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I am planning a 2x2x2 design which includes 1 within factor and 2 between factors.

However, the within factor is a pre-test post-test design, meaning that the order of the two treatments of the within factor is not randomized. From another perspective, it is a normal pretest post-test design with additional 2x2 factors.

The outcome is measured twice.

Therefore the design looks like this. Participants will be assigned to one block (w=within, b1=between 1, b2=between 2) (0=control, 1=treatment) :

         w  b1 b2                   w  b1 b2
Block1   0  1  1                    1  1  1
Block2   0  0  1 > measurement 1 >  1  0  1  > measurement 2
Block3   0  1  0                    1  1  0
Block4   0  0  0                    1  0  0 
Block5   0  1  1                    0  1  1
Block6   0  0  1 > measurement 1 >  0  0  1  > measurement 2
Block7   0  1  0                    0  1  0
Block8   0  0  0                    0  0  0 

My question is, how should I estimate the effects of w, b1, b2 and corresponding two-way interactions?

My gut answer would be a "split-plot" or "mixed between-within" ANOVA.

However, many pre-test post-test studies condition on measurement 1 and treat measurement 2 as the dependent variable because the ordering of the control and treatment conditions is always the same. Would this be preferable to a more standard ANOVA? Would a linear mixed model including measurement 1 as a covariate the best solution?

I am thankful for any help!

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