Bioequivalence and Bioavailability Forum

Dear Kotu!

❝ We are going to conduct one "Two-Stage" bioequivalence study. In this study initially we had fixed the sample size as maximum of 80 subjects (Both stage1 and stage2).

Both Potvin et al. (2007)¹ and Montague et al. (2011)² don’t set a fixed maximum sample size. See the discussion section in Potvin et al.:

More work needs to be done to examine the extension to more than two stages, issues associated with pooling results from two or more stages, inclusion of a futility rule, minimum sample sizes for second stage if performed, and upper limits on the total sample size.

If you are including a fixed sample size in your method you are leaving the validated range covered by the papers. Intuitively I would say you are only jeopardizing power (higher producer’s risk) – but actually I don’t know whether the patient’s risk is unaffected. Since you plan for a maximum sample size of 80 subjects, I guess you expect a high CV (GMR 95%, 80% power)? Have a look at Potvin’s Table II, Method C (36 subjects in Stage 1 because 40 are not given): With a CV of 40% the sample sizes in Stage 2 are 34 (median), with 5 and 95% percentiles of 0 and 76. In other words, 66% of studies proceed to Stage 2; it’s possible that you need more than your maximum sample size to demonstrate BE. If you expect a GMR of 90% and plan for power 80% (Montague, Method D) let’s see another example (Stage 1 36 subjects, CV 30%): 97% chance to proceed to Stage 2, median sample size in Stage 2 58 subjects (5–95%: 22 and 102 subjects).

❝ In stage-1 we include 40 subjects and based on stage1 data we will precede with a maximum sample size of 40 subjects in stage-2 (sample size will be ≤ 40).

See above.

❝ Is there any statistical method to decide sample size in stage2?

You run a conventional estimation of the total sample size (n_total) based on the assumed GMR (fixed at 0.95 – Potvin or 0.90 – Montague), power 80%, adjusted α of 0.0294 (Potvin) or 0.0280 (Montague), and the CV observed in Stage 1. Then

n₂ = n_total – n₁.

I recommend Detlew Labes’ package PowerTOST for R.
Example: 40 subjects dosed in stage 1, 2 drop outs (n₁ 38), fixed GMR 95% (Potvin), CV 43%, adjusted α for the pooled analysis 0.0294 (= 94.12% confidence interval in the pooled analysis)

require(PowerTOST)
sampleN.TOST(alpha=0.0294, targetpower=0.8, theta0=0.95, CV=0.42, design="2x2")
You will get:
+++++++++++ Equivalence test - TOST +++++++++++
            Sample size estimation
-----------------------------------------------
Study design:  2x2 crossover
log-transformed data (multiplicative model)

alpha = 0.0294, target power = 0.8
BE margins        = 0.8 ... 1.25
Null (true) ratio = 0.95,  CV = 0.43

Sample size (total)
 n     power
90   0.805090
Therefore n₂: 90 – 38 = 52.
If you don’t care about leaving the validated range of the method and perform the second stage with only 42 subjects (limited by your maximum sample size 80) power (maybe!) will be 75.2%:
power.TOST(alpha=0.0294, theta0=0.95, CV=0.43, n=38+42, design = "2x2")

For Montague’s Method D use alpha=0.0280 and theta0=0.90 instead.

Other opinions (ElMaestro, Detlew)?

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1. Potvin D, DiLiberti CE, Hauck WW, Parr AF, Schuirmann DJ, Smith RA. Sequential design approaches for bioequivalence studies with crossover designs Pharmaceut Statist. 2008;7(4):245–62. doi:10.1002/pst.294.
   
2. Montague TH, Potvin D, DiLiberti CE, Hauck WW, Parr AF, Schuirmann DJ. Additional results for ‘Sequential design approaches for bioequivalence studies with crossover designs’. Pharmaceut Statist. 2012;11(1):8–13. doi:10.1002/pst.483.