EFFECT OF NON-NORMALITY ON THE POWER FUNCTION OF t-TEST

Student's t-statistic provides a suitable test of significance for the mean when the sample comes from a normal population. The power of the normal theory test has been studied by Neyman (1935); Neyman & Tokarska (1936) and Johnson & Welch (1939). As in many cases the samples appear to belong to populations other than the Gaussian, it is necessary to see how far the normal theory test can be assumed to be valid in controlling the Type I and Type II errors of inference on non-normal samples. The effect of non-normality on the Type I error of Student's t-test was studied experimentally by Pearson & Adyanthaya (1929) and theoretically by Bartlett (1935), Geary (1936, 1947) and Gayen (1949). Assuming the parent population to be specified by the first two terms of the Edgeworth series, Geary (1936) obtained the approximate distribution of t for any sample size, and later this work was extended by Gayen (1949) by including in the distribution the effects of parental kurtosis A4 = fl2-3 and A2 = IA. Apart from the pioneer empirical study by Pearson & Adyanthaya (1929, pp. 276-80), the effect of non-normality on the Type II error (and hence on the power) of the t-test was first studied by Ghurye (1949). He has, however, considered only the effect of skewness of the population, for he started with the joint distribution of the mean and the variance for populations specified by the first two terms of the Edgeworth series (Geary, 1936). In this paper, it has been possible to study the effects of kurtosis and skewness of the parent population which may be assumed to cover a larger range of nonnormality. Gayen's (1949) formulae for the joint distribution of the sample mean and variance for the first four terms of the Edgeworth population have been utilized for the derivation of the corrective terms of the power function. The mnethod followed by Ghurye for the evaluation of the corrective term of the power function due to A3 appears to be satisfactory for derivation of the effects due to higher odd-order cumulants. But for those of the even-order cumulants his method does not appear to be useful, as Ghurye himself encountered some 'analytic difficulties'. In this paper, by a different approach it has been possible to evaluate integrals involved in the power function due to A4 and A3. The non-normal population considered here is supposed to be characterized by non-zero values of the standardized third and fourth cumulants. Since the effects of the higher-order terms depending on A5,6, A3A4, A2, ... are assumed to be negligible, the population covered is only moderately non-normal. Too high values of A3 and A4 can also not be permitted as they will make f(x) negative at one or both tails, and will give rise to subsidiary modes. To ensure a positive definite, unimodal frequency function, A4 should lie roughly between 0 and 2-4 and A3 02 (Barton & Dennis, 1952).t Also it is found possible in this paper to calculate in the non-normal case the critical region