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Adaptive Multi-Arm Multi-Stage Design

Source: R/RcppExports.R
adaptDesign_mams.Rd

Calculates the conditional power for specified incremental information, given the interim results, parameter value, data-dependent changes in treatment selection, the error spending function, and the number and spacing of interim looks. Conversely, calculates the incremental information required to attain a specified conditional power, given the interim results, parameter value, data-dependent changes in treatment selection, the error spending function, and the number and spacing of interim looks.

Usage

adaptDesign_mams(
  betaNew = NA_real_,
  INew = NA_real_,
  M = NA_integer_,
  r = 1,
  corr_known = TRUE,
  L = NA_integer_,
  zL = NA_real_,
  theta = NA_real_,
  IMax = NA_real_,
  kMax = NA_integer_,
  informationRates = NA_real_,
  efficacyStopping = NA_integer_,
  futilityStopping = NA_integer_,
  criticalValues = NULL,
  alpha = 0.025,
  typeAlphaSpending = "sfOF",
  parameterAlphaSpending = NA_real_,
  userAlphaSpending = NA_real_,
  futilityBounds = NULL,
  futilityCP = NULL,
  futilityTheta = NULL,
  spendingTime = NA_real_,
  MullerSchafer = FALSE,
  MNew = NA_integer_,
  selected = NA_integer_,
  rNew = 1,
  kNew = NA_integer_,
  informationRatesNew = NA_real_,
  efficacyStoppingNew = NA_integer_,
  futilityStoppingNew = NA_integer_,
  typeAlphaSpendingNew = "sfOF",
  parameterAlphaSpendingNew = NA_real_,
  futilityBoundsInt = NULL,
  futilityCPInt = NULL,
  futilityThetaInt = NULL,
  typeBetaSpendingNew = "none",
  parameterBetaSpendingNew = NA_real_,
  userBetaSpendingNew = NA_real_,
  spendingTimeNew = NA_real_
)

Arguments

betaNew

The type II error for the secondary trial.

INew

The maximum information for any active arm versus the common control in the secondary trial. Either betaNew or INew should be provided, while the other must be missing.

M

Number of active treatment arms in the primary trial.

r

Randomization ratio of each active arm to the common control in the primary trial.

corr_known

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

L

The interim adaptation look of the primary trial.

zL

The z-test statistics at the interim adaptation look of the primary trial.

theta

A vector of length \(M\) representing the assumed 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 \(i = 1, \ldots, M\).

IMax

Maximum information for any active arm versus the common control for the primary trial. Must be provided.

kMax

The maximum number of stages of the primary trial.

informationRates

The information rates of the primary trial.

efficacyStopping

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

futilityStopping

Indicators of whether futility stopping is allowed at each stage of the primary trial.

criticalValues

The matrix of by-level upper boundaries on the max z-test statistic scale for efficacy stopping for the primary trial. The first column is for level M, the second column is for level M - 1, and so on, with the last column for level 1. If left unspecified, the critical values will be computed based on the specified alpha spending function.

alpha

The significance level of the primary trial. Defaults to 0.025.

typeAlphaSpending

The type of alpha spending for the primary trial. 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 of alpha spending for the primary trial. Corresponds to \(\Delta\) for "WT", \(\rho\) for "sfKD", and \(\gamma\) for "sfHSD".

userAlphaSpending

The user-defined alpha spending for the primary trial. Represents the cumulative alpha spent up to each stage.

futilityBounds

The futility boundaries on the max-z statistic scale for the primary trial. Defaults to rep(-8, kMax-1) if left unspecified.

futilityCP

The conditional power-based futility bounds for the primary trial.

futilityTheta

The parameter value-based futility bounds for the primary trial.

spendingTime

The error spending time of the primary trial. Defaults to missing, in which case it is assumed to be the same as informationRates.

MullerSchafer

Whether to use the Muller and Schafer (2001) method for trial adaptation.

MNew

Number of active treatment arms in the secondary trial.

selected

The indices of the selected active treatment arms for the secondary trial.

rNew

Randomization ratio of each active arm to the common control in the secondary trial.

kNew

The number of looks of the secondary trial.

informationRatesNew

The spacing of looks of the secondary trial.

efficacyStoppingNew

The indicators of whether efficacy stopping is allowed at each look of the secondary trial. Defaults to TRUE if left unspecified.

futilityStoppingNew

The indicators of whether futility stopping is allowed at each look of the secondary trial. Defaults to TRUE if left unspecified.

typeAlphaSpendingNew

The type of alpha spending for the secondary trial. One of the following: "OF" for O'Brien-Fleming 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, and "none" for no early efficacy stopping. Defaults to "sfOF".

parameterAlphaSpendingNew

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

futilityBoundsInt

The futility boundaries on the max-z statistic scale for new stages of the integrated trial.

futilityCPInt

The conditional power-based futility bounds for new stages of the integrated trial.

futilityThetaInt

The parameter value-based futility bounds for the new stages of the integrated trial.

typeBetaSpendingNew

The type of beta spending for the secondary trial. One of the following: "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 futility stopping. Defaults to "none".

parameterBetaSpendingNew

The parameter value of beta spending for the secondary trial. Corresponds to \(\rho\) for "sfKD", and \(\gamma\) for "sfHSD".

userBetaSpendingNew

The user-defined cumulative beta spending. Represents the cumulative beta spent up to each stage of the secondary trial.

spendingTimeNew

The error spending time of the secondary trial. Defaults to missing, in which case it is assumed to be the same as informationRatesNew.

Value

An adaptDesign_mams object with three list components:

  • primaryTrial: A list of selected information for the primary trial, including M, r, corr_known, L, zL, theta, maxInformation, kMax, informationRates, efficacyBounds, futilityBounds, information, alpha, conditionalAlpha, conditionalPower, MullerSchafer, and byLevelBounds, where byLevelBounds is a data frame with columns level, stage, and efficacyBounds, representing the efficacy bounds for each combination of the number of active arms and the stage of analysis in the primary trial.

  • secondaryTrial: A list of selected information for the secondary trial, including overallReject, alpha, M, r, selected, corr_known, kMax, maxInformation, informationRates, cumulativeRejection, cumulativeAlphaSpent, information, typeAlphaSpending, parameterAlphaSpending, typeBetaSpending, parameterBetaSpending, userBetaSpending, spendingTime, and byHypothesisBounds, where byHypothesisBounds is a data frame with columns hypothesis, stage, efficacyBounds, and futilityBounds, representing the efficacy and futility bounds for each hypothesis and each stage of analysis in the secondary trial.

  • integratedTrial: A list of selected information for the integrated trial, including M, r, corr_known, MNew, rNew, selected, L, zL, theta, maxInformation, kMax, informationRates, efficacyBounds, futilityBounds, information, and byIntersectionBounds, where byIntersectionBounds is a data frame with columns intersectionHypothesis, stage, and efficacyBounds, representing the efficacy bounds for each intersection hypothesis and each stage of analysis in the integrated trial.

References

Ping Gao, Yingqiu Li. Adaptive multiple comparison sequential design (AMCSD) for clinical trials. Journal of Biopharmaceutical Statistics, 2024, 34(3), 424-440.

See also

getDesign_mams

Author

Kaifeng Lu, kaifenglu@gmail.com

Examples


# Two active treatment arms are compared with a common control in a
# two-look time-to-event design using O'Brien–Fleming–type alpha spending.
# Suppose each active arm has a true hazard ratio of 0.75 versus control,
# and the total number of events across all three arms at the final analysis
# is 486. This corresponds to approximately 324 events for each active arm
# versus the common control. Under these assumptions, the trial has about
# 80% power to detect the treatment effect in at least one active arm.

(des1 <- getDesign_mams(
  IMax = 324 / 4, theta = c(-log(0.75), -log(0.75)),
  M = 2, r = 1, kMax = 2, informationRates = c(1/2, 1),
  alpha = 0.025, typeAlphaSpending = "OF"))
#>                                                                                
#> Multi-arm multi-stage design                                                   
#> Overall power: 0.7994, overall alpha (1-sided): 0.025                          
#> Number of active arms: 2                                                       
#> Randomization ratio of each active vs. control: 1                              
#> Using correlation for critical value calculation: TRUE                         
#> Number of looks: 2                                                             
#> Max information for pairwise comparion: 81                                     
#> Max information for overall study: 121.5                                       
#> Expected pairwise info under H1: 74.53, expected pairwise info under H0: 80.93 
#> Expected overall info under H1: 111.8, expected overall info under H0: 121.4   
#> Alpha spending: O'Brien-Fleming, beta spending: None                           
#>                                                                                
#>                               Stage 1 Stage 2
#> Information rate              0.500   1.000  
#> Efficacy boundary (Z)         3.142   2.222  
#> Cumulative rejection          0.1598  0.7994 
#> Cumulative alpha spent        0.0016  0.0250 
#> Efficacy boundary (theta)     0.494   0.247  
#> Efficacy boundary (p)         0.0008  0.0132 
#> Information for pairwise comp 40.50   81.00  
#> Information for overall study 60.75   121.50 
#> 
#> By level critical values
#>   Level Stage Boundary (Z)
#> 1     2     1        3.142
#> 2     2     2        2.222
#> 3     1     1        2.797
#> 4     1     2        1.977

# Now assume that, at the interim analysis, the observed hazard ratios for
# the two active arms versus control are 0.91 and 0.78, respectively. Using
# the rule “drop any arm with an observed hazard ratio > 0.9”, arm 1 is
# dropped. We then aim to achieve 80% conditional power to detect a hazard
# ratio of 0.78 for the remaining arm at the final look. The analysis below
# indicates that the required total number of events for arm 2 versus control
# at the final analysis should be increased from 324 to 535.

(des2 <- adaptDesign_mams(
  betaNew = 0.2, M = 2, r = 1, corr_known = FALSE,
  L = 1, zL = c(-log(0.91), -log(0.78)) * sqrt(324 / 4 / 2),
  theta = c(-log(0.91), -log(0.78)),
  IMax = 324 / 4, kMax = 2, informationRates = c(1/2, 1),
  alpha = 0.025, typeAlphaSpending = "OF",
  MNew = 1, selected = 2, rNew = 1))
#>                                                            
#> Primary trial:                                             
#> Multi-arm multi-stage design                               
#> Number of active arms: 2                                   
#> Randomization ratio of each active vs. control: 1          
#> Using correlation for critical value calculation: FALSE    
#> Max information for pairwise comparion: 81                 
#> Number of looks: 2                                         
#> Interim adaptation look: 1, z-statistic value: 0.6, 1.581  
#> theta: 0.094, 0.248                                        
#> Conditional type I error: 0.0597, conditional power: 0.496 
#> Muller & Schafer method for secondary trial: FALSE         
#>                                                            
#>                       Stage 1 Stage 2
#> Information rate      0.500   1.000  
#> Efficacy boundary (Z) 3.179   2.248  
#> Information           40.50   81.00  
#> 
#> By level critical values for primary trial
#>   Level Stage Boundary (Z)
#> 1     2     1        3.179
#> 2     2     2        2.248
#> 3     1     1        2.797
#> 4     1     2        1.977
#>                                                                  
#> Secondary trial:                                                 
#> Multi-arm multi-stage design                                     
#> Number of selected active arms: 1, selected active arms: 2       
#> Randomization ratio of each active vs. control: 1                
#> Maximum information: 93.22                                       
#> Overall power: 0.8, overall significance level (1-sided): 0.0597 
#>                                                                  
#>                        Stage 1
#> Information rate       1.000  
#> Cumulative rejection   0.8000 
#> Cumulative alpha spent 0.0597 
#> Information            93.22  
#> 
#> By hypothesis critical values for secondary trial
#>   Hypothesis Stage Efficacy boundary (Z) Futility boundary (Z)
#> 1          2     1                 1.557                 1.557
#>                                                            
#> Integrated trial:                                          
#> Adaptive multi-arm multi-stage design                      
#> Number of active arms before adaptation: 2                 
#> Number of selected active arms: 1, selected active arms: 2 
#> Total number of looks: 2                                   
#> Interim adaptation look: 1, z-statistic value: 0.6, 1.581  
#>                                                            
#>                     Stage 1 Stage 2
#> Information rate    0.303   1.000  
#> Efficacy bounds (Z) 3.179   2.170  
#> Information         40.50   133.72