Experimental evidence for multivariate stabilizing sexual selection

Stabilizing selection is a fundamental concept in evolutionary biology. In the presence of a single intermediate optimum phenotype (fitness peak) on the fitness surface, stabilizing selection should cause the population to evolve toward such a peak. This prediction has seldom been tested, particularly for suites of correlated traits. The lack of tests for an evolutionary match between population means and adaptive peaks may be due, at least in part, to problems associated with empirically detecting multivariate stabilizing selection and with testing whether population means are at the peak of multivariate fitness surfaces. Here we show how canonical analysis of the fitness surface, combined with the estimation of confidence regions for stationary points on quadratic response surfaces, may be used to define multivariate stabilizing selection on a suite of traits and to establish whether natural populations reside on the multivariate peak. We manufactured artificial advertisement calls of the male cricket Teleogryllus commodus and played them back to females in laboratory phonotaxis trials to estimate the linear and nonlinear sexual selection that female phonotactic choice imposes on male call structure. Significant nonlinear selection on the major axes of the fitness surface was convex in nature and displayed an intermediate optimum, indicating multivariate stabilizing selection. The mean phenotypes of four independent samples of males, from the same population as the females used in phonotaxis trials, were within the 95% confidence region for the fitness peak. These experiments indicate that stabilizing sexual selection may play an important role in the evolution of male call properties in natural populations of T. commodus.     

Seemingly Unrelated Generalized Linear Models

By using deviance standardized residuals, the seemingly unrelated regression estimation procedure is extended to generalized linear models, and fitted by an iterative procedure. The matrix of cross products of standardized residuals is asymptotically multivariate normal, and can be used for further multivariate analyses and for hypothesis testing.     

Estimation of response to selection and utilization of control populations for additional information and accuracy

Problems associated with testing and estimation of response to selection are examined. An alternative procedure with increased power for testing hypotheses is given. The increased power results from a more precise method of estimating the variance about response. The new method is based on a Satterthwaite approximation which combines variance components estimated more precisely by other sources of variation in the analysis of variance. The expected variance about response and expected mean squares for the analysis of variance, used in the Satterthwaite procedure, are given. When intergeneration environmental trends cannot be ruled out, a control population must be used to estimate response to selection. However, if the experimental and control populations do not respond in the same direction and with the same magnitude to common environmental effects, then the usual method of estimating response by deviating the experimental values from the control will result in biased estimates. An alternative procedure, using the control as a covariate to adjust for environmental trends, gives relatively unbiased estimates of response in this situation. Some bias results from measurement error associated with the control. However, this bias is usually minimal.     

Mixed-model analysis of a censored normal distribution with reference to animal breeding

A mixed-model procedure for analysis of censored data assuming a multivariate normal distribution is described. A Bayesian framework is adopted which allows for estimation of fixed effects and variance components and prediction of random effects when records are left-censored. The procedure can be extended to right- and two-tailed censoring. The model employed is a generalized linear model, and the estimation equations resemble those arising in analysis of multivariate normal or categorical data with threshold models. Estimates of variance components are obtained using expressions similar to those employed in the EM algorithm for restricted maximum likelihood (REML) estimation under normality.     

Estimation of the benchmark dose by structural equation models

While epidemiological data typically contain a multivariate response and often also multiple exposure parameters, current methods for safe dose calculations, including the widely used benchmark approach, rely on standard regression techniques. In practice, dose-response modeling and calculation of the exposure limit are often based on the seemingly most sensitive outcome. However, this procedure ignores other available data, is inefficient, and fails to account for multiple testing. Instead, risk assessment could be based on structural equation models, which can accommodate both a multivariate exposure and a multivariate response function. Furthermore, such models will allow for measurement error in the observed variables, which is a requirement for unbiased estimation of the benchmark dose. This methodology is illustrated with the data on neurobehavioral effects in children prenatally exposed to methylmercury, where results based on standard regression models cause an underestimation of the true risk.     

A tale of two matrices: multivariate approaches in evolutionary biology

Two symmetric matrices underlie our understanding of microevolutionary change. The first is the matrix of nonlinear selection gradients (gamma) which describes the individual fitness surface. The second is the genetic variance-covariance matrix (G) that influences the multivariate response to selection. A common approach to the empirical analysis of these matrices is the element-by-element testing of significance, and subsequent biological interpretation of pattern based on these univariate and bivariate parameters. Here, I show why this approach is likely to misrepresent the genetic basis of quantitative traits, and the selection acting on them in many cases. Diagonalization of square matrices is a fundamental aspect of many of the multivariate statistical techniques used by biologists. Applying this, and other related approaches, to the analysis of the structure of gamma and G matrices, gives greater insight into the form and strength of nonlinear selection, and the availability of genetic variance for multiple traits.     

Robust estimation of multivariate covariance components

In many settings, such as interlaboratory testing, small area estimation in sample surveys, and heritability studies, investigators are interested in estimating covariance components for multivariate measurements. However, the presence of outliers can seriously distort estimates obtained using standard procedures such as maximum likelihood. We propose a procedure based on M-estimation for robustly estimating multivariate covariance components in the presence of outliers; the procedure applies to balanced and unbalanced data. We present an algorithm for computing the robust estimates and examine the performance of the estimator through a simulation study. The estimator is used to find covariance components and identify outliers in a study of variability of egg length and breadth measurements of American coots.     

Quadratic inference functions for varying-coefficient models with longitudinal data

Nonparametric smoothing methods are used to model longitudinal data, but the challenge remains to incorporate correlation into nonparametric estimation procedures. In this article, we propose an efficient estimation procedure for varying-coefficient models for longitudinal data. The proposed procedure can easily take into account correlation within subjects and deal directly with both continuous and discrete response longitudinal data under the framework of generalized linear models. The proposed approach yields a more efficient estimator than the generalized estimation equation approach when the working correlation is misspecified. For varying-coefficient models, it is often of interest to test whether coefficient functions are time varying or time invariant. We propose a unified and efficient nonparametric hypothesis testing procedure, and further demonstrate that the resulting test statistics have an asymptotic chi-squared distribution. In addition, the goodness-of-fit test is applied to test whether the model assumption is satisfied. The corresponding test is also useful for choosing basis functions and the number of knots for regression spline models in conjunction with the model selection criterion. We evaluate the finite sample performance of the proposed procedures with Monte Carlo simulation studies. The proposed methodology is illustrated by the analysis of an acquired immune deficiency syndrome (AIDS) data set.     

Multivariate slope ratio assay with repeated measurements

A method is given for analyzing a slope ratio assay in which a test drug is compared with a standard drug, two or more response variates being measured on each subject at each of several successively increased drug doses. The method requires all subjects to receive the same number of doses, all subjects on the same drug to receive the same doses, the ratio of corresponding doses of the two drugs to be constant over the successive increases, and response variables to be measured only once on each subject at each dose with no missing data allowed. The technique is also applicable when doses are randomly assigned, provided there is no carry-over effect between doses. For each of the J response variates, the relative potency of the test drug with respect to the standard is defined and estimated in the usual way; a 100(1-alpha)% confidence region is then obtained for the vector of the J relative potencies. A procedure is given for testing the equality of some or all of the J relative potencies; an estimator of a common relative potency is obtained by a standard multivariate least squares method. A common relative potency is of interest because the multiple outcome variables are often different indicators of a general physiologic response. The procedures in the paper are illustrated by a simple example concerning the effects of two anesthetics on children.     

A Procedure for the Common Estimation of Parameters Corresponding to Several Treatment Groups

In this article we give a procedure for the common estimation of parameters corresponding to several treatment groups. Thereby we assume that the distribution functions of the groups belong to the same family and differ only in the parameter values. The procedure allows the common estimation of some of these parameters. The parameters themselves will be estimated by the maximum likelihood method; the estimators will be calculated iteratively by the Newton-Raphson method. To prove if the common estimation is possible, we propose as a suitable test the maximum likelihood ratio test. Finally we show the application of our procedure in the case of the probit analysis.     

Simultaneous modeling of multivariate heterogeneous responses and heteroskedasticity via a two-stage composite likelihood

Multivariate heterogeneous responses and heteroskedasticity have attracted increasing attention in recent years. In genome-wide association studies, effective simultaneous modeling of multiple phenotypes would improve statistical power and interpretability. However, a flexible common modeling system for heterogeneous data types can pose computational difficulties. Here we build upon a previous method for multivariate probit estimation using a two-stage composite likelihood that exhibits favorable computational time while retaining attractive parameter estimation properties. We extend this approach to incorporate multivariate responses of heterogeneous data types (binary and continuous), and possible heteroskedasticity. Although the approach has wide applications, it would be particularly useful for genomics, precision medicine, or individual biomedical prediction. Using a genomics example, we explore statistical power and confirm that the approach performs well for hypothesis testing and coverage percentages under a wide variety of settings. The approach has the potential to better leverage genomics data and provide interpretable inference for pleiotropy, in which a locus is associated with multiple traits.     

Why the beta-function cannot be used to estimate skewness of species responses

Abstract. The beta-function (β-function) has been suggested for testing the significance of the skewness of species responses along a gradient. However, the location of the optimum and skewness are correlated so that these parameters cannot be estimated independently. The only way for an independent estimation is to let the endpoints of the response curve vary. In that case they would no longer define the range of species occurrence. However, non-linear estimation of endpoints often leads to overwhelming problems in model fitting. Therefore, the beta-function is not suitable to test the shape of species response curves. Hierarchic models proposed by Huisman et al. (1993) seem to be superior to generalized additive models or third-degree polynomials and seem to be the best alternative to study the skewness of responses.     

Estimation of copy number in polyploid plants: the good,the bad,and the ugly

Genetic studies in polyploid plants rely heavily on the collection of data from dominant marker loci. A dominant marker locus is a locus for which only the presence or absence of an observable (dominant) allele is recorded. Before these marker loci can be used for genetic exploration, the number of copies of a dominant allele carried by a parent (copy number) must be determined for each marker locus. Copy number in polyploids is estimated using a hypothesis testing procedure. The performance of this estimation procedure has never been evaluated. In this paper, I quantify whether the highly sought after single-copy markers can be accurately identified, if the performance of the estimation procedure improves with increasing sample size, and whether the estimation procedure is capable of accurately estimating the copy number of high copy markers. I found that the probability of incorrectly estimating copy number is quite low and that more data can actually reduce the accuracy of the estimation procedure when the testing assumptions are violated. Fortunately, when a significant result is obtained, it is almost always correct. The challenge often is in obtaining a significant result.     

Multivariate Dimensionality Reduction Approaches to Identify Gene-Gene and Gene-Environment Interactions Underlying Multiple Complex Traits

The elusive but ubiquitous multifactor interactions represent a stumbling block that urgently needs to be removed in searching for determinants involved in human complex diseases. The dimensionality reduction approaches are a promising tool for this task. Many complex diseases exhibit composite syndromes required to be measured in a cluster of clinical traits with varying correlations and/or are inherently longitudinal in nature (changing over time and measured dynamically at multiple time points). A multivariate approach for detecting interactions is thus greatly needed on the purposes of handling a multifaceted phenotype and longitudinal data, as well as improving statistical power for multiple significance testing via a two-stage testing procedure that involves a multivariate analysis for grouped phenotypes followed by univariate analysis for the phenotypes in the significant group(s). In this article, we propose a multivariate extension of generalized multifactor dimensionality reduction (GMDR) based on multivariate generalized linear, multivariate quasi-likelihood and generalized estimating equations models. Simulations and real data analysis for the cohort from the Study of Addiction: Genetics and Environment are performed to investigate the properties and performance of the proposed method, as compared with the univariate method. The results suggest that the proposed multivariate GMDR substantially boosts statistical power.     

Functional mapping of dynamic traits with robust t-distribution

Functional mapping has been a powerful tool in mapping quantitative trait loci (QTL) underlying dynamic traits of agricultural or biomedical interest. In functional mapping, multivariate normality is often assumed for the underlying data distribution, partially due to the ease of parameter estimation. The normality assumption however could be easily violated in real applications due to various reasons such as heavy tails or extreme observations. Departure from normality has negative effect on testing power and inference for QTL identification. In this work, we relax the normality assumption and propose a robust multivariate t-distribution mapping framework for QTL identification in functional mapping. Simulation studies show increased mapping power and precision with the t distribution than that of a normal distribution. The utility of the method is demonstrated through a real data analysis.     

A Multivariate Procedure for Linear Comparisons of Mean Vectors when Residual Covariance and Regression Coefficient Matrices are Unequal

A multivariate procedure for testing linear comparisons of vectors of adjusted group means of response (dependent) variables when groups differ in residual covariance and regression coefficient matrices is presented. Such disparities have been observed in investigations of trends in contaminant levels in fish. Application of the procedure is illustrated with data on Atlantic cod (Gadus morhua). The procedure is quite general and can be employed to test any linear comparisons.     

Tests for Order Restrictions in Binary Data

In this article, a general procedure is presented for testing for equality of k independent binary response probabilities against any given ordered alternative. The proposed methodology is based on an estimation procedure developed in Hwang and Peddada (1994, Annals of Statistics 22, 67-93) and can be used for a very broad class of order restrictions. The procedure is illustrated through application to two data sets that correspond to three commonly encountered order restrictions: simple tree order, simple order, and down turn order.     

The design and analysis of microarray experiments: applications in parasitology

Microarray experiments can generate enormous amounts of data, but large datasets are usually inherently complex, and the relevant information they contain can be difficult to extract. For the practicing biologist, we provide an overview of what we believe to be the most important issues that need to be addressed when dealing with microarray data. In a microarray experiment we are simply trying to identify which genes are the most "interesting" in terms of our experimental question, and these will usually be those that are either overexpressed or underexpressed (upregulated or downregulated) under the experimental conditions. Analysis of the data to find these genes involves first preprocessing of the raw data for quality control, including filtering of the data (e.g., detection of outlying values) followed by standardization of the data (i.e., making the data uniformly comparable throughout the dataset). This is followed by the formal quantitative analysis of the data, which will involve either statistical hypothesis testing or multivariate pattern recognition. Statistical hypothesis testing is the usual approach to "class comparison," where several experimental groups are being directly compared. The best approach to this problem is to use analysis of variance, although issues related to multiple hypothesis testing and probability estimation still need to be evaluated. Pattern recognition can involve "class prediction," for which a range of supervised multivariate techniques are available, or "class discovery," for which an even broader range of unsupervised multivariate techniques have been developed. Each technique has its own limitations, which need to be kept in mind when making a choice from among them. To put these ideas in context, we provide a detailed examination of two specific examples of the analysis of microarray data, both from parasitology, covering many of the most important points raised.     

A mixed-effects regression model for longitudinal multivariate ordinal data

A mixed-effects item response theory model that allows for three-level multivariate ordinal outcomes and accommodates multiple random subject effects is proposed for analysis of multivariate ordinal outcomes in longitudinal studies. This model allows for the estimation of different item factor loadings (item discrimination parameters) for the multiple outcomes. The covariates in the model do not have to follow the proportional odds assumption and can be at any level. Assuming either a probit or logistic response function, maximum marginal likelihood estimation is proposed utilizing multidimensional Gauss-Hermite quadrature for integration of the random effects. An iterative Fisher scoring solution, which provides standard errors for all model parameters, is used. An analysis of a longitudinal substance use data set, where four items of substance use behavior (cigarette use, alcohol use, marijuana use, and getting drunk or high) are repeatedly measured over time, is used to illustrate application of the proposed model.     

Joint models for a primary endpoint and multiple longitudinal covariate processes

Summary .   Joint models are formulated to investigate the association between a primary endpoint and features of multiple longitudinal processes. In particular, the subject-specific random effects in a multivariate linear random-effects model for multiple longitudinal processes are predictors in a generalized linear model for primary endpoints. Li, Zhang, and Davidian (2004, Biometrics 60 , 1–7) proposed an estimation procedure that makes no distributional assumption on the random effects but assumes independent within-subject measurement errors in the longitudinal covariate process. Based on an asymptotic bias analysis, we found that their estimators can be biased when random effects do not fully explain the within-subject correlations among longitudinal covariate measurements. Specifically, the existing procedure is fairly sensitive to the independent measurement error assumption. To overcome this limitation, we propose new estimation procedures that require neither a distributional or covariance structural assumption on covariate random effects nor an independence assumption on within-subject measurement errors. These new procedures are more flexible, readily cover scenarios that have multivariate longitudinal covariate processes, and can be implemented using available software. Through simulations and an analysis of data from a hypertension study, we evaluate and illustrate the numerical performances of the new estimators.