UnimixedContCont: Fits univariate mixed-effect models to assess surrogacy in...

In Surrogate: Evaluation of Surrogate Endpoints in Clinical Trials

Fits univariate mixed-effect models to assess surrogacy in the meta-analytic multiple-trial setting (continuous-continuous case)

Description

The function UnimixedContCont uses the univariate 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 (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

UnimixedContCont(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), T0T1=seq(-1, 1, by=.2), T0S1=seq(-1, 1, by=.2), 
T1S0=seq(-1, 1, by=.2), S0S1=seq(-1, 1, by=.2), ...)

Arguments

Dataset	A data.frame that should consist of one line per patient. Each line contains (at least) a surrogate value, a true endpoint value, a treatment indicator, a patient ID, and a trial ID.
Surr	The name of the variable in Dataset that contains the surrogate endpoint values.
True	The name of the variable in Dataset that contains the true endpoint values.
Treat	The name of the variable in Dataset that contains the treatment indicators. The treatment indicator should either be coded as 1 for the experimental group and -1 for the control group, or as 1 for the experimental group and 0 for the control group.
Trial.ID	The name of the variable in Dataset that contains the trial ID to which the patient belongs.
Pat.ID	The name of the variable in Dataset that contains the patient's ID.
Model	The type of model that should be fitted, i.e., Model=c("Full"), Model=c("Reduced"), or Model=c("SemiReduced"). See the Details section below. Default Model=c("Full").
Weighted	Logical. If TRUE, then a weighted regression analysis is conducted at stage 2 of the two-stage approach. If FALSE, then an unweighted regression analysis is conducted at stage 2 of the two-stage approach. See the Details section below. Default TRUE.
Min.Trial.Size	The minimum number of patients that a trial should contain to be included in the analysis. If the number of patients in a trial is smaller than the value specified by Min.Trial.Size, the data of the trial are excluded from the analysis. Default 2.
Alpha	The \alpha-level that is used to determine the confidence intervals around R^2_{trial}, R_{trial}, R^2_{indiv}, and R_{indiv}. Default 0.05.
Number.Bootstraps	The confidence intervals for R^2_{indiv} and R_{indiv} are determined as based on a bootstrap procedure. Number.Bootstraps specifies the number of bootstrap samples that are to be used. Default 500.
Seed	The seed to be used in the bootstrap procedure. Default sample(1:1000, size=1).
T0T1	A scalar or vector that contains the correlation(s) between the counterfactuals T0 and T1 that should be considered in the computation of \rho_{\Delta} (ICA). For details, see function ICA.ContCont. Default seq(-1, 1, by=.2).
T0S1	A scalar or vector that contains the correlation(s) between the counterfactuals T0 and S1 that should be considered in the computation of \rho_{\Delta}. Default seq(-1, 1, by=.2).
T1S0	A scalar or vector that contains the correlation(s) between the counterfactuals T1 and S0 that should be considered in the computation of \rho_{\Delta}. Default seq(-1, 1, by=.2).
S0S1	A scalar or vector that contains the correlation(s) between the counterfactuals S0 and S1 that should be considered in the computation of \rho_{\Delta}. Default seq(-1, 1, by=.2).
...	Other arguments to be passed to the function lmer (of the R package lme4) that is used to fit the geralized linear mixed-effect models in the function BimixedContCont.

Details

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 UnimixedContCont implements one such strategy, i.e., it uses a two-stage univariate mixed-effects modelling approach to assess surrogacy. In the first stage of the analysis, two univariate mixed-effects models are fitted to the data. 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 univariate models are 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 error terms \varepsilon_{Sij} and \varepsilon_{Tij} are assumed to be independent.

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

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

T_{ij}=\mu_{T}+(\beta+b_{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 each of the trials). The other parameters are the same as defined above, and \varepsilon_{Sij} and \varepsilon_{Tij} are again assumed to be independent.

An estimate of R^2_{indiv} is computed as r(\varepsilon_{Sij}, \varepsilon_{Tij})^2.

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 models that were fitted in stage 1, i.e., \beta_{i}=\beta+b_{i}, \mu_{Si}=\mu_{S}+m_{Si}, and \alpha_{i}=\alpha+a_{i}.

When a reduced or semi-reduced model is requested by the user (by using the arguments 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 parameters are the same as defined above.

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 UnimixedContCont with components,

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

Author(s)

Wim Van der Elst, Ariel Alonso, & Geert Molenberghs

References

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

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

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

See Also

UnifixedContCont, BifixedContCont, BimixedContCont, plot Meta-Analytic

Examples

## Not run:  #Time consuming code part
# 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="Reduced")

# Fit a reduced univariate mixed-effects model without weighting to assess surrogacy:
Sur <- UnimixedContCont(Dataset=Data.Observed.MTS, Surr=Surr, True=True, Treat=Treat, 
Trial.ID=Trial.ID, Pat.ID=Pat.ID, Model="Reduced", Weighted=FALSE)

# Show a summary and plots of the results:
summary(Sur)
plot(Sur, Weighted=FALSE)
## End(Not run)

Surrogate documentation built on June 22, 2024, 9:16 a.m.