General Linear Model for a Latin Square

In design of experiment, a Latin square is a three-factor experiment in which for each pair of factors in any combination of factor values occurs only once.

Consider the following Latin Square,

where rows correspond to 4 values of factor I, columns – to 4 values of factor II, and letters to 4 values of factor III.

A general linear model for this experiment is

where Y_ijk are observed values of the dependent variable, a_i, i=1,4 are effects corresponding to the 4 values of factor I, b_j – of factor II, g_k – of factor III, e_ijk are random variations.

Note: the Latin square is appropriate only if effects of all three factors are additive – that is, all interaction effects are zero.

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