The genetic structure of female life history in D. melanogaster: comparisons among populations

Two questions were addressed: (1) What is the genetic variance-covariance structure of a suite of four female life history traits in D. melanogaster? and (2) Does the genetic architecture of these traits differ among populations? Three populations of D. melanogaster were studied. Genetic variances and covariances were estimated by sib analysis three times for each population: immediately upon establishment of populations in the laboratory, and subsequently after approximately 6 months and 2 years of laboratory culture. Entire genetic variance-covariance matrices, as well as their individual components, were compared between populations by means of likelihood ratio tests. All traits studied were significantly heritable in at least one-half of estimates. Despite large sample sizes, additive genetic covariances were for the most part not statistically significant, and only two significant negative covariance estimates were obtained throughout the experiments. Therefore, these experiments provide little support for evolutionary life history theories that are based on negative genetic correlations among life history components. Neither do they support the idea that genetic variance for fitness components is maintained by trade-offs. Evidence suggests that the G matrix of one population was initially different from those of the other two populations. Those differences disappeared after 2 years of laboratory culture. At the level of individual (co)variance components, there were relatively few differences among populations, and the overall impression was that the three populations had generally similar genetic architectures for the traits studied.     

The stability of P in coral reef fishes

The constancy of phenotypic variation and covariation is an assumption that underlies most recent investigations of past selective regimes and attempts to predict future responses to selection. Few studies have tested this assumption of constancy despite good reasons to expect that the pattern of phenotypic variation and covariation may vary in space and time. We compared phenotypic variance-covariance matrices (P) estimated for populations of six species of distantly related coral reef fishes sampled at two locations on Australia's Great Barrier Reef separated by more than 1000 km. The intraspecific similarity between these matrices was estimated using two methods: matrix correlation and common principal component analysis. Although there was no evidence of equality between pairs of P, both statistical approaches indicated a high degree of similarity in morphology between the two populations for each species. In general, the hierarchical decomposition of the variance-covariance structure of these populations indicated that all principal components of phenotypic variance-covariance were shared but that they differed in the degree of variation associated with each of these components. The consistency of this pattern is remarkable given the diversity of morphologies and life histories encompassed by these species. Although some phenotypic instability was indicated, these results were consistent with a generally conserved pattern of multivariate selection between populations.     

Optimal design of longitudinal data analysis using generalized estimating equation models

Longitudinal studies are often applied in biomedical research and clinical trials to evaluate the treatment effect. The association pattern within the subject must be considered in both sample size calculation and the analysis. One of the most important approaches to analyze such a study is the generalized estimating equation (GEE) proposed by Liang and Zeger, in which “working correlation structure” is introduced and the association pattern within the subject depends on a vector of association parameters denoted by ρ. The explicit sample size formulas for two‐group comparison in linear and logistic regression models are obtained based on the GEE method by Liu and Liang. For cluster randomized trials (CRTs), researchers proposed the optimal sample sizes at both the cluster and individual level as a function of sampling costs and the intracluster correlation coefficient (ICC). In these approaches, the optimal sample sizes depend strongly on the ICC. However, the ICC is usually unknown for CRTs and multicenter trials. To overcome this shortcoming, Van Breukelen et al. consider a range of possible ICC values identified from literature reviews and present Maximin designs (MMDs) based on relative efficiency (RE) and efficiency under budget and cost constraints. In this paper, the optimal sample size and number of repeated measurements using GEE models with an exchangeable working correlation matrix is proposed under the considerations of fixed budget, where “optimal” refers to maximum power for a given sampling budget. The equations of sample size and number of repeated measurements for a known parameter value ρ are derived and a straightforward algorithm for unknown ρ is developed. Applications in practice are discussed. We also discuss the existence of the optimal design when an AR(1) working correlation matrix is assumed. Our proposed method can be extended under the scenarios when the true and working correlation matrix are different.     

Comparing G matrices: are common principal components informative?

Common principal components (CPC) analysis is a technique for assessing whether variance-covariance matrices from different populations have similar structure. One potential application is to compare additive genetic variance-covariance matrices, G. In this article, the conditions under which G matrices are expected to have common PCs are derived for a two-locus, two-allele model and the model of constrained pleiotropy. The theory demonstrates that whether G matrices are expected to have common PCs is largely determined by whether pleiotropic effects have a modular organization. If two (or more) populations have modules and these modules have the same direction, the G matrices have a common PC, regardless of allele frequencies. In the absence of modules, common PCs exist only for very restricted combinations of allele frequencies. Together, these two results imply that, when populations are evolving, common PCs are expected only when the populations have modules in common. These results have two implications: (1) In general, G matrices will not have common PCs, and (2) when they do, these PCs indicate common modular organization. The interpretation of common PCs identified for estimates of G matrices is discussed in light of these results.     

Multivariate whole genome average interval mapping: QTL analysis for multiple traits and/or environments

A major aim in some plant-based studies is the determination of quantitative trait loci (QTL) for multiple traits or across multiple environments. Understanding these QTL by trait or QTL by environment interactions can be of great value to the plant breeder. A whole genome approach for the analysis of QTL is presented for such multivariate applications. The approach is an extension of whole genome average interval mapping in which all intervals on a linkage map are included in the analysis simultaneously. A random effects working model is proposed for the multivariate (trait or environment) QTL effects for each interval, with a variance-covariance matrix linking the variates in a particular interval. The significance of the variance-covariance matrix for the QTL effects is tested and if significant, an outlier detection technique is used to select a putative QTL. This QTL by variate interaction is transferred to the fixed effects. The process is repeated until the variance-covariance matrix for QTL random effects is not significant; at this point all putative QTL have been selected. Unlinked markers can also be included in the analysis. A simulation study was conducted to examine the performance of the approach and demonstrated the multivariate approach results in increased power for detecting QTL in comparison to univariate methods. The approach is illustrated for data arising from experiments involving two doubled haploid populations. The first involves analysis of two wheat traits, α-amylase activity and height, while the second is concerned with a multi-environment trial for extensibility of flour dough. The method provides an approach for multi-trait and multi-environment QTL analysis in the presence of non-genetic sources of variation.     

QUANTITATIVE GENETICS OF SURVIVAL AND GROWTH IN IMPATIENS CAPENSIS

When variation in life-history characters is caused by many genes of small effect, then quantitative-genetic parameters may quantify constraints on rate and direction of microevolutionary change. I estimated heritabilities and genetic correlations for 16 life-history and morphological characters in two populations of Impatiens capensis, a partially self-pollinating herbaceous annual. The Madison population had little or no additive genetic variance for any of these characters, while the Milwaukee population had significant narrowsense heritabilities and genetic correlations for several traits, including adult size, which is highly correlated with fitness. All genetic correlations among fitness components were positive, hence there is no evidence for antagonistic pleiotropy among these traits. Dissimilarity of heritabilities in the two populations supports theoretical predictions that long-term changes in genetic variance-covariance patterns may occur when population sizes are small and selection is strong, as may occur in many plant species.     

HOMOGENEITY OF THE GENETIC VARIANCE-COVARIANCE MATRIX FOR ANTIPREDATOR TRAITS IN TWO NATURAL POPULATIONS OF THE GARTER SNAKE THAMNOPHIS ORDINOIDES

Quantitative genetic models of evolution rely on the genetic variance-covariance matrix to predict the phenotypic response to selection. Both prospective and retrospective studies of phenotypic evolution across generations rely on assumptions about the constancy of patterns of genetic covariance through time. In the absence of robust theoretical predictions about the stability of genetic covariances, this assumption must be tested with empirical comparisons of genetic parameters among populations and species. Genetic variance-covariance matrices were estimated for a suite of antipredator traits in two populations of the northwestern garter snake, Thamnophis ordinoides. The characters studied include color pattern and antipredator behaviors that interact to facilitate escape from predators. Significant heritabilities for all traits were detected in both populations. Genetic correlations and covariances were found among behaviors in both populations and between color pattern and behavior in one of the populations. Phenotypic means differed among populations, but pairwise comparisons revealed no heterogeneity of genetic parameters between the populations. The structure of the genetic variance-covariance matrix has apparently not changed significantly during the divergence of these two populations.     

Defining the assumptions underlying modeling of epistatic QTL using variance component methods

Variance component models are commonly used to detect quantitative trait loci (QTL) in general pedigrees. The variance-covariance structure of the random QTL effect is given by the identity by descent (IBD) between genotypes. Epistatic effects have previously been modeled, both for unlinked and linked loci, as a random effect with a variance-covariance structure given by the Hadamard product between the IBD matrices of the direct QTL effects. In the original papers, the model was given but not derived. Here, we identify the underlying assumptions of this previously proposed model. It assumes that either an unlinked QTL or a fully informative marker (i.e., all marker alleles are unique in the base generation) is located between the loci. We discuss the need of developing a general algorithm to estimate the variance-covariance structure of the random epistatic effect for linked loci.     

Quantile regression models with multivariate failure time data

As an alternative to the mean regression model, the quantile regression model has been studied extensively with independent failure time data. However, due to natural or artificial clustering, it is common to encounter multivariate failure time data in biomedical research where the intracluster correlation needs to be accounted for appropriately. For right-censored correlated survival data, we investigate the quantile regression model and adapt an estimating equation approach for parameter estimation under the working independence assumption, as well as a weighted version for enhancing the efficiency. We show that the parameter estimates are consistent and asymptotically follow normal distributions. The variance estimation using asymptotic approximation involves nonparametric functional density estimation. We employ the bootstrap and perturbation resampling methods for the estimation of the variance-covariance matrix. We examine the proposed method for finite sample sizes through simulation studies, and illustrate it with data from a clinical trial on otitis media.     

Estimating the responsiveness of an instrument using more than two repeated measures

The responsiveness of a measuring instrument is its ability to detect change over time. A commonly used index of responsiveness is the effect size for paired differences. This paper generalizes the effect size for paired differences to more than two repeated observations per subject. The sampling distribution of the generalized responsiveness statistic, Rt, is simulated for a range of plausible parameter values and for a range of sample sizes varying both the number of subjects (n) and the number of observations per subject (t). The coverage properties of confidence intervals constructed by four methods are compared. Confidence intervals based on jackknife estimates of the standard error and bias of Rt have good coverage properties even when n and t are small. The methods are used to determine which of two standard quality-of-life measures is more responsive to improvements in quality of life following surgery for early-stage breast cancer.     

Genetic architecture of host use in a widely distributed, polyphagous butterfly (Lepidoptera: Papilionidae): adaptive inferences based on comparison of spatio-temporal populations

Genetic variance-covariance structure of larval performance within and among spatio-temporal populations of the widely distributed, polyphagous tiger swallowtail butterfly, Papilio glaucus , is described. Performance traits were assessed for full-sibling families on three host species: Liriodendron tulipifera, Magnolia virginiana and Prunus serotina . Mean performance varied across hosts, indicating these hosts present unique developmental environments. Although full-sibling families significantly differed in plasticity of across-hosts response in three of the five spatio-temporal populations, additive genetic variation was mostly associated with P. serotina or pupal mass. The relative lack of heritable variation in rate and length of larval development on L. tulipifera and M. virginiana was consistent with an earlier study that established host-associated geographic differentiation of P. glaucus populations. Performance appeared relatively independent across hosts and thus genetic constraints cannot be casually invoked to explain persistence of local adaptation and host specialization in the face of extensive gene flow. I promote the hypothesis that gene flow among geographically distant populations is relatively restricted and that previously established, allozyme-based estimates of panmixia are confounded by effects of Pleistocene glaciations. Significant heterogeneity of variance-covariance structure among spatio-temporal P. glaucus populations supports an interpretation of restricted gene flow and relative evolutionary independence. Despite low precision of estimates of genetic parameters, local variance-covariance structure was remarkably consistent with expectations given the presumed evolutionary history of regional populations.     

Using Weed Control Knowledge from Declining Agricultural Communities in Invasive-Species Management

Traditional and small-scale farmers may know of practices that control weedy species. When these species are also problematic in restored or managed areas, a collection of this knowledge might assist control efforts. However, past criticisms of using local ecological knowledge (LEK) from small-scale farmers state that small sample sizes and highly variable responses among informants hinders LEK’s utility in management. Here I document weed-control knowledge held by New Jersey salt-hay farmers to control common reed and adapt strategies to control its invasion in two restoration settings. Accounts indicated that repeated cutting could eradicate the invasive, and subsequent experimental treatments in restoration settings demonstrated this technique to be very effective. However, only one farmer knew of this technique, and this farmer’s accounts seemingly contradicted other farmers’ accounts of cutting. This study demonstrates that small sample sizes and highly variable responses are more problematic to studies of knowledge, per se, than in finding valuable knowledge, which simply must be held by a community member rather than be common or well-distributed.     

A stochastic model to analyze clonal data on multi-type cell populations

This article presents a stochastic model designed to analyze experimental data on the development of cell clones composed of two (or more) distinct types of cells. The proposed model is an extension of the traditional multi-type Bellman-Harris branching stochastic process allowing for nonidentical time-to-transformation distributions defined for different cell types. A simulated pseudo likelihood method has been developed for the parametric statistical inference from experimental data on cell clones under the proposed model. The method uses simulation-based approximations of the means and the variance-covariance matrices of cell counts. The proposed estimator for the vector of unknown parameters is strongly consistent and asymptotically normal under mild regularity conditions, while its variance-covariance matrix is estimated by the parametric bootstrap. A Monte Carlo Wald test is proposed for the test of hypotheses. Finite sample properties of the estimator have been studied by computer simulations. The model and associated methods of parametric inference have been applied to the analysis of proliferation and differentiation of cultured O-2A progenitor cells that play a key role in the development of the central nervous system. It follows from this analysis that the time to division of the progenitor cell and the time to its differentiation (into an oligodendrocyte) are not identically distributed. This biological finding suggests that a molecular event determining the type of cell transformation is more likely to occur at the start rather than at the end of the mitotic cycle.     

PHENOTYPIC CORRELATION STRUCTURE AMONG ELEMENTS OF THE COLOR PATTERN IN PRECIS COENIA (LEPIDOPTERA: NYMPHALIDAE)

The component parts of butterfly wing patterns are arranged in sets of serially homologous pattern elements, repeated from wing cell to wing cell. Measurements were made on the sizes and positions of these elements on two successive, independent, sets of specimens in order to elucidate the phenotypic correlation structure among pattern elements. That portion of the correlation between measures due to overall size variation was accounted for through two alternate methods: multiple regression on two vein length measures, which represent wing size, and a Wright-style factor analysis. The sizes of pattern elements belonging to a homologous series were found to be significantly correlated whereas those of non-homologous elements varied independently. The degree of correlation among homologs varied, and, in the case of eyespot sizes, appeared to be inversely related to the degree of their morphological divergence. Although not correlated in size, the positions of non-homologous elements that lie within the same wing cell are moderately correlated. The results support current developmental models for the ontogeny of butterfly color pattern.     

Effects of sample size on estimates of population growth rates calculated with matrix models

Background

Matrix models are widely used to study the dynamics and demography of populations. An important but overlooked issue is how the number of individuals sampled influences estimates of the population growth rate (λ) calculated with matrix models. Even unbiased estimates of vital rates do not ensure unbiased estimates of λ–Jensen''s Inequality implies that even when the estimates of the vital rates are accurate, small sample sizes lead to biased estimates of λ due to increased sampling variance. We investigated if sampling variability and the distribution of sampling effort among size classes lead to biases in estimates of λ.

Methodology/Principal Findings

Using data from a long-term field study of plant demography, we simulated the effects of sampling variance by drawing vital rates and calculating λ for increasingly larger populations drawn from a total population of 3842 plants. We then compared these estimates of λ with those based on the entire population and calculated the resulting bias. Finally, we conducted a review of the literature to determine the sample sizes typically used when parameterizing matrix models used to study plant demography.

Conclusions/Significance

We found significant bias at small sample sizes when survival was low (survival = 0.5), and that sampling with a more-realistic inverse J-shaped population structure exacerbated this bias. However our simulations also demonstrate that these biases rapidly become negligible with increasing sample sizes or as survival increases. For many of the sample sizes used in demographic studies, matrix models are probably robust to the biases resulting from sampling variance of vital rates. However, this conclusion may depend on the structure of populations or the distribution of sampling effort in ways that are unexplored. We suggest more intensive sampling of populations when individual survival is low and greater sampling of stages with high elasticities.     

Relative efficiency of unequal versus equal cluster sizes in cluster randomized trials using generalized estimating equation models

There is growing interest in conducting cluster randomized trials (CRTs). For simplicity in sample size calculation, the cluster sizes are assumed to be identical across all clusters. However, equal cluster sizes are not guaranteed in practice. Therefore, the relative efficiency (RE) of unequal versus equal cluster sizes has been investigated when testing the treatment effect. One of the most important approaches to analyze a set of correlated data is the generalized estimating equation (GEE) proposed by Liang and Zeger, in which the “working correlation structure” is introduced and the association pattern depends on a vector of association parameters denoted by ρ. In this paper, we utilize GEE models to test the treatment effect in a two‐group comparison for continuous, binary, or count data in CRTs. The variances of the estimator of the treatment effect are derived for the different types of outcome. RE is defined as the ratio of variance of the estimator of the treatment effect for equal to unequal cluster sizes. We discuss a commonly used structure in CRTs—exchangeable, and derive the simpler formula of RE with continuous, binary, and count outcomes. Finally, REs are investigated for several scenarios of cluster size distributions through simulation studies. We propose an adjusted sample size due to efficiency loss. Additionally, we also propose an optimal sample size estimation based on the GEE models under a fixed budget for known and unknown association parameter (ρ) in the working correlation structure within the cluster.     

Genetic Correlations and Maternal Effect Coefficients Obtained from Offspring-Parent Regression

Additive genetic variances and covariances of quantitative characters are necessary to predict the evolutionary response of the mean phenotype vector in a population to natural or artificial selection. Standard formulas for estimating these parameters, from the resemblance between relatives in one or two characters at a time, are biased by natural selection on the parents and by maternal effects. We show how these biases can be removed using a multivariate analysis of offspring-parent regressions. A dynamic model of maternal effects demonstrates that, in addition to the phenotypic variance-covariance matrix of the characters, sufficient parameters for predicting the response of the mean phenotype vector to weak selection are the additive genetic variance-covariance matrix and a set of causal coefficients for maternal effects. These can be simultaneously estimated from offspring-parent regressions alone, in some cases just from the daughter-mother regressions, if all of the important selected and maternal characters have been measured and included in the analysis.     

Weighted normality-based estimator in correcting correlation coefficient estimation between incomplete nutrient measurements

Consider the problem of estimating the correlation between two nutrient measurements, such as the percent energy from fat obtained from a food frequency questionnaire (FFQ) and that from repeated food records or 24-hour recalls. Under a classical additive model for repeated food records, it is known that there is an attenuation effect on the correlation estimation if the sample average of repeated food records for each subject is used to estimate the underlying long-term average. This paper considers the case in which the selection probability of a subject for participation in the calibration study, in which repeated food records are measured, depends on the corresponding FFQ value, and the repeated longitudinal measurement errors have an autoregressive structure. This paper investigates a normality-based estimator and compares it with a simple method of moments. Both methods are consistent if the first two moments of nutrient measurements exist. Furthermore, joint estimating equations are applied to estimate the correlation coefficient and related nuisance parameters simultaneously. This approach provides a simple sandwich formula for the covariance estimation of the estimator. Finite sample performance is examined via a simulation study, and the proposed weighted normality-based estimator performs well under various distributional assumptions. The methods are applied to real data from a dietary assessment study.