Bioequivalence and Bioavailability Forum • Sample size for ref replicate

AB
☆

India,
2012-03-09 13:42
(5214 d 01:13 ago)

Posting: # 8242
Views: 6,119

Sample size for ref replicate [Power / Sample Size]

Dear Helmut

in one of your presentatitions it is mentioned as below

"Replicate designs
Two-sequence three-period
T R T
R T R
Sample size to obtain the same power as a 2×2×2 study: 75%
Two-sequence four-period
T R T R
R T R T
Sample size to obtain the same power as a 2×2×2 study: 50%"

could you provide the rationale to consider the sample size as 75% & 50% (of 2×2×2 study) for 3 period & 4 period replicate designs respectively?

—
Regards,
AB

Helmut
★★★

Vienna, Austria,
2012-03-09 14:32
(5214 d 00:24 ago)

@ AB
Posting: # 8243
Views: 5,274

3-way ~50%, 4-way ~75% of 2×2

Post reply

Dear AB!

THX for bringing this inaccuracy to my attention. I will write in the future ~75% and ~50%. Actual sample sizes depend on the design (3-way / 4-way, number of subjects & sequences / periods ⇒ degrees of freedom). For the dfs see here. Furthermore in designing the study we have to consider complete sequences (which are a multiple of 3 in the partial replicate TRR|RTR|RRT and a multiple of 4 in TRTR|RTRT|TTRR|RRTT – compared to the multiple of 2 in RT|TR).

❝ could you provide the rationale to consider the sample size as 75% & 50% (of 2×2×2 study) for 3 period & 4 period replicate designs respectively?

Power stays roughly the same keeping the number of treatments across designs. Sample sizes obtained by D. Labes’ famous PowerTOST (α 0.05, AR 80–125%, CV 50%, θ₀ 95%, target power 80%):

Design alpha  CV theta0 theta1 theta2 Sample size Achieved power Target power % of 2×2

   2x2  0.05 0.5   0.95    0.8   1.25          98      0.8032172          0.8

 2x2x3  0.05 0.5   0.95    0.8   1.25          74      0.8076842          0.8    75.75

 2x3x3  0.05 0.5   0.95    0.8   1.25          75      0.8128304          0.8    75.53

 2x2x4  0.05 0.5   0.95    0.8   1.25          50      0.8128063          0.8    51.02

 2x4x4  0.05 0.5   0.95    0.8   1.25          52      0.8274094          0.8    53.06

It’s even possible to get the target power with sample sizes <75% or <50% in the case of drop-outs (unequal sequences) – but you cannot plan for that.
First 3-way replicate from above, balanced (37/37) and 1 drop-out (37/36).

power2.TOST(alpha=0.05, logscale=TRUE, CV=0.5, n=c(37,37), design="2x2x3", robust=TRUE)

0.8076842

74 subjects = 75.75% of 2×2

power2.TOST(alpha=0.05, logscale=TRUE, CV=0.5, n=c(36,37), design="2x2x3", robust=TRUE)

0.8023176

73 subjects = 74.49% of 2×2
Or the last design: 1 drop-out in three of the four sequences each:

power2.TOST(alpha=0.05, logscale=TRUE, CV=0.5, n=c(12,12,12,13), design="2x4x4", robust=TRUE)

0.8045673

49 subjects = 50.00% of 2×2

—
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

AB
☆

India,
2012-03-12 09:35
(5211 d 05:20 ago)

@ Helmut
Posting: # 8249
Views: 5,224

3-way ~50%, 4-way ~75% of 2×2

Post reply

Dear HS,

Many thanks for detailed explanation.
But according to Table 1 “Average Bioequivalence Estimated Numbers of Subjects D=0.05” Results for two-period designs use method of Diletti et al. (Diletti 1991) from FDA’s Statistical Approaches to
Establishing Bioequivalence (page 28 of guidance) Sample size of 2 period with CV of 50 and D=0.05 and power 80% is given as 108 and 75% & 50% of the same is 81 & 54.

and when i check the sample size @ 50% CV, 80% power & GMR 0.95 given in the document article (which i came to know from your previous post) is:
30 & 22 for 3 & 4 period studies respectively according to EMA.
24 & 17 for 3 & 4 period studies respectively according to FDA.

Am :confused:

why these figures are very different in the guideline, article to the calculation you illustrated.

Thanks in advance.

—
Regards,
AB

d_labes
★★★

Berlin, Germany,
2012-03-12 16:35
(5210 d 22:21 ago)

@ AB
Posting: # 8253
Views: 5,424

Mixed up something?

Post reply

Dear AB,

❝ But according to Table 1 “Average Bioequivalence Estimated Numbers of Subjects D=0.05” Results for two-period designs use method of Diletti et al. (Diletti 1991) from FDA’s Statistical Approaches to Establishing Bioequivalence (page 28 of guidance) Sample size of 2 period with CV of 50 and D=0.05 and power 80% is given as 108 and 75% & 50% of the same is 81 & 54.

The cited table, as very clearly depicted in the heading (!), is sample size for common average bioequivalence.
Further it is detailed not for any CV but for the standard deviations (square root of variance) of variability decomposed into within-subject variance (assuming equal variances for Test and Reference) and the so-called subject by formulation variance sigma_D.
Additionally this table seems to be obtained using an point estimator of theta₀=1.05 whereas conventionally 0.95 is used because the power is some amount smaller there.
A third difference is the usage of the degrees of freedom = N-seq (called 'robust' df in PowerTOST).

You could obtain the sample sizes of this table considering according to Wang and Chow⁽¹⁾ the formulas relating the intra-subject variance and the subject-by-formulation variance to the variance of the treatment differences (mixed model analysis):

for 2x2 cross-over

var₁=var(µT-µR)=0.5*(s²_D + s²_WT + s²_WR)



for 4-period repl. cross-over

var₂=var(µT-µR)=s²_D + 0.5(s²_WT + s²_WR)



CV_{1 or 2}=sqrt(exp(var_{1 or 2})-1)

and then using CV's for sample size estimation f.i. in PowerTOST.

Here some R-code to play with:

require(PowerTOST)

tp  <- 0.8    # target power

sD  <- 0.01

sWT <- 0.5

sWR <- 0.5

var1 <- 0.5*(sD*sD + sWT*sWT + sWR*sWR)

var2 <- sD*sD + 0.5*(sWT*sWT + sWR*sWR)

CV1  <- sqrt(exp(var1)-1)

CV2  <- sqrt(exp(var2)-1)

sampleN.TOST(CV=CV1, theta0=1.05, targetpower=tp, design="2x2")

sampleN.TOST(CV=CV2, theta0=1.05, targetpower=tp, design="2x2x4", robust=TRUE)

.

This will give you n=108 for design="2x2" and n=54 for design="2x2x4", the numbers shown in Table 1 of the FDA guidance. Note CV1 / CV2 ~0.533 (53.3%).
All other entries of Table 1 can also reproduced with this code (with some minor deviations for power 90%). Try it :cool:

❝ and when i check the sample size @ 50% CV , 80% power & GMR 0.95 given in the document article ...why these figures are very different in the guideline, article to the calculation

The sample size tables of your linked document deals with scaled average bioequivalence evaluated via the methods recommended by EMA (widened acceptance limits) or FDA (95% CI of the scaled ABE criterion).
These methods are completely different from the conventional ABE and can't therefore compared directly :no:

(1)Hansheng Wang and Shein-Chung Chow (2002)
"On Statistical Power for Average Bioequivalence Testing under Replicated Crossover Designs"
J. BIOPHARM. STAT. Vol. 12, No. 3, pp. 295–309, 2002

—
Regards,

Detlew