factorial2x2: Design and Analysis of a 2x2 Factorial Trial with a Time-to-Event Endpoint

Motivation for the factorial2x2 package

The two-by-two factorial design randomizes subjects to receive either treatment A alone, treatment B alone, both treatment A and B (AB), or neither treatment (C). When the combined effect of A and B is less than the sum of the separate A and B effects, called a subadditive interaction, there can be low power to detect the A effect using an overall test which compares the A and AB groups to the C and B groups. In such an instance, a simple effect test of A vs. C may be useful since is not affected by an interaction. However, the simple effect tests can have low power since they use half the subjects of the overall test.

Suppose we are interested benefit of A either by itself, or in combination with B. To exploit the sample size advantage of the overall test and robustness to a subadditive interaction of the simple tests, various combinations of overall and simple effects tests may be considered. We consider three multiple testing procedures:

1. The Equal Allocation 3 procedure which allocates 1/3 of the significance level to testing each of the overall A, simple A, and simple AB effects.

2. The Proportional Allocation 2 which allocates 2/3 of the significance level to testing the overall A effect and the remaining significance level to the simple AB effect.

3. The Equal Allocation 2 which allocates 1/2 of the significance level to testing the simple A effect and 1/2 to the simple AB effect.

As described in Leifer, Troendle, et al. (2020), these multiple testing procedures all use Dunnett-corrected critical values which exploit the correlations between the logrank test statistics. Similar procedures are defined for testing the overall B and simple B effects to be used in conjunction with results for the simple AB effect obtained as described above.

factorial2x2 has two main R functions: fac2x2analyze computes the hazard ratio, 95% confidence interval, and nominal p-value for the overall A, simple A, and simple AB hazard effects. It also performs signficance testing for the three multiple testing procedures. fac2x2design calculates the power for the overall and simple tests of the three multiple testing procedures.

Examples using the fac2x2analyze function

We present two examples using fac2x2analyze.

time <- simdata[, 'time']    # follow-up time
event <- simdata[, 'event']  # event indicator
indA <- simdata[, 'indA']    # treatment A indicator
indB <- simdata[, 'indB']    # treatment B indicator
covmat <- simdata[, 6:10]    # baseline covariates
fac2x2analyze(time, event, indA, indB, covmat, alpha = 0.05, dig = 2, niter = 5)
# simdata[, 6:10] corresponds to the baseline covariates which include
# a history of cardiovascular disease (yes/no) and four indicator
# variables which correspond to which of 5 clinical centers enrolled each of the participants
$loghrAoverall
[1] -0.1170805         # overall A effect log hazard ratio (HR)

$seAoverall
[1] 0.06258749         # standard error of overall A effect log HR

$ZstatAoverall
[1] -1.87067           # Z-statistic for overall A effect

$pvalAoverall
[1] 0.06139083         # nominal p-value for overall A effect

$hrAoverall
[1] 0.8895135          # HR for overall A effect

$ciAoverall
[1] 0.786823 1.005607  # 95% confidence interval for overall A effect HR

$loghrAsimple
[1] -0.2112048         # simple A effect log HR

$seAsimple
[1] 0.08655462         # standard error of simple A effect log HR

$ZstatAsimple
[1] -2.440133          # Z-statistic for simple A effect

$pvalAsimple
[1] 0.01468184         # nominal p-value for simple A effect

$hrAsimple
[1] 0.8096082          # HR for simple A effect

$ciAsimple
[1] 0.6832791 0.9592939 # 95% confidence interval for simple A effect HR

$loghrABsimple
[1] -0.2766681         # simple AB effect log HR

$seABsimple
[1] 0.08738966         # standard error of simple AB effect log HR

$ZstatABsimple
[1] -3.165914          # Z-statistic for simple AB effect

$pvalABsimple
[1] 0.001545967        # nominal p-value for simple AB effect

$hrABsimple
[1] 0.7583061          # HR for simple AB effect 

$ciABsimple
[1] 0.6389355 0.8999785 # 95% confidence interval for simple AB effect HR

$critEA3_A
[1] -2.31              # critical value for Equal Allocation 3 procedure for Family 1 hypotheses

$sigEA3_A
[1] 0.02088815         # significance level corresponding to the critical value

$resultEA3_A
[1] "accept overall A" "reject simple A"  "reject simple AB"  # hypothesis tests results

$critPA2_A
[1] -2.13              # critical value for overall A effect for Proportional Allocation 2 

$sigPA2_A
[1] 0.03317161         # significance level corresponding to the critical value

$critPA2_ab
[1] -2.24              # critical value for simple AB effect for Proportional Allocation 2 

$sigPA2_ab
[1] 0.02509092         # significance level corresponding to the critical value

$resultPA2_A
[1] "accept overall A" "reject simple AB"  # hypothesis tests results

$critEA2_A
[1] -2.22              # critical value for Equal Allocation 2 procedure

$sigEA2_A
[1] 0.02641877         # significance level corresponding to critical value

$resultEA2_A
[1] "reject simple A"  "reject simple AB"  # hypothesis tests results

$corAa
[1] 0.7274961          # correlation between overall A and simple A logrank statistics

$corAab
[1] 0.7164075          # correlation between overall A and simple AB logrank statistics

$coraab
[1] 0.4572905          # correlation between simple A and simple AB logrank statistics

$loghrBoverall         
[1] -0.1664725         # overall B effect log hazard ratio (HR) 

$seBoverall
[1] 0.06273747         # standard error of overall B effect log HR

$ZstatBoverall
[1] -2.653478          # Z-statistic for overall B effect

$pvalBoverall
[1] 0.007966693        # nominal p-value for overll B effect

$hrBoverall
[1] 0.8466461          # HR for overall B effect

$ciBoverall
[1] 0.7486842 0.9574258 # 95% confidence interval for overall B effect HR

$loghrBsimple
[1] -0.2673257         # simple B effect log HR

$seBsimple
[1] 0.08711762         # standard error of simple B effect log HR

$ZstatBsimple
[1] -3.068561          # Z-statistic for simple B effect log HR

$pvalBsimple
[1] 0.002150925        # nominal p-value for simple B effect

$hrBsimple
[1] 0.7654237          # HR for simple B effect

$ciBsimple
[1] 0.6452766 0.9079416 # 95% confidence interval for simple B effect HR

$critEA3_B
[1] -2.32              # critical value for Equal Allocation 3 procedure for Family 2 hypotheses

$sigEA3_B
[1] 0.02034088         # significance level corresponding to the critical value

$resultEA3_B
[1] "reject overall B" "reject simple B"  # hypothesis tests results

$critPA2_B
[1] -2.13              # critical value for overall B effect for Proportional Allocation 2

$sigPA2_B
[1] 0.03317161         # significance level corresponding to critical value

$resultPA2_B
[1] "reject overall B" # hypothesis test result

$critEA2_B
[1] -2.22              # critical value for Equal Allocation 2 procedure

$sigEA2_B
[1] 0.02641877         # significance level corresponding to the critical value

$resultEA2_B
[1] "reject simple B"  # hypothesis test result

$corBb
[1] 0.7149066          # correlation between overall B and simple B logrank statistics

$corBab
[1] 0.7171287          # correlation between overall B and simple AB logrank statistics

$corbab
[1] 0.4575956          # correlation between simple B and simple AB logrank statistics

# read the COMBINE data into an R data frame
Combine <- read.table("c:\\combine_data.txt", header = T, nrows = 1226, na.strings ="", 
                      stringsAsFactors= T)
dim(Combine)
[1] 1226    9

dimnames(Combine)[[2]]
[1] "ID"         "AGE"        "GENDER"     "T0_PDA"     "NALTREXONE" 
[6] "THERAPY"    "site"       "relapse"    "futime"

# create the baseline covariate variables
T0_PDA <- Combine[,"T0_PDA"]            # baseline percentage of days abstinent
site_1 <- Combine[,"site"] == "site_1"  # research site indicator variables
site_2 <- Combine[,"site"] == "site_2"
site_3 <- Combine[,"site"] == "site_3"
site_4 <- Combine[,"site"] == "site_4"
site_5 <- Combine[,"site"] == "site_5"
site_6 <- Combine[,"site"] == "site_6"
site_7 <- Combine[,"site"] == "site_7"
site_8 <- Combine[,"site"] == "site_8"
site_9 <- Combine[,"site"] == "site_9"
site_10 <- Combine[,"site"] == "site_10"

# combine the covariates into a single covariate matrix
CombineCovMat <- cbind(T0_PDA, site_1, site_2, site_3, site_4, site_5, site_6,
                         site_7, site_8, site_9, site_10)

# define the other required variables
relapse <- Combine[,"relapse"]         # heavy drinking relapse indicator
futime <- Combine[,"futime"]           # time to first heavy drinking day or censoring
NALTREXONE <- Combine[,"NALTREXONE"]   # received naltrexone indicator
THERAPY <- Combine[,"THERAPY"]         # received cognitive behavioral intervention (CBI) indicator

# reproduce the COMBINE analysis using fac2x2analyze
fac2x2analyze(futime, relapse, NALTREXONE, THERAPY, CombineCovMat, alpha = 0.025, dig = 4)

$loghrAoverall
[1] -0.0847782              # log hazard rato estimate for the overall effect of naltrexone
                        
$seAoverall
[1] 0.06854294              # std error of the log HR estimate for the overall effect of naltrexone

$ZstatAoverall
[1] -1.236863               # Z-statistic for the overall effect of naltrexone
                            
$pvalAoverall
[1] 0.2161381               # p-value for the overall effect of naltrexone

$hrAoverall
[1] 0.918716                # hazard ratio estimate for the overall effect of naltrexone

$ciAoverall
[1] 0.8032234 1.0508149     # corresponding 95% confidence interval

$loghrAsimple
[1] -0.2517618              # log hazard rato estimate for the simple effect of naltrexone

$seAsimple
[1] 0.09786137              # std error of the log HR estimate for the simple effect of naltrexone

$ZstatAsimple
[1] -2.572637               # Z-statistic for the simple effect of naltrexone

$pvalAsimple
[1] 0.0100927               # p-value for the simple effect of naltrexone

$hrAsimple
[1] 0.7774299               # hazard ratio estimate for the simple effect of naltrexone

$ciAsimple
[1] 0.6417413 0.9418083     # corresponding 95% confidence interval

$loghrABsimple
[1] -0.09132675             # log hazard ratio estimate for the simple effect of naltrexone and CBI

$seABsimple
[1] 0.09553005              # std error of the log HR estimate for the simple effect of naltrexone 
                            # and CBI

$ZstatABsimple
[1] -0.9560003              # Z-statistic for the simple effect of naltrexone and CBI

$pvalABsimple
[1] 0.3390721               # p-value for the simple effect of naltrexone and CBI

$hrABsimple
[1] 0.9127194               # hazard ratio estimate for the simple effect of naltrexone and CBI

$ciABsimple
[1] 0.7568686 1.1006624     # corresponding 95% confidence interval

$critEA3_A
[1] -2.5811                 # critical value for the three tests in Table 4 to provide
                            # two-sided 0.025 familywise error; 
                            # slightly larger in absolute terms than the
                            # critical value -2.573 reported on p.1083 of Lin, Gong, et al.
$sigEA3_A
[1] 0.009848605

$resultEA3_A
[1] "accept overall A" "accept simple A"  "accept simple AB"

Example using the fac2x2design function

Here we use fac2x2design to compute the power for Scenario 5 in Table 2 from Leifer, Troendle, et al. (2019).

  n <- 4600          # total sample size
  rateC <- 0.0445    # one year event rate in the control group
  hrA <- 0.80        # simple A effect hazard ratio
  hrB <- 0.80        # simple B effect hazard ratio
  hrAB <- 0.72       # simple AB effect hazard ratio
  mincens <- 4.0     # minimum censoring time in years
  maxcens <- 8.4     # maximum censoring time in years
  fac2x2design(n, rateC, hrA, hrB, hrAB, mincens, maxcens, dig = 2, alpha = 0.05)
  
$events
[1] 954.8738         # expected number of events
        
$evtprob             # event probabilities for the C, A, B, and AB groups, respectively
    probC     probA     probB    probAB 
0.2446365 0.2012540 0.2012540 0.1831806 

$powerEA3overallA    
[1] 0.5861992        # Equal Allocation 3's power to detect the overall A effect

$powerEA3simpleA    
[1] 0.5817954        # Equal Allocation 3's power to detect the simple A effect

$powerEA3simplAB    
[1] 0.9071236        # Equal Allocation 3's power to detect the simple AB effect

$powerEA3anyA
[1] 0.7060777        # Equal Allocation 3's power to detect either the overall A or simple A effects

$powerPA2overallA
[1] 0.6582819        # Proportional Allocation 2's power to detect the overall A effect

$powerPA2simpleAB
[1] 0.9197286        # Proportional Allocation 2's power to detect the simple AB effect

$powerEA2simpleA
[1] 0.6203837        # Equal Allocation 2's power to detect the simple A effect

$powerEA2simpleAB
[1] 0.9226679        # Equal Allocation 2's power to detect the simple AB effect

$powerA
[1] 0.7182932        # power to detect the overall A effect at the two-sided 0.05 level

References

Leifer, E.S., Troendle, J.F., Kolecki, A., Follmann, D. Joint testing of overall and simple effect for the two-by-two factorial design. 2020. Submitted.

Lin, D-Y., Gong, J., Gallo, P., et al. Simultaneous inference on treatment effects in survival studies with factorial designs. Biometrics. 2016; 72: 1078-1085.

Slud, E.V. Analysis of factorial survival experiments. Biometrics. 1994; 50: 25-38.