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# The Effect of Inequality of Variances on the t-Test

E. M. Carter, C. G. Khatri and M. S. Srivastava
Sankhyā: The Indian Journal of Statistics, Series B (1960-2002)
Vol. 41, No. 3/4 (Dec., 1979), pp. 216-225
Stable URL: http://www.jstor.org/stable/25052152
Page Count: 10
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## Abstract

The effect of inequality of variances on the two sided t-test is studied in this paper. It is found that there is no appreciable effect on the significance level even if the ratio of the variances differs from one by as much as 0.4. The effect starts showing when this difference exceeds 0.4. This effect is neutralized if one is permitted to take a larger number of observations from the population with the larger variance than from the population with the smaller variance. The effect on power is similar. Thus in the above circumstances one could safely use the t-test rather than switch to tests recommended for a Behrens-Fisher situation. Hsu (1938) has considered the effect of the inequality of variances on the significance level of the t-test and he has rejected that test for unequal sample sizes on the grounds that there is a large deviation in this significance level (see Scheffé (1959) for tabulated values). However, if the ratio of the variances is restricted, as is the case here, these large deviations are not obtained, and even the power is not appreciably affected. These results are extended to the multivariate situation. However, only the effect on significance level is considered. The distribution of Hotelling's $T^{2}\text{-statistic}$ is derived when $\Sigma _{1}=\lambda \Sigma _{2}$ and is used to study the effect on the significance level. The results are similar to those of the univariate case and agrees with the Monte Carlo study of Holloway and Dunn (1967) and Hopkins and Clay (1963). That is, if the sample sizes are equal the significance level is not affected. However, it was found here that if the sample sizes are unequal then the larger sample should be taken from the population with the larger dispersion matrix. In this case we have a more conservative test procedure. Even when this is not the case if the dispersion matrices are not too unequal, that is if the ratio of the dispersion matrices is not less than 0.6, then the $T^{2}\text{-test}$ procedure is still valid.

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