Paired t-test is significant but ANOVA between groups is not I have two groups, each received a pre-test and post-test.
T-test for pre- and post-test for group 1 is SIGNIFICANT.
T-test for pre- and post-test for group 2 is NOT significant.
ANOVA for difference between group 1 and 2 pre-tests is NOT significant.
ANOVA for difference between group 1 and 2 post-tests is NOT significant.
Is this possible?
I would think that the ANOVA between post-tests should be significant because of the change in difference of group 1.
 A: In addition to Dave2e's answer, keep in mind that ANOVA and the paired difference t-test are actually assuming different data generating processes for the experiment at hand.
From Wikipedia's article for paired difference test
$Y_{ij} = \mu_j + \alpha_i + \varepsilon_{ij}$
where $\alpha_i$ is a random effect that is shared between the two values in the pair, and $\varepsilon_{ij}$ is a random noise term that is independent across all data points. $\mu_j$ represents the expected value of the measurement for each group being compared.
From Wikipedia's article for One-way ANOVA:
$Y_{ij} = \mu_j + \varepsilon_{ij}$
where the terms $\mu_j$ and $\varepsilon_{ij}$ are defined identically as they are above.
Without going into too much detail, you can probably imagine that the sampling distribution of estimates for $\mu_j$, given the same set of measurements $Y_{ij}$, is going to vary across the two models -- and with different sampling distributions, it is entirely plausible that the estimate for $\mu_j$ is significant under one model, but not the other.
A: Yes, it is possible.  If you ran paired t-test on group 1, this indicates that the change from pre to post was consistent for each individual, the anova between groups is shows no differences between groups.
For example suppose you are testing a new diet.  All members of group 1 lose 2 pounds, that is statistical significant (and practically insignificant)!  Now if you compare post diet for group 1 and group 2, one would not expect the final weights from group 1 to be significant different from group 2.  This is how I interpreted your results about.
Depending on what you are testing, maybe you should compare post-pre test differences from group 1 to group 2.
