recover.B {SurrogateTest}

Recover an estimate of the treatment effect at time t in Study B

Description

Recover an estimate of the treatment effect at time t in Study B

Usage

recover.B(Axzero, Adeltazero, Aszero, Axone, Adeltaone, Asone, Bxzero, Bdeltazero, 
Bszero, Bxone, Bdeltaone, Bsone, t, landmark, extrapolate = T, transform = F)

Arguments

Axzero

observed event times in the control group in Study A

Adeltazero

event/censoring indicators in the control group in Study A

Aszero

surrogate marker values in the control group in Study A, NA for individuals not observable at the time the surrogate marker was measured

Axone

observed event times in the treatment group in Study A

Adeltaone

event/censoring indicators in the treatment group in Study A

Asone

surrogate marker values in the treatment group in Study A, NA for individuals not observable at the time the surrogate marker was measured

Bxzero

observed event times in the control group in Study B

Bdeltazero

event/censoring indicators in the control group in Study B

Bszero

surrogate marker values in the control group in Study B, NA for individuals not observable at the time the surrogate marker was measured

Bxone

observed event times in the treatment group in Study B

Bdeltaone

event/censoring indicators in the treatment group in Study B

Bsone

surrogate marker values in the treatment group in Study B, NA for individuals not observable at the time the surrogate marker was measured

t

time of interest

landmark

landmark time of interest, t0

extrapolate

TRUE or FALSE; indicates whether local constant extrapolation should be used, default is TRUE

transform

TRUE or FALSE; indicates whether a transformation should be used, default is FALSE

Details

Assume there are two randomized studies of a treatment effect, a prior study (Study A) and a current study (Study B). Study A was completed up to some time t, while Study B was stopped at time t_0<t. In both studies, a surrogate marker was measured at time t_0 for individuals still observable at t_0. 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_K^{(1)} and T_K^{(0)} denote the time of the primary outcome of interest, death for example, under the treatment and under the control, respectively, in Study K. Let S_K^{(1)} and S_K^{(0)} denote the surrogate marker measured at time t_0 under the treatment and the control, respectively, in Study K.

The treatment effect quantity of interest, \Delta_K(t), is the difference in survival rates by time t under treatment versus control,

\Delta_K(t)=E\{ I(T_K^{(1)}>t)\} - E\{I(T_K^{(0)}>t)\} = P(T_K^{(1)}>t) - P(T_K^{(0)}>t)

where t>t_0. Here, we recover an estimate of \Delta_B(t) using Study B information (which stopped follow-up at time t_0<t) and Study A information (which has follow-up information through time t). The estimate is obtained as

\hat{\Delta}_{EB}(t,t_0)/ \hat{R}_{SA}(t,t_0)

where \hat{\Delta}_{EB}(t,t_0) is the early treatment effect estimate in Study B, described in the early.delta.test documention, and \hat{R}_{SA}(t,t_0) is the proportion of treatment effect explained by the surrogate marker information at t_0 in Study A. This proportion is calculated as \hat{R}_{SA}(t,t_0) =\hat{\Delta}_{EA}(t,t_0)/\hat{\Delta}_A(t) where

\hat{\Delta}_A(t)=n_{A1}^{-1}\sum_{i=1}^{n_{A1}}\frac{I(X_{Ai}^{(1)}>t)}{\hat{W}_{A1}^C(t)}-n_{A0}^{-1}\sum_{i=1}^{n_{A0}}\frac{I(X_{Ai}^{(0)}>t)}{\hat{W}_{A0}^C(t)},

and \hat{\Delta}_{EA}(t,t_0) is parallel to \hat{\Delta}_{EB}(t,t_0) except replacing n_{A0}^{-1} \sum_{i=1}^{n_{A0}} \hat{r}_A^{(0)}(t|S_{Ai}^{(0)}, t_0) \frac{I(X_{Ai}^{(0)} > t_0)}{\hat{W}_{A0}^C(t_0)} by n_{A0}^{-1}\sum_{i=1}^{n_{A0}}\hat{W}_{A0}^C(t)^{-1}I(X_{Ai}^{(0)}>t), and \hat{W}^C_{Ag}(\cdot) is the Kaplan-Meier estimator of the survival function for C_{A}^{(g)} for g=0,1.

Perturbation resampling is used to provide a standard error estimate for the estimate of \Delta_B(t) and a confidence interval.

Value

recovered.deltaB

The recovered estimate of \Delta_B(t).

sd.recovered.deltaB

The standard error estimate of the recovered estimate of \Delta_B(t).

conf.quantile.recovered.deltaB

A confidence interval for the recovered estimate of \Delta_B(t).

Author(s)

Layla Parast

References

Parast L, Cai T, Tian L (2019). Using a Surrogate Marker for Early Testing of a Treatment Effect. Biometrics, In press.

Parast L, Cai T and Tian L (2017). Evaluating Surrogate Marker Information using Censored Data. Statistics in Medicine, 36(11): 1767-1782.

Examples

data(dataA)
data(dataB)
recover.B(Axzero = dataA$x0, Adeltazero = dataA$delta0, Aszero = dataA$s0, Axone 
= dataA$x1, Adeltaone = dataA$delta1, Asone = dataA$s1, Bxzero = dataB$x0, Bdeltazero
= dataB$delta0, Bszero = dataB$s0, Bxone = dataB$x1, Bdeltaone = dataB$delta1, Bsone 
= dataB$s1, t=1, landmark=0.5,  extrapolate = TRUE)

recover.B(Axzero = dataA$x0, Adeltazero = dataA$delta0, Aszero = dataA$s0, Axone 
= dataA$x1, Adeltaone = dataA$delta1, Asone = dataA$s1, Bxzero = dataB$x0, Bdeltazero
= dataB$delta0, Bszero = dataB$s0, Bxone = dataB$x1, Bdeltaone = dataB$delta1, Bsone 
= dataB$s1, t=0.75, landmark=0.5,  extrapolate = TRUE)