Pseudo-heritability
by
Jeff Wood
The Graduate School, ANU

... co-efficient of heritability, which I regard as one of those unfortunate short-cuts, which have often emerged in biometry for lack of a more thorough analysis of the data. (Fisher, 1951)

Heritability continues to be a popular concept in spite of repeated criticisms, exemplified by the above quotation. However there are many field experiments, such as row and column designs, where classical definitions of heritability cannot be used.

First consider a randomized block experiment with a number of genotypes replicated in r blocks. Assume that the replicate effects are fixed. Suppose that the genotypic and error components of variance are \sigma_g² and \sigma_e². Heritability is defined as the fraction of phenotypic variance which is genetic. For a crop such as wheat it is often impractical or inappropriate to measure individual plants and heritability based on genotype means is defined as

h2=\sigmag2/(\sigmag2+\sigmae2/r). \labeleq:h2

(\theequation)

If effects are centred the ratio of the estimates when genotypes are taken as random to the estimates when they are fixed is h², and the variance of the predicted difference of two effects is

vp=2h2\sigmae2/r.

\sigmae2/r=\sigmag2(1-h2)/h2.

Hence

vp/2=\sigmag2(1-h2),

and

h2=1-vp/(2\sigmag2). \labeleq:h2x

(\theequation)

In experiments with more complex designs it is often not clear what should be substituted for \sigma_e² in Equation , but Equation can be used as a basis for pseudo-heritabilities. Since v_p is usually not the same for all pairs of effects, some compromise value must be used. For example, Cowling et al (1997) used 2[`v]<sub>a</sub> where [`v]_a is the average variance of the predictors of genotype effects.

References

Cowling, W.A. Sweetingham, M.W. Diepeveen, D. and Cullis, B.R. (1997) Heritability of resistance to brown spot and root rot of narrow-leafed lupins caused by Pleichota setosa (Kirchn.) Hughes in field experiments. Plant Breeding, 116, 341-345

Fisher, R.A. (1951) Limits to intensive production in animals. British Agricultural Bulletin, 4, 217-218.

Date received: September 6, 2002