Behrens–Fisher t′ Statistic

The Behrens-Fisher t′ statistic can be employed when one seeks to make inferences about the means of two normal populations without assuming the variances are equal. The statistic was offered first by W. U. Behrens in 1929 and reformulated by Ronald A. Fisher in 1939:

where sample mean

1 and sample variance s21 are obtained from the random sample of size n1 from the normal distribution with mean μ1 and variance has a t distribution with ν1 = n1 − 1 degrees of freedom, the respective quantities with subscript 2 are defined similarly, and or. The distribution of t′ is the Behrens-Fisher distribution. It is, hence, a mixture of the two t distributions. The problem arising when one tries to test the normal population means without making any assumptions about their variances is referred to as the Behrens-Fisher problem or as the two means problem.

Under the usual null hypothesis of H0: μ-1 = μ2, the test statistic t′ can be obtained [Page 76]and compared with the percentage points of the Behrens-Fisher distribution. Tables for the Behrens-Fisher distribution are available, and the table entries are prepared on the basis of the four numbers ν1 = n1 − 1, ν2 = n2 − 1, θ, and the Type I error rate α. For example, Ronald A. Fisher and Frank Yates in 1957 presented significance points of the Behrens-Fisher distribution in two tables, one for ν1 and ν2 = 6,8,12,24, ∞; θ = 0°; 15°, 30°, 45°, 60°, 75°, 90°; and α = .05, .01, and the other for ν1 that is greater than ν2 = 1,2,3,4,5,6,7;θ= 0°, 15°, 30°, 45°, 60°, 75°, 90° and α = .10, .05, .02, .01. Seock-Ho Kim and Allan S. Cohen in 1998 presented significance points of the Behrens-Fisher distribution for ν1 that is greater than ν2 =2,4,6,8,10,12; θ = 0°, 15°, 30°, 45°, 60°, 75°, 90°; and α = .10, .05, .02, .01, and also offered computer programs for obtaining tail areas and percentage values of the Behrens-Fisher distribution.

Using the Behrens-Fisher distribution, one can construct the 100(1 – α)% interval that contains

where the probability that t′ > t′α/2(ν1, ν2,θ) is α/2 or, equivalently, Pr[t′ > t′α/2(ν1, ν2,θ)] = α/2.

This entry first illustrates the statistic with an example. Then related methods are presented, and the methods are compared.

Example

Driving times from a person's house to work were measured for two different routes with n1 = 5 and n2 = 11. The ordered data from the first route are 6.5, 6.8, 7.1, 7.3, 10.2, yielding

1 = 7.580 and s21 = 2.237, and the data from the second route are 5.8, 5.8, 5.9, 6.0, 6.0, 6.0, 6.3, 6.3, 6.4, 6.5, 6.5, yielding

2 = 6.136 and s22 = 0.073. It is assumed that the two independent samples were drawn from two normal distributions having means μ1 and μ2 and variances σ21 and σ22; respectively. A researcher wants to know whether the average driving times differed for the two routes.

The test statistic under the null hypothesis of equal population means is t′ = 2.143 with ν1 = 4, ν2 = 10; and θ = 83.078. From the computer program, Pr(t′ > 2.143) = .049, indicating the null hypothesis cannot be rejected at α = .05 when the alternative hypothesis is nondirectional, Ha: μ1 ≠ μ2, because p = .098. The corresponding 95% interval for the population mean difference is [-0.421,3.308].

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