Fast Kalbfleisch-Prentice average hazard ratio for two groups

Estimates the average hazard ratio of Kalbfleisch and Prentice (1981) between two groups over the time interval from 0 to tau, based on the Kaplan-Meier estimator of each group's survival function. This is the Kaplan-Meier (unweighted) special case of the estimator implemented in the archived AHR package, restricted to two groups and recoded in C++ for use inside simulation loops.

Usage

ahr_fast(
  time,
  event,
  group,
  control,
  side = 2,
  conf.level = 0.95,
  tau = NULL,
  null.ahr = 1,
  presorted = FALSE
)

Arguments

time: vector of right-censored event times
event: 0/1 (or logical) event indicators, 1 for an event
group: vector with exactly two distinct values identifying the groups.
control: the value of group that denotes the reference (control) group. The average hazard ratio is reported for the other group relative to it.
side: 1 for a one-sided test in the direction of treatment benefit (average hazard ratio below null.ahr) or 2 for a two-sided test (default 2).
conf.level: confidence level for the confidence interval (default 0.95)
tau: upper limit of the interval over which the average hazard ratio is computed. If NULL (default) the largest time observed in both groups is used.
null.ahr: value of the average hazard ratio under the null hypothesis used for the Z statistic and p-value (default 1)
presorted: if TRUE, assume time is already sorted in ascending order so that each group's observations are also ascending; this skips the internal sort (default FALSE)

Value

An object of class "ahr_fast", a list with elements ahr (the average hazard ratio, comparison vs reference), log.ahr, se.loghr, lower, upper, conf.level, z and p.value (the primary test on the theta / group-share scale, as in Dormuth et al. 2024 eq. 5), z.loghr and p.value.loghr (the equivalent test on the log(ahr) scale), se.theta (standard error of the tested comparison-group share), null.share, null.ahr, theta (the two group shares), var.theta1, var.theta2, side, tau, n (the two group sizes) and groups.

Details

The estimator works with the group shares of the total hazard. Writing S1 and S2 for the two survival functions, the reference-group share is theta1 = -integral(S2 dS1) / (1 - S1(tau) S2(tau)) over [0, tau], the comparison-group share is theta2 = 1 - theta1, and the average hazard ratio is ahr = theta2 / theta1. Under proportional hazards with hazard ratio psi, ahr estimates psi. A value above 1 indicates higher hazard (worse survival) in the comparison group. The variance of theta1 is the direct Greenwood-based estimator. The primary test is on the theta (group-share) scale, as in Kalbfleisch and Prentice (1981) and Dormuth et al. (2024, eq. 5): the comparison-group share is compared with its null value (0.5 when null.ahr = 1). An equivalent test and a confidence interval on the log(ahr) scale are also reported.

This is distinct from ahsw_fast, which estimates the Uno-Horiguchi average hazard with survival weight.

References

Kalbfleisch, J. D., & Prentice, R. L. (1981). Estimation of the average hazard ratio. Biometrika, 68(1), 105-112.

Dormuth, I., Pauly, M., Rauch, G., & Herrmann, C. (2024). Sample size calculation under nonproportional hazards using average hazard ratios. Biometrical Journal, 66(6), e202300271.

Examples

set.seed(1)
n <- 200
time1 <- rexp(n, 0.1)
time2 <- rexp(n, 0.18)
cens <- rexp(2 * n, 0.05)
obs <- pmin(c(time1, time2), cens)
event <- as.integer(c(time1, time2) <= cens)
group <- rep(c(0, 1), each = n)
ahr_fast(obs, event, group, control = 0, tau = 8)
#> Kalbfleisch-Prentice average hazard ratio (two-group)
#> 
#>   tau = 8,  control = 0
#>   alternative = two.sided
#> 
#>            theta   n
#> control   0.3079 200
#> treatment 0.6921 200
#> 
#>                                             Est. lower 95% upper 95%     z
#> average hazard ratio (treatment / control) 2.248     1.699     2.973 6.317
#>                                            Pr(>|z|)    
#> average hazard ratio (treatment / control) 2.66e-10 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>   (log scale: z = 5.676, p = 1.379e-08)