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
delta.km(tl, dl, treat, tt, var = FALSE, conf.int = FALSE, weight.perturb = NULL)A list is returned:
the estimate of survival at the time of interest for treatment group 1, \(\hat{S}_1(t) = P(T>t | G=1)\)
the estimate of survival at the time of interest for treatment group 0, \(\hat{S}_0(t) = P(T>t | G=0)\)
the estimate of treatment effect at the time of interest
the variance estimate of \(\hat{S}_1(t)\); if var = TRUE or conf.int = TRUE
the variance estimate of \(\hat{S}_0(t)\); if var = TRUE or conf.int = TRUE
the variance estimate of \(\hat{\Delta}(t)\); if var = TRUE or conf.int = TRUE
the p-value from testing \(\Delta(t) = 0\); if var = TRUE or conf.int = TRUE
a vector of size 2; the 95% confidence interval for \(\hat{S}_1(t)\) based on a normal approximation; if conf.int = TRUE
a vector of size 2; the 95% confidence interval for \(\hat{S}_0(t)\) based on a normal approximation; if conf.int = TRUE
a vector of size 2; the 95% confidence interval for \(\hat{\Delta}(t)\) based on a normal approximation; if conf.int = TRUE
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
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
a vector of size 2; the 95% confidence interval for \(\hat{\Delta}(t)\) based on sample quantiles of the perturbed values, described above; if conf.int = TRUE
observed event time of primary outcome, equal to min(T, C) where T is the event time and C is the censoring time.
event indicator, equal to I(T<C) where T is the event time and C is the censoring time.
treatment indicator, should be 0/1.
the time of interest, function estimates the probability of survival past this time
TRUE or FALSE; indicates whether variance estimates for the treatment effect and survival in each group are requested, default is FALSE.
TRUE or FALSE; indicates whether 95% confidence intervals for the treatment effect and survival in each group are requested, default is FALSE.
a (n1+n0) by x matrix of weights where n1 = length of tl for treatment group 1 and n0 = length of tl for treatment group 0; used for perturbation-resampling, default is null. If var or conf.int is TRUE and weight.perturb is not provided, the function generates exponential(1) weights.
Layla Parast
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 \(\delta_{Li} = I(T_{Li}\leq C_{i})\). 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 \(\Delta(t) = S_1(t) - S_0(t)\).
The Kaplan-Meier (KM) estimate of survival at time t for each treatment group is $$ \hat{S}_{KM, j}(t) = \prod _{t_{kj} \leq t} \left [1-\frac{d_{kj}}{y_{kj}}\right ] \mbox{ if } t\geq t_{1j}, \mbox{ or } 1 \mbox{ if } t<t_{1j}$$ 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 \(\hat{\Delta}_{KM}(t) = \hat{S}_{KM, 1}(t) - \hat{S}_{KM, 0}(t)\).
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 \(\hat{\Delta}_{KM} (t)^{(b)}, b=1,...B\). 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.
Kaplan, E. L., & Meier, P. (1958). Nonparametric estimation from incomplete observations. Journal of the American Statistical Association, 53(282), 457-481.
data(example_rct)
delta.km(tl=example_rct$TL, dl = example_rct$DL, treat = example_rct$treat, tt=2)
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