Integer sample size and event counts

Fixed sample size

We present a simple example based on comparing binomial rates with interim analyses after 50% and 75% of events. We assume a 2:1 experimental:control randomization ratio. Note that the sample size is not 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 the beta argument above with an integer sample size that is a multiple of 3 so that we get the desired 2:1 integer sample sizes per arm (432 = 144 control + 288 experimental targeted) 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 size n.fix from above to a 1-sided group sequential design with interims after 50% and 75% of observations. Again, sample size at each analysis is not an integer. We use the Lan-DeMets spending function approximating an O’Brien-Fleming efficacy 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 analyses
xb$n.I

## [1] 219.1621 328.7432 438.3243

Next we convert to integer sample sizes at each analysis. Interim sample sizes are rounded to the nearest integer. The default roundUpFinal = TRUE rounds the final sample size to the nearest 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 analysis
x_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 are slightly different.

# Bound for continuous sample size design
xb$upper$bound

## [1] 2.962588 2.359018 2.014084

# Bound for integer sample size design
x_integer$upper$bound

## [1] 2.974067 2.366106 2.012987

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

# Continuous design sample size fractions at analyses
xb$timing

## [1] 0.50 0.75 1.00

# Integer design sample size fractions at analyses
x_integer$timing

## [1] 0.4965986 0.7460317 1.0000000

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

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

## [1] 0.001525323 0.009649325 0.025000000

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

## [1] 0.001525323 0.009649325 0.025000000

# Integer sample size design
cumsum(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 fraction
x_integer$upper$sf(alpha = x_integer$alpha, t = x_integer$timing, x_integer$upper$param)$spend

## [1] 0.001469404 0.009458454 0.025000000

Finally, we look at cumulative boundary crossing probabilities under the alternate hypothesis for each design. Due to rounding up the final sample size, the integer-based design has slightly higher total power than the specified 80% (Type II error beta = 0.2.). Interim power is slightly lower for the integer-based design since sample size is rounded to the nearest integer rather than rounded up as at the final analysis.

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

## [1] 0.1679704 0.5399906 0.8000000

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

## [1] 0.1649201 0.5374791 0.8025140

Non-binding design

The default test.type = 4 has a non-binding futility bound. We examine behavior of this design next. The futility bound is moderately aggressive and, thus, there is a compensatory increase in sample size to retain power. The parameter delta1 is the natural parameter denoting the difference in response (or failure) rates of 0.2 vs. 0.1 that was specified in the call to nBinomial() 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 design
xnb$n.I

## [1] 231.9610 347.9415 463.9219

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

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

## [1] 232 348 465

# Information fraction based on integer sample sizes
xnbi$timing

## [1] 0.4989247 0.7483871 1.0000000

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

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

## [1] 0.001525323 0.009630324 0.023013764

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

## [1] 0.001507499 0.009553042 0.022999870

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

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

## [1] 0.001507499 0.009571518 0.025000000

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

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

## [1] 0.05360549 0.10853733 0.19921266

# Spending function target is the same at interims, but larger at final
xnbi$lower$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 than 0.2 in the first row above is due to the final sample size powering the trial to greater than 0.8 as seen below.

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

## [1] 0.8007874

Seasonal design with a final zero event-rate period

Seasonal rare-event designs often use piecewise event rates with nonzero rates during seasons and zero rates outside seasons. The example below has three 6-month seasons separated by off-season zero event-rate periods; the final piecewise event-rate period is zero. This is the setting where it is especially important that roundUpFinal = TRUE applies to the final event count. For a complete seasonal exact-binomial monitoring workflow, see vignette("MultiSeasonRareEvents", package = "gsDesign").

seasonal_event_rate_control <- 0.003
season_length <- 0.5
dropout_6mo <- 0.10

season_rate <- -log(1 - seasonal_event_rate_control) / season_length
dropout_rate <- -log(1 - dropout_6mo) / season_length

seasonal_design <- gsSurv(
  k = 3,
  test.type = 4,
  alpha = 0.025,
  beta = 0.1,
  timing = c(1 / 3, 2 / 3),
  sfu = sfHSD,
  sfupar = 1,
  sfl = sfHSD,
  sflpar = -2,
  lambdaC = c(season_rate, 0, season_rate, 0, season_rate, 0),
  S = c(6, 6, 6, 6, 6),
  hr = 0.2,
  hr0 = 0.7,
  eta = dropout_rate,
  gamma = c(1, 0, 1, 0, 1, 0),
  R = c(2, 10, 2, 10, 2, 10),
  T = 42,
  minfup = 6,
  ratio = 3,
  testLower = c(TRUE, FALSE, FALSE)
)

# Continuous event targets and expected enrollment before integer conversion
seasonal_design$n.I

## [1] 11.94976 23.89951 35.84927

rowSums(seasonal_design$eNC + seasonal_design$eNE)

## [1] 2071.587 3442.289 4099.222

seasonal_design$T

## [1] 13.03217 25.03845 42.00000

The final continuous event target is just under 36 events, so with the default roundUpFinal = TRUE the integer event target is 36. The final expected sample size is just over 4099, which rounds up to 4100 for 3:1 randomization. For this design, the rounded sample size supports the rounded-up final event target, so no sample-size feasibility warning is needed. The final analysis time moves later because the rounded-up event target requires additional follow-up under the late zero event-rate period.

seasonal_integer <- toInteger(seasonal_design)

data.frame(
  analysis = seq_len(seasonal_design$k),
  continuous_events = round(seasonal_design$n.I, 3),
  integer_events = seasonal_integer$n.I,
  continuous_time = round(seasonal_design$T, 3),
  integer_time = round(seasonal_integer$T, 3),
  integer_enrollment = round(rowSums(seasonal_integer$eNC + seasonal_integer$eNE), 3)
)

##   analysis continuous_events integer_events
## 1        1            11.950             12
## 2        2            23.900             24
## 3        3            35.849             36
##   continuous_time integer_time integer_enrollment
## 1          13.032       13.032           2080.066
## 2          25.038       25.038           3456.378
## 3          42.000       42.000           4116.000

# Final integer sample size remains a multiple of ratio + 1.
rowSums(seasonal_integer$eNC + seasonal_integer$eNE)[seasonal_integer$k]

## [1] 4116

The important point is that final event-count rounding is not interchangeable with final sample-size rounding. Integer event targets define the information schedule; the enrollment plan and analysis times are then adjusted only as much as needed to keep those integer event targets attainable under the piecewise event-rate model.

The feasibility adjustment is easier to see in a rare-event design where the rounded-up event target is too high for the initially rounded sample size. In the next example, the final event target rounds up to 43, but the rounded final sample size of 5038 cannot ever produce that many expected events. toInteger() increases final sample size by allocation multiples and warns.

rare_design <- gsSurv(
  k = 3,
  test.type = 4,
  alpha = 0.025,
  beta = 0.1,
  timing = c(1 / 3, 2 / 3),
  sfu = sfHSD,
  sfupar = 1,
  sfl = sfHSD,
  sflpar = -2,
  lambdaC = -log(1 - 0.0015) / 0.5,
  hr = 0.2,
  hr0 = 0.7,
  eta = dropout_rate,
  gamma = c(1, 0, 1, 0, 1, 0),
  R = c(2, 10, 2, 10, 2, 10),
  T = 42,
  minfup = 6,
  ratio = 1
)

rare_integer <- toInteger(rare_design)

data.frame(
  continuous_final_events = rare_design$n.I[rare_design$k],
  integer_final_events = rare_integer$n.I[rare_integer$k],
  continuous_final_enrollment = rowSums(rare_design$eNC + rare_design$eNE)[rare_design$k],
  integer_final_enrollment = rowSums(rare_integer$eNC + rare_integer$eNE)[rare_integer$k]
)

##   continuous_final_events integer_final_events
## 1                42.12931                   43
##   continuous_final_enrollment integer_final_enrollment
## 1                    5037.812                     5142