Non-Additivities In A Latin Square Design
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Date
2007
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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 terma of quantities (denoted by S'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.