BBR Session 7: Study Design, and Crossover Experiment Example

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).

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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.

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Ok Professor,
Thank you!