlling the contribution of each random effect and conduct simulations to show that our method has good properties for model selection problems. Finally, we illustrate our methods on data from a clinical trial of patients with bipolar disorder and on data from an environmental study of water disinfection by-products and male reproductive outcomes.

Updating the management of rectal cancer

In the last 25 years rectal cancer has changed from a surgically managed disease into a multidisciplinary treatment model. Accurate staging has a critical role in the decision-making process of patients with rectal cancer. The four most commonly used imaging modalities in the pretherapeutical staging include endoscopic ultrasound, computerized tomography, magnetic resonance imaging and positron emission tomography. Locoregional tumor control in rectal cancer surgery has improved significantly over the last 15 years, after the introduction of total mesorectal excision (TME), which leads to the complete removal of the intact mesorectum including the lymphatics, lymph nodes, nerves, and vascular supply. At the present time, given the improved local control, acute and long-term toxicity profile, and sphincter preservation rate, patients who require combined modality therapy should receive concomitant radiochemotherapy preoperatively. Recently, the novel 'targeted' therapies have been incorporated into a multidisciplinary approach for rectal cancer.

Hypothesis tests for population heterogeneity in meta-analysis

Choice of the appropriate model in meta-analysis is often treated as an empirical question which is answered by examining the amount of variability in the effect sizes. When all of the observed variability in the effect sizes can be accounted for based on sampling error alone, a set of effect sizes is said to be homogeneous and a fixed-effects model is typically adopted. Whether a set of effect sizes is homogeneous or not is usually tested with the so-called Q test. In this paper, a variety of alternative homogeneity tests - the likelihood ratio, Wald and score tests - are compared with the Q test in terms of their Type I error rate and power for four different effect size measures. Monte Carlo simulations show that the Q test kept the tightest control of the Type I error rate, although the results emphasize the importance of large sample sizes within the set of studies. The results also suggest under what conditions the power of the tests can be considered adequate.

Fixed and random effects selection in linear and logistic models

We address the problem of selecting which variables should be included in the fixed and random components of logistic mixed effects models for correlated data. A fully Bayesian variable selection is implemented using a stochastic search Gibbs sampler to estimate the exact model-averaged posterior distribution. This approach automatically identifies subsets of predictors having nonzero fixed effect coefficients or nonzero random effects variance, while allowing uncertainty in the model selection process. Default priors are proposed for the variance components and an efficient parameter expansion Gibbs sampler is developed for posterior computation. The approach is illustrated using simulated data and an epidemiologic example.

A note on permutation tests for variance components in multilevel generalized linear mixed models

In many applications of generalized linear mixed models to multilevel data, it is of interest to test whether a random effects variance component is zero. It is well known that the usual asymptotic chi-square distribution of the likelihood ratio and score statistics under the null does not necessarily hold. In this note we propose a permutation test, based on randomly permuting the indices associated with a given level of the model, that has the correct Type I error rate under the null. Results from a simulation study suggest that it is more powerful than tests based on mixtures of chi-square distributions. The proposed test is illustrated using data on the familial aggregation of sleep disturbance.

Joint analysis of repeatedly observed continuous and ordinal measures of disease severity

In biomedical studies often multiple measures of disease severity are recorded over time. Although correlated, such measures are frequently analysed separately of one another. Joint analysis of the outcomes variables has several potential advantages over separate analyses. However, models for response variables of different types (discrete and continuous) are challenging to define and to fit. Herein we propose correlated probit models for joint analysis of repeated measurements on ordinal and continuous variables measuring the same underlying disease severity over time. We demonstrate how to rewrite the models so that maximum-likelihood estimation and inference can be performed with standard software. Simulation studies are performed to assess efficiency gains in fitting the responses together rather than separately and to guide response variable selection for future studies. Data from a depression clinical trial are used for illustration.