BimixedCbCContCont: Fits a bivariate mixed-effects model using the...

In Surrogate: Evaluation of Surrogate Endpoints in Clinical Trials

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

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

Details

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Value

An object of class BimixedContCont with components,

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

Author(s)

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

References

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

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

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

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

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)
                         

Surrogate documentation built on Sept. 25, 2023, 5:07 p.m.