Surrogate: 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.

Author(s)

Maintainer: Wim Van Der Elst wim.vanderelst@gmail.com

Authors:

• Florian Stijven florian.stijven@kuleuven.be

• Fenny Ong

• Dries De Witte

• Gokce Deliorman

• Paul Meyvisch

• Alvaro Poveda

• Ariel Alonso

• Hannah Ensor

• Christoper Weir

• Geert Molenberghs

See Also

Useful links:

• https://github.com/florianstijven/Surrogate-development

• Report bugs at https://github.com/florianstijven/Surrogate-development/issues

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)

Arguments

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,

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.

See Also

Single.Trial.RE.AA

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 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)

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)

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))

Arguments

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,

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.

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)}

## 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)

Arguments

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,

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.

See Also

BimixedContCont, UnifixedContCont, BifixedContCont, UnimixedContCont

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)
                         

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), ...)

Arguments

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:

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

When the full bivariate mixed-effects approach is used to assess surrogacy in the meta-analytic framework (for details, see Buyse & Molenberghs, 2000), computational issues often occur. Such problems mainly occur when the number of trials is low, the number of patients in the different trials is low, and/or when the trial-level heterogeneity is small (Burzykowski et al., 2000).

In that situation, the use of a simplified model-fitting strategy may be warranted (for details, see Burzykowski et al., 2000; Tibaldi et al., 2003).

For example, a reduced bivariate-mixed effect model can be fitted instead of a full model (by using the Model=c("Reduced") argument in the function call). In the reduced model, the random-effects structure is simplified (i) by assuming that there is no heterogeneity in the random intercepts, or (ii) by assuming that the covariance between the random intercepts and random treatment effects is zero. Note that under this assumption, the computation of the trial-level coefficient of determination (i.e., R^2_{trial}) simplifies to:

R_{trial}^{2}=\frac{d_{ab}^{2}}{d_{aa}d_{bb}}.

Alternatively, the bivariate mixed-effects model may be abandonned and the user may fit a univariate fixed-effects model, a bivariate fixed-effects model, or a univariate mixed-effects model (for details, see Tibaldi et al., 2003). These models are implemented in the functions UnifixedContCont, BifixedContCont, and UnimixedContCont).

Value

An object of class BimixedContCont with components,

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.

See Also

UnifixedContCont, BifixedContCont, UnimixedContCont, plot Meta-Analytic

Examples

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

## Not run:  #Time consuming (>5 sec) code part
# When a reduced bivariate mixed-effect model is used to assess surrogacy, 
# the conditioning number for the D matrix is very high: 
Sur <- BimixedContCont(Dataset=Schizo, Surr=BPRS, True=PANSS, Treat=Treat, Model="Reduced", 
Trial.ID=InvestId, Pat.ID=Id)

# Such problems often occur when the total number of patients, the total number 
# of trials and/or the trial-level heterogeneity
# of the treatment effects is relatively small

# As an alternative approach to assess surrogacy, consider using the functions
# BifixedContCont, UnifixedContCont or UnimixedContCont in the meta-analytic framework,
# or use the information-theoretic approach

## End(Not run)

Bootstrap 95% CI around the maximum-entropy ICA and SPF (surrogate predictive function)

Description

Computes a 95% bootstrap-based CI around the maximum-entropy ICA and SPF (surrogate predictive function) in the binary-binary setting

Usage

Bootstrap.MEP.BinBin(Data, Surr, True, Treat, M=100, Seed=123)

Arguments

Value

Author(s)

Wim Van der Elst, Ariel Alonso, & Geert Molenberghs

References

Alonso, A., & Van der Elst, W. (2015). A maximum-entropy approach for the evluation of surrogate endpoints based on causal inference.

See Also

ICA.BinBin, ICA.BinBin.Grid.Sample, ICA.BinBin.Grid.Full, plot MaxEntSPF BinBin

Examples

## Not run:  # time consuming code part
MEP_CI <- Bootstrap.MEP.BinBin(Data = Schizo_Bin, Surr = "BPRS_Bin", True = "PANSS_Bin",
                     Treat = "Treat", M = 500, Seed=123)
summary(MEP_CI)

## End(Not run)

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.

Description

This function provides a diagram that depicts the medians of the informational coefficients of correlation (or odds ratios) between the counterfactuals for a specified range of values of the individual causal association in the binary-binary setting (R_{H}^{2}).

Usage

CausalDiagramBinBin(x, Values="Corrs", Theta_T0S0, Theta_T1S1, 
Min=0, Max=1, Cex.Letters=3, Cex.Corrs=2, Lines.Rel.Width=TRUE, 
Col.Pos.Neg=TRUE, Monotonicity, Histograms.Correlations=FALSE, 
Densities.Correlations=FALSE)

Arguments

Value

The following components are stored in the fitted object if histograms of the informational correlations are requested in the function call (i.e., if Histograms.Correlations=TRUE and Values="Corrs" in the function call):

Author(s)

Wim Van der Elst, Ariel Alonso, & Geert Molenberghs

References

Alonso, A., Van der Elst, W., Molenberghs, G., Buyse, M., & Burzykowski, T. (submitted). On the relationship between the causal inference and meta-analytic paradigms for the validation of surrogate markers.

Van der Elst, W., Alonso, A., & Molenberghs, G. (submitted). An exploration of the relationship between causal inference and meta-analytic measures of surrogacy.

See Also

ICA.BinBin

Examples

# Compute R2_H given the marginals specified as the pi's
ICA <- ICA.BinBin.Grid.Sample(pi1_1_=0.2619048, pi1_0_=0.2857143, 
pi_1_1=0.6372549, pi_1_0=0.07843137, pi0_1_=0.1349206, pi_0_1=0.127451,
Seed=1, Monotonicity=c("General"), M=1000)

# Obtain a causal diagram that provides the medians of the 
# correlations between the counterfactuals for the range
# of R2_H values between 0.1 and 1
   # Assume no monotonicty 
CausalDiagramBinBin(x=ICA, Min=0.1, Max=1, Monotonicity="No") 

   # Assume monotonicty for S 
CausalDiagramBinBin(x=ICA, Min=0.1, Max=1, Monotonicity="Surr.Endp") 

# Now only consider the results that were obtained when 
# monotonicity was assumed for the true endpoint
CausalDiagramBinBin(x=ICA, Values="ORs", Theta_T0S0=2.156, Theta_T1S1=10, 
Min=0, Max=1,  Monotonicity="True.Endp") 

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

Description

This function provides a diagram that depicts the medians of the correlations between the counterfactuals for a specified range of values of the individual causal association (ICA; \rho_{\Delta}) or the meta-analytic individual causal association (MICA; \rho_{M}).

Usage

CausalDiagramContCont(x, Min=-1, Max=1, Cex.Letters=3, Cex.Corrs=2, 
Lines.Rel.Width=TRUE, Col.Pos.Neg=TRUE, Histograms.Counterfactuals=FALSE)

Arguments

Author(s)

Wim Van der Elst, Ariel Alonso, & Geert Molenberghs

References

Alonso, A., Van der Elst, W., Molenberghs, G., Buyse, M., & Burzykowski, T. (submitted). On the relationship between the causal inference and meta-analytic paradigms for the validation of surrogate markers.

Van der Elst, W., Alonso, A., & Molenberghs, G. (submitted). An exploration of the relationship between causal inference and meta-analytic measures of surrogacy.

See Also

ICA.ContCont, MICA.ContCont

Examples

## Not run:  #Time consuming (>5 sec) code parts
# Generate the vector of ICA values when rho_T0S0=.91, rho_T1S1=.91, and when the
# grid of values {0, .1, ..., 1} is considered for the correlations
# between the counterfactuals:
SurICA <- ICA.ContCont(T0S0=.95, T1S1=.91, T0T1=seq(0, 1, by=.1), T0S1=seq(0, 1, by=.1), 
T1S0=seq(0, 1, by=.1), S0S1=seq(0, 1, by=.1))

#obtain a plot of ICA

# Obtain a causal diagram that provides the medians of the 
# correlations between the counterfactuals for the range
# of ICA values between .9 and 1 (i.e., which assumed 
# correlations between the counterfactuals lead to a 
# high ICA?)
CausalDiagramContCont(SurICA, Min=.9, Max=1)

# Same, for low values of ICA
CausalDiagramContCont(SurICA, Min=0, Max=.5)
## End(Not run)

Confidence interval for the ICA given the unidentifiable parameters

Description

Dvine_ICA_confint() computes the confidence interval for the ICA in the D-vine copula model. The unidentifiable parameters are fixed at the user supplied values.

Usage

Dvine_ICA_confint(
  fitted_model,
  alpha,
  copula_par_unid,
  copula_family2,
  rotation_par_unid,
  n_prec,
  mutinfo_estimator = NULL,
  composite,
  B,
  seed
)

Arguments

Value

(numeric) Vector with the limits of the two-sided 1 - alpha confidence interval.

Apply the Entropy Concentration Theorem

Description

The Entropy Concentration Theorem (ECT; Edwin, 1982) states that if N is large enough, then 100(1-F)% of all \bold{p*} and \Delta H is determined by the upper tail are 1-F of a \chi^2 distribution, with DF = q - m - 1 (which equals 8 in a surrogate evaluation context).

Usage

ECT(Perc=.95, H_Max, N)

Arguments

Value

An object of class ECT with components,

Author(s)

Wim Van der Elst, Paul Meyvisch, & Ariel Alonso

References

Alonso, A., Van der Elst, W., & Molenberghs, G. (2016). Surrogate markers validation: the continuous-binary setting from a causal inference perspective.

See Also

MaxEntICABinBin, ICA.BinBin

Examples

ECT_fit <- ECT(Perc = .05, H_Max = 1.981811, N=454)
summary(ECT_fit)

Evaluate the possibility of finding a good surrogate in the setting where both S and T are binary endpoints

Description

The function Fano.BinBin evaluates the existence of a good surrogate in the single-trial causal-inference framework when both the surrogate and the true endpoints are binary outcomes. See Details below.

Usage

Fano.BinBin(pi1_,  pi_1, rangepi10=c(0,min(pi1_,1-pi_1)), 
fano_delta=c(0.1), M=100, Seed=1)

Arguments

Details

Values for \pi_{10} have to be uniformly sampled from the interval [0,\min(\pi_{1\cdot},\pi_{\cdot0})]. Any sampled value for \pi_{10} will fully determine the bivariate distribution of potential outcomes for the true endpoint. The treatment effect should be positive.

The vector \bold{\pi_{km}} fully determines R^2_{HL}.

Value

An object of class Fano.BinBin with components,

Author(s)

Paul Meyvisch, Wim Van der Elst, Ariel Alonso

References

Alonso, A., Van der Elst, W., & Molenberghs, G. (2014). Validation of surrogate endpoints: the binary-binary setting from a causal inference perspective.

See Also

plot.Fano.BinBin

Examples

# Conduct the analysis assuming no montonicity
# for the true endpoint, using a range of
# upper bounds for prediction errors 
Fano.BinBin(pi1_ = 0.5951 ,  pi_1 = 0.7745, 
fano_delta=c(0.05, 0.1, 0.2), M=1000)


# Conduct the same analysis now sampling from
# a range of values to allow for uncertainty

Fano.BinBin(pi1_ = runif(n=20,min=0.504,max=0.681), 
pi_1 = runif(n=20,min=0.679,max=0.849), 
fano_delta=c(0.05, 0.1, 0.2), M=10, Seed=2)

Fits the first stage model in the two-stage federated data analysis approach.

Description

The function 'FederatedApproachStage1()' fits the first stage model of the two-stage federated data analysis approach to assess surrogacy.

Usage

FederatedApproachStage1(
  Dataset,
  Surr,
  True,
  Treat,
  Trial.ID,
  Min.Treat.Size = 2,
  Alpha = 0.05
)

Arguments

Value

Returns an object of class "FederatedApproachStage1()" that can be used to evaluate surrogacy in the second stage model and contains the following elements:

• Results.Stage.1: a data frame that contains the estimated fixed effects and the elements of \Sigma_i.

• R.i: the variance-covariance matrix of the estimated fixed effects.

Model

The two-stage federated data analysis approach that can be used to assess surrogacy in the meta-analytic multiple-trial setting (Continuous-continuous case), but without the need of sharing data. Instead, each organization conducts separate analyses on their data set using a so-called "first stage" model. The results of these analyses are then aggregated at a central analysis hub, where the aggregated results are analyzed using a "second stage" model and the necessary metrics (R^2_{trial} and R^2_{indiv}) for the validation of the surrogate endpoint are obtained. This function fits the first stage model, where a linear model is fitted, allowing estimation of the fixed effects.

Data Format

The data frame must contain the following columns:

• a column with the true endpoint

• a column with the surrogate endpoint

• a column with the treatment indicator: 0 or 1

• a column with the trial indicator

• a column with the patient indicator

Author(s)

Dries De Witte

References

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

Examples

## Not run: 
#As an example, the federated data analysis approach can be applied to the Schizo data set
data(Schizo)
Schizo <-  Schizo[order(Schizo$InvestId, Schizo$Id),]
#Create separate datasets for each investigator
Schizo_datasets <- list()

for (invest_id in 1:198) {
Schizo_datasets[[invest_id]] <- Schizo[Schizo$InvestId == invest_id, ]
assign(paste0("Schizo", invest_id), Schizo_datasets[[invest_id]])
}
#Fit the first stage model for each dataset separately
results_stage1 <- list()
invest_ids <- list()
i <- 1
for (invest_id in 1:198) {
  dataset <- Schizo_datasets[[invest_id]]

  skip_to_next <- FALSE
  tryCatch(FederatedApproachStage1(dataset, Surr=CGI, True=PANSS, Treat=Treat, Trial.ID = InvestId,
                                   Min.Treat.Size = 5, Alpha = 0.05),
                                   error = function(e) { skip_to_next <<- TRUE})
  #if the trial does not have the minimum required number, skip to the next
  if(skip_to_next) { next }

  results_stage1[[invest_id]] <- FederatedApproachStage1(dataset, Surr=CGI, True=PANSS, Treat=Treat,
                                                         Trial.ID = InvestId, Min.Treat.Size = 5,
                                                         Alpha = 0.05)
  assign(paste0("stage1_invest", invest_id), results_stage1[[invest_id]])
  invest_ids[[i]] <- invest_id #keep a list of ids with datasets with required number of patients
  i <- i+1
}

invest_ids <- unlist(invest_ids)
invest_ids

#Combine the results of the first stage models
for (invest_id in invest_ids) {
  dataset <- results_stage1[[invest_id]]$Results.Stage.1
  if (invest_id == invest_ids[1]) {
    all_results_stage1<- dataset
 } else {
    all_results_stage1 <- rbind(all_results_stage1,dataset)
  }
}

all_results_stage1 #that combines the results of the first stage models

R.list <- list()
i <- 1
for (invest_id in invest_ids) {
  R <- results_stage1[[invest_id]]$R.i
  R.list[[i]] <- as.matrix(R[1:4,1:4])
  i <- i+1
}

R.list #list that combines all the variance-covariance matrices of the fixed effects

fit <- FederatedApproachStage2(Dataset = all_results_stage1, Intercept.S = Intercept.S,
                               alpha = alpha, Intercept.T = Intercept.T, beta = beta,
                               sigma.SS = sigma.SS, sigma.ST = sigma.ST,
                               sigma.TT = sigma.TT, Obs.per.trial = n,
                               Trial.ID = Trial.ID, R.list = R.list)
summary(fit)

## End(Not run)

Fits the second stage model in the two-stage federated data analysis approach.

Description

The function 'FederatedApproachStage2()' fits the second stage model of the two-stage federated data analysis approach to assess surrogacy.

Usage

FederatedApproachStage2(
  Dataset,
  Intercept.S,
  alpha,
  Intercept.T,
  beta,
  sigma.SS,
  sigma.ST,
  sigma.TT,
  Obs.per.trial,
  Trial.ID,
  R.list,
  Alpha = 0.05
)

Arguments

Value

Returns an object of class "FederatedApproachStage2()" that can be used to evaluate surrogacy.

• Indiv.R2: a data frame that contains the R^2_{indiv} and 95% confidence interval to evaluate surrogacy at the trial level.

• Trial.R2: a data frame that contains the R^2_{trial} and 95% confidence interval to evaluate surrogacy at the trial level.

• Fixed.Effects: a data frame that contains the average of the estimated fixed effects.

• D: estimated D matrix.

• Obs.Per.Trial: number of observations in each trial.

Model

The two-stage federated data analysis approach that can be used to assess surrogacy in the meta-analytic multiple-trial setting (Continuous-continuous case), but without the need of sharing data. Instead, each organization conducts separate analyses on their data set using the so-called "first stage" model. The results of these analyses are then aggregated at a central analysis hub, where the aggregated results are analyzed using a "second stage" model and the necessary metrics (R^2_{trial} and R^2_{indiv}) for the validation of the surrogate endpoint are obtained. This function fits the second stage model, where a method-of-moments estimator is used to obtain the variance-covariance matrix D from which the R^2_{trial} can be derived. The R^2_{indiv} is obtained with a weighted average of the elements in \Sigma_i.

Data Format

A data frame that combines the results of the first stage models and contains:

• a column with the trial indicator

• a column with the number of subjects in the trial

• a column with the estimated intercepts for the surrogate

• a column with the estimated treatment effects for the surrogate

• a column with the estimated intercepts for the true endpoint

• a column with the estimated treatment effects for the true endpoint

• a column with the variances of the error term for the surrogate endpoint

• a column with the covariances between the error terms of the surrogate and true endpoint

• a column with the variances of the error term for the true endpoint

A list that combines all the variance-covariance matrices of the fixed effects obtained using the first stage model

Author(s)

Dries De Witte

References

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

Examples

## Not run: 
#As an example, the federated data analysis approach can be applied to the Schizo data set
data(Schizo)
Schizo <-  Schizo[order(Schizo$InvestId, Schizo$Id),]
#Create separate datasets for each investigator
Schizo_datasets <- list()

for (invest_id in 1:198) {
Schizo_datasets[[invest_id]] <- Schizo[Schizo$InvestId == invest_id, ]
assign(paste0("Schizo", invest_id), Schizo_datasets[[invest_id]])
}
#Fit the first stage model for each dataset separately
results_stage1 <- list()
invest_ids <- list()
i <- 1
for (invest_id in 1:198) {
  dataset <- Schizo_datasets[[invest_id]]

  skip_to_next <- FALSE
  tryCatch(FederatedApproachStage1(dataset, Surr=CGI, True=PANSS, Treat=Treat, Trial.ID = InvestId,
                                   Min.Treat.Size = 5, Alpha = 0.05),
                                   error = function(e) { skip_to_next <<- TRUE})
  #if the trial does not have the minimum required number, skip to the next
  if(skip_to_next) { next }

  results_stage1[[invest_id]] <- FederatedApproachStage1(dataset, Surr=CGI, True=PANSS, Treat=Treat,
                                                         Trial.ID = InvestId, Min.Treat.Size = 5,
                                                         Alpha = 0.05)
  assign(paste0("stage1_invest", invest_id), results_stage1[[invest_id]])
  invest_ids[[i]] <- invest_id #keep a list of ids with datasets with required number of patients
  i <- i+1
}

invest_ids <- unlist(invest_ids)
invest_ids

#Combine the results of the first stage models
for (invest_id in invest_ids) {
  dataset <- results_stage1[[invest_id]]$Results.Stage.1
  if (invest_id == invest_ids[1]) {
    all_results_stage1<- dataset
 } else {
    all_results_stage1 <- rbind(all_results_stage1,dataset)
  }
}

all_results_stage1 #that combines the results of the first stage models

R.list <- list()
i <- 1
for (invest_id in invest_ids) {
  R <- results_stage1[[invest_id]]$R.i
  R.list[[i]] <- as.matrix(R[1:4,1:4])
  i <- i+1
}

R.list #list that combines all the variance-covariance matrices of the fixed effects

fit <- FederatedApproachStage2(Dataset = all_results_stage1, Intercept.S = Intercept.S,
                               alpha = alpha, Intercept.T = Intercept.T, beta = beta,
                               sigma.SS = sigma.SS, sigma.ST = sigma.ST,
                               sigma.TT = sigma.TT, Obs.per.trial = n,
                               Trial.ID = Trial.ID, R.list = R.list)
summary(fit)

## End(Not run)

Fits (univariate) fixed-effect models to assess surrogacy in the binary-binary case based on the Information-Theoretic framework

Description

The function FixedBinBinIT uses the information-theoretic approach (Alonso & Molenberghs, 2007) to estimate trial- and individual-level surrogacy based on fixed-effect models when both S and T are binary variables. The user can specify whether a (weighted or unweighted) full, semi-reduced, or reduced model should be fitted. See the Details section below.

Usage

FixedBinBinIT(Dataset, Surr, True, Treat, Trial.ID, Pat.ID, 
Model=c("Full"), Weighted=TRUE, Min.Trial.Size=2, Alpha=.05, 
Number.Bootstraps=50, Seed=sample(1:1000, size=1))

Arguments

Details

Individual-level surrogacy

The following univariate generalised linear models are fitted:

g_{T}(E(T_{ij}))=\mu_{Ti}+\beta_{i}Z_{ij},

g_{T}(E(T_{ij}|S_{ij}))=\gamma_{0i}+\gamma_{1i}Z_{ij}+\gamma_{2i}S_{ij},

where i and j are the trial and subject indicators, g_{T} is an appropriate link function (i.e., a logit link when binary endpoints are considered), S_{ij} and T_{ij} are the surrogate and true endpoint values of subject j in trial i, and Z_{ij} is the treatment indicator for subject j in trial i. \mu_{Ti} and \beta_{i} are the trial-specific intercepts and treatment-effects on the true endpoint in trial i. \gamma_{0i} and \gamma_{1i} are the trial-specific intercepts and treatment-effects on the true endpoint in trial i after accounting for the effect of the surrogate endpoint.

The -2 log likelihood values of the previous models in each of the i trials (i.e., L_{1i} and L_{2i}, respectively) are subsequently used to compute individual-level surrogacy based on the so-called Variance Reduction Factor (VFR; for details, see Alonso & Molenberghs, 2007):

R^2_{h}= 1 - \frac{1}{N} \sum_{i} exp \left(-\frac{L_{2i}-L_{1i}}{n_{i}} \right),

where N is the number of trials and n_{i} is the number of patients within trial i.

When it can be assumed (i) that the treatment-corrected association between the surrogate and the true endpoint is constant across trials, or (ii) when all data come from a single clinical trial (i.e., when N=1), the previous expression simplifies to:

R^2_{h.ind}= 1 - exp \left(-\frac{L_{2}-L_{1}}{N} \right).

The upper bound does not reach to 1 when T is binary, i.e., its maximum is 0.75. Kent (1983) claims that 0.75 is a reasonable upper bound and thus R^2_{h.ind} can usually be interpreted without paying special consideration to the discreteness of T. Alternatively, to address the upper bound problem, a scaled version of the mutual information can be used when both S and T are binary (Joe, 1989):

R^2_{b.ind}= \frac{I(T,S)}{min[H(T), H(S)]},

where the entropy of T and S in the previous expression can be estimated using the log likelihood functions of the GLMs shown above.

Trial-level surrogacy

When a full or semi-reduced model is requested (by using the argument Model=c("Full") or Model=c("SemiReduced") in the function call), trial-level surrogacy is assessed by fitting the following univariate models:

S_{ij}=\mu_{Si}+\alpha_{i}Z_{ij}+\varepsilon_{Sij}, (1)

T_{ij}=\mu_{Ti}+\beta_{i}Z_{ij}+\varepsilon_{Tij}, (1)

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_{Si} and \mu_{Ti} are the fixed trial-specific intercepts for S and T, and \alpha_{i} and \beta_{i} are the fixed trial-specific treatment effects on S and T, respectively. The error terms \varepsilon_{Sij} and \varepsilon_{Tij} are assumed to be independent.

When a reduced model is requested by the user (by using the argument Model=c("Reduced") in the function call), the following univariate models are fitted:

S_{ij}=\mu_{S}+\alpha_{i}Z_{ij}+\varepsilon_{Sij}, (2)

T_{ij}=\mu_{T}+\beta_{i}Z_{ij}+\varepsilon_{Tij}, (2)

where \mu_{S} and \mu_{T} are the common intercepts for S and T. The other parameters are the same as defined above, and \varepsilon_{Sij} and \varepsilon_{Tij} are again assumed to be independent.

When the user requested a full model approach (by using the argument Model=c("Full") in the function call, i.e., when models (1) were fitted), the following model is subsequently fitted:

\widehat{\beta}_{i}=\lambda_{0}+\lambda_{1}\widehat{\mu_{Si}}+\lambda_{2}\widehat{\alpha}_{i}+\varepsilon_{i}, (3)

where the parameter estimates for \beta_i, \mu_{Si}, and \alpha_i are based on models (1) (see above). When a weighted model is requested (using the argument Weighted=TRUE in the function call), model (3) is a weighted regression model (with weights based on the number of observations in trial i). The -2 log likelihood value of the (weighted or unweighted) model (3) (L_1) is subsequently compared to the -2 log likelihood value of an intercept-only model (\widehat{\beta}_{i}=\lambda_{3}; L_0), and R^2_{ht} is computed based based on the Variance Reduction Factor (for details, see Alonso & Molenberghs, 2007):

R^2_{ht}= 1 - exp \left(-\frac{L_1-L_0}{N} \right),

where N is the number of trials.

When a semi-reduced or reduced model is requested (by using the argument Model=c("SemiReduced") or Model=c("Reduced") 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 models (1) when a semi-reduced model is fitted or on models (2) when a reduced model is fitted. The -2 log likelihood value of this (weighted or unweighted) model (L_1) is subsequently compared to the -2 log likelihood value of an intercept-only model (\widehat{\beta}_{i}=\lambda_{3}; L_0), and R^2_{ht} is computed based on the reduction in the likelihood (as described above).

Value

An object of class FixedBinBinIT with components,

Author(s)

Wim Van der Elst, Ariel Alonso, & Geert Molenberghs

References

Alonso, A, & Molenberghs, G. (2007). Surrogate marker evaluation from an information theory perspective. Biometrics, 63, 180-186.

Joe, H. (1989). Relative entropy measures of multivariate dependence. Journal of the American Statistical Association, 84, 157-164.

Kent, T. J. (1983). Information gain as a general measure of correlation. Biometrica, 70, 163-173.

See Also

FixedBinContIT, FixedContBinIT, plot Information-Theoretic BinCombn

Examples

## Not run:  # Time consuming (>5sec) code part
# Generate data with continuous Surr and True
Sim.Data.MTS(N.Total=5000, N.Trial=50, R.Trial.Target=.9, R.Indiv.Target=.9,
             Fixed.Effects=c(0, 0, 0, 0), D.aa=10, D.bb=10, Seed=1,
             Model=c("Full"))
# Dichtomize Surr and True
Surr_Bin <- Data.Observed.MTS$Surr
Surr_Bin[Data.Observed.MTS$Surr>.5] <- 1
Surr_Bin[Data.Observed.MTS$Surr<=.5] <- 0
True_Bin <- Data.Observed.MTS$True
True_Bin[Data.Observed.MTS$True>.15] <- 1
True_Bin[Data.Observed.MTS$True<=.15] <- 0
Data.Observed.MTS$Surr <- Surr_Bin
Data.Observed.MTS$True <- True_Bin

# Assess surrogacy using info-theoretic framework
Fit <- FixedBinBinIT(Dataset = Data.Observed.MTS, Surr = Surr, 
True = True, Treat = Treat, Trial.ID = Trial.ID, 
Pat.ID = Pat.ID, Number.Bootstraps=100)

# Examine results
summary(Fit)
plot(Fit, Trial.Level = FALSE, Indiv.Level.By.Trial=TRUE)
plot(Fit, Trial.Level = TRUE, Indiv.Level.By.Trial=FALSE)

## End(Not run)

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)

Description

The function FixedBinContIT uses the information-theoretic approach (Alonso & Molenberghs, 2007) to estimate trial- and individual-level surrogacy based on fixed-effect models when T is binary and S is continuous. The user can specify whether a (weighted or unweighted) full, semi-reduced, or reduced model should be fitted. See the Details section below.

Usage

FixedBinContIT(Dataset, Surr, True, Treat, Trial.ID, Pat.ID, 
Model=c("Full"), Weighted=TRUE, Min.Trial.Size=2, Alpha=.05, 
Number.Bootstraps=50,Seed=sample(1:1000, size=1))

Arguments

Details

Individual-level surrogacy

The following univariate generalised linear models are fitted:

g_{T}(E(T_{ij}))=\mu_{Ti}+\beta_{i}Z_{ij},

g_{T}(E(T_{ij}|S_{ij}))=\gamma_{0i}+\gamma_{1i}Z_{ij}+\gamma_{2i}S_{ij},

where i and j are the trial and subject indicators, g_{T} is an appropriate link function (i.e., a logit link for binary endpoints and an identity link for normally distributed continuous endpoints), S_{ij} and T_{ij} are the surrogate and true endpoint values of subject j in trial i, and Z_{ij} is the treatment indicator for subject j in trial i. \mu_{Ti} and \beta_{i} are the trial-specific intercepts and treatment-effects on the true endpoint in trial i. \gamma_{0i} and \gamma_{1i} are the trial-specific intercepts and treatment-effects on the true endpoint in trial i after accounting for the effect of the surrogate endpoint.

The -2 log likelihood values of the previous models in each of the i trials (i.e., L_{1i} and L_{2i}, respectively) are subsequently used to compute individual-level surrogacy based on the so-called Variance Reduction Factor (VFR; for details, see Alonso & Molenberghs, 2007):

R^2_{h}= 1 - \frac{1}{N} \sum_{i} exp \left(-\frac{L_{2i}-L_{1i}}{n_{i}} \right),

where N is the number of trials and n_{i} is the number of patients within trial i.

When it can be assumed (i) that the treatment-corrected association between the surrogate and the true endpoint is constant across trials, or (ii) when all data come from a single clinical trial (i.e., when N=1), the previous expression simplifies to:

R^2_{h.ind}= 1 - exp \left(-\frac{L_{2}-L_{1}}{N} \right).

The upper bound does not reach to 1 when T is binary, i.e., its maximum is 0.75. Kent (1983) claims that 0.75 is a reasonable upper bound and thus R^2_{h.ind} can usually be interpreted without paying special consideration to the discreteness of T. Alternatively, to address the upper bound problem, a scaled version of the mutual information can be used when both S and T are binary (Joe, 1989):

R^2_{b.ind}= \frac{I(T,S)}{min[H(T), H(S)]},

where the entropy of T and S in the previous expression can be estimated using the log likelihood functions of the GLMs shown above.

Trial-level surrogacy

When a full or semi-reduced model is requested (by using the argument Model=c("Full") or Model=c("SemiReduced") in the function call), trial-level surrogacy is assessed by fitting the following univariate models:

S_{ij}=\mu_{Si}+\alpha_{i}Z_{ij}+\varepsilon_{Sij}, (1)

T_{ij}=\mu_{Ti}+\beta_{i}Z_{ij}+\varepsilon_{Tij}, (1)

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_{Si} and \mu_{Ti} are the fixed trial-specific intercepts for S and T, and \alpha_{i} and \beta_{i} are the fixed trial-specific treatment effects on S and T, respectively. The error terms \varepsilon_{Sij} and \varepsilon_{Tij} are assumed to be independent.

When a reduced model is requested by the user (by using the argument Model=c("Reduced") in the function call), the following univariate models are fitted:

S_{ij}=\mu_{S}+\alpha_{i}Z_{ij}+\varepsilon_{Sij}, (2)

T_{ij}=\mu_{T}+\beta_{i}Z_{ij}+\varepsilon_{Tij}, (2)

where \mu_{S} and \mu_{T} are the common intercepts for S and T. The other parameters are the same as defined above, and \varepsilon_{Sij} and \varepsilon_{Tij} are again assumed to be independent.

When the user requested a full model approach (by using the argument Model=c("Full") in the function call, i.e., when models (1) were fitted), the following model is subsequently fitted:

\widehat{\beta}_{i}=\lambda_{0}+\lambda_{1}\widehat{\mu_{Si}}+\lambda_{2}\widehat{\alpha}_{i}+\varepsilon_{i}, (3)

where the parameter estimates for \beta_i, \mu_{Si}, and \alpha_i are based on models (1) (see above). When a weighted model is requested (using the argument Weighted=TRUE in the function call), model (3) is a weighted regression model (with weights based on the number of observations in trial i). The -2 log likelihood value of the (weighted or unweighted) model (3) (L_1) is subsequently compared to the -2 log likelihood value of an intercept-only model (\widehat{\beta}_{i}=\lambda_{3}; L_0), and R^2_{ht} is computed based based on the Variance Reduction Factor (for details, see Alonso & Molenberghs, 2007):

R^2_{ht}= 1 - exp \left(-\frac{L_1-L_0}{N} \right),

where N is the number of trials.

When a semi-reduced or reduced model is requested (by using the argument Model=c("SemiReduced") or Model=c("Reduced") 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 models (1) when a semi-reduced model is fitted or on models (2) when a reduced model is fitted. The -2 log likelihood value of this (weighted or unweighted) model (L_1) is subsequently compared to the -2 log likelihood value of an intercept-only model (\widehat{\beta}_{i}=\lambda_{3}; L_0), and R^2_{ht} is computed based on the reduction in the likelihood (as described above).

Value

An object of class FixedBinContIT with components,

Author(s)

Wim Van der Elst, Ariel Alonso, & Geert Molenberghs

References

Alonso, A, & Molenberghs, G. (2007). Surrogate marker evaluation from an information theory perspective. Biometrics, 63, 180-186.

Joe, H. (1989). Relative entropy measures of multivariate dependence. Journal of the American Statistical Association, 84, 157-164.

Kent, T. J. (1983). Information gain as a general measure of correlation. Biometrica, 70, 163-173.

See Also

FixedBinBinIT, FixedContBinIT, plot Information-Theoretic BinCombn

Examples

## Not run:  # Time consuming (>5sec) code part
# Generate data with continuous Surr and True
Sim.Data.MTS(N.Total=2000, N.Trial=100, R.Trial.Target=.8, 
R.Indiv.Target=.8, Seed=123, Model="Full")

# Make T binary
Data.Observed.MTS$True_Bin <- Data.Observed.MTS$True
Data.Observed.MTS$True_Bin[Data.Observed.MTS$True>=0] <- 1
Data.Observed.MTS$True_Bin[Data.Observed.MTS$True<0] <- 0

# Analyze data
Fit <- FixedBinContIT(Dataset = Data.Observed.MTS, Surr = Surr, 
True = True_Bin, Treat = Treat, Trial.ID = Trial.ID, Pat.ID = Pat.ID, 
Model = "Full", Number.Bootstraps=50)

# Examine results
summary(Fit)
plot(Fit, Trial.Level = FALSE, Indiv.Level.By.Trial=TRUE)
plot(Fit, Trial.Level = TRUE, Indiv.Level.By.Trial=FALSE)

## End(Not run)

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)

Description

The function FixedContBinIT uses the information-theoretic approach (Alonso & Molenberghs, 2007) to estimate trial- and individual-level surrogacy based on fixed-effect models when T is continuous normally distributed and S is binary. The user can specify whether a (weighted or unweighted) full, semi-reduced, or reduced model should be fitted. See the Details section below.

Usage

FixedContBinIT(Dataset, Surr, True, Treat, Trial.ID, Pat.ID, 
Model=c("Full"), Weighted=TRUE, Min.Trial.Size=2, Alpha=.05, 
Number.Bootstraps=50,Seed=sample(1:1000, size=1))

Arguments

Details

Individual-level surrogacy

The following univariate generalised linear models are fitted:

g_{T}(E(T_{ij}))=\mu_{Ti}+\beta_{i}Z_{ij},

g_{T}(E(T_{ij}|S_{ij}))=\gamma_{0i}+\gamma_{1i}Z_{ij}+\gamma_{2i}S_{ij},

where i and j are the trial and subject indicators, g_{T} is an appropriate link function (i.e., a logit link for binary endpoints and an identity link for normally distributed continuous endpoints), S_{ij} and T_{ij} are the surrogate and true endpoint values of subject j in trial i, and Z_{ij} is the treatment indicator for subject j in trial i. \mu_{Ti} and \beta_{i} are the trial-specific intercepts and treatment-effects on the true endpoint in trial i. \gamma_{0i} and \gamma_{1i} are the trial-specific intercepts and treatment-effects on the true endpoint in trial i after accounting for the effect of the surrogate endpoint.

The -2 log likelihood values of the previous models in each of the i trials (i.e., L_{1i} and L_{2i}, respectively) are subsequently used to compute individual-level surrogacy based on the so-called Variance Reduction Factor (VFR; for details, see Alonso & Molenberghs, 2007):

R^2_{h}= 1 - \frac{1}{N} \sum_{i} exp \left(-\frac{L_{2i}-L_{1i}}{n_{i}} \right),

where N is the number of trials and n_{i} is the number of patients within trial i.

When it can be assumed (i) that the treatment-corrected association between the surrogate and the true endpoint is constant across trials, or (ii) when all data come from a single clinical trial (i.e., when N=1), the previous expression simplifies to:

R^2_{h.ind}= 1 - exp \left(-\frac{L_{2}-L_{1}}{N} \right).

Trial-level surrogacy

When a full or semi-reduced model is requested (by using the argument Model=c("Full") or Model=c("SemiReduced") in the function call), trial-level surrogacy is assessed by fitting the following univariate models:

S_{ij}=\mu_{Si}+\alpha_{i}Z_{ij}+\varepsilon_{Sij}, (1)

T_{ij}=\mu_{Ti}+\beta_{i}Z_{ij}+\varepsilon_{Tij}, (1)

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_{Si} and \mu_{Ti} are the fixed trial-specific intercepts for S and T, and \alpha_{i} and \beta_{i} are the fixed trial-specific treatment effects on S and T, respectively. The error terms \varepsilon_{Sij} and \varepsilon_{Tij} are assumed to be independent.

When a reduced model is requested by the user (by using the argument Model=c("Reduced") in the function call), the following univariate models are fitted:

S_{ij}=\mu_{S}+\alpha_{i}Z_{ij}+\varepsilon_{Sij}, (2)

T_{ij}=\mu_{T}+\beta_{i}Z_{ij}+\varepsilon_{Tij}, (2)

where \mu_{S} and \mu_{T} are the common intercepts for S and T. The other parameters are the same as defined above, and \varepsilon_{Sij} and \varepsilon_{Tij} are again assumed to be independent.

When the user requested a full model approach (by using the argument Model=c("Full") in the function call, i.e., when models (1) were fitted), the following model is subsequently fitted:

\widehat{\beta}_{i}=\lambda_{0}+\lambda_{1}\widehat{\mu_{Si}}+\lambda_{2}\widehat{\alpha}_{i}+\varepsilon_{i}, (3)

where the parameter estimates for \beta_i, \mu_{Si}, and \alpha_i are based on models (1) (see above). When a weighted model is requested (using the argument Weighted=TRUE in the function call), model (3) is a weighted regression model (with weights based on the number of observations in trial i). The -2 log likelihood value of the (weighted or unweighted) model (3) (L_1) is subsequently compared to the -2 log likelihood value of an intercept-only model (\widehat{\beta}_{i}=\lambda_{3}; L_0), and R^2_{ht} is computed based based on the Variance Reduction Factor (for details, see Alonso & Molenberghs, 2007):

R^2_{ht}= 1 - exp \left(-\frac{L_1-L_0}{N} \right),

where N is the number of trials.

When a semi-reduced or reduced model is requested (by using the argument Model=c("SemiReduced") or Model=c("Reduced") 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 models (1) when a semi-reduced model is fitted or on models (2) when a reduced model is fitted. The -2 log likelihood value of this (weighted or unweighted) model (L_1) is subsequently compared to the -2 log likelihood value of an intercept-only model (\widehat{\beta}_{i}=\lambda_{3}; L_0), and R^2_{ht} is computed based on the reduction in the likelihood (as described above).

Value

An object of class FixedContBinIT with components,

Author(s)

Wim Van der Elst, Ariel Alonso, & Geert Molenberghs

References

Alonso, A, & Molenberghs, G. (2007). Surrogate marker evaluation from an information theory perspective. Biometrics, 63, 180-186.

See Also

FixedBinBinIT, FixedBinContIT, plot Information-Theoretic BinCombn

Examples

## Not run:  # Time consuming (>5sec) code part
# Generate data with continuous Surr and True
Sim.Data.MTS(N.Total=2000, N.Trial=100, R.Trial.Target=.8, 
R.Indiv.Target=.8, Seed=123, Model="Full")

# Make S binary
Data.Observed.MTS$Surr_Bin <- Data.Observed.MTS$Surr
Data.Observed.MTS$Surr_Bin[Data.Observed.MTS$Surr>=0] <- 1
Data.Observed.MTS$Surr_Bin[Data.Observed.MTS$Surr<0] <- 0

# Analyze data
Fit <- FixedContBinIT(Dataset = Data.Observed.MTS, Surr = Surr_Bin, 
True = True, Treat = Treat, Trial.ID = Trial.ID, Pat.ID = Pat.ID, 
Model = "Full", Number.Bootstraps=50)

# Examine results
summary(Fit)
plot(Fit, Trial.Level = FALSE, Indiv.Level.By.Trial=TRUE)
plot(Fit, Trial.Level = TRUE, Indiv.Level.By.Trial=FALSE)

## End(Not run)

Fits (univariate) fixed-effect models to assess surrogacy in the continuous-continuous case based on the Information-Theoretic framework

Description

The function FixedContContIT uses the information-theoretic approach (Alonso & Molenberghs, 2007) to estimate trial- and individual-level surrogacy based on fixed-effect models when both S and T are continuous variables. The user can specify whether a (weighted or unweighted) full, semi-reduced, or reduced model should be fitted. See the Details section below.

Usage

FixedContContIT(Dataset, Surr, True, Treat, Trial.ID, Pat.ID, 
Model=c("Full"), Weighted=TRUE, Min.Trial.Size=2, 
Alpha=.05, Number.Bootstraps=500, Seed=sample(1:1000, size=1))

Arguments

Details

Individual-level surrogacy

The following univariate generalised linear models are fitted:

g_{T}(E(T_{ij}))=\mu_{Ti}+\beta_{i}Z_{ij},

g_{T}(E(T_{ij}|S_{ij}))=\gamma_{0i}+\gamma_{1i}Z_{ij}+\gamma_{2i}S_{ij},

where i and j are the trial and subject indicators, g_{T} is an appropriate link function (i.e., an identity link when a continuous true endpoint is considered), S_{ij} and T_{ij} are the surrogate and true endpoint values of subject j in trial i, and Z_{ij} is the treatment indicator for subject j in trial i. \mu_{Ti} and \beta_{i} are the trial-specific intercepts and treatment-effects on the true endpoint in trial i. \gamma_{0i} and \gamma_{1i} are the trial-specific intercepts and treatment-effects on the true endpoint in trial i after accounting for the effect of the surrogate endpoint.

The -2 log likelihood values of the previous models in each of the i trials (i.e., L_{1i} and L_{2i}, respectively) are subsequently used to compute individual-level surrogacy based on the so-called Variance Reduction Factor (VFR; for details, see Alonso & Molenberghs, 2007):

R^2_{h.ind}= 1 - \frac{1}{N} \sum_{i} exp \left(-\frac{L_{2i}-L_{1i}}{n_{i}} \right),

where N is the number of trials and n_{i} is the number of patients within trial i.

When it can be assumed (i) that the treatment-corrected association between the surrogate and the true endpoint is constant across trials, or (ii) when all data come from a single clinical trial (i.e., when N=1), the previous expression simplifies to:

R^2_{h.ind.clust}= 1 - exp \left(-\frac{L_{2}-L_{1}}{N} \right).

Trial-level surrogacy

When a full or semi-reduced model is requested (by using the argument Model=c("Full") or Model=c("SemiReduced") in the function call), trial-level surrogacy is assessed by fitting the following univariate models:

S_{ij}=\mu_{Si}+\alpha_{i}Z_{ij}+\varepsilon_{Sij}, (1)

T_{ij}=\mu_{Ti}+\beta_{i}Z_{ij}+\varepsilon_{Tij}, (1)

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_{Si} and \mu_{Ti} are the fixed trial-specific intercepts for S and T, and \alpha_{i} and \beta_{i} are the fixed trial-specific treatment effects on S and T, respectively. The error terms \varepsilon_{Sij} and \varepsilon_{Tij} are assumed to be independent.

When a reduced model is requested by the user (by using the argument Model=c("Reduced") in the function call), the following univariate models are fitted:

S_{ij}=\mu_{S}+\alpha_{i}Z_{ij}+\varepsilon_{Sij}, (2)

T_{ij}=\mu_{T}+\beta_{i}Z_{ij}+\varepsilon_{Tij}, (2)

where \mu_{S} and \mu_{T} are the common intercepts for S and T. The other parameters are the same as defined above, and \varepsilon_{Sij} and \varepsilon_{Tij} are again assumed to be independent.

When the user requested a full model approach (by using the argument Model=c("Full") in the function call, i.e., when models (1) were fitted), the following model is subsequently fitted:

\widehat{\beta}_{i}=\lambda_{0}+\lambda_{1}\widehat{\mu_{Si}}+\lambda_{2}\widehat{\alpha}_{i}+\varepsilon_{i}, (3)

where the parameter estimates for \beta_i, \mu_{Si}, and \alpha_i are based on models (1) (see above). When a weighted model is requested (using the argument Weighted=TRUE in the function call), model (3) is a weighted regression model (with weights based on the number of observations in trial i). The -2 log likelihood value of the (weighted or unweighted) model (3) (L_1) is subsequently compared to the -2 log likelihood value of an intercept-only model (\widehat{\beta}_{i}=\lambda_{3}; L_0), and R^2_{ht} is computed based based on the Variance Reduction Factor (for details, see Alonso & Molenberghs, 2007):

R^2_{ht}= 1 - exp \left(-\frac{L_1-L_0}{N} \right),

where N is the number of trials.

When a semi-reduced or reduced model is requested (by using the argument Model=c("SemiReduced") or Model=c("Reduced") 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 models (1) when a semi-reduced model is fitted or on models (2) when a reduced model is fitted. The -2 log likelihood value of this (weighted or unweighted) model (L_1) is subsequently compared to the -2 log likelihood value of an intercept-only model (\widehat{\beta}_{i}=\lambda_{3}; L_0), and R^2_{ht} is computed based on the reduction in the likelihood (as described above).

Value

An object of class FixedContContIT with components,

Author(s)

Wim Van der Elst, Ariel Alonso, & Geert Molenberghs

References

Alonso, A, & Molenberghs, G. (2007). Surrogate marker evaluation from an information theory perspective. Biometrics, 63, 180-186.

See Also

MixedContContIT, FixedContBinIT, FixedBinContIT, FixedBinBinIT, plot Information-Theoretic

Examples

# Example 1
# Based on the ARMD data

data(ARMD)
# Assess surrogacy based on a full fixed-effect model
# in the information-theoretic framework:
Sur <- FixedContContIT(Dataset=ARMD, Surr=Diff24, True=Diff52, Treat=Treat, Trial.ID=Center,
Pat.ID=Id, Model="Full", Number.Bootstraps=50)
# Obtain a summary of the results:
summary(Sur)

## Not run:  #time consuming code
# Example 2
# Conduct an 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="Full")
# Assess surrogacy based on a full fixed-effect model
# in the information-theoretic framework:
Sur2 <- FixedContContIT(Dataset=Data.Observed.MTS, Surr=Surr, True=True, Treat=Treat,
Trial.ID=Trial.ID, Pat.ID=Pat.ID, Model="Full", Number.Bootstraps=50)

# Show a summary of the results:
summary(Sur2)
## End(Not run)

Investigates surrogacy for binary or ordinal outcomes using the Information Theoretic framework

Description

The function FixedDiscrDiscrIT uses the information theoretic approach (Alonso and Molenberghs 2007) to estimate trial and individual level surrogacy based on fixed-effects models when the surrogate is binary and the true outcome is ordinal, the converse case or when both outcomes are ordinal (the user must specify which form the data is in). The user can specify whether a weighted or unweighted analysis is required at the trial level. The penalized likelihood approach of Firth (1993) is applied to resolve issues of separation in discrete outcomes for particular trials. Requires packages OrdinalLogisticBiplot and logistf.

Usage

FixedDiscrDiscrIT(Dataset, Surr, True, Treat, Trial.ID,
Weighted = TRUE, Setting = c("binord"))

Arguments

Details

Individual level surrogacy

The following univariate logistic regression models are fitted when Setting=c("ordbin"):

logit(P(T_{ij}=1))=\mu_{Ti}+\beta_{i}Z_{ij}, (1)

logit(P(T_{ij}=1|S_{ij}=s))=\gamma_{0i}+\gamma_{1i}Z_{ij}+\gamma_{2i}S_{ij}, (1)

where: i and j are the trial and subject indicators; S_{ij} and T_{ij} are the surrogate and true outcome values of subject j in trial i; and Z_{ij} is the treatment indicator for subject j in trial i; \mu_{Ti} and \beta_{i} are the trial-specific intercepts and treatment-effects on the true endpoint in trial i; and \gamma_{0i} and \gamma_{1i} are the trial-specific intercepts and treatment-effects on the true endpoint in trial i after accounting for the effect of the surrogate endpoint. The -2 log likelihood values of the previous models in each of the i trials (i.e., L_{1i} and L_{2i}, respectively) are subsequently used to compute individual-level surrogacy based on the so-called Likelihood Reduction Factor (LRF; for details, see Alonso & Molenberghs, 2006):

R^2_{h}= 1 - \frac{1}{N} \sum_{i} exp \left(-\frac{L_{2i}-L_{1i}}{n_{i}} \right),

where N is the number of trials and n_{i} is the number of patients within trial i.

At the individual level in the discrete case R^2_{h} is bounded above by a number strictly less than one and is re-scaled (see Alonso & Molenberghs (2007)):

\widehat{R^2_{h}}= \frac{R^2_{h}}{1-e^{-2L_{0}}},

where L_{0} is the log-likelihood of the intercept only model of the true outcome (logit(P(T_{ij}=1)=\gamma_{3}).

In the case of Setting=c("binord") or Setting=c("ordord") proportional odds models in (1) are used to accommodate the ordinal true response outcome, in all other respects the calculation of R^2_{h} would proceed in the same manner.

Trial-level surrogacy

When Setting=c("ordbin") trial-level surrogacy is assessed by fitting the following univariate logistic regression and proportional odds models for the ordinal surrogate and binary true response variables regressed on treatment for each trial i:

logit(P(S_{ij} \leq W))=\mu_{S_{wi}}+\alpha_{i}Z_{ij}, (2)

logit(P(T_{ij}=1))=\mu_{Ti}+\beta_{i}Z_{ij}, (2)

where: i and j are the trial and subject indicators; S_{ij} and T_{ij} are the surrogate and true outcome values of subject j in trial i; Z_{ij} is the treatment indicator for subject j in trial i; \mu_{S_{wi}} are the trial-specific intercept values for each cut point w, where w=1,..,W-1, of the ordinal surrogate outcome; \mu_{Ti} are the fixed trial-specific intercepts for T; and \alpha_{i} and \beta_{i} are the fixed trial-specific treatment effects on S and T, respectively. The mean trial-specific intercepts for the surrogate are calculated, \overline{\mu}_{S_{wi}}.The following model is subsequently fitted:

\widehat{\beta}_{i}=\lambda_{0}+\lambda_{1}\widehat{\overline{\mu}}_{S_{wi}}+\lambda_{2}\widehat{\alpha}_{i}+\varepsilon_{i}, (3)

where the parameter estimates for \beta_i, \overline{\mu}_{S_{wi}}, and \alpha_i are based on models (2) (see above). When a weighted model is requested (using the argument Weighted=TRUE in the function call), model (2) is a weighted regression model (with weights based on the number of observations in trial i). The -2 log likelihood value of the (weighted or unweighted) model (2) (L_1) is subsequently compared to the -2 log likelihood value of an intercept-only model (\widehat{\beta}_{i}=\lambda_{3}; L_0), and R^2_{ht} is computed based on the Likelihood Reduction Factor (for details, see Alonso & Molenberghs, 2006):

R^2_{ht}= 1 - exp \left(-\frac{L_1-L_0}{N} \right),

where N is the number of trials.

When separation (the presence of zero cells) occurs in the cross tabs of treatment and the true or surrogate outcome for a particular trial in models (2) extreme bias can occur in R^2_{ht}. Under separation there are no unique maximum likelihood for parameters \beta_i, \overline{\mu}_{S_{wi}} and \alpha_i, in (2), for the affected trial i. This typically leads to extreme bias in the estimation of these parameters and hence outlying influential points in model (3), bias in R^2_{ht} inevitably follows.

To resolve the issue of separation the penalized likelihood approach of Firth (1993) is applied. This approach adds an asymptotically negligible component to the score function to allow unbiased estimation of \beta_i, \overline{\mu}_{S_{wi}}, and \alpha_i and in turn R^2_{ht}. The penalized likelihood R function logitf from the package of the same name is applied in the case of binary separation (Heinze and Schemper, 2002). The function pordlogistf from the package OrdinalLogisticBioplot is applied in the case of ordinal separation (Hern'andez, 2013). All instances of separation are reported.

In the case of Setting=c("binord") or Setting=c("ordord") the appropriate models (either logistic regression or a proportional odds models) are fitted in (2) to accommodate the form (either binary or ordinal) of the true or surrogate response variable. The rest of the analysis would proceed in a similar manner as that described above.

Value

An object of class FixedDiscrDiscrIT with components,

Author(s)

Hannah M. Ensor & Christopher J. Weir

References

Alonso, A, & Molenberghs, G. (2007). Surrogate marker evaluation from an information theory perspective. Biometrics, 63, 180-186.

Alonso, A, & Molenberghs, G., Geys, H., Buyse, M. & Vangeneugden, T. (2006). A unifying approach for surrogate marker validation based on Prentice's criteria. Statistics in medicine, 25, 205-221.

Firth, D. (1993). Bias reduction of maximum likelihood estimates. Biometrika, 80, 27-38.

Heinze, G. & Schemper, M. 2002. A solution to the problem of separation in logistic regression. Statistics in medicine, 21, 2409-2419.

Hern'andez, J. C. V.-V. O., J. L. 2013. OrdinalLogisticBiplot: Biplot representations of ordinal variables. R.

See Also

FixedContContIT, plot Information-Theoretic

Examples

## Not run:  # Time consuming (>5sec) code part
# Example 1
# Conduct an 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="Full")

# create a binary true and ordinal surrogate outcome
Data.Observed.MTS$True<-findInterval(Data.Observed.MTS$True,
c(quantile(Data.Observed.MTS$True,0.5)))
Data.Observed.MTS$Surr<-findInterval(Data.Observed.MTS$Surr,
c(quantile(Data.Observed.MTS$Surr,0.333),quantile(Data.Observed.MTS$Surr,0.666)))

# Assess surrogacy based on a full fixed-effect model
# in the information-theoretic framework for a binary surrogate and ordinal true outcome:
SurEval <- FixedDiscrDiscrIT(Dataset=Data.Observed.MTS, Surr=Surr, True=True, Treat=Treat,
Trial.ID=Trial.ID, Setting="ordbin")

# Show a summary of the results:
summary(SurEval)
SurEval$Trial.Spec.Results
SurEval$R2h
SurEval$R2ht

## End(Not run)

Assess surrogacy in the causal-inference single-trial setting in the binary-binary case

Description

The function ICA.BinBin quantifies surrogacy in the single-trial causal-inference framework (individual causal association and causal concordance) when both the surrogate and the true endpoints are binary outcomes. See Details below.

Usage

ICA.BinBin(pi1_1_, pi1_0_, pi_1_1, pi_1_0, pi0_1_, pi_0_1,
Monotonicity=c("General"), Sum_Pi_f = seq(from=0.01, to=0.99, by=.01),
M=10000, Volume.Perc=0, Seed=sample(1:100000, size=1))

Arguments

Details

In the continuous normal setting, surroagacy can be assessed by studying the association between the individual causal effects on S and T (see ICA.ContCont). In that setting, the Pearson correlation is the obvious measure of association.

When S and T are binary endpoints, multiple alternatives exist. Alonso et al. (2014) proposed the individual causal association (ICA; R_{H}^{2}), which captures the association between the individual causal effects of the treatment on S (\Delta_S) and T (\Delta_T) using information-theoretic principles.

The function ICA.BinBin computes R_{H}^{2} based on plausible values of the potential outcomes. Denote by \bold{Y}'=(T_0,T_1,S_0,S_1) the vector of potential outcomes. The vector \bold{Y} can take 16 values and the set of parameters \pi_{ijpq}=P(T_0=i,T_1=j,S_0=p,S_1=q) (with i,j,p,q=0/1) fully characterizes its distribution.

However, the parameters in \pi_{ijpq} are not all functionally independent, e.g., 1=\pi_{\cdot\cdot\cdot\cdot}. When no assumptions regarding monotonicity are made, the data impose a total of 7 restrictions, and thus only 9 proabilities in \pi_{ijpq} are allowed to vary freely (for details, see Alonso et al., 2014). Based on the data and assuming SUTVA, the marginal probabilites \pi_{1 \cdot 1 \cdot}, \pi_{1 \cdot 0 \cdot}, \pi_{\cdot 1 \cdot 1}, \pi_{\cdot 1 \cdot 0}, \pi_{0 \cdot 1 \cdot}, and \pi_{\cdot 0 \cdot 1} can be computed (by hand or using the function MarginalProbs). Define the vector

\bold{b}'=(1, \pi_{1 \cdot 1 \cdot}, \pi_{1 \cdot 0 \cdot}, \pi_{\cdot 1 \cdot 1}, \pi_{\cdot 1 \cdot 0}, \pi_{0 \cdot 1 \cdot}, \pi_{\cdot 0 \cdot 1})

and \bold{A} is a contrast matrix such that the identified restrictions can be written as a system of linear equation

\bold{A \pi} = \bold{b}.

The matrix \bold{A} has rank 7 and can be partitioned as \bold{A=(A_r | A_f)}, and similarly the vector \bold{\pi} can be partitioned as \bold{\pi^{'}=(\pi_r^{'} | \pi_f^{'})} (where f refers to the submatrix/vector given by the 9 last columns/components of \bold{A/\pi}). Using these partitions the previous system of linear equations can be rewritten as

\bold{A_r \pi_r + A_f \pi_f = b}.

The following algorithm is used to generate plausible distributions for \bold{Y}. First, select a value of the specified grid of values (specified using Sum_Pi_f in the function call). For k=1 to M (specified using M in the function call), generate a vector \pi_f that contains 9 components that are uniformly sampled from hyperplane subject to the restriction that the sum of the generated components equals Sum_Pi_f (the function RandVec, which uses the randfixedsum algorithm written by Roger Stafford, is used to obtain these components). Next, \bold{\pi_r=A_r^{-1}(b - A_f \pi_f)} is computed and the \pi_r vectors where all components are in the [0;\:1] range are retained. This procedure is repeated for each of the Sum_Pi_f values. Based on these results, R_H^2 is estimated. The obtained values can be used to conduct a sensitivity analysis during the validation exercise.

The previous developments hold when no monotonicity is assumed. When monotonicity for S, T, or for S and T is assumed, some of the probabilities of \pi are zero. For example, when montonicity is assumed for T, then P(T_0 <= T_1)=1, or equivantly, \pi_{1000}=\pi_{1010}=\pi_{1001}=\pi_{1011}=0. When monotonicity is assumed, the procedure described above is modified accordingly (for details, see Alonso et al., 2014). When a general analysis is requested (using Monotonicity=c("General") in the function call), all settings are considered (no monotonicity, monotonicity for S alone, for T alone, and for both for S and T.)

To account for the uncertainty in the estimation of the marginal probabilities, a vector of values can be specified from which a random draw is made in each run (see Examples below).

Value

An object of class ICA.BinBin with components,

Author(s)

Wim Van der Elst, Paul Meyvisch, Ariel Alonso & Geert Molenberghs

References

Alonso, A., Van der Elst, W., & Molenberghs, G. (2015). Validation of surrogate endpoints: the binary-binary setting from a causal inference perspective.

See Also

ICA.ContCont, MICA.ContCont

Examples

## Not run: # Time consuming code part
# Compute R2_H given the marginals specified as the pi's, making no
# assumptions regarding monotonicity (general case)
ICA <- ICA.BinBin(pi1_1_=0.2619048, pi1_0_=0.2857143, pi_1_1=0.6372549,
pi_1_0=0.07843137, pi0_1_=0.1349206, pi_0_1=0.127451, Seed=1,
Monotonicity=c("General"), Sum_Pi_f = seq(from=0.01, to=.99, by=.01), M=10000)

# obtain plot of the results
plot(ICA, R2_H=TRUE)

# Example 2 where the uncertainty in the estimation
# of the marginals is taken into account
ICA_BINBIN2 <- ICA.BinBin(pi1_1_=runif(10000, 0.2573, 0.4252),
pi1_0_=runif(10000, 0.1769, 0.3310),
pi_1_1=runif(10000, 0.5947, 0.7779),
pi_1_0=runif(10000, 0.0322, 0.1442),
pi0_1_=runif(10000, 0.0617, 0.1764),
pi_0_1=runif(10000, 0.0254, 0.1315),
Monotonicity=c("General"),
Sum_Pi_f = seq(from=0.01, to=0.99, by=.01),
M=50000, Seed=1)

# Plot results
plot(ICA_BINBIN2)

## End(Not run)

ICA (binary-binary setting) that is obtaied when the counterfactual correlations are assumed to fall within some prespecified ranges.

Description

Shows the results of ICA (binary-binary setting) in the subgroup of results where the counterfactual correlations are assumed to fall within some prespecified ranges.

Usage

ICA.BinBin.CounterAssum(x, r2_h_S0S1_min, r2_h_S0S1_max, r2_h_S0T1_min, 
r2_h_S0T1_max, r2_h_T0T1_min, r2_h_T0T1_max, r2_h_T0S1_min, r2_h_T0S1_max, 
Monotonicity="General", Type="Freq", MainPlot=" ", Cex.Legend=1, 
Cex.Position="topright", ...)

Arguments

Author(s)

Wim Van der Elst, Ariel Alonso, & Geert Molenberghs

References

Alonso, A., Van der Elst, W., Molenberghs, G., Buyse, M., & Burzykowski, T. (submitted). On the relationship between the causal inference and meta-analytic paradigms for the validation of surrogate markers.

Van der Elst, W., Alonso, A., & Molenberghs, G. (submitted). An exploration of the relationship between causal inference and meta-analytic measures of surrogacy.

See Also

ICA.BinBin

Examples

## Not run:  #Time consuming (>5 sec) code part
# Compute R2_H given the marginals specified as the pi's, making no 
# assumptions regarding monotonicity (general case)
ICA <- ICA.BinBin.Grid.Sample(pi1_1_=0.261, pi1_0_=0.285, 
pi_1_1=0.637, pi_1_0=0.078, pi0_1_=0.134, pi_0_1=0.127,  
Monotonicity=c("General"), M=5000, Seed=1)

# Obtain a density plot of R2_H, assuming that 
# r2_h_S0S1>=.2, r2_h_S0T1>=0, r2_h_T0T1>=.2, and r2_h_T0S1>=0
ICA.BinBin.CounterAssum(ICA, r2_h_S0S1_min=.2, r2_h_S0S1_max=1, 
r2_h_S0T1_min=0, r2_h_S0T1_max=1, r2_h_T0T1_min=0.2, r2_h_T0T1_max=1, 
r2_h_T0S1_min=0, r2_h_T0S1_max=1, Monotonicity="General",
Type="Density") 

# Now show the densities of R2_H under the different 
# monotonicity assumptions 
ICA.BinBin.CounterAssum(ICA, r2_h_S0S1_min=.2, r2_h_S0S1_max=1, 
r2_h_S0T1_min=0, r2_h_S0T1_max=1, r2_h_T0T1_min=0.2, r2_h_T0T1_max=1, 
r2_h_T0S1_min=0, r2_h_T0S1_max=1, Monotonicity="General",
Type="All.Densities", MainPlot=" ", Cex.Legend=1, 
Cex.Position="topright", ylim=c(0, 20)) 

## End(Not run)

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

Description

The function ICA.BinBin.Grid.Full quantifies surrogacy in the single-trial causal-inference framework (individual causal association and causal concordance) when both the surrogate and the true endpoints are binary outcomes. This method provides an alternative for ICA.BinBin and ICA.BinBin.Grid.Sample. It uses an alternative strategy to identify plausible values for \pi. See Details below.

Usage

ICA.BinBin.Grid.Full(pi1_1_, pi1_0_, pi_1_1, pi_1_0, pi0_1_, pi_0_1, 
Monotonicity=c("General"), pi_1001=seq(0, 1, by=.02), 
pi_1110=seq(0, 1, by=.02), pi_1101=seq(0, 1, by=.02),
pi_1011=seq(0, 1, by=.02), pi_1111=seq(0, 1, by=.02), 
pi_0110=seq(0, 1, by=.02), pi_0011=seq(0, 1, by=.02), 
pi_0111=seq(0, 1, by=.02), pi_1100=seq(0, 1, by=.02), 
Seed=sample(1:100000, size=1))

Arguments

Details

In the continuous normal setting, surroagacy can be assessed by studying the association between the individual causal effects on S and T (see ICA.ContCont). In that setting, the Pearson correlation is the obvious measure of association.

When S and T are binary endpoints, multiple alternatives exist. Alonso et al. (2014) proposed the individual causal association (ICA; R_{H}^{2}), which captures the association between the individual causal effects of the treatment on S (\Delta_S) and T (\Delta_T) using information-theoretic principles.

The function ICA.BinBin.Grid.Full computes R_{H}^{2} using a grid-based approach where all possible combinations of the specified grids for the parameters that are allowed that are allowed to vary freely are considered. When it is not assumed that monotonicity holds for both S and T, the computationally less demanding algorithm ICA.BinBin.Grid.Sample may be preferred.

Value

An object of class ICA.BinBin with components,

Author(s)

Wim Van der Elst, Paul Meyvisch, Ariel Alonso & Geert Molenberghs

References

Alonso, A., Van der Elst, W., & Molenberghs, G. (2014). Validation of surrogate endpoints: the binary-binary setting from a causal inference perspective.

Buyse, M., Burzykowski, T., Aloso, A., & Molenberghs, G. (2014). Direct estimation of joint counterfactual probabilities, with application to surrogate marker validation.

See Also

ICA.ContCont, MICA.ContCont, ICA.BinBin, ICA.BinBin.Grid.Sample

Examples

## Not run:  # time consuming code part
# Compute R2_H given the marginals, 
# assuming monotonicity for S and T and grids
# pi_0111=seq(0, 1, by=.001) and 
# pi_1100=seq(0, 1, by=.001)
ICA <- ICA.BinBin.Grid.Full(pi1_1_=0.2619048, pi1_0_=0.2857143, pi_1_1=0.6372549, 
pi_1_0=0.07843137, pi0_1_=0.1349206, pi_0_1=0.127451,  
pi_0111=seq(0, 1, by=.01), pi_1100=seq(0, 1, by=.01), Seed=1)

# obtain plot of R2_H
plot(ICA, R2_H=TRUE)

## End(Not run)

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

Description

The function ICA.BinBin.Grid.Sample quantifies surrogacy in the single-trial causal-inference framework (individual causal association and causal concordance) when both the surrogate and the true endpoints are binary outcomes. This method provides an alternative for ICA.BinBin and ICA.BinBin.Grid.Full. It uses an alternative strategy to identify plausible values for \pi. See Details below.

Usage

ICA.BinBin.Grid.Sample(pi1_1_, pi1_0_, pi_1_1, pi_1_0, pi0_1_,
pi_0_1, Monotonicity=c("General"), M=100000,
Volume.Perc=0, Seed=sample(1:100000, size=1))

Arguments

Details

In the continuous normal setting, surroagacy can be assessed by studying the association between the individual causal effects on S and T (see ICA.ContCont). In that setting, the Pearson correlation is the obvious measure of association.

When S and T are binary endpoints, multiple alternatives exist. Alonso et al. (2014) proposed the individual causal association (ICA; R_{H}^{2}), which captures the association between the individual causal effects of the treatment on S (\Delta_S) and T (\Delta_T) using information-theoretic principles.

The function ICA.BinBin.Grid.Full computes R_{H}^{2} using a grid-based approach where all possible combinations of the specified grids for the parameters that are allowed that are allowed to vary freely are considered. When it is not assumed that monotonicity holds for both S and T, the number of possible combinations become very high. The function ICA.BinBin.Grid.Sample considers a random sample of all possible combinations.

Value

An object of class ICA.BinBin with components,

Author(s)

Wim Van der Elst, Paul Meyvisch, Ariel Alonso & Geert Molenberghs

References

Alonso, A., Van der Elst, W., & Molenberghs, G. (2014). Validation of surrogate endpoints: the binary-binary setting from a causal inference perspective.

Buyse, M., Burzykowski, T., Aloso, A., & Molenberghs, G. (2014). Direct estimation of joint counterfactual probabilities, with application to surrogate marker validation.

See Also

ICA.ContCont, MICA.ContCont, ICA.BinBin, ICA.BinBin.Grid.Sample

Examples

## Not run:  #time-consuming code parts
# Compute R2_H given the marginals,
# assuming monotonicity for S and T and grids
# pi_0111=seq(0, 1, by=.001) and
# pi_1100=seq(0, 1, by=.001)
ICA <- ICA.BinBin.Grid.Sample(pi1_1_=0.261, pi1_0_=0.285,
pi_1_1=0.637, pi_1_0=0.078, pi0_1_=0.134, pi_0_1=0.127,
Monotonicity=c("Surr.True.Endp"), M=2500, Seed=1)

# obtain plot of R2_H
plot(ICA, R2_H=TRUE)

## End(Not run)

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.

Description

The function ICA.BinBin.Grid.Sample.Uncert quantifies surrogacy in the single-trial causal-inference framework (individual causal association and causal concordance) when both the surrogate and the true endpoints are binary outcomes. This method provides an alternative for ICA.BinBin and ICA.BinBin.Grid.Full. It uses an alternative strategy to identify plausible values for \pi. The function allows to account for sampling variability in the marginal \pi. See Details below.

Usage

ICA.BinBin.Grid.Sample.Uncert(pi1_1_, pi1_0_, pi_1_1, pi_1_0, pi0_1_,
pi_0_1, Monotonicity=c("General"), M=100000,
Volume.Perc=0, Seed=sample(1:100000, size=1))

Arguments

Details

In the continuous normal setting, surroagacy can be assessed by studying the association between the individual causal effects on S and T (see ICA.ContCont). In that setting, the Pearson correlation is the obvious measure of association.

When S and T are binary endpoints, multiple alternatives exist. Alonso et al. (2014) proposed the individual causal association (ICA; R_{H}^{2}), which captures the association between the individual causal effects of the treatment on S (\Delta_S) and T (\Delta_T) using information-theoretic principles.

The function ICA.BinBin.Grid.Full computes R_{H}^{2} using a grid-based approach where all possible combinations of the specified grids for the parameters that are allowed that are allowed to vary freely are considered. When it is not assumed that monotonicity holds for both S and T, the number of possible combinations become very high. The function ICA.BinBin.Grid.Sample.Uncert considers a random sample of all possible combinations.

Value

An object of class ICA.BinBin with components,

Author(s)

Wim Van der Elst, Paul Meyvisch, Ariel Alonso & Geert Molenberghs

References

Alonso, A., Van der Elst, W., & Molenberghs, G. (2014). Validation of surrogate endpoints: the binary-binary setting from a causal inference perspective.

Buyse, M., Burzykowski, T., Aloso, A., & Molenberghs, G. (2014). Direct estimation of joint counterfactual probabilities, with application to surrogate marker validation.

See Also

ICA.ContCont, MICA.ContCont, ICA.BinBin, ICA.BinBin.Grid.Sample.Uncert

Examples

# Compute R2_H given the marginals (sample from uniform),
# assuming no monotonicity
ICA_No2 <- ICA.BinBin.Grid.Sample.Uncert(pi1_1_=runif(10000, 0.3562, 0.4868),
pi0_1_=runif(10000, 0.0240, 0.0837), pi1_0_=runif(10000, 0.0240, 0.0837),
pi_1_1=runif(10000, 0.4434, 0.5742), pi_1_0=runif(10000, 0.0081, 0.0533),
pi_0_1=runif(10000, 0.0202, 0.0763), Seed=1, Monotonicity=c("No"), M=1000)

summary(ICA_No2)

# obtain plot of R2_H
plot(ICA_No2)

Assess surrogacy in the causal-inference single-trial setting in the binary-continuous case

Description

The function ICA.BinCont quantifies surrogacy in the single-trial setting within the causal-inference framework (individual causal association) when the surrogate endpoint is continuous (normally distributed) and the true endpoint is a binary outcome. For details, see Alonso Abad et al. (2023).

Usage

ICA.BinCont(Dataset, Surr, True, Treat, 
  BS=FALSE,
  G_pi_10=c(0,1), 
  G_rho_01_00=c(-1,1), 
  G_rho_01_01=c(-1,1), 
  G_rho_01_10=c(-1,1), 
  G_rho_01_11=c(-1,1), 
  Theta.S_0, 
  Theta.S_1, 
  M=1000, Seed=123, 
  Monotonicity=FALSE,
  Independence=FALSE,
  HAA=FALSE,
  Cond_ind=FALSE,
  Plots=TRUE, Save.Plots="No", Show.Details=FALSE)

Arguments

Value

An object of class ICA.BinCont with components,

Author(s)

Wim Van der Elst, Fenny Ong, Ariel Alonso, and Geert Molenberghs

References

Alonso Abad, A., Ong, F., Stijven, F., Van der Elst, W., Molenberghs, G., Van Keilegom, I., Verbeke, G., & Callegaro, A. (2023). An information-theoretic approach for the assessment of a continuous outcome as a surrogate for a binary true endpoint based on causal inference: Application to vaccine evaluation.

See Also

ICA.ContCont, MICA.ContCont, ICA.BinBin

Examples

## Not run: # Time consuming code part
data(Schizo)
Fit <- ICA.BinCont(Dataset = Schizo, Surr = BPRS, True = PANSS_Bin, 
Theta.S_0=c(-10,-5,5,10,10,10,10,10), Theta.S_1=c(-10,-5,5,10,10,10,10,10), 
Treat=Treat, M=50, Seed=1)

summary(Fit)
plot(Fit)

## End(Not run)

Assess surrogacy in the causal-inference single-trial setting in the binary-continuous case with an additional bootstrap procedure before the assessment

Description

The function ICA.BinCont.BS quantifies surrogacy in the single-trial setting within the causal-inference framework (individual causal association) when the surrogate endpoint is continuous (normally distributed) and the true endpoint is a binary outcome. This function also allows for an additional bootstrap procedure before the assessment to take the imprecision due to finite sample size into account. For details, see Alonso Abad et al. (2023).

Usage

ICA.BinCont.BS(Dataset, Surr, True, Treat, 
  BS=TRUE,
  nb=300,
  G_pi_10=c(0,1), 
  G_rho_01_00=c(-1,1), 
  G_rho_01_01=c(-1,1), 
  G_rho_01_10=c(-1,1), 
  G_rho_01_11=c(-1,1), 
  Theta.S_0, 
  Theta.S_1, 
  M=1000, Seed=123, 
  Monotonicity=FALSE,
  Independence=FALSE,
  HAA=FALSE,
  Cond_ind=FALSE,
  Plots=TRUE, Save.Plots="No", Show.Details=FALSE)

Arguments

Value

An object of class ICA.BinCont with components,

Author(s)

Wim Van der Elst, Fenny Ong, Ariel Alonso, and Geert Molenberghs

References

Alonso Abad, A., Ong, F., Stijven, F., Van der Elst, W., Molenberghs, G., Van Keilegom, I., Verbeke, G., & Callegaro, A. (2023). An information-theoretic approach for the assessment of a continuous outcome as a surrogate for a binary true endpoint based on causal inference: Application to vaccine evaluation.

See Also

ICA.BinCont

Examples

## Not run: # Time consuming code part
data(Schizo)
Fit <- ICA.BinCont.BS(Dataset = Schizo, Surr = BPRS, True = PANSS_Bin, nb = 10, 
Theta.S_0=c(-10,-5,5,10,10,10,10,10), Theta.S_1=c(-10,-5,5,10,10,10,10,10), 
Treat=Treat, M=50, Seed=1)

summary(Fit)
plot(Fit)

## End(Not run)

Assess surrogacy in the causal-inference single-trial setting (Individual Causal Association, ICA) in the Continuous-continuous case

Description

The function ICA.ContCont quantifies surrogacy in the single-trial causal-inference framework. See Details below.

Usage

ICA.ContCont(T0S0, T1S1, T0T0=1, T1T1=1, S0S0=1, S1S1=1, T0T1=seq(-1, 1, by=.1), 
T0S1=seq(-1, 1, by=.1), T1S0=seq(-1, 1, by=.1), S0S1=seq(-1, 1, by=.1))

Arguments

Details

Based on the causal-inference framework, it is assumed that each subject j has four counterfactuals (or potential outcomes), i.e., T_{0j}, T_{1j}, S_{0j}, and S_{1j}. Let T_{0j} and T_{1j} denote the counterfactuals for the true endpoint (T) under the control (Z=0) and the experimental (Z=1) treatments of subject j, respectively. Similarly, S_{0j} and S_{1j} denote the corresponding counterfactuals for the surrogate endpoint (S) under the control and experimental treatments, respectively. The individual causal effects of Z on T and S for a given subject j are then defined as \Delta_{T_{j}}=T_{1j}-T_{0j} and \Delta_{S_{j}}=S_{1j}-S_{0j}, respectively.

In the single-trial causal-inference framework, surrogacy can be quantified as the correlation between the individual causal effects of Z on S and T (for details, see Alonso et al., submitted):

\rho_{\Delta}=\rho(\Delta_{T_{j}},\:\Delta_{S_{j}})=\frac{\sqrt{\sigma_{S_{0}S_{0}}\sigma_{T_{0}T_{0}}}\rho_{S_{0}T_{0}}+\sqrt{\sigma_{S_{1}S_{1}}\sigma_{T_{1}T_{1}}}\rho_{S_{1}T_{1}}-\sqrt{\sigma_{S_{0}S_{0}}\sigma_{T_{1}T_{1}}}\rho_{S_{0}T_{1}}-\sqrt{\sigma_{S_{1}S_{1}}\sigma_{T_{0}T_{0}}}\rho_{S_{1}T_{0}}}{\sqrt{(\sigma_{T_{0}T_{0}}+\sigma_{T_{1}T_{1}}-2\sqrt{\sigma_{T_{0}T_{0}}\sigma_{T_{1}T_{1}}}\rho_{T_{0}T_{1}})(\sigma_{S_{0}S_{0}}+\sigma_{S_{1}S_{1}}-2\sqrt{\sigma_{S_{0}S_{0}}\sigma_{S_{1}S_{1}}}\rho_{S_{0}S_{1}})}},

where the correlations \rho_{S_{0}T_{1}}, \rho_{S_{1}T_{0}}, \rho_{T_{0}T_{1}}, and \rho_{S_{0}S_{1}} are not estimable. It is thus warranted to conduct a sensitivity analysis (by considering vectors of possible values for the correlations between the counterfactuals – rather than point estimates).

When the user specifies a vector of values that should be considered for one or more of the counterfactual correlations in the above expression, the function ICA.ContCont constructs all possible matrices that can be formed as based on these values, identifies the matrices that are positive definite (i.e., valid correlation matrices), and computes \rho_{\Delta} for each of these matrices. The obtained vector of \rho_{\Delta} values can subsequently be used to examine (i) the impact of different assumptions regarding the correlations between the counterfactuals on the results (see also plot Causal-Inference ContCont), and (ii) the extent to which proponents of the causal-inference and meta-analytic frameworks will reach the same conclusion with respect to the appropriateness of the candidate surrogate at hand.

The function ICA.ContCont also generates output that is useful to examine the plausibility of finding a good surrogate endpoint (see GoodSurr in the Value section below). For details, see Alonso et al. (submitted).

Notes

A single \rho_{\Delta} value is obtained when all correlations in the function call are scalars.

Value

An object of class ICA.ContCont with components,

Author(s)

Wim Van der Elst, Ariel Alonso, & Geert Molenberghs

References

Alonso, A., Van der Elst, W., Molenberghs, G., Buyse, M., & Burzykowski, T. (submitted). On the relationship between the causal-inference and meta-analytic paradigms for the validation of surrogate markers.

See Also

MICA.ContCont, ICA.Sample.ContCont, Single.Trial.RE.AA, plot Causal-Inference ContCont

Examples

## Not run:  #time-consuming code parts
# Generate the vector of ICA.ContCont values when rho_T0S0=rho_T1S1=.95, 
# sigma_T0T0=90, sigma_T1T1=100,sigma_ S0S0=10, sigma_S1S1=15, and  
# the grid of values {0, .2, ..., 1} is considered for the correlations
# between the counterfactuals:
SurICA <- ICA.ContCont(T0S0=.95, T1S1=.95, T0T0=90, T1T1=100, S0S0=10, S1S1=15,
T0T1=seq(0, 1, by=.2), T0S1=seq(0, 1, by=.2), T1S0=seq(0, 1, by=.2), 
S0S1=seq(0, 1, by=.2))

# Examine and plot the vector of generated ICA values:
summary(SurICA)
plot(SurICA)

# Obtain the positive definite matrices than can be formed as based on the 
# specified (vectors) of the correlations (these matrices are used to 
# compute the ICA values)
SurICA$Pos.Def

# Same, but specify vectors for rho_T0S0 and rho_T1S1: Sample from
# normal with mean .95 and SD=.05 (to account for uncertainty 
# in estimation) 
SurICA2 <- ICA.ContCont(T0S0=rnorm(n=10000000, mean=.95, sd=.05), 
T1S1=rnorm(n=10000000, mean=.95, sd=.05), 
T0T0=90, T1T1=100, S0S0=10, S1S1=15,
T0T1=seq(0, 1, by=.2), T0S1=seq(0, 1, by=.2), T1S0=seq(0, 1, by=.2), 
S0S1=seq(0, 1, by=.2))

# Examine results
summary(SurICA2)
plot(SurICA2)

## End(Not run)

Assess surrogacy in the causal-inference single-trial setting (Individual Causal Association, ICA) using a continuous univariate T and multiple continuous S

Description

The function ICA.ContCont.MultS quantifies surrogacy in the single-trial causal-inference framework where T is continuous and there are multiple continuous S.

Usage

ICA.ContCont.MultS(M = 500, N, Sigma, 
G = seq(from=-1, to=1, by = .00001),
Seed=c(123), Show.Progress=FALSE)

Arguments

Details

The multivariate ICA (R^2_{H}) is not identifiable because the individual causal treatment effects on T, S_1, ..., S_k cannot be observed. A simulation-based sensitivity analysis is therefore conducted in which the multivariate ICA (R^2_{H}) is estimated across a set of plausible values for the unidentifiable correlations. To this end, consider the variance covariance matrix of the potential outcomes \boldsymbol{\Sigma} (0 and 1 subscripts refer to the control and experimental treatments, respectively):

\boldsymbol{\Sigma} = \left(\begin{array}{ccccccccc} \sigma_{T_{0}T_{0}}\\ \sigma_{T_{0}T_{1}} & \sigma_{T_{1}T_{1}}\\ \sigma_{T_{0}S1_{0}} & \sigma_{T_{1}S1_{0}} & \sigma_{S1_{0}S1_{0}}\\ \sigma_{T_{0}S1_{1}} & \sigma_{T_{1}S1_{1}} & \sigma_{S1_{0}S1_{1}} & \sigma_{S1_{1}S1_{1}}\\ \sigma_{T_{0}S2_{0}} & \sigma_{T_{1}S2_{0}} & \sigma_{S1_{0}S2_{0}} & \sigma_{S1_{1}S2_{0}} & \sigma_{S2_{0}S2_{0}}\\ \sigma_{T_{0}S2_{1}} & \sigma_{T_{1}S2_{1}} & \sigma_{S1_{0}S2_{1}} & \sigma_{S1_{1}S2_{1}} & \sigma_{S2_{0}S2_{1}} & \sigma_{S2_{1}S2_{1}}\\ ... & ... & ... & ... & ... & ... & \ddots\\ \sigma_{T_{0}Sk_{0}} & \sigma_{T_{1}Sk_{0}} & \sigma_{S1_{0}Sk_{0}} & \sigma_{S1_{1}Sk_{0}} & \sigma_{S2_{0}Sk_{0}} & \sigma_{S2_{1}Sk_{0}} & ... & \sigma_{Sk_{0}Sk_{0}}\\ \sigma_{T_{0}Sk_{1}} & \sigma_{T_{1}Sk_{1}} & \sigma_{S1_{0}Sk_{1}} & \sigma_{S1_{1}Sk_{1}} & \sigma_{S2_{0}Sk_{1}} & \sigma_{S2_{1}Sk_{1}} & ... & \sigma_{Sk_{0}Sk_{1}} & \sigma_{Sk_{1}Sk_{1}}. \end{array}\right)

The ICA.ContCont.MultS function requires the user to specify a distribution G for the unidentified correlations. Next, the identifiable correlations are fixed at their estimated values and the unidentifiable correlations are independently and randomly sampled from G. In the function call, the unidentifiable correlations are marked by specifying NA in the Sigma matrix (see example section below). The algorithm generates a large number of 'completed' matrices, and only those that are positive definite are retained (the number of positive definite matrices that should be obtained is specified by the M= argument in the function call). Based on the identifiable variances, these positive definite correlation matrices are converted to covariance matrices \boldsymbol{\Sigma} and the multiple-surrogate ICA are estimated.

An issue with this approach (i.e., substituting unidentified correlations by random and independent samples from G) is that the probability of obtaining a positive definite matrix is very low when the dimensionality of the matrix increases. One approach to increase the efficiency of the algorithm is to build-up the correlation matrix in a gradual way. In particular, the property that a \left(k \times k\right) matrix is positive definite if and only if all principal minors are positive (i.e., Sylvester's criterion) can be used. In other words, a \left(k \times k\right) matrix is positive definite when the determinants of the upper-left \left(2 \times 2\right), \left(3 \times 3\right), ..., \left(k \times k\right) submatrices all have a positive determinant. Thus, when a positive definite \left(k \times k\right) matrix has to be generated, one can start with the upper-left \left(2 \times 2\right) submatrix and randomly sample a value from the unidentified correlation (here: \rho_{T_0T_0}) from G. When the determinant is positive (which will always be the case for a \left(2 \times 2\right) matrix), the same procedure is used for the upper-left \left(3 \times 3\right) submatrix, and so on. When a particular draw from G for a particular submatrix does not give a positive determinant, new values are sampled for the unidentified correlations until a positive determinant is obtained. In this way, it can be guaranteed that the final \left(k \times k\right) submatrix will be positive definite. The latter approach is used in the current function. This procedure is used to generate many positive definite matrices. Based on these matrices, \boldsymbol{\Sigma_{\Delta}} is generated and the multivariate ICA (R^2_{H}) is computed (for details, see Van der Elst et al., 2017).

Value

An object of class ICA.ContCont.MultS with components,

Author(s)

Wim Van der Elst, Ariel Alonso, & Geert Molenberghs

References

Van der Elst, W., Alonso, A. A., & Molenberghs, G. (2017). Univariate versus multivariate surrogate endpoints.

See Also

MICA.ContCont, ICA.ContCont, Single.Trial.RE.AA, plot Causal-Inference ContCont, ICA.ContCont.MultS_alt

Examples

## Not run:  #time-consuming code parts
# Specify matrix Sigma (var-cavar matrix T_0, T_1, S1_0, S1_1, ...)
# here for 1 true endpoint and 3 surrogates

s<-matrix(rep(NA, times=64),8)
s[1,1] <- 450; s[2,2] <- 413.5; s[3,3] <- 174.2; s[4,4] <- 157.5; 
s[5,5] <- 244.0; s[6,6] <- 229.99; s[7,7] <- 294.2; s[8,8] <- 302.5
s[3,1] <- 160.8; s[5,1] <- 208.5; s[7,1] <- 268.4 
s[4,2] <- 124.6; s[6,2] <- 212.3; s[8,2] <- 287.1
s[5,3] <- 160.3; s[7,3] <- 142.8 
s[6,4] <- 134.3; s[8,4] <- 130.4 
s[7,5] <- 209.3; 
s[8,6] <- 214.7 
s[upper.tri(s)] = t(s)[upper.tri(s)]

# Marix looks like (NA indicates unidentified covariances):
#            T_0    T_1  S1_0  S1_1  S2_0   S2_1  S2_0  S2_1
#            [,1]  [,2]  [,3]  [,4]  [,5]   [,6]  [,7]  [,8]
# T_0  [1,] 450.0    NA 160.8    NA 208.5     NA 268.4    NA
# T_1  [2,]    NA 413.5    NA 124.6    NA 212.30    NA 287.1
# S1_0 [3,] 160.8    NA 174.2    NA 160.3     NA 142.8    NA
# S1_1 [4,]    NA 124.6    NA 157.5    NA 134.30    NA 130.4
# S2_0 [5,] 208.5    NA 160.3    NA 244.0     NA 209.3    NA
# S2_1 [6,]    NA 212.3    NA 134.3    NA 229.99    NA 214.7
# S3_0 [7,] 268.4    NA 142.8    NA 209.3     NA 294.2    NA
# S3_1 [8,]    NA 287.1    NA 130.4    NA 214.70    NA 302.5

# Conduct analysis
ICA <- ICA.ContCont.MultS(M=100, N=200, Show.Progress = TRUE,
  Sigma=s, G = seq(from=-1, to=1, by = .00001), Seed=c(123))

# Explore results
summary(ICA)
plot(ICA)

## End(Not run)

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

Description

The function ICA.ContCont.MultS.MPC quantifies surragacy in the single-trial causal-inference framework in which the true endpoint (T) and multiple surrogates (S) are continuous. This function is a modification of the ICA.ContCont.MultS.PC algorithm based on partial correlations. it mitigates the effect of non-informative surrogates and effectively explores the PD space to capture the ICA range (Florez, et al. 2021).

Usage

ICA.ContCont.MultS.MPC(M=1000,N,Sigma,prob = NULL,Seed=123,
Save.Corr=F, Show.Progress=FALSE)

Arguments

Details

The multivariate ICA (R^2_{H}) is not identifiable because the individual causal treatment effects on T, S_1, ..., S_k cannot be observed. A simulation-based sensitivity analysis is therefore conducted in which the multivariate ICA (R^2_{H}) is estimated across a set of plausible values for the unidentifiable correlations. To this end, consider the variance covariance matrix of the potential outcomes \boldsymbol{\Sigma} (0 and 1 subscripts refer to the control and experimental treatments, respectively):

\boldsymbol{\Sigma} = \left(\begin{array}{ccccccccc} \sigma_{T_{0}T_{0}}\\ \sigma_{T_{0}T_{1}} & \sigma_{T_{1}T_{1}}\\ \sigma_{T_{0}S1_{0}} & \sigma_{T_{1}S1_{0}} & \sigma_{S1_{0}S1_{0}}\\ \sigma_{T_{0}S1_{1}} & \sigma_{T_{1}S1_{1}} & \sigma_{S1_{0}S1_{1}} & \sigma_{S1_{1}S1_{1}}\\ \sigma_{T_{0}S2_{0}} & \sigma_{T_{1}S2_{0}} & \sigma_{S1_{0}S2_{0}} & \sigma_{S1_{1}S2_{0}} & \sigma_{S2_{0}S2_{0}}\\ \sigma_{T_{0}S2_{1}} & \sigma_{T_{1}S2_{1}} & \sigma_{S1_{0}S2_{1}} & \sigma_{S1_{1}S2_{1}} & \sigma_{S2_{0}S2_{1}} & \sigma_{S2_{1}S2_{1}}\\ ... & ... & ... & ... & ... & ... & \ddots\\ \sigma_{T_{0}Sk_{0}} & \sigma_{T_{1}Sk_{0}} & \sigma_{S1_{0}Sk_{0}} & \sigma_{S1_{1}Sk_{0}} & \sigma_{S2_{0}Sk_{0}} & \sigma_{S2_{1}Sk_{0}} & ... & \sigma_{Sk_{0}Sk_{0}}\\ \sigma_{T_{0}Sk_{1}} & \sigma_{T_{1}Sk_{1}} & \sigma_{S1_{0}Sk_{1}} & \sigma_{S1_{1}Sk_{1}} & \sigma_{S2_{0}Sk_{1}} & \sigma_{S2_{1}Sk_{1}} & ... & \sigma_{Sk_{0}Sk_{1}} & \sigma_{Sk_{1}Sk_{1}}. \end{array}\right)

The identifiable correlations are fixed at their estimated values and the unidentifiable correlations are independently and randomly sampled using a modification of an algorithm based on partial correlations (PC), called modified partial correlation (MPC) algorithm. In the function call, the unidentifiable correlations are marked by specifying NA in the Sigma matrix (see example section below).

The PC algorithm generate each correlation matrix progressively based on parameterization of terms of the correlations \rho_{i,i+1}, for i=1,\ldots,d-1, and the partial correlations \rho_{i,j|i+1,\ldots,j-1}, for j-i>2 (for details, see Joe, 2006 and Florez et al., 2018). The MPC algorithm randomly fixed some of the unidentifiable correlations to zero in order to explore the PD, which is coherent with the estimable entries of the correlation matrix, to capture the ICA range more efficiently.

Based on the identifiable variances, these correlation matrices are converted to covariance matrices \boldsymbol{\Sigma} and the multiple-surrogate ICA are estimated (for details, see Van der Elst et al., 2017).

This approach to simulate the unidentifiable parameters of \boldsymbol{\Sigma} is computationally more efficient than the one used in the function ICA.ContCont.MultS.

Value

An object of class ICA.ContCont.MultS.PC with components,

Author(s)

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

References

Florez, A., Molenberghs, G., Van der Elst, W., Alonso, A. A. (2021). An efficient algorithm for causally assessing surrogacy in a multivariate setting.

Florez, A., Alonso, A. A., Molenberghs, G. & Van der Elst, W. (2020). Generating random correlation matrices with fixed values: An application to the evaluation of multivariate surrogate endpoints. Computational Statistics & Data Analysis 142.

Joe, H. (2006). Generating random correlation matrices based on partial correlations. Journal of Multivariate Analysis, 97(10):2177-2189.

Van der Elst, W., Alonso, A. A., & Molenberghs, G. (2017). Univariate versus multivariate surrogate endpoints.

See Also

MICA.ContCont, ICA.ContCont, Single.Trial.RE.AA, plot Causal-Inference ContCont, ICA.ContCont.MultS, ICA.ContCont.MultS_alt

Examples

## Not run:  
# Specify matrix Sigma (var-cavar matrix T_0, T_1, S1_0, S1_1, ...)
# here we have 1 true endpoint and 10 surrogates (8 of these are non-informative)

Sigma = ks::invvech(
  c(25, NA, 17.8, NA, -10.6, NA, 0, NA, 0, NA, 0, NA, 0, NA, 0, NA, 0, NA, 0, NA, 0, NA, 
    4, NA, -0.32, NA, -1.32, NA, 0, NA, 0, NA, 0, NA, 0, NA, 0, NA, 0, NA, 0, NA, 0, 16, 
    NA, -4, NA, 0, NA, 0, NA, 0, NA, 0, NA, 0, NA, 0, NA, 0, NA, 0, NA, 1, NA, 0.48, NA, 
    0, NA, 0, NA, 0, NA, 0, NA, 0, NA, 0, NA, 0, NA, 0, 16, NA, 0, NA, 0, NA, 0, NA, 0, 
    NA, 0, NA, 0, NA, 0, NA, 0, NA, 1, NA, 0, NA, 0, NA, 0, NA, 0, NA, 0, NA, 0, NA, 0, 
    NA, 0, 16, NA, 8, NA, 8, NA, 8, NA, 8, NA, 8, NA, 8, NA, 8, NA, 1, NA, 0.5, NA, 0.5, 
    NA, 0.5, NA, 0.5, NA, 0.5, NA, 0.5, NA, 0.5, 16, NA, 8, NA, 8, NA, 8, NA, 8, NA, 8, 
    NA, 8, NA, 1, NA, 0.5, NA, 0.5, NA, 0.5, NA, 0.5, NA, 0.5, NA, 0.5, 16, NA, 8, NA, 
    8, NA, 8, NA, 8, NA, 8, NA, 1,NA,0.5,NA,0.5,NA,0.5,NA,0.5,NA,0.5, 16, NA, 8, NA, 8, 
    NA, 8, NA, 8, NA, 1, NA, 0.5, NA, 0.5, NA, 0.5, NA, 0.5, 16, NA, 8, NA, 8, NA, 8, NA,
    1, NA, 0.5, NA, 0.5, NA, 0.5, 16, NA, 8, NA, 8, NA, 1, NA, 0.5, NA, 0.5, 16, NA, 8, NA,
    1, NA, 0.5, 16, NA, 1)) 

# Conduct analysis using the PC and MPC algorithm 
## first evaluating two surrogates
ICA.PC.2 = ICA.ContCont.MultS.PC(M = 30000, N=200, Sigma[1:6,1:6], Seed = 123) 
ICA.MPC.2 = ICA.ContCont.MultS.MPC(M = 30000, N=200, Sigma[1:6,1:6],prob=NULL, 
Seed = 123, Save.Corr=T, Show.Progress = TRUE) 

## later evaluating two surrogates
ICA.PC.10 = ICA.ContCont.MultS.PC(M = 150000, N=200, Sigma, Seed = 123) 
ICA.MPC.10 = ICA.ContCont.MultS.MPC(M = 150000, N=200, Sigma,prob=NULL, 
Seed = 123, Save.Corr=T, Show.Progress = TRUE) 

# Explore results
range(ICA.PC.2$R2_H)
range(ICA.PC.10$R2_H)

range(ICA.MPC.2$R2_H)
range(ICA.MPC.10$R2_H)
## as we observe, the MPC algorithm displays a wider interval of possible values for the ICA

## End(Not run)

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

Description

The function ICA.ContCont.MultS quantifies surrogacy in the single-trial causal-inference framework where T is continuous and there are multiple continuous S. This function provides an alternative for ICA.ContCont.MultS.

Usage

ICA.ContCont.MultS.PC(M=1000,N,Sigma,Seed=123,Show.Progress=FALSE)

Arguments

Details

The multivariate ICA (R^2_{H}) is not identifiable because the individual causal treatment effects on T, S_1, ..., S_k cannot be observed. A simulation-based sensitivity analysis is therefore conducted in which the multivariate ICA (R^2_{H}) is estimated across a set of plausible values for the unidentifiable correlations. To this end, consider the variance covariance matrix of the potential outcomes \boldsymbol{\Sigma} (0 and 1 subscripts refer to the control and experimental treatments, respectively):

\boldsymbol{\Sigma} = \left(\begin{array}{ccccccccc} \sigma_{T_{0}T_{0}}\\ \sigma_{T_{0}T_{1}} & \sigma_{T_{1}T_{1}}\\ \sigma_{T_{0}S1_{0}} & \sigma_{T_{1}S1_{0}} & \sigma_{S1_{0}S1_{0}}\\ \sigma_{T_{0}S1_{1}} & \sigma_{T_{1}S1_{1}} & \sigma_{S1_{0}S1_{1}} & \sigma_{S1_{1}S1_{1}}\\ \sigma_{T_{0}S2_{0}} & \sigma_{T_{1}S2_{0}} & \sigma_{S1_{0}S2_{0}} & \sigma_{S1_{1}S2_{0}} & \sigma_{S2_{0}S2_{0}}\\ \sigma_{T_{0}S2_{1}} & \sigma_{T_{1}S2_{1}} & \sigma_{S1_{0}S2_{1}} & \sigma_{S1_{1}S2_{1}} & \sigma_{S2_{0}S2_{1}} & \sigma_{S2_{1}S2_{1}}\\ ... & ... & ... & ... & ... & ... & \ddots\\ \sigma_{T_{0}Sk_{0}} & \sigma_{T_{1}Sk_{0}} & \sigma_{S1_{0}Sk_{0}} & \sigma_{S1_{1}Sk_{0}} & \sigma_{S2_{0}Sk_{0}} & \sigma_{S2_{1}Sk_{0}} & ... & \sigma_{Sk_{0}Sk_{0}}\\ \sigma_{T_{0}Sk_{1}} & \sigma_{T_{1}Sk_{1}} & \sigma_{S1_{0}Sk_{1}} & \sigma_{S1_{1}Sk_{1}} & \sigma_{S2_{0}Sk_{1}} & \sigma_{S2_{1}Sk_{1}} & ... & \sigma_{Sk_{0}Sk_{1}} & \sigma_{Sk_{1}Sk_{1}}. \end{array}\right)

The identifiable correlations are fixed at their estimated values and the unidentifiable correlations are independently and randomly sampled using an algorithm based on partial correlations (PC). In the function call, the unidentifiable correlations are marked by specifying NA in the Sigma matrix (see example section below). The PC algorithm generate each correlation matrix progressively based on parameterization of terms of the correlations \rho_{i,i+1}, for i=1,\ldots,d-1, and the partial correlations \rho_{i,j|i+1,\ldots,j-1}, for j-i>2 (for details, see Joe, 2006 and Florez et al., 2018). Based on the identifiable variances, these correlation matrices are converted to covariance matrices \boldsymbol{\Sigma} and the multiple-surrogate ICA are estimated (for details, see Van der Elst et al., 2017).

This approach to simulate the unidentifiable parameters of \boldsymbol{\Sigma} is computationally more efficient than the one used in the function ICA.ContCont.MultS.

Value

An object of class ICA.ContCont.MultS.PC with components,

Author(s)

Alvaro Florez

References

Florez, A., Alonso, A. A., Molenberghs, G. & Van der Elst, W. (2018). Simulation of random correlation matrices with fixed values: comparison of algorithms and application on multiple surrogates assessment.

Joe, H. (2006). Generating random correlation matrices based on partial correlations. Journal of Multivariate Analysis, 97(10):2177-2189.

Van der Elst, W., Alonso, A. A., & Molenberghs, G. (2017). Univariate versus multivariate surrogate endpoints.

See Also

MICA.ContCont, ICA.ContCont, Single.Trial.RE.AA, plot Causal-Inference ContCont, ICA.ContCont.MultS, ICA.ContCont.MultS_alt

Examples

## Not run:  
# Specify matrix Sigma (var-cavar matrix T_0, T_1, S1_0, S1_1, ...)
# here for 1 true endpoint and 3 surrogates

s<-matrix(rep(NA, times=64),8)
s[1,1] <- 450; s[2,2] <- 413.5; s[3,3] <- 174.2; s[4,4] <- 157.5; 
s[5,5] <- 244.0; s[6,6] <- 229.99; s[7,7] <- 294.2; s[8,8] <- 302.5
s[3,1] <- 160.8; s[5,1] <- 208.5; s[7,1] <- 268.4 
s[4,2] <- 124.6; s[6,2] <- 212.3; s[8,2] <- 287.1
s[5,3] <- 160.3; s[7,3] <- 142.8 
s[6,4] <- 134.3; s[8,4] <- 130.4 
s[7,5] <- 209.3; 
s[8,6] <- 214.7 
s[upper.tri(s)] = t(s)[upper.tri(s)]

# Marix looks like (NA indicates unidentified covariances):
#            T_0    T_1  S1_0  S1_1  S2_0   S2_1  S2_0  S2_1
#            [,1]  [,2]  [,3]  [,4]  [,5]   [,6]  [,7]  [,8]
# T_0  [1,] 450.0    NA 160.8    NA 208.5     NA 268.4    NA
# T_1  [2,]    NA 413.5    NA 124.6    NA 212.30    NA 287.1
# S1_0 [3,] 160.8    NA 174.2    NA 160.3     NA 142.8    NA
# S1_1 [4,]    NA 124.6    NA 157.5    NA 134.30    NA 130.4
# S2_0 [5,] 208.5    NA 160.3    NA 244.0     NA 209.3    NA
# S2_1 [6,]    NA 212.3    NA 134.3    NA 229.99    NA 214.7
# S3_0 [7,] 268.4    NA 142.8    NA 209.3     NA 294.2    NA
# S3_1 [8,]    NA 287.1    NA 130.4    NA 214.70    NA 302.5

# Conduct analysis
ICA <- ICA.ContCont.MultS.PC(M=1000, N=200, Show.Progress = TRUE,
Sigma=s, Seed=c(123))

# Explore results
summary(ICA)
plot(ICA)

## End(Not run)

Assess surrogacy in the causal-inference single-trial setting (Individual Causal Association, ICA) using a continuous univariate T and multiple continuous S, alternative approach

Description

The function ICA.ContCont.MultS_alt quantifies surrogacy in the single-trial causal-inference framework where T is continuous and there are multiple continuous S. This function provides an alternative for ICA.ContCont.MultS.

Usage

ICA.ContCont.MultS_alt(M = 500, N, Sigma, 
G = seq(from=-1, to=1, by = .00001),
Seed=c(123), Model = "Delta_T ~ Delta_S1 + Delta_S2", 
Show.Progress=FALSE)

Arguments

Details

The multivariate ICA (R^2_{H}) is not identifiable because the individual causal treatment effects on T, S_1, ..., S_k cannot be observed. A simulation-based sensitivity analysis is therefore conducted in which the multivariate ICA (R^2_{H}) is estimated across a set of plausible values for the unidentifiable correlations. To this end, consider the variance covariance matrix of the potential outcomes \boldsymbol{\Sigma} (0 and 1 subscripts refer to the control and experimental treatments, respectively):

\boldsymbol{\Sigma} = \left(\begin{array}{ccccccccc} \sigma_{T_{0}T_{0}}\\ \sigma_{T_{0}T_{1}} & \sigma_{T_{1}T_{1}}\\ \sigma_{T_{0}S1_{0}} & \sigma_{T_{1}S1_{0}} & \sigma_{S1_{0}S1_{0}}\\ \sigma_{T_{0}S1_{1}} & \sigma_{T_{1}S1_{1}} & \sigma_{S1_{0}S1_{1}} & \sigma_{S1_{1}S1_{1}}\\ \sigma_{T_{0}S2_{0}} & \sigma_{T_{1}S2_{0}} & \sigma_{S1_{0}S2_{0}} & \sigma_{S1_{1}S2_{0}} & \sigma_{S2_{0}S2_{0}}\\ \sigma_{T_{0}S2_{1}} & \sigma_{T_{1}S2_{1}} & \sigma_{S1_{0}S2_{1}} & \sigma_{S1_{1}S2_{1}} & \sigma_{S2_{0}S2_{1}} & \sigma_{S2_{1}S2_{1}}\\ ... & ... & ... & ... & ... & ... & \ddots\\ \sigma_{T_{0}Sk_{0}} & \sigma_{T_{1}Sk_{0}} & \sigma_{S1_{0}Sk_{0}} & \sigma_{S1_{1}Sk_{0}} & \sigma_{S2_{0}Sk_{0}} & \sigma_{S2_{1}Sk_{0}} & ... & \sigma_{Sk_{0}Sk_{0}}\\ \sigma_{T_{0}Sk_{1}} & \sigma_{T_{1}Sk_{1}} & \sigma_{S1_{0}Sk_{1}} & \sigma_{S1_{1}Sk_{1}} & \sigma_{S2_{0}Sk_{1}} & \sigma_{S2_{1}Sk_{1}} & ... & \sigma_{Sk_{0}Sk_{1}} & \sigma_{Sk_{1}Sk_{1}}. \end{array}\right)

The ICA.ContCont.MultS_alt function requires the user to specify a distribution G for the unidentified correlations. Next, the identifiable correlations are fixed at their estimated values and the unidentifiable correlations are independently and randomly sampled from G. In the function call, the unidentifiable correlations are marked by specifying NA in the Sigma matrix (see example section below). The algorithm generates a large number of 'completed' matrices, and only those that are positive definite are retained (the number of positive definite matrices that should be obtained is specified by the M= argument in the function call). Based on the identifiable variances, these positive definite correlation matrices are converted to covariance matrices \boldsymbol{\Sigma} and the multiple-surrogate ICA are estimated.

An issue with this approach (i.e., substituting unidentified correlations by random and independent samples from G) is that the probability of obtaining a positive definite matrix is very low when the dimensionality of the matrix increases. One approach to increase the efficiency of the algorithm is to build-up the correlation matrix in a gradual way. In particular, the property that a \left(k \times k\right) matrix is positive definite if and only if all principal minors are positive (i.e., Sylvester's criterion) can be used. In other words, a \left(k \times k\right) matrix is positive definite when the determinants of the upper-left \left(2 \times 2\right), \left(3 \times 3\right), ..., \left(k \times k\right) submatrices all have a positive determinant. Thus, when a positive definite \left(k \times k\right) matrix has to be generated, one can start with the upper-left \left(2 \times 2\right) submatrix and randomly sample a value from the unidentified correlation (here: \rho_{T_0T_0}) from G. When the determinant is positive (which will always be the case for a \left(2 \times 2\right) matrix), the same procedure is used for the upper-left \left(3 \times 3\right) submatrix, and so on. When a particular draw from G for a particular submatrix does not give a positive determinant, new values are sampled for the unidentified correlations until a positive determinant is obtained. In this way, it can be guaranteed that the final \left(k \times k\right) submatrix will be positive definite. The latter approach is used in the current function. This procedure is used to generate many positive definite matrices. These positive definite matrices are used to generate M datasets which contain \Delta T, \Delta S_1, \Delta S_2, ..., \Delta S_k. Finally, the multivariate ICA (R^2_{H}) is estimated by regressing \Delta T on \Delta S_1, \Delta S_2, ..., \Delta S_k and computing the multiple coefficient of determination.

Value

An object of class ICA.ContCont.MultS_alt with components,

Author(s)

Wim Van der Elst, Ariel Alonso, & Geert Molenberghs

References

Van der Elst, W., Alonso, A. A., & Molenberghs, G. (2017). Univariate versus multivariate surrogate endpoints.

See Also

MICA.ContCont, ICA.ContCont, Single.Trial.RE.AA, plot Causal-Inference ContCont

Examples

## Not run:  #time-consuming code parts
# Specify matrix Sigma (var-cavar matrix T_0, T_1, S1_0, S1_1, ...)
# here for 1 true endpoint and 3 surrogates

s<-matrix(rep(NA, times=64),8)
s[1,1] <- 450; s[2,2] <- 413.5; s[3,3] <- 174.2; s[4,4] <- 157.5; 
s[5,5] <- 244.0; s[6,6] <- 229.99; s[7,7] <- 294.2; s[8,8] <- 302.5
s[3,1] <- 160.8; s[5,1] <- 208.5; s[7,1] <- 268.4 
s[4,2] <- 124.6; s[6,2] <- 212.3; s[8,2] <- 287.1
s[5,3] <- 160.3; s[7,3] <- 142.8 
s[6,4] <- 134.3; s[8,4] <- 130.4 
s[7,5] <- 209.3; 
s[8,6] <- 214.7 
s[upper.tri(s)] = t(s)[upper.tri(s)]

# Marix looks like (NA indicates unidentified covariances):
#            T_0    T_1  S1_0  S1_1  S2_0   S2_1  S2_0  S2_1
#            [,1]  [,2]  [,3]  [,4]  [,5]   [,6]  [,7]  [,8]
# T_0  [1,] 450.0    NA 160.8    NA 208.5     NA 268.4    NA
# T_1  [2,]    NA 413.5    NA 124.6    NA 212.30    NA 287.1
# S1_0 [3,] 160.8    NA 174.2    NA 160.3     NA 142.8    NA
# S1_1 [4,]    NA 124.6    NA 157.5    NA 134.30    NA 130.4
# S2_0 [5,] 208.5    NA 160.3    NA 244.0     NA 209.3    NA
# S2_1 [6,]    NA 212.3    NA 134.3    NA 229.99    NA 214.7
# S3_0 [7,] 268.4    NA 142.8    NA 209.3     NA 294.2    NA
# S3_1 [8,]    NA 287.1    NA 130.4    NA 214.70    NA 302.5

# Conduct analysis
ICA <- ICA.ContCont.MultS_alt(M=100, N=200, Show.Progress = TRUE,
  Sigma=s, G = seq(from=-1, to=1, by = .00001), Seed=c(123), 
  Model = "Delta_T ~ Delta_S1 + Delta_S2 + Delta_S3")

# Explore results
summary(ICA)
plot(ICA)

## End(Not run)

Assess surrogacy in the causal-inference single-trial setting (Individual Causal Association, ICA) in the Continuous-continuous case using the grid-based sample approach

Description

The function ICA.Sample.ContCont quantifies surrogacy in the single-trial causal-inference framework. It provides a faster alternative for ICA.ContCont. See Details below.

Usage

ICA.Sample.ContCont(T0S0, T1S1, T0T0=1, T1T1=1, S0S0=1, S1S1=1, T0T1=seq(-1, 1, by=.001), 
T0S1=seq(-1, 1, by=.001), T1S0=seq(-1, 1, by=.001), S0S1=seq(-1, 1, by=.001), M=50000)

Arguments

Details

Based on the causal-inference framework, it is assumed that each subject j has four counterfactuals (or potential outcomes), i.e., T_{0j}, T_{1j}, S_{0j}, and S_{1j}. Let T_{0j} and T_{1j} denote the counterfactuals for the true endpoint (T) under the control (Z=0) and the experimental (Z=1) treatments of subject j, respectively. Similarly, S_{0j} and S_{1j} denote the corresponding counterfactuals for the surrogate endpoint (S) under the control and experimental treatments, respectively. The individual causal effects of Z on T and S for a given subject j are then defined as \Delta_{T_{j}}=T_{1j}-T_{0j} and \Delta_{S_{j}}=S_{1j}-S_{0j}, respectively.

In the single-trial causal-inference framework, surrogacy can be quantified as the correlation between the individual causal effects of Z on S and T (for details, see Alonso et al., submitted):

\rho_{\Delta}=\rho(\Delta_{T_{j}},\:\Delta_{S_{j}})=\frac{\sqrt{\sigma_{S_{0}S_{0}}\sigma_{T_{0}T_{0}}}\rho_{S_{0}T_{0}}+\sqrt{\sigma_{S_{1}S_{1}}\sigma_{T_{1}T_{1}}}\rho_{S_{1}T_{1}}-\sqrt{\sigma_{S_{0}S_{0}}\sigma_{T_{1}T_{1}}}\rho_{S_{0}T_{1}}-\sqrt{\sigma_{S_{1}S_{1}}\sigma_{T_{0}T_{0}}}\rho_{S_{1}T_{0}}}{\sqrt{(\sigma_{T_{0}T_{0}}+\sigma_{T_{1}T_{1}}-2\sqrt{\sigma_{T_{0}T_{0}}\sigma_{T_{1}T_{1}}}\rho_{T_{0}T_{1}})(\sigma_{S_{0}S_{0}}+\sigma_{S_{1}S_{1}}-2\sqrt{\sigma_{S_{0}S_{0}}\sigma_{S_{1}S_{1}}}\rho_{S_{0}S_{1}})}},

where the correlations \rho_{S_{0}T_{1}}, \rho_{S_{1}T_{0}}, \rho_{T_{0}T_{1}}, and \rho_{S_{0}S_{1}} are not estimable. It is thus warranted to conduct a sensitivity analysis.

The function ICA.ContCont constructs all possible matrices that can be formed based on the specified vectors for \rho_{S_{0}T_{1}}, \rho_{S_{1}T_{0}}, \rho_{T_{0}T_{1}}, and \rho_{S_{0}S_{1}}, and retains the positive definite ones for the computation of \rho_{\Delta}.

In contrast, the function ICA.ContCont samples random values for \rho_{S_{0}T_{1}}, \rho_{S_{1}T_{0}}, \rho_{T_{0}T_{1}}, and \rho_{S_{0}S_{1}} based on a uniform distribution with user-specified minimum and maximum values, and retains the positive definite ones for the computation of \rho_{\Delta}.

The obtained vector of \rho_{\Delta} values can subsequently be used to examine (i) the impact of different assumptions regarding the correlations between the counterfactuals on the results (see also plot Causal-Inference ContCont), and (ii) the extent to which proponents of the causal-inference and meta-analytic frameworks will reach the same conclusion with respect to the appropriateness of the candidate surrogate at hand.

The function ICA.Sample.ContCont also generates output that is useful to examine the plausibility of finding a good surrogate endpoint (see GoodSurr in the Value section below). For details, see Alonso et al. (submitted).

Notes

A single \rho_{\Delta} value is obtained when all correlations in the function call are scalars.

Value

An object of class ICA.ContCont with components,

Author(s)

Wim Van der Elst, Ariel Alonso, & Geert Molenberghs

References

Alonso, A., Van der Elst, W., Molenberghs, G., Buyse, M., & Burzykowski, T. (submitted). On the relationship between the causal-inference and meta-analytic paradigms for the validation of surrogate markers.

See Also

MICA.ContCont, ICA.ContCont, Single.Trial.RE.AA, plot Causal-Inference ContCont

Examples

# Generate the vector of ICA values when rho_T0S0=rho_T1S1=.95, 
# sigma_T0T0=90, sigma_T1T1=100,sigma_ S0S0=10, sigma_S1S1=15, and  
# min=-1 max=1 is considered for the correlations
# between the counterfactuals:
SurICA2 <- ICA.Sample.ContCont(T0S0=.95, T1S1=.95, T0T0=90, T1T1=100, S0S0=10, 
S1S1=15, M=5000)

# Examine and plot the vector of generated ICA values:
summary(SurICA2)
plot(SurICA2)

Assess surrogacy in the causal-inference single-trial setting (Individual Causal Association, ICA) in the Continuous-continuous case using the grid-based sample approach when data is only avalable for the control treatment

Description

The function ICA.Sample.ControlTreat quantifies surrogacy in the single-trial causal-inference framework when data is only avalable for the control treatment.

Usage

ICA.Sample.ControlTreat(T0S0, T1S1=seq(-1, 1, by = 0.001), 
T0T0=1, T1T1=1, S0S0=1, S1S1=1, T0T1=seq(-1, 1, by=.001), 
T0S1=seq(-1, 1, by=.001), T1S0=seq(-1, 1, by=.001), S0S1=seq(-1, 1, by=.001), 
M=50000, M.Target=NA)

Arguments

Value

An object of class ICA.ContCont with components,

Author(s)

Wim Van der Elst, Ariel Alonso, & Geert Molenberghs

References

Van der Elst, W. et al. (submitted). On the Early Identification of Promising Surrogate Endpoints Using Causal Inference.

See Also

MICA.ContCont, ICA.ContCont, Single.Trial.RE.AA, plot Causal-Inference ContCont

Examples

# Generate the vector of ICA values when rho_T0S0=.95, 
# sigma_T0T0=90, sigma_T1T1=100,sigma_ S0S0=10, sigma_S1S1=15, and  
# min=-1 max=1 is considered for the correlations
# between the counterfactuals and rho_T1S1:
SurICA2 <- ICA.Sample.ControlTreat(T0S0=.95, T0T0=90, T1T1=100, S0S0=10, 
S1S1=15, M=5000)

# Examine and plot the vector of generated ICA values:
summary(SurICA2)
plot(SurICA2)

Assess surrogacy using a Rényi divergence based family of metrics in the causal-inference single-trial setting in normal case

Description

The function ICA_alpha_ContCont() is a set of metrics to evaluate surrogacy. ICA_alpha have the similar mathematical properties with ICA.ContCont().

Usage

ICA_alpha_ContCont(
  alpha = numeric(),
  T0S0,
  T1S1,
  T0T0 = 1,
  T1T1 = 1,
  S0S0 = 1,
  S1S1 = 1,
  T0T1 = seq(-1, 1, by = 0.1),
  T0S1 = seq(-1, 1, by = 0.1),
  T1S0 = seq(-1, 1, by = 0.1),
  S0S1 = seq(-1, 1, by = 0.1)
)

Arguments

Value

• Total.Num.Matrices: An object of class numeric that contains the total number of matrices that can be formed as based on the user-specified correlations in the function call.

• Pos.Def: A data.frame that contains the positive definite matrices that can be formed based on the user-specified correlations. These matrices are used to compute the vector of the \rho_{\Delta} values.

• rho: A scalar or vector that contains the individual causal association \rho_{\Delta}

• ICA: A scalar or vector that contains the individual causal association \rho_{\Delta}^2=ICA

• ICA_alpha: A scalar or vector that contains the individual causal association ICA_{\alpha}

• Sigmas: A data.frame that contains the \sigma_{\Delta T} and \sigma_{\Delta S}

Constructor for the function that returns that ICA as a function of the identifiable parameters

Description

ICA_given_model_constructor() returns a function fixes the unidentifiable parameters at user-specified values and takes the identifiable parameters as argument.

Usage

ICA_given_model_constructor(
  fitted_model,
  copula_par_unid,
  copula_family2,
  rotation_par_unid,
  n_prec,
  measure = "ICA",
  mutinfo_estimator = NULL,
  ICA_estimator = NULL,
  seed,
  composite = NULL,
  restr_time = +Inf
)

Arguments

Value

A function that computes the ICA as a function of the identifiable parameters. In this computation, the unidentifiable parameters are fixed at the values supplied as arguments to ICA_given_model_constructor_SurvSurv() or ICA_given_model_constructor().

Constructor for the function that returns that ICA as a function of the identifiable parameters for survival-survival

Description

ICA_given_model_constructor_SurvSurv() returns a function fixes the unidentifiable parameters at user-specified values and takes the identifiable parameters as argument.

Usage

ICA_given_model_constructor_SurvSurv(
  fitted_model,
  copula_par_unid,
  copula_family2,
  rotation_par_unid,
  n_prec,
  measure = "ICA",
  mutinfo_estimator,
  composite,
  seed,
  restr_time = +Inf
)

Arguments

Value

A function that computes the ICA as a function of the identifiable parameters. In this computation, the unidentifiable parameters are fixed at the values supplied as arguments to ICA_given_model_constructor_SurvSurv() or ICA_given_model_constructor().

The function ICA_t() is to evaluate surrogacy in the single-trial causal-inference framework.

Description

The function ICA_t() is to evaluate surrogacy in the single-trial causal-inference framework.

Usage

ICA_t(
  df,
  T0S0,
  T1S1,
  T0T0 = 1,
  T1T1 = 1,
  S0S0 = 1,
  S1S1 = 1,
  T0T1 = seq(-1, 1, by = 0.1),
  T0S1 = seq(-1, 1, by = 0.1),
  T1S0 = seq(-1, 1, by = 0.1),
  S0S1 = seq(-1, 1, by = 0.1)
)

Arguments

Value

• Total.Num.Matrices: An object of class numeric that contains the total number of matrices that can be formed as based on the user-specified correlations in the function call.

• Pos.Def: A data.frame that contains the positive definite matrices that can be formed based on the user-specified correlations. These matrices are used to compute the vector of the \rho_{\Delta} values.

• rho: A scalar or vector that contains the individual causal association \rho_{\Delta}

• ICA: A scalar or vector that contains the individual causal association \rho_{\Delta}^2=ICA

• ICA_t: A scalar or vector that contains the individual causal association ICA_{t}

• Sigmas: A data.frame that contains the \sigma_{\Delta T} and \sigma_{\Delta S}

Individual-level surrogate threshold effect for continuous normally distributed surrogate and true endpoints.

Description

Computes the individual-level surrogate threshold effect in the causal-inference single-trial setting where both the surrogate and the true endpoint are continuous normally distributed variables. For details, see paper in the references section.

Usage

ISTE.ContCont(Mean_T1, Mean_T0, Mean_S1, Mean_S0, N, Delta_S=c(-10, 0, 10), 
zeta.PI=0.05, PI.Bound=0, PI.Lower=TRUE, Show.Prediction.Plots=TRUE, Save.Plots="No", 
T0S0, T1S1, T0T0=1, T1T1=1, S0S0=1, S1S1=1, T0T1=seq(-1, 1, by=.001), 
T0S1=seq(-1, 1, by=.001), T1S0=seq(-1, 1, by=.001),
S0S1=seq(-1, 1, by=.001), M.PosDef=500, Seed=123)

Arguments

Details

See paper in the references section.

Value

An object of class ICA.ContCont with components,

Author(s)

Wim Van der Elst, Ariel Alonso, & Geert Molenberghs

References

Van der Elst, W., Alonso, A. A., and Molenberghs, G. (submitted). The individual-level surrogate threshold effect in a causal-inference setting.

See Also

ICA.ContCont

Examples

# Define input for analysis using the Schizo dataset, 
# with S=BPRS and T = PANSS. 
# For each of the identifiable quantities,
# uncertainty is accounted for by specifying a uniform
# distribution with min, max values corresponding to
# the 95% confidence interval of the quantity.
T0S0 <- runif(min = 0.9524, max = 0.9659, n = 1000)
T1S1 <- runif(min = 0.9608, max = 0.9677, n = 1000)

S0S0 <- runif(min=160.811, max=204.5009, n=1000)
S1S1 <- runif(min=168.989, max = 194.219, n=1000)
T0T0 <- runif(min=484.462, max = 616.082, n=1000)
T1T1 <- runif(min=514.279, max = 591.062, n=1000)

Mean_T0 <- runif(min=-13.455, max=-9.489, n=1000)
Mean_T1 <- runif(min=-17.17, max=-14.86, n=1000)
Mean_S0 <- runif(min=-7.789, max=-5.503, n=1000)
Mean_S1 <- runif(min=-9.600, max=-8.276, n=1000)

# Do the ISTE analysis
## Not run: 
ISTE <- ISTE.ContCont(Mean_T1=Mean_T1, Mean_T0=Mean_T0, 
 Mean_S1=Mean_S1, Mean_S0=Mean_S0, N=2128, Delta_S=c(-50:50), 
 zeta.PI=0.05, PI.Bound=0, Show.Prediction.Plots=TRUE,
 Save.Plots="No", T0S0=T0S0, T1S1=T1S1, T0T0=T0T0, T1T1=T1T1, 
 S0S0=S0S0, S1S1=S1S1)

# Examine results:
summary(ISTE)

# Plots of results. 
  # Plot ISTE
plot(ISTE)
  # Other plots, see plot.ISTE.ContCont for details
plot(ISTE, Outcome="MSE")
plot(ISTE, Outcome="gamma0")
plot(ISTE, Outcome="gamma1")
plot(ISTE, Outcome="Exp.DeltaT")
plot(ISTE, Outcome="Exp.DeltaT.Low.PI")
plot(ISTE, Outcome="Exp.DeltaT.Up.PI")

## End(Not run)

Reshapes a dataset from the 'long' format (i.e., multiple lines per patient) into the 'wide' format (i.e., one line per patient)

Description

Reshapes a dataset that is in the 'long' format into the 'wide' format. The dataset should contain a single surrogate endpoint and a single true endpoint value per subject.

Usage

LongToWide(Dataset, OutcomeIndicator, IdIndicator, TreatIndicator, OutcomeValue)

Arguments

Value

A data.frame in the 'wide' format, i.e., a data.frame that contains one line per subject. Each line contains a surrogate value, a true endpoint value, a treatment indicator, a patient ID, and a trial ID.

Author(s)

Wim Van der Elst, Ariel Alonso, and Geert Molenberghs

Examples

# Generate a dataset in the 'long' format that contains 
# S and T values for 100 patients
Outcome <- rep(x=c(0, 1), times=100)
ID <- rep(seq(1:100), each=2)
Treat <- rep(seq(c(0,1)), each=100)
Outcomes <- as.numeric(matrix(rnorm(1*200, mean=100, sd=10), 
                                      ncol=200))
Data <- data.frame(cbind(Outcome, ID, Treat, Outcomes))

# Reshapes the Data object 
LongToWide(Dataset=Data, OutcomeIndicator=Outcome, IdIndicator=ID, 
           TreatIndicator=Treat, OutcomeValue=Outcomes)

Assess surrogacy in the causal-inference multiple-trial setting (Meta-analytic Individual Causal Association; MICA) in the continuous-continuous case

Description

The function MICA.ContCont quantifies surrogacy in the multiple-trial causal-inference framework. See Details below.

Usage

MICA.ContCont(Trial.R, D.aa, D.bb, T0S0, T1S1, T0T0=1, T1T1=1, S0S0=1, S1S1=1,
T0T1=seq(-1, 1, by=.1), T0S1=seq(-1, 1, by=.1), T1S0=seq(-1, 1, by=.1),
S0S1=seq(-1, 1, by=.1))

Arguments

Details

Based on the causal-inference framework, it is assumed that each subject j in trial i has four counterfactuals (or potential outcomes), i.e., T_{0ij}, T_{1ij}, S_{0ij}, and S_{1ij}. Let T_{0ij} and T_{1ij} denote the counterfactuals for the true endpoint (T) under the control (Z=0) and the experimental (Z=1) treatments of subject j in trial i, respectively. Similarly, S_{0ij} and S_{1ij} denote the corresponding counterfactuals for the surrogate endpoint (S) under the control and experimental treatments of subject j in trial i, respectively. The individual causal effects of Z on T and S for a given subject j in trial i are then defined as \Delta_{T_{ij}}=T_{1ij}-T_{0ij} and \Delta_{S_{ij}}=S_{1ij}-S_{0ij}, respectively.

In the multiple-trial causal-inference framework, surrogacy can be quantified as the correlation between the individual causal effects of Z on S and T (for details, see Alonso et al., submitted):

\rho_{M}=\rho(\Delta_{Tij},\:\Delta_{Sij})=\frac{\sqrt{d_{bb}d_{aa}}R_{trial}+\sqrt{V(\varepsilon_{\Delta Tij})V(\varepsilon_{\Delta Sij})}\rho_{\Delta}}{\sqrt{V(\Delta_{Tij})V(\Delta_{Sij})}},

where

V(\varepsilon_{\Delta Tij})=\sigma_{T_{0}T_{0}}+\sigma_{T_{1}T_{1}}-2\sqrt{\sigma_{T_{0}T_{0}}\sigma_{T_{1}T_{1}}}\rho_{T_{0}T_{1}},

V(\varepsilon_{\Delta Sij})=\sigma_{S_{0}S_{0}}+\sigma_{S_{1}S_{1}}-2\sqrt{\sigma_{S_{0}S_{0}}\sigma_{S_{1}S_{1}}}\rho_{S_{0}S_{1}},

V(\Delta_{Tij})=d_{bb}+\sigma_{T_{0}T_{0}}+\sigma_{T_{1}T_{1}}-2\sqrt{\sigma_{T_{0}T_{0}}\sigma_{T_{1}T_{1}}}\rho_{T_{0}T_{1}},

V(\Delta_{Sij})=d_{aa}+\sigma_{S_{0}S_{0}}+\sigma_{S_{1}S_{1}}-2\sqrt{\sigma_{S_{0}S_{0}}\sigma_{S_{1}S_{1}}}\rho_{S_{0}S_{1}}.

The correlations between the counterfactuals (i.e., \rho_{S_{0}T_{1}}, \rho_{S_{1}T_{0}}, \rho_{T_{0}T_{1}}, and \rho_{S_{0}S_{1}}) are not identifiable from the data. It is thus warranted to conduct a sensitivity analysis (by considering vectors of possible values for the correlations between the counterfactuals – rather than point estimates).

When the user specifies a vector of values that should be considered for one or more of the correlations that are involved in the computation of \rho_{M}, the function MICA.ContCont constructs all possible matrices that can be formed as based on the specified values, identifies the matrices that are positive definite (i.e., valid correlation matrices), and computes \rho_{M} for each of these matrices. An examination of the vector of the obtained \rho_{M} values allows for a straightforward examination of the impact of different assumptions regarding the correlations between the counterfactuals on the results (see also plot Causal-Inference ContCont), and the extent to which proponents of the causal-inference and meta-analytic frameworks will reach the same conclusion with respect to the appropriateness of the candidate surrogate at hand.

Notes A single \rho_{M} value is obtained when all correlations in the function call are scalars.

Value

An object of class MICA.ContCont with components,

Warning

The theory that relates the causal-inference and the meta-analytic frameworks in the multiple-trial setting (as developped in Alonso et al., submitted) assumes that a reduced or semi-reduced modelling approach is used in the meta-analytic framework. Thus R_{trial}, d_{aa} and d_{bb} should be estimated based on a reduced model (i.e., using the Model=c("Reduced") argument in the functions UnifixedContCont, UnimixedContCont, BifixedContCont, or BimixedContCont) or based on a semi-reduced model (i.e., using the Model=c("SemiReduced") argument in the functions UnifixedContCont, UnimixedContCont, or BifixedContCont).

Author(s)

Wim Van der Elst, Ariel Alonso, & Geert Molenberghs

References

Alonso, A., Van der Elst, W., Molenberghs, G., Buyse, M., & Burzykowski, T. (submitted). On the relationship between the causal-inference and meta-analytic paradigms for the validation of surrogate markers.

See Also

ICA.ContCont, MICA.Sample.ContCont, plot Causal-Inference ContCont, UnifixedContCont, UnimixedContCont, BifixedContCont, BimixedContCont

Examples

## Not run:  #time-consuming code parts
# Generate the vector of MICA values when R_trial=.8, rho_T0S0=rho_T1S1=.8,
# sigma_T0T0=90, sigma_T1T1=100,sigma_ S0S0=10, sigma_S1S1=15, D.aa=5, D.bb=10,
# and when the grid of values {0, .2, ..., 1} is considered for the
# correlations between the counterfactuals:
SurMICA <- MICA.ContCont(Trial.R=.80, D.aa=5, D.bb=10, T0S0=.8, T1S1=.8,
T0T0=90, T1T1=100, S0S0=10, S1S1=15, T0T1=seq(0, 1, by=.2),
T0S1=seq(0, 1, by=.2), T1S0=seq(0, 1, by=.2), S0S1=seq(0, 1, by=.2))

# Examine and plot the vector of the generated MICA values:
summary(SurMICA)
plot(SurMICA)


# Same analysis, but now assume that D.aa=.5 and D.bb=.1:
SurMICA <- MICA.ContCont(Trial.R=.80, D.aa=.5, D.bb=.1, T0S0=.8, T1S1=.8,
T0T0=90, T1T1=100, S0S0=10, S1S1=15, T0T1=seq(0, 1, by=.2),
T0S1=seq(0, 1, by=.2), T1S0=seq(0, 1, by=.2), S0S1=seq(0, 1, by=.2))

# Examine and plot the vector of the generated MICA values:
summary(SurMICA)
plot(SurMICA)


# Same as first analysis, but specify vectors for rho_T0S0 and rho_T1S1:
# Sample from normal with mean .8 and SD=.1 (to account for uncertainty
# in estimation)
SurMICA <- MICA.ContCont(Trial.R=.80, D.aa=5, D.bb=10,
T0S0=rnorm(n=10000000, mean=.8, sd=.1),
T1S1=rnorm(n=10000000, mean=.8, sd=.1),
T0T0=90, T1T1=100, S0S0=10, S1S1=15, T0T1=seq(0, 1, by=.2),
T0S1=seq(0, 1, by=.2), T1S0=seq(0, 1, by=.2), S0S1=seq(0, 1, by=.2))

## End(Not run)

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

Description

The function MICA.Sample.ContCont quantifies surrogacy in the multiple-trial causal-inference framework. It provides a faster alternative for MICA.ContCont. See Details below.

Usage

MICA.Sample.ContCont(Trial.R, D.aa, D.bb, T0S0, T1S1, T0T0=1, T1T1=1, S0S0=1, S1S1=1,
T0T1=seq(-1, 1, by=.001), T0S1=seq(-1, 1, by=.001), T1S0=seq(-1, 1, by=.001),
S0S1=seq(-1, 1, by=.001), M=50000)

Arguments

Details

Based on the causal-inference framework, it is assumed that each subject j in trial i has four counterfactuals (or potential outcomes), i.e., T_{0ij}, T_{1ij}, S_{0ij}, and S_{1ij}. Let T_{0ij} and T_{1ij} denote the counterfactuals for the true endpoint (T) under the control (Z=0) and the experimental (Z=1) treatments of subject j in trial i, respectively. Similarly, S_{0ij} and S_{1ij} denote the corresponding counterfactuals for the surrogate endpoint (S) under the control and experimental treatments of subject j in trial i, respectively. The individual causal effects of Z on T and S for a given subject j in trial i are then defined as \Delta_{T_{ij}}=T_{1ij}-T_{0ij} and \Delta_{S_{ij}}=S_{1ij}-S_{0ij}, respectively.

In the multiple-trial causal-inference framework, surrogacy can be quantified as the correlation between the individual causal effects of Z on S and T (for details, see Alonso et al., submitted):

\rho_{M}=\rho(\Delta_{Tij},\:\Delta_{Sij})=\frac{\sqrt{d_{bb}d_{aa}}R_{trial}+\sqrt{V(\varepsilon_{\Delta Tij})V(\varepsilon_{\Delta Sij})}\rho_{\Delta}}{\sqrt{V(\Delta_{Tij})V(\Delta_{Sij})}},

where

V(\varepsilon_{\Delta Tij})=\sigma_{T_{0}T_{0}}+\sigma_{T_{1}T_{1}}-2\sqrt{\sigma_{T_{0}T_{0}}\sigma_{T_{1}T_{1}}}\rho_{T_{0}T_{1}},

V(\varepsilon_{\Delta Sij})=\sigma_{S_{0}S_{0}}+\sigma_{S_{1}S_{1}}-2\sqrt{\sigma_{S_{0}S_{0}}\sigma_{S_{1}S_{1}}}\rho_{S_{0}S_{1}},

V(\Delta_{Tij})=d_{bb}+\sigma_{T_{0}T_{0}}+\sigma_{T_{1}T_{1}}-2\sqrt{\sigma_{T_{0}T_{0}}\sigma_{T_{1}T_{1}}}\rho_{T_{0}T_{1}},

V(\Delta_{Sij})=d_{aa}+\sigma_{S_{0}S_{0}}+\sigma_{S_{1}S_{1}}-2\sqrt{\sigma_{S_{0}S_{0}}\sigma_{S_{1}S_{1}}}\rho_{S_{0}S_{1}}.

The correlations between the counterfactuals (i.e., \rho_{S_{0}T_{1}}, \rho_{S_{1}T_{0}}, \rho_{T_{0}T_{1}}, and \rho_{S_{0}S_{1}}) are not identifiable from the data. It is thus warranted to conduct a sensitivity analysis (by considering vectors of possible values for the correlations between the counterfactuals – rather than point estimates).

When the user specifies a vector of values that should be considered for one or more of the correlations that are involved in the computation of \rho_{M}, the function MICA.ContCont constructs all possible matrices that can be formed as based on the specified values, and retains the positive definite ones for the computation of \rho_{M}.

In contrast, the function MICA.Sample.ContCont samples random values for \rho_{S_{0}T_{1}}, \rho_{S_{1}T_{0}}, \rho_{T_{0}T_{1}}, and \rho_{S_{0}S_{1}} based on a uniform distribution with user-specified minimum and maximum values, and retains the positive definite ones for the computation of \rho_{M}.

An examination of the vector of the obtained \rho_{M} values allows for a straightforward examination of the impact of different assumptions regarding the correlations between the counterfactuals on the results (see also plot Causal-Inference ContCont), and the extent to which proponents of the causal-inference and meta-analytic frameworks will reach the same conclusion with respect to the appropriateness of the candidate surrogate at hand.

Notes A single \rho_{M} value is obtained when all correlations in the function call are scalars.

Value

An object of class MICA.ContCont with components,

Warning

The theory that relates the causal-inference and the meta-analytic frameworks in the multiple-trial setting (as developped in Alonso et al., submitted) assumes that a reduced or semi-reduced modelling approach is used in the meta-analytic framework. Thus R_{trial}, d_{aa} and d_{bb} should be estimated based on a reduced model (i.e., using the Model=c("Reduced") argument in the functions UnifixedContCont, UnimixedContCont, BifixedContCont, or BimixedContCont) or based on a semi-reduced model (i.e., using the Model=c("SemiReduced") argument in the functions UnifixedContCont, UnimixedContCont, or BifixedContCont).

Author(s)

Wim Van der Elst, Ariel Alonso, & Geert Molenberghs

References

Alonso, A., Van der Elst, W., Molenberghs, G., Buyse, M., & Burzykowski, T. (submitted). On the relationship between the causal-inference and meta-analytic paradigms for the validation of surrogate markers.

See Also

ICA.ContCont, MICA.ContCont, plot Causal-Inference ContCont, UnifixedContCont, UnimixedContCont, BifixedContCont, BimixedContCont

Examples

## Not run:  #Time consuming (>5 sec) code part
# Generate the vector of MICA values when R_trial=.8, rho_T0S0=rho_T1S1=.8,
# sigma_T0T0=90, sigma_T1T1=100,sigma_ S0S0=10, sigma_S1S1=15, D.aa=5, D.bb=10,
# and when the grid of values {-1, -0.999, ..., 1} is considered for the
# correlations between the counterfactuals:
SurMICA <- MICA.Sample.ContCont(Trial.R=.80, D.aa=5, D.bb=10, T0S0=.8, T1S1=.8,
T0T0=90, T1T1=100, S0S0=10, S1S1=15, T0T1=seq(-1, 1, by=.001),
T0S1=seq(-1, 1, by=.001), T1S0=seq(-1, 1, by=.001),
S0S1=seq(-1, 1, by=.001), M=10000)

# Examine and plot the vector of the generated MICA values:
summary(SurMICA)
plot(SurMICA, ICA=FALSE, MICA=TRUE)


# Same analysis, but now assume that D.aa=.5 and D.bb=.1:
SurMICA <- MICA.Sample.ContCont(Trial.R=.80, D.aa=.5, D.bb=.1, T0S0=.8, T1S1=.8,
T0T0=90, T1T1=100, S0S0=10, S1S1=15, T0T1=seq(-1, 1, by=.001),
T0S1=seq(-1, 1, by=.001), T1S0=seq(-1, 1, by=.001),
S0S1=seq(-1, 1, by=.001), M=10000)

# Examine and plot the vector of the generated MICA values:
summary(SurMICA)
plot(SurMICA)

## End(Not run)

Computes marginal probabilities for a dataset where the surrogate and true endpoints are binary

Description

This function computes the marginal probabilities associated with the distribution of the potential outcomes for the true and surrogate endpoint.

Usage

MarginalProbs(Dataset=Dataset, Surr=Surr, True=True, Treat=Treat)

Arguments

Value

Author(s)

Wim Van der Elst, Ariel Alonso, & Geert Molenberghs

See Also

ICA.BinBin

Examples

# Open the ARMD dataset and recode Diff24 and Diff52 as 1
# when the original value is above 0, and 0 otherwise
data(ARMD)
ARMD$Diff24_Dich <- ifelse(ARMD$Diff24>0, 1, 0)
ARMD$Diff52_Dich <- ifelse(ARMD$Diff52>0, 1, 0)

# Obtain marginal probabilities and ORs
MarginalProbs(Dataset=ARMD, Surr=Diff24_Dich, True=Diff52_Dich, 
Treat=Treat)

Use the maximum-entropy approach to compute ICA in the continuous-continuous sinlge-trial setting

Description

In a surrogate evaluation setting where both S and T are continuous endpoints, a sensitivity-based approach where multiple 'plausible values' for ICA are retained can be used (see functions ICA.ContCont). The function MaxEntContCont identifies the estimate which has the maximuum entropy.

Usage

MaxEntContCont(x, T0T0, T1T1, S0S0, S1S1)

Arguments

Value

Author(s)

Wim Van der Elst, Ariel Alonso, Paul Meyvisch, & Geert Molenberghs

References

Add

See Also

ICA.ContCont, MaxEntICABinBin

Examples

## Not run:  #time-consuming code parts
# Compute ICA for ARMD dataset, using the grid  
# G={-1, -.80, ..., 1} for the undidentifiable correlations

ICA <- ICA.ContCont(T0S0 = 0.769, T1S1 = 0.712, S0S0 = 188.926, 
S1S1 = 132.638, T0T0 = 264.797, T1T1 = 231.771, 
T0T1 = seq(-1, 1, by = 0.2), T0S1 = seq(-1, 1, by = 0.2), 
T1S0 = seq(-1, 1, by = 0.2), S0S1 = seq(-1, 1, by = 0.2))

# Identify the maximum entropy ICA
MaxEnt_ARMD <- MaxEntContCont(x = ICA, S0S0 = 188.926, 
S1S1 = 132.638, T0T0 = 264.797, T1T1 = 231.771)

  # Explore results using summary() and plot() functions
summary(MaxEnt_ARMD)
plot(MaxEnt_ARMD)
plot(MaxEnt_ARMD, Entropy.By.ICA = TRUE)

## End(Not run)

Use the maximum-entropy approach to compute ICA in the binary-binary setting

Description

In a surrogate evaluation setting where both S and T are binary endpoints, a sensitivity-based approach where multiple 'plausible values' for ICA are retained can be used (see functions ICA.BinBin, ICA.BinBin.Grid.Full, or ICA.BinBin.Grid.Sample). Alternatively, the maximum entropy distribution of the vector of potential outcomes can be considered, based upon which ICA is subsequently computed. The use of the distribution that maximizes the entropy can be justified based on the fact that any other distribution would necessarily (i) assume information that we do not have, or (ii) contradict information that we do have. The function MaxEntICABinBin implements the latter approach.

Usage

MaxEntICABinBin(pi1_1_, pi1_0_, pi_1_1,
pi_1_0, pi0_1_, pi_0_1, Method="BFGS", 
Fitted.ICA=NULL)

Arguments

Value