Confidence Intervals for a Variance Ratio, or for Heritability, in an Unbalanced Mixed Linear Model

Abstract

A procedure is presented for constructing an exact confidence interval for the ratio of the two variance components in a possibly unbalanced mixed linear model that contains a single set of m random effects. This procedure can be used in animal and plant breeding problems to obtain an exact confidence interval for a heritability. The confidence interval can be defined in terms of the output of a least squares analysis. It can be computed by a graphical or iterative technique requiring the diagonalization of an m X m matrix or, alternatively, the inversion of a number of m X m matrices. Confidence intervals that are approximate can be obtained with much less computational burden, using either of two approaches. The various confidence interval procedures can be extended to some problems in which the mixed linear model contains more than one set of random effects. Corresponding to each interval procedure is a significance test and one or more estimators.

Some Bayesian and Non-Bayesian Procedures for the Analysis of Comparative Experiments and for Small-Area Estimation: Computational Aspects, Frequentist Properties, and Relationships

“…These expressions, which are due to Dempster, Selwyn, Patel, and Roth (1984), Harville and Fenech (1985), Call an an (1989), andHarville (1989), can be used to advantage in the Bayesian approach, which is to be discussed in Chapter 3, as well as in the frequentist approach.…”

“…Note that if a set of covariates is available for each of the castings, or for the individual bars, we might instead employ a model of the form (1.5). Harville and Fenech (1985), and Harville (1989) discussed a prediction problem that arises in animal breeding. They presented data, from five distinct population lines of sheep, consisting of the birth weights of 62 singlebirth male lambs (see Table 1 Note that this prediction problem is very similar to the small-area estimation problem.…”

Some Bayesian and Non-Bayesian Procedures for the Analysis of Comparative Experiments and for Small-Area Estimation: Computational Aspects, Frequentist Properties, and Relationships

“…The sum of squares Ss can be reexpressed as Ss = and hence Se can be reexpressed as Se = y'{I -P~ ^1=1 (Harville and Fenech, 1985, sec 3.8).…”

Reliable confidence intervals in quantitative genetics: narrow-sense heritability

“…The sampling variance of broad-sense or familymean heritability, both calculated from two variance components, has received considerable attention, especially for balanced data (Osborne and Paterson 1952;Knapp et al 1985Knapp et al , 1989Knapp and Bridges 1987;Koots and Gibson 1996;Visscher 1998;Burch and Harris 2005). For unbalanced data and under normality assumptions, Harville and Fenech (1985) developed a method to calculate exact confidence intervals on a ratio of two variance components, which allows one to give exact confidence intervals for broad-sense heritability. Graybill and Wang (1979) described a method for calculating confidence intervals for a ratio involving three variance components, based on balanced data, but it has been used only rarely to calculate confidence intervals for narrow-sense heritability (but see Collaku and Harrison 2005).…”

“…Therefore the variance of a trait can be partitioned into three components, referred to as male r M 2 , female r F 2 , and residual r R 2 (in the animal breeding literature commonly called sire, dam, and residual). Statistical analysis of such twofold nested data must take all three variance components into account; methods for computation of confidence intervals involving just two variance components such as that of Harville and Fenech (1985) are not applicable here.…”