## Purpose of the study? [Power / Sample Size]

Hi Sveta,

» Let’s say, we are planning a 2x2 crossover study for BE, and the intra-subject coefficient of variation for our test compound, from a prior study, is 23%. The assumed geometric mean ratio is 1.00, …

Which PE did you find in the prior study? Its sample size would be helpful as well.

» … alpha is 0.05, equivalence limits are 80%-125%, and we desire power of 80%. These criteria result in an overall sample size of 20 subjects, or 10 per sequence.

Correct – if you are a believer of the ‘carved in stone’ approach (

» Subsequently, …

What do you mean by ‘subsequently’?

» … we want to add 2 extra dose levels of our test drug, resulting in a 4x4 crossover trial.

What is the purpose of the study – dose proportionality? If yes, that’s another cup of tea.

» Let’s assume that %CV and GMR are not changing, as no further data is available.

Rather strong assumptions, right?

» For this new scenario, is it appropriate to: 1) use the original sample size of N=20 and simply divide it over 4 sequences (5 per sequence), or 2) take the original per sequence sample size of 10 and multiply by 4 to get 40 subjects overall needed?

It depends whether you want to show dose-normalized equivalence (

Say we have four formulations (

You randomize subjects either to the Latin Square

» Let’s say, we are planning a 2x2 crossover study for BE, and the intra-subject coefficient of variation for our test compound, from a prior study, is 23%. The assumed geometric mean ratio is 1.00, …

Which PE did you find in the prior study? Its sample size would be helpful as well.

» … alpha is 0.05, equivalence limits are 80%-125%, and we desire power of 80%. These criteria result in an overall sample size of 20 subjects, or 10 per sequence.

Correct – if you are a believer of the ‘carved in stone’ approach (

*i.e.*, that in the planned study the CV will be*exactly*23% and the PE*exactly*1). I suggest to have a look at Example 3 of the ABE-Vignette of the R package`PowerTOST`

to reconsider your assumptions. See also there (slide 8 and followings).» Subsequently, …

What do you mean by ‘subsequently’?

» … we want to add 2 extra dose levels of our test drug, resulting in a 4x4 crossover trial.

What is the purpose of the study – dose proportionality? If yes, that’s another cup of tea.

» Let’s assume that %CV and GMR are not changing, as no further data is available.

Rather strong assumptions, right?

» For this new scenario, is it appropriate to: 1) use the original sample size of N=20 and simply divide it over 4 sequences (5 per sequence), or 2) take the original per sequence sample size of 10 and multiply by 4 to get 40 subjects overall needed?

It depends whether you want to show dose-normalized equivalence (

*i.e.*, strict dose normality) or dose proportionality by the power model \(E[Y]=\alpha \cdot D^{\; \beta}\).Say we have four formulations (

`A`

, the reference `R`

, `B`

, and `C`

) and three dose levels (*x*,*y*,*z*), where`A`

= `R`

= *x*,`B`

= *y*,`C`

= *z*.You randomize subjects either to the Latin Square

`ARBC|RBCA|BCAR|CARB`

or any one of the six Williams’ designs `ACBR|RBCA|BARC|CRAB`

, `ARBC|RCAB|BACR|CBRA`

, `ACRB|RABC|BRCA|CBAR`

, `ABRC|RACB|BCAR|CRBA`

, `ABCR|RCBA|BRAC|CARB`

, `ARCB|RBAC|BCRA|CABR`

.- If you want to assess dose-normalized equivalence, follow the ‘Two‐at‐a‐Time Principle’,
*i.e.*, perform pairwise comparisons whilst excluding the others.^{1}Do not use a pooled ANOVA because it may give biased estimates and/or inflate the Type I Error.^{2,3}

That means your first scenario is correct (you estimate the sample size like a 2×2×2 crossover) and get three incomplete block designs to evaluate:`A`

↔`R`

,*x*`B`

/*y*↔`R`

, and*x*`C`

/*z*↔`R`

.

If you insist in a pooled ANOVA for any reasons, you end up with 20 subjects as well (though with slightly higher power due to the higher degrees of freedom 3*n*–6 compared to the*n*–2 in the 2×2×2):

`library(PowerTOST)`

x <- data.frame(design = c("2x2x2", "4x4"), n = NA, power.pct = NA)

x[1, 2:3] <- sampleN.TOST(CV = 0.23, theta0 = 1, design = "2x2x2",

print = FALSE)[7:8]

x[2, 2:3] <- sampleN.TOST(CV = 0.23, theta0 = 1, design = "4x4",

print = FALSE)[7:8]

x[3] <- round(100*x[3], 2)

print(x, row.names = FALSE)

design n power.pct

2x2x2 20 82.08

4x4 20 84.55

- If you want to assess dose proportionality, the acceptance range depends on the dose-range (the wider the dose-range the narrower it will be). For the sample size estimation you need the dose-range and an assumed slope of the power model β. Furthermore, it depends whether you plan a confirmatory
^{4}or exploratory^{5}study.

- Schuirmann D.
*Two at a Time? Or All at Once?*Pittsburgh: International Biometric Society, Eastern North American Region, Spring Meeting; March 28–31, 2005. Abstract.

- European Medicines Agency, CHMP.
*Guideline on the Investigation of Bioequivalence.*London; 20 January 2010. Doc. Ref. CPMP/EWP/QWP/1401/98 Rev. 1/ Corr **.

- D’Angelo P.
*Testing for Bioequivalence in Higher‐Order Crossover Designs: Two‐at‐a‐Time Principle Versus Pooled ANOVA*. Rockville: 2^{nd}Workshop of the Global Bioequivalence Harmonisation Initiative; 15–16 September, 2016. Some of her slides in this post.

- Smith BP, Vandenhende FR, DeSante KA, Farid NA, Welch PA, Callaghan JT, Forgue ST.
*Confidence Interval Criteria for Assessment of Dose Proportionality*. Pharm Res. 2000; 17(19): 1278–83. doi:10.1023/a:1026451721686.

- Hummel J, McKendrick S, Brindley C, French R.
*Exploratory assessment of dose proportionality: review of current approaches and proposal for a practical criterion.*Pharm. Stat. 2009; 8(1): 38–49. doi:10.1002/pst.326.

—

Helmut Schütz

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

Science Quotes

*Dif-tor heh smusma*🖖Helmut Schütz

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

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### Complete thread:

- Higher-order crossovers sweiner 2020-01-22 20:22 [Power / Sample Size]
- Purpose of the study?Helmut 2020-01-23 13:48
- Purpose of the study? Ihababdallah 2020-09-30 15:17
- Purpose of the study? Helmut 2020-09-30 18:08
- Purpose of the study? Ihababdallah 2020-09-30 19:31
- Purpose of the study? Helmut 2020-10-01 10:28

- Purpose of the study? Ihababdallah 2020-09-30 19:31

- Purpose of the study? Helmut 2020-09-30 18:08

- Purpose of the study? Ihababdallah 2020-09-30 15:17
- Higher-order crossovers sweiner 2020-01-26 16:50

- Purpose of the study?Helmut 2020-01-23 13:48