Departure of Bartlett's distribution for the homogeneity of variances for unequal sample sizes from that of equal sample sizes

Summary

A rigorous study is carried out for the comparison of the exact Bartlett distribution for the test of the homogeneity of variances for unequal sample sizes with that of equal sample size. A detailed estimate is derived to the correction term of the relevant distributions. LetM ¹=−Σν _j ln(S ² _j /S ²), wherek denotes the number of (normal) populations and ν _j denotes the degree of freedom of thej-th sample. LetM ² be the corresponding variable when allv _j =v. In general, letv=minv _j and δ=max(v _j -v)/v _j . We then obtain a bound$|P [M^1 /C_1 \geqslant x] - P [M^2 /C_2 \geqslant x]| \leqslant \bar \varepsilon (\delta ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{v} ,k)$, uniformly inx, whereC ₁=C ₁(k,v _j ),C ₂=C ₂(k,v) are suitable scaling factors. For sufficiently small$\bar \varepsilon$, the critical values for unequal sample sizes may be then taken to beM ¹ _C =(C ₁/C ₂)M ² _C , where the equal sample sizeM ² _C values are well known. A rate of decrease of$\bar \varepsilon (\delta ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{v} ,k)$ for δ→0 and δ→0 is also given. An example is given to illustrate the practical usefulness of our estimate.