Does T-test's independence assumption apply across comparison groups?

Question

The independence assumption of T-test states "The data (scores) are independent of each other (that is, scores of one participant are not systematically related to scores of the other participants)."

Say I have two groups

                   Group-A     Group-B
Mean score             8.9         6.3
Standard deviation     1.0         1.1

Group-A contains 100 people, and Group-B contains 150 people. However, Group-A is part of Group-B, meaning that 100 people of Group-B came from Group-A, and the additional 50 people came elsewhere.

The measures were a snapshot, and not a pre and post for the same people (N=100). Is it still valid to conduct a T-test to test the difference between Group-A and Group-B?

Individuals are independent within group, but related across groups as explained above.

Answer 1

There are two ways in which correlation may arise in a two group setting: intragroup correlation and crossgroup correlation:

Intragroup correlation: The independence assumption for the T-test with equal variance assumption means that everyone in the sample is conditionally dependent of each other conditional on the group from which they came. For instance, if I compare adolescent women to men for weight/height, women may on average be a bit taller, a bit heavier because of early puberty onset. Aside from that, however, no woman is more correlated with another woman than she is with a man because the mean is a sufficient description of the group difference(s). This is of course an untenable assumption. Using the unequal variance assumption allows for some extent of correlation within groups.

Crossgroup correlation: The unequal variance assumption will actually provide consistent inference despite the cross correlations provided a few assumptions can be made about the 50 new entrants. They must not in some sense be "different" from those who participated in the study. This assumption cannot really be verified, regrettably.

For instance, suppose I measure pre/post change in a SAT prep course. An index cohort of students from suburban schools participated in the program to provide baseline (pre) and follow-up (post) test performance and demonstrated modest improvement, a subsequent cohort of inner city students were rushed to take the course, and their post performance was comparable to the original cohort. It is not reasonable to assume that the pre-performance would have been the same. There may, in fact, be effect modification by school setting, and even the marginal results are biased as a consequence.

In summary, because of design and setting, the assumptions needed to justify the pre/post comparisons with imbalanced clusters is usually not warranted. Whenever possible, conduct a complete cluster analysis.