Bioequivalence and Bioavailability Forum

Dear D. Labes!

❝ Ok, this thread could also go into the category "R for BE/BA". But this here is Helmut's favorite .

Right. But if ties are considered, you are stirring up a hornets' nest!

❝ Since in case of tmax there are usually some/many equal values (ties in the non-parametric methods speak) the used software should be able to handle such.

Well, cough…

Summary of your results (T-R) - changed signs if necessary and divided by 2:
program                   PE      90.26% CI
library(exactRankTests)  +0.125  -0.25 +0.75
library(coin)            +0.25   -0.25 +0.75
SAS (asymptotic)         +0.25   -0.25 +0.75
SAS (exact)              +0.25   -0.25 +0.75

My quick and dirty solution:
PKt$pdiff    <- (PKt$P1 - PKt$P2)/2
PKt$sequence <- relevel(PKt$sequence, ref="TR")
PKt$sequence
 [1] TR RT TR RT TR RT TR RT TR RT TR RT TR RT
Levels: TR RT
require(exactRankTests)
 t1 <- wilcox.exact(pdiff ~ sequence, data=PKt, conf.int=TRUE, conf.level=0.9)
print(t1)

        Exact Wilcoxon rank sum test

data:  pdiff by sequence
W = 31, p-value = 0.4149
alternative hypothesis: true mu is not equal to 0
90 percent confidence interval:
 -0.50  0.75
sample estimates:
difference in location
                 0.125

require(coin)
t2 <- wilcox_test(pdiff ~ sequence, data=PKt, conf.int=TRUE, conf.level=0.9,
                  distribution="exact")
print(t2)

        Exact Wilcoxon Mann-Whitney Rank Sum Test

data:  pdiff by sequence (TR, RT)
Z = 0.8465, p-value = 0.4149
alternative hypothesis: true mu is not equal to 0
90 percent confidence interval:
 -0.25  0.75
sample estimates:
difference in location
                  0.25

❝ Any body out there who knows whats going on here?

Hhm; don't use library(exactRankTests)?

Hauschke et al. (1990) don't speak about ties; they refer to Hollander and Wolfe.¹ In my edition the word 'tie' is not even mentioned in the subject index… With my mid-1990 implementation in Rocky Mountain BASIC (not corrected for ties) I get:
                          PE      90.26% CI
                         +0.25   -0.25 +0.75

Now for StatXact 6.1. using the coding according to Senn (2002) I get:
                          PE      90.26% CI
asymptotic               +0.50   -0.25 +1.00
exact                    +0.50   -0.25 +1.00
What the heck?

Phoenix 6.1 (asymptotic), according to Koch² (no ties!):
                          PE      90.26% CI
                         +0.25   -0.25 +0.75

EquivTest/PK (asymptotic):
                          PE      90.26% CI
                         +0.25   -0.25 +0.75

Not a single word about ties in the manuals of Phoenix and EquivTest/PK.
StatXact states:

[…] zero differences are ignored. We assume initially that there are no ties in the observed values of the D_i's. We will show subsequently how to adjust for the possibility of ties. […] By suitably adapting the reasoning in Lehmann (1975, pages 181 to 182) so as to accommodate ties one can show that the probability that the interval [λ_*, λ^*] excludes λ is at most α. This ensures that the Hodges-Lehmann confidence intervals preserve the desired coverage probability.

In a more recent textbook³ the common procedure of using mid-ranks for ties is given. According to this reference StatXact adjusts for ties. But I'm not so sure anymore. Open questions:

• Signs… (sigh!) I've checked the coding in all programs numerous times, but don't know what's going on here. Fishbrain. I guess the reason might be that you are comparing period differences rather than treatment differences. Since we have a balanced case here, it might explain the opposite signs.
  
• In my understanding wilcox_test() does not correct for ties. I tried ties.method="mid-ranks" and ties.method="average-scores" and got:
  Warning:
In independence_test.IndependenceProblem(object, teststat = "scalar",  :
  additional arguments ties.method will be ignored
  
• Apart from the signs, the numerical values are identical with the exception of StatXact's results and the PE in R's exactRankTests. We have four out of 14 cases with a zero difference. If I manually exclude these cases from the dataset I get exactly the same results as for the full dataset.
  
• If I exclude subjects with a zero difference (1, 3, 104, 14) I get StatXact's results in all (!) other programs… So what do we have here? One program (maybe!) adjusts for ties, but excludes zero differences.
  

❝ Sorry for that lengthy post.

So am I. You gave me a nice excuse not to spend the afternoon struggling with my income tax return.

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1. Holander M, Wolfe DA. Nonparametric Statistical Methods. New York: Wiley; 1973.
   
2. Koch GG. The Use of Non-Parametric Methods in the Statistical Analysis of the Two-Period Change-Over Design. Biometrics. 1972: 28; 578–85. PMID 4556704.
   
3. Sprent P, Smeeton NC. Applied Nonparametric Statistical Methods. Boca Raton: Chapman & Hall/CRC; 4^th ed. 2007.