library(gsDesign)

x<-gsSurv(ratio =1,hr = .74)y<- x|>toInteger()# Continuous event countsx$n.I

# Rounded event counts at interims, rounded up at final analysisy$n.I

# Continuous sample size at interim and final analysesas.numeric(x$eNE+ x$eNC)

# Rounded up to even final sample size given that x$ratio = 1# and rounding to multiple of x$ratio + 1as.numeric(y$eNE+ y$eNC)

# With roundUpFinal = FALSE, final sample size rounded to nearest integerz<- x|>toInteger(roundUpFinal =FALSE)as.numeric(z$eNE+ z$eNC)

Fixed sample size

We present a simple example based on comparing binomial rates withinterim analyses after 50% and 75% of events. We assume a 2:1experimental:control randomization ratio. Note that the sample size isnot an integer. We target 80% power (beta = .2).

n.fix<-nBinomial(p1 = .2,p2 = .1,alpha = .025,beta = .2,ratio =2)n.fix

## [1] 429.8846

If we replace thebeta argument above with a integersample size that is a multiple of 3 so that we get the desired 2:1integer sample sizes per arm (432 = 144 control + 288 experimentaltargeted) we get slightly larger than the targeted 80% power:

nBinomial(p1 = .2,p2 = .1,alpha = .025,n =432,ratio =2)

## [1] 0.801814

1-sided design

Now we convert the fixed sample sizen.fix from above toa 1-sided group sequential design with interims after 50% and 75% ofobservations. Again, sample size at each analysis is not an integer. Weuse the Lan-DeMets spending function approximating an O’Brien-Flemingefficacy bound.

# 1-sided design (efficacy bound only; test.type = 1)xb<-gsDesign(alpha = .025,beta = .2,n.fix = n.fix,test.type =1,sfu = sfLDOF,timing =c(.5, .75))# Continuous sample size (non-integer) at planned analysesxb$n.I

## [1] 219.1621 328.7432 438.3243

Next we convert to integer sample sizes at each analysis. Interimsample sizes are rounded to the nearest integer. The defaultroundUpFinal = TRUE rounds the final sample size to thenearest integer to 1 + the experimental:control randomization ratio.Thus, the final sample size of 441 below is a multiple of 3.

# Convert to integer sample size with even multiple of ratio + 1# i.e., multiple of 3 in this case at final analysisx_integer<-toInteger(xb,ratio =2)x_integer$n.I

## [1] 219 329 441

Next we examine the efficacy bound of the 2 designs as they areslightly different.

# Bound for continuous sample size designxb$upper$bound

## [1] 2.962588 2.359018 2.014084

# Bound for integer sample size designx_integer$upper$bound

## [1] 2.974067 2.366106 2.012987

The differences are associated with slightly different timing of theanalyses associated with the different sample sizes noted above:

# Continuous design sample size fractions at analysesxb$timing

## [1] 0.50 0.75 1.00

# Integer design sample size fractions at analysesx_integer$timing

## [1] 0.4965986 0.7460317 1.0000000

These differences also make a difference in the cumulative Type Ierror associated with each analysis as shown below.

# Continuous sample size designcumsum(xb$upper$prob[,1])

## [1] 0.001525323 0.009649325 0.025000000

# Specified spending based on the spending functionxb$upper$sf(alpha = xb$alpha,t = xb$timing, xb$upper$param)$spend

## [1] 0.001525323 0.009649325 0.025000000

# Integer sample size designcumsum(x_integer$upper$prob[,1])

## [1] 0.001469404 0.009458454 0.025000000

# Specified spending based on the spending function# Slightly different from continuous design due to slightly different information fractionx$upper$sf(alpha = x_integer$alpha,t = x_integer$timing, x_integer$upper$param)$spend

## [1] 0.01547975 0.02079331 0.02500000

Finally, we look at cumulative boundary crossing probabilities underthe alternate hypothesis for each design. Due to rounding up the finalsample size, the integer-based design has slightly higher total powerthan the specified 80% (Type II errorbeta = 0.2.). Interimpower is slightly lower for the integer-based design since sample sizeis rounded to the nearest integer rather than rounded up as at the finalanalysis.

# Cumulative upper boundary crossing probability under alternate by analysis# under alternate hypothesis for continuous sample sizecumsum(xb$upper$prob[,2])

## [1] 0.1679704 0.5399906 0.8000000

# Same for integer sample sizes at each analysiscumsum(x_integer$upper$prob[,2])

## [1] 0.1649201 0.5374791 0.8025140

Non-binding design

The defaulttest.type = 4 has a non-binding futilitybound. We examine behavior of this design next. The futility bound ismoderately aggressive and, thus, there is a compensatory increase insample size to retain power. The parameterdelta1 is thenatural parameter denoting the difference in response (or failure) ratesof 0.2 vs. 0.1 that was specified in the call tonBinomial() above.

# 2-sided asymmetric design with non-binding futility bound (test.type = 4)xnb<-gsDesign(alpha = .025,beta = .2,n.fix = n.fix,test.type =4,sfu = sfLDOF,sfl = sfHSD,sflpar =-2,timing =c(.5, .75),delta1 = .1)# Continuous sample size for non-binding designxnb$n.I

## [1] 231.9610 347.9415 463.9219

As before, we convert to integer sample sizes at each analysis andsee the slight deviations from the interim timing of 0.5 and 0.75.

xnbi<-toInteger(xnb,ratio =2)# Integer design sample size at each analysisxnbi$n.I

## [1] 232 348 465

# Information fraction based on integer sample sizesxnbi$timing

## [1] 0.4989247 0.7483871 1.0000000

These differences also make a difference in the Type I errorassociated with each analysis

# Type I error, continuous designcumsum(xnb$upper$prob[,1])

## [1] 0.001525323 0.009630324 0.023013764

# Type I error, integer designcumsum(xnbi$upper$prob[,1])

## [1] 0.001507499 0.009553042 0.022999870

The Type I error ignoring the futility bounds just shown does not usethe full targeted 0.025 as the calculations assume the trial stops forfutility if an interim futility bound is crossed. The non-binding Type Ierror assuming the trial does not stop for futility is:

# Type I error for integer design ignoring futility boundcumsum(xnbi$falseposnb)

## [1] 0.001507499 0.009571518 0.025000000

Finally, we look at cumulative lower boundary crossing probabilitiesunder the alternate hypothesis for the integer-based design and compareto the planned\(\beta\)-spending. Wenote that the final Type II error spending is slightly lower than thetargeted 0.2 due to rounding up the final sample size.

# Actual cumulative beta spent at each analysiscumsum(xnbi$lower$prob[,2])

## [1] 0.05360549 0.10853733 0.19921266

# Spending function target is the same at interims, but larger at finalxnbi$lower$sf(alpha = xnbi$beta,t = xnbi$n.I/max(xnbi$n.I),param = xnbi$lower$param)$spend

## [1] 0.05360549 0.10853733 0.20000000

The\(\beta\)-spending lower than0.2 in the first row above is due to the final sample size powering thetrial to greater than 0.8 as seen below.

# beta-spendingsum(xnbi$upper$prob[,2])

## [1] 0.8007874