Calculates the augmented estimator of the proportion of treatment effect explained by the surrogate marker information measured at a specified time and primary outcome information up to that specified time

Description

This function calculates the augmented version of the proportion of treatment effect on the primary outcome explained by the surrogate marker information measured at t_0 and primary outcome information up to t_0. Variance estimates and 95 % confidence intervals for the augmented estimates are provided automatically; three versions of the confidence interval are provided: a normal approximation based interval, a quantile based interval and Fieller's confidence interval, all using perturbation-resampling. The user can also request an estimate of the incremental value of surrogate marker information.

Usage

Aug.R.s.surv.estimate(xone, xzero, deltaone, deltazero, sone, szero, t, 
weight.perturb = NULL, landmark, extrapolate = FALSE, transform = FALSE, 
basis.delta.one, basis.delta.zero, basis.delta.s.one = NULL, 
basis.delta.s.zero = NULL, incremental.value = FALSE, approx = T)

Arguments

Details

Please see R.s.surv.estimate documention for details about the estimates before augmentation is performed. Recent work has shown that augmentation can lead to improvements in efficiency by taking advantage of the association between baseline information, denoted here as Z, and the primary outcome. This function calculates the augmented estimates of the quantities of interest. For example, the augmented version of \hat{\Delta}(t) is defined as:

\hat{\Delta}(t)^{AUG} = \hat{\Delta}(t) + \gamma \{n_1^{-1}\sum_{i=1}^{n_1}h(Z_{1i})-n_0^{-1}\sum_{i=1}^{n_0}h(Z_{0i}) \}

where Z_{gi}, i=1, 2, \cdots, n_g are i.i.d. random vectors of baseline covariates from treatment group g and h(\cdot) is a basis transformation given a priori. Due to treatment randomization, \{n_1^{-1}\sum_{i=1}^{n_1}h(Z_{1i})-n_0^{-1}\sum_{i=1}^{n_0}h(Z_{0i}) \} converges to zero in probability as the sample size goes to infinity and thus the augmented estimator converges to the same limit as the original counterparts. The quantity \gamma is selected such that the variance of \hat{\Delta}(t)^{AUG} is minimized. That is, \gamma = (\Xi_{12}) ( \Xi_{22} ) ^{-1} where

\Xi_{12} = \mbox{cov} \{ \hat{\Delta}(t), n_1^{-1}\sum_{i=1}^{n_1}h(Z_{1i})-n_0^{-1}\sum_{i=1}^{n_0}h(Z_{0i}) \},

\Xi_{22} = \mbox{var} \{n_1^{-1}\sum_{i=1}^{n_1}h(Z_{1i})-n_0^{-1}\sum_{i=1}^{n_0}h(Z_{0i})\}

and thus we can obtain \hat{\Delta}(t)^{AUG} by replacing \gamma with a consistent estimator, \hat{\gamma} obtained using perturbation-resampling. A similar approach is used to obtain \hat{\Delta}_S(t)^{AUG} and thus construct

\hat{R}_S(t,t_0)^{AUG}=1-\frac{\hat{\Delta}_S(t,t_0)^{AUG}}{\hat{\Delta}(t)^{AUG}}.

When only a single Z_{gi} is provided in the basis argument, the following basis is used in this function: h(Z_{gi}) = (1, Z_{gi}, Z_{gi}^2)'.

Value

A list is returned:

Note

If the treatment effect is not significant, the user will receive the following message: "Warning: it looks like the treatment effect is not significant; may be difficult to interpret the residual treatment effect in this setting". If the observed support of the surrogate marker for the control group is outside the observed support of the surrogate marker for the treatment group, the user will receive the following message: "Warning: observed supports do not appear equal, may need to consider a transformation or extrapolation".

Author(s)

Layla Parast

References

Tian L, Cai T, Zhao L,Wei L. On the covariate-adjusted estimation for an overall treatment difference with data from a randomized comparative clinical trial. Biostatistics 2012; 13(2): 256-273.

Garcia TP, Ma Y, Yin G. Efficiency improvement in a class of survival models through model-free covariate incorporation. Lifetime Data Analysis 2011; 17(4): 552-565.

Zhang M, Tsiatis AA, Davidian M. Improving efficiency of inferences in randomized clinical trials using auxiliary covariates. Biometrics 2008; 64(3): 707-715.

Parast, L., Cai, T., & Tian, L. (2017). Evaluating surrogate marker information using censored data. Statistics in Medicine, 36(11), 1767-1782.

Examples

#computationally intensive
#Aug.R.s.surv.estimate(xone = d_example_surv$x1, xzero = d_example_surv$x0,  
#deltaone = d_example_surv$delta1, deltazero = d_example_surv$delta0, 
#sone = d_example_surv$s1, szero = d_example_surv$s0, t=3, landmark = 1, 
#basis.delta.one = d_example_surv$z1 , basis.delta.zero = d_example_surv$z0)

Calculates kernel matrix

Description

Helper function; this calculates the kernel matrix

Usage

Kern.FUN(zz, zi, bw)

Arguments

Value

the kernel matrix

Author(s)

Layla Parast

Calculates the proportion of treatment effect explained by multiple surrogate markers measured at a specified time and primary outcome information up to that specified time

Description

This function calculates the proportion of treatment effect on the primary outcome explained by multiple surrogate markers measured at t_0 and primary outcome information up to t_0. The user can also request a variance estimate, estimated using perturbating-resampling, and a 95% confidence interval. If a confidence interval is requested three versions are provided: a normal approximation based interval, a quantile based interval and Fieller's confidence interval, all using perturbation-resampling. The user can also request an estimate of the incremental value of the surrogate marker information.

Usage

R.multiple.surv(xone, xzero, deltaone, deltazero, sone, szero, type =1, t, 
weight.perturb = NULL, landmark, extrapolate = FALSE, transform = FALSE, 
conf.int = FALSE, var = FALSE, incremental.value = FALSE, approx = T)

Arguments

Details

Let G \in \{A, B\} be the binary treatment indicator and we assume that subjects are randomly assigned to either treatment group A or B at baseline. Let T denote the time to the occurrence of the primary outcome, death for example, and S = (S_1,S_2,...,S_k)^{T} denote the vector of k surrogate markers measured at a given time t_0. Let T^{(g)} and S^{(g)} denote the counterfactual event time and surrogate marker measurements under treatment G = g for g \in \{A, B\}. In practice, we only observe (T, S)=(T^{(A)}, S^{(A)}) or (T^{(B)}, S^{(B)}) depending on whether G=A or B. The treatment effect, \Delta(t), is the treatment difference in survival rates at time t > t_0, \Delta(t)=E\{ I(T^{(A)}>t)\} - E\{I(T^{(B)}>t)\} = P(T^{(A)}>t) - P(T^{(B)}>t) where I(\cdot) is the indicator function. For individuals who are censored or experience the primary outcome before t_0, we assume that their S information is not available.

The surrogate marker information at time t_0 is defined as a combination of the observed information on I(T >t_0) and the observed S at t_0, denoted by Q_{t_0} = \{Q_{t_0}^{(g)}, g = A, B\}, where Q_{t_0} ^{(g)} = \{ T ^{(g)} \wedge t_0, S ^{(g)} I(T ^{(g)} > t_0)\}. With information on Q_{t_0}, the residual treatment effect is defined as: \Delta_{S}(t,t_0) = E\{ I(T ^{(A)} > t) - I(T ^{(B)}>t) \mid Q_{t_0}^{(A)} = Q_{t_0}^{(B)} \} =P(T^{(B)} > t_0) \int_{S} \psi_A(t \mid S, t_0) dF_{B}(S \mid t_0) -P(T^{(B)}> t) where S = (s_1, ..., s_k)^{T}, F_{g}(S \mid t_0) = P(S^{(g)}\le S \mid T^{(g)}>t_0), \psi_g(t\mid S,t_0) = P(T^{(g)}> t\mid S^{(g)}=S, T^{(g)}> t_0). The proportion of the treatment effect on the primary outcome that is explained by the treatment effect on Q_{t_0} is R_{S}(t,t_0)=1-\Delta_{S}(t,t_0)/\Delta. This function provides 6 different estimators for R_{S}(t,t_0) using censored data.

Due to censoring, the observed data consist of n observations \{(G_i, X_{i}, \delta_{i}, S_{i}I(X_{i} > t_0)), i=1,...,n\} from the two treatment groups, where X_{i} = \min(T_{i}, C_{i}), \delta_{i} = I(T_{i} < C_{i}), and C_{i} denotes the censoring time for the ith subject. We assume independent censoring i.e., (T_i, S_i) \perp C_i \mid G_i. For ease of notation, we also let \{(X_{gi}, \delta_{gi}, S_{gi}I(X_{gi} > t_0)), i=1,...,n_g\} denote n_g = \sum_{i=1}^n I(G_i = g) observations from treatment group g \in \{A,B\}, where X_{gi}=\min(T_{gi}, C_{gi}) and \delta_{gi}=I(T_{gi}<C_{gi}). Without loss of generality, we assume that \bar{\pi}_g = n_g/n \to \pi_g \in (0,1) as n\to \infty. Throughout, we estimate the treatment effect \Delta(t) =P(T^{(A)}>t) - P(T^{(B)}>t) as \hat{\Delta}(t) = n_A^{-1} \sum_{i=1}^{n_A} \frac{I(X_{Ai}>t)}{\hat{W}^C_A(t)} - n_B^{-1} \sum_{i=1}^{n_B} \frac{I(X_{Bi}>t)}{\hat{W}^C_B(t)} where \hat{W}^C_g(t) is the Kaplan-Meier estimator of W^C_g(t) = P(C_{g} > t).

We first describe the two-stage robust estimator which involves a two-stage procedure combining the use of a working model and a nonparametric estimation procedure for \Delta_{S}(t,t_0). The idea is simply to summarize S into a univariate score U and then construct a nonparametric estimator for R_S(t,t_0) treating U as S. To construct U, we approximate the conditional distribution of T^{(A)} \mid S^{(A)}, T^{(A)} > t_0 by using a working semiparametric model such as the landmark proportional hazards model q_A(S) = P(T^{(A)} >t \mid T^{(A)} >t_0, S^{(A)}) = \exp\{-\Lambda_0(t|t_0) \exp(\beta ^{T} S^{(A)}) \}, t>t_0, where \Lambda_0(t|t_0) is the unspecified baseline cumulative hazard function for T^{(A)} conditional on \{T^{(A)} > t_0\} and \beta is an unknown vector of coefficients. Let \hat{\beta} be the maximizer of the corresponding log partial likelihood function and \hat{\Lambda}_0(t|t_0) be the Breslow-type estimate of baseline hazard. If one were to assume that this working model is correctly specified, then a consistent estimate of \Delta_{S}(t,t_0) would simply be: \hat{\Delta}_{S}^M=n_B^{-1} \sum_{i=1}^{n_B} [ \exp \{ -\hat{\Lambda}_0(t|t_0)\exp(\hat{\beta} ^{T} S_{Bi}) \} \frac{I(X_{Bi} > t_0)}{\hat{W}^C_B(t_0)} - \frac{I(X_{Bi} > t)}{\hat{W}^C_B(t)}]. We refer to this estimate as the two-stage model-based estimator (option 4 for type). Instead of relying on correct specification of this model, we use the resulting score U = \beta_0^{T}S as a univariate “pseudo-marker" to summarize the k surrogates. In the second stage, to estimate \Delta_{S}(t,t_0), we apply a nonparametric approach with S represented by the univariate marker U. Specifically, we use a kernel Nelson-Aalen estimator to nonparametrically estimate \phi_A(t|u, t_0)=P(T^{(A)}> t\mid U^{(A)}=u, T^{(A)}> t_0) = \exp\{-\Lambda_A(t|u, t_0 )\} as \hat \phi_A(t|u, t_0) = \exp\{-\hat{\Lambda}_A(t|u, t_0) \}, where \hat{\Lambda}_A(t|u, t_0) = \int_{t_0}^t \frac{\sum_{i=1}^{n_A} I(X_{Ai}>t_0) K_h\{\gamma(\hat{U}_{Ai}) - \gamma(u)\}dN_{Ai}(z)}{\sum_{i=1}^{n_A} I(X_{Ai}>t_0) K_h\{\gamma(\hat{U}_{Ai}) - \gamma(u)\} Y_{Ai}(z)}, \hat{U}_{Ai} = \hat{\beta} ^{T} S_{Ai}, \hat{U}_{Bi} = \hat{\beta} ^{T} S_{Bi}, Y_{Ai} = I(X_{Ai} \geq t), N_{Ai}(t) = I(X_{Ai} \leq t) \delta_{Ai}, K(\cdot) is a smooth symmetric density function, K_h(x) = K(x/h)/h, and \gamma(\cdot) is a given monotone transformation function. We then estimate \Delta_{S}(t,t_0) as \hat{\Delta}_{S}(t,t_0) = n_B^{-1} \sum_{i=1}^{n_B} [\hat{\phi}_A(t|\hat{U}_{Bi},t_0) \frac{I(X_{Bi} > t_0)}{\hat{W}^C_B(t_0)} - \frac{I(X_{Bi} > t)}{\hat{W}^C_B(t)}] and \hat{R}_{S}(t,t_0) =1- \hat{\Delta}_{S}(t,t_0)/\hat{\Delta}(t). We refer to this estimate as the two-stage robust estimator (option 1 for type).

The next estimator borrows ideas from the extensive causal inference literature focusing on double robust estimators two-stage weighted estimator with a propensity score weight explicitly balancing the two treatment groups with respect to the distribution of S. The weighting enables us to “adjust" the distribution of S^{(A)} before constructing the conditional survival estimate \hat \phi_A(t|u, t_0). This approach results in a double-robust estimator of \Delta_{S}(t, t_0), which is consistent when either U^{(A)} captures all the information about the relationship between I(T^{(A)}\ge t) and S^{(A)} or the propensity score model for \pi(S,t_0)=P(G_i=B|S_i=S, T_{i} > t_0) is correctly specified. While \pi(S,t_0) depends on t_0, for simplicity, we drop t_0 from our notation and simply use \pi(S).

Regression models can be imposed to obtain estimates for \pi(S). For example, a simple logistic regression model can be imposed for \tilde{\pi}(S)=P(G_i=B|S_i=S, X_{i}>t_0) with logit\{\tilde{\pi}(S)\} = \alpha_0 + \alpha_1 ^{T} S, where \alpha_0 and \alpha_1 are estimated only among those with X_{gi} > t_0 to account for censoring. The propensity score of interest, \pi(S), can be derived from \tilde{\pi}(S) directly since logit\{\pi(S)\}=logit\{\tilde{\pi}(S)\} + \log\{W_A^C(t_0)/W_B^C(t_0)\}, which follows from the assumption that (T_{gi}, S_{gi}) \perp C_{gi}. We then modify the above expression by weighting observations with the estimated L(S_{Ai})=\pi(S_{Ai})/\{1-\pi(S_{Ai})\} and obtain

\hat{\Lambda}^w_A(t|u, t_0) = \int_{t_0}^t \frac{\sum_{i=1}^{n_A} \hat{L}(S_{Ai}) I(X_{Ai}>t_0) K_h\{\gamma(\hat{U}_{Ai}) - \gamma(u)\}dN_{Ai}(z)}{\sum_{i=1}^{n_A} \hat{L}(S_{Ai}) I(X_{Ai}>t_0) K_h\{\gamma(\hat{U}_{Ai}) - \gamma(u)\} Y_{Ai}(z)},

, where \hat{L}(S_{gi}) = \exp(\hat{\alpha}_0+\hat{\alpha}_1^{T} S_{gi})\hat{W}^C_B(t_0)/\hat{W}^C_A(t_0).

Subsequently, we define \hat{\Delta}^w_{S}(t,t_0) = n_B^{-1} \sum_{i=1}^{n_B} [\hat{\phi}^w_A(t|\hat{U}_{Bi},t_0) \frac{I(X_{Bi} > t_0)}{\hat{W}^C_B(t_0)} - \frac{I(X_{Bi} > t)}{\hat{W}^C_B(t)}] and \hat{R}^w_{S}(t,t_0) =1- \hat{\Delta}^w_{S}(t,t_0)/\hat{\Delta}(t) where \hat \phi^w_A(t|t_0,u) = \exp\{-\hat{\Lambda}^w_A(t|t_0,u) \}. We refer to this estimate as the weighted two-stage robust estimator (option 2 for type).

While the two-stage weighted estimator reflects one way to enhance the robustness of an initial estimator, the idea of combining a propensity-score type model and a regression-type model has certainly been extensively studied in the causal inference literature and a more familiar double-robust estimator can be constructed as: \hat{\Delta}_{S}^{DR}(t,t_0) = n^{-1} [\sum_{i=1}^{n_A}\frac{I(X_{Ai}>t)}{\hat{W}_{A}^C(t)\bar{\pi}_B} \hat{L}(S_{Ai}) - \sum_{i=1}^{n_B} \frac{I(X_{Bi}>t)}{\hat{W}_{B}^C(t)\bar{\pi}_B} ] - n^{-1} [ \sum_{i=1}^{n_A} \frac{ \hat{\phi}_A(t\mid\hat{U}_{Ai},t_0) I(X_{Ai} > t_0) }{\hat{W}_{A}^C(t_0)\bar{\pi}_B} \hat{L}(S_{Ai}) - \sum_{i=1}^{n_B}\frac{ \hat{\phi}_A(t\mid \hat{U}_{Bi},t_0) I(X_{Bi} > t_0) }{\hat{W}_{B}^C(t_0)\bar{\pi}_B} ] and \hat{R}_{S}^{DR}(t,t_0) =1- \hat{\Delta}_{S}^{DR}(t,t_0)/\hat{\Delta}(t), where \hat{\phi}_A(t|\hat{U}_{gi},t_0) is the (unweighted) estimate of \phi_A(t\mid u,t_0) used in \hat{\Delta}_{S}(t,t_0). We refer to this estimate as the double robust estimator (option 3 for type).

The weighted estimator (option type 5) is defined as: \hat{\Delta}^{PS}_{S}(t,t_0) = n^{-1} \sum_{i=1}^n \{\frac{I(X_{i}>t)}{\hat{W}_{G_i}^C(t)\bar{\pi}_B} [ I(G_i = A)\hat{L}(S_{Ai}) - I(G_i = B ) ] \} and \hat{R}^{PS}_{S}(t,t_0) =1- \hat{\Delta}^{PS}_{S}(t,t_0)/\hat{\Delta}(t). This estimator completely relies on the correct specification of \pi(S). The double-robust model-based estimator (option 6 for type) is defined as \hat{\Delta}_{S}^{DR2}(t,t_0) and \hat{R}_{S}^{DR2}(t,t_0) =1- \hat{\Delta}_{S}^{DR2}(t,t_0)/\hat{\Delta}(t) which are constructed parallel to the construction of \hat{\Delta}_{S}^{DR}(t,t_0) i.e., a combination of \hat{\Delta}_{S}^M(t,t_0) and \hat{R}^{PS}_{S}(t,t_0).

Variance estimates are obtained using perturbation resampling. If a confidence interval is requested three versions are provided: a normal approximation based interval, a quantile based interval and Fieller's confidence interval, all using perturbation-resampling. An estimate of the incremental value of the surrogate marker information can also be requested; this essentially compared the proportion explained by the surrogate information vs. the proportion explained by T alone up to t_0. Details can be found in Parast, L., Cai, T., & Tian, L. (2021). Evaluating multiple surrogate markers with censored data. Biometrics, 77(4), 1315-1327.

Value

A list is returned:

Note

If the treatment effect is not significant, the user will receive the following message: "Warning: it looks like the treatment effect is not significant; may be difficult to interpret the residual treatment effect in this setting".

Author(s)

Layla Parast

References

Parast, L., Cai, T., & Tian, L. (2017). Evaluating surrogate marker information using censored data. Statistics in Medicine, 36(11), 1767-1782.

Parast, L., Cai, T., & Tian, L. (2021). Evaluating multiple surrogate markers with censored data. Biometrics, 77(4), 1315-1327.

Examples

data(d_example_multiple)
names(d_example_multiple)
## Not run: 
R.multiple.surv(xone = d_example_multiple$x1, xzero = d_example_multiple$x0, deltaone = 
d_example_multiple$delta1, deltazero = d_example_multiple$delta0, sone = 
as.matrix(d_example_multiple$s1), szero = as.matrix(d_example_multiple$s0), 
type =1, t = 1, landmark=0.5)
R.multiple.surv(xone = d_example_multiple$x1, xzero = d_example_multiple$x0, deltaone = 
d_example_multiple$delta1, deltazero = d_example_multiple$delta0, sone = 
as.matrix(d_example_multiple$s1), szero = as.matrix(d_example_multiple$s0), 
type =1, t = 1, landmark=0.5, conf.int = T)	
R.multiple.surv(xone = d_example_multiple$x1, xzero = d_example_multiple$x0, deltaone = 
d_example_multiple$delta1, deltazero = d_example_multiple$delta0, sone = 
as.matrix(d_example_multiple$s1), szero = as.matrix(d_example_multiple$s0), 
type =3, t = 1, landmark=0.5)

## End(Not run)

Calculates the proportion of treatment effect explained

Description

This function calculates the proportion of treatment effect on the primary outcome explained by the treatment effect on the surrogate marker(s). This function is intended to be used for a fully observed continuous outcome. The user can also request a variance estimate and a 95% confidence interval, both estimated using perturbating-resampling. If a confidence interval is requested three versions are provided: a normal approximation based interval, a quantile based interval, and Fieller's confidence interval.

Usage

R.s.estimate(sone, szero, yone, yzero, var = FALSE, conf.int = FALSE, 
weight.perturb = NULL, number = "single", type = "robust",extrapolate = FALSE, 
transform = FALSE,warn.te = FALSE, warn.support = FALSE)

Arguments

Details

Let Y^{(1)} and Y^{(0)} denote the primary outcome under the treatment and primary outcome under the control,respectively. Let S^{(1)} and S^{(0)} denote the surrogate marker under the treatment and the surrogate marker under the control,respectively. The residual treatment effect is defined as

\Delta_S=\int_{-\infty}^{\infty} E(Y^{(1)}|S^{(1)}=s) dF_0(s) - \int_{-\infty}^{\infty} E(Y^{(0)}|S^{(0)}=s) dF_0(s),

where \Delta_S(s)= E(Y^{(1)}|S^{(1)}=s)-E(Y^{(0)}|S^{(0)}=s) and F_0(\cdot) is the marginal cumulative distribution function of S^{(0)}, the surrogate marker measure under the control. The proportion of treatment effect explained by the surrogate marker, which we denote by R_S, can be expressed using a contrast between \Delta_S and \Delta:

R_S=\{\Delta-\Delta_S\}/\Delta=1-\Delta_S/\Delta.

The definition and estimation of \Delta is described in the delta.estimate documentation.

A flexible model-based approach to estimate \Delta_S in the single marker setting is to specify:

E(S^{(0)})=\alpha_0 \quad\mbox{and}\quad E(S^{(1)})-E(S^{(0)}) = \alpha_1,

E(Y^{(0)} | S^{(0)}) = \beta_0 + \beta_1 S^{(0)} \quad \mbox{and} \quad E(Y^{(1)} | S^{(1)}) = (\beta_0 +\beta_2)+ (\beta_1+\beta_3) S^{(1)}.

It can be shown that when these models hold, \Delta_S = \beta_2 + \beta_3 \alpha_0. Thus, reasonable estimates for \Delta_S and R_S using this approach would be \hat{\Delta}_S = \hat{\beta}_2 + \hat{\beta}_3 \hat{\alpha}_0 and \hat{R}_S = 1-\hat{\Delta}_S / \hat{\Delta}.

For robust estimation of \Delta_S in the single marker setting, we estimate \mu_1(s) = E(Y^{(1)}|S^{(1)}=s) nonparametrically using kernel smoothing:

\hat{\mu}_1(s) = \frac{\sum_{i=1}^{n_1} K_h\left (S_{1i}-s \right ) Y_{1i} }{\sum_{i=1}^{n_1} K_h\left (S_{1i}-s \right )}

where S_{1i} is the observed S^{(1)} for person i, Y_{1i} is the observed Y^{(1)} for person i, K(\cdot) is a smooth symmetric density function with finite support, K_h(\cdot)=K(\cdot/h)/h and h is a specified bandwidth. As in most nonparametric functional estimation procedures, the choice of the smoothing parameter h is critical. To eliminate the impact of the bias of the conditional mean function on the resulting estimator, we require the standard undersmoothing assumption of h=O(n_1^{-\delta}) with \delta \in (1/4,1/3). To obtain an appropriate h we first use bw.nrd to obtain h_{opt}; and then we let h = h_{opt}n_1^{-c_0} with c_0 = 0.25. We then estimate \Delta_S as

\hat{\Delta}_S= \sum_{i=1}^{n_0} \frac{\hat{\mu}_1(S_{0i})- Y_{0i}}{n_0}

where S_{0i} is the observed S^{(0)} for person i and Y_{0i} is the observed Y^{(0)} for person i. Lastly, we estimate R_S as \hat{R}_S = 1-\hat{\Delta}_S/\hat{\Delta}.

This function also allows for estimation of R_S using Freedman's approach. Let Y denote the primary outcome, S denote the surrogate marker, and G denote the treatment group (0 for control, 1 for treatment). Freedman's approach to calculating the proportion of treatment effect explained by the surrogate marker is to fit the following two regression models:

E(Y|G) = \gamma_0 + \gamma_1 I(G=1) \quad \mbox{and} \quad E(Y|G, S) = \gamma_{0S} + \gamma_{1S}I(G=1) + \gamma_{2S} S

and estimating the proportion of treatment effect explained, denoted by R_S, as 1-\hat{\gamma}_{1S}/\hat{\gamma}_1.

This function also estimates R_S in a multiple marking setting. A flexible model-based approach to estimate \Delta_S in the multiple marker setting is to specify models for E(Y|G, S) and E(S_j | G) for each S_j in S = \{S_1,...S_p\} (where p is the number of surrogate markers). Without loss of generality, consider the case where there are three surrogate markers, S = \{S_1, S_2, S_3\} and one specifies the following linear models:

E(Y^{(0)} | S^{(0)}) = \beta_0 + \beta_1 S_1^{(0)} + \beta_2 S_2^{(0)} + \beta_3 S_3^{(0)}

E(Y^{(1)} | S^{(1)}) = (\beta_0+\beta_4) + (\beta_1+\beta_5) S_1^{(1)} + (\beta_2+\beta_6) S_2^{(1)} + (\beta_3+\beta_7) S_3^{(1)}

E(S_j^{(0)}) = \alpha_j, ~~~~j=1,2,3.

It can be shown that when these models hold

\Delta_{S} = \beta_4 + \beta_5\alpha_1 + \beta_6 \alpha_2 + \beta_7 \alpha_3.

Thus, reasonable estimates for \Delta_{S} and R_{S} here would be easily obtained by replacing the unknown regression coefficients in the models above by their consistent estimators.

For robust estimation of S \Delta_S in the multiple marker setting, we use a two-stage procedure combining the model-based approach and the nonparametric estimation procedure from the single marker setting. Specifically, we use a working semiparametric model:

E(Y^{(1)}|S^{(1)}=S)=\beta_0 + \beta_1 S_1^{(1)} + \beta_2 S_2^{(1)} + \beta_3 S_3^{(1)}

and define Q^{(1)} = \hat{\beta}_0 + \hat{\beta}_1 S_1^{(1)} + \hat{\beta}_2 S_2^{(1)} + \hat{\beta}_3 S_3^{(1)} and Q^{(0)} = \hat{\beta}_0 + \hat{\beta}_1 S_1^{(0)} + \hat{\beta}_2 S_2^{(0)} + \hat{\beta}_3 S_3^{(0)} to reduce the dimension of S in the first stage and in the second stage, we apply the robust approach used in the single marker setting to estimate its surrogacy.

To use Freedman's approach in the presence of multiple markers, the markers are simply additively entered into the second regression model.

Variance estimation and confidence interval construction are performed using perturbation-resampling. Specifically, let \left \{ V^{(b)} = (V_{11}^{(b)}, ...V_{1n_1}^{(b)}, V_{01}^{(b)}, ...V_{0n_0}^{(b)})^T, b=1,....,D \right \} be n \times D independent copies of a positive random variables V from a known distribution with unit mean and unit variance. Let

\hat{\Delta}^{(b)} = \frac{ \sum_{i=1}^{n_1} V_{1i}^{(b)} Y_{1i}}{ \sum_{i=1}^{n_1} V_{1i}^{(b)}} - \frac{ \sum_{i=1}^{n_0} V_{0i}^{(b)} Y_{0i}}{ \sum_{i=1}^{n_0} V_{0i}^{(b)}}.

The variance of \hat{\Delta} is obtained as the empirical variance of \{\hat{\Delta}^{(b)}, b = 1,...,D\}. In this package, we use weights generated from an Exponential(1) distribution and use D=500. Variance estimates for \hat{\Delta}_S and \hat{R}_S are calculated similarly. We construct two versions of the 95\% confidence interval for each estimate: one based on a normal approximation confidence interval using the estimated variance and another taking the 2.5th and 97.5th empirical percentile of the perturbed quantities. In addition, we use Fieller's method to obtain a third confidence interval for R_S as

\left\{1-r: \frac{(\hat{\Delta}_S-r\hat{\Delta})^2}{\hat{\sigma}_{11}-2r\hat\sigma_{12}+r^2\hat\sigma_{22}} \le c_{\alpha}\right\},

where \hat{\Sigma}=(\hat\sigma_{ij})_{1\le i,j\le 2} and c_\alpha is the (1-\alpha)th percentile of

\left\{\frac{\{\hat{\Delta}^{(b)}_S-(1-\hat R_S)\hat{\Delta}^{(b)}\}^2}{\hat{\sigma}_{11}-2(1-\hat R_S)\hat\sigma_{12}+(1-\hat R_S)^2\hat\sigma_{22}}, b=1, \cdots, C\right\}

where \alpha=0.05.

Note that if the observed supports for S are not the same, then \hat{\mu}_1(s) for S_{0i} = s outside the support of S_{1i} may return NA (depending on the bandwidth). If extrapolation = TRUE, then the \hat{\mu}_1(s) values for these surrogate values are set to the closest non-NA value. If transform = TRUE, then S_{1i} and S_{0i} are transformed such that the new transformed values, S^{tr}_{1i} and S^{tr}_{0i} are defined as: S^{tr}_{gi} = F([S_{gi} - \mu]/\sigma) for g=0,1 where F(\cdot) is the cumulative distribution function for a standard normal random variable, and \mu and \sigma are the sample mean and standard deviation, respectively, of (S_{1i}, S_{0i})^T.

Value

A list is returned:

For all options other then "freedman", the following are also returned:

Note

If the treatment effect is not significant, the user will receive the following message: "Warning: it looks like the treatment effect is not significant; may be difficult to interpret the proportion of treatment effect explained in this setting". In the single marker case with the robust estimation approach, if the observed support of the surrogate marker for the control group is outside the observed support of the surrogate marker for the treatment group, the user will receive the following message: "Warning: observed supports do not appear equal, may need to consider a transformation or extrapolation"

Author(s)

Layla Parast

References

Freedman, L. S., Graubard, B. I., & Schatzkin, A. (1992). Statistical validation of intermediate endpoints for chronic diseases. Statistics in medicine, 11(2), 167-178.

Parast, L., McDermott, M., Tian, L. (2016). Robust estimation of the proportion of treatment effect explained by surrogate marker information. Statistics in Medicine, 35(10):1637-1653.

Wang, Y., & Taylor, J. M. (2002). A measure of the proportion of treatment effect explained by a surrogate marker. Biometrics, 58(4), 803-812.

Fieller, Edgar C. (1954). Some problems in interval estimation. Journal of the Royal Statistical Society. Series B (Methodological), 175-185.

Fieller, E. C. (1940). The biological standardization of insulin. Supplement to the Journal of the Royal Statistical Society, 1-64.

Examples

data(d_example)
names(d_example)
R.s.estimate(yone=d_example$y1, yzero=d_example$y0, sone=d_example$s1.a, szero=d_example$s0.a, 
number = "single", type = "robust")
R.s.estimate(yone=d_example$y1, yzero=d_example$y0, sone=cbind(d_example$s1.a,d_example$s1.b, 
d_example$s1.c), szero=cbind(d_example$s0.a, d_example$s0.b, d_example$s0.c), 
number = "multiple", type = "model")

Calculates the proportion of treatment effect explained correcting for measurement error in the surrogate marker

Description

This function calculates the proportion of treatment effect on the primary outcome explained by the treatment effect on a surrogate marker, correcting for measurement error in the surrogate marker. This function is intended to be used for a fully observed continuous outcome. The user must specify what type of estimation they would like (parametric or nonparametric estimation of the proportion explained, denoted by R) and what estimator they would like (see below for details).

Usage

R.s.estimate.me(sone, szero, yone, yzero, parametric = FALSE, estimator = "n", 
me.variance, extrapolate = TRUE, transform = FALSE, naive = FALSE, Ronly = TRUE)

Arguments

Details

While there are many methods available to quantify the value of a surrogate marker, most assume that the marker is measured without error. This function calculates the proportion of treatment effect on the primary outcome explained by the treatment effect on a surrogate marker, correcting for measurement error in the surrogate marker. The user can choose either the parametric framework or nonparametric framework for estmation. Within the parametric framework there are three options for measurement error correction: the disattenuated estimator, the SIMEX estimator with quadratic extrapolation, and the SIMEX estimator with nonlinear extrapolation. Within the nonparametric framework there are two options for measurement error correction: the SIMEX estimator with quadratic extrapolation and the SIMEX estimator with nonlinear extrapolation. We describe each below.

Let G be the binary treatment indicator with G=1 indicating treatment and G=0 indicating control (or placebo). We assume throughout that subjects are randomly assigned to treatment or control at baseline. Let Y and S denote the continuous primary outcome and continuous surrogate marker, respectively, where S is measured post-baseline and is assumed to be a biomarker, clinical measurement, psychological test score, or other physiological measurement. In the absence of measurement error, the observed data consists of \{Y_i, S_i, G_i\} for i \in \{1,...,n\}. With measurement error, instead of observing S we observe W = S + U, where E(U|S) = 0 and the variance of U is \sigma_u^2. Such measurement error may be attributable to, for example, laboratory error. Thus, our observed data will consist of \{Y_i, W_i, G_i\} for i \in \{1,...,n\}. Throughout, we assume that \sigma_u^2 is known. Here, we are interested in estimating the proprtion of the treatment effect on the primary outcome that is explained by the treatment effect on the surrogate marker, denoted as R_S.

To estimate R_S parametrically, we assume the following models E(Y|G) = \beta_0 + \beta_1 G and E(Y|G,S) = \beta_0^* + \beta_1^*G + \beta_2^* S. It can be shown that if these models hold, R_S=1-\beta_1^*/\beta_1. When W = S+U is available instead of S, this measurement error does not affect estimation of \beta_1, but it does affect estimation of \beta_1^*, and \beta_2^*. Since estimation of R_S relies on estimation of \beta_1 and \beta_1^*, we focus on the effect of measurement error on \beta_1^* estimation. The attenuation bias for \hat \beta_1^* and \hat R can be written out in closed form when the proportion of treatment effect is parametrically estimated as described above, when these specified models hold, and when the surrogate marker S is measured with error. There exist two methods to eliminate this bias when estimating R_S. Taking advantage of the fact that we can express the attenuation bias in closed form, the first is a straightforward disattenuated estimator: \hat \beta _{1A} = \hat{\beta}_1^* - \frac{ \hat{\beta}_2^* \{\Omega^2_{W} \Omega_{GW}-\Omega_{GW}(\Omega^2_{W} - \sigma_u^2)\}}{\Omega^2_{G}(\Omega^2_{W} - \sigma_u^2)-\Omega_{GW}\Omega_{GW}} and \hat{R}_{A} = 1- \left [ \hat{\beta}_1^* - \frac{ \hat{\beta}_2^* \{\Omega^2_{W} \Omega_{GW}-\Omega_{GW}(\Omega^2_{W} - \sigma_u^2)\}}{\Omega^2_{G}(\Omega^2_{W} - \sigma_u^2)-\Omega_{GW}\Omega_{GW}} \right] / \hat{\beta}_1 where \Omega^2 denotes the sample variance or covariance.

The second method to eliminate this bias uses Simulation Extrapolation (SIMEX) estimation, which is a simulation-based method that involves first generating additional measurement error and observing how it affects the bias of the parameter estimate of interest, and then extrapolating this information to a setting with no measurement error. To incorporate SIMEX estimation within our surrogate marker framework, we define W_{b,i}(\lambda) = W_i + \lambda^{1/2} \sigma_u \epsilon_{i,b} for b=1,...,B where B=50, \epsilon_{i,b} \sim N(0,1), \sigma_u is assumed known, and \lambda \in (0,0.25,0.5,0.75,1.0, 1.25,1.5,1.75,2.0) and for each iteration b and \lambda value, obtaining \hat \beta_{1b}^*(\lambda) by fitting the regression model: E(Y \mid W_b(\lambda),S) = \beta_{0b}^* + \beta_{1b}^* W_{b}(\lambda) + \beta_{2b}^* S. We then calculate the average estimate for each quantity over the iterations b=1,...,B for each \lambda value, denoted as \hat \beta^*_{1,S,\sigma^2_u(1+\lambda)} = \sum_{b=1}^B \hat \beta_{1b}^*(\lambda). The second step, extrapolation, takes these average estimates for each \lambda value and extrapolates using a function G(\Gamma, \lambda) to obtain the estimated quantity if \lambda=-1. For the extrapolation step, we use both a quadratic extrapolation and nonlinear extrapolation i.e., we solve for \Gamma = (\alpha_0, \alpha_1, \alpha_2)^T in \hat \beta^*_{1,S,\sigma^2_u(1+\lambda)} = \alpha_0 + \alpha_1 \lambda + \alpha_2 \lambda^2 and \hat \beta^*_{1,S,\sigma^2_u(1+\lambda)}= \alpha_0 + \alpha_1 /( \alpha_2 + \lambda), respectively. Using the estimates of \alpha_0, \alpha_1, \alpha_2, we calculate the predicted \hat \beta^*_{1,S,\sigma^2_u(1+\lambda)} when \lambda = -1. In essence, the simulations add successively larger measurement errors of size (1+\lambda)\sigma^2_u and then extrapolate to the case when \lambda = -1 such that the measurement error is 0. We denote the resulting estimator of \beta_1^* as \hat{\beta}^*_{1,SIMEX} = G(\hat \Gamma, -1) and define \hat{R}_{SIMEX} = 1- \hat{\beta}^*_{1,SIMEX}/ \hat \beta_1.

While the parametric approach to estimate the proportion of treatment effect explained by S is most commonly used in clinical practice, previous work has demonstrated biased results when the assumed models are not correctly specified. An alternative approach involves estimating the treatment effect, \Delta, and residual treatment effect, \Delta_S, as R_S is defined as 1-\Delta/\Delta_S. The quantity \Delta can be estimated simply by \hat{\Delta} = n_1^{-1}\sum_{i=1}^{n} Y_i I(G_i = 1) - n_0^{-1}\sum_{i=1}^{n} Y_i I(G_i = 0), where n_1 and n_0 denote the number of individuals in the treatment and control groups, respectively. The quantity \Delta_S can be estimated nonparametrically using kernel smoothing as \hat{\Delta}_S = n_0^{-1} \sum_{i: G_i = 0}\hat{\mu}_1(S_i) - n_0^{-1}\sum_{i=1}^{n} Y_i I(G_i = 0) where \hat{\mu}_1(s) = \{ \sum_{j: G_j = 1} K_h(S_j - s)Y_j \}/ \{\sum_{j:G_j = 1} K_h(S_j - s)\}, K(\cdot) is a smooth symmetric density function with finite support, K_h(\cdot)=K(\cdot/h)/h and h is a specified bandwidth such that h=O(n_1^{-\nu}) with \nu \in (1/4,1/2).

When W = S + U is available instead of S, estimation of \Delta is not affected whereas estimation of \Delta_S is affected and thus, the nonparametric estimation procedure described above results in a biased estimate of R_S. Unlike the parametric approach, the attenuation bias cannot be expressed in closed form. Within this nonparametric framework, SIMEX estimation can be used to correct for measurement error. We implement the estimation procedure as described above where we first generate additional measurement error to obtain W_{b,i}(\lambda) and for each iteration b and \lambda values obtain \hat{\Delta}_{S,b}(\lambda) = n_0^{-1} \sum_{i: G_i = 0} \left \{ \frac{\sum_{j: G_j = 1} K_h(W_{b,j}(\lambda) - W_{b,i}(\lambda))Y_j}{\sum_{j:G_j = 1} K_h(W_{b,j}(\lambda)- W_{b,i}(\lambda))} \right \} - n_0^{-1}\sum_{i=1}^{n} Y_i I(G_i = 0). We then calculate the average estimate for each quantity over the iterations b=1,...,B for each \lambda value, denoted as \hat{\Delta}_{S,\sigma_u^2(1+\lambda)} = \sum_{b=1}^B \hat{\Delta}_{S,b}(\lambda) and extrapolate using a function G(\Gamma, \lambda); we specifically use the quadratic and nonlinear functions as in the parametric setting. We denote the resulting estimator of \Delta_S as \hat{\Delta}_{S,SIMEX} = G(\hat \Gamma, -1) and define \hat{R}_{S,SIMEX} = 1- \hat{\Delta}_{S,SIMEX} / \hat \Delta.

In this function, parametric estimation is equivalent to Freedman's approach in the R.s.estimate documentation; nonparametric estimation is equivalent to the robust approach in the R.s.estimate documentation. Variance estimates for all estimators are calculated in this function based on derived closed form variance expressions. For all approaches, confidence intervals for \Delta_S can be constructed using a normal approximation; confidence intervals for R_S can be constructed using either a normal approximation or using Fieller's method, all of which are provided in this function. Details regarding the asymptotic properties of these estimators and closed form variance calculation can be found in: Parast, L., Garcia, T. P., Prentice, R. L., & Carroll, R. J. (2022). Robust methods to correct for measurement error when evaluating a surrogate marker. Biometrics, 78(1), 9-23.

Value

A list is returned:

Author(s)

Layla Parast

References

Parast, L., Garcia, T. P., Prentice, R. L., & Carroll, R. J. (2022). Robust methods to correct for measurement error when evaluating a surrogate marker. Biometrics, 78(1), 9-23.

Examples

data(d_example_me)
names(d_example_me)
R.s.estimate.me(yone=d_example_me$y1, yzero=d_example_me$y0, sone=d_example_me$s1, 
szero=d_example_me$s0, parametric = TRUE, estimator = "d", me.variance = 0.5, 
naive= TRUE, Ronly = FALSE)
R.s.estimate.me(yone=d_example_me$y1, yzero=d_example_me$y0, sone=d_example_me$s1, 
szero=d_example_me$s0, parametric = TRUE, estimator = "q", me.variance = 0.5, 
naive= FALSE, Ronly = TRUE)

#estimating measurement error variance with replicates
replicates = rbind(cbind(d_example_me$s1_rep1, d_example_me$s1_rep2, 
d_example_me$s1_rep3), cbind(d_example_me$s0_rep1, d_example_me$s0_rep2, 
d_example_me$s0_rep3))
mean.i = apply(replicates,1,mean, na.rm = TRUE)
num.i = apply(replicates,1,function(x) sum(!is.na(x)))
var.u = sum((replicates-mean.i)^2, na.rm = TRUE)/sum(num.i)
var.u
R.s.estimate.me(yone=d_example_me$y1, yzero=d_example_me$y0, sone=d_example_me$s1, 
szero=d_example_me$s0, parametric = TRUE, estimator = "d", me.variance = var.u, 
naive= TRUE, Ronly = FALSE)

R.s.estimate.me(yone=d_example_me$y1, yzero=d_example_me$y0, 
sone=d_example_me$s1, szero=d_example_me$s0, parametric = FALSE, estimator = "q", 
me.variance = 0.5, naive= FALSE, Ronly = TRUE)

Calculates the proportion of treatment effect explained by the surrogate marker information measured at a specified time and primary outcome information up to that specified time

Description

This function calculates the proportion of treatment effect on the primary outcome explained by the surrogate marker information measured at t_0 and primary outcome information up to t_0. The user can also request a variance estimate, estimated using perturbating-resampling, and a 95% confidence interval. If a confidence interval is requested three versions are provided: a normal approximation based interval, a quantile based interval and Fieller's confidence interval, all using perturbation-resampling. The user can also request an estimate of the incremental value of surrogate marker information.

Usage

R.s.surv.estimate(xone, xzero, deltaone, deltazero, sone, szero, t, 
weight.perturb = NULL, landmark, extrapolate = FALSE, transform = FALSE, 
conf.int = FALSE, var = FALSE, incremental.value = FALSE, approx = T)

Arguments

Details

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^{(1)} and T^{(0)} denote the time of the primary outcome of interest, death for example, under the treatment and under the control, respectively. Let S^{(1)} and S^{(0)} denote the surrogate marker measured at time t_0 under the treatment and the control, respectively.

The residual treatment effect is defined as

\Delta_S(t,t_0) = P(T^{(0)} > t_0) \left \{\int \psi_1(t | s, t_0) dF_0(s | t_0)-P(T^{(0)}> t| T^{(0)}> t_0) \right \}

where F_0(\cdot | t_0) is the cumulative distribution function of S^{(0)} conditional on T^{(0)}> t_0 and \psi_1(t | s,t_0) = P(T^{(1)}> t | S^{(1)}=s, T ^{(1)} > t_0). The proportion of treatment effect explained by the surrogate marker information measured at t_0 and primary outcome information up to t_0, which we denote by R_S(t,t_0), can be expressed using a contrast between \Delta_S(t,t_0) and \Delta(t):

R_S(t,t_0)=\{\Delta(t)-\Delta_S(t,t_0)\}/\Delta(t)=1-\Delta_S(t,t_0)/\Delta(t).

The definition and estimation of \Delta(t) is described in the delta.surv.estimate documentation.

Due to censoring, our data consist of n_1 observations \{(X_{1i}, \delta_{1i}, S_{1i}), i=1,...,n_1\} from the treatment group G=1 and n_0 observations \{(X_{0i}, \delta_{0i}, S_{0i}), i=1,...,n_0\} from the control group G=0 where X_{gi} = \min(T_{gi}, C_{ gi}), \delta_{gi} = I(T_{gi} < C_{gi}), C_{gi} denotes the censoring time, and S_{gi} denotes the surrogate marker information measured at time t_0, for g= 1,0, for individual i. Note that if X_{gi} < t_0, then S_{gi} should be NA (not available).

To estimate \Delta_S(t,t_0), we use a nonparametric kernel Nelson-Aalen estimator to estimate \psi_1(t | s,t_0) as \hat{\psi}_1(t| s,t_0) = \exp\{-\hat{\Lambda}_1(t| s,t_0) \}, where

\hat{\Lambda}_1(t| s,t_0) = \int_{t_0}^t \frac{\sum_{i=1}^{n_1} I(X_{1i}>t_0) K_h\{\gamma(S_{1i}) - \gamma(s)\}dN_{1i}(z)}{\sum_{i=1}^{n_1} I(X_{1i}>t_0) K_h\{\gamma(S_{1i}) - \gamma(s)\} Y_{1i}(z)},

is a consistent estimate of \Lambda_1(t| s,t_0 ) = -\log [\psi_1(t| s,t_0)], Y_{1i}(t) = I(X_{1i} \geq t), N_{1i}(t) = I(X_{1i} \leq t) \delta_i, K(\cdot) is a smooth symmetric density function, K_h(x) = K(x/h)/h, \gamma(\cdot) is a given monotone transformation function, and h is a specified bandwidth. To obtain an appropriate h we first use bw.nrd to obtain h_{opt}; and then we let h = h_{opt}n_1^{-c_0} with c_0 = 0.11.

Since F_0(s | t_0) = P(S_{0i} \le s | X_{0i} > t_0), we empirically estimate F_0(s | t_0) using all subjects with X_{0i} > t_0 as

\hat{F}_0(s | t_0) = \frac{\sum_{i=1}^{n_0} I(S_{0i} \le s, X_{0i} > t_0)}{\sum_{i=1}^{n_0} I(X_{0i} > t_0)}.

Subsequently, we construct an estimator for \Delta_{S}(t,t_0) as

\hat{\Delta}_S(t,t_0) = n_0^{-1} \sum_{i=1}^{n_0} \left[\hat{\psi}_1(t| S_{0i},t_0) \frac{I(X_{0i} > t_0)}{\hat{W}^C_0(t_0)} - \frac{I(X_{0i} > t)}{\hat{W}^C_0(t)}\right]

where \hat{W}^C_g(\cdot) is the Kaplan-Meier estimator of survival for censoring for g=1,0. Finally, we estimate R_S(t,t_0) as \hat{R}_S(t,t_0) =1- \hat{\Delta}_S(t,t_0)/\hat{\Delta}(t).

Variance estimation and confidence interval construction are performed using perturbation-resampling. Specifically, let \left \{ V^{(b)} = (V_{11}^{(b)}, ...V_{1n_1}^{(b)}, V_{01}^{(b)}, ...V_{0n_0}^{(b)})^T, b=1,....,D \right \} be n \times D independent copies of a positive random variables V from a known distribution with unit mean and unit variance. Let

\hat{\Delta}^{(b)}(t) = \frac{ \sum_{i=1}^{n_1} V_{1i}^{(b)} I(X_{1i}>t)}{ \sum_{i=1}^{n_1} V_{1i}^{(b)} \hat{W}_1^{C(b)}(t)} -\frac{ \sum_{i=1}^{n_0} V_{0i}^{(b)} I(X_{0i}>t)}{ \sum_{i=1}^{n_0} V_{0i}^{(b)} \hat{W}_0^{C(b)}(t)}.

In this package, we use weights generated from an Exponential(1) distribution and use D=500. The variance of \hat{\Delta}(t) is obtained as the empirical variance of \{\hat{\Delta}(t)^{(b)}, b = 1,...,D\}. Variance estimates for \hat{\Delta}_S(t,t_0) and \hat{R}_S(t,t_0) are calculated similarly. We construct two versions of the 95\% confidence interval for each estimate: one based on a normal approximation confidence interval using the estimated variance and another taking the 2.5th and 97.5th empirical percentile of the perturbed quantities. In addition, we use Fieller's method to obtain a third confidence interval for R_S(t,t_0) as

\left\{1-r: \frac{(\hat{\Delta}_S(t,t_0)-r\hat{\Delta}(t))^2}{\hat{\sigma}_{11}-2r\hat\sigma_{12}+r^2\hat\sigma_{22}} \le c_{\alpha}\right\},

where \hat{\Sigma}=(\hat\sigma_{ij})_{1\le i,j\le 2} and c_\alpha is the (1-\alpha)th percentile of

\left\{\frac{\{\hat{\Delta}^{(b)}_S(t)-(1-\hat R_S(t,t_0))\hat{\Delta}(t)^{(b)}\}^2}{\hat{\sigma}_{11}-2(1-\hat R_S(t,t_0))\hat\sigma_{12}+(1-\hat R_S(t,t_0))^2\hat\sigma_{22}}, b=1, \cdots, C\right\}

where \alpha=0.05.

Since the definition of R_S(t,t_0) considers the surrogate information as a combination of both S information and T information up to t_0, a logical inquiry would be how to assess the incremental value of the S information in terms of the proportion of treatment effect explained, when added to T information up to t_0. The proportion of treatment effect explained by T information up to t_0 only is denoted as R_T(t,t_0) and is described in the documentation for R.t.surv.estimate. The incremental value of S information is defined as:

IV_S(t,t_0) = R_S(t,t_0) - R_T(t,t_0) = \frac{\Delta_T(t,t_0) - \Delta_S(t,t_0)}{\Delta (t)}.

For estimation of R_T(t_0), see documentation for R.t.surv.estimate. The quantity IV_S(t,t_0) is then estimated by \hat{IV}_S(t,t_0) = \hat{R}_S(t,t_0) - \hat{R}_T(t,t_0). Perturbation-resampling is used for variance estimation and confidence interval construction for this quantity, similar to the other quantities in this package.

Note that if the observed supports for S are not the same, then \hat{\Lambda}_1(t| s,t_0) for S_{0i} = s outside the support of S_{1i} may return NA (depending on the bandwidth). If extrapolation = TRUE, then the \hat{\Lambda}_1(t| s,t_0) values for these surrogate values are set to the closest non-NA value. If transform = TRUE, then S_{1i} and S_{0i} are transformed such that the new transformed values, S^{tr}_{1i} and S^{tr}_{0i} are defined as: S^{tr}_{gi} = F([S_{gi} - \mu]/\sigma) for g=0,1 where F(\cdot) is the cumulative distribution function for a standard normal random variable, and \mu and \sigma are the sample mean and standard deviation, respectively, of \{(S_{1i}, S_{0i})^T, i \quad s.t. X_{gi} > t_0\}.

Value

A list is returned:

Note

If the treatment effect is not significant, the user will receive the following message: "Warning: it looks like the treatment effect is not significant; may be difficult to interpret the residual treatment effect in this setting". If the observed support of the surrogate marker for the control group is outside the observed support of the surrogate marker for the treatment group, the user will receive the following message: "Warning: observed supports do not appear equal, may need to consider a transformation or extrapolation".

Author(s)

Layla Parast

References

Parast, L., Cai, T., & Tian, L. (2017). Evaluating surrogate marker information using censored data. Statistics in Medicine, 36(11), 1767-1782.

Examples

data(d_example_surv)
names(d_example_surv)

Calculates the proportion of treatment effect explained by the primary outcome information up to a specified time

Description

This function calculates the proportion of treatment effect on the primary outcome explained by the treatment effect on the primary outcome up to t_0. The user can also request a variance estimate, estimated using perturbating-resampling, and a 95% confidence interval. If a confidence interval is requested three versions are provided: a normal approximation based interval, a quantile based interval and Fieller's confidence interval, all using perturbation-resampling.

Usage

R.t.surv.estimate(xone, xzero, deltaone, deltazero, t, weight.perturb = NULL, 
landmark, var = FALSE, conf.int = FALSE, approx = T)

Arguments

Details

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 denote the time of the primary outcome of interest, death for example. We use potential outcomes notation such that T^{(g)} denotes the time of the primary outcome under treatment G = g. The proportion of treatment effect explained by T observed up to t_0 only is R_T(t,t_0) = 1-\Delta_T(t,t_0)/\Delta(t) where

\Delta_T(t, t_0) = P(T^{(0)}>t_0)P(T^{(1)}>t\mid T^{(1)}>t_0)-P(T^{(0)}>t).

To estimate R_T(t,t_0), we use the estimator \hat{R}_T(t,t_0) = 1-\hat{\Delta}_T(t,t_0)/\hat{\Delta}(t) where \hat{\Delta}_T(t,t_0) = \hat{\phi}_0(t_0)\hat{\phi}_1(t)/\hat{\phi}_1(t_0) - \hat{\phi}_0(t) and \hat{\phi}_g(u) = n_g^{-1} \sum_{i=1}^{n_g} \frac{I(X_{gi}>u)}{\hat{W}^C_g(u)} for g=1,0 where \widehat{W}^C_g(\cdot) is the Kaplan-Meier estimator of survival for censoring for g=1,0.

Value

A list is returned:

Note

If the treatment effect is not significant, the user will receive the following message: "Warning: it looks like the treatment effect is not significant; may be difficult to interpret the residual treatment effect in this setting".

Author(s)

Layla Parast

References

Parast, L., Cai, T., & Tian, L. (2017). Evaluating surrogate marker information using censored data. Statistics in Medicine, 36(11), 1767-1782.

Examples

data(d_example_surv)
names(d_example_surv)

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

Augmentation function

Description

Augmentation function

Usage

augment.est.vector(point.delta, perturb.delta, treat.ind, basis, weights)

Arguments

Author(s)

Layla Parast

calculates closed form variance estimate for R

Description

calculates closed form variance estimate for R; used in R function that corrects for measurement error

Usage

calculate.var.np(s1, s0, y1, y0, extrapolate = TRUE)

Arguments

Value

Author(s)

Layla Parast

References

Parast, L., McDermott, M., Tian, L. (2016). Robust estimation of the proportion of treatment effect explained by surrogate marker information. Statistics in Medicine, 35(10):1637-1653.

Parast, L., Garcia, T. P., Prentice, R. L., & Carroll, R. J. (2022). Robust methods to correct for measurement error when evaluating a surrogate marker. Biometrics, 78(1), 9-23.

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, approx = T)

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

Helper function

Description

Helper function; should not be called directly by user.

Usage

cumsum2(mydat)

Arguments

Value

Author(s)

Layla Parast

Hypothetical data

Description

Hypothetical data to be used in examples.

Usage

data(d_example)

Format

A list with 8 elements representing 500 observations from a control group and 500 observations from a treatment group:

s1.a

First surrogate marker measurement for treated observations.

s1.b

Second surrogate marker measurement for treated observations.

s1.c

Third surrogate marker measurement for treated observations.

y1

Primary outcome for treated observations.

s0.a

First surrogate marker measurement for control observations.

s0.b

Second surrogate marker measurement for control observations.

s0.c

Third surrogate marker measurement for control observations.

y0

Primary outcome for control observations.

Examples

data(d_example)
names(d_example)

Hypothetical data with replicate measurements

Description

Hypothetical data to be used in measurement error example.

Usage

data(d_example_me)

Format

A list with 10 elements representing 500 observations from a control group and 500 observations from a treatment group:

y1

Primary outcome for treated observations.

s1

Surrogate marker for treated observations.

s1_rep1

Replicate measurement of the surrogate marker for treated observations.

s1_rep2

Replicate measurement of the surrogate marker for treated observations.

s1_rep3

Replicate measurement of the surrogate marker for treated observations.

y0

Primary outcome for control observations.

s0

Surrogate marker for control observations.

s0_rep1

Replicate measurement of the surrogate marker for control observations.

s0_rep2

Replicate measurement of the surrogate marker for control observations.

s0_rep3

Replicate measurement of the surrogate marker for control observations.

Examples

data(d_example_me)
names(d_example_me)

Hypothetical survival data with multiple surrogate markers

Description

Hypothetical survival data with multiple surrogate markers to be used in examples.

Usage

data(d_example_multiple)

Format

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

s1

Surrogate marker measurements for treated observations; these markers are measured at time = 0.5. For observations that experience the primary outcome or are censored before 0.5, the surrogate values are 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 measurements for control observations; these markers are measured at time = 0.5. For observations that experience the primary outcome or are censored before 0.5, the surrogate values are 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.

Examples

data(d_example_multiple)
names(d_example_multiple)

Hypothetical survival data

Description

Hypothetical survival data to be used in examples.

Usage

data(d_example_surv)

Format

A list with 8 elements representing 500 observations from a control group and 500 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.

z1

A baseline covariate value for treated observations.

z0

A baseline covariate value for control observations.

Examples

data(d_example_surv)
names(d_example_surv)

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

Calculates treatment effect

Description

This function calculates the treatment effect estimate, the difference in the average outcome in the treatment group minus the control group. This function is intended to be used for a fully observed continuous outcome. The user can also request a variance estimate, estimated using perturbating-resampling, and a 95% confidence interval. If a confidence interval is requested two versions are provided: a normal approximation based interval and a quantile based interval, both use perturbation-resampling.

Usage

delta.estimate(yone,yzero, var = FALSE, conf.int = FALSE, weight = NULL, 
weight.perturb = NULL)

Arguments

Details

Let Y^{(1)} and Y^{(0)} denote the primary outcome under the treatment and primary outcome under the control,respectively. The treatment effect, \Delta, is the expected difference in Y^{(1)} compared to Y^{(0)}, \Delta=E(Y^{(1)}-Y^{(0)}). We estimate \Delta as

\hat{\Delta} = n_1^{-1} \sum_{i=1}^{n_1} Y_{1i} - n_0^{-1} \sum_{i=1}^{n_0} Y_{0i}

where Y_{1i} is the observed primary outcome for person i in the treated group, Y_{0i} is the observed primary outcome for person i in the control group, and n_1 and n_0 are the number of individuals in the treatment and control group, respectively. Randomized treatment assignment is assumed throughout this package.

Variance estimation and confidence interval construction are performed using perturbation-resampling. Specifically, let \left \{ V^{(b)} = (V_{11}^{(b)}, ...V_{1n_1}^{(b)}, V_{01}^{(b)}, ...V_{0n_0}^{(b)})^T, b=1,....,D \right \} be n \times D independent copies of a positive random variables V from a known distribution with unit mean and unit variance. Let

\hat{\Delta}^{(b)} = \frac{ \sum_{i=1}^{n_1} V_{1i}^{(b)} Y_{1i}}{ \sum_{i=1}^{n_1} V_{1i}^{(b)}} - \frac{ \sum_{i=1}^{n_0} V_{0i}^{(b)} Y_{0i}}{ \sum_{i=1}^{n_0} V_{0i}^{(b)}}.

The variance of \hat{\Delta} is obtained as the empirical variance of \{\hat{\Delta}^{(b)}, b = 1,...,D\}. In this package, we use weights generated from an Exponential(1) distribution and use D=500. We construct two versions of the 95\% confidence interval for \hat{\Delta}: one based on a normal approximation confidence interval using the estimated variance and another taking the 2.5th and 97.5th empirical percentiles of \hat{\Delta}^{(b)}.

Value

A list is returned:

Author(s)

Layla Parast

Examples

data(d_example)
names(d_example)
delta.estimate(yone=d_example$y1, yzero=d_example$y0)

Calculates robust residual treatment effect accounting for multiple surrogate markers at a specified time and primary outcome information up to that specified time

Description

This function calculates the robust estimate of the residual treatment effect accounting for multiple surrogate markers measured at t_0 and primary outcome information up to t_0 i.e. the hypothetical treatment effect if both the surrogate marker distributions at t_0 and survival up to t_0 in the treatment group look like the surrogate marker distributions and survival up to t_0 in the control group. Ideally this function is only used as a helper function and is not directly called.

Usage

delta.multiple.surv(xone, xzero, deltaone, deltazero, sone, szero, type =1, t, 
weight.perturb = NULL, landmark, extrapolate = FALSE, transform = FALSE,
approx = T)

Arguments

Details

Details are included in the documentation for R.multiple.surv.

Value

\hat{\Delta}_S(t,t_0), the residual treatment effect estimate accounting for multiple surrogarte markers measured at t_0 and primary outcome information up to t_0.

Note

If the treatment effect is not significant, the user will receive the following message: "Warning: it looks like the treatment effect is not significant; may be difficult to interpret the residual treatment effect in this setting".

Author(s)

Layla Parast

References

Parast, L., Cai, T., & Tian, L. (2021). Evaluating multiple surrogate markers with censored data. Biometrics, 77(4), 1315-1327.

Examples

data(d_example_multiple)
names(d_example_multiple)
## Not run: 
delta.multiple.surv(xone = d_example_multiple$x1, xzero = d_example_multiple$x0, deltaone =
 d_example_multiple$delta1, deltazero = d_example_multiple$delta0, sone = 
 as.matrix(d_example_multiple$s1), szero = as.matrix(d_example_multiple$s0), 
 type =1, t = 1, landmark=0.5)

## End(Not run)

Calculates model-based or robust residual treatment effect

Description

This function calculates the model-based or robust estimate of the residual treatment effect i.e. the hypothetical treatment effect if the distribution of the surrogate in the treatment group looks like the distribution of the surrogate in the control group. Ideally, this function is only used as a helper function and is not directly called.

Usage

delta.s.estimate(sone, szero, yone, yzero, weight.perturb = NULL, number="single",
type="robust", warn.te = FALSE, warn.support = FALSE, extrapolate = FALSE, 
transform = FALSE)

Arguments

Details

Details are included in the documentation for R.s.estimate.

Value

\hat{\Delta}_S, the model-based or robust residual treatment effect estimate.

Note

If the treatment effect is not significant, the user will receive the following message: "Warning: it looks like the treatment effect is not significant; may be difficult to interpret the residual treatment effect in this setting". In the single marker case with the robust estimation approach, if the observed support of the surrogate marker for the control group is outside the observed support of the surrogate marker for the treatment group, the user will receive the following message: "Warning: observed supports do not appear equal, may need to consider a transformation or extrapolation".

Author(s)

Layla Parast

References

Parast, L., McDermott, M., Tian, L. (2015). Robust estimation of the proportion of treatment effect explained by surrogate marker information. Statistics in Medicine, 35(10):1637-1653.

Wang, Y., & Taylor, J. M. (2002). A measure of the proportion of treatment effect explained by a surrogate marker. Biometrics, 58(4), 803-812.

Examples

data(d_example)
names(d_example)
delta.s.estimate(yone=d_example$y1, yzero=d_example$y0, sone=d_example$s1.a, szero=
d_example$s0.a, number = "single", type = "robust")
delta.s.estimate(yone=d_example$y1, yzero=d_example$y0, sone=d_example$s1.a, szero=
d_example$s0.a, number = "single", type = "model")
delta.s.estimate(yone=d_example$y1, yzero=d_example$y0, sone=cbind(d_example$s1.a, 
d_example$s1.b, d_example$s1.c), szero=cbind(d_example$s0.a, d_example$s0.b, d_example$s0.c), 
number = "multiple", type = "robust")
delta.s.estimate(yone=d_example$y1, yzero=d_example$y0, sone=cbind(d_example$s1.a, 
d_example$s1.b, d_example$s1.c), szero=cbind(d_example$s0.a, d_example$s0.b, d_example$s0.c),
number = "multiple", type = "model")

Calculates robust residual treatment effect accounting for surrogate marker information measured at a specified time and primary outcome information up to that specified time

Description

This function calculates the robust estimate of the residual treatment effect accounting for surrogate marker information measured at t_0 and primary outcome information up to t_0 i.e. the hypothetical treatment effect if both the surrogate marker distribution at t_0 and survival up to t_0 in the treatment group look like the surrogate marker distribution and survival up to t_0 in the control group. Ideally this function is only used as a helper function and is not directly called.

Usage

delta.s.surv.estimate(xone, xzero, deltaone, deltazero, sone, szero, t, 
weight.perturb = NULL, landmark, extrapolate = FALSE, transform = FALSE,
approx = T, warn.te = FALSE, warn.support = FALSE)

Arguments

Details

Details are included in the documentation for R.s.surv.estimate.

Value

\hat{\Delta}_S(t,t_0), the robust residual treatment effect estimate accounting for surrogate marker information measured at t_0 and primary outcome information up to t_0.

Note

If the treatment effect is not significant, the user will receive the following message: "Warning: it looks like the treatment effect is not significant; may be difficult to interpret the residual treatment effect in this setting". If the observed support of the surrogate marker for the control group is outside the observed support of the surrogate marker for the treatment group, the user will receive the following message: "Warning: observed supports do not appear equal, may need to consider a transformation or extrapolation".

Author(s)

Layla Parast

References

Parast, L., Cai, T., & Tian, L. (2017). Evaluating surrogate marker information using censored data. Statistics in Medicine, 36(11), 1767-1782.

Examples

data(d_example_surv)
names(d_example_surv)

Calculates treatment effect in a survival setting

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 user can also request a variance estimate, estimated using perturbating-resampling, and a 95% confidence interval. If a confidence interval is requested two versions are provided: a normal approximation based interval and a quantile based interval, both use perturbation-resampling.

Usage

delta.surv.estimate(xone, xzero, deltaone, deltazero, t, var = FALSE, conf.int 
= FALSE, weight = NULL, weight.perturb = NULL, approx = T)

Arguments

Details

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 denote the time of the primary outcome of interest, death for example. We use potential outcomes notation such that T^{(g)} denotes the time of the primary outcome under treatment G = g. We define the treatment effect, \Delta(t), as the difference in survival rates by time t under treatment versus control,

\Delta(t)=E\{ I(T^{(1)}>t)\} - E\{I(T^{(0)}>t)\} = P(T^{(1)}>t) - P(T^{(0)}>t)

where t>t_0

Due to censoring, our data consist of n_1 observations \{(X_{1i}, \delta_{1i}), i=1,...,n_1\} from the treatment group G=1 and n_0 observations \{(X_{0i}, \delta_{0i}), i=1,...,n_0\} from the control group G=0 where X_{gi} = \min(T_{gi}, C_{ gi}), \delta_{gi} = I(T_{gi} < C_{gi}), and C_{gi} denotes the censoring time for g= 1,0, for individual i. Throughout, we estimate the treatment effect \Delta(t) as

\hat{\Delta}(t) = n_1^{-1} \sum_{i=1}^{n_1} \frac{I(X_{1i}>t)}{\hat{W}^C_1(t)} - n_0^{-1} \sum_{i=1}^{n_0} \frac{I(X_{0i}>t)}{\hat{W}^C_0(t)}

where \hat{W}^C_g(\cdot) is the Kaplan-Meier estimator of survival for censoring for g=1,0.

Variance estimation and confidence interval construction are performed using perturbation-resampling. Specifically, let \left \{ V^{(b)} = (V_{11}^{(b)}, ...V_{1n_1}^{(b)}, V_{01}^{(b)}, ...V_{0n_0}^{(b)})^T, b=1,....,D \right \} be n \times D independent copies of a positive random variables V from a known distribution with unit mean and unit variance. Let

\hat{\Delta}^{(b)} (t) = \frac{ \sum_{i=1}^{n_1} V_{1i}^{(b)} I(X_{1i}>t)}{ \sum_{i=1}^{n_1} V_{1i}^{(b)} \hat{W}_1^{C(b)}(t)} -\frac{ \sum_{i=1}^{n_0} V_{0i}^{(b)} I(X_{0i}>t)}{ \sum_{i=1}^{n_0} V_{0i}^{(b)} \hat{W}_0^{C(b)}(t)}.

In this package, we use weights generated from an Exponential(1) distribution and use D=500. The variance of \hat{\Delta}(t) is obtained as the empirical variance of \{\hat{\Delta}(t)^{(b)}, b = 1,...,D\}. We construct two versions of the 95\%confidence interval for \hat{\Delta}(t): one based on a normal approximation confidence interval using the estimated variance and another taking the 2.5th and 97.5th empirical percentiles of \hat{\Delta}(t)^{(b)}.

Value

A list is returned:

Author(s)

Layla Parast

Examples

data(d_example_surv)
names(d_example_surv)
delta.surv.estimate(xone = d_example_surv$x1, xzero = d_example_surv$x0,  
deltaone = d_example_surv$delta1, deltazero = d_example_surv$delta0, t = 3)

Calculates robust residual treatment effect accounting only for primary outcome information up to a specified time

Description

This function calculates the robust estimate of the residual treatment effect accounting only for primary outcome information up to t_0 i.e. the hypothetical treatment effect if survival up to t_0 in the treatment group looks like survival up to t_0 in the control group. Ideally this function is only used as a helper function and is not directly called.

Usage

delta.t.surv.estimate(xone, xzero, deltaone, deltazero, t, weight.perturb = NULL,
landmark, approx = T)

Arguments

Details

Details are included in the documentation for R.t.surv.estimate.

Value

\hat{\Delta}_T(t,t_0), the robust residual treatment effect estimate accounting only for survival up to t_0.

Note

If the treatment effect is not significant, the user will receive the following message: "Warning: it looks like the treatment effect is not significant; may be difficult to interpret the residual treatment effect in this setting".

Author(s)

Layla Parast

References

Parast, L., Cai, T., & Tian, L. (2017). Evaluating surrogate marker information using censored data. Statistics in Medicine, 36(11), 1767-1782.

Examples

data(d_example_surv)
names(d_example_surv)

Constructs Fieller's confidence interval.

Description

Constructs Fieller's confidence interval - closed form, not using perturbation.

Usage

fieller.calculate.me(a, B, var.mat)

Arguments

Details

Produces a confidence interval for 1-a/B when closed form variance and covariance estimates are available. See documention for R.s.estimate.me for more detail.

Value

Returns a vector of length 2, lower bound of the 95% confidence interval and upper bound of the 95% confidence interval.

Author(s)

Layla Parast

References

Parast, L., Garcia, T. P., Prentice, R. L., & Carroll, R. J. (2022). Robust methods to correct for measurement error when evaluating a surrogate marker. Biometrics, 78(1), 9-23.

Fieller, Edgar C. (1954). Some problems in interval estimation. Journal of the Royal Statistical Society. Series B (Methodological), 175-185.

Fieller, E. C. (1940). The biological standardization of insulin. Supplement to the Journal of the Royal Statistical Society, 1-64.

Constructs Fieller's confidence interval.

Description

Constructs Fieller's confidence interval.

Usage

fieller.ci(perturb.delta.s, perturb.delta, delta.s, delta)

Arguments

Details

See documention for R.s.estimate for more detail.

Value

Returns a vector of length 2, lower bound of the 95% confidence interval and upper bound of the 95% confidence interval.

Author(s)

Layla Parast

References

Fieller, Edgar C. (1954). Some problems in interval estimation. Journal of the Royal Statistical Society. Series B (Methodological), 175-185.

Fieller, E. C. (1940). The biological standardization of insulin. Supplement to the Journal of the Royal Statistical Society, 1-64.

Parast, L., McDermott, M., Tian, L. (2016). Robust estimation of the proportion of treatment effect explained by surrogate marker information. Statistics in Medicine, 35(10):1637-1653.

Estimates measurement error variance given replicate data.

Description

Estimates measurement error variance given replicate data using a simple components of variance analysis.

Usage

me.variance.estimate(replicates)

Arguments

Details

Estimates measurement error variance given replicate data using a simple components of variance analysis.

Value

estimate of measurement error variance

Author(s)

Layla Parast

References

Carroll, R. J., Ruppert, D., Crainiceanu, C. M., and Stefanski, L. A. (2006). Measurement error in nonlinear models: a modern perspective. Chapman and Hall/CRC.

Parast, L., Garcia, T. P., Prentice, R. L., & Carroll, R. J. (2022). Robust methods to correct for measurement error when evaluating a surrogate marker. Biometrics, 78(1), 9-23.

Helper for augmentation function

Description

Helper for augmentation function

Usage

perturb.nu.vector(mat, weights)

Arguments

Calculates expected outcome for control group surrogate values

Description

Helper function; calculates the expected outcome for control group surrogate values using treatment group surrogate and outcome information.

Usage

pred.smooth(zz, zi.one, bw = NULL, y1, weight = NULL)

Arguments

Details

Details are included in the documentation for R.s.estimate

Value

expected outcome for each surrogate value in the control group

Author(s)

Layla Parast

References

Parast, L., McDermott, M., Tian, L. (2016). Robust estimation of the proportion of treatment effect explained by surrogate marker information. Statistics in medicine, 35(10):1637-1653.

Calculates the conditional probability of survival for control group values

Description

Helper function; calculates the estimated probability of survival for control group for control group surrogate values using treatment group surrogate and outcome information.

Usage

pred.smooth.surv(xone.f, deltaone.f, sone.f, szero.one, myt, weight.pred, 
extrapolate, transform, ps.weight = NULL, warn.support = FALSE)

Arguments

Details

Details are included in the documentation for R.s.surv.estimate and R.multiple.surv

Value

conditional probability of survival past t for control group

Author(s)

Layla Parast