R: Estimates survival and treatment effect using Kaplan-Meier...

delta.km

R Documentation

Estimates survival and treatment effect using Kaplan-Meier estimation

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

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<C) where T is the event time and C is the censoring time.

treat

treatment indicator, should be 0/1.

tt

the time of interest, function estimates the probability of survival past this time

var

TRUE or FALSE; indicates whether variance estimates for the treatment effect and survival in each group are requested, default is FALSE.

conf.int

TRUE or FALSE; indicates whether 95% confidence intervals for the treatment effect and survival in each group are requested, default is FALSE.

weight.perturb

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.

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}}
ight ] 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

Author(s)

Layla Parast

References

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