Bioequivalence and Bioavailability Forum

martin
★★

Austria,
2012-11-15 18:29
(4599 d 21:28 ago)

Posting: # 9534
Views: 6,115

Power with Fieller [General Statistics]

Dear all!

We recently encountered an unexpected behavior (at least I was not aware of it) with Fieller confidence intervals. It seems that the power for showing equivalence is not maximized at delta = 1 for rather large variances. Is there an error in the code or is this behavior well known?

best regards

martin


library(pairwiseCI) 

set.seed(7750)

nsim <- 1E5

sd <- 0.25

res1 <- res2 <- res3 <- res4 <- matrix(NA,ncol=2,nrow=nsim) 

n <- 10 

for(i in 1:nsim){

   a1 <- rnorm(n,1,sd=sd)

   a2 <- rnorm(n,0.975,sd=sd)

   a3 <- rnorm(n,1.025,sd=sd)

   a4 <- rnorm(n,1.050,sd=sd)

   b <- rnorm(n,1,sd=sd)

   res1[i,] <- as.vector(Param.ratio(a1,b,conf.level=0.9,var.equal=F)$conf.int)

   res2[i,] <- as.vector(Param.ratio(a2,b,conf.level=0.9,var.equal=F)$conf.int)

   res3[i,] <- as.vector(Param.ratio(a3,b,conf.level=0.9,var.equal=F)$conf.int)

   res4[i,] <- as.vector(Param.ratio(a4,b,conf.level=0.9,var.equal=F)$conf.int)

}



# delta=1: expected to have the largest power 

mean(res1[,1]>0.8 & res1[,2]<1.25)

# delta=0.975

mean(res2[,1]>0.8 & res2[,2]<1.25)

# delta=1.025

mean(res3[,1]>0.8 & res3[,2]<1.25)

# delta=1.05

mean(res4[,1]>0.8 & res4[,2]<1.25)

Helmut
★★★

Vienna, Austria,
2012-11-16 02:28
(4599 d 13:30 ago)

@ martin
Posting: # 9535
Views: 4,980

Power with Fieller

Post reply

Hi Martin,

the term “fiducial probability” always scares me shitless. Let’s see:

       var.equal=F¹  var.equal=T²    PowerTOST¹

0.975    0.20600       0.20765       0.2104624

1.000    0.22331       0.22842       0.2288324

1.025    0.23282       0.23900       0.2382360

← larger than at 1 (all methods)
1.950 0.23159 0.23411 0.2365934
Not sure whether it is correct to simply plug in your sd 0.25 as CV into PowerTOST (though the mean is 1): power.RatioF(alpha=0.05, theta1=0.8, theta2=1.25, theta0=1, CV=0.25, n=20, design="parallel"). However, the same tendency.
Reminds me on the good ol’ days where people started to analyse log-transformed data in BE, but kept the acceptance range at ±20%. Maximum power at 0.9798. ;-)

I’m not familiar with this kind of stuff, but doesn’t the analysis work with untransformed data, i.e., Δ ±0.2 → 0.8–1.20?

Homogenous variances
Fieller EC. Some problems in interval estimation. J Royal Stat Soc B. 1954:16(2);175–85. online
Heterogenous variances
Tamhane AC, Logan BR. Finding the maximum safe dose level for heteroscedastic data. J Biopharm Stat. 2004:14(4);843–56. doi 10.1081/BIP-200035413
Hasler M, Vonk R, Hothorn LA. Assessing non-inferiority of a new treatment in a three arm trial in the presence of heteroscedasticity. Stat Med. 2008:27(4);490–503. doi 10.1002/sim.3052

—
Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz

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

d_labes
★★★

Berlin, Germany,
2012-11-16 10:03
(4599 d 05:55 ago)

@ martin
Posting: # 9537
Views: 4,831

'Expected' Power with Fieller

Post reply

Dear Martin, dear Helmut!

❝ We recently encountered an unexpected behavior (at least I was not aware of it) with Fieller confidence intervals. It seems that the power for showing equivalence is not maximized at delta = 1 for rather large variances. Is there an error in the code or is this behavior well known?

This behaviour is not unexpected. See the various power curves in:

Hauschke, Steinijans, Pigeot
"Bioequivalence Studies in Drug Development"
Wiley, Chichester (2007)
Chapter 10: Equivalence assessment for clinical endpoints /
10.3 Power and sample size calculation

They all show maximum power apart from ratio 1.

This has only partly to do with the fact that we use the 'unsymmetrical' acceptance ranges [theta1=0.8, theta2=1/theta1=1.25] ('unsymmetrical' in the original domain):

library(PowerTOST)

power.RatioF(alpha=0.05, theta1=0.8, theta2=1.2, theta0=0.98, CV=0.25, n=20, design="parallel")

theta0   power

0.98    0.123749

1       0.130499

1.01    0.132072

1.02    0.1328319

1.03    0.1321316

1.04    0.1310061

—
Regards,

Detlew

Helmut
★★★

Vienna, Austria,
2012-11-16 14:54
(4599 d 01:03 ago)

@ d_labes
Posting: # 9541
Views: 4,786

'Expected' Power with Fieller

Post reply

Dear Detlew & Martin!

❝ This behaviour is not unexpected. See the various power curves in:

❝

Hauschke, Steinijans, Pigeot

❝ "Bioequivalence Studies in Drug Development"

❝ Wiley, Chichester (2007)

❝ Chapter 10: Equivalence assessment for clinical endpoints /

❝ 10.3 Power and sample size calculation

They all show maximum power apart from ratio 1.

I see.

❝ This has only partly to do with the fact that we use the 'unsymmetrical' acceptance ranges [theta1=0.8, theta2=1/theta1=1.25] ('unsymmetrical' in the original domain):

Noticed that yesterday, but was too tired to explore it further.

Forget my idea. The acceptance range is asymmetrical in the original domain (θ₁, θ₂=1/θ₁).
Interesting the last sentence on page 284: “Additionally, it can be shown that, only for equivalence limits so defined, the maximum power stabilizes at the point of equality, i.e., μ_T∕μ_R=1, as the sample size increases.*”
(my emphasis)

❝ ❝ […] is this behavior well known?

Yes – at least for equal variances. Martin, your sims show a similar behavior for heterogenous variances. Increase the sample size and see what’s happening.

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

—
Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz

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

Jack ☆ Lancaster, United Kingdom, 2012-11-19 11:35 (4596 d 04:22 ago) @ Helmut Posting: # 9549 Views: 4,748	'Expected' Power with Fieller Post reply
	Thanks all for the input and in particular the reference. I was completely unaware of this and got really worried Anyway to contribute something substantial as well: Yes the curves do center once the sample size increases. Jack

Power with Fieller [General Sta­tis­tics]

Power with Fieller

'Expected' Power with Fieller

'Expected' Power with Fieller

'Expected' Power with Fieller

Power with Fieller [General Statistics]