R.q.event {SurrogateOutcome} | R Documentation |
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.
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")
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, |
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". |
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}, \delta_{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}
, \delta_{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,
\Delta(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)} \le t_0)\}', g=1, 0
and define the residual treatment effect after accounting for the treatment effect on the surrogate outcome information as:
\Delta_Q(t,t_0) = P ^{(0)}_{t_0,2}\int_0^{t_0} \phi_1(t|t_0,s)dF_0(s) + P^{(0)}_{t_0,3}\psi_1(t|t_0) - P(T ^{(0)}> t_0) \nu_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)
, \psi_1(t \mid t_0) = E(T^{(1)}\wedge t \mid T^{(1)}> t_0, S^{(1)} > t_0)
,
\phi_1(t\mid t_0,s) = E(T^{(1)}\wedge t \mid T ^{(1)}> t_0, S ^{(1)} = s), \quad \nu_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 \Delta(t)
and \Delta_Q(t,t_0)
:
R_Q(t,t_0) = \{\Delta(t) - \Delta_Q(t,t_0) \} / \Delta(t) = 1- \Delta_Q(t,t_0) / \Delta(t).
The quantity \Delta(t)
is estimated using inverse probability of censoring weights:
\hat{\Delta}(t) = n_1^{-1} \sum_{i=1}^{n_1} \hat{M}_{1i}(t)- n_0^{-1} \sum_{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}\delta_{gi}/\hat{W}^C_g(X_{gi})
and \hat{W}^C_g(t)
is the Kaplan-Meier estimator of P(C_{gi} \ge t).
The residual treatment effect \Delta_Q(t,t_0)
can be estimated nonparametrically or semi-parametrically. For nonparametric estimation, \psi_{1}(t|t_0)
is estimated by \hat{\psi}_{1}(t|t_0) = \sum_{i=1}^{n_1}\frac{ { \hat{W}^C_1(t_0)} I(S_{1i}>t_0, X_{1i} > t_0) }{ \sum_{i=1}^{n_1}I(S_{1i}>t_0, X_{1i} > t_0)} \hat{M}_{1i}(t)
, and \phi_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 \Lambda_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 \phi_1(t \mid t_0,s) = t_0+\int_ {t_0}^t \exp\{-\hat{\Lambda}_1(t\mid t_0,s) \}dt,
where
\hat{\Lambda}_1(t\mid t_0,s) = \int_{t_0}^t \frac{\sum_{i=1}^{n_1} I(X_{1i}>t_0, {S_{1i} < t_0}) K_h\{\gamma(S_{1i}) - \gamma(s)\}dN_{1i}(z)}{\sum_{i=1}^{n_1} I(X_{1i}>t_0, {S_{1i} < t_0}) K_h\{\gamma(S_{1i}) - \gamma(s)\} Y_{1i}(z)},
is a consistent estimate of \Lambda_1(t\mid t_0,s ),
Y_{1i}(t) = I(X_{1i} \geq t)
, N_{1i}(t) = I(X_{1i} \leq t) \delta_i, K(\cdot)
is a smooth symmetric density function, K_h(x) = K(x/h)/h
, \gamma(\cdot)
is a given monotone transformation function, and h=O(n_1^{-\eta})
is a specified bandwidth with \eta \in (1/2,1/4)
. Finally, we let
\hat{\nu}_{0}(t|t_0) = \sum_{i=1}^{n_0}\frac{ {\hat{W}^C_0(t_0)}I(X_{0i} > t_0) }{ \sum_{i=1}^{n_0}I(X_{0i} > t_0)} \hat{M}_{0i}(t).
We then estimate \Delta_{Q}(t,t_0)
as \hat{\Delta}_{Q}(t,t_0)
defined as
n_0^{-1} \sum_{i=1}^{n_0} \left \{ \frac{I_{t_0,2}(X_{0i}, S_{0i})\hat{\phi}_1(t\mid t_0, S_{0i}) + I_{t_0,3}(X_{0i}, S_{0i})\hat{\psi}_1(t\mid t_0) - I_{t_0}(X_{0i})\hat{\nu}(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{\Delta}_Q(t,t_0)/\hat{\Delta}(t).
For the semi-parametric estimate, \hat \phi_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 \{ -\Lambda_0(t|t_0)\exp(\beta_0S ^{(1)})\}
where \Lambda_0(t|t_0)
is the unspecified baseline cumulative hazard among \Omega_{t_0} = \{T^{(1)}> t_0, S^{(1)} < t_0\}
and \beta_0
is unknown. That is, let \tilde \phi_1(t| t_0,s) = t_0+\int_{t_0}^{t}\exp \{ -\hat{\Lambda}_0(t|t_0)\exp(\hat{\beta}s)\} dt,
where \hat{\beta}
is estimated by fitting a Cox model to the subpopulation \Omega_{t_0}
with a single predictor S
and \hat{\Lambda}_0(\cdot|t_0)
is the corresponding Breslow estimator. Then the semiparametric estimator for
\Delta_{Q}(t,t_0)
is \tilde{\Delta}_{Q}(t,t_0)
defined as
n_0^{-1} \sum_{i=1}^{n_0} \left \{ \frac{I_{t_0,2}(X_{0i}, S_{0i})\tilde{\phi}_1(t\mid t_0, S_{0i}) + I_{t_0,3}(X_{0i}, S_{0i})\hat{\psi}_1(t\mid t_0) - I_{t_0}(X_{0i})\hat{\nu}(t|t_0) }{\hat{W}^C_0(t_0)} \right \}
and \tilde{R}_Q(t,t_0) =1- \tilde{\Delta}_Q(t,t_0)/\hat{\Delta}(t).
A list is returned:
delta |
the estimate, |
delta.q |
the estimate, |
R.q |
the estimate, |
delta.sd |
the standard error estimate of |
delta.mad |
the standard error estimate of |
delta.q.sd |
the standard error estimate of |
delta.q.mad |
the standard error estimate of |
R.q.sd |
the standard error estimate of |
R.q.mad |
the standard error estimate of |
conf.int.delta |
a vector of size 2; the 95% confidence interval for |
conf.int.delta.q |
a vector of size 2; the 95% confidence interval for |
conf.int.R.q |
a vector of size 2; the 95% confidence interval for |
Layla Parast
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.
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)