logrank.maxtest: Maximum log-rank test

Description

Calculates a test for the comparison of two groups based on the maximum of test statistics of a set of weighted log-rank tests

Usage

logrank.maxtest(
  time,
  event,
  group,
  alternative = c("two.sided", "less", "greater"),
  rho = c(0, 0, 1),
  gamma = c(0, 1, 0),
  weights = NULL
)

Arguments

time

Vector of observed event/censored times

event

logical vector indicating if an event was observed (TRUE) or the time is censored (FALSE)

group

Vector of group allocations

alternative

Either of "two.sided","less" or "greater", specifies if two-sided or respective one-sided p-values are calculated. In any case the z test statistic of each included weighted log-rank test is based on the (weighted) sum of expected minus observed events in the group corresponding to the first factor level of group. Hence a small value of the test statistic corresponds to a lower (weighted average) hazard rate in the first group.

rho

Vector of parameter values rho for a set of weighting functions in the rho-gamma family

gamma

Vector of parameter values gamma for a set of weighting functions in the rho-gamma family

weights

Optional matrix, each column containing a different weighting vector for the data

Value

A list with elements:

pmult: The two sided p-value for the null hypothesis of equal hazard functions in both groups, based on the multivariate normal approximation for the z-statistics of differently weighted log-rank tests.
p.Bonf: The two sided p-value for the null hypothesis of equal hazard functions in both groups, based on a Bonferroni multiplicity adjustment for differently weighted log-rank tests.
tests: Data frame with z-statistics and two-sided unadjusted p-values of the individual weighted log-rank tests
korr: Estimated correlation matrix for the z-statistics of the differently weighted log-rank tests.

Details

To perform a maximum-type combination test, a set of $m$ different weight functions $w_1(t), \dots, w_m(t)$ is specified and the correspondingly weighted logrank statistics $z_1,\dots,z_m$ are calculated. The maximum test statistic is $z_{max}=\max_{i=1,\dots,m} z_i$. If at least one of the selected weight functions results in high power, we may expect a large value of $z_{max}$. Under the least favorable configuration in $H_0$, approximately $(Z_1,\dots,Z_m)\sim N_m({0},{\Sigma})$. The p-value of the maximum test, $P_{H_0}(Z_{max}>z_{max})=1-P(Z_1 \leq z_{max},\dots,Z_m \leq z_{max})$, is calculated based on this multivariate normal approximation via numeric integration.

This approach automatically corrects for multiple testing with different weights and does so efficiently since the correlation between the different tests is incorporated in ${\Sigma}$. For actual calculations, ${\Sigma}$ is replaced by an estimate. Note that $cov (w_i(t)d_{t,ctr},w_j(t)d_{t,ctr})=w_i(t)w_j(t) var(d_{t,ctr})$, at least approximately assuming weights are converging in probability to a non-random function. Thus the $i,j$-the element of ${\Sigma}$ is estimated as $$\hat{cov}(Z_i,Z_j)=\sum_{t \in \mathcal{D}} w_i(t)w_j(t)var(d_{t,ctr})/ \sqrt{\sum_{t \in \mathcal{D}} w_i^2(t) var(d_{t,ctr}) \sum_{t \in \mathcal{D}} w_j^2(t) var(d_{t,ctr})}$$

Examples

Run this code

# NOT RUN {
A <- pop_pchaz(Tint = c(0, 90, 1500),
  lambdaMat1 = matrix(c(0.2, 0.1, 0.4, 0.1), 2, 2) / 365,
 lambdaMat2 = matrix(c(0.5, 0.2, 0.6, 0.2), 2, 2) / 365,
 lambdaProg = matrix(c(0.5, 0.5, 0.4, 0.4), 2, 2) / 365,
 p = c(0.8, 0.2), 
 timezero = FALSE, discrete_approximation = TRUE)
B <- pop_pchaz(Tint = c(0, 90, 1500),
  lambdaMat1 = matrix(c(0.2, 0.1, 0.4, 0.1), 2, 2) / 365,
 lambdaMat2 = matrix(c(0.5, 0.1, 0.6, 0.1), 2, 2) / 365,
 lambdaProg = matrix(c(0.5, 0.5, 0.04, 0.04), 2, 2) / 365,
 p = c(0.8, 0.2), 
 timezero = FALSE, discrete_approximation = TRUE)
dat <- sample_fun(A, B, r0 = 0.5, eventEnd = 30,
  lambdaRecr = 0.5, lambdaCens = 0.25 / 365,
 maxRecrCalendarTime = 2 * 365,
 maxCalendar = 4 * 365)
logrank.maxtest(dat$y, dat$event, dat$group)

# }