| 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, |
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)