Independence of errors assumption in paired & independent samples t-test

Question

first of all, this is a great site and I've already learned a lot from threads here wile searching for answers to my stats questions on the web. Yet here's another one:

I understand that the independence of errors assumption in the t-test means that you shouldn't be able to predict the value of the outcome variable in any observation from the values in the other observations.

1. This refers to observations regardless of group membership (as in group 1 and group 2 of the t-test), right? That is, you should neither be able to predict the outcome variable based on other observations of the outcome variable WITHIN one group, nor BETWEEN the groups?

2. And is paired samples t-test necessary because of the specific violation of this assumption in the sense that you could predict observations in one group from those in the other?

3. And finally, then, the paired t-test should still require independence of observations within the 2 groups, right?

If I understand correctly, the first answer in the thread below basically confirms my presumptions, but since I'm not quite sure, I figured I'd ask about it explicitly.

Question about independence assumption for ANOVA, t-test, and non-parametric tests

Thanks and cheers, Hans

Answer 1

The t-test is a special case of regression, so I'll discuss this in that context. The assumption of independence isn't that it's not possible to predict the observations, but that you can't predict the residuals. Strictly speaking this isn't true either, because with finite degrees of freedom in any model, the last few residuals can always be predicted from the rest of the information available. However, outside of that, the residuals should be impossible to predict above chance. The paired samples t-test, and mixed-effects models more generally, are required because otherwise the residuals would be able to be predicted to a large degree.