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 withinclasslevel 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 onesample 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 confidence interval for the mean difference is






The t value for the pooled test is computed as
The pvalue of the test is computed as
Under the assumption of unequal variances (the BehrensFisher 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 confidence interval for the mean difference is






The t value for the unpooled Satterthwaite test is computed as
The pvalue 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 pvalue 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 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 onesample 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 twotailed F test because you do not specify which variance you expect to be larger. The pvalue gives the probability of a greater F value under the null hypothesis that . Note that this 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 logtransforming 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 onesample 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 withinclasslevel 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 twoindependentsample 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 twoindependentsample design. If (which occurs when is too close to zero), then the pooled twosided Fieller confidence interval for does not exist. If , then the interval is
For the onesided intervals, let




which differ from and only in the use of in place of . If , then the pooled onesided 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 twoindependentsample 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 pvalue of the test is computed as
Under the assumption of unequal variances, the unpooled Satterthwaitebased confidence interval for the mean ratio is computed according to the method in Dilba, Schaarschmidt, and Hothorn (2006), extended to accommodate weights. The degrees of freedom are computed as
Note that the estimate is used in . Let






where and are the withinclasslevel standard deviations defined in the section Normal Difference (DIST=NORMAL TEST=DIFF) for the twoindependentsample design. If (which occurs when is too close to zero), then the unpooled Satterthwaitebased twosided 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 twoindependentsample design is conducted on the original values () and transformed values of
with a null difference of 0. The degrees of freedom used in the unpooled t test differs from the used in the unpooled confidence interval. The mean ratio under the null hypothesis is used in place of the estimate :
The t value for the Satterthwaitebased unpooled test is computed as
The pvalue of the test is computed as