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A Permutation Approach to the Behrens-Fisher Problem
Michael A. Proschan,Dean A. Follmann 한국통계학회 2004 Journal of the Korean Statistical Society Vol.33 No.1
We propose a permutation approach to the classic Behrens-Fisher prob-lem of comparing two means in the presence of unequal variances. It ismotivated by the observation that a paired test is valid whether or not thevariances are equal. Rather than using a single arbitrary pairing of the data,we average over all possible pairings. We do this in both a parametric andnonparametric setting. When the sample sizes are equal, the parametricversion is equivalent to referral of the unpairedt-statistic to at-table withhalf the usual degrees of freedom. The derivation provides an interestingrepresentation of the unpairedt-statistic in terms of all possible pairwiset-statistics. The nonparametric version uses the same idea of consideringall dierent pairings of data from the two groups, but applies it to a per-mutation test setting. Each pairing gives rise to a permutation distributionobtained by relabeling treatment and control within pairs. The totality ofdierent mean dierences across all possible pairings and relabelings formsthe null distribution upon which the p-value is based. The conservatism ofthis procedure diminishes as the disparity in variances increases, disappear-ing completely when the ratio of the smaller to larger variance approaches0. The nonparametric procedure behaves increasingly like a pairedt-test asthe sample sizes increase.