Two-Stage Least Squares Analysis of Hierarchical Linear Models

This paper has two major objectives. The first is to present a two-stage least squares procedure for estimation of the parameters in a linear model whose parameters are in themselves linear functions of some hyperparameters. The second, and perhaps more important point, is that the new estimator can be shown to be generally more precise than either the Bayesian or the generalized single-stage least squares estimator reported by LINDLEY and SMITH (1072).     

A Monte Carlo Test for Variance Homogeneity in Linear Models

A Monte Carlo procedure is proposed for testing homogeneity of variances in linear models. The method is applicable to a variety of common experimental designs. It is valid when errors are independently normally distributed. Under nonnormality the test is expected to behave robust in a similar fashion as Levene's test. Three examples are given to demonstrate the method.     

二阶段等距抽样

本文提出了二阶段等距抽样的概念．分第二阶段样本为等距抽样和SRSWOR两种情况讨论了总体平均值的估计问题．在第二阶段抽样为SRSWOR情况下,讨论了辅助信息的利用问题,提出了总体平均值的比型估计和回归型估计．并通过一个实际计算的例子,对文中提出的不同估计量进行了比较．     

Linear Discriminant Functions and Problems of Hybridization

A discriminant analysis method for frequency data for hybridization based on weighted multivariate analysis of variance is given for allotting an individual to one of groups.     

Models to estimate maternally controlled genetic variation in quantitative seed characters

Summary Estimating quantitative contributions to specific traits can be accomplished from a variety of genetic models (Mather 1949; Mather and Jinks 1971; Falconer 1981). Residual genetic effects, those beyond main and interaction effects of the embryo genotype, are often pooled under a single classification, termed maternal effects. Maternal contributions to seed-related traits can originate from various maternal sources (e.g., endosperm, testa and cytoplasm). Quantitative contributions of a maternal nature are not predictable from parental performance and effects are largely non-persistent over generations (Jinks et al. 1972). The methods used to determine maternal effects in quantitative traits often do not measure quantitative genetic parameters, while those that do are either complex or partially resolve potential contributions of individual sources of maternal effects. We present simple genetic models for estimating quantitative genetic parameters which take into account maternal effects expressed in the major seed tissues of higher plants.     

The General Linear Model as a Framework for Models in Configural Frequency Analysis

The present paper discusses models of Configural Frequency Analysis (CFA). For most models of CFA maximum likelihood estimators are given. For all of these models least squares estimators are also given. These estimators are equivalent to each other if quasiparametric conditions prevail. Using the second approach, the general linear model can be used to systematize CFA models. Numerical examples are given, using both artificial and psychiatric data.     

A New Algorithm for Explicit Determination of Variance Component Estimations in Mixed Models and Mixed Classifications of Balanced ANOVA

This paper deals with the balanced case of the analysis of variance. The use of a classification function leads to an easy determination of all possible sources of variation of any mixed classification. For mixed models a new method is derived, which allows to represent explicit the ANOVA-estimations of the variance components respectively the estimation of the mean sum of squares of the fixed effects for all sources of variation. Thereby the corresponding F-quotients and the approximate confidence intervals of variance components are received in a simple way.     

A Booststrap Adaptive Test for Two-way Analysis of Variance

We propose a method to construct adaptive tests based on a bootstrap technique. The procedure leads to a nearly exact adaptive test depending on the size of the sample. With the use of the estimated Pitman's relative efficacy as selector statistic, we show that the adaptive test has a power that is asymptotically equal to the power of it's better component. We apply the idea to construct an adaptive test for two-way analysis of variance model. Finally, we use simulations to observe the behaviour of the method for small sample sizes.     

A Two-Stage Approximation for Analysis of Mixture Genetic Models in Large Pedigrees

Information from cosegregation of marker and QTL alleles, in addition to linkage disequilibrium (LD), can improve genomic selection. Variance components linear models have been proposed for this purpose, but accommodating dominance and epistasis is not straightforward with them. A full-Bayesian analysis of a mixture genetic model is favorable in this respect, but is computationally infeasible for whole-genome analyses. Thus, we propose an approximate two-step approach that neglects information from trait phenotypes in inferring ordered genotypes and segregation indicators of markers. Quantitative trait loci (QTL) fine-mapping scenarios, using high-density markers and pedigrees of five generations without genotyped females, were simulated to test this strategy against an exact full-Bayesian approach. The latter performed better in estimating QTL genotypes, but precision of QTL location and accuracy of genomic breeding values (GEBVs) did not differ for the two methods at realistically low LD. If, however, LD was higher, the exact approach resulted in a slightly higher accuracy of GEBVs. In conclusion, the two-step approach makes mixture genetic models computationally feasible for high-density markers and large pedigrees. Furthermore, markers need to be sampled only once and results can be used for the analysis of all traits. Further research is needed to evaluate the two-step approach for complex pedigrees and to analyze alternative strategies for modeling LD between QTL and markers.DUE to advances in molecular genetics, high-density single-nucleotide polymorphisms (SNPs) are becoming available in animal and plant breeding. These can be used for whole-genome analyses such as prediction of genomic breeding values (GEBVs) and fine mapping of quantitative trait loci (QTL). Genomic selection (GS) (Meuwissen et al. 2001) is promising to improve response to selection by exploiting linkage disequilibrium (LD) between SNPs and QTL (Hayes et al. 2009; Vanraden et al. 2009), but the accuracy of GEBVs depends on additive-genetic relationships between the individuals used to estimate SNP effects and selection candidates (Habier et al. 2007, 2010). Use of cosegregation information, in addition to LD, may reduce this dependency and improve GS. Calus et al. (2008) used a variance components linear model for this purpose in which random QTL effects are modeled conditional on marker haplotypes. The covariance between founder haplotypes allows one to include LD (Meuwissen and Goddard 2000), and the covariance between nonfounder haplotypes computed as in Fernando and Grossman (1989) allows one to include cosegregation. The resulting covariance matrices, however, can be nonpositive definite, which necessitates bending with the effect that information can be lost (Legarra and Fernando 2009). Furthermore, accommodating dominance and epistasis is not straightforward with linear models, especially for crossbred data. In contrast with mixture genetic models, genetic covariance matrices do not enter into the analysis, and accommodating dominance and epistasis is more straightforward (Goddard 1998; Pong-Wong et al. 1998; Stricker and Fernando 1998; Du et al. 1999; Du and Hoeschele 2000; Hoeschele 2001; Yi and Xu 2002; Pérez-Enciso 2003; Yi et al. 2003, 2005).Mixture model analyses, however, are more computationally demanding because the unknowns to be estimated in these analyses include the effects of unobservable QTL genotypes. In linear model analyses, in contrast, it is effects of observable marker genotypes that are estimated. The mixture model analysis can be thought of as a weighted sum of linear model analyses corresponding to each possible state for the unobservable QTL genotypes, where the weights are the probabilities of the QTL genotype states conditional on the observed marker genotypes and trait phenotypes. In practice, the analysis needs to consider all possible haplotypes at the markers also because even when all marker genotypes are observed, some of the marker haplotypes may not be known. As a result, the computational burden of these analyses stems from the number of unknown genotype and haplotype states that need to be summed over being exponentially related to the number of individuals in the pedigree and the number of loci.It can be shown that conditional on the genotypes of their parents, genotypes of offspring are independent of the genotypes of all their ancestors. This conditional independence can be exploited to efficiently compute the weighted summation in the mixture model analysis, provided the pedigree is not too complex (Lauritzen and Sheehan 2003). In genetics, this strategy is called peeling (Elston and Stewart 1971; Cannings et al. 1978) and is equivalent to variable elimination in graphical models (Lauritzen and Sheehan 2003). This approach, however, becomes infeasible when the pedigree is complex and the number of loci is large. Another strategy for analysis of mixture models is based on using Markov chain Monte Carlo (MCMC) theory to draw samples of QTL genotypes and marker haplotypes conditional on the observed marker genotypes and trait phenotypes. Pérez-Enciso (2003) developed an MCMC-based Bayesian analysis for a mixture genetic model that uses information from both LD and cosegregation to fine map a single QTL, but this approach becomes computationally infeasible for whole-genome analyses without approximations.In this article, we investigate a two-stage, approximate analysis that uses information from both LD and cosegregation. In the first stage, ordered genotypes of markers are sampled conditional only on the observed, unordered marker genotypes, ignoring information from the trait phenotypes. These samples are drawn using a Gibbs sampler with overlapping blocks (Thomas et al. 2000; Abraham et al. 2007) in which peeling is performed within a block while conditioning on variables outside the block. From these samples, founder haplotype probabilities and segregation probabilities for the QTL, also called probabilities of descent of QTL (PDQs) alleles, are calculated. In the second stage, these probabilities are used to sample QTL genotypes conditional on the trait phenotypes. In this analysis, information from LD is incorporated by allowing the QTL allele frequencies in founders to be dependent on the marker haplotypes, and information from cosegregation is incorporated by using the PDQs from the first stage to sample QTL alleles in nonfounders. The approximation comes from ignoring trait phenotypes in sampling ordered marker genotypes. A major advantage of the two-step approach is that markers have to be sampled only once and can then be used to analyze all quantitative traits with a mixture model.The objective of this study is to test the hypothesis that this approximation is negligible given high-density SNPs. To test this hypothesis, results from the two-stage, approximate analysis are compared to a full-Bayesian analysis that does not ignore the information from the trait phenotypes in sampling the ordered marker genotypes. The full-Bayesian approach was selected, because it is considered to be the ideal statistical model as it accounts for all uncertainties (Hoeschele 2001). Because the full-Bayesian approach is computationally too demanding for application to GS, the approximate and full-Bayesian analyses are used to fine map within a simulated chromosomal region that is known to contain a QTL to make the comparison computationally feasible. If the consequences of ignoring trait phenotypes to sample ordered marker genotypes are negligible, further research to apply mixture genetic models to GS and comparisons with linear models are justifiable.     

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.     

Efficient Test Molecule Sampling in Molecular Simulation

Abstract

Excluded volume map sampling (EVMS) is a particularly efficient means of performing test molecule sampling to estimate or impose chemical potential in molecular simulations. This paper discusses the motivation and applications of excluded volume map sampling, presents computer code demonstrating its implementation, and gives an example of its application.     

Inference on Collapsibility in Generalized Linear Models

GREENLAND and MICKEY (1988) derived a closed-form collapsibility test and confidence interval for IxJxK contingency tables with qualitative factors, and presented a small simulation study of its performance. We show how their method can be extended to regression models linear in the natural parameter of a one-parameter exponential family, in which the parameter of interest is the difference of “crude” and “adjusted” regression coefficients. A simplification of the method yields a generalization of the test for omitted covariates given by HAUSMAN (1978) for ordinary linear regression. We present an application to a study of coffee use and myocardial infarction, and a simulation study which indicates that the simplified test performs adequately in typical epidemiologic settings.     

Gene Level Meta-Analysis of Quantitative Traits by Functional Linear Models

Meta-analysis of genetic data must account for differences among studies including study designs, markers genotyped, and covariates. The effects of genetic variants may differ from population to population, i.e., heterogeneity. Thus, meta-analysis of combining data of multiple studies is difficult. Novel statistical methods for meta-analysis are needed. In this article, functional linear models are developed for meta-analyses that connect genetic data to quantitative traits, adjusting for covariates. The models can be used to analyze rare variants, common variants, or a combination of the two. Both likelihood-ratio test (LRT) and F-distributed statistics are introduced to test association between quantitative traits and multiple variants in one genetic region. Extensive simulations are performed to evaluate empirical type I error rates and power performance of the proposed tests. The proposed LRT and F-distributed statistics control the type I error very well and have higher power than the existing methods of the meta-analysis sequence kernel association test (MetaSKAT). We analyze four blood lipid levels in data from a meta-analysis of eight European studies. The proposed methods detect more significant associations than MetaSKAT and the P-values of the proposed LRT and F-distributed statistics are usually much smaller than those of MetaSKAT. The functional linear models and related test statistics can be useful in whole-genome and whole-exome association studies.