Assessing Adaptive Strategies through Simulation

Gernot Wassmer

January 14, 2026

The Promising Zone Approach

Chen, DeMets, and Lan (2004) have shown that, roughly speaking, if at interim the conditional power calculated for the observed effect exceeds 50%, then the test statistic for the group sequential test needs not to be adjusted and the sample size might be increased
Based on this result (and some refinements) Mehta and Pocock (2011) proposed the promising zone approach
Sample size recalculation methods based on conditional power were proposed from the very beginning of adaptive designs (Bauer and Köhne (1994))
A “constrained” and “optimal” promising zone design was proposed by Hsiao, Liu, and Mehta (2019)
This design is essentially using the inverse normal combination test because the sample size recalculation procedure is defined in a different way
We consider this design and show how it can be implemented with rpact.

A Motivating Example from Hsiao et al. (2019)

Efficacy endpoint PFS
Assumed hazard ratio = 0.67, \(\alpha = 0.025\) and \(\beta = 0.10\) requires 263 events

getSampleSizeSurvival(
   alpha = 0.025, 
   beta = 0.1, 
   hazardRatio = 0.67
)$maxNumberOfEvents

[1] 262.0594

280 PFS events yields power 91.8 %

getPowerSurvival(
   alpha = 0.025, 
   hazardRatio = 0.67, 
   directionUpper = FALSE, 
   maxNumberOfEvents = 280, 
   maxNumberOfSubjects = 350,      
) |> fetch(overallReject)

overallReject 
    0.9178375

Note: maxNumberOfSubjects has no influence on the power calculation

If 350 patients are enrolled over 28 months with a median PFS time of 8.5 months in the control group, the final analysis is expected to be after an additional follow-up of about 12 months:

getPowerSurvival(
    alpha = 0.025, 
    hazardRatio = c(0.67, 1), 
    directionUpper = FALSE, 
    maxNumberOfEvents = 280, 
    maxNumberOfSubjects = 350,
    median2 = 8.5,
    accrualTime = 28
) |> summary()

Power calculation for a survival endpoint

Fixed sample analysis, one-sided significance level 2.5%. The results were calculated for a two-sample logrank test, H0: hazard ratio = 1, power directed towards smaller values, H1: hazard ratio as specified, control median(2) = 8.5, number of subjects = 350, number of events = 280, accrual time = 28, accrual intensity = 12.5.

Stage	Fixed
Stage level (one-sided)	0.0250
Efficacy boundary (z-value scale)	1.960
Efficacy boundary (t)	0.791
Power, HR = 0.67	0.9178
Power, HR = 1	0.0250
Number of subjects, HR = 0.67	350.0
Number of subjects, HR = 1	350.0
Number of events, HR = 0.67	280.0
Number of events, HR = 1	280.0
Analysis time, HR = 0.67	40.52
Analysis time, HR = 1	36.29
Expected study duration under H1, HR = 0.67	40.52
Expected study duration under H1, HR = 1	36.29

Legend:

(t): treatment effect scale
HR: hazard ratio

508 PFS events are needed to have 90% power at HR = 0.75:

getSampleSizeSurvival(
   alpha = 0.025, 
   beta = 0.1, 
   hazardRatio = 0.75
) |> fetch(maxNumberOfEvents)

maxNumberOfEvents 
         507.8443

Clearly, more patients are needed, and a different expected follow-up is calculated.

“Milestone-based” investment:

Two-stage approach with interim after 140 events
Enough power for detecting HR = 0.67
If the conditional power (rejecting the null at given interim data) falls in a “promising zone”, an additional investment would be made that allows the trial to remain open until 420 PFS events were obtained
Otherwise, no additional investment is made, stick to the originally planned event number (= 280) is based on assumed minimum clinical relevant effect HR = 0.75.

Constrained Promising Zone Design

Number of additional events for the second stage between 140 and 280
If conditional power for 280 additional events at HR = 0.75 is smaller than \(cp_{min}\), set number of additional events = 140 (non-promising case)
If conditional power for 140 additional events at HR = 0.75 exceeds \(cp_{max}\), set number of additional events = 140, otherwise calculate event number according to \[CP_{HR = 0.75} = cp_{max}\] (promising case)
This defines a promising zone for HR within the sample size may be modified.
E.g.,
- \(cp_{min} = 0.80\)
- \(cp_{max} = 0.90\)

Constrained Promising Zone Design

How do I obtain these plots?

Constrained Promising Zone Design Using `rpact`

First, define the design

myDesign <- getDesignInverseNormal(
  kMax = 2, 
  alpha = 0.025,
  typeOfDesign = "noEarlyEfficacy"
)

Check by simulation

getSimulationSurvival(
   design = myDesign,
   hazardRatio = 0.67, 
   directionUpper = FALSE, 
   plannedEvents = c(140, 280), 
   maxNumberOfSubjects = 350,
   median2 = 8.5,
   accrualTime = 28,
   maxNumberOfIterations = 1000,
   seed = 12
) |> summary()

Simulation of a survival endpoint

Sequential analysis with a maximum of 2 looks (inverse normal combination test design), one-sided overall significance level 2.5%. The results were simulated for a two-sample logrank test, H0: hazard ratio = 1, power directed towards smaller values, H1: hazard ratio = 0.67, control median(2) = 8.5, planned cumulative events = c(140, 280), maximum number of subjects = 350, accrual time = 28, accrual intensity = 12.5, simulation runs = 1000, seed = 12.

Stage	1	2
Fixed weight	0.707	0.707
Cumulative alpha spent	0	0.0250
Stage levels (one-sided)	0	0.0250
Efficacy boundary (z-value scale)	Inf	1.960
Cumulative power	0	0.9100
Number of subjects	285.9	350.0
Expected number of subjects under H1		350.0
Cumulative number of events	140.0	280.0
Expected number of events under H1		280.0
Analysis time	22.91	40.48
Expected study duration under H1		40.48
Conditional power (achieved)		0.8186
Exit probability for efficacy	0

Define the event number calculation function `myEventSizeCalculationFunction()`

# Define promising zone event size function
myEventSizeCalculationFunction <- function(..., 
   stage,
   plannedEvents,
   conditionalPower,
   minNumberOfEventsPerStage,
   maxNumberOfEventsPerStage,
   conditionalCriticalValue,
   estimatedTheta
) {
    calculateStageEvents <- function(cp) {
        4 * max(0, conditionalCriticalValue + qnorm(cp))^2 / 
            log(max(1 + 1e-12, estimatedTheta))^2
    }
    #  Note: estimatedTheta is 1 / hazardRatio if directionUpper = FALSE
    
    # Calculate events required to reach maximum desired conditional power
    # cp_max (provided as argument conditionalPower)
    stageEventsCPmax <- ceiling(calculateStageEvents(cp = conditionalPower))
    
    # Calculate events required to reach minimum desired conditional power
    # cp_min, manually to be set = 0.80
    stageEventsCPmin <- ceiling(calculateStageEvents(cp = 0.80))
    
    # Define stageEvents
    stageEvents <- min(max(minNumberOfEventsPerStage[stage], 
                           stageEventsCPmax),
                       maxNumberOfEventsPerStage[stage])
    
    # Set stageEvents to minimal sample size in case minimum conditional 
    # power cannot be reached with available sample size
    if (stageEventsCPmin > maxNumberOfEventsPerStage[stage]) {
        stageEvents <- minNumberOfEventsPerStage[stage]
        
    }
    # return overall events for second stage 
    return(plannedEvents[1] + stageEvents)
}

Run the Simulation

by specifying calcEventsFunction = myEventSizeCalculationFunction and a range of assumed true hazard ratios

hazardRatioSeq <- seq(0.65, 0.85, by = 0.01)
maxNumberOfIterations <- 1000
simSurvPromZone <- getSimulationSurvival(
    design = myDesign,
    hazardRatio = hazardRatioSeq,
    directionUpper = FALSE, 
    plannedEvents = c(140, 280), 
    median2 = 8.5,
    minNumberOfEventsPerStage = c(NA, 140),
    maxNumberOfEventsPerStage = c(NA, 280),
    thetaH1 = 0.75,
    conditionalPower = 0.9,
    accrualTime = 36, 
    calcEventsFunction = myEventSizeCalculationFunction,
    maxNumberOfIterations = maxNumberOfIterations,
    maxNumberOfSubjects = 1000
)

“Usual” Conditional Power Approach

Specify calcEventsFunction = NULL

simSurvCondPower <- getSimulationSurvival(
    design = myDesign,
    hazardRatio = hazardRatioSeq, 
    directionUpper = FALSE, 
    plannedEvents = c(140, 280), 
    median2 = 8.5,
    minNumberOfEventsPerStage = c(NA, 140),
    maxNumberOfEventsPerStage = c(NA, 280),
    thetaH1 = 0.75,
    conditionalPower = 0.9,
    accrualTime = 36, 
    calcEventsFunction = NULL,
    maxNumberOfIterations = maxNumberOfIterations,
    maxNumberOfSubjects = 1000
)

Comparison of Approaches

aggSimCondPower <- getData(simSurvCondPower)
sumCP <- summarize(
    aggSimCondPower, 
    .by = c(iterationNumber, hazardRatio), 
    design = "Event re-calculation for cp = 90%",
    totalEvents = sum(eventsPerStage), 
    Z1 = testStatistic[1], 
    conditionalPower = conditionalPowerAchieved[2]
)
aggSimPromZone <- getData(simSurvPromZone)
sumCPZ <- summarize(
    aggSimPromZone, 
    .by = c(iterationNumber, hazardRatio), 
    design = "Constrained Promising Zone (CPZ) with cpmin = 80%",
    totalEvents = sum(eventsPerStage), 
    Z1 = testStatistic[1], 
    conditionalPower = conditionalPowerAchieved[2]
)
sumBoth <- rbind(sumCP, sumCPZ) |> 
  filter(Z1 > -1, Z1 < 4)

What am I Doing Here (20 Simulations)?

What am I Doing Here (20 Simulations)?

aggSimCondPower (excerpt)

    iterationNumber stageNumber hazardRatio testStatistic numberOfSubjects eventsPerStage conditionalPowerAchieved
1                 1           1        0.65   3.055749872              299            140                       NA
2                 1           2        0.65   2.147946985              462            140               0.97647736
3                 2           1        0.65   2.791088080              286            140                       NA
4                 2           2        0.65   4.604784960              484            140               0.95739550
5                 3           1        0.65   2.273360339              308            140                       NA
6                 3           2        0.65   2.354898141              500            154               0.90000000
7                 4           1        0.65   0.339945318              301            140                       NA
8                 4           2        0.65   3.394222098              500            280               0.49005084
9                 5           1        0.65   3.214083674              292            140                       NA
10                5           2        0.65   3.867095531              475            140               0.98399261
11                6           1        0.65   1.927766223              310            140                       NA
12                6           2        0.65   3.421863309              500            219               0.90000000
13                7           1        0.65   3.852722577              307            140                       NA
14                7           2        0.65   4.474589607              465            140               0.99730594
15                8           1        0.65   1.550133199              292            140                       NA
16                8           2        0.65   4.449574478              500            280               0.88203999
17                9           1        0.65   1.486578277              311            140                       NA
18                9           2        0.65   3.808874199              500            280               0.86900319
19               10           1        0.65   2.703908473              304            140                       NA
20               10           2        0.65   3.532846072              477            140               0.94887592
21               11           1        0.65   2.091238294              298            140                       NA
22               11           2        0.65   4.722070782              500            186               0.90000000
23               12           1        0.65   2.943605275              296            140                       NA
24               12           2        0.65   4.716717792              465            140               0.96951740
25               13           1        0.65  -0.499247196              295            140                       NA
26               13           2        0.65   2.432593895              500            280               0.19375716
27               14           1        0.65   2.610837957              289            140                       NA
28               14           2        0.65   1.969946441              466            140               0.93833922
29               15           1        0.65   2.371736530              295            140                       NA
30               15           2        0.65   5.372958420              470            140               0.90352112
31               16           1        0.65   1.975357793              287            140                       NA
32               16           2        0.65   3.401824564              500            209               0.90000000
33               17           1        0.65   3.026869996              293            140                       NA
34               17           2        0.65   3.782438202              469            140               0.97482700
35               18           1        0.65   2.796080259              307            140                       NA
36               18           2        0.65   3.783345012              470            140               0.95784632
37               19           1        0.65   2.524581930              302            140                       NA
38               19           2        0.65   3.172294909              480            140               0.92712721
39               20           1        0.65   1.196936971              299            140                       NA
40               20           2        0.65   3.570700190              500            280               0.79730970
41                1           1        0.75   2.470085788              279            140                       NA
42                1           2        0.75   2.236430987              431            140               0.91927751
43                2           1        0.75   2.204008347              292            140                       NA
44                2           2        0.75   2.367671894              490            165               0.90000000
45                3           1        0.75   2.794543191              282            140                       NA
46                3           2        0.75   4.490879678              450            140               0.95770793
47                4           1        0.75   2.395092736              281            140                       NA
48                4           2        0.75   2.227229288              459            140               0.90745342
49                5           1        0.75   2.700123248              289            140                       NA
50                5           2        0.75   4.312460477              464            140               0.94847732
51                6           1        0.75   2.194758267              289            140                       NA
52                6           2        0.75   1.855574258              473            166               0.90000000
53                7           1        0.75   0.216062465              293            140                       NA
54                7           2        0.75   1.153246569              500            280               0.44084615
55                8           1        0.75   0.747564676              303            140                       NA
56                8           2        0.75   2.117224473              500            280               0.64902071
57                9           1        0.75   0.501316144              286            140                       NA
58                9           2        0.75   2.739216708              500            280               0.55425908
59               10           1        0.75   1.213772140              312            140                       NA
60               10           2        0.75   1.477944963              500            280               0.80202747
61               11           1        0.75   0.567341869              302            140                       NA
62               11           2        0.75   2.938394897              500            280               0.58021953
63               12           1        0.75   0.530193610              290            140                       NA
64               12           2        0.75   1.395018013              500            280               0.56564877
65               13           1        0.75  -1.264623177              281            140                       NA
66               13           2        0.75   1.448231116              500            280               0.05160256
67               14           1        0.75   0.378436071              293            140                       NA
68               14           2        0.75   0.883424560              500            280               0.50540523
69               15           1        0.75   1.857994818              297            140                       NA
70               15           2        0.75   2.633081703              500            232               0.90000000
71               16           1        0.75   1.654564506              278            140                       NA
72               16           2        0.75   3.205090849              500            279               0.90000000
73               17           1        0.75   0.647792034              296            140                       NA
74               17           2        0.75   1.996768361              500            280               0.61137528
75               18           1        0.75   1.703834619              285            140                       NA
76               18           2        0.75   1.740330927              500            266               0.90000000
77               19           1        0.75   1.192409648              290            140                       NA
78               19           2        0.75   3.279819482              500            280               0.79602963
79               20           1        0.75   1.504144023              271            140                       NA
80               20           2        0.75   3.640154342              500            280               0.87270207
81                1           1        0.85   0.174592408              276            140                       NA
82                1           2        0.85   0.968212625              500            280               0.42453925
83                2           1        0.85  -0.426483621              289            140                       NA
84                2           2        0.85   1.314187066              500            280               0.21436395
85                3           1        0.85   2.621495325              294            140                       NA
86                3           2        0.85   2.320747472              446            140               0.93962554
87                4           1        0.85   0.003110406              284            140                       NA
88                4           2        0.85   1.375511813              500            280               0.35875959
89                5           1        0.85   1.945365188              290            140                       NA
90                5           2        0.85   3.133356546              500            215               0.90000000
91                6           1        0.85  -1.011310210              288            140                       NA
92                6           2        0.85   0.789394462              500            280               0.08438033
93                7           1        0.85   2.081112613              289            140                       NA
94                7           2        0.85   2.993868715              500            187               0.90000000
95                8           1        0.85   1.191204214              287            140                       NA
96                8           2        0.85   2.607254610              500            280               0.79568799
97                9           1        0.85   0.600912682              284            140                       NA
98                9           2        0.85   2.301541734              500            280               0.59329373
99               10           1        0.85   0.331164152              282            140                       NA
100              10           2        0.85   1.526268947              500            280               0.48654918
101              11           1        0.85   3.065198083              301            140                       NA
102              11           2        0.85   3.308201059              437            140               0.97699712
103              12           1        0.85   0.319882727              286            140                       NA
104              12           2        0.85   2.584600483              500            280               0.48205205
105              13           1        0.85   1.173937794              279            140                       NA
106              13           2        0.85   1.983222291              500            280               0.79075711
107              14           1        0.85   1.098982145              302            140                       NA
108              14           2        0.85   1.347918579              500            280               0.76855470
109              15           1        0.85   1.883605591              288            140                       NA
110              15           2        0.85   2.736588475              500            228               0.90000000
111              16           1        0.85  -0.340007504              291            140                       NA
112              16           2        0.85   0.836723409              500            280               0.24043802
113              17           1        0.85   1.570731736              281            140                       NA
114              17           2        0.85   1.005634342              500            280               0.88606137
115              18           1        0.85   2.448170467              283            140                       NA
116              18           2        0.85   2.329024192              446            140               0.91594665
117              19           1        0.85  -0.577379568              297            140                       NA
118              19           2        0.85   0.228266524              500            280               0.17302816
119              20           1        0.85  -0.332651632              279            140                       NA
120              20           2        0.85   1.610059349              500            280               0.24273297

sumCP

sumCPMini <- summarize(
    aggSimCondPowerMini, 
    .by = c(iterationNumber, hazardRatio), 
    design = "Event re-calculation for cp = 90%",
    totalEvents = sum(eventsPerStage), 
    Z1 = testStatistic[1], 
    condPower = conditionalPowerAchieved[2]
)
sumCPMini

   iterationNumber hazardRatio                            design totalEvents           Z1  condPower
1                1        0.65 Event re-calculation for cp = 90%         280  3.055749872 0.97647736
2                2        0.65 Event re-calculation for cp = 90%         280  2.791088080 0.95739550
3                3        0.65 Event re-calculation for cp = 90%         294  2.273360339 0.90000000
4                4        0.65 Event re-calculation for cp = 90%         420  0.339945318 0.49005084
5                5        0.65 Event re-calculation for cp = 90%         280  3.214083674 0.98399261
6                6        0.65 Event re-calculation for cp = 90%         359  1.927766223 0.90000000
7                7        0.65 Event re-calculation for cp = 90%         280  3.852722577 0.99730594
8                8        0.65 Event re-calculation for cp = 90%         420  1.550133199 0.88203999
9                9        0.65 Event re-calculation for cp = 90%         420  1.486578277 0.86900319
10              10        0.65 Event re-calculation for cp = 90%         280  2.703908473 0.94887592
11              11        0.65 Event re-calculation for cp = 90%         326  2.091238294 0.90000000
12              12        0.65 Event re-calculation for cp = 90%         280  2.943605275 0.96951740
13              13        0.65 Event re-calculation for cp = 90%         420 -0.499247196 0.19375716
14              14        0.65 Event re-calculation for cp = 90%         280  2.610837957 0.93833922
15              15        0.65 Event re-calculation for cp = 90%         280  2.371736530 0.90352112
16              16        0.65 Event re-calculation for cp = 90%         349  1.975357793 0.90000000
17              17        0.65 Event re-calculation for cp = 90%         280  3.026869996 0.97482700
18              18        0.65 Event re-calculation for cp = 90%         280  2.796080259 0.95784632
19              19        0.65 Event re-calculation for cp = 90%         280  2.524581930 0.92712721
20              20        0.65 Event re-calculation for cp = 90%         420  1.196936971 0.79730970
21               1        0.75 Event re-calculation for cp = 90%         280  2.470085788 0.91927751
22               2        0.75 Event re-calculation for cp = 90%         305  2.204008347 0.90000000
23               3        0.75 Event re-calculation for cp = 90%         280  2.794543191 0.95770793
24               4        0.75 Event re-calculation for cp = 90%         280  2.395092736 0.90745342
25               5        0.75 Event re-calculation for cp = 90%         280  2.700123248 0.94847732
26               6        0.75 Event re-calculation for cp = 90%         306  2.194758267 0.90000000
27               7        0.75 Event re-calculation for cp = 90%         420  0.216062465 0.44084615
28               8        0.75 Event re-calculation for cp = 90%         420  0.747564676 0.64902071
29               9        0.75 Event re-calculation for cp = 90%         420  0.501316144 0.55425908
30              10        0.75 Event re-calculation for cp = 90%         420  1.213772140 0.80202747
31              11        0.75 Event re-calculation for cp = 90%         420  0.567341869 0.58021953
32              12        0.75 Event re-calculation for cp = 90%         420  0.530193610 0.56564877
33              13        0.75 Event re-calculation for cp = 90%         420 -1.264623177 0.05160256
34              14        0.75 Event re-calculation for cp = 90%         420  0.378436071 0.50540523
35              15        0.75 Event re-calculation for cp = 90%         372  1.857994818 0.90000000
36              16        0.75 Event re-calculation for cp = 90%         419  1.654564506 0.90000000
37              17        0.75 Event re-calculation for cp = 90%         420  0.647792034 0.61137528
38              18        0.75 Event re-calculation for cp = 90%         406  1.703834619 0.90000000
39              19        0.75 Event re-calculation for cp = 90%         420  1.192409648 0.79602963
40              20        0.75 Event re-calculation for cp = 90%         420  1.504144023 0.87270207
41               1        0.85 Event re-calculation for cp = 90%         420  0.174592408 0.42453925
42               2        0.85 Event re-calculation for cp = 90%         420 -0.426483621 0.21436395
43               3        0.85 Event re-calculation for cp = 90%         280  2.621495325 0.93962554
44               4        0.85 Event re-calculation for cp = 90%         420  0.003110406 0.35875959
45               5        0.85 Event re-calculation for cp = 90%         355  1.945365188 0.90000000
46               6        0.85 Event re-calculation for cp = 90%         420 -1.011310210 0.08438033
47               7        0.85 Event re-calculation for cp = 90%         327  2.081112613 0.90000000
48               8        0.85 Event re-calculation for cp = 90%         420  1.191204214 0.79568799
49               9        0.85 Event re-calculation for cp = 90%         420  0.600912682 0.59329373
50              10        0.85 Event re-calculation for cp = 90%         420  0.331164152 0.48654918
51              11        0.85 Event re-calculation for cp = 90%         280  3.065198083 0.97699712
52              12        0.85 Event re-calculation for cp = 90%         420  0.319882727 0.48205205
53              13        0.85 Event re-calculation for cp = 90%         420  1.173937794 0.79075711
54              14        0.85 Event re-calculation for cp = 90%         420  1.098982145 0.76855470
55              15        0.85 Event re-calculation for cp = 90%         368  1.883605591 0.90000000
56              16        0.85 Event re-calculation for cp = 90%         420 -0.340007504 0.24043802
57              17        0.85 Event re-calculation for cp = 90%         420  1.570731736 0.88606137
58              18        0.85 Event re-calculation for cp = 90%         280  2.448170467 0.91594665
59              19        0.85 Event re-calculation for cp = 90%         420 -0.577379568 0.17302816
60              20        0.85 Event re-calculation for cp = 90%         420 -0.332651632 0.24273297

Plot it

ggplot(data = sumBoth, 
  aes(Z1, totalEvents, col = design,  group = design)) +
  geom_line(aes(linetype = design), lwd = 1.2) +
  theme_classic() +
  geom_line(aes(Z1, 280 + 200*dnorm(Z1, log(0.75*sqrt(140)/2))), 
        color = "black") +  
  grids(linetype = "dashed") + 
  scale_x_continuous(name = "Z-score at interim analysis") +
  scale_y_continuous(name = "Re-calculated number of events", 
        limits = c(280, 450)) +
  scale_color_manual(values = c("red", "orange"))

ggplot(data = sumBoth, 
  aes(Z1, conditionalPower, col = design, group = design)) +
  geom_line(aes(linetype = design), lwd = 1.2) +
  theme_classic() +
  grids(linetype = "dashed") + 
  geom_line(aes(Z1, dnorm(Z1, log(0.75*sqrt(140)/2))), 
        color = "black") +
  scale_x_continuous(name = "Z-score at interim analysis") +
  scale_y_continuous(
    breaks = seq(0, 1, by = 0.1),
    name = "Conditional power at re-calculated event size"
  ) +
  scale_color_manual(values = c("red", "orange"))

Don’t Increase for, e.g., p = 0.15?

Compare Power and Expected Sample Size

# Pool datasets from simulations (and fixed designs)
simCondPowerData <- with(as.list(simSurvCondPower),
    data.frame(
      design = "Events re-calculation with cp = 90%",
      hazardRatio = hazardRatio, 
      power = overallReject,
      expectedNumberOfEvents = expectedNumberOfEvents
    )
)    
simPromZoneData <- with(as.list(simSurvPromZone),
    data.frame(
      design = "Constrained Promising Zone (CPZ)",
      hazardRatio = hazardRatio, 
      power = overallReject,
      expectedNumberOfEvents = expectedNumberOfEvents
    )
)
simFixed280 <- data.frame(
    design = "Fixed events = 280",
    hazardRatio = hazardRatioSeq,
    power = getPowerSurvival(alpha = 0.025,
          directionUpper = FALSE, 
          maxNumberOfEvents = 280, 
          median2 = 8.5,
          accrualTime = 28, 
          maxNumberOfSubjects = 500,
          hazardRatio = hazardRatioSeq
    )$overallReject,
    expectedNumberOfEvents = 280
)
simFixed420 <- data.frame(
  design = "Fixed events = 420",
  hazardRatio = hazardRatioSeq,
  power = getPowerSurvival(alpha = 0.025,
         directionUpper = FALSE, 
         maxNumberOfEvents = 420, 
         median2 = 8.5,
         accrualTime = 28, 
         maxNumberOfSubjects = 500,
         hazardRatio = hazardRatioSeq
  )$overallReject,
  expectedNumberOfEvents = 420
)
simdata <- rbind(simCondPowerData, simPromZoneData, simFixed280, simFixed420)
simdata$design <- factor(
  simdata$design,
  levels = c(
   "Fixed events = 280", 
   "Fixed events = 420",
   "Events re-calculation with cp = 90%", 
   "Constrained Promising Zone (CPZ)"
  )
)

Difference in Power

Difference in Expected Sample Size

Summary

Easy implementation in rpact
Simulation very fast
Consideration of efficacy or futility stops straightforward
Trade-off between overall expected sample size and power
Change of \(cp_{min}\)?
Usage of combination test (or equivalent) theoretically mandatory
Adaptations based on test statistic only in survival designs.
See also our vignette Promising Zone Design with rpact

References

Bauer, P., and K. Köhne. 1994. “Evaluation of Experiments with Adaptive Interim Analyses.” Biometrics 50: 1029–41, correction in Biometrics 1996, 52: 380.

Chen, Y H, David L DeMets, and K K G Lan. 2004. “Increasing the Sample Size When the Unblinded Interim Result Is Promising.” Statistics in Medicine 23 (7): 1023–38.

Hsiao, Samuel T, Lingyun Liu, and Cyrus R Mehta. 2019. “Optimal Promising Zone Designs.” Biometrical Journal 61 (5): 1175–86.

Mehta, Cyrus R, and Stuart J Pocock. 2011. “Adaptive Increase in Sample Size When Interim Results Are Promising: A Practical Guide with Examples.” Statistics in Medicine 30 (28): 3267–84.

Assessing Adaptive Strategies through Simulation

The Promising Zone Approach

A Motivating Example from Hsiao et al. (2019)

“Milestone-based” investment:

Constrained Promising Zone Design

Constrained Promising Zone Design

Constrained Promising Zone Design Using rpact

Define the event number calculation function myEventSizeCalculationFunction()

Run the Simulation

“Usual” Conditional Power Approach

Comparison of Approaches

What am I Doing Here (20 Simulations)?

Plot it

Don’t Increase for, e.g., p = 0.15?

Compare Power and Expected Sample Size

Difference in Power

Difference in Expected Sample Size

Summary

References

Constrained Promising Zone Design Using `rpact`

Define the event number calculation function `myEventSizeCalculationFunction()`