Hierarchical Bayesian modeling of heterogeneous cluster‐ and subject‐level associations between continuous and binary outcomes in dairy production

The augmentation of categorical outcomes with underlying Gaussian variables in bivariate generalized mixed effects models has facilitated the joint modeling of continuous and binary response variables. These models typically assume that random effects and residual effects (co)variances are homogeneous across all clusters and subjects, respectively. Motivated by conflicting evidence about the association between performance outcomes in dairy production systems, we consider the situation where these (co)variance parameters may themselves be functions of systematic and/or random effects. We present a hierarchical Bayesian extension of bivariate generalized linear models whereby functions of the (co)variance matrices are specified as linear combinations of fixed and random effects following a square‐root‐free Cholesky reparameterization that ensures necessary positive semidefinite constraints. We test the proposed model by simulation and apply it to the analysis of a dairy cattle data set in which the random herd‐level and residual cow‐level effects (co)variances between a continuous production trait and binary reproduction trait are modeled as functions of fixed management effects and random cluster effects.     

TYPE I ERROR RATES FOR TESTING GENETIC DRIFT WITH PHENOTYPIC COVARIANCE MATRICES: A SIMULATION STUDY

Studies of evolutionary divergence using quantitative genetic methods are centered on the additive genetic variance–covariance matrix ( G ) of correlated traits. However, estimating G properly requires large samples and complicated experimental designs. Multivariate tests for neutral evolution commonly replace average G by the pooled phenotypic within‐group variance–covariance matrix ( W ) for evolutionary inferences, but this approach has been criticized due to the lack of exact proportionality between genetic and phenotypic matrices. In this study, we examined the consequence, in terms of type I error rates, of replacing average G by W in a test of neutral evolution that measures the regression slope between among‐population variances and within‐population eigenvalues (the Ackermann and Cheverud [AC] test) using a simulation approach to generate random observations under genetic drift. Our results indicate that the type I error rates for the genetic drift test are acceptable when using W instead of average G when the matrix correlation between the ancestral G and P is higher than 0.6, the average character heritability is above 0.7, and the matrices share principal components. For less‐similar G and P matrices, the type I error rates would still be acceptable if the ratio between the number of generations since divergence and the effective population size (t/N_e) is smaller than 0.01 (large populations that diverged recently). When G is not known in real data, a simulation approach to estimate expected slopes for the AC test under genetic drift is discussed.     

Functional mixed effects models

In this article, a new class of functional models in which smoothing splines are used to model fixed effects as well as random effects is introduced. The linear mixed effects models are extended to nonparametric mixed effects models by introducing functional random effects, which are modeled as realizations of zero-mean stochastic processes. The fixed functional effects and the random functional effects are modeled in the same functional space, which guarantee the population-average and subject-specific curves have the same smoothness property. These models inherit the flexibility of the linear mixed effects models in handling complex designs and correlation structures, can include continuous covariates as well as dummy factors in both the fixed or random design matrices, and include the nested curves models as special cases. Two estimation procedures are proposed. The first estimation procedure exploits the connection between linear mixed effects models and smoothing splines and can be fitted using existing software. The second procedure is a sequential estimation procedure using Kalman filtering. This algorithm avoids inversion of large dimensional matrices and therefore can be applied to large data sets. A generalized maximum likelihood (GML) ratio test is proposed for inference and model selection. An application to comparison of cortisol profiles is used as an illustration.     

Random main effects of treatment: A case study with a network meta‐analysis

If the number of treatments in a network meta‐analysis is large, it may be possible and useful to model the main effect of treatment as random, that is to say as random realizations from a normal distribution of possible treatment effects. This then constitutes a third sort of random effect that may be considered in connection with such analyses. The first and most common models treatment‐by‐trial interaction as being random and the second, rather rarer, models the main effects of trial as being random and thus permits the recovery of intertrial information. Taking the example of a network meta‐analysis of 44 similar treatments in 10 trials, we illustrate how a hierarchical approach to modeling a random main effect of treatment can be used to produce shrunk (toward the overall mean) estimates of effects for individual treatments. As a related problem, we also consider the issue of using a random‐effect model for the within‐trial variances from trial to trial. We provide a number of possible graphical representations of the results and discuss the advantages and disadvantages of such an approach.     

Limited plasticity in the phenotypic variance‐covariance matrix for male advertisement calls in the black field cricket,Teleogryllus commodus

Phenotypic integration and plasticity are central to our understanding of how complex phenotypic traits evolve. Evolutionary change in complex quantitative traits can be predicted using the multivariate breeders’ equation, but such predictions are only accurate if the matrices involved are stable over evolutionary time. Recent study, however, suggests that these matrices are temporally plastic, spatially variable and themselves evolvable. The data available on phenotypic variance‐covariance matrix ( P ) stability are sparse, and largely focused on morphological traits. Here, we compared P for the structure of the complex sexual advertisement call of six divergent allopatric populations of the Australian black field cricket, Teleogryllus commodus. We measured a subset of calls from wild‐caught crickets from each of the populations and then a second subset after rearing crickets under common‐garden conditions for three generations. In a second experiment, crickets from each population were reared in the laboratory on high‐ and low‐nutrient diets and their calls recorded. In both experiments, we estimated P for call traits and used multiple methods to compare them statistically (Flury hierarchy, geometric subspace comparisons and random skewers). Despite considerable variation in means and variances of individual call traits, the structure of P was largely conserved among populations, across generations and between our rearing diets. Our finding that P remains largely stable, among populations and between environmental conditions, suggests that selection has preserved the structure of call traits in order that they can function as an integrated unit.     

Rapid and unpredictable changes of the G‐matrix in a natural bird population over 25 years

Knowledge of the genetic variances and covariances of traits (the G ‐matrix) is fundamental for the understanding of evolutionary dynamics of populations. Despite its essential importance in evolutionary studies, empirical tests of the temporal stability of the G ‐matrix in natural populations are few. We used a 25‐year‐long individual‐based field study on almost 7000 breeding attempts of the collared flycatcher (Ficedula albicollis) to estimate the stability of the G‐matrix over time. Using animal models to estimate G for several time periods, we show that the structure of the time‐specific G‐matrices changed significantly over time. The temporal changes in the G‐matrix were unpredictable, and the structure at one time period was not indicative of the structure at the next time period. Moreover, we show that the changes in the time‐specific G‐matrices were not related to changes in mean trait values or due to genetic drift. Selection, differences in acquisition/allocation patterns or environment‐dependent allelic effects are therefore likely explanations for the patterns observed, probably in combination. Our result cautions against assuming constancy of the G ‐matrix and indicates that even short‐term evolutionary predictions in natural populations can be very challenging.     

Comparison of genetic (co)variance matrices within and between Scabiosa canescens and S. columbaria

In the current study, we used bootstrap analyses and the common principal component (CPC) method of Flury (1988) to estimate and compare the G ‐matrix of Scabiosa columbaria and S. canescens populations. We found three major patterns in the G ‐matrices: (i) the magnitude of the (co)variances was more variable among characters than among populations, (ii) different populations showed high (co)variance for different characters, and (iii) there was a tendency for S. canescens to have higher genetic (co)variances than S. columbaria. The hypothesis of equal G ‐matrices was rejected in all comparisons and there was no evidence that the matrices differed by a proportional constant in any of the analyses. The two ‘species matrices’ were found to be unrelated, both for raw data and data standardized over populations, and there was significant between‐population variation in the G ‐matrix in both species. Populations of S. canescens showed conservation of structure (principal components) in their G ‐matrices, contrasting with the lack of common structure among the S. columbaria matrices. Given these observations and the results from previous studies, we propose that selection may be responsible for some of the variation between the G ‐matrices, at least in S. columbaria and at the between‐species level.     

Longitudinal analysis of pre- and post-treatment measurements with equal baseline assumptions in randomized trials

For continuous variables of randomized controlled trials, recently, longitudinal analysis of pre- and posttreatment measurements as bivariate responses is one of analytical methods to compare two treatment groups. Under random allocation, means and variances of pretreatment measurements are expected to be equal between groups, but covariances and posttreatment variances are not. Under random allocation with unequal covariances and posttreatment variances, we compared asymptotic variances of the treatment effect estimators in three longitudinal models. The data-generating model has equal baseline means and variances, and unequal covariances and posttreatment variances. The model with equal baseline means and unequal variance–covariance matrices has a redundant parameter. In large sample sizes, these two models keep a nominal type I error rate and have high efficiency. The model with equal baseline means and equal variance–covariance matrices wrongly assumes equal covariances and posttreatment variances. Only under equal sample sizes, this model keeps a nominal type I error rate. This model has the same high efficiency with the data-generating model under equal sample sizes. In conclusion, longitudinal analysis with equal baseline means performed well in large sample sizes. We also compared asymptotic properties of longitudinal models with those of the analysis of covariance (ANCOVA) and t-test.     

Discrete and morphometric traits reveal contrasting patterns and processes in the macroevolutionary history of a clade of scorpions

Many palaeontological studies have investigated the evolution of entire body plans, generally relying on discrete character‐taxon matrices. In contrast, macroevolutionary studies performed by neontologists have mostly focused on morphometric traits. Although these data types are very different, some studies have suggested that they capture common patterns. Nonetheless, the tests employed to support this claim have not explicitly incorporated a phylogenetic framework and may therefore be susceptible to confounding effects due to the presence of common phylogenetic structure. We address this question using the scorpion genus Brachistosternus Pocock 1893 as case study. We make use of a time‐calibrated multilocus molecular phylogeny, and compile discrete and traditional morphometric data sets, both capturing the overall morphology of the organisms. We find that morphospaces derived from these matrices are significantly different, and that the degree of discordance cannot be replicated by simulations of random character evolution. Moreover, we find strong support for contrasting modes of evolution, with discrete characters being congruent with an ‘early burst’ scenario whereas morphometric traits suggest species‐specific adaptations to have driven morphological evolution. The inferred macroevolutionary dynamics are therefore contingent on the choice of character type. Finally, we confirm that metrics of correlation fail to detect these profound differences given common phylogenetic structure in both data sets, and that methods incorporating a phylogenetic framework and accounting for expected covariance should be favoured.     

STATISTICAL ANALYSES FOR R-INDEX

R‐index is an important statistic for testing and measuring product effects. The validation and merits of R‐index are to a great extent due to the fact that it is closely related to the famous Mann–Whitney U statistic. Based on this fortunate relationship, statistical analyses for R‐index are explored. The statistical analyses include estimations of R‐index and its null and nonconditional variances with and without assuming continuity of data; difference and similarity tests using R‐index; powers and sample sizes for the tests; linking R‐index with Thurstonian δ (or d′). The new techniques developed in the paper extend greatly the original R‐index analysis for categorical ratings data. It is expected that the recognition of the profound theoretical origin of R‐index and the available statistical analyses for R‐index will provide the impetus for the resurgence of interest in using R‐index in sensory and consumer researches.     

Estimating genetic variance contributed by a quantitative trait locus: A random model approach

Detecting quantitative trait loci (QTL) and estimating QTL variances (represented by the squared QTL effects) are two main goals of QTL mapping and genome-wide association studies (GWAS). However, there are issues associated with estimated QTL variances and such issues have not attracted much attention from the QTL mapping community. Estimated QTL variances are usually biased upwards due to estimation being associated with significance tests. The phenomenon is called the Beavis effect. However, estimated variances of QTL without significance tests can also be biased upwards, which cannot be explained by the Beavis effect; rather, this bias is due to the fact that QTL variances are often estimated as the squares of the estimated QTL effects. The parameters are the QTL effects and the estimated QTL variances are obtained by squaring the estimated QTL effects. This square transformation failed to incorporate the errors of estimated QTL effects into the transformation. The consequence is biases in estimated QTL variances. To correct the biases, we can either reformulate the QTL model by treating the QTL effect as random and directly estimate the QTL variance (as a variance component) or adjust the bias by taking into account the error of the estimated QTL effect. A moment method of estimation has been proposed to correct the bias. The method has been validated via Monte Carlo simulation studies. The method has been applied to QTL mapping for the 10-week-body-weight trait from an F₂ mouse population.     

Comparison of asymptotic population growth rates with heterogeneous variances

This report explores how the heterogeneity of variances affects randomization tests used to evaluate differences in the asymptotic population growth rate, λ. The probability of Type I error was calculated in four scenarios for populations with identical λ but different variance of λ: (1) Populations have different projection matrices: the same λ may be obtained from different sets of vital rates, which gives room for different variances of λ. (2) Populations have identical projection matrices but reproductive schemes differ and fecundity in one of the populations has a larger associated variance. The two other scenarios evaluate a sampling artifact as responsible for heterogeneity of variances. The same population is sampled twice, (3) with the same sampling design, or (4) with different sampling effort for different stages. Randomization tests were done with increasing differences in sample size between the two populations. This implies additional differences in the variance of λ. The probability of Type I error keeps at the nominal significance level (α = .05) in Scenario 3 and with identical sample sizes in the others. Tests were too liberal, or conservative, under a combination of variance heterogeneity and different sample sizes. Increased differences in sample size exacerbated the difference between observed Type I error and the nominal significance level. Type I error increases or decreases depending on which population has a larger sample size, the population with the smallest or the largest variance. However, by their own, sample size is not responsible for changes in Type I errors.     

Sex differences in the genetic architecture of aggressiveness in a sexually dimorphic spider

Sex differences in the genetic architecture of behavioral traits can offer critical insight into the processes of sex‐specific selection and sexual conflict dynamics. Here, we assess genetic variances and cross‐sex genetic correlations of two personality traits, aggression and activity, in a sexually size‐dimorphic spider, Nuctenea umbratica. Using a quantitative genetic approach, we show that both traits are heritable. Males have higher heritability estimates for aggressiveness compared to females, whereas the coefficient of additive genetic variation and evolvability did not differ between the sexes. Furthermore, we found sex differences in the coefficient of residual variance in aggressiveness with females exhibiting higher estimates. In contrast, the quantitative genetic estimates for activity suggest no significant differentiation between males and females. We interpret these results with caution as the estimates of additive genetic variances may be inflated by nonadditive genetic effects. The mean cross‐sex genetic correlations for aggression and activity were 0.5 and 0.6, respectively. Nonetheless, credible intervals of both estimates were broad, implying high uncertainty for these estimates. Future work using larger sample sizes would be needed to draw firmer conclusions on how sexual selection shapes sex differences in the genetic architecture of behavioral traits.     

A Bayesian multinomial logistic exposure model for estimating probabilities of competing sources of nest failure

Understanding causes of nest loss is critical for the management of endangered bird populations. Available methods for estimating nest loss probabilities to competing sources do not allow for random effects and covariation among sources, and there are few data simulation methods or goodness‐of‐fit (GOF) tests for such models. We developed a Bayesian multinomial extension of the widely used logistic exposure (LE) nest survival model which can incorporate multiple random effects and fixed‐effect covariates for each nest loss category. We investigated the performance of this model and the accompanying GOF test by analysing simulated nest fate datasets with and without age‐biased discovery probability, and by comparing the estimates with those of traditional fixed‐effects estimators. We then exemplify the use of the multinomial LE model and GOF test by analysing Piping Plover Charadrius melodus nest fate data (n = 443) to explore the effects of wire cages (exclosures) constructed around nests, which are used to protect nests from predation but can lead to increased nest abandonment rates. Mean parameter estimates of the random‐effects multinomial LE model were all within 1 sd of the true values used to simulate the datasets. Age‐biased discovery probability did not result in biased parameter estimates. Traditional fixed‐effects models provided estimates with a high bias of up to 43% with a mean of 71% smaller standard deviations. The GOF test identified models that were a poor fit to the simulated data. For the Piping Plover dataset, the fixed‐effects model was less well‐supported than the random‐effects model and underestimated the risk of exclosure use by 16%. The random‐effects model estimated a range of 1–6% probability of abandonment for nests not protected by exclosures across sites and 5–41% probability of abandonment for nests with exclosures, suggesting that the magnitude of exclosure‐related abandonment is site‐specific. Our results demonstrate that unmodelled heterogeneity can result in biased estimates potentially leading to incorrect management recommendations. The Bayesian multinomial LE model offers a flexible method of incorporating random effects into an analysis of nest failure and is robust to age‐biased nest discovery probability. This model can be generalized to other staggered‐entry, time‐to‐hazard situations.     

Taxon sampling and seed plant phylogeny

We investigated the effects of taxon sampling on phylogenetic inference by exchanging terminals in two sizes of rbcL matrices for seed plants, applying parsimony and bayesian analyses to ten 38‐taxon matrices and ten 80‐taxon matrices. In comparing tree topologies we concentrated on the position of the Gnetales, an important group whose placement has long been disputed. With either method, trees obtained from different taxon samples could be mutually contradictory and even disagree on groups that seemed strongly supported. Adding terminals improved the consistency of results for unweighted parsimony, but not for parsimony with third positions excluded and not for bayesian analysis, particularly when the general time‐reversible model was employed. This suggests that attempting to resolve deep relationships using only a few taxa can lead to spurious conclusions, groupings unlikely to be repeatable with different taxon samplings or larger data sets. The effect of taxon sampling has not generally been recognized, and phylogenetic studies of seed plants have often been based on few taxa. Such insufficient sampling may help explain the variety of phylogenetic hypotheses for seed plants proposed in recent years. We recommend that restricted data sets such as single‐gene subsets of multigene studies should be reanalyzed with alternative selections of terminals to assess topological consistency.     

Hierarchical Bayesian modeling of random and residual variance–covariance matrices in bivariate mixed effects models

Bivariate mixed effects models are often used to jointly infer upon covariance matrices for both random effects ( u ) and residuals ( e ) between two different phenotypes in order to investigate the architecture of their relationship. However, these (co)variances themselves may additionally depend upon covariates as well as additional sets of exchangeable random effects that facilitate borrowing of strength across a large number of clusters. We propose a hierarchical Bayesian extension of the classical bivariate mixed effects model by embedding additional levels of mixed effects modeling of reparameterizations of u‐ level and e ‐level (co)variances between two traits. These parameters are based upon a recently popularized square‐root‐free Cholesky decomposition and are readily interpretable, each conveniently facilitating a generalized linear model characterization. Using Markov Chain Monte Carlo methods, we validate our model based on a simulation study and apply it to a joint analysis of milk yield and calving interval phenotypes in Michigan dairy cows. This analysis indicates that the e ‐level relationship between the two traits is highly heterogeneous across herds and depends upon systematic herd management factors.     

Rejoinder to “A Note on Type II Error Under Random Effects Misspecification in Generalized Linear Mixed Models”

Summary In this rejoinder, we discuss the impact of misspecifying the random effects distribution on inferences obtained from generalized linear mixed models (GLMMs). Special attention is paid to the power of the tests for the fixed‐effect parameters. To study this misspecification, researchers often use simulation designs in which several choices for the true underlying random‐effects distribution are considered, while the assumed distribution is kept fixed. Neuhaus, McCulloch, and Boylan (2010, Biometrics 00 , 000–000) argue that a logically correct approach should consist of varying the assumed, fitted distribution, while holding the true fixed. We argue that both simulation designs can bring valuable insights into the impact of the misspecification. Furthermore, using both designs, we illustrate that the power associated with the tests for the fixed‐effect parameters in GLMM may be affected by misspecifying the random‐effects distribution.     

Partitioning the effects of species loss on community variability using multi‐level selection theory

The temporal variability of ecological communities may depend on species richness and composition due to a variety of statistical and ecological mechanisms. However, ecologists currently lack a general, unified theoretical framework within which to compare the effects of these mechanisms. Developing such a framework is difficult because community variability depends not just on how species vary, but also how they covary, making it unclear how to isolate the contributions of individual species to community variability. Here I develop such a theoretical framework using the multi‐level Price equation, originally developed in evolutionary biology to partition the effects of group selection and individual selection. I show how the variability of a community can be related to the properties of the individual species comprising it, just as the properties of an evolving group can be related to the properties of the individual organisms comprising it. I show that effects of species loss on community variability can be partitioned into effects of species richness (random loss of species), effects of species composition (non‐random loss of species with respect to their variances and covariances), and effects of context dependence (post‐loss changes in species’ variances and covariances). I illustrate the application of this framework using data from the Biodiversity II experiment, and show that it leads to new conceptual and empirical insights. For instance, effects of species richness on community variability necessarily occur, but often are swamped by other effects, particularly context dependence.     

THE MIXED-MODEL ANALYSIS OF VARIANCE APPLIED TO QUANTITATIVE GENETICS: BIOLOGICAL MEANING OF THE PARAMETERS

The mixed-model factorial analysis of variance has been used in many recent studies in evolutionary quantitative genetics. Two competing formulations of the mixed-model ANOVA are commonly used, the “Scheffe” model and the “SAS” model; these models differ in both their assumptions and in the way in which variance components due to the main effect of random factors are defined. The biological meanings of the two variance component definitions have often been unappreciated, however. A full understanding of these meanings leads to the conclusion that the mixed-model ANOVA could have been used to much greater effect by many recent authors. The variance component due to the random main effect under the two-way SAS model is the covariance in true means associated with a level of the random factor (e.g., families) across levels of the fixed factor (e.g., environments). Therefore the SAS model has a natural application for estimating the genetic correlation between a character expressed in different environments and testing whether it differs from zero. The variance component due to the random main effect under the two-way Scheffe model is the variance in marginal means (i.e., means over levels of the fixed factor) among levels of the random factor. Therefore the Scheffe model has a natural application for estimating genetic variances and heritabilities in populations using a defined mixture of environments. Procedures and assumptions necessary for these applications of the models are discussed. While exact significance tests under the SAS model require balanced data and the assumptions that family effects are normally distributed with equal variances in the different environments, the model can be useful even when these conditions are not met (e.g., for providing an unbiased estimate of the across-environment genetic covariance). Contrary to statements in a recent paper, exact significance tests regarding the variance in marginal means as well as unbiased estimates can be readily obtained from unbalanced designs with no restrictive assumptions about the distributions or variance-covariance structure of family effects.     

Diallel analysis for sex-linked and maternal effects

Genetic models including sex-linked and maternal effects as well as autosomal gene effects are described. Monte Carlo simulations were conducted to compare efficiencies of estimation by minimum norm quadratic unbiased estimation (MINQUE) and restricted maximum likelihood (REML) methods. MINQUE(1), which has 1 for all prior values, has a similar efficiency to MINQUE(), which requires prior estimates of parameter values. MINQUE(1) has the advantage over REML of unbiased estimation and convenient computation. An adjusted unbiased prediction (AUP) method is developed for predicting random genetic effects. AUP is desirable for its easy computation and unbiasedness of both mean and variance of predictors. The jackknife procedure is appropriate for estimating the sampling variances of estimated variances (or covariances) and of predicted genetic effects. A t-test based on jackknife variances is applicable for detecting significance of variation. Worked examples from mice and silkworm data are given in order to demonstrate variance and covariance estimation and genetic effect prediction.