[378] HETEROGENEOUS ERROR VARIANCES IN SPLIT-PLOT EXPERIMENTS

SUMMARY. In a split-plot experiment, the oommon assumption is that the same error variance applies to all subplot treatmente. This paper is concerned with tests of significance for the departure from equality of the variances for different subplot treatments, and with the estimation of the ratio of a pair of such variances. The methods used may be regarded as extensions of those described by Morgan (1939) and Pitman (1939) in connexion with the problem of comparing the variances of two correlated and normally distributed measurements. In particular, Pitman's method is extended to provide confidence limits for the variance ratio. Suppose that visual acuities were measured, for right eye and left eye separately, on a sample of men, and that the possibility of a difference between of and of, the variances for the two eyes, was to be investigated. If the distribution of the original measuremente is bivariate normal, then so is that of their sum and their difference. Morgan and Pitman noted that the correlation coefficient in the latter distribution depends upon the difference (o ^ —of), being zero if and only if 07 = of. Consequently, they proposed to test the hypothesis of equal variance by the standard significance test for a sample correlation coefficient. In an agricultural experiment of split-plot design the variance for a subplot may depend upon the treatment it receives. In particular this is liable to happen if one subplot treatment increases the yields by a considerable amount. The simple procedure for testing the equality of two variances outlined above would allow for the correlation between the subplots of each whole-plot, but is no longer applicable, because of treatment and block classifications. The more general methods of this paper enable tests of significance and confidence limits to be calculated. An example of a simple split-plot experiment is given, with a discussion of the standard errors for the various treatment comparisons. 1