Mixed Model Formulations for Multi-Environment Trials

The objectives of this paper are to discuss the statisti- cal and genetic issues concerned with using the mixed When studying genotype environment interaction in multi-envi- linear model in a plant breeding context, illustrate the ronment trials, plant breeders and geneticists often consider one of the effects, environments or genotypes, to be fixed and the other to application of the two formulations using a wheat (Triti- be random. However, there are two main formulations for variance cum aestivum L.) breeding example (with balanced data), component estimation for the mixed model situation, referred to as and make some recommendations and conclusions. the unconstrained-parameters (UP) and constrained-parameters (CP) formulations. These formulations give different estimates of genetic AGRICULTURAL BACKGROUND correlation and heritability as well as different tests of significance for the random effects factor. The definition of main effects and interac- Cooper et al. (1995) hypothesized that regional testing tions and the consequences of such definitions should be clearly under- strategies in a plant breeding program could be im- stood, and the selected formulation should be consistent for both proved by accommodating the effects of genotype fixed and random effects. A discussion of the practical outcomes of environment interactions to maximize the response to using the two formulations in the analysis of balanced data from multi- selection. They argued that one way of doing this was environment trials is presented. It is recommended that the CP formu- to identify the set of selection environments most rele- lation be used because of the meaning of its parameters and the vant to the future production environments. If these test corresponding variance components. When managed (fixed) environ- ments are considered, users will have more confidence in prediction for environments can be repeated from year to year, confi- them but will not be overconfident in prediction in the target (random) dence in predicting response in future environments environments. Genetic gain (predicted response to selection in the would be increased. They therefore assessed the scope for target environments from the managed environments) is independent managing environmental conditions at a restricted num- of formulation. ber of sites to provide discrimination among wheat lines for grain yield that matches that in target production en- vironments. W hen studying genotype environment interac- In analyzing data from such a multi-environment test- tion, breeders and geneticists often consider one ing regime, the genotypes can be considered to be a ran- of the two factors to be fixed and the other to be random. dom sample of the lines from the relevant stage of the This results in a linear mixed model. For the fixed effect, breeding program. The managed environments can be all of the levels in the population of parameters are pres- considered to be fixed as they can be repeated over years ent while for the random factor, only a random sample and locations. Hence, a mixed model for the genotype- from the population of levels is obtained. The experi- environment system will be appropriate. However, the menter often wishes to obtain estimates of variance interpretation of experimental results and any inference components to compute genetic correlation, heritability from selection will apply to the target or production estimates, repeatability estimates, genetic advance esti- environments that could be considered to be random. mates, and other related statistics. Several discussions Cooper et al. (1995) argued that a successful breeding of variance component estimation in the mixed model strategy is one that gives a high indirect response to situation have appeared in the literature (Federer, 1955; selection for average yield over the production environ- Cornfield and Tukey, 1956; Scheffe, 1956, 1959; Hock- ments and quantified this using the genetic correlation, ing, 1973; Ayres and Thomas, 1990; Samuels et al., 1991; which measured the similarity of line discrimination be- Fry, 1992; Schwarz, 1993; Searle et al., 1992; Nelder, tween the managed-environment selection regime and 1998; Voss, 1999). Different formulations have been pro- that for average performance in the production envi- posed, with two of these being used most frequently. ronments. This poses a dilemma for the breeder and geneticist as to which formulation to use as they give different esti-