For continuous variables you might use the more robust and efficient Wilcoxon signed-rank test and its associated point estimate the pseudomedian (median of all possible pairs of differences from pre to post). For binary variables McNemar’s test does seem applicable, and get a confidence interval for a probability that is part of this test. If memory serves, McNemar’s is based on the following if X is pre and Y is post: P(Y > X | Y \neq X) = P(\Delta > 0 | \Delta \neq 0) where \Delta = Y - X. You can compute Wilson’s confidence interval on this probability after removing all the tied observations. You can also get confidence intervals on unconditional probabilities e.g. P(Y = X).

Hi Professor @f2harrell,

Apologies for the mistake, sometimes it’s not clear to me where to post my questions. I hope this is now placed in the correct topic. Please let me know if that’s not the case.

50 subjects received 2 interventions. The order of the interventions was randomized. The subjects were assessed before each intervention (Pre), immediately after (Post_1), and then one more time (Post_2). Each experimental session (Pre + Intervention + Post_1 + Post_2) lasted about 1 hour and took place on a different day (2 days total). The attached Plot shows the mean response (with a 95% nonparametric confidence band) on each time for each intervention.

The interventions are the same, but one is a “stronger” version of the other.

I’m a bit confused with this design as it includes 2 within-subjects factors, namely, intervention and assessment time.

The goals are:

- To test for an intervention effect from Pre to Post_1;
- To test for a different intervention effect between the 2 interventions;
- To test for a “recovery” effect from Post_1 to Post_2 (i.e., what value the response returns to);
- To test for a different “recovery” effect between the 2 interventions;

I tried to model these data using the approach suggested here: Biostatistics for Biomedical Research - 7 Nonparametric Statistical Tests

For exemple:

`f <- ols(response ~ intervention * time + id)`

Where “intervention” and “time” are modeled as categorical variables.

But I’m not sure this is correct, because I have 3 measurements per subject for each intervention, whereas in the BBR example each subject has only 1 measurement per condition (drug).

I think I could reduce the 3 measurements per subject to only 1, using for instance the AUC for each subject’s curve. But then I don’t know how I could address the goals from above. By comparing the mean AUC between the 2 interventions I would simply be testing for an overall difference between the interventions. I think this would be a limitation of any summary measure.

Does anyone have any suggestion?

Thank you.

Dear Professor @f2harrell,

**Section 7.9** of **BBR** says that Regression can be used to analyze *multi-period crossover studies*. Are there any examples available on **BBR**, **RMS**, or here? I’m struggling with the case above.

No examples. Take a look at Stephen Senn’s book on crossover studies. I’ll bet the prevailing way to handle that is using subject random intercepts just as you can do with 2 periods.

Ok Professor,

Thank you!

Dear Professor @f2harrell,

I’m a bit confused with regards to the analysis presented in **Section 7.8** of **BBR**. There we have a case of a *PO Model* being applied to a type of experiment where one of the factors (surface) seems to be a within-subjects factor. However, the model does not include ‘subject’ as a predictor and thus it does not seem to be taking into account the repeated measurements within subject:

`f <- orm(y ~ sex + surface, data=d, x=TRUE, y=TRUE)`

Am I missing something?

Thank you for your time.

I was assuming that the two surfaces yield uncorrelated observations.