sequentialPValue computes a sequential p-value for a group sequential design using a spending function as described in
Maurer and Bretz (2013) and previously defined by Liu and Anderson (2008).
It is the minimum of repeated p-values computed at each analysis (Jennison and Turnbull, 2000).
This is particularly useful for multiplicity methods such as the graphical method for group sequential designs
where sequential p-values for multiple hypotheses can be used as nominal p-values to plug into a multiplicity graph.
A sequential p-value is described as the minimum alpha level at which a one-sided group sequential bound would
be rejected given interim and final observed results.
It is meaningful for both one-sided designs and designs with non-binding futility bounds (test.type 1, 4, 6),
but not for 2-sided designs with binding futility bounds (test.type 2, 3 or 5).
Mild restrictions are required on spending functions used, but these are satisfied for commonly used spending functions
such as the Lan-DeMets spending function approximating an O'Brien-Fleming bound or a Hwang-Shih-DeCani spending function; see Maurer and Bretz (2013).
Arguments
- gsD
Group sequential design generated by
gsDesignorgsSurv.- n.I
Event counts (for time-to-event outcomes) or sample size (for most other designs); numeric vector with increasing, positive values with at most one value greater than or equal to largest value in
gsD$n.I; NOTE: if NULL, planned n.I will be used (gsD$n.I).- Z
Z-value tests corresponding to analyses in
n.I; positive values indicate a positive finding; must have the same length asn.I.- usTime
Spending time for upper bound at specified analyses; specify default:
NULLif this is to be based on information fraction; if notNULL, must have the same length asn.I; increasing positive values with at most 1 greater than or equal to 1.- interval
Interval for search to derive p-value; Default:
c(1e-05, 0.9999). Lower end of interval must be >0 and upper end must be < 1. The primary reason to not use the defaults would likely be if a test were vs a Type I error <0.0001.
Value
Sequential p-value (single numeric one-sided p-value between 0 and 1). Note that if the sequential p-value is less than the lower end of the input interval, the lower of interval will be returned. Similarly, if the sequential p-value is greater than the upper end of the input interval, then the upper end of interval is returned.
Details
Solution is found with a search using uniroot.
This finds the maximum alpha-level for which an efficacy bound is crossed,
completely ignoring any futility bound.
References
Jennison C and Turnbull BW (2000), Group Sequential Methods with Applications to Clinical Trials. Boca Raton: Chapman and Hall.
Liu, Qing, and Keaven M. Anderson. "On adaptive extensions of group sequential trials for clinical investigations." Journal of the American Statistical Association 103.484 (2008): 1621-1630.
Maurer, Willi, and Frank Bretz. "Multiple testing in group sequential trials using graphical approaches." Statistics in Biopharmaceutical Research 5.4 (2013): 311-320.;
Examples
# Derive Group Sequential Design
x <- gsSurv(k = 4, alpha = 0.025, beta = 0.1, timing = c(.5,.65,.8), sfu = sfLDOF,
sfl = sfHSD, sflpar = 2, lambdaC = log(2)/6, hr = 0.6,
eta = 0.01 , gamma = c(2.5,5,7.5,10), R = c( 2,2,2,6 ),
T = 30 , minfup = 18)
x$n.I
#> [1] 109.2757 142.0585 174.8412 218.5515
# Analysis at IA2
sequentialPValue(gsD=x,n.I=c(100,160),Z=c(1.5,2))
#> [1] 0.05363369
# Use planned spending instead of information fraction; do final analysis
seqp <- sequentialPValue(gsD=x,n.I=c(100,160,190,230),Z=c(1.5,2,2.5,3),usTime=x$timing)
seqp
#> [1] 0.001452684
# Check bounds for updated design to verify at least one was crossed
xupdate <- gsDesign(maxn.IPlan=max(x$n.I),n.I=c(100,160,190,230),usTime=x$timing,
delta=x$delta,delta1=x$delta1,k=4,alpha=x$alpha,test.type=1,
sfu=x$upper$sf,sfupar=x$upper$param)
gsBoundSummary(xupdate,logdelta=TRUE,Nname="Events",deltaname="HR")
#> Analysis Value Efficacy
#> IA 1: 46% Z 2.9626
#> Events: 100 p (1-sided) 0.0015
#> ~HR at bound 0.5529
#> P(Cross) if HR=1 0.0015
#> P(Cross) if HR=0.6 0.3456
#> IA 2: 73% Z 2.6020
#> Events: 160 p (1-sided) 0.0046
#> ~HR at bound 0.6627
#> P(Cross) if HR=1 0.0054
#> P(Cross) if HR=0.6 0.7459
#> IA 3: 87% Z 2.3051
#> Events: 190 p (1-sided) 0.0106
#> ~HR at bound 0.7157
#> P(Cross) if HR=1 0.0122
#> P(Cross) if HR=0.6 0.8953
#> Final Z 2.0241
#> Events: 230 p (1-sided) 0.0215
#> ~HR at bound 0.7657
#> P(Cross) if HR=1 0.0250
#> P(Cross) if HR=0.6 0.9709
# Now update bound using sequential p-value as alpha level
xupdate <- gsDesign(maxn.IPlan=max(x$n.I),n.I=c(100,160,190,230),usTime=x$timing,
delta=x$delta,delta1=x$delta1,k=4,alpha=seqp,test.type=1,
sfu=x$upper$sf,sfupar=x$upper$param)
# Check that we are at bound for 1 (or more) analysis and did not exceed bound for others
gsBoundSummary(xupdate,logdelta=TRUE,Nname="Events",deltaname="HR")
#> Analysis Value Efficacy
#> IA 1: 46% Z 4.3533
#> Events: 100 p (1-sided) 0.0000
#> ~HR at bound 0.4187
#> P(Cross) if HR=1 0.0000
#> P(Cross) if HR=0.6 0.0369
#> IA 2: 73% Z 3.7935
#> Events: 160 p (1-sided) 0.0001
#> ~HR at bound 0.5489
#> P(Cross) if HR=1 0.0001
#> P(Cross) if HR=0.6 0.2930
#> IA 3: 87% Z 3.3900
#> Events: 190 p (1-sided) 0.0003
#> ~HR at bound 0.6115
#> P(Cross) if HR=1 0.0004
#> P(Cross) if HR=0.6 0.5612
#> Final Z 3.0000
#> Events: 230 p (1-sided) 0.0013
#> ~HR at bound 0.6733
#> P(Cross) if HR=1 0.0015
#> P(Cross) if HR=0.6 0.8161