Least Significant Differences for Combined Analyses of Experiments with Two‐ or Three‐Factor Treatment Designs

Abstract Many agronomic experiments with complete factorial treatment designs are conducted in two or more environments. If at least one of the treatment factors is qualitative, the combined analysis may appropriately include pairwise comparisons of various treatment means averaged over environments. Heretofore, however, formulae for estimation of the variances of pairwise mean differences, which are needed for the calculation of least significant differences (or other mean separation procedures), have not been available in either the agronomic or statistical literature. The main purpose of this paper is to provide the formulae for estimation of these variances so that researchers can use them in testing the significance of differences among various treatment means averaged over environments. Emphasis is given to combined analyses of series of experiments in which the effects of experiments are considered to be random; individual experiments are assumed to have complete factorial treatment designs involving either two or three factors of which at least one is qualitative. All treatment effects, including main effects and interactions, are considered to be fixed effects. Formulae applicable to commonly used experimental designs are presented for the estimation of variances for the five categories of pairprise mean differences in two‐factor treatment designs and the 19 categories of pairwise mean differences in three‐factor treatment designs. Experimental design considerations and the implications of assuming random (versus fixed) experiment effects as well as the structure of the treatment × experiment interactions are studied. Pooling portions of the treatment × experiment interaction with error mean squares are considered, and suggestions are given for cases where some of the treatment factors are quantitative rather than qualitative.