Define the following notation:

Observations at the first class level are assumed to be distributed as , and observations at the second class level are assumed to be distributed as , where , , , and are unknown.

The within-class-level mean estimates ( and ), standard deviation estimates ( and ), standard errors ( and ), and confidence limits for means and standard deviations are computed in the same way as for the one-sample design in the section Normal Data (DIST=NORMAL).

The mean difference is estimated by

Under the assumption of equal variances (), the pooled estimate of the common standard deviation is

The pooled standard error (the estimated standard deviation of assuming equal variances) is

The pooled 100(1 – )% confidence interval for the mean difference is

The t value for the pooled test is computed as

The p-value of the test is computed as

Under the assumption of unequal variances (the Behrens-Fisher problem), the unpooled standard error is computed as

Satterthwaite’s (1946) approximation for the degrees of freedom, extended to accommodate weights, is computed as

The unpooled Satterthwaite 100(1 – )% confidence interval for the mean difference is

The t value for the unpooled Satterthwaite test is computed as

The p-value of the unpooled Satterthwaite test is computed as

When the COCHRAN option is specified in the PROC TTEST statement, the Cochran and Cox (1950) approximation of the p-value of the statistic is the value of p such that

where and are the critical values of the t distribution corresponding to a significance level of p and sample sizes of and , respectively. The number of degrees of freedom is undefined when . In general, the Cochran and Cox test tends to be conservative (Lee and Gurland 1975).

The 100(1 – )% CI= EQUAL and CI= UMPU confidence intervals for the common population standard deviation assuming equal variances are computed as discussed in the section Normal Data (DIST=NORMAL) for the one-sample design, except replacing by and by .

The folded form of the F statistic, , tests the hypothesis that the variances are equal (Steel and Torrie 1980), where

A test of is a two-tailed F test because you do not specify which variance you expect to be larger. The p-value (Steel and Torrie 1980) is equal-tailed and is computed as

where , , , and are the degrees of freedom that correspond to , , , and , respectively.

Note that the p-value is similar to the probability of a greater value under the null hypothesis that ,

The test is not very robust to violations of the assumption that the data are normally distributed, and thus it is not recommended without confidence in the normality assumption.

The DIST= LOGNORMAL analysis is handled by log-transforming the data and null value, performing a DIST= NORMAL analysis, and then transforming the results back to the original scale. See the section Normal Data (DIST=NORMAL) for the one-sample design for details on how the DIST= NORMAL computations for means and standard deviations are transformed into the DIST= LOGNORMAL results for geometric means and CVs. As mentioned in the section Coefficient of Variation, the assumption of equal CVs on the lognormal scale is analogous to the assumption of equal variances on the normal scale.

The distributional assumptions, equality of variances test, and within-class-level mean estimates ( and ), standard deviation estimates ( and ), standard errors ( and ), and confidence limits for means and standard deviations are the same as in the section Normal Difference (DIST=NORMAL TEST=DIFF) for the two-independent-sample design.

The mean ratio is estimated by

No estimates or confidence intervals for the ratio of standard deviations are computed.

Under the assumption of equal variances (), the pooled confidence interval for the mean ratio is the Fieller (1954) confidence interval, extended to accommodate weights. Let

where is the pooled standard deviation defined in the section Normal Difference (DIST=NORMAL TEST=DIFF) for the two-independent-sample design. If (which occurs when is too close to zero), then the pooled two-sided 100(1 – )% Fieller confidence interval for does not exist. If , then the interval is

For the one-sided intervals, let

which differ from and only in the use of in place of . If , then the pooled one-sided 100(1 – )% Fieller confidence intervals for do not exist. If , then the intervals are

The pooled t test assuming equal variances is the Sasabuchi (1988a, 1988b) test. The hypothesis is rewritten as , and the pooled t test in the section Normal Difference (DIST=NORMAL TEST=DIFF) for the two-independent-sample design is conducted on the original values () and transformed values of

with a null difference of 0. The t value for the Sasabuchi pooled test is computed as

The p-value of the test is computed as

Under the assumption of unequal variances, the unpooled Satterthwaite-based confidence interval for the mean ratio is computed according to the method in Dilba, Schaarschmidt, and Hothorn (2007, the section "Two-sample Problem" on page 20), extended to accommodate weights. The degrees of freedom for the confidence interval are based on the same approximation as in Tamhane and Logan (2004) for the unpooled t test but with the null mean ratio replaced by the maximum likelihood estimate :

Let

where and are the within-class-level standard deviations defined in the section Normal Difference (DIST=NORMAL TEST=DIFF) for the two-independent-sample design. If (which occurs when is too close to zero), then the unpooled Satterthwaite-based two-sided 100(1 – )% confidence interval for does not exist. If , then the interval is

The t test assuming unequal variances is the test derived in Tamhane and Logan (2004). The hypothesis is rewritten as , and the Satterthwaite t test in the section Normal Difference (DIST=NORMAL TEST=DIFF) for the two-independent-sample design is conducted on the original values () and transformed values of

with a null difference of 0. The degrees of freedom are computed as

The t value for the Satterthwaite-based unpooled test is computed as

The p-value of the test is computed as