Dr Andrew Leary
★

Ireland,
2009-01-26 18:18
(4946 d 15:07 ago)

Posting: # 3127
Views: 14,716

## Hodges-Lehmann Point Estimate [Nonparametrics]

Does anyone know where we can find a set of extended critical values for Wilcoxons test statistic? This is the set of tables used in the calculation of the Hodges-Lehmann Point Estimate. We've recently completed two studies with a sample size of 52 so we're looking for ones that deal with n1=n2=26. [Worse yet, we're soon to run a study with n=60.] In the past we've only ever run studies with 48 subjects or less and our current references for Wilcoxons test statistic give values up to n1=n2=25 only.

Kind regards

Andrew Leary
Helmut
★★★

Vienna, Austria,
2009-01-26 19:43
(4946 d 13:41 ago)

@ Dr Andrew Leary
Posting: # 3128
Views: 13,430

## Moses CI

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 – Z0.05 × √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).

1. Pairwise averages: (Xi+Xj)/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.

Dif-tor heh smusma 🖖
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes
Dr Andrew Leary
★

Ireland,
2009-01-26 19:51
(4946 d 13:34 ago)

@ Helmut
Posting: # 3129
Views: 13,109

## Moses CI

Many thanks, Helmut. I expect that my statistician should be able to make sense of this!

All the best. Keep fighting those regulators.

Andrew

--
Edit: Full quote removed. Please see this post! [Helmut]
Helmut
★★★

Vienna, Austria,
2009-01-26 20:08
(4946 d 13:16 ago)

@ Dr Andrew Leary
Posting: # 3130
Views: 13,139

## Moses CI

Dear Andrew!

» […] Keep fighting those regulators.

Oh, some of them are quite nice guys.
I don’t know the context of your study, but at least in the drafted BE-guideline they have removed nonparametrics at all.

Dif-tor heh smusma 🖖
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes
martin
★★

Austria,
2009-01-27 11:25
(4945 d 22:00 ago)

@ Helmut
Posting: # 3135
Views: 13,034

## Moses CI

dear HS and andrew!

the HL-estimate for the difference and the corresponding CI is discussed for example in this Introductionary book (which I can strongly recommend):

Altman D. G., Machin D., Bryant T. N., Gardener M. J. (2000). Statistics with Confidence. Brit. Med. J. Books, 2nd ed., JW Arrowsmith Ltd., Bristol.

you may find the function HL.diff(x, y, conf.level=0.95, alternative="two.sided", ...) of R package pairwiseCI (Schaarschmidt and Gerhard, 2008) of interest. in the case that you have to use SAS, you may find this link of interest.

best regards
martin
Dr Andrew Leary
★

Ireland,
2009-01-27 11:41
(4945 d 21:44 ago)

@ martin
Posting: # 3136
Views: 13,002

## Moses CI

Thank you both for your responses.

I had not picked up that point during my hasty reading of the new (draft) guideline, Helmut, but saw your extensive discussions of same when looking at the forum yesterday evening.

At present all our BE protocols stick with the old guideline and require non-parametric assessment of Tmax. I can foresee a time in 2 years when we'll have to repeat part of every analysis to provide parametric results. Are you sure that this aspect of the new guideline will stick? I'm a clinician not a statistician, but it seems to me that perfroming parametric stats on a small amount of categorical data flies in the face of traditional teaching.

Kind regards

Andrew

--
Edit: Full quote removed. Please see this post! [Helmut]
Helmut
★★★

Vienna, Austria,
2009-01-27 12:04
(4945 d 21:21 ago)

@ Dr Andrew Leary
Posting: # 3138
Views: 13,188

## Moses CI

Dear Andrew!

» At present all our BE protocols stick with the old guideline and require non-parametric assessment of Tmax.

Even according to the current guideline assessment of tmax was mandatory only if
• either a clinical claim was made (e.g., rapid onset like for some analgetics),
• or based on safety grounds (e.g., IR nifedipine).
Therefore tmax should only be reported in the majority of cases - not compared and assessed for BE.

» I can foresee a time in 2 years when we'll have to repeat part of every analysis to provide parametric results.

Theoretically it should be evident to any regulator that the study was performed according to the current NfG on BA/BE and the Q&A-document. We have no time machine at our disposal for planing our studies - so they should this take into account…
In my experience it will be almost impossible to show BE retrospectively by the partial AUC method proposed in the drafted guideline. The intrasubject-variability of this metric is just scary. I worked like a Trojan at the recent EUFEPS workshop to convince the PK group that it simply doesn’t make sense. Kamal Midha showed nice slides, there are papers published on the topic,… But: If we don’t like it, we have to send comments to EMEA. The clock is ticking. Deadline is this Saturday, 11:59 p.m.

» Are you sure that this aspect of the new guideline will stick?

You never can be sure...
I’ll give you my personal impressions. I guess there are two reasons why the drafted guideline suggested a switch from nonparametric testing of tmax to parametric testing of partial AUC truncated at median tmax of the reference.
• “Harmonization” with the FDA and Canada (you may also read this as copy-and-paste).
• A deep dislike of nonparametrics by UK’s statisticians.

Dif-tor heh smusma 🖖
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes
d_labes
★★★

Berlin, Germany,
2009-01-27 12:10
(4945 d 21:15 ago)

@ martin
Posting: # 3139
Views: 13,031

## Moses CI in the "power to know"

dear Martin, dear all

» ... in the case that you have to use SAS, you may find this link of interest.

The latest version of SAS 9.2 (released 2008! sic, in Germany) now has built in the Moses (Hodges-Lehmann) CI in Proc NPAR1WAY, large sample and exact.
See the HL option and the EXACT statement of this procedure.

The "Power to Know", like wine, improves with age (42 years old, 1966 vintage).
Late, eventually to late for BE studies .

Regards,

Detlew
Dr Andrew Leary
★

Ireland,
2009-01-27 12:33
(4945 d 20:52 ago)

@ d_labes
Posting: # 3141
Views: 13,217

## Moses CI in the "power to know"

» The "Power to Know", like wine, improves with age (42 years old, 1966 vintage).
» Late, eventually to late for BE studies .

Thanks again Helmut, and also Martin for your input. My vintage is 1965 and I'm concerned that this wine was at its best at least decade ago. Now possibly only good for keeping in the cellar rather than drinking.

Regarding Tmax we've been making a BE comparison as standard practice but of course ignoring this for the purposes of declaring bioequivalence/bioinequivalence. It is merely a calculation that sits deeply hidden in the stats appendices.

The English are a strange bunch who take pleasure in disagreeing with everyone else. I have no doubt that their statisticians believe that they invented the science. I don't know much about the Canadians, but I'm told that the Americans have a bureacracy that makes Europe look ungoverned. We're all doomed.

--
Edit: Full quote removed. Please see this post! [Helmut]
Helmut
★★★

Vienna, Austria,
2009-01-27 12:39
(4945 d 20:46 ago)

@ martin
Posting: # 3142
Views: 13,005

## Moses CI

Dear Martin!

» in the case that you have to use SAS, you may find this link of interest.

Haha,
To quote the reference:
Cα is an integer that approximates the ordered value of the lower confidence interval. […] In general the value […] is not an integer, so round to the closest integer and use that in the confidence interval equation above.

Nice, but wrong. Cα from the normal approximation should be rounded to the next lower integer, not to the next closest integer. In the code calpha=round() should be replaced by calpha=int(). Comparing values obtained with the formula from the reference with the 900 (n=m=3 to n=m=32) exact values, one would get the correct value in 815 (90.56%) of cases, a conservative value in 81 (9.00%) of cases, but also in 4 (0.44%) of cases a liberal value (α ≥0.05).
Examples:
1. n=m=12, Cα 43.51, next lower integer 43, next closest integer 44, exact 43.
2. n=30, m=29, Cα 326.51, next lower integer 326, next closest integer 327, exact 326.
If int() is used, the confidence interval is always conservative (≤ nominal α), which may not be the case if round() is used.

The reference also states
For large samples (>30) Cα is a integer approximated by […]
but uses the approximation irrespective of the sample size.

Dif-tor heh smusma 🖖
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes
d_labes
★★★

Berlin, Germany,
2009-01-27 13:22
(4945 d 20:03 ago)

@ Helmut
Posting: # 3144
Views: 12,987

## Deckerian CI

Dear Helmut!

» Haha,

Thanks for clarifying aspects of the SAS code otherwise overlooked by someone (Me not, I'm an initiate ).
Note that this reference is a users contribution to a SAS user group.

It is not "The power to know" but "The power to author", also sometimes called Deckerian power.

This reference is an impressive example of using "The power to know" if the users are lacking the power to know.

But this applies to all software .

Regards,

Detlew
martin
★★

Austria,
2009-01-27 19:10
(4945 d 14:15 ago)

@ d_labes
Posting: # 3147
Views: 12,867

## Deckerian CI

dear HS and d_labes !

fascinating – thank you for pointing this out.

by the way this is one reason why I re-calculate at least the results for the primary endpoint with at least one different implementation / software.

best regards
martin

PS.: You can also calculate a CI for the HL-estimate for the ratio. Just use log-transformed data and anti-log the estimator and the confidence limits so determined (on assumption of a location shift model with equal coefficient of variations). Have a look at the function HL.ratio() in the R package pairwiseCI.
d_labes
★★★

Berlin, Germany,
2009-01-28 15:32
(4944 d 17:53 ago)

@ martin
Posting: # 3157
Views: 12,855

## Transformers

Dear martin,

» PS.: You can also calculate a CI for the HL-estimate for the ratio. Just use log-transformed data and anti-log the estimator and the confidence limits so determined [...]

Since a monotone transformation does not alter the ranks you can achieve the same without any log- and back-transformation. Just start the whole calculations with the individual ratios and use the pairwise geometric ratios as order statistics.

Regards,

Detlew