Calculates kernel matrix

Description

Helper function; this calculates the kernel matrix

Usage

Kern.FUN(zz, zi, bw, kern0)

Arguments

Value

the kernel matrix

Author(s)

Layla Parast

Repeats a row.

Description

Helper function; this function creates a matrix that repeats vc, dm times where each row is equal to the vc vector.

Usage

VTM(vc, dm)

Arguments

Value

a matrix that repeats vc, dm times where each row is equal to the vc vector

Calculates censoring probability for weighting

Description

Helper function; calculates censoring probability needed for inverse probability of censoring weighting

Usage

censor.weight(data.x, data.delta, t, weight = NULL)

Arguments

Details

Computes the Kaplan Meier estimate of survival for the censoring random variable at the specified time

Value

Kaplan Meier estimate of survival for censoring at time t

Author(s)

Layla Parast

Examples

data(dataA)
censor.weight(data.x = dataA$x1, data.delta = dataA$delta1, t=0.5)

Helper function

Description

Helper function; should not be called directly by user.

Usage

cumsum2(mydat)

Arguments

Value

Author(s)

Layla Parast

Hypothetical Study A data

Description

Hypothetical Study A data to be used in examples; t=1 and the landmark time = 0.50.

Usage

data(dataA)

Format

A list with 6 elements representing 1000 observations from a control group and 1000 observations from a treatment group:

s1

Surrogate marker measurement for treated observations; this marker is measured at time = 0.5. For observations that experience the primary outcome or are censored before 0.5, this value is NA.

x1

The observed event or censoring time for treated observations; X = min(T, C) where T is the time of the primary outcome and C is the censoring time.

delta1

The indicator identifying whether the treated observation was observed to have the event or was censored; D =1*(T<C) where T is the time of the primary outcome and C is the censoring time.

s0

Surrogate marker measurement for control observations; this marker is measured at time = 0.5. For observations that experience the primary outcome or are censored before 0.5, this value is NA.

x0

The observed event or censoring time for control observations; X = min(T, C) where T is the time of the primary outcome and C is the censoring time.

delta0

The indicator identifying whether the control observation was observed to have the event or was censored; D =1*(T<C) where T is the time of the primary outcome and C is the censoring time.

Details

Note that if the observation is censored or experienced the primary outcome before the landmark time of 0.50, the surrogate marker measurement is not observed and coded NA.

Examples

data(dataA)
names(dataA)

Hypothetical Study B data

Description

Hypothetical Study B data to be used in examples; landmark time = 0.50.

Usage

data(dataB)

Format

A list with 6 elements representing 800 observations from a control group and 800 observations from a treatment group:

s1

Surrogate marker measurement for treated observations; this marker is measured at time = 0.5. For observations that experience the primary outcome or are censored before 0.5, this value is NA.

x1

The observed event or censoring time for treated observations; X = min(T, C) where T is the time of the primary outcome and C is the censoring time. This time is administratively censored at 0.55 (see details).

delta1

The indicator identifying whether the treated observation was observed to have the event or was censored; D =1*(T<C) where T is the time of the primary outcome and C is the censoring time.

s0

Surrogate marker measurement for control observations; this marker is measured at time = 0.5. For observations that experience the primary outcome or are censored before 0.5, this value is NA.

x0

The observed event or censoring time for control observations; X = min(T, C) where T is the time of the primary outcome and C is the censoring time. This time is administratively censored at 0.55 (see details).

delta0

The indicator identifying whether the control observation was observed to have the event or was censored; D =1*(T<C) where T is the time of the primary outcome and C is the censoring time.

Details

Note that if the observation is censored or experienced the primary outcome before the landmark time of 0.50, the surrogate marker measurement is not observed and coded NA. In addition, Study B data is only observed up to the landmark time plus some epsilon, here epsilon=0.05 such that all observations are essentially adminstratively censored at time=0.55.

Examples

data(dataB)
names(dataB)

Calculates the early treatment effect estimate in Study A

Description

Calculates the early treatment effect estimate in Study B, generally not called directly by the user

Usage

delta.ea.single(Axzero, Adeltazero, Aszero, Bxzero, Bdeltazero, 
Bszero, Bxone, Bdeltaone, Bsone, t, landmark, weightA = NULL, 
weightB = NULL, weight.both = NULL, extrapolate)

Arguments

Details

Details are included in the documentation for design.study and recover.B

Value

early treatment effect estimate

Author(s)

Layla Parast

Examples

data(dataA)

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

Calculates the early treatment effect estimate in Study B

Description

Calculates the early treatment effect estimate in Study B, generally not called directly by the user

Usage

delta.eb.single(Axzero, Adeltazero, Aszero, Bxzero, Bdeltazero, 
Bszero, Bxone, Bdeltaone, Bsone, t, landmark, weightA = NULL, 
weightB = NULL, weight.both = NULL, extrapolate)

Arguments

Details

Details are included in the documentation for early.delta.test

Value

early treatment effect estimate

Author(s)

Layla Parast

Examples

data(dataA)
data(dataB)

delta.eb.single(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,  extrapolate = TRUE)

Calculates the treatment effect, the difference in survival at time t

Description

This function calculates the treatment effect in the survival setting i.e. the difference in survival at time t between the treatment group and the control group. The inverse probability of censoring weighted estimate of survival within each treatment group is used; there is an option to use the Kaplan-Meier estimate instead. This function is generally not expected to be used directly by the user, it is called by the recover.B function.

Usage

delta.estimate(xone, xzero, deltaone, deltazero, t, weight = NULL, KM = FALSE)

Arguments

Value

the difference in survival at time t (treatment group minus control group)

Author(s)

Layla Parast

Examples

data(dataA)
delta.estimate(xone = dataA$x1, xzero = dataA$x0, deltaone = dataA$delta1, deltazero = 
dataA$delta0, t=1)

delta.estimate(xone = dataA$x1, xzero = dataA$x0, deltaone = dataA$delta1, deltazero = 
dataA$delta0, t=0.5)

Power and sample size calculation for designing a future study

Description

Power and sample size calculation for designing a future study

Usage

design.study(Axzero, Adeltazero, Aszero, Axone = NULL, Adeltaone = NULL, Asone = 
NULL, delta.ea = NULL, psi = NULL, R.A.given = NULL, t, landmark, extrapolate = T, 
adjustment = F, n = NULL, power = NULL, pi.1 = 0.5, pi.0 = 0.5, cens.rate, transform = F)

Arguments

Details

Assume information is available on a prior study, Study A, examining the effectiveness of a treatment up to some time of interest, t. The aim is to plan a future study, Study B, that would be conducted only up to time t_0<t and a test for a treatment effect would occur at t_0. In both studies, we assume a surrogate marker is/will be 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 null and alternative hypotheses of interest are:

H_0: \Delta_B(t) \equiv P(T_B^{(1)}>t) - P(T_B^{(0)}>t) = 0

H_1: \Delta_B(t) = \psi >0

Here, we plan to test H_0 in Study B 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)}

(see early.delta.test documentation). The estimated power at a type I error rate of 0.05 is thus

1 - \Phi \left\{1.96 - \frac{\sqrt{n_B}\hat{R}_{SA}(t, t_0)\psi }{ \hat{\sigma}_{EB0}(t,t_0\mid \hat{r}_A^{(0)}, W_{B}^{C})} \right \}

where \hat{R}_{SA}(t,t_0) =\hat{\Delta}_{EA}(t,t_0)/\hat{\Delta}_A(t), and

\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. In addition, \hat{\sigma}_{EB0}(t, t_0| \hat{r}_A^{(0)}, W_{B}^{C})^2 =

\frac{1}{\pi_{B0}\pi_{B1}}\left[ \frac{\hat\mu_{AB2}^{(0)}(t, t_0, \mid \hat r_A^{(0)})}{W_{B}^{C}(t_0)}-\hat\mu_{AB1}^{(0)}(t, t_0, \mid \hat r_A^{(0)})^2\left\{1+\int_0^{t_0}\frac{\lambda_{B}^{C}(u)du}{\hat{W}_{A0}^{T}(u)W_{B}^{C}(u)}\right\}\right]

assuming that the survival function of the censoring distribution is W_{B}^{C}(t) in both arms, where \pi_{Bg}=n_{Bg}/n_B and \hat{W}_{A0}^{T}(\cdot) is the Kaplan-Meier estimator of the survival function of T_A^{(0)} based on the observations from Study A, and

\hat\mu_{ABm}^{(0)}(t, t_0, \mid \hat r_A^{(0)})=n_{A0}^{-1}\sum_{i=1}^{n_{A0}}\frac{\hat{r}_A^{(0)}(t|S_{Ai}^{(0)}, t_0)^mI(X_{Ai}^{(0)}>t_0)}{\hat{W}_{A0}^{C}(t_0)}

where \hat{r}_A^{(0)}(t|s, t_0) is provided in the early.delta.test documentation.

This can be re-arranged to calculate the sample size needed in Study B to achieve a power of 100(1-\beta)\%:

n_B=\left \{ \hat{\sigma}_{EB0}(t,t_0\mid \hat{r}_A^{(0)},W_{B}^{C}) \left (\frac{1.96 - \Phi^{-1}(\beta)}{\hat{R}_{SA}(t,t_0)\psi } \right ) \right \}^2.

When the outcome rate is low (i.e., survival rate at t is high), an adjustment to the variance calculation is needed. This is automatically implemented if the survival rate at t in either arm is 0.90 or higher.

Value

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)
design.study(Axzero = dataA$x0, Adeltazero = dataA$delta0, Aszero = dataA$s0, 
Axone = dataA$x1, Adeltaone = dataA$delta1, Asone = dataA$s1, t=1, landmark=0.5, 
power = 0.80, cens.rate=0.5)

design.study(Axzero = dataA$x0, Adeltazero = dataA$delta0, Aszero = dataA$s0, 
Axone = dataA$x1, Adeltaone = dataA$delta1, Asone = dataA$s1, t=1, landmark=0.5, 
n=2500, cens.rate=0.5)

design.study(Axzero = dataA$x0, Adeltazero = dataA$delta0, Aszero = dataA$s0, 
Axone = dataA$x1, Adeltaone = dataA$delta1, Asone = dataA$s1, t=1, landmark=0.5, 
power = 0.80, cens.rate=0.5, psi = 0.05)

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

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

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)	

Helper function

Description

Helper function; should not be called directly by user.

Usage

helper.si(yy,FUN,Yi,Vi=NULL)

Arguments

Value

Author(s)

Layla Parast

Helper function

Description

Helper function; should not be called directly by user.

Usage

int.hazc(landmark, delta, x)

Arguments

Value

value

Author(s)

Layla Parast

Helper function

Description

Helper function; should not be called directly by user.

Usage

int.hazc.plan(landmark, delta, x, cens.rate, B = 10000)

Arguments

Value

value

Author(s)

Layla Parast

Calculates the conditional probability of survival to time t

Description

Helper function; calculates the estimated conditional probability of survival to time t given survival to the landmark time and given surrogate marker information.

Usage

pred.smooth.surv.new(Axzero.f, Adeltazero.f, Aszero.f, Bsnew.f, 
Bsnew2.f = NULL, myt, bw = NULL, weight.pred, extrapolate)

Arguments

Details

Details are included in the documentation for early.delta.test

Value

conditional probability of survival past t

Author(s)

Layla Parast

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

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

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)

Variance estimation

Description

Variance estimation, generally not called directly by the user

Usage

var.delta.eb(Axzero, Adeltazero, Aszero, Bxone, Bdeltaone, 
Bsone, Bxzero, Bdeltazero, Bszero, t, landmark = landmark, 
extrapolate)

Arguments

Details

Variance estimation using the closed form expression under the null hypothesis of no treatment effect; more details are included in the documentation for early.delta.test.

Value

Variance estimate for \sqrt{n_{B}}\hat{\Delta}_{EB}(t,t_0)

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)

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