Single.Trial.ContCont: Conducts a surrogacy analysis in the single-trial setting for...
In Surrogate: Evaluation of Surrogate Endpoints in Clinical Trials
View source: R/Single.Trial.ContCont.R
Single.Trial.ContCont R Documentation
Conducts a surrogacy analysis in the single-trial setting for Continuous S and T
Description
The function Single.Trial.ContCont conducts a surrogacy analysis in the single-trial setting for Continuous S and T. See Details below.
Usage
Single.Trial.ContCont(Dataset, Surr, True, Treat, Pat.ID, Alpha=.05,
Number.Bootstraps=500, Seed=12345)
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 patient ID.
Surr
The name of the variable in Dataset that contains the surrogate 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. The -1/1 coding is recommended.
Pat.ID
The name of the variable in Dataset that contains the patient's ID.
Alpha
The \alpha-level that is used to determine the confidence intervals around Alpha (which is a parameter estimate of a model where the surrogate is regressed on the treatment indicator, see Details below), Beta, RE, and \gamma. Default 0.05.
Number.Bootstraps
The number of bootstrap samples that are used to obtain the bootstrap-based confidence intervals for RE, PE and the adjusted association (\gamma). Default 500. A non-parametric bootstrap is used.
Seed
The seed that is used to generate the bootstrap samples. Default Seed=12345.
Details
Prentice criteria
The Prentice criteria are examined by fitting the following regression models:
S_{j}=\mu_{S}+\alpha Z_{j}+\varepsilon_{Sj}, (1)
T_{j}=\mu_{T}+\beta Z_{j}+\varepsilon_{Tj}, (2)
T_{j}=\mu+\gamma Z_{j}+\varepsilon_{j}, (3)
T_{j}=\tilde{\mu}_{T}+\beta_{S} Z_{j}+\gamma_{Z} S_{j}+\tilde{\varepsilon}_{Tj}, (4)
where the error terms of (1) and (2) have a joint zero-mean normal distribution with variance-covariance matrix
\boldsymbol{\Sigma}=\left(\begin{array}{cc}
\sigma_{SS}\\
\sigma_{ST} & \sigma_{TT}
\end{array}\right)
,
and where j is the subject indicator, S_{j} and T_{j} are the surrogate and true endpoint values of subject j, and Z_{j} is the treatment indicator for subject j.
To be in line with the Prentice criteria, Z should have a significant effect on S in model 1 (Prentice criterion 1), Z should have a significant effect on T in model 2 (Prentice criterion 2), S should have a significant effect on T in model 3 (Prentice criterion criterion 3), and the effect of Z on T should be fully captured by S in model 4 (Prentice criterion 4).
Proportion Explained
In view of the problems with the Prentice criteria (see the Limitations section below), Freedman et al. (1992) proposed to quantify surrogacy as the proportion of the effect of the treatment on T that is explained by S (the proportion explained, PE)
PE(T,\: S,\: Z)=\dfrac{\beta-\beta_{S}}{\beta}=1-\dfrac{\beta_{S}}{\beta},
where \beta is the estimate of the effect of the treatment on T without correction for S and \beta_{S} is the estimate of the effect of the treatment on T with correction for S (see above). The intuition behind the PE is that, if all treatment effect is mediated by S (i.e., if \beta_{S}=0), then PE=1. On the other hand, if there is no mediation at all (i.e., if \beta=\beta_{S}), then PE=0. Note that the fourth Prentice criterion is equivalent to the requirement that PE=1.
Relative Effect and Adjusted Association
The Relative Effect (RE) and the adjusted association (\gamma) are based on the bivariate regression model (1) and (2) and its variance-covariance matrix
\boldsymbol{\Sigma}. The parameter estimates of the fitted regression model and the variance-covariance matrix of the residuals are used to compute RE and the adjusted association (\gamma), respectively:
RE=\frac{\beta}{\alpha},
\gamma=\frac{\sigma_{ST}}{\sqrt{\sigma_{SS}\sigma_{TT}}}.
Limitations
The Prentice approach to assess surrogacy has some fundamental limitations. For example, the fourth Prentice criterion requires that the statistical test for the \beta_S in model 4 is non-significant. This criterion is useful to reject a poor surrogate, but it is not suitable to validate a good surrogate (i.e., a non-significant result may always be attributable to a lack of statistical power). Even when lack of power would not be an issue, the result of the statistical test to evaluate the fourth Prentice criterion cannot prove that the effect of the treatment on the true endpoint is fully captured by the surrogate.
In general, the single-trial framework is hampered by a number of issues (Burzykowski et al., 2005). For example, a key motivation to validate a surrogate endpoint is to be able to predict the effect of Z on T as based on the effect of Z on S in a new clinical trial where T is not (yet) observed. The RE allows for such a prediction, but this requires the assumption that the relation between \alpha and \beta can be described by a linear regression model that goes through the origin. In other words, it has to be assumed that the RE remains constant across clinical trials. The constant RE assumption is unverifiable in a single-trial setting, but a way out of this problem is to combine the information of multiple clinical trials and generalize the RE concept to a multiple-trial setting (as is done in the multiple-trial meta-analytic approach, see UnifixedContCont, BifixedContCont, UnimixedContCont, and BimixedContCont).
Value
An object of class Single.Trial.ContCont 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. Data.Analyze is the dataset on which the surrogacy analysis was conducted.
Prentice.Model.1
An object of class lm that contains the fitted model 1 (using the Prentice approach).
Prentice.Model.2
An object of class lm that contains the fitted model 2 (using the Prentice approach).
Prentice.Model.3
An object of class lm that contains the fitted model 3 (using the Prentice approach).
Prentice.Model.4
An object of class lm that contains the fitted model 4 (using the Prentice approach).
Prentice.Passed
Logical. If all four Prentice criteria are fulfilled, Prentice.Passed=TRUE. If at least one criterion is not fulfilled, Prentice.Passed=FALSE.
Alpha
An object of class data.frame that contains the parameter estimate for \alpha, its standard error, and its confidence interval. Note that Alpha is not to be confused with the Alpha argument in the function call, which specifies the \alpha-level of the confidence intervals of the parameters.
Beta
An object of class data.frame that contains the parameter estimate for \beta, its standard error, and its confidence interval.
RE.Delta
An object of class data.frame that contains the estimated RE, its standard error, and its confidence interval (based on the Delta method).
RE.Boot
An object of class data.frame that contains the estimated RE, its standard error, and its confidence interval (based on a non-parametric bootstra).
AA
An object of class data.frame that contains the adjusted association (i.e., \gamma), its standard error, and its confidence interval (based on the Fisher-Z transformation procedure).
AA.Boot
An object of class data.frame that contains the adjusted association (i.e., \gamma), its standard error, and its confidence interval (based on a bootstrap procedure).
RE.Boot.Samples
A vector that contains the RE values that were generated during the bootstrap procedure.
AA.Boot.Samples
A vector that contains the adjusted association (i.e., \gamma) values that were generated during the bootstrap procedure.
Cor.Endpoints
A data.frame that contains the correlations between the surrogate and the true endpoint in the control treatment group (i.e., \rho_{T0T1}) and in the experimental treatment group (i.e., \rho_{T1S1}), their standard errors and their confidence intervals.
Residuals
A data.frame that contains the residuals for the surrogate and true endpoints that are obtained when the surrogate and the true endpoint are regressed on the treatment indicator.
Beta_S
An object of class data.frame that contains the parameter estimate for \beta_S, its standard error, and its confidence interval.
PE.Delta
An object of class data.frame that contains the estimated PE, its standard error, and its confidence interval (based on the Delta method).
PE.Boot
An object of class data.frame that contains the estimated PE, its standard error, and its confidence interval (based on a bootstrap).
PE.Boot.Samples
A vector that contains the PE values that were generated during the bootstrap procedure.
PE.Fieller
An object of class data.frame that contains the estimated PE and its confidence interval (based on the Fieller approach).
RE.Fieller
An object of class data.frame that contains the estimated RE and its confidence interval (based on the Fieller approach).
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. (1998). The validation of surrogate endpoints in randomized experiments. Biometrics, 54, 1014-1029.
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.
See Also
UnifixedContCont, BifixedContCont, UnimixedContCont, BimixedContCont, ICA.ContCont
Examples
## Not run: # time consuming code part
# Example 1, based on the ARMD data:
data(ARMD)
# Assess surrogacy based on the single-trial meta-analytic approach:
Sur <- Single.Trial.ContCont(Dataset=ARMD, Surr=Diff24, True=Diff52, Treat=Treat, Pat.ID=Id)
# Obtain a summary and plot of the results
summary(Sur)
plot(Sur)
# Example 2
# Conduct an analysis based on a simulated dataset with 2000 patients
# and Rindiv=.8
# Simulate the data:
Sim.Data.STS(N.Total=2000, R.Indiv.Target=.8, Seed=123)
# Assess surrogacy:
Sur2 <- Single.Trial.ContCont(Dataset=Data.Observed.STS, Surr=Surr, True=True, Treat=Treat,
Pat.ID=Pat.ID)
# Show a summary and plots of results
summary(Sur2)
plot(Sur2)
## End(Not run)
Surrogate documentation built on Oct. 16, 2026, 9:07 a.m.
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View source: R/Single.Trial.ContCont.R
| Single.Trial.ContCont | R Documentation |
Conducts a surrogacy analysis in the single-trial setting for Continuous S and T
Description
The function Single.Trial.ContCont conducts a surrogacy analysis in the single-trial setting for Continuous S and T. See Details below.
Usage
Single.Trial.ContCont(Dataset, Surr, True, Treat, Pat.ID, Alpha=.05,
Number.Bootstraps=500, Seed=12345)
Arguments
Dataset |
A |
Surr |
The name of the variable in |
True |
The name of the variable in |
Treat |
The name of the variable in |
Pat.ID |
The name of the variable in |
Alpha |
The |
Number.Bootstraps |
The number of bootstrap samples that are used to obtain the bootstrap-based confidence intervals for RE, PE and the adjusted association ( |
Seed |
The seed that is used to generate the bootstrap samples. Default |
Details
Prentice criteria The Prentice criteria are examined by fitting the following regression models:
S_{j}=\mu_{S}+\alpha Z_{j}+\varepsilon_{Sj}, (1)
T_{j}=\mu_{T}+\beta Z_{j}+\varepsilon_{Tj}, (2)
T_{j}=\mu+\gamma Z_{j}+\varepsilon_{j}, (3)
T_{j}=\tilde{\mu}_{T}+\beta_{S} Z_{j}+\gamma_{Z} S_{j}+\tilde{\varepsilon}_{Tj}, (4)
where the error terms of (1) and (2) have a joint zero-mean normal distribution with variance-covariance matrix
\boldsymbol{\Sigma}=\left(\begin{array}{cc}
\sigma_{SS}\\
\sigma_{ST} & \sigma_{TT}
\end{array}\right)
,
and where j is the subject indicator, S_{j} and T_{j} are the surrogate and true endpoint values of subject j, and Z_{j} is the treatment indicator for subject j.
To be in line with the Prentice criteria, Z should have a significant effect on S in model 1 (Prentice criterion 1), Z should have a significant effect on T in model 2 (Prentice criterion 2), S should have a significant effect on T in model 3 (Prentice criterion criterion 3), and the effect of Z on T should be fully captured by S in model 4 (Prentice criterion 4).
Proportion Explained
In view of the problems with the Prentice criteria (see the Limitations section below), Freedman et al. (1992) proposed to quantify surrogacy as the proportion of the effect of the treatment on T that is explained by S (the proportion explained, PE)
PE(T,\: S,\: Z)=\dfrac{\beta-\beta_{S}}{\beta}=1-\dfrac{\beta_{S}}{\beta},
where \beta is the estimate of the effect of the treatment on T without correction for S and \beta_{S} is the estimate of the effect of the treatment on T with correction for S (see above). The intuition behind the PE is that, if all treatment effect is mediated by S (i.e., if \beta_{S}=0), then PE=1. On the other hand, if there is no mediation at all (i.e., if \beta=\beta_{S}), then PE=0. Note that the fourth Prentice criterion is equivalent to the requirement that PE=1.
Relative Effect and Adjusted Association
The Relative Effect (RE) and the adjusted association (\gamma) are based on the bivariate regression model (1) and (2) and its variance-covariance matrix
\boldsymbol{\Sigma}. The parameter estimates of the fitted regression model and the variance-covariance matrix of the residuals are used to compute RE and the adjusted association (\gamma), respectively:
RE=\frac{\beta}{\alpha},
\gamma=\frac{\sigma_{ST}}{\sqrt{\sigma_{SS}\sigma_{TT}}}.
Limitations
The Prentice approach to assess surrogacy has some fundamental limitations. For example, the fourth Prentice criterion requires that the statistical test for the \beta_S in model 4 is non-significant. This criterion is useful to reject a poor surrogate, but it is not suitable to validate a good surrogate (i.e., a non-significant result may always be attributable to a lack of statistical power). Even when lack of power would not be an issue, the result of the statistical test to evaluate the fourth Prentice criterion cannot prove that the effect of the treatment on the true endpoint is fully captured by the surrogate.
In general, the single-trial framework is hampered by a number of issues (Burzykowski et al., 2005). For example, a key motivation to validate a surrogate endpoint is to be able to predict the effect of Z on T as based on the effect of Z on S in a new clinical trial where T is not (yet) observed. The RE allows for such a prediction, but this requires the assumption that the relation between \alpha and \beta can be described by a linear regression model that goes through the origin. In other words, it has to be assumed that the RE remains constant across clinical trials. The constant RE assumption is unverifiable in a single-trial setting, but a way out of this problem is to combine the information of multiple clinical trials and generalize the RE concept to a multiple-trial setting (as is done in the multiple-trial meta-analytic approach, see UnifixedContCont, BifixedContCont, UnimixedContCont, and BimixedContCont).
Value
An object of class Single.Trial.ContCont 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. |
Prentice.Model.1 |
An object of class |
Prentice.Model.2 |
An object of class |
Prentice.Model.3 |
An object of class |
Prentice.Model.4 |
An object of class |
Prentice.Passed |
|
Alpha |
An object of class |
Beta |
An object of class |
RE.Delta |
An object of class |
RE.Boot |
An object of class |
AA |
An object of class |
AA.Boot |
An object of class |
RE.Boot.Samples |
A vector that contains the RE values that were generated during the bootstrap procedure. |
AA.Boot.Samples |
A vector that contains the adjusted association (i.e., |
Cor.Endpoints |
A |
Residuals |
A |
Beta_S |
An object of class |
PE.Delta |
An object of class |
PE.Boot |
An object of class |
PE.Boot.Samples |
A vector that contains the PE values that were generated during the bootstrap procedure. |
PE.Fieller |
An object of class |
RE.Fieller |
An object of class |
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. (1998). The validation of surrogate endpoints in randomized experiments. Biometrics, 54, 1014-1029.
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.
See Also
UnifixedContCont, BifixedContCont, UnimixedContCont, BimixedContCont, ICA.ContCont
Examples
## Not run: # time consuming code part
# Example 1, based on the ARMD data:
data(ARMD)
# Assess surrogacy based on the single-trial meta-analytic approach:
Sur <- Single.Trial.ContCont(Dataset=ARMD, Surr=Diff24, True=Diff52, Treat=Treat, Pat.ID=Id)
# Obtain a summary and plot of the results
summary(Sur)
plot(Sur)
# Example 2
# Conduct an analysis based on a simulated dataset with 2000 patients
# and Rindiv=.8
# Simulate the data:
Sim.Data.STS(N.Total=2000, R.Indiv.Target=.8, Seed=123)
# Assess surrogacy:
Sur2 <- Single.Trial.ContCont(Dataset=Data.Observed.STS, Surr=Surr, True=True, Treat=Treat,
Pat.ID=Pat.ID)
# Show a summary and plots of results
summary(Sur2)
plot(Sur2)
## End(Not run)
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