## Higher-order crossovers [Power / Sample Size]

Helmut, thank you for the detailed response, examples in R, and references! I put the answers to the questions below (I tried to change color on responses, but not sure if that worked). Thanks. Sveta

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

This is more a hypothetical example, so really not known

» » … 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’?

‘Subsequently’ here refers to the next study on somewhat different population (where we expect smaller value of CV), with higher order of crossover

» » … 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.

The purpose is to show dose-normalized equivalence

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

» Rather strong assumptions, right?

We actually expect a smaller value of %CV, not sure on GMR though

» » 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 (

» You randomize subjects either to the Latin Square

» — If you want to assess dose-normalized equivalence, follow the ‘Two‐at‐a‐Time Principle’,

» 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:

» 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

»

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This is what we are trying to do, so this is exactly the response I was looking for.

I used the following example in R with CV=1.04 and got similar overall SS from either 2x2 or 4x4, so then the overall SS will just need to be split to higher order crossover – about 67 subjects per sequence for 2x2 and about 34 per sequence for 4x4. I assume that if we had to do even higher order crossover, then we would follow similar approach. R does not support 5x5 for example, but if we had to use 5x5 it would be about 26 subjects per sequence, it seems to be,is it correct?

CV=1.04, design = 2x2:

CV=1.04, design = 4x4

Edit: Standard quotes restored; see also this post #8. [Helmut]

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

This is more a hypothetical example, so really not known

» » … 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’?

‘Subsequently’ here refers to the next study on somewhat different population (where we expect smaller value of CV), with higher order of crossover

» » … 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.

The purpose is to show dose-normalized equivalence

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

» Rather strong assumptions, right?

We actually expect a smaller value of %CV, not sure on GMR though

» » 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`

This is what we are trying to do, so this is exactly the response I was looking for.

I used the following example in R with CV=1.04 and got similar overall SS from either 2x2 or 4x4, so then the overall SS will just need to be split to higher order crossover – about 67 subjects per sequence for 2x2 and about 34 per sequence for 4x4. I assume that if we had to do even higher order crossover, then we would follow similar approach. R does not support 5x5 for example, but if we had to use 5x5 it would be about 26 subjects per sequence, it seems to be,is it correct?

CV=1.04, design = 2x2:

```
sampleN.TOST(CV = 1.04, theta1=0.67, theta2 = 1.5, theta0 = 1.1, design = "2x2", print = FALSE, logscale=TRUE, targetpower=0.9)[7:8]
```

Sample size Achieved power

1 134 0.9029653

CV=1.04, design = 4x4

```
sampleN.TOST(CV = 1.04, theta1=0.67, theta2 = 1.5, theta0 = 1.1, design = "4x4", print = FALSE, logscale=TRUE, targetpower=0.9)[7:8]
```

Sample size Achieved power

1 132 0.9008229

Edit: Standard quotes restored; see also this post #8. [Helmut]

### Complete thread:

- Higher-order crossovers sweiner 2020-01-22 20:22
- 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 crossoverssweiner 2020-01-26 16:50

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