## TIE = p(BE) at expanded limits [RSABE / ABEL]

Hi Yura,

» Did I understand correctly, when constructing a confidence interval for T-R differences, use the adjusted alpha for R-R?

Exactly – now you got it!

In a nutshell, the Type I Error is the probability of falsely claiming BE. The TIE can be estimated by setting theta0 to one of the limits of the acceptance range. Easy for ABE (since an explicit solution exists).

library(PowerTOST) CV  <- 0.3 des <- "2x2x4" n   <- 34 U   <- 1.25 power.TOST(CV=CV, n=n, theta0=U, design=des) # [1] 0.05

Alternatively you can perform simulations.

power.TOST.sim(CV=CV, n=n, theta0=U, design=des, nsims=1e7) # [1] 0.0500178

The convergence is lousy. If you have a lot of time increase the number of simulations.
When it comes to reference-scaling no explicit formula for power exists. Hence, we need simulations. There is a complication: ABEL is a framework of decisions where the Null-hypothesis is constructed in face of the data. In other words we don’t know the expanded limits until we have calculated CVwR. Unlike in ABE the limits are random variables themselves.

CVwR <- 0.3 reg  <- "EMA" U    <- scABEL(CV=CVwR, regulator=reg)[["upper"]] power.scABEL(CV=CVwR, n=n, theta0=U, design=des,              regulator=reg, nsims=1e6) # [1] 0.081626

If you don’t like (or trust in) simulating the underlying statistics, with the latest version of PowerTOST you can simulate subject data as well.

power.scABEL.sdsims(CV=CVwR, n=n, theta0=U, design=des,                     regulator=reg, nsims=1e6) # [1] 0.081602

Note that in the final evaluation (i.e., the PE and its CI) the EMA’s model assumes that CVwT = CVwR which is strong meat. To get the adjusted α, and the TIEs for nominal and adjusted α:

res  <- scABEL.ad(CV=CVwR, n=n, design=des, regulator=reg, print=FALSE) TIE0 <- res[["TIE.unadj"]] adj  <- res[["alpha.adj"]] TIE1 <- res[["TIE.adj"]] cat(TIE0, adj, TIE1, "\n") # 0.081626 0.028572 0.05

Now imagine a study which passed with the nominal α 0.05 (90% CI):

round(100*CI.BE(alpha=0.05, pe=0.9, CV=CV, n=n, design=des), 2) # lower upper # 82.78 97.85

The study would still pass with the adjusted α 0.028572 (94.2856% CI) but it is a more close shave:

round(100*CI.BE(alpha=adj, pe=0.9, CV=CV, n=n, design=des), 2) # lower  upper # 81.69  99.16

If you want to avoid surprises increase the sample size.

res <- sampleN.scABEL.ad(CV=CVwR, theta0=0.9, design=des,                          regulator=reg, print=FALSE) adj <- res[["alpha.adj"]] n.a <- res[["Sample size"]] cat(adj, n.a, "\n") # 0.028311 42

Note that the TIE depends also on the sample size. Therefore, slightly more adjustment is needed (n=34: 0.028572, n=42: 0.028311). With 42 subjects you should be on the safe side.

round(100*CI.BE(alpha=adj, pe=0.9, CV=CV, n=n.a, design=des), 2) # lower upper # 82.49 98.20

Dif-tor heh smusma 🖖
Helmut Schütz

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