## Exact TSD methods: Example [Two-Stage / GS Designs]

answering my own post in order to keep it short.

In the following an example. We have a guesstimate of the CV (0.20), assume a GMR of 0.95, and aim at power 0.80. No futility criteria. Some regulatory statisticians told me to prefer a first stage as estimated for a fixed-sample design (

*i.e.*, the second stage is solely a ‘safety net’).

`library(PowerTOST)`

library(Power2Stage)

CV0 <- 0.20

n0 <- sampleN.TOST(CV=CV0, details=FALSE, print=FALSE)[["Sample size"]]

n.tot <- power.tsd.in(CV=CV0, n1=n0, fCrit="No", npct=0.5)$nperc[["50%"]]

w <- c(n0, n.tot - n0) / n.tot

if (w[1] == 1) w <- w + c(-1, +1) * 1e-6

In the method the weights have to be pre-specified, stated in the SAP, and used throughout subsequent steps (irrespective of the re-estimated n

_{2}). In the fixed-sample design we would need 20 subjects. How to set the weights? An intuitive way is to use the x̃ (20) of the total sample size based on simulations. This would give us weights of [1, 0]. Great. But weights have to be >0 and <1. Hence, I tweaked them a little to [0.999999, 0.000001]. What can we expect if we run the study with n

_{1}20?

`power.tsd.in(CV=CV0, n1=n0, fCrit="No", weight=w,`

npct=c(0.05, 0.25, 0.50, 0.75, 0.95))

TSD with 2x2 crossover

Inverse Normal approach

- maximum combination test (weights = 0.999999 1e-06)

- alpha (s1/s2) = 0.02531 0.02531

- critical value (s1/s2) = 1.95463 1.95463

- with conditional error rates and conditional power

Overall target power = 0.8

Threshold in power monitoring step for futility = 0.8

Power calculation via non-central t approx.

CV1 and GMR = 0.95 in sample size est. used

No futility criterion regarding PE, CI or Nmax

Minimum sample size in stage 2 = 4

BE acceptance range = 0.8 ... 1.25

CV = 0.2; n(stage 1) = 20; GMR = 0.95

1e+05 sims at theta0 = 0.95 (p(BE) = 'power').

p(BE) = 0.84868

p(BE) s1 = 0.72513

Studies in stage 2 = 21.76%

Distribution of n(total)

- mean (range) = 23.4 (20 ... 86)

- percentiles

5% 25% 50% 75% 95%

20 20 20 20 42

Fine. If everything turns out as expected we have to be unlucky to need a second stage. Power in the first is already 0.73 and stage 2 sample sizes are not shocking. As common in TSDs the overall power is generally higher than in a fixed-sample design.

We perform the first stage and get GMR 0.91 and CV 0.25. Oops! Both are worse than assumed. Especially the GMR is painful.

`n1 <- n0`

GMR1 <- 0.91

CV1 <- 0.25

res <- interim.tsd.in(GMR1=GMR1, CV1=CV1, n1=n1, fCrit="No", weight=w)

res

TSD with 2x2 crossover

Inverse Normal approach

- maximum combination test with weights for stage 1 = 1 0

- significance levels (s1/s2) = 0.02531 0.02531

- critical values (s1/s2) = 1.95463 1.95463

- BE acceptance range = 0.8 ... 1.25

- Observed point estimate from stage 1 is not used for SSR

- with conditional error rates and conditional (estimated target) power

Interim analysis of first stage

- Derived key statistics:

z1 = 1.57468, z2 = 3.38674,

Repeated CI = (0.77306, 1.07120)

- No futility criterion met

- Test for BE not positive (not considering any futility rule)

- Calculated n2 = 24

- Decision: Continue to stage 2 with 24 subjects

We fail to show BE (lower CL 77.31%) and should initiate the second stage with 24 subjects.

How would a ‘Type 1’ TSD perform?

`Interim analysis (specified α1 0.0294)`

───────────────────────────────────────────────────

94.12% CI: 77.77–106.48% (failed to demonstrate BE)

Power : 0.5092 (approx. via non-central t)

Second stage with 14 subjects (N=34) is justified.

Pretty similar though a lower n

_{2}is suggested.

OK, we perform the second stage and get GMR 0.93 and CV 0.21. Both are slightly better than what we got in the first stage but again worse than assumed.

`n2 <- res$n2`

GMR2 <- 0.93

CV2 <- 0.21

final.tsd.in(GMR1=GMR1, CV1=CV1, n1=n1,

GMR2=GMR2, CV2=CV2, n2=n2, weight=w)

TSD with 2x2 crossover

Inverse Normal approach

- maximum combination test with weights for stage 1 = 1 0

- significance levels (s1/s2) = 0.02531 0.02531

- critical values (s1/s2) = 1.95463 1.95463

- BE acceptance range = 0.8 ... 1.25

Final analysis of second stage

- Derived key statistics:

z1 = 2.32999, z2 = 4.00748,

Repeated CI = (0.82162, 1.05264)

Median unbiased estimate = 0.8997

- Decision: BE achieved

We survived.

In a ‘Type 1’ TSD we would get:

`Final analysis of pooled data (specified α2 0.0294) `

═══════════════════════════════════════════════════

94.12% CI: 83.86–101.12% (BE concluded)

Pretty similar again.

If we state it in the protocol, we could also aim for higher power in the second stage if the GMR in the first doesn’t look nice. If we switch to 0.90 we would run the second stage with 36 subjects.

`Final analysis of second stage`

- Derived key statistics:

z1 = 2.86939, z2 = 4.94730,

Repeated CI = (0.84220, 1.02693)

Median unbiased estimate = 0.9053

- Decision: BE achieved

Helps. Another option would be to adjust for GMR1 by using the argument

`usePE=TRUE`

in `interim.tsd.in()`

. For power 0.80 that would mean 40 subjects in the second stage and for 0.90 already 62…Cheers,

Helmut Schütz

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

Science Quotes

### Complete thread:

- Finally: Exact TSD methods for 2×2 crossover designs Helmut 2018-04-21 17:17
- Exact TSD methods: ExampleHelmut 2018-04-21 20:33
- Finally: Exact TSD methods for 2×2 crossover designs ElMaestro 2018-04-21 20:49
- Flow chart (without details) Helmut 2018-04-21 21:41
- naive questions regarding new functions in Power2Stage mittyri 2018-04-28 15:54
- Some answers Helmut 2018-04-28 17:29
- Some more "answers" d_labes 2018-04-29 21:11
- clarification regarding user Power2Stage guides mittyri 2018-04-30 13:41

- naive questions regarding new functions in Power2Stage mittyri 2018-04-28 15:54

- Flow chart (without details) Helmut 2018-04-21 21:41
- Technicality: Weigths for the inverse normal approach d_labes 2018-04-25 14:19
- Selection of w and w* Helmut 2018-04-26 09:51
- Selection of w and w* d_labes 2018-04-26 20:02
- Now what? w & w* examples d_labes 2018-05-09 13:53
- Now what? w & w* examples Ben 2018-06-10 20:12
- Now what? w & w* examples Helmut 2018-06-11 13:57
- Now what? w & w* examples Ben 2018-06-12 19:14

- a bug in interim.tsd.in()? mittyri 2018-06-11 23:27
- a bug in interim.tsd.in()? Ben 2018-06-12 19:32
- Nonbinding futility rule d_labes 2018-06-13 16:59
- Bad weather? Helmut 2018-06-13 19:23
- NLYW? d_labes 2018-06-14 10:18

- Nonbinding futility rule Ben 2018-06-13 20:26
- Nonbinding futility rule d_labes 2018-06-14 10:47
- Nonbinding futility rule Ben 2018-06-15 17:58
- Binding / Nonbinding futility rule - alpha control d_labes 2018-06-16 19:42
- Binding / Nonbinding futility rule - alpha control Ben 2019-03-30 09:52

- Binding / Nonbinding futility rule - alpha control d_labes 2018-06-16 19:42

- Nonbinding futility rule Ben 2018-06-15 17:58

- Nonbinding futility rule d_labes 2018-06-14 10:47

- Bad weather? Helmut 2018-06-13 19:23

- Nonbinding futility rule d_labes 2018-06-13 16:59

- a bug in interim.tsd.in()? Ben 2018-06-12 19:32

- Now what? w & w* examples Helmut 2018-06-11 13:57

- Now what? w & w* examples Ben 2018-06-10 20:12

- Selection of w and w* Helmut 2018-04-26 09:51