## Tmax in Partial replicate design [Nonparametrics]

Dear Joshua, dear All!

❝ Can anyone kindly guide on how to do the non-parametric analysis for tmax for a two treatment, three period, three sequence (partial replicate) design.

Beside Helmut's answer there may be some reasons to do an non-parametric analysis of tmax, among them:
• The world is sometimes greater then North America and Europe
(Myself got a deficiency letter from a small country in Asia requesting a p-value for differences in tmax)
• Not satisfied with the in the discussion section of the report
• Scientific credibility
• Fun in developing/programming ambitious analysis methods

To my knowledge there is no explicit method described for calculating non-parametric CIs in case of a partial replicate design in the literature out there. But from the known literature (known by the small brain of mine, for me as agnostic lazy reader) I would suggest two methods.

Method 1:

Calculate the differences T-R for each subject and do a Wilcoxon signed rank test with these intra-subject differences, including confidence intervals based on that test.
Implementation should not so hard in R because wilcox.test() implementing the asymptotic variant is contained in the basic installation, package stats. For exact calculations considering also ties see package exactRankTests.
In SAS® itself out of the box there is no implementation of the CIs based on Wilcoxon signed rank test, only the test statistic and p-value can be acquired from Proc Univariate. Additional expensive software Proc StatXact is needed to do it. Else you have to roll your own. And this may be hard for the uninitiated . But see here.

This method of course assumes that no period effects occur (see the expectations of the intra-subject differences below). If period effects are present, the CI's calculated are wider then with taking the period effects into account. But I wouldn't dare too much about that.
First: Usually there are no equivalence limits constrained upon tmax. Thus wider intervals are of no consequences.
Second: In balanced partial replicate designs (equal number of subjects in the sequence groups) the period effects cancel out.
To see this lets look at the intra-subject contrasts and their expectation (assuming only treatment, period and subject effects for the sake of simplicity).
Let Yijkl the tmax value of subject i in sequence j having administered treatment k in period l.
Sequence   Contrast             Expectation 1  TRR   YiT1-0.5(YiR2+YiR3)  µ(T-R) + p1-0.5(p2+p3) 2  RTR   YiT2-0.5(YiR1+YiR3)  µ(T-R) + p2-0.5(p1+p3) 3  RRT   YiT3-0.5(YiR1+YiR2)  µ(T-R) + p3-0.5(p1+p3) (index j omitted)
Homework: Show that the mean of the expectations over sequences is µ(T-R). One subject per sequence .

Method 2:

Not satisfied with the neglection of period effects?
Then borrow the ideas from:

Oehrvik J.
"Nonparametric Methods in Crossover Trials"
Biometrical Journal, 40, Issue 7, pages 771–789, November 1998

to get rid of the period effects. This method (but only the test statistics and elaborated for a 3-period crossover with 3 treatments) is also described in:

Mary E. Putt and Vernon M. Chinchilli
"Nonparametric Approaches to the Analysis of Crossover Studies"
Statist. Sci. Volume 19, Number 4 (2004), 712-719.

which can be found online here.

The al-Gore-Rhythm proceeds as following (cook book manner):
1. Calculate a 'mean' function for the values in each period. This function has expectation pl+µ where µ is the mean over treatment effects, pl is the period effect of period l. 'Align' the tmax values by subtracting the corresponding period 'mean'. Note that the period effects vanish by this operation but the aligned values are lowered by µ.
2. Use the aligned values to calculate the intra-subject contrasts to estimate µ(T-R) as above. Note that µ cancels out by this step.
3. Use the intra-subject contrasts to perform a Wilcoxon signed rank test with calculation of CIs based on it.
This algorithm and related ones are sometimes called Aligned-rank test.
The 'mean' function could be the ordinary mean, but more robust (and more in the spirit of nonparametrics) would be the median or, as Oehrvik proposed, the Hodges-Lehmann point estimator (median of the order statistics, the pairwise Walsh averages of the intra-subject differences).

Hope this make sense for you.
BTW: Can post some R code upon request. But is a little bit lengthy.

Regards,

Detlew

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