ElMaestro
★★★

Belgium?,
2019-11-29 03:40

Posting: # 20893
Views: 548

## On CI calculation, untransformed metrics [General Sta­tis­tics]

Hi all,

I never did much work on untransformed metrics for BE calculation, but I am facing a situation where it is mandated by an authority, and what I need is the CI for the ratio (not per se the difference). I'd like to be well prepared.

A relevant publication, at least for sample size, is Hauschke et al. from 1999.

The ratio of two normal distributions is not itself a normal distribution. How is the calculation of the CI for the ratio actually done when the upper and lower limits are percentages of mu(R) ?

I think Hauschke's paper is silent on the matter, as is Chow & Liu.

Note also that powerTOST's nomenclature seems to differ a bit (?) from Hauschke's in that it uses theta1 and theta2 where Hauscke would use f1 and f2. In powerTOST theta1 defaults to 0.8 when the limit for the ratio is actually 0.8*mu(R), or so I am reading it. I may be quite wrong?!?

Anyhow, the important part of this post is how the CI for the ratio is actually derived.

I could be wrong, but...
Best regards,
ElMaestro
Helmut
★★★

Vienna, Austria,
2019-11-29 10:03

@ ElMaestro
Posting: # 20894
Views: 282

## Fieller’s (‘fiducial’) confidence interval

Hi ElMaestro,

» […] powerTOST's nomenclature seems to differ a bit (?) from Hauschke's in that it uses theta1 and theta2 where Hauscke would use f1 and f2. In powerTOST theta1 defaults to 0.8 when the limit for the ratio is actually 0.8*mu(R), or so I am reading it.

Nope. Hauschke et al.1 use $$H_0:\frac{\mu_T}{\mu_R}\leqslant\theta_1\,\textrm{or}\,\frac{\mu_T}{\mu_R}\geqslant\theta_2\;\textrm{versus}\;H_1:\theta_1<\frac{\mu_T}{\mu_R}<\theta_2\tag{3}$$and $$(\theta_1,\theta_2)=(0.8,1.25)$$ as well (see the figures and paragraphs below them).

» […] the important part of this post is how the CI for the ratio is actually derived.

It never hurts to read the primary document.2

1. Hauschke D, Kieser M, Diletti E, Burke M. Sample size determination for proving equivalence based on the ratio of two means for normally distributed data. Stat Med. 1999;18(1):93–105. doi:10.1002/(SICI)1097-0258(19990115)18:1<93::AID-SIM992>3.0.CO;2-8.
2. Fieller EC. Some Problems in Interval Estimation. J Royal Stat Soc B. 1954;16(2):175–85. JSTOR:2984043.

Cheers,
Helmut Schütz

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ElMaestro
★★★

Belgium?,
2019-11-29 15:53

@ Helmut
Posting: # 20897
Views: 243

## Fieller’s (‘fiducial’) confidence interval

Hi Hötzi,

» Nope.

For such cases we are setting logscale to False, right?
PowerTOST says about theta1: "Defaults to 0.8 if logscale=TRUE or to -0.2 if logscale=FALSE. "
So as I read it, when logscale=F theta1 defaults to -.2 or we set it to something negative. This is how I think Hauchke's f1 and theta1 get confused.

I will read up on the other stuff. Still not sure how to derive the CI. it is a really interesting problem, though. Feels like going back into a discussion which was closed decades ago before I knew anything about BE

Thanks PharmCat for the ref. below.

I could be wrong, but...
Best regards,
ElMaestro
d_labes
★★★

Berlin, Germany,
2019-11-29 17:20

@ ElMaestro
Posting: # 20898
Views: 233

## Fieller’s (‘fiducial’) confidence interval

Dear ElMaestro,

» For such cases we are setting logscale to False, right?

Correct in so far if we use the approximation that the estimate of µR is (statistically) greater than zero. A very reasonable assumption for the usual metrics AUC and Cmax IMHO.
But this has than nothing to do with Fieller’s (‘fiducial’) confidence interval, a more correct method for deriving a confidence interval for the ratio of untransformed PK metrics.
PowerTOST has for this case the functions
- CI.RatioF() - power.RatioF() - sampleN.RatioF()
Have a look into the man pages of these functions and notice that these functions don't have a logscale argument.

» I will read up on the other stuff. Still not sure how to derive the CI. ...
Eventually the book

Hauschke D, Steinijans VW, Pigeot I.
Bioequivalence Studies in Drug Development
Chichester: John Wiley; 2007

may help.
Especially chapter 10 Equivalence assessment for clinical endpoints.
If you don't own this book, gimme a hint.

Regards,

Detlew
ElMaestro
★★★

Belgium?,
2019-11-30 05:11

@ d_labes
Posting: # 20899
Views: 203

## Fieller’s (‘fiducial’) confidence interval

Dear d_labes,

you left me baffled.

» » For such cases we are setting logscale to False, right?
»
» Correct in so far if we use the approximation that the estimate of µR is (statistically) greater than zero. A very reasonable assumption for the usual metrics AUC and Cmax IMHO.

Please explain then what exactly it is that power.TOST calculates when I use logscale=F.
Does it calculate power for a hypothesis based on a difference or for a ratio?
Which difference? Which ratio?

» But this has than nothing to do with Fieller’s (‘fiducial’) confidence interval, a more correct method for deriving a confidence interval for the ratio of untransformed PK metrics.

The mention of Fieller was not mine. I am quite confused now, what it is power.TOST tries to calculate when I do logscale=F.

I am convinced the assuming theta1=-0.2 by default when logscale=F is a misnomer. theta1 is elsewhere understood as an equivalence margin expressed as a ratio and that can't realistically be negative. If powerTOST tries to emulate Hauschke's paper then -.2 is f1, not a theta.
We need to be careful here about f, delta and theta.

I could be wrong, but...
Best regards,
ElMaestro
d_labes
★★★

Berlin, Germany,
2019-11-30 14:01

@ ElMaestro
Posting: # 20900
Views: 185

## power.TOST with logscale=FALSE

Dear ElMaestro,

» Please explain then what exactly it is that power.TOST calculates when I use logscale=F.
» Does it calculate power for a hypothesis based on a difference or for a ratio?
» Which difference? Which ratio?

Take the reference you mentioned above, Hauschke et al. Statist. Med 1999 and use equation (2) for the hypotheses tested, hypotheses based on the difference µT-µR.
These hypotheses can be reformulated with ratios as written in equation (3) by division with µR and by adding 1. But this then had the implicit constraint that µR has to be >0.

» ...
» I am convinced the assuming theta1=-0.2 by default when logscale=F is a misnomer. theta1 is elsewhere understood as an equivalence margin expressed as a ratio and that can't realistically be negative. If powerTOST tries to emulate Hauschke's paper then -.2 is f1, not a theta.

PowerTOST does not emulate any paper. And it does not use the argument theta1 solely as equivalence margin of a ratio. See the man page of power.TOST() how theta1, theta2 and theta0 have to be set for logscale = TRUE or logscale=FALSE.
But you are right: compared to Hauschke's paper -.2 is f1.

Please consider the rest of the Hauschke paper introducing the Fieller CI as exact method.

Regards,

Detlew
PharmCat
★

Russia,
2019-11-29 12:05

@ ElMaestro
Posting: # 20896
Views: 266