Package 'Surrogate'

Title:	Evaluation of Surrogate Endpoints in Clinical Trials
Description:	In a clinical trial, it frequently occurs that the most credible outcome to evaluate the effectiveness of a new therapy (the true endpoint) is difficult to measure. In such a situation, it can be an effective strategy to replace the true endpoint by a (bio)marker that is easier to measure and that allows for a prediction of the treatment effect on the true endpoint (a surrogate endpoint). The package 'Surrogate' allows for an evaluation of the appropriateness of a candidate surrogate endpoint based on the meta-analytic, information-theoretic, and causal-inference frameworks. Part of this software has been developed using funding provided from the European Union's Seventh Framework Programme for research, technological development and demonstration (Grant Agreement no 602552), the Special Research Fund (BOF) of Hasselt University (BOF-number: BOF2OCPO3), GlaxoSmithKline Biologicals, Baekeland Mandaat (HBC.2022.0145), and Johnson & Johnson Innovative Medicine.
Authors:	Wim Van Der Elst [cre, aut], Florian Stijven [aut], Fenny Ong [aut], Dries De Witte [aut], Paul Meyvisch [aut], Alvaro Poveda [aut], Ariel Alonso [aut], Hannah Ensor [aut], Christoper Weir [aut], Geert Molenberghs [aut]
Maintainer:	Wim Van Der Elst <[email protected]>
License:	GPL (>= 2)
Version:	3.3.0.9000
Built:	2024-11-06 05:29:22 UTC
Source:	https://github.com/florianstijven/surrogate-development

Help Index

Compute the multiple-surrogate adjusted association
Data of the Age-Related Macular Degeneration Study
Data of the Age-Related Macular Degeneration Study with multiple candidate surrogates
Fits a bivariate fixed-effects model to assess surrogacy in the meta-analytic multiple-trial setting (Continuous-continuous case)
Fits a bivariate mixed-effects model using the cluster-by-cluster (CbC) estimator to assess surrogacy in the meta-analytic multiple-trial setting (Continuous-continuous case)
Fits a bivariate mixed-effects model to assess surrogacy in the meta-analytic multiple-trial setting (Continuous-continuous case)
Loglikelihood function for binary-continuous copula model
Bootstrap 95% CI around the maximum-entropy ICA and SPF (surrogate predictive function)
Draws a causal diagram depicting the median informational coefficients of correlation (or odds ratios) between the counterfactuals for a specified range of values of the ICA in the binary-binary setting.
Draws a causal diagram depicting the median correlations between the counterfactuals for a specified range of values of ICA or MICA in the continuous-continuous setting
Function factory for distribution functions
Loglikelihood on the Copula Scale for the Clayton Copula
The Colorectal dataset with a binary surrogate.
The Colorectal dataset with an ordinal surrogate.
Assesses the surrogate predictive value of each of the 27 prediction functions in the setting where both S and T are binary endpoints
Compute Individual Causal Association for a given D-vine copula model in the Binary-Continuous Setting
Compute Individual Causal Association for a given D-vine copula model in the Survival-Survival Setting
Variance of log-mutual information based on the delta method
Confidence interval for the ICA given the unidentifiable parameters
Apply the Entropy Concentration Theorem
Estimate ICA in Binary-Continuous Setting
Estimate the Mutual Information in the Survival-Survival Setting
Evaluate the possibility of finding a good surrogate in the setting where both S and T are binary endpoints
Fits the first stage model in the two-stage federated data analysis approach.
Fits the second stage model in the two-stage federated data analysis approach.
Fit copula model for binary true endpoint and continuous surrogate endpoint
Fit binary-continuous copula submodel
Fit Survival-Survival model
Fits (univariate) fixed-effect models to assess surrogacy in the binary-binary case based on the Information-Theoretic framework
Fits (univariate) fixed-effect models to assess surrogacy in the case where the true endpoint is binary and the surrogate endpoint is continuous (based on the Information-Theoretic framework)
Fits (univariate) fixed-effect models to assess surrogacy in the case where the true endpoint is continuous and the surrogate endpoint is binary (based on the Information-Theoretic framework)
Fits (univariate) fixed-effect models to assess surrogacy in the continuous-continuous case based on the Information-Theoretic framework
Investigates surrogacy for binary or ordinal outcomes using the Information Theoretic framework
Loglikelihood on the Copula Scale for the Frank Copula
Loglikelihood on the Copula Scale for the Gaussian Copula
Loglikelihood on the Copula Scale for the Gumbel Copula
Constructor for the function that returns that ICA as a function of the identifiable parameters
Assess surrogacy in the causal-inference single-trial setting in the binary-binary case
ICA (binary-binary setting) that is obtaied when the counterfactual correlations are assumed to fall within some prespecified ranges.
Assess surrogacy in the causal-inference single-trial setting in the binary-binary case when monotonicity for S and T is assumed using the full grid-based approach
Assess surrogacy in the causal-inference single-trial setting in the binary-binary case when monotonicity for S and T is assumed using the grid-based sample approach
Assess surrogacy in the causal-inference single-trial setting in the binary-binary case when monotonicity for S and T is assumed using the grid-based sample approach, accounting for sampling variability in the marginal \pi.
Assess surrogacy in the causal-inference single-trial setting in the binary-continuous case
Assess surrogacy in the causal-inference single-trial setting in the binary-continuous case with an additional bootstrap procedure before the assessment
Assess surrogacy in the causal-inference single-trial setting (Individual Causal Association, ICA) in the Continuous-continuous case
Assess surrogacy in the causal-inference single-trial setting (Individual Causal Association, ICA) using a continuous univariate T and multiple continuous S
Assess surrogacy in the causal-inference single-trial setting (Individual Causal Association, ICA) using a continuous univariate T and multiple continuous S, alternative approach
Assess surrogacy in the causal-inference single-trial setting (Individual Causal Association, ICA) using a continuous univariate T and multiple continuous S, by simulating correlation matrices using a modified algorithm based on partial correlations
Assess surrogacy in the causal-inference single-trial setting (Individual Causal Association, ICA) using a continuous univariate T and multiple continuous S, by simulating correlation matrices using an algorithm based on partial correlations
Assess surrogacy in the causal-inference single-trial setting (Individual Causal Association, ICA) in the Continuous-continuous case using the grid-based sample approach
Individual-level surrogate threshold effect for continuous normally distributed surrogate and true endpoints.
Computes loglikelihood for a given copula model
Loglikelihood on the Copula Scale
Reshapes a dataset from the 'long' format (i.e., multiple lines per patient) into the 'wide' format (i.e., one line per patient)
Fit marginal distribution
Marginal survival function goodness of fit
Goodness-of-fit plot for the marginal survival functions
Computes marginal probabilities for a dataset where the surrogate and true endpoints are binary
Use the maximum-entropy approach to compute ICA in the continuous-continuous sinlge-trial setting
Use the maximum-entropy approach to compute ICA in the binary-binary setting
Use the maximum-entropy approach to compute SPF (surrogate predictive function) in the binary-binary setting
Goodness of fit plot for the fitted copula
Compute surrogacy measures for a binary surrogate and a time-to-event true endpoint in the meta-analytic multiple-trial setting.
Compute surrogacy measures for a categorical (ordinal) surrogate and a time-to-event true endpoint in the meta-analytic multiple-trial setting.
Compute surrogacy measures for a continuous (normally-distributed) surrogate and a time-to-event true endpoint in the meta-analytic multiple-trial setting.
Compute surrogacy measures for a time-to-event surrogate and a time-to-event true endpoint in the meta-analytic multiple-trial setting.
Assess surrogacy in the causal-inference multiple-trial setting (Meta-analytic Individual Causal Association; MICA) in the continuous-continuous case
Assess surrogacy in the causal-inference multiple-trial setting (Meta-analytic Individual Causal Association; MICA) in the continuous-continuous case using the grid-based sample approach
Examine the plausibility of finding a good surrogate endpoint in the Continuous-continuous case
Fits (univariate) mixed-effect models to assess surrogacy in the continuous-continuous case based on the Information-Theoretic framework
Goodness of fit information for survival-survival model
Fits a multivariate fixed-effects model to assess surrogacy in the meta-analytic multiple-trial setting (Continuous-continuous case with multiple surrogates)
Fits a multivariate mixed-effects model to assess surrogacy in the meta-analytic multiple-trial setting (Continuous-continuous case with multiple surrogates)
Constructor for vine copula model
The Ovarian dataset
PANSS subscales and total score based on the data of five clinical trials in schizophrenia
Function factory for density functions
Plots the (Meta-Analytic) Individual Causal Association and related metrics when S and T are binary outcomes
Plots the (Meta-Analytic) Individual Causal Association when S and T are continuous outcomes
Provides plots of trial-level surrogacy in the Information-Theoretic framework
Plots the Individual Causal Association in the setting where there are multiple continuous S and a continuous T
Provides plots of trial- and individual-level surrogacy in the Information-Theoretic framework
Provides plots of trial- and individual-level surrogacy in the Information-Theoretic framework when both S and T are binary, or when S is binary and T is continuous (or vice versa)
Plots the individual-level surrogate threshold effect (STE) values and related metrics
Plots the sensitivity-based and maximum entropy based Individual Causal Association when S and T are continuous outcomes in the single-trial setting
Plots the sensitivity-based and maximum entropy based Individual Causal Association when S and T are binary outcomes
Plots the sensitivity-based and maximum entropy based surrogate predictive function (SPF) when S and T are binary outcomes.
Provides plots of trial- and individual-level surrogacy in the meta-analytic framework
Graphically illustrates the theoretical plausibility of finding a good surrogate endpoint in the continuous-continuous case
Plots the expected treatment effect on the true endpoint in a new trial (when both S and T are normally distributed continuous endpoints)
Plots the surrogate predictive function (SPF) in the binary-binary settinf.
Provides a plots of trial-level surrogacy in the information-theoretic framework based on the output of the TrialLevelIT() function
Provides a plots of trial-level surrogacy in the meta-analytic framework based on the output of the TrialLevelMA() function
Plots trial-level surrogacy in the meta-analytic framework when two survival endpoints are considered.
Plots the distribution of prediction error functions in decreasing order of appearance.
Plots the distribution of R^2_{HL} either as a density or as function of \pi_{10} in the setting where both S and T are binary endpoints
Plot the individual causal association (ICA) in the causal-inference single-trial setting in the binary-continuous case.
Generates a plot of the estimated treatment effects for the surrogate endpoint versus the estimated treatment effects for the true endpoint for an object fitted with the 'MetaAnalyticSurvBin()' function.
Generates a plot of the estimated treatment effects for the surrogate endpoint versus the estimated treatment effects for the true endpoint for an object fitted with the 'MetaAnalyticSurvCat()' function.
Generates a plot of the estimated treatment effects for the surrogate endpoint versus the estimated treatment effects for the true endpoint for an object fitted with the 'MetaAnalyticSurvCont()' function.
Generates a plot of the estimated treatment effects for the surrogate endpoint versus the estimated treatment effects for the true endpoint for an object fitted with the 'MetaAnalyticSurvSurv()' function.
Plots the distribution of either PPE, RPE or R^2_{H} either as a density or as a histogram in the setting where both S and T are binary endpoints
Plot the surrogate predictive function (SPF) in the causal-inference single-trial setting in the binary-continuous case.
Provides plots of trial- and individual-level surrogacy in the Information-Theoretic framework when both S and T are time-to-event endpoints
Generate 4 by 4 correlation matrices and flag the positive definite ones
Evaluate a surrogate predictive value based on the minimum probability of a prediction error in the setting where both S and T are binary endpoints
Compute the expected treatment effect on the true endpoint in a new trial (when both S and T are normally distributed continuous endpoints)
Evaluates surrogacy based on the Prentice criteria for continuous endpoints (single-trial setting)
Prints all the elements of an object fitted with the 'MetaAnalyticSurvBin()' function.
Prints all the elements of an object fitted with the 'MetaAnalyticSurvCat()' function.
Prints all the elements of an object fitted with the 'MetaAnalyticSurvCont()' function.
Prints all the elements of an object fitted with the 'MetaAnalyticSurvSurv()' function.
Evaluate the individual causal association (ICA) and reduction in probability of a prediction error (RPE) in the setting where both S and T are binary endpoints
The prostate dataset with a continuous surrogate.
Generate random vectors with a fixed sum
Examine restrictions in \bold{\pi}_{f} under different montonicity assumptions for binary S and T
Sample Unidentifiable Copula Parameters
Sample individual casual treatment effects from given D-vine copula model in binary continuous setting
Sample copula data from a given four-dimensional D-vine copula
Data of five clinical trials in schizophrenia
Data of a clinical trial in Schizophrenia (with binary outcomes).
Data of a clinical trial in schizophrenia, with binary and continuous endpoints
Longitudinal PANSS data of five clinical trials in schizophrenia
Perform Sensitivity Analysis for the Individual Causal Association with a Continuous Surrogate and Binary True Endpoint
Sensitivity analysis for individual causal association
Compute Sensitivity Intervals
Simulate a dataset that contains counterfactuals
Simulate a dataset that contains counterfactuals for binary endpoints
Simulates a dataset that can be used to assess surrogacy in the multiple-trial setting
Simulates a dataset that can be used to assess surrogacy in the single-trial setting
Simulates a dataset that can be used to assess surrogacy in the single trial setting when S and T are binary endpoints
Conducts a surrogacy analysis based on the single-trial meta-analytic framework
Evaluate the surrogate predictive function (SPF) in the binary-binary setting (sensitivity-analysis based approach)
Evaluate the surrogate predictive function (SPF) in the causal-inference single-trial setting in the binary-continuous case
Bootstrap based on the multivariate normal sampling distribution
Provides a summary of the surrogacy measures for an object fitted with the 'FederatedApproachStage2()' function.
Provides a summary of the surrogacy measures for an object fitted with the 'MetaAnalyticSurvBin()' function.
Provides a summary of the surrogacy measures for an object fitted with the 'MetaAnalyticSurvCat()' function.
Provides a summary of the surrogacy measures for an object fitted with the 'MetaAnalyticSurvCont()' function.
Provides a summary of the surrogacy measures for an object fitted with the 'MetaAnalyticSurvSurv()' function.
Assess surrogacy for two survival endpoints based on information theory and a two-stage approach
Test whether the data are compatible with monotonicity for S and/or T (binary endpoints)
Estimates trial-level surrogacy in the information-theoretic framework
Estimates trial-level surrogacy in the meta-analytic framework
Assess trial-level surrogacy for two survival endpoints using a two-stage approach
Fit binary-continuous copula submodel with two-step estimator
Fit survival-survival copula submodel with two-step estimator
Fits univariate fixed-effect models to assess surrogacy in the meta-analytic multiple-trial setting (continuous-continuous case)
Fits univariate mixed-effect models to assess surrogacy in the meta-analytic multiple-trial setting (continuous-continuous case)

Compute the multiple-surrogate adjusted association

Description

The function AA.MultS computes the multiple-surrogate adjusted correlation. This is a generalisation of the adjusted association proposed by Buyse & Molenberghs (1998) (see Single.Trial.RE.AA) to the setting where there are multiple endpoints. See Details below.

Usage

AA.MultS(Sigma_gamma, N, Alpha=0.05)
AA.MultS(Sigma_gamma, N, Alpha=0.05)

Arguments

`Sigma_gamma`	The variance covariance matrix of the residuals of regression models in which the true endpoint ( $T$ ) is regressed on the treatment ( $Z$ ), the first surrogate ( $S1$ ) is regressed on $Z$ , ..., and the $k$ -th surrogate ( $Sk$ ) is regressed on $Z$ . See Details below.
`N`	The sample size (needed to compute a CI around the multiple adjusted association; $\gamma_M$ )
`Alpha`	The $\alpha$ -level that is used to determine the confidence interval around $\gamma_M$ . Default $0.05$ .

Details

The multiple-surrogate adjusted association ( $\gamma_M$ ) is obtained by regressing $T$ , $S1$ , $S2$ , ..., $Sk$ on the treatment ( $Z$ ):

$T_{j}=\mu_{T}+\beta Z_{j}+\varepsilon_{Tj},$

$S1_{j}=\mu_{S1}+\alpha_{1}Z_{j}+\varepsilon_{S1j},$

$\ldots,$

$Sk_{j}=\mu_{Sk}+\alpha_{k}Z_{j}+\varepsilon_{Skj},$

where the error terms have a joint zero-mean normal distribution with variance-covariance matrix:

${\boldsymbol{\Sigma}=\left(\begin{array}{cc} \sigma_{TT} & \Sigma_{\boldsymbol{S}T}\\ \Sigma^{'}_{\boldsymbol{S}T} & \Sigma_{\boldsymbol{SS}} \\ \end{array}\right).}$

The multiple adjusted association is then computed as

$\gamma_M = \sqrt(\frac{\left(\Sigma^{'}_{ST} \Sigma^{-1}_{SS} \Sigma_{ST}\right)}{\sigma_{TT}})$

Value

An object of class AA.MultS with components,

`Gamma.Delta`	An object of class `data.frame` that contains the multiple-surrogate adjusted association (i.e., $\gamma_M$ ), its standard error, and its confidence interval (based on the Fisher-Z transformation procedure).
`Corr.Gamma.Delta`	An object of class `data.frame` that contains the bias-corrected multiple-surrogate adjusted association (i.e., corrected $\gamma_M$ ), its standard error, and its confidence interval (based on the Fisher-Z transformation procedure).
`Sigma_gamma`	The variance covariance matrix of the residuals of regression models in which $T$ is regressed on $Z$ , $S1$ is regressed on $Z$ , ..., and $Sk$ is regressed on $Z$ .
`N`	The sample size (used to compute a CI around the multiple adjusted association; $\gamma_M$ )
`Alpha`	The $\alpha$ -level that is used to determine the confidence interval around $\gamma_M$ .

Author(s)

Wim Van der Elst, Ariel Alonso, & Geert Molenberghs

References

Buyse, M., & Molenberghs, G. (1998). The validation of surrogate endpoints in randomized experiments. Biometrics, 54, 1014-1029.

Van der Elst, W., Alonso, A. A., & Molenberghs, G. (2017). A causal inference-based approach to evaluate surrogacy using multiple surrogates.

Examples

data(ARMD.MultS)

# Regress T on Z, S1 on Z, ..., Sk on Z 
# (to compute the covariance matrix of the residuals)
Res_T <- residuals(lm(Diff52~Treat, data=ARMD.MultS))
Res_S1 <- residuals(lm(Diff4~Treat, data=ARMD.MultS))
Res_S2 <- residuals(lm(Diff12~Treat, data=ARMD.MultS))
Res_S3 <- residuals(lm(Diff24~Treat, data=ARMD.MultS))
Residuals <- cbind(Res_T, Res_S1, Res_S2, Res_S3)

# Make covariance matrix of residuals, Sigma_gamma
Sigma_gamma <- cov(Residuals)

# Conduct analysis
Result <- AA.MultS(Sigma_gamma = Sigma_gamma, N = 188, Alpha = .05)

# Explore results
summary(Result)
data(ARMD.MultS)

# Regress T on Z, S1 on Z, ..., Sk on Z 
# (to compute the covariance matrix of the residuals)
Res_T <- residuals(lm(Diff52~Treat, data=ARMD.MultS))
Res_S1 <- residuals(lm(Diff4~Treat, data=ARMD.MultS))
Res_S2 <- residuals(lm(Diff12~Treat, data=ARMD.MultS))
Res_S3 <- residuals(lm(Diff24~Treat, data=ARMD.MultS))
Residuals <- cbind(Res_T, Res_S1, Res_S2, Res_S3)

# Make covariance matrix of residuals, Sigma_gamma
Sigma_gamma <- cov(Residuals)

# Conduct analysis
Result <- AA.MultS(Sigma_gamma = Sigma_gamma, N = 188, Alpha = .05)

# Explore results
summary(Result)

Data of the Age-Related Macular Degeneration Study

Description

These are the data of a clinical trial involving patients suffering from age-related macular degeneration (ARMD), a condition that involves a progressive loss of vision. A total of $181$ patients from $36$ centers participated in the trial. Patients' visual acuity was assessed using standardized vision charts. There were two treatment conditions (placebo and interferon- $\alpha$ ). The potential surrogate endpoint is the change in the visual acuity at $24$ weeks ( $6$ months) after starting treatment. The true endpoint is the change in the visual acuity at $52$ weeks.

Usage

data(ARMD)data(ARMD)

Format

A data.frame with $181$ observations on $5$ variables.

Id: The Patient ID.
Center: The center in which the patient was treated.
Treat: The treatment indicator, coded as $-1$ = placebo and $1$ = interferon- $\alpha$ .
Diff24: The change in the visual acuity at $24$ weeks after starting treatment. This endpoint is a potential surrogate for Diff52.
Diff52: The change in the visual acuity at $52$ weeks after starting treatment. This outcome serves as the true endpoint.

Data of the Age-Related Macular Degeneration Study with multiple candidate surrogates

Description

These are the data of a clinical trial involving patients suffering from age-related macular degeneration (ARMD), a condition that involves a progressive loss of vision. A total of $181$ patients participated in the trial. Patients' visual acuity was assessed using standardized vision charts. There were two treatment conditions (placebo and interferon- $\alpha$ ). The potential surrogate endpoints are the changes in the visual acuity at $4$ , $12$ , and $24$ weeks after starting treatment. The true endpoint is the change in the visual acuity at $52$ weeks.

Usage

data(ARMD.MultS)data(ARMD.MultS)

Format

A data.frame with $181$ observations on $6$ variables.

Id: The Patient ID.
Diff4: The change in the visual acuity at $4$ weeks after starting treatment. This endpoint is a potential surrogate for Diff52.
Diff12: The change in the visual acuity at $12$ weeks after starting treatment. This endpoint is a potential surrogate for Diff52.
Diff24: The change in the visual acuity at $24$ weeks after starting treatment. This endpoint is a potential surrogate for Diff52.
Diff52: The change in the visual acuity at $52$ weeks after starting treatment. This outcome serves as the true endpoint.
Treat: The treatment indicator, coded as $-1$ = placebo and $1$ = interferon- $\alpha$ .

Fits a bivariate fixed-effects model to assess surrogacy in the meta-analytic multiple-trial setting (Continuous-continuous case)

Description

The function BifixedContCont uses the bivariate fixed-effects approach to estimate trial- and individual-level surrogacy when the data of multiple clinical trials are available. The user can specify whether a (weighted or unweighted) full, semi-reduced, or reduced model should be fitted. See the Details section below. Further, the Individual Causal Association (ICA) is computed.

Usage

BifixedContCont(Dataset, Surr, True, Treat, Trial.ID, Pat.ID, Model=c("Full"), 
Weighted=TRUE, Min.Trial.Size=2, Alpha=.05, T0T1=seq(-1, 1, by=.2), 
T0S1=seq(-1, 1, by=.2), T1S0=seq(-1, 1, by=.2), S0S1=seq(-1, 1, by=.2))
BifixedContCont(Dataset, Surr, True, Treat, Trial.ID, Pat.ID, Model=c("Full"), 
Weighted=TRUE, Min.Trial.Size=2, Alpha=.05, T0T1=seq(-1, 1, by=.2), 
T0S1=seq(-1, 1, by=.2), T1S0=seq(-1, 1, by=.2), S0S1=seq(-1, 1, by=.2))

Arguments

`Dataset`	A `data.frame` that should consist of one line per patient. Each line contains (at least) a surrogate value, a true endpoint value, a treatment indicator, a patient ID, and a trial ID.
`Surr`	The name of the variable in `Dataset` that contains the surrogate endpoint values.
`True`	The name of the variable in `Dataset` that contains the true endpoint values.
`Treat`	The name of the variable in `Dataset` that contains the treatment indicators. The treatment indicator should either be coded as $1$ for the experimental group and $-1$ for the control group, or as $1$ for the experimental group and $0$ for the control group.
`Trial.ID`	The name of the variable in `Dataset` that contains the trial ID to which the patient belongs.
`Pat.ID`	The name of the variable in `Dataset` that contains the patient's ID.
`Model`	The type of model that should be fitted, i.e., `Model=c("Full")`, `Model=c("Reduced")`, or `Model=c("SemiReduced")`. See the Details section below. Default `Model=c("Full")`.
`Weighted`	Logical. If `TRUE`, then a weighted regression analysis is conducted at stage 2 of the two-stage approach. If `FALSE`, then an unweighted regression analysis is conducted at stage 2 of the two-stage approach. See the Details section below. Default `TRUE`.
`Min.Trial.Size`	The minimum number of patients that a trial should contain in order to be included in the analysis. If the number of patients in a trial is smaller than the value specified by `Min.Trial.Size`, the data of the trial are excluded from the analysis. Default $2$ .
`Alpha`	The $\alpha$ -level that is used to determine the confidence intervals around $R^2_{trial}$ , $R_{trial}$ , $R^2_{indiv}$ and $R_{indiv}$ . Default $0.05$ .
`T0T1`	A scalar or vector that contains the correlation(s) between the counterfactuals T0 and T1 that should be considered in the computation of $\rho_{\Delta}$ (ICA). For details, see function `ICA.ContCont`. Default `seq(-1, 1, by=.2)`.
`T0S1`	A scalar or vector that contains the correlation(s) between the counterfactuals T0 and S1 that should be considered in the computation of $\rho_{\Delta}$ . Default `seq(-1, 1, by=.2)`.
`T1S0`	A scalar or vector that contains the correlation(s) between the counterfactuals T1 and S0 that should be considered in the computation of $\rho_{\Delta}$ . Default `seq(-1, 1, by=.2)`.
`S0S1`	A scalar or vector that contains the correlation(s) between the counterfactuals S0 and S1 that should be considered in the computation of $\rho_{\Delta}$ . Default `seq(-1, 1, by=.2)`.

Details

When the full bivariate mixed-effects model is fitted to assess surrogacy in the meta-analytic framework (for details, Buyse & Molenberghs, 2000), computational issues often occur. In that situation, the use of simplified model-fitting strategies may be warranted (for details, see Burzykowski et al., 2005; Tibaldi et al., 2003).

The function BifixedContCont implements one such strategy, i.e., it uses a two-stage bivariate fixed-effects modelling approach to assess surrogacy. In the first stage of the analysis, a bivariate linear regression model is fitted. When a full or semi-reduced model is requested (by using the argument Model=c("Full") or Model=c("SemiReduced") in the function call), the following bivariate model is fitted:

$S_{ij}=\mu_{Si}+\alpha_{i}Z_{ij}+\varepsilon_{Sij},$

$T_{ij}=\mu_{Ti}+\beta_{i}Z_{ij}+\varepsilon_{Tij},$

where $S_{ij}$ and $T_{ij}$ are the surrogate and true endpoint values of subject $j$ in trial $i$ , $Z_{ij}$ is the treatment indicator for subject $j$ in trial $i$ , $\mu_{Si}$ and $\mu_{Ti}$ are the fixed trial-specific intercepts for S and T, and $\alpha_{i}$ and $\beta_{i}$ are the trial-specific treatment effects on S and T, respectively. When a reduced model is requested (by using the argument Model=c("Reduced") in the function call), the following bivariate model is fitted:

$S_{ij}=\mu_{S}+\alpha_{i}Z_{ij}+\varepsilon_{Sij},$

$T_{ij}=\mu_{T}+\beta_{i}Z_{ij}+\varepsilon_{Tij},$

where $\mu_{S}$ and $\mu_{T}$ are the common intercepts for S and T (i.e., it is assumed that the intercepts for the surrogate and the true endpoints are identical in all trials). The other parameters are the same as defined above.

In the above models, the error terms $\varepsilon_{Sij}$ and $\varepsilon_{Tij}$ are assumed to be mean-zero normally distributed with variance-covariance matrix $\bold{\Sigma}$ :

$\bold{\Sigma}=\left(\begin{array}{cc}\sigma_{SS}\\\sigma_{ST} & \sigma_{TT}\end{array}\right).$

Based on $\bold{\Sigma}$ , individual-level surrogacy is quantified as:

$R_{indiv}^{2}=\frac{\sigma_{ST}^{2}}{\sigma_{SS}\sigma_{TT}}.$

Next, the second stage of the analysis is conducted. When a full model is requested by the user (by using the argument Model=c("Full") in the function call), the following model is fitted:

$\widehat{\beta_{i}}=\lambda_{0}+\lambda_{1}\widehat{\mu_{Si}}+\lambda_{2}\widehat{\alpha_{i}}+\varepsilon_{i},$

where the parameter estimates for $\beta_i$ , $\mu_{Si}$ , and $\alpha_i$ are based on the full model that was fitted in stage 1.

When a reduced or semi-reduced model is requested by the user (by using the arguments Model=c("Reduced") or Model=c("SemiReduced") in the function call), the following model is fitted:

$\widehat{\beta_{i}}=\lambda_{0}+\lambda_{1}\widehat{\alpha_{i}}+\varepsilon_{i}.$

where the parameter estimates for $\beta_i$ and $\alpha_i$ are based on the semi-reduced or reduced model that was fitted in stage 1.

When the argument Weighted=FALSE is used in the function call, the model that is fitted in stage 2 is an unweighted linear regression model. When a weighted model is requested (using the argument Weighted=TRUE in the function call), the information that is obtained in stage 1 is weighted according to the number of patients in a trial.

The classical coefficient of determination of the fitted stage 2 model provides an estimate of $R^2_{trial}$ .

Value

An object of class BifixedContCont with components,

`Data.Analyze`	Prior to conducting the surrogacy analysis, data of patients who have a missing value for the surrogate and/or the true endpoint are excluded. In addition, the data of trials (i) in which only one type of the treatment was administered, and (ii) in which either the surrogate or the true endpoint was a constant (i.e., all patients within a trial had the same surrogate and/or true endpoint value) are excluded. In addition, the user can specify the minimum number of patients that a trial should contain in order to include the trial in the analysis. If the number of patients in a trial is smaller than the value specified by `Min.Trial.Size`, the data of the trial are excluded. `Data.Analyze` is the dataset on which the surrogacy analysis was conducted.
`Obs.Per.Trial`	A `data.frame` that contains the total number of patients per trial and the number of patients who were administered the control treatment and the experimental treatment in each of the trials (in `Data.Analyze`).
`Results.Stage.1`	The results of stage 1 of the two-stage model fitting approach: a `data.frame` that contains the trial-specific intercepts and treatment effects for the surrogate and the true endpoints (when a full or semi-reduced model is requested), or the trial-specific treatment effects for the surrogate and the true endpoints (when a reduced model is requested).
`Residuals.Stage.1`	A `data.frame` that contains the residuals for the surrogate and true endpoints that are obtained in stage 1 of the analysis ( $\varepsilon_{Sij}$ and $\varepsilon_{Tij}$ ).
`Results.Stage.2`	An object of class `lm` (linear model) that contains the parameter estimates of the regression model that is fitted in stage 2 of the analysis.
`Trial.R2`	A `data.frame` that contains the trial-level coefficient of determination ( $R^2_{trial}$ ), its standard error and confidence interval.
`Indiv.R2`	A `data.frame` that contains the individual-level coefficient of determination ( $R^2_{indiv}$ ), its standard error and confidence interval.
`Trial.R`	A `data.frame` that contains the trial-level correlation coefficient ( $R_{trial}$ ), its standard error and confidence interval.
`Indiv.R`	A `data.frame` that contains the individual-level correlation coefficient ( $R_{indiv}$ ), its standard error and confidence interval.
`Cor.Endpoints`	A `data.frame` that contains the correlations between the surrogate and the true endpoint in the control treatment group (i.e., $\rho_{T0S0}$ ) and in the experimental treatment group (i.e., $\rho_{T1S1}$ ), their standard errors and their confidence intervals.
`D.Equiv`	The variance-covariance matrix of the trial-specific intercept and treatment effects for the surrogate and true endpoints (when a full or semi-reduced model is fitted, i.e., when `Model=c("Full")` or `Model=c("SemiReduced")` is used in the function call), or the variance-covariance matrix of the trial-specific treatment effects for the surrogate and true endpoints (when a reduced model is fitted, i.e., when `Model=c("Reduced")` is used in the function call). The variance-covariance matrix `D.Equiv` is equivalent to the $\bold{D}$ matrix that would be obtained when a (full or reduced) bivariate mixed-effect approach is used; see function `BimixedContCont`).
`Sigma`	The $2$ by $2$ variance-covariance matrix of the residuals ( $\varepsilon_{Sij}$ and $\varepsilon_{Tij}$ ).
`ICA`	A fitted object of class `ICA.ContCont`.
`T0T0`	The variance of the true endpoint in the control treatment condition.
`T1T1`	The variance of the true endpoint in the experimental treatment condition.
`S0S0`	The variance of the surrogate endpoint in the control treatment condition.
`S1S1`	The variance of the surrogate endpoint in the experimental treatment condition.

Author(s)

Wim Van der Elst, Ariel Alonso, & Geert Molenberghs

References

Burzykowski, T., Molenberghs, G., & Buyse, M. (2005). The evaluation of surrogate endpoints. New York: Springer-Verlag.

Buyse, M., Molenberghs, G., Burzykowski, T., Renard, D., & Geys, H. (2000). The validation of surrogate endpoints in meta-analysis of randomized experiments. Biostatistics, 1, 49-67.

Tibaldi, F., Abrahantes, J. C., Molenberghs, G., Renard, D., Burzykowski, T., Buyse, M., Parmar, M., et al., (2003). Simplified hierarchical linear models for the evaluation of surrogate endpoints. Journal of Statistical Computation and Simulation, 73, 643-658.

Examples

## Not run:  # time consuming code part
# Example 1, based on the ARMD data
data(ARMD)

# Fit a full bivariate fixed-effects model with weighting according to the  
# number of patients in stage 2 of the two stage approach to assess surrogacy:
Sur <- BifixedContCont(Dataset=ARMD, Surr=Diff24, True=Diff52, Treat=Treat, Trial.ID=Center, 
Pat.ID=Id, Model="Full", Weighted=TRUE)

# Obtain a summary of the results
summary(Sur)

# Obtain a graphical representation of the trial- and individual-level surrogacy
plot(Sur)


# Example 2
# Conduct a surrogacy analysis based on a simulated dataset with 2000 patients, 
# 100 trials, and Rindiv=Rtrial=.8
# Simulate the data:
Sim.Data.MTS(N.Total=2000, N.Trial=100, R.Trial.Target=.8, R.Indiv.Target=.8,
Seed=123, Model="Reduced")

# Fit a reduced bivariate fixed-effects model with no weighting according to the 
# number of patients in stage 2 of the two stage approach to assess surrogacy:
\dontrun{ #time-consuming code parts
Sur2 <- BifixedContCont(Dataset=Data.Observed.MTS, Surr=Surr, True=True, Treat=Treat, 
Trial.ID=Trial.ID, Pat.ID=Pat.ID, , Model="Reduced", Weighted=FALSE)

# Show summary and plots of results:
summary(Sur2)
plot(Sur2, Weighted=FALSE)}

## End(Not run)
## Not run:  # time consuming code part
# Example 1, based on the ARMD data
data(ARMD)

# Fit a full bivariate fixed-effects model with weighting according to the  
# number of patients in stage 2 of the two stage approach to assess surrogacy:
Sur <- BifixedContCont(Dataset=ARMD, Surr=Diff24, True=Diff52, Treat=Treat, Trial.ID=Center, 
Pat.ID=Id, Model="Full", Weighted=TRUE)

# Obtain a summary of the results
summary(Sur)

# Obtain a graphical representation of the trial- and individual-level surrogacy
plot(Sur)


# Example 2
# Conduct a surrogacy analysis based on a simulated dataset with 2000 patients, 
# 100 trials, and Rindiv=Rtrial=.8
# Simulate the data:
Sim.Data.MTS(N.Total=2000, N.Trial=100, R.Trial.Target=.8, R.Indiv.Target=.8,
Seed=123, Model="Reduced")

# Fit a reduced bivariate fixed-effects model with no weighting according to the 
# number of patients in stage 2 of the two stage approach to assess surrogacy:
\dontrun{ #time-consuming code parts
Sur2 <- BifixedContCont(Dataset=Data.Observed.MTS, Surr=Surr, True=True, Treat=Treat, 
Trial.ID=Trial.ID, Pat.ID=Pat.ID, , Model="Reduced", Weighted=FALSE)

# Show summary and plots of results:
summary(Sur2)
plot(Sur2, Weighted=FALSE)}

## End(Not run)

Fits a bivariate mixed-effects model using the cluster-by-cluster (CbC) estimator to assess surrogacy in the meta-analytic multiple-trial setting (Continuous-continuous case)

Description

The function BimixedCbCContCont uses the cluster-by-cluster (CbC) estimator of the bivariate mixed-effects to estimate trial- and individual-level surrogacy when the data of multiple clinical trials are available. See the Details section below.

Usage

BimixedCbCContCont(Dataset, Surr, True, Treat, Trial.ID,Min.Treat.Size=2,Alpha=0.05)
BimixedCbCContCont(Dataset, Surr, True, Treat, Trial.ID,Min.Treat.Size=2,Alpha=0.05)

Arguments

`Dataset`	A `data.frame` that should consist of one line per patient. Each line contains (at least) a surrogate value, a true endpoint value, a treatment indicator, and a trial ID.
`Surr`	The name of the variable in `Dataset` that contains the surrogate endpoint values.
`True`	The name of the variable in `Dataset` that contains the true endpoint values.
`Treat`	The name of the variable in `Dataset` that contains the treatment indicators. The treatment indicator should either be coded as $1$ for the experimental group and $-1$ for the control group, or as $1$ for the experimental group and $0$ for the control group.
`Trial.ID`	The name of the variable in `Dataset` that contains the trial ID to which the patient belongs.
`Min.Treat.Size`	The minimum number of patients in each group (control or experimental) that a trial should contain to be included in the analysis. If the number of patients in a group of a trial is smaller than the value specified by `Min.Treat.Size`, the data of the trial are excluded from the analysis. Default $2$ .
`Alpha`	The $\alpha$ -level that is used to determine the confidence intervals around $R^2_{trial}$ and $R^2_{indiv}$ . Default $0.05$ .

Details

The function BimixedContCont fits a bivariate mixed-effects model using the CbC estimator (for details, see Florez et al., 2019) to assess surrogacy (for details, see Buyse et al., 2000). In particular, the following mixed-effects model is fitted:

$S_{ij}=\mu_{S}+m_{Si}+(\alpha+a_{i})Z_{ij}+\varepsilon_{Sij},$

$T_{ij}=\mu_{T}+m_{Ti}+(\beta+b_{i})Z_{ij}+\varepsilon_{Tij},$

where $i$ and $j$ are the trial and subject indicators, $S_{ij}$ and $T_{ij}$ are the surrogate and true endpoint values of subject $j$ in trial $i$ , $Z_{ij}$ is the treatment indicator for subject $j$ in trial $i$ , $\mu_{S}$ and $\mu_{T}$ are the fixed intercepts for S and T, $m_{Si}$ and $m_{Ti}$ are the corresponding random intercepts, $\alpha$ and $\beta$ are the fixed treatment effects for S and T, and $a_{i}$ and $b_{i}$ are the corresponding random treatment effects, respectively.

The vector of the random effects (i.e., $m_{Si}$ , $m_{Ti}$ , $a_{i}$ and $b_{i}$ ) is assumed to be mean-zero normally distributed with variance-covariance matrix $\bold{D}$ :

$\bold{D}=\left(\begin{array}{cccc} d_{SS}\\ d_{ST} & d_{TT}\\ d_{Sa} & d_{Ta} & d_{aa}\\ d_{Sb} & d_{Tb} & d_{ab} & d_{bb} \end{array}\right).$

The trial-level coefficient of determination (i.e., $R^2_{trial}$ ) is quantified as:

$R_{trial}^{2}=\frac{\left(\begin{array}{c} d_{Sb}\\ d_{ab} \end{array}\right)^{'}\left(\begin{array}{cc} d_{SS} & d_{Sa}\\ d_{Sa} & d_{aa} \end{array}\right)^{-1}\left(\begin{array}{c} d_{Sb}\\ d_{ab} \end{array}\right)}{d_{bb}}.$

The error terms $\varepsilon_{Sij}$ and $\varepsilon_{Tij}$ are assumed to be mean-zero normally distributed with variance-covariance matrix $\bold{\Sigma}$ :

$\bold{\Sigma}=\left(\begin{array}{cc}\sigma_{SS}\\\sigma_{ST} & \sigma_{TT}\end{array}\right).$

Based on $\bold{\Sigma}$ , individual-level surrogacy is quantified as:

$R_{indiv}^{2}=\frac{\sigma_{ST}^{2}}{\sigma_{SS}\sigma_{TT}}.$

Note The CbC estimator for the full bivariate mixed-effects model is closed-form (for details, see Florez et al., 2019). Therefore, it is fast. Furthermore, it is recommended when computational issues occur with the full maximum likelihood estimator (implemented in function BimixedContCont).

The CbC estimator is performed in two stages: (1) a linear model is fitted in each trial. Evidently, it is require that the design matrix ( $X_i$ ) is full column rank within each trial, allowing estimation of the fixed effects. When $X_i$ is not full rank, trial i is excluded from the analysis. (2) a global estimator of the fixed effects ( $\beta$ ) is obtained by weighted averaging the sets of estimates of each trial, and $\bold{D}$ is estimated using a method-of-moments estimator. Optimal weights (for details, see Molenberghs et al., 2018) are used as a weighting scheme.

The estimator of $\bold{D}$ might lead to a non-positive-definite solution. Therefore, the eigenvalue method (for details, see Rousseeuw and Molenberghs, 1993) is used for non-positive-definiteness adjustment.

Value

An object of class BimixedContCont with components,

`Obs.Per.Trial`	A `data.frame` that contains the total number of patients per trial and the number of patients who were administered the control treatment and the experimental treatment in each of the trials (after excluding clusters). Clusters are excluded for two reasons: (i) the number of patients is smaller than the value especified by `Min.Trial.Size`, and (ii) the design matrix ( $X_i$ ) is not full rank.
`Trial.removed`	Number of trials excluded from the analysis
`Fixed.Effects`	A `data.frame` that contains the fixed intercept and treatment effects for the true and the surrogate endpoints (i.e., $\mu_{S}$ , $\mu_{T}$ , $\alpha$ , and $\beta$ ) and their corresponding standard error.
`Trial.R2`	A `data.frame` that contains the trial-level coefficient of determination ( $R^2_{trial}$ ), its standard error and confidence interval.
`Indiv.R2`	A `data.frame` that contains the individual-level coefficient of determination ( $R^2_{indiv}$ ), its standard error and confidence interval.
`D`	The variance-covariance matrix of the random effects (the $\bold{D}$ matrix), i.e., a $4$ by $4$ variance-covariance matrix of the random intercept and treatment effects.
`DH.pd`	`DH.pd=TRUE` if an adjustment for non-positive definiteness was not needed to estimate $\bold{D}$ . `DH.pd=FALSE` if this adjustment was required.
`Sigma`	The $2$ by $2$ variance-covariance matrix of the residuals ( $\varepsilon_{Sij}$ and $\varepsilon_{Tij}$ ).

Author(s)

Alvaro J. Florez, Wim Van der Elst, Ariel Alonso, & Geert Molenberghs

References

Buyse, M., Molenberghs, G., Burzykowski, T., Renard, D., & Geys, H. (2000). The validation of surrogate endpoints in meta-analysis of randomized experiments. Biostatistics, 1, 49-67.

Florez, A. J., Molenberghs G, Verbeke G, Alonso, A. (2019). A closed-form estimator for meta-analysis and surrogate markers evaluation. Journal of Biopharmaceutical Statistics, 29(2) 318-332.

Molenberghs, G., Hermans, L., Nassiri, V., Kenward, M., Van der Elst, W., Aerts, M. and Verbeke, G. (2018). Clusters with random size: maximum likelihood versus weighted estimation. Statistica Sinica, 28, 1107-1132.

Rousseeuw, P. J. and Molenberghs, G. (1993) Transformation of non positive semidefinite correlation matrices. Communications in Statistics, Theory and Methods, 22, 965-984.

Examples

# Open the Schizo dataset (clinial trial in schizophrenic patients)
data(Schizo)

# Fit a full bivariate random-effects model by the cluster-by-cluster (CbC) estimator
# a minimum of 2 subjects per group are allowed in each trial
  fit <- BimixedCbCContCont(Dataset=Schizo, Surr=BPRS, True=PANSS, Treat=Treat,Trial.ID=InvestId,
                              Alpha=0.05, Min.Treat.Size = 10)
# Note that an adjustment for non-positive definiteness was requiered and 113 trials were removed.

# Obtain a summary of the results
summary(fit)
                         
# Open the Schizo dataset (clinial trial in schizophrenic patients)
data(Schizo)

# Fit a full bivariate random-effects model by the cluster-by-cluster (CbC) estimator
# a minimum of 2 subjects per group are allowed in each trial
  fit <- BimixedCbCContCont(Dataset=Schizo, Surr=BPRS, True=PANSS, Treat=Treat,Trial.ID=InvestId,
                              Alpha=0.05, Min.Treat.Size = 10)
# Note that an adjustment for non-positive definiteness was requiered and 113 trials were removed.

# Obtain a summary of the results
summary(fit)

Fits a bivariate mixed-effects model to assess surrogacy in the meta-analytic multiple-trial setting (Continuous-continuous case)

Description

The function BimixedContCont uses the bivariate mixed-effects approach to estimate trial- and individual-level surrogacy when the data of multiple clinical trials are available. The user can specify whether a full or reduced model should be fitted. See the Details section below. Further, the Individual Causal Association (ICA) is computed.

Usage

BimixedContCont(Dataset, Surr, True, Treat, Trial.ID, Pat.ID, Model=c("Full"), 
Min.Trial.Size=2, Alpha=.05, T0T1=seq(-1, 1, by=.2), T0S1=seq(-1, 1, by=.2), 
T1S0=seq(-1, 1, by=.2), S0S1=seq(-1, 1, by=.2), ...)
BimixedContCont(Dataset, Surr, True, Treat, Trial.ID, Pat.ID, Model=c("Full"), 
Min.Trial.Size=2, Alpha=.05, T0T1=seq(-1, 1, by=.2), T0S1=seq(-1, 1, by=.2), 
T1S0=seq(-1, 1, by=.2), S0S1=seq(-1, 1, by=.2), ...)

Arguments

`Dataset`	A `data.frame` that should consist of one line per patient. Each line contains (at least) a surrogate value, a true endpoint value, a treatment indicator, a patient ID, and a trial ID.
`Surr`	The name of the variable in `Dataset` that contains the surrogate endpoint values.
`True`	The name of the variable in `Dataset` that contains the true endpoint values.
`Treat`	The name of the variable in `Dataset` that contains the treatment indicators. The treatment indicator should either be coded as $1$ for the experimental group and $-1$ for the control group, or as $1$ for the experimental group and $0$ for the control group.
`Trial.ID`	The name of the variable in `Dataset` that contains the trial ID to which the patient belongs.
`Pat.ID`	The name of the variable in `Dataset` that contains the patient's ID.
`Model`	The type of model that should be fitted, i.e., `Model=c("Full")` or `Model=c("Reduced")`. See the Details section below. Default `Model=c("Full")`.
`Min.Trial.Size`	The minimum number of patients that a trial should contain to be included in the analysis. If the number of patients in a trial is smaller than the value specified by `Min.Trial.Size`, the data of the trial are excluded from the analysis. Default $2$ .
`Alpha`	The $\alpha$ -level that is used to determine the confidence intervals around $R^2_{trial}$ , $R_{trial}$ , $R^2_{indiv}$ and $R_{indiv}$ . Default $0.05$ .
`T0T1`	A scalar or vector that contains the correlation(s) between the counterfactuals T0 and T1 that should be considered in the computation of $\rho_{\Delta}$ (ICA). For details, see function `ICA.ContCont`. Default `seq(-1, 1, by=.2)`.
`T0S1`	A scalar or vector that contains the correlation(s) between the counterfactuals T0 and S1 that should be considered in the computation of $\rho_{\Delta}$ . Default `seq(-1, 1, by=.2)`.
`T1S0`	A scalar or vector that contains the correlation(s) between the counterfactuals T1 and S0 that should be considered in the computation of $\rho_{\Delta}$ . Default `seq(-1, 1, by=.2)`.
`S0S1`	A scalar or vector that contains the correlation(s) between the counterfactuals S0 and S1 that should be considered in the computation of $\rho_{\Delta}$ . Default `seq(-1, 1, by=.2)`.
`...`	Other arguments to be passed to the function `lmer` (of the R package `lme4`) that is used to fit the geralized linear mixed-effect models in the function `BimixedContCont`.

Details

The function BimixedContCont fits a bivariate mixed-effects model to assess surrogacy (for details, see Buyse et al., 2000). In particular, the following mixed-effects model is fitted: