Evaluation of Simple Methods for Estimating Broad-Sense Heritability in Stands of Randomly Planted Genotypes S. E. Smith,* R. O. Kuehl, I. M. Ray, R. Hui, and D. Soleri ABSTRACT hande, 1957; Nyquist, 1991). This approach is based on the assumption that both random and systematic Inexpensive estimates of broad-sense heritability (BSH) may be environmental variation within a planting containing a valuable in plant breeding. This research evaluated two methods for estimating BSH with data from stands of equidistantly spaced geno- single genotype follows an inverse logarithmic function types. Both methods depend on the assumption that genetic and of the number of individual plants within a plot (x ), environmental contributions to plot variance (plot = group of contigu- where a plot is a group of contiguous plants (Smith, ous plants) change at different rates as plot size changes if genetic 1936). Given a measure of environmental variation on variation is distributed randomly. For the method proposed by V.J. an individual basis (V 1 ), Smith (1936) showed that the Shrikhande, variances among plot means are computed and least- pattern of environmental variation on a plot mean basis squares regression used to estimate environmental and genetic vari- (V x ) in such a planting could be represented by the ances and the change in a plot variance with changes in plot size. The regression coefficient b (termed the heterogeneity coef- other method involves the same approach, but uses a two-parameter ficient) of the function (Smith’s Law): model suggested by G.H. Freeman but not previously used to estimate BSH. Both methods produce biased BSH estimates since genotypic and genotypic  environmental components of variation are insepara- V x = V 1 x b [1] ble. Our objectives were to: (i) develop software to calculate variances for the methods, and (ii) compare BSH estimates generated using In a planting where environmental variation changes these methods with each other and with those from analysis of variance abruptly, phenotypic correlations between neighboring (ANOVA) of data from families grown in the same environment. plants and mean plot variance would decrease rapidly Data were from a perennial herb, Sphaeralcea emoryi Torr. grown as plot size increases and b approaches zero. More con- in Tucson, AZ. Shrikhande’s method produced parameter estimates sistent environmental effects would be associated with with large variances and BSH estimates that averaged 3.6 times larger a more gradual decline in plot variance and b values than those from Freeman’s method. Only BSH estimates from Free- approaching one. man’s method agreed well with those from ANOVA for most traits. Freeman’s method may be useful for rapidly and inexpensively gener- In the case where genetic variation is present and it ating BSH estimates in a variety of situations, especially when tradi- is distributed randomly within a planting, Shrikhande tional genetic analysis are difficult. (1957) suggested that genetic variation among plots within a planting would follow a direct inverse function of plot size and could change at a rate different than G enerating estimates of heritability in plants typi- environmental variation as plot size changed. Therefore, cally involves the evaluation of progenies or clones Shrikhande (1957) expressed phenotypic variance on a of known genetic relationship from populations of inter- plot mean basis for plots of x individuals (V x ) as: est (Namkoong, 1979; Nyquist, 1991). In most crop plants, this usually requires controlled matings and the V x = V g x + V e x b [2] establishment and evaluation of progenies in field trials. In many situations, it may be difficult to produce the where V g and V e represent estimates of genetic and needed genotypes, or to justify the time and expense environmental variances, respectively. To generate required to generate highly accurate estimates. It would estimates of broad-sense heritability (BSH = be useful to have methods that could be used to estimate V g /[V g + V e ]) by Shrikhande’s method, means for traits heritability using data from standard evaluation trials of interest are calculated for all plots of size x = 1 to where samples of genotypes produced from open polli- n (with n typically equal to a maximum of half the total nation are grown. number of plants in the stand) from a planting with Forest geneticists developed a method for generating equidistant spacing among plants. The variance between estimates of broad-sense heritability using data from these plot means is then calculated for each plot size. plantings where the distance between neighboring indi- The parameters V g , V e , and b can be estimated iteratively vidual plants in contiguous rows is equal to that between by Eq. 2 and least-squares procedures (Namkoong and plants within a row (‘‘equidistant spacing’’) (Shrik- Squillace, 1970). This technique for estimating BSH is referred to here as “Shrikhande’s method.” Since geno- S.E. Smith, Dep. of Plant Sciences, R.O. Kuehl, Dep. of Agricultural typic and genotypic  environmental components of and Resource Economics, and D. Soleri, Program in Arid Lands variation cannot be separated by this technique, these Resource Sciences, Office of Arid Land Studies, Univ. of Arizona, BSH estimates may be biased upward and therefore Tucson, AZ 85721; I.M. Ray, Dep. of Agronomy and Horticulture, New Mexico State Univ., Las Cruces, NM 88003; R. Hui, Zephyr should be seen as maximum estimates for heritability Development Corporation, 11 Greenway Plaza, Suite 520, Houston, (Nyquist, 1991). TX 77046. Received 5 June 1997. *Corresponding author (azalfalf@ Shrikhande’s method has been used to estimate BSH ag.arizona.edu). Abbreviations: BSH, broad-sense heritability. Published in Crop Sci. 38:1125–1129 (1998). 1125