Calculates the proportion of treatment effect explained by the surrogate marker information measured at a specified time and primary outcome information up to that specified time
Calculates the proportion of treatment effect explained by the surrogate marker information measured at a specified time and primary outcome information up to that specified time
This function calculates the proportion of treatment effect on the primary outcome explained by the surrogate marker information measured at t0 and primary outcome information up to t0. The user can also request a variance estimate, estimated using perturbating-resampling, and a 95% confidence interval. If a confidence interval is requested three versions are provided: a normal approximation based interval, a quantile based interval and Fieller's confidence interval, all using perturbation-resampling. The user can also request an estimate of the incremental value of surrogate marker information.
xone: numeric vector, the observed event times in the treatment group, X = min(T,C) where T is the time of the primary outcome and C is the censoring time.
xzero: numeric vector, the observed event times in the control group, X = min(T,C) where T is the time of the primary outcome and C is the censoring time.
deltaone: numeric vector, the event indicators for the treatment group, D = I(T<C) where T is the time of the primary outcome and C is the censoring time.
deltazero: numeric vector, the event indicators for the control group, D = I(T<C) where T is the time of the primary outcome and C is the censoring time.
sone: numeric vector; surrogate marker measurement at t0 for treated observations, assumed to be continuous. If X1i<t0, then the surrogate marker measurement should be NA.
szero: numeric vector; surrogate marker measurement at t0 for control observations, assumed to be continuous. If X1i<t0, then the surrogate marker measurement should be NA.
t: the time of interest.
weight.perturb: weights used for perturbation resampling.
landmark: the landmark time t0 or time of surrogate marker measurement.
extrapolate: TRUE or FALSE; indicates whether the user wants to use extrapolation.
transform: TRUE or FALSE; indicates whether the user wants to use a transformation for the surrogate marker.
conf.int: TRUE or FALSE; indicates whether a 95% confidence interval for delta is requested, default is FALSE.
var: TRUE or FALSE; indicates whether a variance estimate is requested, default is FALSE.
incremental.value: TRUE or FALSE; indicates whether the user would like to see the incremental value of the surrogate marker information, default is FALSE.
approx: TRUE or FALSE indicating whether an approximation should be used when calculating the probability of censoring; most relevant in settings where the survival time of interest for the primary outcome is greater than the last observed event but before the last censored case, default is TRUE.
Details
Let G be the binary treatment indicator with G=1 for treatment and G=0 for control and we assume throughout that subjects are randomly assigned to a treatment group at baseline. Let T(1) and T(0) denote the time of the primary outcome of interest, death for example, under the treatment and under the control, respectively. Let S(1) and S(0) denote the surrogate marker measured at time t0 under the treatment and the control, respectively.
where F0(⋅∣t0) is the cumulative distribution function of S(0) conditional on T(0)>t0 and ψ1(t∣s,t0)=P(T(1)>t∣S(1)=s,T(1)>t0). The proportion of treatment effect explained by the surrogate marker information measured at t0 and primary outcome information up to t0, which we denote by RS(t,t0), can be expressed using a contrast between ΔS(t,t0) and Δ(t):
The definition and estimation of Δ(t) is described in the delta.surv.estimate documentation.
Due to censoring, our data consist of n1 observations {(X1i,δ1i,S1i),i=1,...,n1} from the treatment group G=1 and n0 observations {(X0i,δ0i,S0i),i=1,...,n0} from the control group G=0 where Xgi=min(Tgi,Cgi), δgi=I(Tgi<Cgi), Cgi denotes the censoring time, and Sgi denotes the surrogate marker information measured at time t0, for g=1,0, for individual i. Note that if Xgi<t0, then Sgi should be NA (not available).
To estimate ΔS(t,t0), we use a nonparametric kernel Nelson-Aalen estimator to estimate ψ1(t∣s,t0) as ψ^1(t∣s,t0)=exp{−Λ^1(t∣s,t0)}, where
is a consistent estimate of Λ1(t∣s,t0)=−log[ψ1(t∣s,t0)],Y1i(t)=I(X1i≥t),N1i(t)=I(X1i≤t)δi,K(⋅) is a smooth symmetric density function, Kh(x)=K(x/h)/h,γ(⋅) is a given monotone transformation function, and h is a specified bandwidth. To obtain an appropriate h we first use bw.nrd to obtain hopt; and then we let h=hoptn1−c0 with c0=0.11.
Since F0(s∣t0)=P(S0i≤s∣X0i>t0), we empirically estimate F0(s∣t0) using all subjects with X0i>t0 as
where W^gC(⋅) is the Kaplan-Meier estimator of survival for censoring for g=1,0. Finally, we estimate RS(t,t0) as R^S(t,t0)=1−Δ^S(t,t0)/Δ^(t).
Variance estimation and confidence interval construction are performed using perturbation-resampling. Specifically, let {V(b)=(V11(b),...V1n1(b),V01(b),...V0n0(b))T,b=1,....,D} be n×D independent copies of a positive random variables V from a known distribution with unit mean and unit variance. Let
In this package, we use weights generated from an Exponential(1) distribution and use D=500. The variance of Δ^(t) is obtained as the empirical variance of {Δ^(t)(b),b=1,...,D}. Variance estimates for Δ^S(t,t0) and R^S(t,t0) are calculated similarly. We construct two versions of the 95% confidence interval for each estimate: one based on a normal approximation confidence interval using the estimated variance and another taking the 2.5th and 97.5th empirical percentile of the perturbed quantities. In addition, we use Fieller's method to obtain a third confidence interval for RS(t,t0) as
Since the definition of RS(t,t0) considers the surrogate information as a combination of both S information and T information up to t0, a logical inquiry would be how to assess the incremental value of the S information in terms of the proportion of treatment effect explained, when added to T information up to t0. The proportion of treatment effect explained by T information up to t0 only is denoted as RT(t,t0) and is described in the documentation for R.t.surv.estimate. The incremental value of S information is defined as:
For estimation of RT(t0), see documentation for R.t.surv.estimate. The quantity IVS(t,t0) is then estimated by IV^S(t,t0)=R^S(t,t0)−R^T(t,t0). Perturbation-resampling is used for variance estimation and confidence interval construction for this quantity, similar to the other quantities in this package.
Note that if the observed supports for S are not the same, then Λ^1(t∣s,t0) for S0i=s outside the support of S1i may return NA (depending on the bandwidth). If extrapolation = TRUE, then the Λ^1(t∣s,t0) values for these surrogate values are set to the closest non-NA value. If transform = TRUE, then S1i and S0i are transformed such that the new transformed values, S1itr and S0itr are defined as: Sgitr=F([Sgi−μ]/σ) for g=0,1 where F(⋅) is the cumulative distribution function for a standard normal random variable, and μ and σ are the sample mean and standard deviation, respectively, of {(S1i,S0i)T,is.t.Xgi>t0}.
Returns
A list is returned: - delta: the estimate, Δ^(t), described in delta.estimate documentation.
delta.s: the estimate, Δ^S(t,t0), described above.
R.s: the estimate, R^S(t,t0), described above.
delta.var: the variance estimate of Δ^(t); if var = TRUE or conf.int = TRUE.
delta.s.var: the variance estimate of Δ^S(t,t0); if var = TRUE or conf.int = TRUE.
R.s.var: the variance estimate of R^S(t,t0); if var = TRUE or conf.int = TRUE.
conf.int.normal.delta: a vector of size 2; the 95% confidence interval for Δ^(t) based on a normal approximation; if conf.int = TRUE.
conf.int.quantile.delta: a vector of size 2; the 95% confidence interval for Δ^(t) based on sample quantiles of the perturbed values, described above; if conf.int = TRUE.
conf.int.normal.delta.s: a vector of size 2; the 95% confidence interval for Δ^S(t,t0) based on a normal approximation; if conf.int = TRUE.
conf.int.quantile.delta.s: a vector of size 2; the 95% confidence interval for Δ^S(t,t0) based on sample quantiles of the perturbed values, described above; if conf.int = TRUE.
conf.int.normal.R.s: a vector of size 2; the 95% confidence interval for R^S(t,t0) based on a normal approximation; if conf.int = TRUE.
conf.int.quantile.R.s: a vector of size 2; the 95% confidence interval for R^S(t,t0) based on sample quantiles of the perturbed values, described above; if conf.int = TRUE.
conf.int.fieller.R.s: a vector of size 2; the 95% confidence interval for R^S(t,t0) based on Fieller's approach, described above; if conf.int = TRUE.
delta.t: the estimate, Δ^T(t,t0), described above; if incremental.vaue = TRUE.
R.t: the estimate, R^T(t,t0), described above; if incremental.vaue = TRUE.
incremental.value: the estimate, IV^S(t,t0), described above; if incremental.vaue = TRUE.
delta.t.var: the variance estimate of Δ^T(t,t0); if var = TRUE or conf.int = TRUE and incremental.vaue = TRUE.
R.t.var: the variance estimate of R^T(t,t0); if var = TRUE or conf.int = TRUE and incremental.vaue = TRUE.
incremental.value.var: the variance estimate of IV^S(t,t0); if var = TRUE or conf.int = TRUE and incremental.vaue = TRUE.
conf.int.normal.delta.t: a vector of size 2; the 95% confidence interval for Δ^T(t,t0) based on a normal approximation; if conf.int = TRUE and incremental.vaue = TRUE.
conf.int.quantile.delta.t: a vector of size 2; the 95% confidence interval for Δ^T(t,t0) based on sample quantiles of the perturbed values, described above; if conf.int = TRUE and incremental.vaue = TRUE.
conf.int.normal.R.t: a vector of size 2; the 95% confidence interval for R^T(t,t0) based on a normal approximation; if conf.int = TRUE and incremental.vaue = TRUE.
conf.int.quantile.R.t: a vector of size 2; the 95% confidence interval for R^T(t,t0) based on sample quantiles of the perturbed values, described above; if conf.int = TRUE and incremental.vaue = TRUE.
conf.int.fieller.R.t: a vector of size 2; the 95% confidence interval for R^T(t,t0) based on Fieller's approach, described above; if conf.int = TRUE and incremental.vaue = TRUE.
conf.int.normal.iv: a vector of size 2; the 95% confidence interval for IV^S(t,t0) based on a normal approximation; if conf.int = TRUE and incremental.vaue = TRUE.
conf.int.quantile.iv: a vector of size 2; the 95% confidence interval for IV^S(t,t0) based on sample quantiles of the perturbed values, described above; if conf.int = TRUE and incremental.vaue = TRUE.
References
Parast, L., Cai, T., & Tian, L. (2017). Evaluating surrogate marker information using censored data. Statistics in Medicine, 36(11), 1767-1782.
Author(s)
Layla Parast
Note
If the treatment effect is not significant, the user will receive the following message: "Warning: it looks like the treatment effect is not significant; may be difficult to interpret the residual treatment effect in this setting". If the observed support of the surrogate marker for the control group is outside the observed support of the surrogate marker for the treatment group, the user will receive the following message: "Warning: observed supports do not appear equal, may need to consider a transformation or extrapolation".