design.study {SurrogateTest}

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

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; optional (user must provide either (1) data from treatment arm in Study A or (2) hypothesized values for delta.ea (or R.A.given)and psi or (3) data from treatment arm in Study A and hypothesized psi (if different from observed treatment effect at t in Study A))

Adeltaone

event/censoring indicators in the treatment group in Study A; optional (user must provide either (1) data from treatment arm in Study A or (2) hypothesized values for delta.ea (or R.A.given)and psi or (3) data from treatment arm in Study A and hypothesized psi (if different from observed treatment effect at t 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; optional (user must provide either (1) data from treatment arm in Study A or (2) hypothesized values for delta.ea (or R.A.given) and psi or (3) data from treatment arm in Study A and hypothesized psi (if different from observed treatment effect at t in Study A))

delta.ea

hypothesized value for the early treatment effect at time t0; optional (user must provide either (1) data from treatment arm in Study A or (2) hypothesized values for delta.ea (or R.A.given) and psi or (3) data from treatment arm in Study A and hypothesized psi (if different from observed treatment effect at t in Study A)), if not given then it is assumed that this quantity equals the osberved early treatment effect at time t0 in Study A

psi

hypothesized value for the treatment effect at time t; optional (user must provide either (1) data from treatment arm in Study A or (2) hypothesized values for delta.ea (or R.A.given) and psi or (3) data from treatment arm in Study A and hypothesized psi (if different from observed treatment effect at t in Study A)), if not given then it is assumed that this quantity equals the osberved treatment effect at time t in Study A

R.A.given

hypothesized value for the proportion of treatment effect on the primary outcome explained by surrogate information at t0 in Study A; optional (user must provide either (1) data from treatment arm in Study A or (2) hypothesized values for delta.ea (or R.A.given) and psi or (3) data from treatment arm in Study A and hypothesized psi (if different from observed treatment effect at t in Study A))

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

adjustment

TRUE or FALSE; indicates whether adjustment that is needed when survival past time t is high should be used, default is FALSE if survival past t0 is < 0.90 in both arms arm of Study A, otherwise default is true if survival past t0 is >= 0.90 in either arm of Study A

n

total sample size for future study (Study B); optional (user needs to provide either n or power)

power

desired power for testing at time t0 for future study (Study B); optional (user needs to provide either n or power)

pi.1

proportion of total sample size in future study (Study B) that would be assigned to the treatment group, default is 0.5

pi.0

proportion of total sample size in future study (Study B) that would be assigned to the treatment group, default is 0.5

cens.rate

censoring in the future study (Study B) is assumed to follow an exponential distribution with censoring rate equal to this specificed value

transform

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

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

n

Total sample size needed for Study B at the given power (if power is provided by user).

power

Estimated power for Study B at the given sample size (if sample size is provided by user).

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)