Kaplan-Meier estimator with left truncation (delayed entry)

Under delayed entry a subject is not at risk before its entry time, so the risk set at t counts only those with entry < t <= observed time. Ignoring the entry times inflates the early risk sets and biases survival upwards. The variance is Greenwood's, accumulated over the same truncated risk sets.

Usage

Lftrt(entry, time, event, alpha = 0.05)

Arguments

entry: Left-truncation (entry) times.
time: Observed event or censoring times.
event: Event indicator, 1 = event, 0 = censored.
alpha: Significance level for the confidence interval.

Value

List with times, survival, se, ci_lower, ci_upper, n_obs, n_events.

Details

R arm of the existing Python lftrt module.

Formula: S(t) = prod_{t_j <= t} (1 - d_j / n_j) with n_j = #{i : entry_i < t_j <= time_i}.

References

Klein, J. P. and Moeschberger, M. L. (2003). Survival Analysis: Techniques for Censored and Truncated Data, 2nd ed. Springer, section 4.2 and chapter 3.

Examples

Lftrt(c(0, 0, 1, 2), c(5, 6, 7, 8), c(1, 1, 0, 1))
#> $times
#> [1] 5 6 8
#> 
#> $survival
#> [1] 0.75 0.50 0.00
#> 
#> $se
#> [1] 0.2165064 0.2500000 0.0000000
#> 
#> $ci_lower
#> [1] 0.3256553 0.0100090 0.0000000
#> 
#> $ci_upper
#> [1] 1.000000 0.989991 0.000000
#> 
#> $n_obs
#> [1] 4
#> 
#> $n_events
#> [1] 3
#>