Setting confidence limits to genetic parameters estimated by restricted maximum likelihood analysis of North Carolina design II experiments

Received 10 November 1995 Heredity 77 (1996) 476—487 Setting confidence limits to genetic parameters estimated by restricted maximum likelihood analysis of North Carolina design II experiments TREVOR HOHLS Department of Genetics, University of Natal, Private Bag X01, Scottsville, 3209, South Africa The properties of the sampling distributions of additive and nonadditive genetic variances and narrow-sense heritabilities were investigated by means of a computer simulation study. Commonly used formulae for the standard errors of these quantities have been tested by means of the coverage probabilities of confidence intervals set using them. Approximate confidence intervals were set assuming a Normal distribution. Formulae that are based on the 'delta' technique, and are shown to produce reliable estimates of the standard errors of additive and nonadditive genetic variances and narrow-sense heritabilities, are provided. Confidence intervals for estimates of the narrow-sense heritability, based on the F-distribution, were tested and found to be conservative. A worked example provides an empirical appraisal of the methods of setting confidence limits to genetic parameters estimated by restricted maximum likelihood analysis of North Carolina design II experiments. Suitable statistical software is suggested for obtaining the standard errors of the genetic parameters by the 'delta' technique. Keywords: computer simulation, error rates, genetic variance components, narrow-sense heritability, quantitative genetics, statistical software procedures. from selection. The narrow-sense heritability (h2) of Introduction The North Carolina designs I and II (Comstock & Robinson, 1948) are frequently used in plant breeding experiments to obtain estimates of genetic variance components and heritability. The popularity of these designs is because of their amenability to standard statistical procedures and the ease with plants may be estimated according to several methods, depending on the genotypic structure of the population to which the plants belong, the type of artificial selection used, and the linear model (Hanson, 1963; Knapp et al., 1985; Nyquist, 1991). Heritabilities also differ, depending on whether the reference unit is an individual plant or a plot mean, i.e. heritability on a progeny mean basis (Hanson, which interpretations of variance components can be made in terms of covariances of relatives (Nyquist, 1963). Appropriate confidence intervals must be given to indicate the accuracy of the estimators of the genetic variance components and h2. Standard errors (SEs) and confidence limits are frequently misleading or 1991). In the North Carolina design II (Comstock & Robinson, 1948) or factorial mating design, a.b fullsib families are produced by crossing a females to b males. The North Carolina design II produces both half-sib and full-sib families. Half-sib relationships omitted (Knapp et al., 1985). Hallauer & Miranda (1988) provide formulae for the variance of additive exist for individuals within each female and male and nonadditive genetic variance components and h2 parental array, whereas each single-cross is a full-sib. on an entry mean basis. These formulae are based Narrow-sense heritability, defined as the ratio of additive genetic variance (VA) to the phenotypic variance (Vp) of individuals in the population, provides a means of predicting the response resulting on Satterthwaite's (1946) approximation for the variance of a variance component. The 'delta' technique (Taylor approximation) has also been used to obtain the standard deviation of a heritability estimate 476 1996 The Genetical Society of Great Britain. CONFIDENCE INTERVALS FOR PARAMETERS OF NC II DESIGNS 477 (Gordon et al., 1972; Magnussen, 1992). Exact confi- dence intervals have been developed for h2 calculated on a full-sib progeny mean basis from a North Carolina design II experiment (Knapp, 1986). The exact confidence intervals are based on the on the 'delta' technique, and SEs derived from formulae adapted from those given by Hallauer & Miranda (1988). Coverage probabilities have been determined for confidence intervals calculated assuming a Normal distribution. The accuracy of F-distribution because the independent ratios of the observed variance components (used in calculating h2) and their expectations, are distributed as mutually independent chi-square variables (Knapp et al., 1985; Knapp, 1986). Magnussen (1992) has questioned the accuracy of this method of setting confidence limits, because a covariance exists between the mean squares used in obtaining h2 when there are shared environments and unbalanced data, and because the degrees of freedom of the chi-squared confidence intervals of h2 based upon the F-distribution has also been evaluated. distributions are approximations (Satterthwaite, Y,yk = +f, + m + (mf) + eij 1946). Maximum likelihood estimates of variance components are biased downwards because they disregard the degrees of freedom used in estimating treatment effects (Thompson & Welham, 1993). Patterson & Thompson (1971) developed the method of restricted maximum likelihood (REML), also known as residual maximum likelihood, to avoid this bias in variance component estimates. REML allows for the analysis of unbalanced data or data with more than one source of variation and is widely used for the analysis of agricultural experiments (Robinson, 1987). Materials and methods The simulation study was based on a 7 x 7 North Carolina design II experiment with 10 observations per cross. The model upon which the analysis was based was the following (1) where: Y1k = kth phenotypic measurement of the cross between inbreds i and j for trait Y; p = population mean; f, = effect of the ith female, f, are N(pf, m = effect of the jth male, m are N(!im, ); (mf)q = interaction effect obtained in the cross between lines i and j, (mf) are N(Umf, a,f), and = within-family variation, N(0, cr). Under the e,Jk random effects model, p, Pm and /2mf (population means of the female and male main effects and interaction effects, respectively) may be taken as zero. For full-sibs (FS), In this article, computer simulations have been used to obtain the sampling distributions of VA, VD and h2, estimated through REML analysis of a 7 x 7 North Carolina design II experiment. The properties of the sampling distributions have been investigated and the empirical SEs of the estimates have been compared with SEs obtained from formulae based (2) and for half-sibs (HS), coy (YJk, Yimi) = = (3) Parameters of particular interest are (assuming completely inbred parents, i.e. an inbreeding coeffi- Table 1 ANOVA showing expected MS for the North Carolina design II experiment used in the simulation study Source d.f. MS Males and Females Females Males Interaction Within-family variation Total (a+b—2) (a—i) Mm+f (b_i) (a — 1)(b — 1) ab(r—1) Mf Mm Mm1 Expected MS +rr+r[(a+b)I (...truncated)