Comparison of the sensitivities of similar independent and non-independent experiments

Abstract

SUMMARY The comparison of sensitivities is considered in detail for a random effects model in which the different methods of measurement are applied to the same experimental units. An exact distribution is obtained and tabulated and approximations to it considered. Other applications of the results are outlined. The problem of comparing two or more methods of measuring the effects of experimental treatments arises in various fields of research. Cochran (1943) discussed this problem in considerable detail under the assumption that analysis of variance techniques were applicable. He suggested that a comparison of the sensitivities of two scales of measurement, or two experimental techniques, should depend both on the magnitudes of treatment effects and the experimental errors associated with the measurement scales, or experimental techniques. A method of measurement for which the differences between treatment effects show up well, and which, at the same time, has a relatively small error variance, will be considered a sensitive method. In this paper we distinguish between three types of situation in experimentation: (i) Type 1. The different methods of measurement are applied to the same experimental units. This can be done whenever not more than one method affects the experimental units. The results of the methods will be correlated. (ii) Type 11. The different methods of measurement are applied to separate subsamples of the same experimental units. Here only the 'treatment effects' for the methods will be correlated. (iii) Type 111. The different methods of measurement are applied to independent experimental units. Schumann & Bradley (1957, 1959) considered the type III situation. Following Cochran's line of thought, they effected a comparison of the sensitivities of two experiments through a comparison of two variance ratios. Their work was restricted to independent experiments and they were mainly concerned with similar experiments in which the variance ratios compared had identical degrees of freedom. They calculated critical values of the ratios of two independent F's to be used in hypothesis tests to compare sensitivities. Schumann & Bradley assumed that analysis of variance techniques were applicable and considered both model I (fixed effects model) and model II (random effects model) of the analysis of variance. Dar (1962) derived a normal approximation to the Schumann & Bradley density function for the ratio of two F's. He showed that the logarithm of the ratio was approximately normally distributed for large degrees of freedom. He also extended the theory to obtain an approximate chi-square test for the comparison of the sensitivities of several