R.q.event: Calculates the proportion of the treatment effect (the...

In SurrogateOutcome: Estimation of the Proportion of Treatment Effect Explained by Surrogate Outcome Information

Description

Calculates the proportion of the treatment effect (the difference in restriced mean survival time at time t) explained by surrogate outcome information observed up to the landmark time; also provides standard error estimate and confidence interval.

Usage

R.q.event(xone, xzero, deltaone, deltazero, sone, szero, t, landmark, number = 40, 
transform = FALSE, extrapolate = TRUE, std = FALSE, conf.int = FALSE, 
weight.perturb = NULL, type = "np")

Arguments

xone	numeric vector, observed event times for the primary outcome in the treatment group.
xzero	numeric vector, observed event times for the primary outcome in the control group.
deltaone	numeric vector, event/censoring indicators for the primary outcome in the treatment group.
deltazero	numeric vector, event/censoring indicators for the primary outcome in the control group.
sone	numeric vector, observed event times for the surrogate outcome in the treatment group.
szero	numeric vector, observed event times for the surrogate outcome in the control group.
t	time of interest for treatment effect.
landmark	landmark time of interest, t_0.
number	number of points for RMST calculation, default is 40.
transform	TRUE or FALSE; indicates whether a transformation should be used, default is FALSE.
extrapolate	TRUE or FALSE; indicates whether local constant extrapolation should be used, default is FALSE.
std	TRUE or FALSE; indicates whether standard error estimates should be provided, default is FALSE. Estimates are calculated using perturbation-resampling. Two versions are provided: one that takes the standard deviation of the perturbed estimates (denoted as "sd") and one that takes the median absolute deviation (denoted as "mad").
conf.int	TRUE or FALSE; indicates whether 95% confidence intervals should be provided. Confidence intervals are calculated using the percentiles of perturbed estimates, default is FALSE. If this is TRUE, standard error estimates are automatically provided.
weight.perturb	weights used for perturbation resampling.
type	Type of estimate that should be provided; options are "np" for the nonparametric estimate or "semi" for the semiparametric estimate, default is "np".

Details

Let G \in \{1,0\} be the randomized treatment indicator, T denote the time of the primary outcome of interest, and S denote the time of the surrogate outcome. We use potential outcomes notation such that T^{(G)} and S^{(G)} denote the respective times of the primary and surrogate outcomes under treatment G, for G \in \{1, 0\}. In the absence of censoring, we only observe (T, S)=(T^{(1)}, S^{(1)}) or (T^{(0)}, S^{(0)}) for each individual depending on whether G=1 or 0. Due to censoring, data consist of n = n_1 + n_0 independent observations \{X_{gi}, δ_{gi}, I(S_{gi}< t_0)I(X_{gi} > t_0), S_{gi}\wedge t_0 I(X_{gi} > t_0), i=1,...,n_g, g = 1,0\}, where X_{gi} = T_{gi}\wedge C_{ gi}, δ_{gi} = I(T_{gi} < C_{gi}), C_{gi} denotes the censoring time, T_{gi} denotes the time of the primary outcome, S_{gi} denotes the time of the surrogate outcome, \{(T_{gi}, C_{gi}, S_{gi}), i = 1, ..., n_g\} are identically distributed within treatment group, and t_0 is the landmark time of interest.

We define the treatment effect as the difference in restricted mean survival time up to a fixed time t under treatment 1 versus under treatment 0,

Δ(t)=E\{T^{(1)}\wedge t\} - E\{T^{(0)}\wedge t \}

where \wedge indicates the minimum. To define the proportion of treatment effect explained by the surrogate outcome information, let

Q_{t_0} ^{(g)} = (Q_{t_01}, Q_{t_02})'=\{S ^{(g)} \wedge t_0I(T ^{(g)} > t_0), T^{(g)} I(T^{(g)} ≤ t_0)\}', g=1, 0

and define the residual treatment effect after accounting for the treatment effect on the surrogate outcome information as:

Δ_Q(t,t_0) = P ^{(0)}_{t_0,2}\int_0^{t_0} φ_1(t|t_0,s)dF_0(s) + P^{(0)}_{t_0,3}ψ_1(t|t_0) - P(T ^{(0)}> t_0) ν_0(t|t_0)

where P^{(0)}_{t_0,2} = P(T^{(0)} > {t_0}, S ^{(0)} < t_0) and P^{(0)}_{t_0,3} = P(T^{(0)} > {t_0}, S ^{(0)} > t_0), ψ_1(t \mid t_0) = E(T^{(1)}\wedge t \mid T^{(1)}> t_0, S^{(1)} > t_0), φ_1(t\mid t_0,s) = E(T^{(1)}\wedge t \mid T ^{(1)}> t_0, S ^{(1)} = s), \quad ν_0(t|t_0) = E(T ^{(0)} \wedge t | T ^{(0)}> t_0), and F_0(\cdot\mid t_0) is the cumulative distribution function of S^{(0)} conditional on T ^{(0)}> t_0 and S ^{(0)} < t_0. Then, the proportion of treatment effect on the primary outcome that is explained by surrogate information up to t_0, Q_{t_0}, can be expressed as a contrast between Δ(t) and Δ_Q(t,t_0):

R_Q(t,t_0) = \{Δ(t) - Δ_Q(t,t_0) \} / Δ(t) = 1- Δ_Q(t,t_0) / Δ(t).

The quantity Δ(t) is estimated using inverse probability of censoring weights:

\hat{Δ}(t) = n_1^{-1} ∑_{i=1}^{n_1} \hat{M}_{1i}(t)- n_0^{-1} ∑_{i=1}^{n_0} \hat{M}_{0i}(t)

where \hat{M}_{gi}(t) = I(X_{gi} > t)t/\hat{W}^C_g(t) + I(X_{gi} < t)X_{gi}δ_{gi}/\hat{W}^C_g(X_{gi}) and \hat{W}^C_g(t) is the Kaplan-Meier estimator of P(C_{gi} ≥ t). The residual treatment effect Δ_Q(t,t_0) can be estimated nonparametrically or semi-parametrically. For nonparametric estimation, ψ_{1}(t|t_0) is estimated by \hat{ψ}_{1}(t|t_0) = ∑_{i=1}^{n_1}\frac{ { \hat{W}^C_1(t_0)} I(S_{1i}>t_0, X_{1i} > t_0) }{ ∑_{i=1}^{n_1}I(S_{1i}>t_0, X_{1i} > t_0)} \hat{M}_{1i}(t), and φ_1(t \mid t_0,s) = E(T^{(1)}\wedge t\mid X^{(1)}> t_0, S ^{(1)} = s) is estimated using a nonparametric kernel Nelson-Aalen estimator for Λ_1(t\mid t_0,s ), the cumulative hazard function of T^{(1)} conditional on S^{(1)}=s and T^{(1)}>t_0, as

\hat φ_1(t \mid t_0,s) = t_0+\int_ {t_0}^t \exp\{-\hat{Λ}_1(t\mid t_0,s) \}dt,

where

\hat{Λ}_1(t\mid t_0,s) = \int_{t_0}^t \frac{∑_{i=1}^{n_1} I(X_{1i}>t_0, {S_{1i} < t_0}) K_h\{γ(S_{1i}) - γ(s)\}dN_{1i}(z)}{∑_{i=1}^{n_1} I(X_{1i}>t_0, {S_{1i} < t_0}) K_h\{γ(S_{1i}) - γ(s)\} Y_{1i}(z)},

is a consistent estimate of Λ_1(t\mid t_0,s ), Y_{1i}(t) = I(X_{1i} ≥q t), N_{1i}(t) = I(X_{1i} ≤q t) δ_i, K(\cdot) is a smooth symmetric density function, K_h(x) = K(x/h)/h, γ(\cdot) is a given monotone transformation function, and h=O(n_1^{-η}) is a specified bandwidth with η \in (1/2,1/4). Finally, we let

\hat{ν}_{0}(t|t_0) = ∑_{i=1}^{n_0}\frac{ {\hat{W}^C_0(t_0)}I(X_{0i} > t_0) }{ ∑_{i=1}^{n_0}I(X_{0i} > t_0)} \hat{M}_{0i}(t).

We then estimate Δ_{Q}(t,t_0) as \hat{Δ}_{Q}(t,t_0) defined as

n_0^{-1} ∑_{i=1}^{n_0} ≤ft \{ \frac{I_{t_0,2}(X_{0i}, S_{0i})\hat{φ}_1(t\mid t_0, S_{0i}) + I_{t_0,3}(X_{0i}, S_{0i})\hat{ψ}_1(t\mid t_0) - I_{t_0}(X_{0i})\hat{ν}(t|t_0) }{\hat{W}^C_0(t_0)} \right \}

where I_{t_0,2}(x, s) = I(x > {t_0}, s < t_0) and I_{t_0,3}(x, s) = I(x > {t_0}, s > t_0) and I_{t_0}(x)=I(x > {t_0}) and thus, \hat{R}_Q(t,t_0) =1- \hat{Δ}_Q(t,t_0)/\hat{Δ}(t).

For the semi-parametric estimate, \hat φ_1(t| t_0,s) is replaced with an estimate obtained using a landmark Cox proportional hazards model

P(T^{(1)}> t\mid T^{(1)}> t_0, S^{(1)} < t_0, S ^{(1)}) = \exp \{ -Λ_0(t|t_0)\exp(β_0S ^{(1)})\}

where Λ_0(t|t_0) is the unspecified baseline cumulative hazard among Ω_{t_0} = \{T^{(1)}> t_0, S^{(1)} < t_0\} and β_0 is unknown. That is, let \tilde φ_1(t| t_0,s) = t_0+\int_{t_0}^{t}\exp \{ -\hat{Λ}_0(t|t_0)\exp(\hat{β}s)\} dt, where \hat{β} is estimated by fitting a Cox model to the subpopulation Ω_{t_0} with a single predictor S and \hat{Λ}_0(\cdot|t_0) is the corresponding Breslow estimator. Then the semiparametric estimator for Δ_{Q}(t,t_0) is \tilde{Δ}_{Q}(t,t_0) defined as

n_0^{-1} ∑_{i=1}^{n_0} ≤ft \{ \frac{I_{t_0,2}(X_{0i}, S_{0i})\tilde{φ}_1(t\mid t_0, S_{0i}) + I_{t_0,3}(X_{0i}, S_{0i})\hat{ψ}_1(t\mid t_0) - I_{t_0}(X_{0i})\hat{ν}(t|t_0) }{\hat{W}^C_0(t_0)} \right \}

and \tilde{R}_Q(t,t_0) =1- \tilde{Δ}_Q(t,t_0)/\hat{Δ}(t).

Value

A list is returned:

delta	the estimate, \hat{Δ}(t), described in delta.estimate documentation.
delta.q	the estimate, \hat{Δ}_Q(t,t_0), described above.
R.q	the estimate, \hat{R}_Q(t,t_0), described above.
delta.sd	the standard error estimate of \hat{Δ}(t); if std = TRUE or conf.int = TRUE.
delta.mad	the standard error estimate of \hat{Δ}(t) using the median absolute deviation; if std = TRUE or conf.int = TRUE.
delta.q.sd	the standard error estimate of \hat{Δ}_Q(t,t_0); if std = TRUE or conf.int = TRUE.
delta.q.mad	the standard error estimate of \hat{Δ}_Q(t,t_0) using the median absolute deviation; if std = TRUE or conf.int = TRUE.
R.q.sd	the standard error estimate of \hat{R}_Q(t,t_0); if std = TRUE or conf.int = TRUE.
R.q.mad	the standard error estimate of \hat{R}_Q(t,t_0) using the median absolute deviation; if std = TRUE or conf.int = TRUE.
conf.int.delta	a vector of size 2; the 95% confidence interval for \hat{Δ}(t) based on sample quantiles of the perturbed values; if conf.int = TRUE.
conf.int.delta.q	a vector of size 2; the 95% confidence interval for \hat{Δ}_Q(t,t_0) based on sample quantiles of the perturbed values; if conf.int = TRUE.
conf.int.R.q	a vector of size 2; the 95% confidence interval for \hat{R}_Q(t,t_0) based on sample quantiles of the perturbed values; if conf.int = TRUE.

Author(s)

Layla Parast

References

Parast L, Tian L, and Cai T (2020). Assessing the Value of a Censored Surrogate Outcome. Lifetime Data Analysis, 26(2):245-265.

Parast, L and Cai, T (2013). Landmark risk prediction of residual life for breast cancer survival. Statistics in Medicine, 32(20), 3459-3471.

Examples

data(ExampleData)
names(ExampleData)


R.q.event(xone = ExampleData$x1, xzero = ExampleData$x0, deltaone = ExampleData$delta1, 
deltazero = ExampleData$delta0, sone = ExampleData$s1, szero = ExampleData$s0, t = 5, 
landmark=2, type = "np")
R.q.event(xone = ExampleData$x1, xzero = ExampleData$x0, deltaone = ExampleData$delta1, 
deltazero = ExampleData$delta0, sone = ExampleData$s1, szero = ExampleData$s0, t = 5, 
landmark=2, type = "semi")
R.q.event(xone = ExampleData$x1, xzero = ExampleData$x0, deltaone = ExampleData$delta1, 
deltazero = ExampleData$delta0, sone = ExampleData$s1, szero = ExampleData$s0, t = 5, 
landmark=2, type = "np", std = TRUE, conf.int = TRUE)

SurrogateOutcome documentation built on Nov. 15, 2021, 5:08 p.m.