A Simple Method for the Elimination of Individual Trends in the Analysis of Balanced Sets of Latin Squares

Abstract

The usual model for the analysis of Latin squares assumes that there are no interactions between rows, columns and treatments. In biological or psychological experiments where several treatments are given in sequence to each human or animal subject, Latin squares have been used to eliminate variation of response with time as well as inter-subject variation. If row X column interactions are present the estimates of treatment effects are still valid because of randomization but the precision of the estimates is reduced (Gaito [1958]). In particular, the assumption that change of response with time is the same for all subjects is usually very doubtful. C. P. Cox [1958a] has proposed an alternative model which fits individual polynomial regressions on time for the response of each subject. This is an approach which is likely to be useful in a variety of situations but the model has two disadvantages. The computation is considerably more arduous than for the usual Latin square and treatment differences are not estimated with equal precision. If extra replication is possible, the present paper shows how the computations are considerably simplified and equal precision is obtained (except for 6 X 6 squares) when a balanced set of squares is used. Cox [1958b] has stated that the same principle of fitting individual regressions can be applied when the Latin square design restriction is relaxed and that the use of balanced treatment sequences simplifies the analysis. In the present paper suitable sets of squares up to 7 X 7 with the information necessary for computation are tabulated and a numerical example is given to demonstrate the computational procedure.