Some Consequences When the Assumptions for the Analysis of Variance are not Satisfied

(i) To estimate certain differences that are of interest. In this statement both the words treatment and are used in rather a loose sense: e.g., a difference might be the difference between the mean yields of two varieties in a plant-breeding trial, or the relative toxicity of an unknown to a standard poison in a dosage-mortality experiment. We want such estimates to be efficient. That is, speaking roughly, we want the difference between the estimate arid the true value to have as small a variance as can be attained from the data that are being analyzed. (ii) To obtain some idea of the accuracy of our estimates, e.g., by attaching to them estimated standard errors, fiducial or confidence limits, etc. Such standard errors, etc., should be reasonably free from bias. The usual property of the analysis of variance, when all assumptions are fulfilled, is that estimated variances are unbiased. (iii) To perform tests of significance. The most common are the F-test of the null hypothesis that a group of means all have the same true value, and the t-test of the null hypothesis that a difference is zero or has some known value. We should like such tests to be valid, in the sense that if the table shows a significance probability of, say, 0.023, the chance of getting the observed result or a more discordant one on the null hypothesis should really be 0.023 or something near it. Further, such tests should be sensitive or powerful, meaning that they should detect the presence of real differences as often as possible. Theobject of this paper is to describe what happens to these desirable properties of the analysis of variance when the assumptions required for the technique do not hold. Obviously, any practical value of the paper will be increased if advice can also be given on how to detect failure of the assumptions and how to avoid the more serious consequences.