Factorial Experiments for Random Effects and Mixed Models In a two-factor experiment with random effects, we have the model for i = 1, 2, …, a; j = 1, 2, …, b; and k = 1, 2, …, n, where the A , B ,

(AB) , and ϵ are independent random variables with means 0 i j ij ijk and variances , and σ , respectively. The sums of squares for random effects experiments are computed in exactly the same way as for fixed effects experiments. We are now interested in testing hypotheses of the form where the denominator in the f-ratio is not necessarily the mean square error. The appropriate denominator can be determined by examining the expected values of the various mean squares. These are shown in Table 14.14.

Table 14.14: Expected Mean Squares for a Two-Factor Random Effects Experiment 2

From Table 14.14 we see that and are tested by using in the denominator of the f-ratio, whereas is tested using s in the denominator. The unbiased estimates of the variance components are Table 14.15: Expected Mean Squares for a Three-Factor Random Effects Experiment The expected mean squares for the three-factor experiment with random effects in a completely randomized design are shown in Table 14.15. It is evident from the expected mean 2

squares of Table 14.15 that one can form appropriate f-ratios for testing all two-factor and three-factor interaction variance components. However, to test a hypothesis of the form there appears to be no appropriate f-ratio unless we have found one or more of the two-factor interaction variance components not significant. Suppose, for example, that we have compared

(mean square AC) with (mean square ABC) and found to be negligible. We could then argue that the term should be dropped from all the expected mean squares of Table 14.15;

then the ratio provides a test for the significance of the variance component . Therefore, if we are to test hypotheses concerning the variance components of the main effects, it is necessary first to investigate the significance of the two-factor interaction components. An approximate test derived by Satterthwaite (1946; see the Bibliography) may be used when certain two-factor interaction variance components are found to be significant and hence must remain a part of the expected mean square.