Paul L. Cornelius

Approximate F-tests of multiple degree of freedom hypotheses in generalized least squares analyses of unbalanced split-plot experiments

Approximate t-tests of single degree of freedom hypotheses in generalized least squares analyses (GLS) of mixed linear models using restricted maximum likelihood (REML) estimates of variance components have been previously developed by Giesbrecht and Burns (GB), and by Jeske and Harville (JH), using method of moment approximations for the degrees of freedom (df) for the tstatistics. This paper proposes approximate Fstatistics for tests of multiple df hypotheses using one-moment and two-moment approximations which may be viewed as extensions of the GB and JH methods. The paper focuses specifically on tests of hypotheses concerning the main-plot treatment factor in split-plot experiments with missing data. Simulation results indicate usually satisfactory control of Type I error rates.

Using the shifted multiplicative model to search for “separability” in crop cultivar trials

SummaryThe shifted multiplicative model (SHMM) is used in an exploratory step-down method for identifying subsets of environments in which genotypic effects are “separable” from environmental effects. Subsets of environments are chosen on the basis of a SHMM analysis of the entire data set. SHMM analyses of the subsets may indicate a need for further subdivision and/or suggest that a different subdivision at the previous stage should be tried. The process continues until SHMM analysis indicates that a SHMM with only one multiplicative term and its “point of concurrence” outside (left or right) of the cluster of data points adequately fits the data in all subsets. The method is first illustrated with a simple example using a small data set from the statistical literature. Then results obtained in an international maize (Zea mays L.) yield trial with 20 sites and nine cultivars is presented and discussed.

A shifted multiplicative model cluster analysis for grouping environments without genotypic rank change

SummaryThe shifted multiplicative model (SHMM) is used with a cluster method to identify subsets of sites in an international maize (Zea mays L.) trial without genotypic rank-change. For cluster analysis, distance between two sites is defined as the residual sum of squares after fitting SHMM with one multiplicative term (SHMM1) if SHMM1 does not show genotypic rank-change. However, if SHMM1 does show genotypic rank-change, the distance between two sites is defined as the smaller of the sums of squares owing to genotypes within each of the two sites. Calculation of distance between two sites is facilitated by using the site regression model with one multiplicative term (SREG1), which can be reparameterized as SHMM1 when only two sites are considered. The dichotomous splitting procedure, used on the dendrogram obtained from cluster analysis, will first perform SHMM analyses on each of the last two cluster groups to join (end of the dendrogram). If SHMM1 does not give an adequate fit, the next step is to move down the branches of the tree until groups of sites (clusters) are found to which SHMM1 provides an adequate fit and primary effects of sites are all of the same sign. Five final groups of sites to which SHMM1 provides an adequate fit and primary effects of sites are all of the same sign were obtained. The procedure appears to be useful in identifying subsets of sites in which genotypic rank-change interactions are negligible.

Shifted multiplicative models for nonadditive two-way tables

This paper presents analysis of a two-way table of data using a shifted multiplicative model (SHMM) of the form The least squares estimates of the multiplicative terms are obtained from singular value decomposition of the matrix where but the least squares estimate of the shift parameter depends on estimates of parameters contained in the multiplicative terms. The sum of squares can be minimized as a function only of for which iterative Newton-Raphson and generalized En algorithms are developed. Expectations of sums of squares owing to sequentially increasing the number, t, of multiplicative terms (presented in an AMOVA format) were obtained by Monte Carlo simulation for the case where errors are i.i.d.N(0,σ2) and all The analysis is illustrated with several examples from the literature.

Estimation of general linear-bilinear models for two-way tables

We define the General Linear-Bilinear Model (GLBM) for data arranged as a r×c table as . This includes linear-bilinear models known as Additive Main Effects and Multiplicative Interaction, Rows Regression, Columns Regression, and Shifted Multiplicative models as special cases, but further allows for inclusion of regression on covariates as additional linear terms and for estimation of missing cells. A GLBM is defined as “balanced” if least squares estimates of its linear effects are free of the bilinear effects. A closed form least squares solution exists if the GLBM is balanced or if and is of rank one for all k, where q is the number of linear effects fitted within each (and every) row. In all GLBMs, the least squares estimates of the multiplicative terms are obtained by singular value decomposition of the matrix A of deviations but, if the GLBM is unbalanced, solutions for the depend on the decomposition to be obtained. For such cases, iterative Newton-Raphson and generalized EM algorithms are develope...

Properties of components of covariance of inbred relatives and their estimates in a maize population

SummaryProperties of three parameterizations, denoted as the C-model, D-model and Q-model, for covariances of inbred relatives under assumptions of no linkage or epistasis are explored and compared. Additive variance in an inbred population with inbreeding coefficient F, σ2AF=(1+F)σ2A where σ2A is additive variance in a panmictic population, if Q-model parameters Qxx and Qxy are both zero. Conditions sufficient for this to hold are presented in terms of gene frequencies and dominance contrasts (homozygotes vs. heterozygotes). Some other properties and potential uses of estimates of components in the models are also discussed. Estimates of components in the D-model and Q-model were calculated from a maize (Zea mays L.) study from which estimates of components in the C-model were previously published. Of particular interest were the covariance (Qxy) of effects of alleles at complete homozygosity with “inbreeding depression effects”, the covariance (D1) of additive effects at panmixia with inbreeding depression effects and the within-locus variance (D2, alias Qxx) of inbreeding depression effects. Estimates of Qxy, D1, and D2 were small and nonsignificant in most cases. For ear height in the second year of the study, D2 appeared to be a major component. In some cases, results were obtained which had contradictory implications (negative D2 coupled with positive Qxy or D1, and positive D2 coupled with negative σ2D). A negative estimate of one or the other of σ2D or σ2A was obtained in one of the two within-year analyses for every character. Problems in getting realistic results were thought to be owing to excessive multicollinearity among the coefficients of the components in the expectations of the covariances of the kinds of relatives included in the study. Implications for future studies of this kind are discussed.

Using linear-bilinear models for studying gene expression × treatment interaction in microarray experiments

In microarray experiments, the global and the specific gene expression in the two-way table of gene x treatments (or tissues) can be studied using linear-bilinear models that incorporate the main effects of genes (G), treatment (T), and gene x treatment interaction (G x T). The plot of the first two axes obtained from the singular value decomposition of the bilinear (multiplicative) term of these models (biplot) facilitates the interpretation of the gene expression patterns. In this study, two microarray datasets were used to illustrate how two linear-bilinear models, the additive main effect and multiplicative interaction (AMMI) and the treatment regression model (TREG) and their biplots can be used to determine the overall gene expression pattern across treatments (or tissues) and for specific treatments. Dataset 1 had 5,339 genes and the objective was to identify genes with modified expression during maize (Zea mays) seed development in response to different parental ploidy levels. In Dataset 2, the aim was to study gene expression in 15 tissue samples with different levels of development of breast cancer when compared with the expression of the genes in noninfected tissues. The results from the analyses of Dataset 1 showed that the biplots of the AMMI and TREG models allow identification of subsets of genes and treatments with noncrossover G x T interaction or with important levels of crossover G x T. Results from Dataset 2 showed that the TREG model and its biplot facilitates the identification of genes with high expression in all tumor cells. Also, the TREG biplots allowed identification of subsets of genes with a low level of gene x tissue crossover interaction.

Hypothesis testing for components of the shifted multiplicative model for a nonadditive two-way table

To determine the number of multiplicative components in a shifted multiplicative model (SHMM) for a r × c table (c≤r) given by likelihood ratio tests of Ho: λt = 0 versus Ha:λt≠0, λt+1=0 are developed in a stepwise procedure with t successively set equal to 1,…,p − 1, where p=min(r−1,c). For t = 1, the empirical distribution and percentage points of the likelihood ratio test statistic are obtained through a Monte Carlo simulation study. Distribution of the likelihood ratio test statistic is also approximated by a Beta distribution. For t ≥ 2, a method analogous to one developed by Marasinghe (1985) and Schott (1986) for the additive main effect and multiplicative interaction (AMMI) model is used, namely, that the tth term is tested as thought it is the first term for a (r−t+1)×(c−t+1)table. Type I error rates for tests of λ2 and λ3 agreed closely with nominal significant levels 0.01,0.05 and 0.10 if previous λ values were sufficiently large, but tended to be conservative otherwise. The procedure is illust...

Comparison of the isozyme variation in tall fescue parents and their somaclones

SummaryThis study was conducted using the isozymes ACP-1, ADH-1, GOT-2, GOT-3, MDH, 6-PGD-1 and PGI-2 to: a) compare isozyme banding patterns of tall fescue somaclones with parents and b) correlate tissue culture-induced chromosome abnormalities with variant banding patterns. The 174 somaclones were grouped into seven categories based on their meiotic analyses and time of regeneration from culture. Differences in isozyme frequency between categories compared by chi-square tests were greatest for MDH, 6-PGD-1 and PGI-2, and least for ACP-1. The most significant differences in frequency were found between somaclones and parents. In comparisons of somaclone categories, the most different isozyme distributions were between the early vs. late regenerated somaclones. No significant differences in isozyme frequencies were found between all 42-chromosome somaclones vs. aneuploid somaclones and the three somaclone groups (42-normal, 42-abnormal, aneuploid) compared to each other. This study suggests that culture-induced isozyme variation alters the distribution of the isozyme phenotypes, but is not directly correlated with chromosome abnormalities.

Genotype × Environment Interacion in multi-environment Trials using shrinkage factors for ammi models

Shrinkage factors applied to the additive main effects and multiplicative interaction (AMMI) models improve prediction of cultivar responses in multi-environment trials (MET). Estimates of shrinkage factors based on the eigenvalue partition (EVP) method may get a further improvement in the predictions of cell means. Objectives of this work were: (1) to compare the EVP-based shrinkage method with unshrunken AMMI, best linear unbiased predictor (BLUP) and other shrunken method (herein named CCC), when they were applied to five maize MET and simulation data; (2) to assess by cross validation the equation which estimates the standard error of predicted means (SEPM) based on the EVP theory; (3) to estimate the genotype × environment interaction (GEI) variance components after applying the EVP shrinkage method to the five maize MET. Empirical data of five maize MET and simulation data were used for cross validation of the methods using the root mean square predictive difference (RMSPD) criterion. The RMSPD of the shrunken EVP predicted cell means was generally smaller than those of the other methods, suggesting that the EVP method was generally better predictor than the other methods. The truncated AMMI was the worst among the four methods studied. The EVP-based equation, which predicts the SEPM, was a good predictor as determined by the RMSPD cross validation criterion, with the advantage that it does not need one replication for validation. Estimates of mean squares, and GEI and error variances associated with the GEI effects were smaller for the shrunken EVP predicted effects than for the original data.