Origin and Robustness of "Equal Variances Not Assumed" Independent Samples T-Test
I would like some information on the origins and limits of use of the "Equal Variances Not Assumed" test that is produced when one runs the Independent Samples T-test in SPSS Statistics. For some of my analyses, the two groups are extremely different in size. In some of these analyses, the very small groupmay have a variance of 0, whereas the larger group does have variance. Do such extreme differences in group size and invariance invalidate the "Equal Variance Not Assumed" t-test results?
When the groups in an independent samples t-test have unequal variances, the option that SPSS gives is simply called a t-test with a Satterthwaite approximation for the degrees of freedom. There is apparently no formal name for this test. Both the Brown-Forsythe and Welch tests, which are available in the ONEWAY procedure as tests that are robust for unequal variance, collapse to this t-test when there are only two groups, but this modification of the standard t-test predates both of these, and thus doesn't generally carry either name. The interpretation of the test results is the same as with a t-test for equality of means with equal variances assumed. (Note that the Brown-Forsythe and Welch tests give the same significance level as the unequal variance t-test. when ONEWAY is run with 2 groups. However, when one group has 0 variance, ONEWAY will not calculate the Brown-Forsythe and Welch tests, whereas T-TEST will calculate the unequal variance t-test significance.)
We do not have material at hand on the robustness of the unequal-variance t-test to very small or very unbalanced sample sizes or suggestions for a minimum size. Winer (1971) has a discussion on the test on pages 41-44 with an example (although not with the extreme difference in sample size that you discussed. However, the references there may provide more detail. The third edition(s) of Winer may have additional and more recent material.
Winer, B.J. (1971). Statistical Principles in Experimental Design (2nd Ed.). New York: McGraw Hill.