early.delta.test: Estimate and test the early treatment effect

In SurrogateTest: Early Testing for a Treatment Effect using Surrogate Marker Information

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, Δ_K(t), is the difference in survival rates by time t under treatment versus control,

Δ_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,

Δ_{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)} ≤ s \mid T_B^{(g)} > t_0).

To test the null hypothesis that Δ_B(t) = 0, we test the null hypothesis Δ_{EB}(t,t_0) = 0 using the test statistic

Z_{EB}(t,t_0) = √{n_B}\frac{\hat{Δ}_{EB}(t,t_0)}{\hat{σ}_{EB}(t,t_0)}

where \hat{Δ}_{EB}(t,t_0) is a consistent estimate of Δ_{EB}(t,t_0) and \hat{σ}_{EB}(t,t_0) is the estimated standard error of √{n_B}\{\hat{Δ}_{EB}(t,t_0)-Δ_{EB}(t, t_0)\}. We reject the null hypothesis when |Z_{EB}(t,t_0) | > Φ^{-1}(1-α/2) where α is the Type 1 error rate.

To obtain \hat{Δ}_{EB}(t,t_0), we use

\hat{Δ}_{EB}(t,t_0) = n_{B1}^{-1} ∑_{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} ∑_{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{Λ}_A^{(0)}(t\mid s,t_0) \}, where

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

is a consistent estimate of Λ_A^{(0)}(t\mid t_0,s ) = -\log [r_A^{(0)}(t\mid t_0,s)], Y_{Ai}^{(0)}(t) = I(X_{Ai}^{(0)} ≥q t), N_{Ai}^{(0)}(t) = I(X_{Ai}^{(0)} ≤q t) δ_{Ai}^{(0)}, K(\cdot) is a smooth symmetric density function, K_h(x) = K(x/h)/h and γ(\cdot) is a given monotone transformation function. For the bandwidth h, we require the standard undersmoothing assumption of h=O(n_g^{-γ}) with γ \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{σ}_{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{Δ}_{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)	

SurrogateTest documentation built on Nov. 16, 2021, 9:10 a.m.