What statistical tests can I use? Repeated measures design, two groups, each is control and experimental at different points in time I need to determine if an intervention had an effect. In the experimental design, measures were taken at three points in time: pre-test, post-test #1 and post-test #2. One group (group A) received the intervention between pre-test and post-test #1. The other group (group B) received it between post-test #1 and post-test #2. Thus, both groups served as control at different points in time.
In other words, the experimental design can be summarised as follow:


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*Group A: pre-test -> treatment -> post-test #1 -> no treatment -> post-test #2

*Group B: pre-test -> no treatment -> post-test #1 -> treatment -> post-test #2
The expected effect is that group A will progress between pre-test and post-test #1, and will maintain its gain at post-test #2. However, group B, having only underwent the intervention between post-test #1 and post-test 2, will remain stable between pre-test and post-test #1, but will progress between post-test #1 and post-test #2.
Given these conditions, which statistical tests can I use? I have seen repeated measures ANOVA being recommended but all the use case examples I have seen involved two or more different treatments being administrated in different orders, which is not the case here, unless "no treatment" can be regarded as a kind of treatment (which I am not sure of).
 A: This seems like a classic AB/BA cross-over study, but where you have problem with carry-over effects. Typically when administring an AB/BA cross-over study (as this one seems to be), a long wash-out period is implemented between the two periods to ensure that the treatment effect from the first period is not lingering in the second period. 
The problem with the carry-over effect can be illustrated by considering the case where the treatment in period one completely cures the disease studied for all individuals. Then those who are treated in the first period will give no information about the relative effect of treatment to placebo, since they will have the  same outcome at the end of both periods. This will bias your estimates of the treatment effect.
If the carry-over effect is strong, as you indicated that it is in your data, then perhaps the best option is to ignore the second treatment occasion and just use the first treatment occasion data, thus turning the study into a cross-sectional study instead of a longitudinal one. The main issues of that approach being that you lose power from discarding data, and it also might be unethical to take measurements on the individuals and then not use the data in the analysis.
I would recommend reading Senn (1994) for an introduction to the AB/BA cross-over design, or the books by Senn (2002) or Jones and Kenward (2015) for a more thorough covering of the topic.
References


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*Jones, B. and Kenward, M. G. (2015). Design and analysis of cross-over trials.  CRC Press LLC, Boca Raton, FL. Third edition.

*Senn, S. (1994). The AB/BA crossover: past, present and future?, Statistical Methods in Medical Research, 3: 303-324.

*Senn, S. (2002). Cross-over trials in clinical research. John Wiley & Sons, Chichester. Second edition.

A: Fit a linear model with treatment, group and their interaction as covariates. Need to add the random effects to the model to take care of the correlation between the measurements from the same patient. Given your data structure is very good, instead of using random effect, you can specify the correlation between the measurements from the same patient directly. 
After model fitting, you can estimate and test any combination of 6 means that can be derived from model.
