Helmut
★★★

Vienna, Austria,
2012-03-21 17:18
(4028 d 13:11 ago)

Posting: # 8306
Views: 7,879

Empiric distributions [PK / PD]

Dear all,

Detlew wrote here:

❝ The whole discussion up to here depends on the assumption of log-normality. If this is a reasonable assumption for measures of fluctuation like PTF or swing is left to you.

Good question. For Cmax and AUC there’s a consensus that these metrics are lognormal. See also the empiric distribution of AUCs of methylphenidate in 405 subjects in this presentation which agrees with pharmacokinetic considerations and lead to a multiplicative model (or additive on logtransformed data). Steinijans et al.* state about MR theophylline:

The choice of the multiplicative model and hence the ratio analysis for the plateu time T75% Cmax is motivated by the distribution analysis of the plateau time for the test formulation Euphylong® in n = 102 healthy subjects and n = 85 patients presenting with obstructive pulmonary diseases. The logarithmic normal distribution was superior to the normal distribution (data on file).
(my emphasis)

Cmax/AUC, %PTF, and Swing were also analysed based on logtransformed data without further justification.

I looked at my MPH MR data (10–60 mg, SD/MD; linear PK). Data of 303 subjects (HVD, MRT, Cmax/AUC) and 276 subjects (t75%):

Interesting, isn’t it?

• Steinjans VW, Sauter R, Diletti E. Shape Analysis in Single- and Multiple-Dose Studies of Modified Release Products. In: Blume HH and KK Midha, editors. Bio-International 2. Stuttgart: medpharm Scientific Pub­li­shers; 1995. p. 193–206.

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Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
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ElMaestro
★★★

Denmark,
2012-03-21 18:32
(4028 d 11:58 ago)

@ Helmut
Posting: # 8307
Views: 6,887

nitpicking

Dear HS,

[nitpick]

❝ For Cmax and AUC there’s a consensus that these metrics are lognormal.

No. There is consensus that you have to analyse the data as though they (the residuals) were normal. It does not mean it is assumed that they are. Regulators, some of them at least, know that normality sometimes really, truly isn't the case but they are willing to ignore this at the evaluation stage.

[/nitpick]

Anyways, as usual I have no qualified response to any of the core stuff you have raised in this thread. Have a good day.

Pass or fail!
ElMaestro
Helmut
★★★

Vienna, Austria,
2012-03-22 17:06
(4027 d 13:23 ago)

@ ElMaestro
Posting: # 8317
Views: 6,784

nitpicking

Dear ElMaestro!

❝ [nitpick]

❝ No. There is consensus that you have to analyse the data as though they (the residuals) were normal.

❝ [/nitpick]

Nitpicking is one of my hobbies. Working on the residuals now.

❝ Anyways, as usual I have no qualified response to any of the core stuff you have raised in this thread.

I’m interested in underlying distributions. Should give me a clue whether to transform or not.

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Helmut Schütz

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

Austria,
2012-03-21 22:16
(4028 d 08:13 ago)

@ Helmut
Posting: # 8308
Views: 6,909

Empiric distributions

dear HS!

pretty interesting; thank you very much for sharing this information. If you find some time I would be happy to see the Q-Q plots and the Q-Q plots of the log-transformed PK parameters.

best regards

martin

PS.: have you also investigated the distribution of residuals of the applied statistical model?
Helmut
★★★

Vienna, Austria,
2012-03-22 02:17
(4028 d 04:12 ago)

@ martin
Posting: # 8309
Views: 7,042

QQ-Plots

Dear Martin!

❝ If you find some time I would be happy to see the Q-Q plots and the Q-Q plots of the log-transformed PK parameters.

Here you are:

❝ PS.: have you also investigated the distribution of residuals of the applied statistical model?

I used to analyze HVD, t75%, and MRT untransformed and Cmax/AUC logtransformed, so I have these residuals on file. Will analyze them the other way ’round time allowing.

First impressions (5 studies, 76 subjects):

Which one would you prefer?

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Helmut Schütz

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d_labes
★★★

Berlin, Germany,
2012-03-22 10:31
(4027 d 19:58 ago)

@ Helmut
Posting: # 8310
Views: 6,861

Transform or not transform

Dear Helmut!

First many thanks for sharing this information.
Your pictures are a good example of the inherent difficulties in determining the distribution form empirical.

Thus I'm a fan of the arguments for the log-normal distribution via theoretical PK considerations (@EM: even if regulators have written down them in guidances ).

❝ I used to analyze HVD, t75%, and MRT untransformed and Cmax/AUC logtransformed ...

IMHO this is a good choice .
Although the residuals (and these count at least I think) don't show a very distinct picture.

BTW: Since these metrics (the ones you have shown) are usually not primaries then the question of their (their residuals) distribution is not so much of concern I think. I would handle them only in a descriptive way (mean, sd, median and ... and ...). Or do you analyze those metrics also via ANOVA and (1-2*alpha) CI's in a standard fashion?

My originally question was more in the direction of swing metrics. Do you have similar data for PTF or swing? As for ratios of two terms deemed as log-normally distributed I at least questioning a log-normal distribution. On the other hand one may argue with your results for Cmax/AUC ...

Geary 1947, Biometrika
Normality is a myth; there never was, and never will be, a normal distribution.

Regards,

Detlew
Helmut
★★★

Vienna, Austria,
2012-03-22 14:27
(4027 d 16:03 ago)

@ d_labes
Posting: # 8316
Views: 6,937

Transform or not transform

Dear Detlew!

❝ Your pictures are a good example of the inherent difficulties in determining the distribution form empirical.

I would rather say: Impossibility in any given study due to limited sample size, but good chances if a wealth of historical data is available.

❝ Thus I'm a fan of the arguments for the log-normal distribution via theoretical PK considerations.

Absolutely. I have no problems with AUC (and Cmax as well) but fail to derive a reasonable justification for others. Walnut brain. At the first Bio-International there was this poll amongst participants about transformations. Result was ⅓ always, ⅓ never, ⅓ case-by-case. My idea was to study the empirical distributions of less common metrics to derive a suggestion (of course applicable only to a specific drug/formulation).

❝ ❝ I used to analyze HVD, t75%, and MRT untransformed and Cmax/AUC logtransformed ...

❝ IMHO this is a good choice .

OK, but how did you conclude that?

❝ Although the residuals (and these count at least I think) don't show a very distinct picture.

Yes, the sample size I assessed is yet inconclusive. See here for the final outcome.

❝ BTW: Since these metrics (the ones you have shown) are usually not primaries then the question of their (their residuals) distribution is not so much of concern I think. I would handle them only in a descriptive way (mean, sd, median and ... and ...). Or do you analyze those metrics also via ANOVA and (1-2*alpha) CI's in a standard fashion?

These data come from MR products. I calculated the CIs – but only descriptively.

❝ My originally question was more in the direction of swing metrics. Do you have similar data for PTF or swing? As for ratios of two terms deemed as log-normally distributed I at least questioning a log-normal distribution. On the other hand one may argue with your results for Cmax/AUC ...

Well, that’s the reason I started a new thread. I never calculated Swing. Though I have a lot of MD studies the pooled sample size / drug is too small to assess the empiric distributions. I would expect PTF to have a similar distribution as Cmax/AUC (at least if Cmin ⇒ LLOQ).

Normality is a myth; there never was, and never will be, a normal distribution.

Nice quote.

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Helmut Schütz

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ElMaestro
★★★

Denmark,
2012-03-22 19:16
(4027 d 11:13 ago)

@ Helmut
Posting: # 8318
Views: 6,791

Empiric distributions

Hi HS,

this thread gave me an idea (sorry for going a bit off topic now):
The whole general use of log transformation is quite unjustified from a biological perspective, but has a lot of appeal from a mathematical perspective in that it rectifies an assumption that must hold for normal dists and which can strictly be said to be violated with untransformed values. But other empirical transformations might have appeal as well.

Someone should therefore make an Al Gore Rhythm which could find the 'best' transformation among a set of possible transformations (this would by nature have to be a limited number of known transformations) on a limited number of datasets and see if there's anything useful, like consensus, coming out for whichever parameters are of interest.

Of course, there's trouble ahead: It might not be easy to define an objective function that gives a clearcut winner. I am myself in uncharted territory here but I would perhaps naïvely start with something like the Shapiro-Wilks statistic or a regression goodness-of-fit statistic from the QQ plot, and then make sure not to compare it between datasets but only within.

Just another useless idea from ElMaestro... I should perhaps just stick to poetry?!?

Pass or fail!
ElMaestro
Helmut
★★★

Vienna, Austria,
2012-03-23 02:50
(4027 d 03:39 ago)

@ ElMaestro
Posting: # 8319
Views: 6,938

Empiric distributions

Dear ElMaestro!

❝ this thread gave me an idea (sorry for going a bit off topic now):

That’s perfect on topic!

❝ The whole general use of log transformation is quite unjustified from a biological perspective, but has a lot of appeal from a mathematical perspective in that it rectifies an assumption that must hold for normal dists and which can strictly be said to be violated with untransformed values. But other empirical transformations might have appeal as well.

Yep.

❝ Someone should therefore make an Al Gore Rhythm which could find the 'best' transformation among a set of possible transformations (this would by nature have to be a limited number of known transformations) on a limited number of datasets and see if there's anything useful, like consensus, coming out for whichever parameters are of interest.

Right. $$1/\sqrt{x}$$ is a nice one. I think there are different parties out there:
• What the heck are you talking about?
• Not being aware about assumptions and/or stick to what the majority does.
• Don’t test assumptions, although: “Sponsors and/or applicants are not encouraged to test for normality of error distribution after log-transformation, nor should they use normality of error distribution as a reason for carrying out the statistical analysis on the original scale. Justification should be provided if sponsors or applicants believe that their BE study data should be statistically analyzed on the original rather than on the log scale. (FDA 2001)
• I don’t care. Asymptotics will save me.
• Log-transformation is based on PK grounds. Reasonable for AUC (and Cmax?) – but other metrics?
• Log-transform or not (if justified) or even go with a nonparametric method (Japan 2006)

❝ Of course, there's trouble ahead: It might not be easy to define an objective function that gives a clearcut winner.

Yep. Generally the sample size is much too small. Whilst log-transform is a clear winner in the 405 subjects (slide 43) I would not bet on the SW’s in the next slide (12 subjects; p 0.29668 vs. 0.85764). Normality tests are no decision tools between two transformations.

❝ I am myself in uncharted territory here but I would perhaps naïvely start with something like the Shapiro-Wilks statistic or a regression goodness-of-fit statistic from the QQ plot, and then make sure not to compare it between datasets but only within.

What do you mean by “only within”? If we run them on the single studies, power will be too low (see the example above) and inconclusive. That’s why Volker Steinijans looked at the distribution of historical data and defined the method of a particular study a priori. I think that’s the way to go. But: Requires data, data, data.

So here we are – 10 studies on MR methylphenidate formulations (designs 2×2, 6×3, 4×4; 10–60 mg, 12–24 subjects/study, 174 subjects total; t75%, HVD, MRTt, Cmax/AUCt):

Tails are amazing. We use a pretty sensitive method; LLOQ <1% of Cmax, residual AUC <5%, residual AUMC <15%…

❝ Just another useless idea from ElMaestro... I should perhaps just stick to poetry?!?

Have you tried that before? Far more complicated.

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Helmut Schütz

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