Bioequivalence and Bioavailability Forum

Dear Andrew,

the Hodges-Lehmann estimator is just the median of the Walsh averages.¹ But obviously you are able to calculate the cumulative distribution function – I guess you are more interested in obtaining the confidence interval (according to Moses)? I used an old reference* (in FORTRAN) to calculate the critical values (α 0.05) for up to 64 subjects (m=n=32) and the exact error probabilities as well.
I uploaded two files (in CSV-format, variable separator semicolon, decimal separator period):

1. Critical values
   
2. Error probabilies
   I take no responsibilities about correctness whatsover!

I would suggest looking for suitable software (Cytel’s StatXact, SAS PROC StatXact,…), or go with the normal approximation – at least to check the outcome.

For your example (m=26, n=26) the lower critical value according to the first table is 248 and upper one is calculated according to m × n - 248 + 1 with 429.
The normal approximation is calculated according to
m × n/2 – Z_0.05 × √m × n × (m+n+1)/12 (rounded to the next lower integer) with 248.
The normal approximation is always conservative (α ≤0.05); 57.3% of the 900 critical values match the exact ones, the remaining 42.7% would calculate one rank lower than the exact one (hence the CI will be wider). Although some textbooks state that the approximation should be used only if m≥8, n≥8 I can’t see any pattern (i.e., an improvement towards the exact value for higher m,n).

According to the second table the exact error probability for m=n=26 is 0.0498 (1 – 2α = 0.9004).

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1. Pairwise averages: (X_i+X_j)/2 for all i≤j.
   
2. Dinneen LC, Blakesley BC. Algorithm AS 62: A Generator for the Sampling Distribution of the Mann-Whitney U Statistic. Appl Stat. 1973;22:269–73.