Levene tests of homogeneity of variance for general block and treatment designs

This article develops a weighted least squares version of Levene's test of homogeneity of variance for a general design, available both for univariate and multivariate situations. When the design is balanced, the univariate and two common multivariate test statistics turn out to be proportional to the corresponding ordinary least squares test statistics obtained from an analysis of variance of the absolute values of the standardized mean-based residuals from the original analysis of the data. The constant of proportionality is simply a design-dependent multiplier (which does not necessarily tend to unity). Explicit results are presented for randomized block and Latin square designs and are illustrated for factorial treatment designs and split-plot experiments. The distribution of the univariate test statistic is close to a standard F-distribution, although it can be slightly underdispersed. For a complex design, the test assesses homogeneity of variance across blocks, treatments, or treatment factors and offers an objective interpretation of residual plots.     

Environmental and sibling resemblance components of variance and covariance in traits of early child development

The multiple regression analyses were undertaken to elucidate the significance and relative importance of different potential determinants of several child development traits (i.e. age at which child 1. turns himself over; 2. sits up; 3. stands up; 4. walks, and 5. cuts his first tooth). Despite the fact that we used a relatively wide gamut of potential determinants (such as parental age, geographic origin, occupation, current residence, or family size) of trait variability, the results of multiple regression analysis (N = 300 families) indicate an almost complete absence of significant factors and studied variables of development. Sibling resemblance component of variance and covariance for five mentioned traits of development and some morphological characters (weight, length and head circumference at day of birth and at 16 months) were studied in 66 families. The transmissibility ("heritability") values for the development traits were small, ranging from 0 ("Turn") to 0.42 ("Walked"). The Findings suggest that there are significant inverse relationship between some of the developmental traits and inborn morphological characters.     

Evolution of the environmental component of the phenotypic variance: stabilizing selection in changing environments and the cost of homogeneity

Quantitative traits show abundant genetic, environmental, and phenotypic variance, yet if they are subject to stabilizing selection for an optimal phenotype, both the genetic and environmental components are expected to decline. The mechanisms that determine the level and maintenance of phenotypic variance are not yet fully understood. While there has been extensive study of mechanisms maintaining genetic variability, it has generally been assumed that environmental variance is not dependent on the genotype and therefore not subject to change. However, accumulating data suggest that the environmental variance is under some degree of genetic control. In this study, it is assumed accordingly that both the genotypic value (i.e., mean phenotypic value) and the variance of phenotypic value given genotypic value depend on the genotype. Two models are investigated as potentially able to explain the protected maintenance of environmental variance of quantitative traits under stabilizing selection. One is varying environment among generations, such that both the optimal phenotype and the strength of the stabilizing selection vary between generations. The other is the cost of homogeneity, which is based on an assumption of an engineering cost of minimizing variability in development. It is shown that a small homogeneity cost is enough to maintain the observed levels of environmental variance, whereas a large amount of temporal variation in the optimal phenotype and the strength of selection would be necessary.     

Sampling variance of the correlation coefficients estimated from analyses of variance and covariance

Summary A generalized sampling variance of correlation coefficients is derived for phenotypic, genetic and en vironmental correlations estimated from nested analyses of variance and covarianee for the equal number case. A numerical example is presented to estimate the sampling variance for the genetic correlation coefficient based on the relationship among full sibs using unequal subclass numbers.Journal Paper No. 3472 of the Purdue University Agricultural Experiment Station. This research was supported, in part, by NIH Biometry Training Grant, GM-00024.     

Aspects of variance and covariance analysis with cultural inheritance

Linear Gaussian models of genotypic, phenotypic, and environmental transmission are studied. The nature of equilibrium assumptions under various modes of assortative mating are discussed with particular emphasis on expected correlations between relatives. Assorting models based only on phenotype, or only on environment, are compared with those in which the mating correlation structure is more complex. Explicit values in terms of transmission parameters and within individual covariances are given for the usual correlations between relatives. The possibility of decomposing these in terms of correlations involving adoptive families is indicated.     

The effect of genetic drift on the variance/covariance components generated by multilocus additive x additive epistatic systems

The effect of population bottlenecks on the components of the genetic variance/covariance generated by n neutral independent additive x additive loci has been studied theoretically. In its simplest version, this situation can be modelled by specifying the allele frequencies and homozygous effects at each locus, and an additional factor measuring the strength of the n-th order epistatic interaction. The variance/covariance components in an infinitely large panmictic population (ancestral components) were compared with their expected values at equilibrium over replicates randomly derived from the base population, after t bottlenecks of size N (derived components). Formulae were obtained giving the derived components (and the between-line variance) as functions of the ancestral ones (alternatively, in terms of allele frequencies and effects) and the corresponding inbreeding coefficient F(t). The n-th order derived component of the genetic variance/covariance is continuously eroded by inbreeding, but the remaining components may increase initially until a critical F(t) value is attained, which is inversely related to the order of the pertinent component, and subsequently decline to zero. These changes can be assigned to the between-line variances/covariances of gene substitution and epistatic effects induced by drift. Numerical examples indicate that: (1) the derived additive variance/covariance component will generally exceed its ancestral value unless epistasis is weak; (2) the derived epistatic variance/covariance components will generally exceed their ancestral values unless allele frequencies are extreme; (3) for systems showing equal ancestral additive and total non-additive variance/covariance components, those including a smaller number of epistatic loci may generate a larger excess in additive variance/covariance after bottlenecks than others involving a larger number of loci, provided that F(t) is low. Our results indicate that it is unlikely that the rate of evolution may be significantly accelerated after population bottlenecks, in spite of occasional increments of the derived additive variance over its ancestral value.     

The genetic covariance among clinal environments after adaptation to an environmental gradient in Drosophila serrata

We examined the genetic basis of clinal adaptation by determining the evolutionary response of life-history traits to laboratory natural selection along a gradient of thermal stress in Drosophila serrata. A gradient of heat stress was created by exposing larvae to a heat stress of 36 degrees for 4 hr for 0, 1, 2, 3, 4, or 5 days of larval development, with the remainder of development taking place at 25 degrees. Replicated lines were exposed to each level of this stress every second generation for 30 generations. At the end of selection, we conducted a complete reciprocal transfer experiment where all populations were raised in all environments, to estimate the realized additive genetic covariance matrix among clinal environments in three life-history traits. Visualization of the genetic covariance functions of the life-history traits revealed that the genetic correlation between environments generally declined as environments became more different and even became negative between the most different environments in some cases. One exception to this general pattern was a life-history trait representing the classic trade-off between development time and body size, which responded to selection in a similar genetic fashion across all environments. Adaptation to clinal environments may involve a number of distinct genetic effects along the length of the cline, the complexity of which may not be fully revealed by focusing primarily on populations at the ends of the cline.     

Estimation of variance components based on diallel model

The problem of estimation of variance components based on diallel model for unbalanced data has been addressed. The least squares approach to quadratic estimation has been adopted in obtaining the explicit solutions for the design and genetic components of variance.     

Genetic variance in fitness and its cross-sex covariance predict adaptation during experimental evolution

The additive genetic variation (V_A) of fitness in a population is of particular importance to quantify its adaptive potential and predict its response to rapid environmental change. Recent statistical advances in quantitative genetics and the use of new molecular tools have fostered great interest in estimating fitness V_A in wild populations. However, the value of V_A for fitness in predicting evolutionary changes over several generations remains mostly unknown. In our study, we addressed this question by combining classical quantitative genetics with experimental evolution in the model organism Tribolium castaneum (red flour beetle) in three new environmental conditions (Dry, Hot, Hot-Dry). We tested for potential constraints that might limit adaptation, including environmental and sex genetic antagonisms captured by negative genetic covariance between environments and female and male fitness, respectively. Observed fitness changes after 20 generations mainly matched our predictions. Given that body size is commonly used as a proxy for fitness, we also tested how this trait and its genetic variance (including nonadditive genetic variance) were impacted by environmental stress. In both traits, genetic variances were sex and condition dependent, but they differed in their variance composition, cross-sex and cross-environment genetic covariances, as well as in the environmental impact on V_A.     

Analysis of variance components in gene expression data

MOTIVATION: A microarray experiment is a multi-step process, and each step is a potential source of variation. There are two major sources of variation: biological variation and technical variation. This study presents a variance-components approach to investigating animal-to-animal, between-array, within-array and day-to-day variations for two data sets. The first data set involved estimation of technical variances for pooled control and pooled treated RNA samples. The variance components included between-array, and two nested within-array variances: between-section (the upper- and lower-sections of the array are replicates) and within-section (two adjacent spots of the same gene are printed within each section). The second experiment was conducted on four different weeks. Each week there were reference and test samples with a dye-flip replicate in two hybridization days. The variance components included week-to-week, animal-to-animal and between-array and within-array variances. RESULTS: We applied the linear mixed-effects model to quantify different sources of variation. In the first data set, we found that the between-array variance is greater than the between-section variance, which, in turn, is greater than the within-section variance. In the second data set, for the reference samples, the week-to-week variance is larger than the between-array variance, which, in turn, is slightly larger than the within-array variance. For the test samples, the week-to-week variance has the largest variation. The animal-to-animal variance is slightly larger than the between-array and within-array variances. However, in a gene-by-gene analysis, the animal-to-animal variance is smaller than the between-array variance in four out of five housekeeping genes. In summary, the largest variation observed is the week-to-week effect. Another important source of variability is the animal-to-animal variation. Finally, we describe the use of variance-component estimates to determine optimal numbers of animals, arrays per animal and sections per array in planning microarray experiments.     

fullfact: an R package for the analysis of genetic and maternal variance components from full factorial mating designs

Full factorial breeding designs are useful for quantifying the amount of additive genetic, nonadditive genetic, and maternal variance that explain phenotypic traits. Such variance estimates are important for examining evolutionary potential. Traditionally, full factorial mating designs have been analyzed using a two‐way analysis of variance, which may produce negative variance values and is not suited for unbalanced designs. Mixed‐effects models do not produce negative variance values and are suited for unbalanced designs. However, extracting the variance components, calculating significance values, and estimating confidence intervals and/or power values for the components are not straightforward using traditional analytic methods. We introduce fullfact – an R package that addresses these issues and facilitates the analysis of full factorial mating designs with mixed‐effects models. Here, we summarize the functions of the fullfact package. The observed data functions extract the variance explained by random and fixed effects and provide their significance. We then calculate the additive genetic, nonadditive genetic, and maternal variance components explaining the phenotype. In particular, we integrate nonnormal error structures for estimating these components for nonnormal data types. The resampled data functions are used to produce bootstrap‐t confidence intervals, which can then be plotted using a simple function. We explore the fullfact package through a worked example. This package will facilitate the analyses of full factorial mating designs in R, especially for the analysis of binary, proportion, and/or count data types and for the ability to incorporate additional random and fixed effects and power analyses.     

Estimating synaptic parameters from mean, variance, and covariance in trains of synaptic responses

Fluctuation analysis of synaptic transmission using the variance-mean approach has been restricted in the past to steady-state responses. Here we extend this method to short repetitive trains of synaptic responses, during which the response amplitudes are not stationary. We consider intervals between trains, long enough so that the system is in the same average state at the beginning of each train. This allows analysis of ensemble means and variances for each response in a train separately. Thus, modifications in synaptic efficacy during short-term plasticity can be attributed to changes in synaptic parameters. In addition, we provide practical guidelines for the analysis of the covariance between successive responses in trains. Explicit algorithms to estimate synaptic parameters are derived and tested by Monte Carlo simulations on the basis of a binomial model of synaptic transmission, allowing for quantal variability, heterogeneity in the release probability, and postsynaptic receptor saturation and desensitization. We find that the combined analysis of variance and covariance is advantageous in yielding an estimate for the number of release sites, which is independent of heterogeneity in the release probability under certain conditions. Furthermore, it allows one to calculate the apparent quantal size for each response in a sequence of stimuli.