Surrogate (version 2.0)

BifixedContCont: 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))

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 $R_{trial}^2$, $R_{trial}$, $R_{indiv}^2$ 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 ${\rho }_{\mathrm{\Delta }}$ (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 ${\rho }_{\mathrm{\Delta }}$. 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 ${\rho }_{\mathrm{\Delta }}$. 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 ${\rho }_{\mathrm{\Delta }}$. Default seq(-1, 1, by=.2).

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 (${\epsilon }_{Sij}$ and ${\epsilon }_{Tij}$).

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 ($R_{trial}^2$), its standard error and confidence interval.

Indiv.R2

A data.frame that contains the individual-level coefficient of determination ($R_{indiv}^2$), 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 $\text{bold}D$ 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 (${\epsilon }_{Sij}$ and ${\epsilon }_{Tij}$).

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.

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:

$$S_{ij}={\mu }_{Si}+{\alpha }_i{Z}_{ij}+{\epsilon }_{Sij},$$ $$T_{ij}={\mu }_{Ti}+{\beta }_i{Z}_{ij}+{\epsilon }_{Tij},$$

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:

$$S_{ij}={\mu }_S+{\alpha }_i{Z}_{ij}+{\epsilon }_{Sij},$$ $$T_{ij}={\mu }_T+{\beta }_i{Z}_{ij}+{\epsilon }_{Tij},$$

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 ${\epsilon }_{Sij}$ and ${\epsilon }_{Tij}$ are assumed to be mean-zero normally distributed with variance-covariance matrix $\text{bold}\mathrm{\Sigma }$:

$$\text{bold}\mathrm{\Sigma }=\left(\begin{array}{c}{\sigma }_{SS}\\ {\sigma }_{ST}& {\sigma }_{TT}\end{array}\right).$$

Based on $\text{bold}\mathrm{\Sigma }$, 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:

$$\widehat{{\beta }_i}={\lambda }_0+{\lambda }_1\widehat{{\mu }_{Si}}+{\lambda }_2\widehat{{\alpha }_i}+{\epsilon }_i,$$

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:

$$\widehat{{\beta }_i}={\lambda }_0+{\lambda }_1\widehat{{\alpha }_i}+{\epsilon }_i.$$

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 $R_{trial}^2$.

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.

See Also

UnifixedContCont, UnimixedContCont, BimixedContCont, plot Meta-Analytic

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)}
# }