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I have data from an experiment testing a new treatment. Each subject has 4 data points, 2 of the new treatment and 2 of the control. It is like a paired analysis, but instead having 1 observation of treatment and 1 of control, I have 2 of each. Which design is it (complete block, ....). How should I analyze this data ? A mixed model (response is continuous) ? How bad will it be if I average the 2 of each treatment per each subject ?

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This is a repeated measures design with replicates. Given the simple nature of the design (only one treatment effect that is applied to each subject and no between subject effects), you could also analyse this as a randomized block ANOVA, where each subject is a block. If you have a small number of subjects, I would recommend randomized block. However, if you have a large number of subjects whose attributes are randomly distributed, you can analyse the data as a mixed model with subject as random effect.

If your research question is simply to compare treatment and control, a fixed effects, randomized block analysis is fine. If you want to estimate the response for a typical, random person under the new treatment, then you will want the mixed model.

However ... since you have a balanced design with only two treatments, there is no reason not to average the data at the start. Suppose you observe $T_1,T_2,C_1,C_2$ for each subject (2 treatments and 2 controls) ... just calculate $T_1+T_2-C_1-C_2$ for each subject and analyse the outcome. You would have a simple one-sample t-test where the null hypothesis is that the mean is 0. The p-value would be the same as what you would get from the fixed effects, randomized block analysis.

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