early.delta.test {SurrogateTest}R Documentation

Estimate and test the early treatment effect

Description

Estimates the early treatment effect estimate and provides two versions of the standard error; tests the null hypothesis that this treatment effect is equal to 0

Usage

early.delta.test(Axzero, Adeltazero, Aszero, Bxzero, Bdeltazero, Bszero, Bxone, 
Bdeltaone, Bsone, t, landmark, perturb = T, 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

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

perturb

TRUE or FALSE; indicates whether the standard error estimate obtained using perturbation resampling should be calculated

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 estimate an early treatment effect quantity using surrogate marker information defined as,

\Delta_{EB}(t,t_0) = P( T_B^{(1)} > t_0) \int r(t|s,t_0) dF_B^{(1)} (s|t_0) - P( T_B^{(0)} > t_0) \int r(t|s,t_0) dF_B^{(0)} (s|t_0)

where r(t|s,t_0) = P(T_{A}^{(0)} > t | T_{A}^{(0)} > t_0, S_{A}^{(0)}=s) and F_B^{(g)}(s|t_0) = P(S_B^{(g)} \le s \mid T_B^{(g)} > t_0).

To test the null hypothesis that \Delta_B(t) = 0, we test the null hypothesis \Delta_{EB}(t,t_0) = 0 using the test statistic

Z_{EB}(t,t_0) = \sqrt{n_B}\frac{\hat{\Delta}_{EB}(t,t_0)}{\hat{\sigma}_{EB}(t,t_0)}

where \hat{\Delta}_{EB}(t,t_0) is a consistent estimate of \Delta_{EB}(t,t_0) and \hat{\sigma}_{EB}(t,t_0) is the estimated standard error of \sqrt{n_B}\{\hat{\Delta}_{EB}(t,t_0)-\Delta_{EB}(t, t_0)\}. We reject the null hypothesis when |Z_{EB}(t,t_0) | > \Phi^{-1}(1-\alpha/2) where \alpha is the Type 1 error rate.

To obtain \hat{\Delta}_{EB}(t,t_0), we use

\hat{\Delta}_{EB}(t,t_0) = n_{B1}^{-1} \sum_{i=1}^{n_{B1}} \hat{r}_A^{(0)}(t|S_{Bi}^{(1)}, t_0) \frac{I(X_{Bi}^{(1)} > t_0)}{\hat{W}_{B1}^C(t_0)} - n_{B0}^{-1} \sum_{i=1}^{n_{B0}} \hat{r}_A^{(0)}(t|S_{Bi}^{(0)}, t_0) \frac{I(X_{Bi}^{(0)} > t_0)}{\hat{W}_{B0}^C(t_0)}

where \hat{W}^C_{k g}(u) is the Kaplan-Meier estimator of W_{k g}^{C}(u)=P(C_{k}^{(g)} > u) and \hat{r}_A^{(0)}(t|s,t_0) = \exp\{-\hat{\Lambda}_A^{(0)}(t\mid s,t_0) \}, where

\hat{\Lambda}_A^{(0)}(t \mid t_0,s) = \int_{t_0}^t \frac{\sum_{i=1}^{n_{A0}} I(X_{Ai}^{(0)}>t_0) K_h\{\gamma(S_{Ai}^{(0)}) - \gamma(s)\}dN_{Ai}^{(0)} (z)}{\sum_{i=1}^{n_{A0}} K_h\{\gamma(S_{Ai}^{(0)}) - \gamma(s)\} Y_{Ai}^{(0)}(z)}

is a consistent estimate of \Lambda_A^{(0)}(t\mid t_0,s ) = -\log [r_A^{(0)}(t\mid t_0,s)], Y_{Ai}^{(0)}(t) = I(X_{Ai}^{(0)} \geq t), N_{Ai}^{(0)}(t) = I(X_{Ai}^{(0)} \leq t) \delta_{Ai}^{(0)}, K(\cdot) is a smooth symmetric density function, K_h(x) = K(x/h)/h and \gamma(\cdot) is a given monotone transformation function. For the bandwidth h, we require the standard undersmoothing assumption of h=O(n_g^{-\gamma}) with \gamma \in (1/4,1/2) in order to eliminate the impact of the bias of the conditional survival function on the resulting estimator.

The quantity \hat{\sigma}_{EB}(t,t_0) is obtained using either a closed form expression under the null or a perturbation resampling approach. If a confidence interval is desired, perturbation resampling is required.

Value

delta.eb

The estimate early treatment effect, \hat{\Delta}_{EB}(t,t_0).

se.closed

The standard error estimate of the early treatment effect using the closed form expression under the null.

Z.closed

The test statistic using the closed form standard error expression.

p.value.closed

The p-value using the closed form standard error expression.

conf.closed.norm

The confidence interval for the early treatment effect, using a normal approximation and using the closed form standard error expression.

se.perturb

The standard error estimate of the early treatment effect using perturbation resampling, if perturb = T.

Z.perturb

The test statistic using the perturbed standard error estimate, if perturb = T.

p.value.perturb

The p-value using the perturbed standard error estimate, if perturb = T.

conf.perturb.norm

The confidence interval for the early treatment effect, using a normal approximation and using the perturbed standard error expression, if perturb = T.

delta.eb.CI

The confidence interval for the early treatment effect, using the quantiles of the perturbed estimates, if perturb = 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, 75(4):1253-1263.

Examples

data(dataA)
data(dataB)
early.delta.test(Axzero = dataA$x0, Adeltazero = dataA$delta0, Aszero = dataA$s0, 
Bxzero = dataB$x0, Bdeltazero = dataB$delta0, Bszero = dataB$s0, Bxone = dataB$x1, 
Bdeltaone = dataB$delta1, Bsone = dataB$s1, t=1, landmark=0.5, perturb = FALSE, 
extrapolate = TRUE)

early.delta.test(Axzero = dataA$x0, Adeltazero = dataA$delta0, Aszero = dataA$s0, 
Bxzero = dataB$x0, Bdeltazero = dataB$delta0, Bszero = dataB$s0, Bxone = dataB$x1, 
Bdeltaone = dataB$delta1, Bsone = dataB$s1, t=0.75, landmark=0.5, perturb = FALSE, 
extrapolate = TRUE)


early.delta.test(Axzero = dataA$x0, Adeltazero = dataA$delta0, Aszero = dataA$s0, 
Bxzero = dataB$x0, Bdeltazero = dataB$delta0, Bszero = dataB$s0, Bxone = dataB$x1, 
Bdeltaone = dataB$delta1, Bsone = dataB$s1, t=1, landmark=0.5, perturb = TRUE, 
extrapolate = TRUE)	



[Package SurrogateTest version 1.3 Index]