Hall-Wellner simultaneous confidence band for a Kaplan-Meier curve

With sigma^2(t) the Greenwood sum, the band over the observed range is S(t) +/- h_alpha n^{-1/2} (1 + n sigma^2(t)) S(t), where h_alpha is the upper alpha point of the supremum of a Brownian bridge. h is found by bisection on the Kolmogorov series rather than read from a table, so any alpha may be used; at alpha = 0.05 it reproduces the tabulated 1.3581. Unlike the pointwise Greenwood interval the band holds simultaneously over t, so it is strictly wider.

Usage

Kpmsmp(fit, alpha)

Arguments

fit

Risk table with time, n_risk, n_event.

alpha

Simultaneous error rate.

Value

List with estimate (half-width at the last time), time, surv, half_width, sigma2, lower, upper, h, alpha, n_times, n_risk_start, n, method.

Details

Formula: P(sup|B| > h) = 2 sum_k (-1)^{k+1} exp(-2 k^2 h^2) = alpha.

References

Hall and Wellner (1980), Confidence bands for a survival curve from censored data, Biometrika 67(1):133-143. doi:10.1093/biomet/67.1.133

Examples

fit <- list(time = c(2, 3, 5, 8), n_risk = c(10, 9, 7, 4), n_event = c(1, 1, 2, 1))
Kpmsmp(fit, alpha = 0.05)[c("surv", "lower", "upper", "h")]
#> $surv
#> [1] 0.9000000 0.8000000 0.5714286 0.4285714
#> 
#> $lower
#> [1] 0.4705315 0.3705315 0.1244307 0.0000000
#> 
#> $upper
#> [1] 1.0000000 1.0000000 1.0000000 0.9172014
#> 
#> $h
#> [1] 1.358099
#>