Non-Additivities in a Latin Square Design

Abstract

A brief history of the Latin square design is given. A generalization of the design to the case in which the rows, columns and treatments represented in the experiment are samples from populations of rows, columns and treatments respectively is studied. A possible frame of reference for the interpretation of the experimental results is described.The leads to what is termed a "population model," various population parameters, means and components of variation which are of interest to the experimenter. No assumptions are made about additivity of experimental units and treatments. Results on expectations of mean squares in the analysis of variance are given in terms of quantities (denoted by 2:'s) which result in a concise description and in terms of the components of variation. Biases in the estimation of components of variation by means of the analysis of variance are discussed and assessed. Comparisons of randomized block designs and Latin square designs are given for the general case of non-additive treatments for random, mixed, or fixed population of rows and columns. A generalization of the design is discussed. The mathematical machinery which is used to derive the results is presented briefly. Finally linear estimates and errors of estimates are discussed. One main conclusion of the study is that the Latin square analysis of variance may overestimate the error of treatment comparisons and underestimate the component of variation associated with treatment main effects.

Chapter III: Research Methods: Experimental Design

“…Wilk and Kempthorne (80) couched this in relatively easy language and symbolism, presented finite-model expectations of mean squares from which EMS's for the other models can be derived quickly, and concluded that analyses of variance for Latin square designs may overestimate the error term for treatment comparisons and underestimate the component of variance due to treatment main effects. Nevertheless, when both the Latin square and the randomized block designs are reasonable for a proposed experiment, they recommend the former, tho with caution, because its error term for the treatment mean square will usually be too large.…”

Evaluating Subject-Treatment Interaction When Comparing Two Treatments

“…Earlier results in the literature on this topic deal with testing for the presence of a subject-treatment interaction (i.e., nonadditivity) by testing for its observable consequences (10,11) or with finding transformations of the data so that subject-treatment additivity appears to hold on the transformed scale (12). Earlier results in the literature on this topic deal with testing for the presence of a subject-treatment interaction (i.e., nonadditivity) by testing for its observable consequences (10,11) or with finding transformations of the data so that subject-treatment additivity appears to hold on the transformed scale (12).…”

Inference from Randomized (Factorial) Experiments

“…Wilk and Kempthorne (1957) developed this argument further. In 1957, Kempthorne chaired a six-week I.M.S. Summer Institute on the topic at Boulder, Colorado.…”