J. T. Webster

Latent Root Regression Analysis

Least squares estimates of parameters of a multiple linear regression model are known to be highly variable when the matrix of independent variables is near singular. Using the latent roots and latent vectors of the “correlation matrix” of the dependent and independent variables a modified least squares estimation procedure is introduced. This technique enables one to determine whether the near singularity has predictive value and examine alternate prediction equations in which the effect of the near singrtlarity has been removed from the estimates of the regression coefficients. In addition a method for performing backward elimination of variables using standard least squares or the modified procedure is presented.

Regression analysis and problems of multicollinearity

Multicollinearity or linear dependence among the vectors of regressor variables in a multiple linear regression analysis can have sever effects on the estimation of parameters and on variables selection techniques. This expository paper examines the sources of multicollinearity and discusses some of its harmful affects. Several methods proposed in the literature for detecting multicollinearity and dealing with the associated problems are also presented and discussed.

The Use of an F-Statistic in Stepwise Regression Procedures

This is an expository paper, pointing out explicitly the pseudoness of the “F-statistic” used in stepwise procedures for determining the independent variables to be used in a linear prediction equation. Unfortunately this pseudoness prevents one from obtaining any probabilistic measure of the goodness of the final prediction equation. The use of the distribution of an order statistic is discussed as an aid to understanding the problem as well as furnishing a “better” (although still unsatisfactory) approach.

A Comparison of Least Squares and Latent Root Regression Estimators

Miilticollinesrity among the columns of regressor variables is known to cause severe distortion of the least squares estimates of the parameters in a multiple linear regression model. An alternate method of estimating the parameters which was proposed by the authors in a previous paper is Latent Root Regression Analysis. In this article several comparisons between the two methods of estimation are presented. The improvement of Latent Root Regression over ordinary least squares is shown to depend on the orientation of the parameter vector with respect to a vector defining the multicollinearity. Despite this dependence on orientation, the authors conclude that witch multicollinear data Latent Root, Regression Analysis is preferable to ordinary least squares for parameter estimation and variable selectJion.

Density functions of the bivariate chi-square distribution

The purpose of this paper is to present a practical approach to solving simultaneous testing and estimation problems faced by the experimenter. New forms of the bivariate Chi-square distribution are introduced which afford straightforward computer programming. An approximation which reduces the general type of dependency to a specific form is suggested, reinforced by strong theoretical justification. Finally, the bivariate Chi-square density function is used to find the density function of a linear combination of independent Chi-square random variables

A Comparison of Some Approximate F-tests

When an experiment is run with a factorial layout and some of the factors are random effects, there may not be an exact test for some effect of interest. This paper considers three statistics that could be used to test that hypothesis using a ratio of linear combinations of independent Chi-square statistics. The common case, utilizing four Chi-square statistics, is examined for a number of con-figurations of nuisance parameters. Both the power and the probability of a type I error are used in the comparison. Two statistics appear to be equally good over a large region and, in certain situations, the statistics involving the subtraction of Chi-square statistics is shown to be more stable.

On Estimating Approximate Degrees of Freedom

Abstract Inference in linear models with multiple random effects is often complicated by variance estimators with distributions that are not exact multiples of chi-squared variates. Using a result attributed to Satterthwaite, one can approximate these estimators to a chi-square with appropriate degrees of freedom. These degrees of freedom, which can be used to find percentiles for tests of hypotheses and confidence intervals, are generally functions of the unknown variance components and hence must be estimated. This article investigates approximate degrees of freedom estimators and proposes a class of alternatives to the one in general use.

MINQE for the one-way classification

The minimum norm quadratic estimator (MINQE), without the condition of unbiasedness, is given for the effect variance of a one-way classification. The computational form is relatively simple with no iteration necessary, and the prior weight is given as a function of the harmonic mean of the numbers of readings per classification. A comparison is made of the mean squared error (MSE) of MINQE and the estimators of Swallow and Monahan (1984) for the layouts of that article. The MSE of MINQE is shown to be smaller than the MSE of these estimators when the effect variance is greater than the error variance. A discussion is also given illustrating a desirable property of a smaller MSE even in the presence of nontrivial bias.

The Singular-Value Decomposition as a Tool for Solving Estimability Problems

Abstract The intent of this article is to present a straightforward method of investigating the estimability problems of linear models. As a teaching tool the value is twofold: (a) the problems can be structured and solved by standard matrix multiplication, and (b) the uniqueness (or absence of uniqueness) of a solution is explicitly demonstrated to the student. The approach is a direct application of the singular-value decomposition of matrix. As an intermediate step, a useful representation of the generalized inverse of a matrix is formulated.

A Comparison of Four Designs for Estimating the Variances of Random Effects in a Two-Way Classification

Foltr designs are compared for estimating the variances of random effects in a two-wa) classification without interaction. The four designs are the Disjoint Rectangle, the Generalized L-shaped, the Generalized Staggered and the Balanced Incomplete Block. A small variance of the estimate is the criterion of goodness and an estimate whose variance is a function of only the error variance and the variance of that effect is used. This paper shows that the choice of type of design (once the L-shaped is eliminated) is of minor importance as compared to the choice of parameters within the design. The optimum selection will depend upon the ratio of the effect variance to the error variance. This ratio will rarely be known, however some idea as to the region can lead to a satisfactory choice of design.