delta.land.rct: Estimates survival and treatment effect using landmark... In landest: Landmark Estimation of Survival and Treatment Effect

 delta.land.rct R Documentation

Estimates survival and treatment effect using landmark 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 landmark estimation for a randomized trial setting

Usage

delta.land.rct(tl, dl, treat, tt, landmark, short = NULL, z.cov = NULL,
var = FALSE, conf.int = FALSE, weight.perturb = NULL, bw = 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, T_{Si} denote the time of the available intermediate event(s), C_i denote the censoring time, Z_{i} denote the vector of baseline (pretreatment) covariates, 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}) and X_{Si}= min(T_{Si}, C_{i}) and \delta_{Si} = I(T_{Si}\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).

To derive these estimates using landmark estimation, we first decompose the quantity into two components S_j (t)= S_j(t|t_0) S_j(t_0) using a landmark time t_0 and estimate each component separately. Intermediate event information is used in estimation of the conditional component S_j(t|t_0),

S_j(t|t_0)= P(T_L>t |T_L> t_0,G=j)=E[E[I(T_L>t | T_L> t_0,G=j,H)]]=E[S_{j,H} (t|t_0)]

where S_{j,H}(t|t_0) = P(T_L>t | T_L> t_0,G=j,H) and H = \{Z, I(T_S \leq t_0), min(T_S, t_0) \}. Then S_{j,H}(t|t_0) is estimated in two stages: 1) fitting the Cox proportional hazards model among individuals with X_L> t_0 to obtain an estimate of \beta, denoted as \hat{\beta},

S_{j,H}(t|t_0)=\exp \{-\Lambda_{j,0} (t|t_0) \exp(\beta^{T} H) \}

where \Lambda_{j,0} (t|t_0) is the cumulative baseline hazard in group j and then 2) using a nonparametric kernel Nelson-Aalen estimator to obtain a local constant estimator for the conditional hazard \Lambda_{j,u}(t|t_0) = -\log [S_{j,u}(t|t_0)] as

\hat{\Lambda}_{j,u}(t|t_0) = \int_{t_0}^t \frac{\sum_i K_h(\hat{U}_i - u) dN_i(z)}{\sum_i K_h(\hat{U}_i - u) Y_i(z)}

where S_{j,u}(t|t_0)=P(T_L>t | T_L> t_0,G=j,\hat{U}=u), \hat{U} = \hat{\beta}^{T} H, Y_i(t)=I(T_L \geq t),N_i (t)=I(T_L\leq t)I(T_L<C),K(\cdot) is a smooth symmetric density function, K_h (x/h)/h, h=O(n^{-v}) is a bandwidth with 1/2 > v > 1/4, and the summation is over all individuals with G=j and X_L>t_0. The resulting estimate for S_{j,u}(t|t_0) is \hat{S}_{j,u}(t|t_0) = \exp \{-\hat{\Lambda}_{j,u}(t|t_0)\}, and the final estimate

\hat{S}_j(t|t_0) =\frac{n^{-1} \sum_{i =1}^n \hat{S}_j(t|t_0, H_i) I(G_i=1)I(X_{Li} > t_0)}{n^{-1} \sum_{i =1}^n I(G_i=1)I(X_{Li} > t_0) }

is a consistent estimate of S_j(t|t_0).

Estimation of S_j(t_0) uses a similar two-stage approach but using only baseline covariates, to obtain \hat{S}_j(t_0). The final overall estimate of survival at time t is, \hat{S}_{LM,j} (t)= \hat{S}_j(t|t_0) \hat{S}_j(t_0). The treatment effect in terms of the difference in survival at time t is estimated as \hat{\Delta}_{LM}(t) = \hat{S}_{LM,1}(t) - \hat{S}_{LM,0}(t). To obtain an appropriate h we first use the bandwidth selection procedure given by Scott(1992) to obtain h_{opt}; and then we let h = h_{opt}n^{-0.10}.

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}_{LM,0} (t)^{(b)}, \hat{S}_{LM,1} (t)^{(b)}, and \hat{\Delta}_{LM} (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{\Delta}(t); if var = TRUE or conf.int = TRUE p.value the p-value from testing \Delta(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{\Delta}(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{\Delta}(t) based on sample quantiles of the perturbed values, described above; if conf.int = TRUE

Layla Parast

References

Parast, L., Tian, L., & Cai, T. (2014). Landmark Estimation of Survival and Treatment Effect in a Randomized Clinical Trial. Journal of the American Statistical Association, 109(505), 384-394.

Beran, R. (1981). Nonparametric regression with randomly censored survival data. Technical report, University of California Berkeley.

Scott, D. (1992). Multivariate density estimation. Wiley.

Examples

data(example_rct)
#executable but takes time
#delta.land.rct(tl=example_rct$TL, dl = example_rct$DL, treat = example_rct$treat, tt=2, #landmark = 1, short = cbind(example_rct$TS,example_rct$DS), z.cov = as.matrix(example_rct$Z))



landest documentation built on Aug. 26, 2023, 1:08 a.m.