Were finalizing an RCT with two intervention groups (n=13, n=11). Both samples are evaluated pre vs post treatment for pain (VAS), and also against each other (group vs group). However, there are four subjects lost to post treatment in one of the groups (n13 --> n=9). Which statistic model/test would be appropriate to use?
Equal sample sizes is not an assumption or requirement of the t-test. Therefore, you can perform them.
The t-test is sensitive to unequal sample sizes. Unequal sample sizes tend to reduce power relative to having the total N divided equally between the groups. Furthermore, while the t-test is robust to unequal variances for alpha, this is NOT the case when the sample sizes are unequal. Then unequal variances can influence alpha quite a lot.
While not necessarily avoiding the potential issues above, which would be minor in the case of small deviations from equal sample sizes, in this case you should probably be doing an ANOVA instead of 3 t-tests. There are a number of ways to handle before-after designs, ANCOVA and factorial ANOVA would be the most popular. I'd recommend the latter. You could test your within effects, between effects, and interaction in a single mixed ANOVA.