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The basic two-fixed-factor classification model is:
, where i and j are the levels of the factors and r is the repeated data for the factor ( i, j) combination. The factor (i, j ) combination is commonly called the cell.
, is the estimator for the response for the cell ( i, j), averaged over the r-repeated runs.
The usual assumption is that the cell averages for levels of one factor, averaged over the levels of the other factor, are all equal. Alternatively, the difference between any pair of cell means (a contrast) is hypothesized to be zero. Whenever a cell has no data (the data is not complete), the comparison cannot be made and the hypothesis cannot be satisfied. If there are a different number of repeated runs for each cell, the variance is not uniform and the design is unbalanced. Interaction effects are evaluated by contrasts between pairs of pairs. The total number of independent contrasts that may be formed cannot exceed the number of degrees of freedom (runs - 1) available for the estimates.