sievePH (version 1.0.4)

testIndepTimeMark: Kolmogorov-Smirnov-Type Test of Conditional Independence between the Time-to-Event and a Multivariate Mark Given Treatment

Description

A nonparametric Komogorov-Smirnov-type test of the null hypothesis that the time-to-event \(T\) and a possibly multivariate mark \(V\) are conditionally independent given treatment \(Z\) as described in Juraska and Gilbert (2013). The conditional independence is a necessary assumption for parameter identifiability in the time-independent density ratio model. A bootstrap algorithm is used to compute the p-value.

Usage

testIndepTimeMark(data, iter = 1000)

Value

Returns the bootstrap p-value from the test of conditional independence between \(T\) and \(V\) given \(Z\).

Arguments

data

a data frame restricted to subjects in a given treatment group with the following columns (in this order): the observed right-censored time to the event of interest, the event indicator (1 if event, 0 if right-censored), and the mark variable (one column for each component, if multivariate)

iter

the number of bootstrap iterations (1000 by default) used for computing the p-value

Details

The test statistic is the supremum of the difference between the estimated conditional joint cumulative distribution function (cdf) of \((T,V)\) given \(Z\) and the product of the estimated conditional cdfs of \(T\) and \(V\) given \(Z\). The joint cdf is estimated by the nonparametric maximum likelihood estimator developed by Huang and Louis (1998). The marginal cdf of \(T\) is estimated as one minus the Kaplan-Meier estimator for the conditional survival function of \(T\), and the cdf of \(V\) is estimated as the empirical cdf of the observed values of \(V\). A bootstrap algorithm is used to compute the p-value.

References

Juraska, M. and Gilbert, P. B. (2013), Mark-specific hazard ratio model with multivariate continuous marks: an application to vaccine efficacy. Biometrics 69(2):328–337.

Huang, Y. and Louis, T. (1998), Nonparametric estimation of the joint distribution of survival time and mark variables. Biometrika 85, 785–798.

Examples

n <- 500
tx <- rep(0:1, each=n/2)
tm <- c(rexp(n/2, 0.2), rexp(n/2, 0.2 * exp(-0.4)))
cens <- runif(n, 0, 15)
eventTime <- pmin(tm, cens, 3)
eventInd <- as.numeric(tm <= pmin(cens, 3))
mark1 <- ifelse(eventInd==1, c(rbeta(n/2, 2, 5), rbeta(n/2, 2, 2)), NA)
mark2 <- ifelse(eventInd==1, c(rbeta(n/2, 1, 3), rbeta(n/2, 5, 1)), NA)

# perform the test for a univariate mark in the placebo group
testIndepTimeMark(data.frame(eventTime, eventInd, mark1)[tx==0, ], iter=20)

# perform the test for a bivariate mark in the placebo group
testIndepTimeMark(data.frame(eventTime, eventInd, mark1, mark2)[tx==0, ], iter=20)