Computation of variance components by the MINQUE method

One of the main concerns in the family studies of complex diseases is the effect that ascertainment and correction for it may have on test procedures and estimators. Elston and Sobel [1979] and Hopper and Mathews [1982] proposed two ways to correct for ascertainment in the study of quantitative trait data. For single ascertainment, using a variance components approach, we present results of simulation studies comparing estimates from these two methods for different selection criteria. We also show results from simulations when ascertained families are analyzed either at random or by correcting for ascertainment. For discordant sibpairs, we compare a variance components model that incorporates ascertainment correction with the extreme discordant sib pairs (EDSP) design proposed by Risch and Zhang [1995]. Our results show that there is minimal difference between the two methods of ascertainment correction. In the presence of effects from a large genetic background and the segregation of a rare gene, both ascertainment affected the polygenic and environmental components of variance but had rather little impact on the estimate of the linked major gene component of variance. The results also show the EDSP is slightly more powerful the variance components procedures for common alleles, and the variance components procedure is much more powerful than using EDSP when there is a rare allele segregating in the population.

: Exact tests are derived for testing hypotheses concerning the variance components of the main effects in an unbalanced random two-way crossed classification with interaction model. The tests are based on four sums of squares which are distributed independently as scalar multiples of chi-squared variates. These sums of squares can also be used to find an exact test concerning the interaction variance component, and to obtain simultaneous confidence intervals on all continuous functions of the model's variance components. A study is made concerning the power of the proposed tests, including a comparison with other approximate tests. (Author)

Pulse pressure (PP) is an independent risk factor for cardiovascular disease. PP rises with age, more so in women. We examined sex differences in the correlations and variance components of PP in adult subjects from 767 nuclear families, enriched with those containing twins, from the Victorian Family Heart Study. After adjusting for age, we found no significant differences in the means or variances of PP in males and females. Under the assumption of no sex differences, the proportions of variance due to shared genes, shared environment, and individual-specific environment were 20%, 23% and 57%, respectively. However, same-sex relative pairs had significantly higher correlations than opposite-sex pairs (P = 0.005), implying the existence of sex-dependent effects. Extensions to the simple variance components model suggested three possible explanations for these differences: smaller genetic correlation between opposite-sex pairs (rhoG,MF = 0.45, P = 0.007); smaller environmental correlation between opposite-sex pairs (P = 0.0003); or different environmental and genetic correlations obtained by estimating genetic, environmental, and individual variance components separately for males and females (not nested, Akaike's Information Criterion (AIC) smaller by 6.69). Under the last model, the genetic component of PP variance is greater for males (1.62 vs 0.33) while the environmental component is greater for females (1.84 vs 0), which would have implications for the planning of gene discovery studies, since heritability would be higher in males. However, the second (environmental) approach best fits the data according to the AIC. Genetic explanations for sex differences in phenotypic correlations may be misleading unless shared environmental factors are also considered. PP illustrates a phenotype in which sex dependency represents an important component of phenotypic determination that can be revealed by detailed variance components modelling.

A note on non-negative minimum bias MINQE in variance components model

Variance components modeling has emerged as a powerful method for quantitative trait loci (QTL) linkage analysis. However, the power to detect a gene of minor effect is low. Many complex traits are affected by environmental as well as genetic factors, and one strategy to increase power is to reduce nongenetic phenotypic variance by adjusting for environmental covariates. In this paper, we investigate the power of three approaches to covariate adjustment in variance components linkage analysis: (i) incorporate covariates in the means model, (ii) incorporate covariates in the covariance matrix, and (iii) perform analysis on residual statistics. These approaches are compared to an analysis without adjustment. The results show that in the absence of correlation between the covariate and the QTL effect, adjusting for covariates indeed increases power to detect an underlying QTL. As this correlation increases, however, the power decreases. In the presence of a causal association between QTL and covariates, not adjusting for covariates appeared to be more powerful. The three approaches for adjusting for covariate: residual statistics, the means model, and the covariance model, had equal power to detect a QTL.

In this paper, we study the existence of the uniformly minimum risk equivariant (UMRE) estimators of parameters in a class of normal linear models, which include the normal variance components model, the growth curve model, the extended growth curve model, and the seemingly unrelated regression equations model, and so on. The necessary and sufficient conditions are given for the existence of UMRE estimators of the estimable linear functions of regression coefficients, the covariance matrix V and (tr V ) α , where α > 0 is known, in the models under an affine group of transformations for quadratic losses and matrix losses, respectively. Under the (extended) growth curve model and the seemingly unrelated regression equations model, the conclusions given in literature for estimating regression coefficients can be derived by applying the general results in this paper, and the sufficient conditions for non-existence of UMRE estimators of V and tr( V ) are expanded to be necessary and sufficient conditions. In addition, the necessary and sufficient conditions that there exist UMRE estimators of parameters in the variance components model are obtained for the first time.

SUMMARY The optimum properties of the popular estimate of the interclass variance in components of variance models are known if there are an equal number of observations in the subclasses. In contrast to this, it is shown that unbiased quadratic estimates having uniformly minimum variance do not exist if the design is not balanced.

Estimation of Variance Components of Quantitative Traits in Inbred Populations

Models used for investigation of variance components in analysis of variance are moderately restrictive for point estimation and quite restrictive, assuming normality, when tests and confidence regions are developed. However, the same results as occur for these models can be obtained for generalizations that are much less restrictive. In particular, results based on normality for the random variables of the model, and on their independence, remain valid for situations where each observation can have a different nonnormal distribution and dependence between each two observations can be different. The level of generalization for a model depends on what is being investigated and decreases as the number of variance components being investigated increases. Considered are the N-way classification models of various kinds and some specific examples. Similar extensions can be made for virtually all types of variance components models. The verification consists in showing that the statistics involved have the same joint distribution for the extended model as they do for the original model. Some ways of investigating outliers of nested designs, which also apply to the generalizations, are discussed.

Variance components models were used to analyze total IgE levels in families ascertained though the Collaborative Study of the Genetics of Asthma (CSGA) using a genome-wide array of polymorphic markers. While IgE levels are known to be associated with clinical asthma and recognized to be under strong genetic control (here the heritability was estimated at 44-60% in the three racial groups), specific genes influencing this trait are still largely unknown. Multipoint analysis of 323 markers yielded little indication of specific regions containing a trait locus controlling total serum IgE levels (adjusted for age and gender). Although a number of regions showed LOD statistics above 1.5 in Caucasian families (chromosome 4) and in African-American families (chromosomes 2 and 4), none yielded consistent evidence in all three racial groups. Analysis of total IgE adjusted for gender, age and Allergy Index (a quantitative score of skin test sensitivity to 14 common aeroallergens) was conducted on these data. In this analysis, a much stronger signal for a trait locus controlling adjusted log[total IgE] was seen on the telomeric end of chromosome 18, but only in Caucasian families. This region accounted for most of the genetic variation in log[total IgE], and may represent a quantitative trait locus for IgE levels independent of atopic response. Oligogenic analysis accounting simultaneously for the contribution of this locus on chromosome 18 and other chromosomal regions showing some evidence of linkage in these Caucasian families (on chromosomes 2, 4 and 20) failed to yield significant evidence for interaction.

This study aims to expand our understanding of the genetic architecture of crown morphology in the human diphyodont dentition. Here, we present bivariate genetic correlation estimates for deciduous and permanent molar traits and evaluate the patterns of pleiotropy within (e.g., m1–m2) and between (e.g., m2–M1) dentitions. Morphology was observed and scored from dental models representing participants of an Australian twin and family study (deciduous n = 290, permanent n = 339). Data collection followed Arizona State University Dental Anthropology System standards. Genetic correlation estimates were generated using maximum likelihood variance components analysis in SOLAR v.8.1.1. Approximately 23% of deciduous variance components models and 30% of permanent variance components models yielded significant genetic correlation estimates. By comparison, over half (56%) of deciduous–permanent homologues (e.g., m2 hypocone–M1 hypocone) were significantly genetically correlated. It is generally assumed that the deciduous and permanent molars represent members of a meristic molar field emerging from the primary dental lamina. However, stronger genetic integration among m2–M1/M2 homologues than among paired deciduous traits suggests the m2 represents the anterior-most member of a “true” molar field. The results indicate genetic factors act at distinct points throughout development to generate homologous molar form, starting with the m2, which is later replaced by a permanent premolariform crown.

We used variance components analysis to investigate the underlying determinants of the quantitative phenotypes (Q1-Q5) and their interrelationships in replicate 42 of the Genetic Analysis Workshop 12 simulated general population. Variance components models were fitted using Gibbs sampling in WinBUGS v1.3. Sigma-squared-A-random-effects (SSARs) were estimated for each phenotype, and were used as derived phenotypes in subsequent linkage analyses. Whole-genome, multipoint linkage analyses were based upon a new Haseman-Elston identity-by descent sib-pair method that takes a weighted combination of the trait-sum and trait-difference. The five quantitative traits simulated were closely correlated with each other and with affection status. The whole-genome screen of quantitative traits associated with the simulated complex disease suggested that one or more major loci regulating Q1 localizes to chromosome 2p and that one or more major loci regulating Q5 may localize to chromosome 1p.

Traditional methods for detection and mapping of quantitative trait loci (QTL) in dairy cattle populations are usually based on daughter design (DD) or granddaughter design (GDD). Although these designs are well established and usually successful in detecting QTL, they consider sire families independently of each other, thereby ignoring relationships among other animals in the population and consequently, reducing the power of QTL detection. In this study we compared a traditional GDD with a general pedigree design (GPD) and assessed the precision and power of both methods for detecting and locating QTL in a simulated complex pedigree. QTL analyses were performed under the variance component model containing a random QTL and a random polygenic effect. The covariance matrix of the polygenic effect was a standard additive relationship matrix. The (co)variance matrix of the random QTL effect contained probabilities that QTL alleles shared by two individuals were identical by descent (IBD). In the GDD analysis, IBD probabilities were calculated using sires’ and daughters’ marker genotypes. In the GPD analysis, IBD probabilities were obtained using a deterministic approach. The estimation of QTL position and variance components was conducted using REML algorithm. Although both methods were able to locate the region of the QTL properly, the GPD method showed better precision of QTL position estimates in most cases and significantly higher power than the GDD method.  

Interval estimation of certain functions of variance components is of interest to research workers in all fields of applications in which the variance component model is used. Confidence intervals for linear functions and ratios of variance components have been proposed by several authors. For the most part, these intervals are approximate with unknown exact probabilities associated with their coverage. In this article a technique is given for the construction of simultaneous confidence intervals for the values of all continuous functions of the variance components in a balanced, general random linear model. These confidence intervals are conservative; that is, the actual confidence level cannot be less than any preset value. The proposed technique is easy to apply as it only requires the optimization of a given continuous function of the variance components over a bounded region. Several examples of continuous functions are considered, including linear functions and ratios of variance components. Furthermore, the technique can be applied to unbalanced, random two-way classification models.

A procedure for finding exact tests for some hypotheses on variance components in unbalanced models is proposed. It is based on F-distributed statistics got by an orthogonal decomposition of the sample space.