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Power and Sample Size for Two-Stage Seamless Sequential Design (TSSSD)

Source: R/RcppExports.R
getDesign_tsssd.Rd

Computes either the maximum information and stopping boundaries for a generic two-stage seamless sequential design, or the achieved power when the maximum information and stopping boundaries are provided.

Usage

getDesign_tsssd(
  beta = NA_real_,
  IMax = NA_real_,
  theta = NA_real_,
  M = NA_integer_,
  r = 1,
  corr_known = TRUE,
  K = 1L,
  informationRates = NA_real_,
  efficacyStopping = NA_integer_,
  criticalValues = NA_real_,
  alpha = 0.025,
  typeAlphaSpending = "sfOF",
  parameterAlphaSpending = NA_real_,
  userAlphaSpending = NA_real_,
  spendingTime = NA_real_,
  varianceRatio = 1
)

Arguments

beta

Type II error rate. Provide either beta or IMax; the other should be missing.

IMax

Maximum information for any active arm versus the common control. Provide either IMax or beta; the other should be missing.

theta

A vector of length \(M\) representing the true treatment effects for each active arm versus the common control. The global null is \(\theta_i = 0\) for all \(i\), and alternatives are one-sided: \(\theta_i > 0\) for at least one \(i = 1, \ldots, M\).

M

Number of active treatment arms in Phase 2.

r

Randomization ratio of each active arm to the common control in Phase 2.

corr_known

Logical. If TRUE, the correlation between Wald statistics in Phase 2 is derived from the randomization ratio r as \(r / (r + 1)\). If FALSE, a conservative correlation of 0 is used.

K

Number of sequential looks in Phase 3.

informationRates

A numeric vector of information rates fixed before the trial. If unspecified, defaults to \((1:(K+1)) / (K+1)\).

efficacyStopping

Indicators of whether efficacy stopping is allowed at each stage. Defaults to TRUE if left unspecified.

criticalValues

Upper boundaries on the z-test statistic scale for stopping for efficacy.

alpha

The significance level. Defaults to 0.025.

typeAlphaSpending

The type of alpha spending. One of the following: "OF" for O'Brien-Fleming boundaries, "P" for Pocock boundaries, "WT" for Wang & Tsiatis boundaries, "sfOF" for O'Brien-Fleming type spending function, "sfP" for Pocock type spending function, "sfKD" for Kim & DeMets spending function, "sfHSD" for Hwang, Shi & DeCani spending function, "user" for user defined spending, and "none" for no early efficacy stopping. Defaults to "sfOF".

parameterAlphaSpending

The parameter value for the alpha spending. Corresponds to \(\Delta\) for "WT", \(\rho\) for "sfKD", and \(\gamma\) for "sfHSD".

userAlphaSpending

The user defined alpha spending. Cumulative alpha spent up to each stage.

spendingTime

A numeric vector of length \(K+1\) specifying the error spending time at each analysis. Values must be strictly increasing and ends at 1. If omitted, defaults to informationRates.

varianceRatio

Ratio of the variance under \(H_0\) to the variance under \(H_1\).

Value

An S3 object of class tsssd with these components:

  • overallResults: A data frame containing:

    • overallReject: Overall probability of rejecting the null hypothesis.

    • alpha: Overall significance level.

    • M: Number of active arms in phase 2.

    • r: Randomization ratio per active arm versus control in phase 2.

    • corr_known: Whether the phase-2 correlation was assumed known.

    • K: Number of stages in phase 3.

    • information: Maximum information for any active arm versus control.

  • byStageResults: A data frame containing:

    • informationRates: Information rates at each analysis.

    • efficacyBounds: Efficacy boundaries on the Z-scale.

    • rejectPerStage: Probability of efficacy stopping at each stage.

    • cumulativeRejection: Cumulative probability of efficacy stopping.

    • cumulativeAlphaSpent: Cumulative alpha spent.

    • efficacyTheta: Efficacy boundaries on the parameter scale.

    • efficacyP: Efficacy boundaries on the p-value scale.

    • information: Cumulative information for any active arm versus control at each analysis.

    • efficacyStopping: Indicator of whether efficacy stopping is permitted.

  • byArmResults: A data frame containing:

    • theta: Parameter values for the active arms.

    • selectAsBest: Probability an arm is selected as best in at the end of phase 2.

    • powerByArm: Probability of rejecting the null for each arm by trial end.

    • condPowerByArm: Conditional power for each arm given it was selected as best at the end of phase 2.

  • settings: A list of input settings:

    • typeAlphaSpending: Type of alpha spending function.

    • parameterAlphaSpending: Parameter value for the chosen alpha spending function.

    • userAlphaSpending: User-specified alpha spending values.

    • spendingTime: Error-spending times at each analysis.

    • varianceRatio: Ratio of variance under \(H_0\) to variance under \(H_1\).

Details

If corr_known is FALSE, critical boundaries are computed assuming independence among the phase-2 Wald statistics (a conservative assumption). Power calculations, however, use the correlation implied by the randomization ratio \(r\).

References

Ping Gao, Yingqiu Li. Adaptive two-stage seamless sequential design for clinical trials. Journal of Biopharmaceutical Statistics, 2025, 35(4), 565-587.

Author

Kaifeng Lu, kaifenglu@gmail.com

Examples


# Example 1: obtain the maximum information given power
(design1 <- getDesign_tsssd(
  beta = 0.1, theta = c(0.3, 0.5), M = 2, r = 1.0,
  K = 2, informationRates = seq(1, 3)/3,
  alpha = 0.025, typeAlphaSpending = "OF"))
#>                                                              
#> Phase 2/3 seamless group-sequential design for log-rank test 
#> Overall power: 0.9, overall alpha (1-sided): 0.025           
#> Number of active arms in phase 2: 2                          
#> Randomization ratio of each active vs. control: 1            
#> Using correlation for critical value calculation: TRUE       
#> Number of looks in phase 3: 2                                
#> Alpha spending: O'Brien-Fleming                              
#>                                                              
#>                           Stage 1 Stage 2 Stage 3
#> Information rate          0.333   0.667   1.000  
#> Efficacy boundary (Z)     3.777   2.670   2.180  
#> Cumulative rejection      0.0541  0.6198  0.9000 
#> Cumulative alpha spent    0.0002  0.0066  0.0250 
#> Efficacy boundary (theta) 0.885   0.442   0.295  
#> Efficacy boundary (p)     0.0001  0.0038  0.0146 
#> Information               18.22   36.44   54.67  
#> 
#>                           Arm 1  Arm 2 
#> Treatment effect (theta)  0.300  0.500 
#> Being the best in phase 2 0.1966 0.8033
#> Power                     0.1353 0.7646
#> Conditional power         0.6883 0.9518

# Example 2: obtain power given the maximum information
(design2 <- getDesign_tsssd(
  IMax = 110/(2*1^2), theta = c(0.3, 0.5), M = 2, r = 1.0,
  K = 2, informationRates = seq(1, 3)/3,
  alpha = 0.025, typeAlphaSpending = "OF"))
#>                                                              
#> Phase 2/3 seamless group-sequential design for log-rank test 
#> Overall power: 0.9016, overall alpha (1-sided): 0.025        
#> Number of active arms in phase 2: 2                          
#> Randomization ratio of each active vs. control: 1            
#> Using correlation for critical value calculation: TRUE       
#> Number of looks in phase 3: 2                                
#> Alpha spending: O'Brien-Fleming                              
#>                                                              
#>                           Stage 1 Stage 2 Stage 3
#> Information rate          0.333   0.667   1.000  
#> Efficacy boundary (Z)     3.777   2.670   2.180  
#> Cumulative rejection      0.0548  0.6231  0.9016 
#> Cumulative alpha spent    0.0002  0.0066  0.0250 
#> Efficacy boundary (theta) 0.882   0.441   0.294  
#> Efficacy boundary (p)     0.0001  0.0038  0.0146 
#> Information               18.33   36.67   55.00  
#> 
#>                           Arm 1  Arm 2 
#> Treatment effect (theta)  0.300  0.500 
#> Being the best in phase 2 0.1959 0.8040
#> Power                     0.1354 0.7662
#> Conditional power         0.6909 0.9529