Behrens-Fisher problem

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In statistics, the Behrens-Fisher problem is the problem of interval estimation and hypothesis testing concerning the difference between the means of two normally distributed populations when the variances of the two populations are not assumed to be equal, based on two independent samples.

Ronald Fisher in 1935 introduced fiducial inference in order to apply it to this problem. He referred to an earlier paper by W. V. Behrens from 1929. Behrens and Fisher proposed to find the probability distribution of

$\tau \equiv {\bar x_1 - \bar x_2 \over \sqrt{s_1^2/n_1 + s_2^2/n_2}}$

where $\bar x_1$ and $\bar x_2$ are the two sample means, and $s 1$ and $s 2$ are their standard deviations. Fisher approximated the distribution of this by ignoring the random variation of the relative sizes of the standard deviations, ${s_1 / \sqrt{n_1} \over \sqrt{s_1^2/n_1 + s_2^2/n_2}}$ . Fisher's solution provoked controversy because it did not have the property that the hypothesis of equal means would be rejected with probability α if the means were in fact equal. Many other methods of treating the problem have been proposed since.

Perhaps the most widely used method (for example in Microsoft Excel) is that of B. L. Welch (1939), who, like Fisher, was at University College London. He approximated the distribution of $\sqrt{s_1^2/n_1 + s_2^2/n_2}$ with a Type III Pearson distribution. The latter is like a chi-squared distribution with a non-integer number of degrees of freedom, which can be estimated from the sizes of the two samples and their sample variances:

$d.f. = {(a_1 + a_2)^2 \over a_1^2/(n_1-1) + a_2^2/(n_2-1)} \,\!$

with $a_i = s_i^2/n_i$ . This results in a Student's t distribution for $τ$ with a non-integer number of degrees of freedom. This method also does not give exactly the nominal false positive rate, but is generally not too far off. However, if the population variances can be assumed equal, it is more accurate to use the standard method for that case, which is the two-sample t-test.

[edit] References and external links

W. V. Behrens, "Ein beitrag zur Fehlerberechnung bei wenigen Beobachtungen", 'Landwirtschaftliche Jahrbuecher' 68, pp. 807-37.
"On the Behrens-Fisher Problem: A Review", by Seock-Ho Kim and Allan Cohen, University of Wisconsin, 1995. Paper presented at the annual meeting of the Psychometric Society, Minneapolis.