Two Normally Distributed Endpoints

The Meta-Analytic Framework

In the Surrogate package, there are four main functions that can be used to evaluate the appropriateness of a candidate surrogate endpoint based on the meta-analytic framework in the setting where both endpoints are normally distributed variables: the functions BimixedContCont(), UnimixedContCont(), BifixedContCont(), and UnifixedContCont(). The first part of the function name refers to the endpoint dimension of the model, i.e., whether a Univariate or a Bivariate modeling approach is used (for details, see Section 4.3.3). The second part of the function name refers to the trial dimension, i.e., whether a fixed- or mixed-effects model should be fitted (for details, see Section 4.3.1). The third part of the function name refers to the fact that both S and T are assumed to be normally distributed Continuous endpoints. Further, the arguments Model="Full" or Model="Reduced" can be used in the function call to specify whether a full or a reduced model should be fitted, respectively (for details, see Section 4.3.2). Finally, the arguments Weighted=TRUE or Weighted=FALSE can be used in the function call to specify whether measurement error should be accounted for or not, respectively (for details, see 4.3.4). Note that the latter argument is not used in the BimixedContCont() function because it is irrelevant when the bivariate mixed-effects (hierarchical) approach is used, i.e., measurement error is automatically accounted for in that approach. A summary of the different models that can be fitted using the BimixedContCont(),UnimixedContCont(), Bif ixedContCont(), and UnifixedContCont() functions is provided in Table 13.1.

Main function arguments The functions

  • • BimixedContCont(), • UnimixedContCont(), • BifixedContCont(), and
  • • UnifixedContCont() require the following arguments:
  • • Dataset=: The name of a data.frame that should consist of one line per patient. Each line should contain (at least) a surrogate value, a true endpoint value, a treatment indicator, a patient ID, and a trial ID.
  • • Surr=, True=, Trial.ID=, Pat.ID=: The names of the variables in the dataset that contain the surrogate and true endpoint values, the trial indicator, and the patient indicator, respectively.
  • • Treat=: The name of the variable in the dataset that contains the treatment indicator. The treatment indicator should be coded as 1 for the experimental

TABLE 13.1

Surrogate package. Overview of the functions that can be used to evaluate surrogacy in the meta-analytic framework when both S and T are normally distributed endpoints.

Note. The indicator (. . .) refers to a number of required function arguments; for details see Section 13.2.1.

Full

Reduced

Mixed-effects approach

Bivariate

(Un) weighted

(Un) weighted

BimixedContCont( . . . , Model="Full")

BimixedContCont(..., Model="Reduced")

Univariate

Unweighted

Weighted

Unweighted

Weighted

UnimixedContCont(..., Model="Full", Weighted=FALSE)

UnimixedContCont(..., Model="Full", Weighted=TRUE)

UnimixedContCont(..., Model="Reduced", Weighted=FALSE)

UnimixedContCont(..., Model="Reduced", Weighted=TRUE)

Fixed-effects approach

Bivariate

Unweighted

Weighted

Unweighted

Weighted

BifixedContCont(..., Model="Full", Weighted=FALSE)

BifixedContCont(..., Model="Full", Weighted=TRUE)

BifixedContCont(..., Model="Reduced", Weighted=FALSE)

BifixedContCont(..., Model="Reduced", Weighted=TRUE)

Univariate

Unweighted

Weighted

Unweighted

Weighted

UnifixedContCont(..., Model="Full", Weighted=FALSE)

UnifixedContCont(..., Model="Full", Weighted=TRUE)

UnifixedContCont(..., Model="Reduced", Weighted=FALSE)

UnifixedContCont(..., Model="Reduced", Weighted=TRUE)

group or —1 for the control group, or as 1 for the experimental group or 0 for the control group. Notice that the choice for a 0/1 or —1/1 coding of treatment is relevant and may impact the results when a hierarchical (bivariate mixed-effects) modeling strategy is used (see Section 4.4).

  • • Model=: The type of model that should be fitted, i.e., Model=c("Full") or Model=c("Reduced"). For details, see Section 4.3.2. If Model= is not specified, Model=c("Full") is used by default.
  • • Weighted=: A logical indicator that specifies whether a weighted (argument Weighted=TRUE) or unweighted (argument Weighted=FALSE) model should be fitted in Stage 2 (for details, see Section 4.3.4). Weighting accounts for the heterogeneity in information content between the different trial-level contributions. It is mainly relevant to take this heterogeneity into account when there are large differences in the number of patients in the different trials. If the argument Weighted= is not specified, by default Weighted=TRUE is used. Note that the Weighted= argument is only used in the UnimixedContCont(), BifixedContCont(), and UnifixedContCont() functions, because the heterogeneity in information content is automatically accounted for when the bivariate mixed-effects approach (implemented in the BimixedContCont() function) is used.
  • • Alpha=: The а-level that should be used to establish confidence intervals around Rrial and R?ndiv. When this argument is not specified, Alpha=0.05 is used by default.
  • • Min.Trial.Size=: The minimum number of patients that a trial-level unit should contain in order to be included in the analysis. If the number of patients in a trial is smaller than the value specified by Min.Trial.Size=, the data of the trial are excluded from the analysis. When this argument is not specified, Min.Trial.Size=2 is used by default.

Depending on the function at hand, additional arguments can be used. For example, the functions UnimixedContCont() and UnifixedContCont() use bootstrapping to establish a confidence interval around R?ndiv. In these functions, the argument Number.Bootstraps= can be used to specify the number of bootstrap samples that should be used (by default Number. Bootstraps=500 is used) and the argument Seed= can be used to specify a seed (for reproducibility of the results). Full details with respect to the arguments that can be used in the different functions are provided in the Surrogate manual.

 
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