delta.land.rct {landest}

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<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

landmark

the landmark time

short

a matrix of intermediate event information, there should be two columns for each intermediate event, the first column contains the observed intermediate event time, equal to min(TS, C) where TS is the event time and C is the censoring time, and the second column contains the event indicator, equal to I(TS<C)

z.cov

matrix of baseline covariate information

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.

bw

bandwidth used for kernel estimation, default is NULL

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

Author(s)

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))