# delta.km: Estimates survival and treatment effect using Kaplan-Meier... In landest: Landmark Estimation of Survival and Treatment Effect

## Description

Estimates the probability of survival past some specified time and the treatment effect, defined as the difference in survival at the specified time, using Kaplan-Meier estimation

## Usage

 1 delta.km(tl, dl, treat, tt, var = FALSE, conf.int = FALSE, weight.perturb = NULL)

## Arguments

 tl observed event time of primary outcome, equal to min(T, C) where T is the event time and C is the censoring time. dl event indicator, equal to I(T

## Details

Let T_{Li} denote the time of the primary event of interest for person i, C_i denote the censoring time and G_i be the treatment group indicator such that G_i = 1 indicates treatment and G_i = 0 indicates control. Due to censoring, we observe X_{Li}= min(T_{Li}, C_{i}) and δ_{Li} = I(T_{Li}≤q C_{i}). This function estimates survival at time t within each treatment group, S_j(t) = P(T_{L} > t | G = j) for j = 1,0 and the treatment effect defined as Δ(t) = S_1(t) - S_0(t).

The Kaplan-Meier (KM) estimate of survival at time t for each treatment group is

\hat{S}_{KM, j}(t) = ∏ _{t_{kj} ≤q t} ≤ft [1-\frac{d_{kj}}{y_{kj}}\right ] \mbox{ if } t≥q t_{1j}, \mbox{ or } 1 \mbox{ if } t<t_{1j}

where t_{1j},...,t_{Dj} are the distinct observed event times of the primary outcome in treatment group j, d_{kj} is the number of events at time t_{kj} in treatment group j, and y_{kj} is the number of patients at risk at t_{kj} in treatment group j. The Kaplan-Meier (KM) estimate of treatment effect at time t is \hat{Δ}_{KM}(t) = \hat{S}_{KM, 1}(t) - \hat{S}_{KM, 0}(t).

To obtain variance estimates and construct confidence intervals, we use a perturbation-resampling method. Specifically, let \{V^{(b)}=(V_1^{(b)}, . . . ,V_n^{(b)})^{T}, b=1,...B\} be n\times B independent copies of a positive random variable U from a known distribution with unit mean and unit variance such as an Exp(1) distribution. To estimate the variance of our estimates, we appropriately weight the estimates using these perturbation weights to obtain perturbed values: \hat{S}_{KM,0} (t)^{(b)}, \hat{S}_{KM,1} (t)^{(b)}, and \hat{Δ}_{KM} (t)^{(b)}, b=1,...B. We then estimate the variance of each estimate as the empirical variance of the perturbed quantities. To construct confidence intervals, one can either use the empirical percentiles of the perturbed samples or a normal approximation.

## Value

A list is returned:

 S.estimate.1 the estimate of survival at the time of interest for treatment group 1, \hat{S}_1(t) = P(T>t | G=1) S.estimate.0 the estimate of survival at the time of interest for treatment group 0, \hat{S}_0(t) = P(T>t | G=0) delta.estimate the estimate of treatment effect at the time of interest S.var.1 the variance estimate of \hat{S}_1(t); if var = TRUE or conf.int = TRUE S.var.0 the variance estimate of \hat{S}_0(t); if var = TRUE or conf.int = TRUE delta.var the variance estimate of \hat{Δ}(t); if var = TRUE or conf.int = TRUE p.value the p-value from testing Δ(t) = 0; if var = TRUE or conf.int = TRUE conf.int.normal.S.1 a vector of size 2; the 95% confidence interval for \hat{S}_1(t) based on a normal approximation; if conf.int = TRUE conf.int.normal.S.0 a vector of size 2; the 95% confidence interval for \hat{S}_0(t) based on a normal approximation; if conf.int = TRUE conf.int.normal.delta a vector of size 2; the 95% confidence interval for \hat{Δ}(t) based on a normal approximation; if conf.int = TRUE conf.int.quantile.S.1 a vector of size 2; the 95% confidence interval for \hat{S}_1(t) based on sample quantiles of the perturbed values, described above; if conf.int = TRUE conf.int.quantile.S.0 a vector of size 2; the 95% confidence interval for \hat{S}_0(t) based on sample quantiles of the perturbed values, described above; if conf.int = TRUE conf.int.quantile.delta a vector of size 2; the 95% confidence interval for \hat{Δ}(t) based on sample quantiles of the perturbed values, described above; if conf.int = TRUE

Layla Parast

## References

Kaplan, E. L., & Meier, P. (1958). Nonparametric estimation from incomplete observations. Journal of the American Statistical Association, 53(282), 457-481.

## Examples

 1 2 data(example_rct) delta.km(tl=example_rct$TL, dl = example_rct$DL, treat = example_rct\$treat, tt=2)

landest documentation built on Nov. 16, 2021, 9:24 a.m.