Description Usage Arguments Details Value Author(s) References Examples
View source: R/Functions_SurrogateTest.R
Power and sample size calculation for designing a future study
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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. |
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: Δ_B(t) \equiv P(T_B^{(1)}>t) - P(T_B^{(0)}>t) = 0
H_1: Δ_B(t) = ψ >0
Here, we plan to test H_0 in Study B using the test statistic
Z_{EB}(t,t_0) = √{n_B}\frac{\hat{Δ}_{EB}(t,t_0)}{\hat{σ}_{EB}(t,t_0)}
(see early.delta.test documentation). The estimated power at a type I error rate of 0.05 is thus
1 - Φ ≤ft\{1.96 - \frac{√{n_B}\hat{R}_{SA}(t, t_0)ψ }{ \hat{σ}_{EB0}(t,t_0\mid \hat{r}_A^{(0)}, W_{B}^{C})} \right \}
where \hat{R}_{SA}(t,t_0) =\hat{Δ}_{EA}(t,t_0)/\hat{Δ}_A(t), and
\hat{Δ}_A(t)=n_{A1}^{-1}∑_{i=1}^{n_{A1}}\frac{I(X_{Ai}^{(1)}>t)}{\hat{W}_{A1}^C(t)}-n_{A0}^{-1}∑_{i=1}^{n_{A0}}\frac{I(X_{Ai}^{(0)}>t)}{\hat{W}_{A0}^C(t)},
and \hat{Δ}_{EA}(t,t_0) is parallel to \hat{Δ}_{EB}(t,t_0) except replacing n_{A0}^{-1} ∑_{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}∑_{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{σ}_{EB0}(t, t_0| \hat{r}_A^{(0)}, W_{B}^{C})^2 =
\frac{1}{π_{B0}π_{B1}}≤ft[ \frac{\hatμ_{AB2}^{(0)}(t, t_0, \mid \hat r_A^{(0)})}{W_{B}^{C}(t_0)}-\hatμ_{AB1}^{(0)}(t, t_0, \mid \hat r_A^{(0)})^2≤ft\{1+\int_0^{t_0}\frac{λ_{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 π_{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
\hatμ_{ABm}^{(0)}(t, t_0, \mid \hat r_A^{(0)})=n_{A0}^{-1}∑_{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-β)\%:
n_B=≤ft \{ \hat{σ}_{EB0}(t,t_0\mid \hat{r}_A^{(0)},W_{B}^{C}) ≤ft (\frac{1.96 - Φ^{-1}(β)}{\hat{R}_{SA}(t,t_0)ψ } \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.
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). |
Layla Parast
Parast L, Cai T, Tian L (2019). Using a Surrogate Marker for Early Testing of a Treatment Effect. Biometrics, 75(4):1253-1263.
1 2 3 4 5 6 7 8 9 10 11 12 | 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)
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