Surrogate (version 1.1)

TrialLevelIT: Estimates trial-level surrogacy in the information-theoretic framework

Description

The function TrialLevelIT estimates trial-level surrogacy based on the vectors of treatment effects on \(S\) (i.e., \(\alpha_{i}\)), intercepts on \(S\) (i.e., \(\mu_{i}\)) and \(T\) (i.e., \(\beta_{i}\)) in the different trials. See the Details section below.

Usage

TrialLevelIT(Alpha.Vector, Mu_S.Vector=NULL, 
Beta.Vector, N.Trial, Model="Reduced", Alpha=.05)

Arguments

Alpha.Vector

The vector of treatment effects on \(S\) in the different trials, i.e., \(\alpha_{i}\).

Mu_S.Vector

The vector of intercepts for \(S\) in the different trials, i.e., \(\mu_{Si}\). Only required when a full model is requested.

Beta.Vector

The vector of treatment effects on \(T\) in the different trials, i.e., \(\beta_{i}\).

N.Trial

The total number of available trials.

Model

The type of model that should be fitted, i.e., Model=c("Full") or Model=c("Reduced"). See the Details section below. Default Model=c("Reduced").

Alpha

The \(\alpha\)-level that is used to determine the confidence intervals around \(R^2_{trial}\) and \(R_{trial}\). Default \(0.05\).

Value

An object of class TrialLevelIT with components,

Alpha.Vector

The vector of treatment effects on \(S\) in the different trials.

Beta.Vector

The vector of treatment effects on \(T\) in the different trials.

N.Trial

The total number of trials.

R2.ht

A data.frame that contains the trial-level coefficient of determination (\(R^2_{ht}\)), its standard error and confidence interval.

Details

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

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

where \(\beta_i\) = the trial-specific treatment effects on \(T\), \(\mu_{Si}\) = the trial-specific intercepts for \(S\), and \(\alpha_i\) = the trial-specific treatment effects on \(S\). The \(-2\) log likelihood value of model (1) (\(L_1\)) is subsequently compared to the \(-2\) log likelihood value of an intercept-only model (\({\beta}_{i}=\lambda_{3}\); \(L_0\)), 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 reduced model is requested (by using the argument Model=c("Reduced") in the function call), the following model is fitted:

$${\beta}_{i}=\lambda_{0}+\lambda_{1}{\alpha}_{i}+\varepsilon_{i}.$$

The \(-2\) log likelihood value of this model (\(L_1\) for the reduced model) is subsequently compared to the \(-2\) log likelihood value of an intercept-only model (\({\beta}_{i}=\lambda_{3}\); \(L_0\)), and \(R^2_{ht}\) is computed based on the reduction in the likelihood (as described above).

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.

See Also

UnimixedContCont, UnifixedContCont, BifixedContCont, BimixedContCont, plot.TrialLevelIT

Examples

# NOT RUN {
# Generate vector treatment effects on S
set.seed(seed = 1)
Alpha.Vector <- seq(from = 5, to = 10, by=.1) + runif(min = -.5, max = .5, n = 51)

# Generate vector treatment effects on T
set.seed(seed=2)
Beta.Vector <- (Alpha.Vector * 3) + runif(min = -5, max = 5, n = 51)

# Apply the function to estimate R^2_{h.t}
Fit <- TrialLevelIT(Alpha.Vector=Alpha.Vector,
Beta.Vector=Beta.Vector, N.Trial=50, Model="Reduced")

summary(Fit)
plot(Fit)
# }