ICA_contcont_long_galecki: Assess surrogacy in the information-theoretic...

In Surrogate: Evaluation of Surrogate Endpoints in Clinical Trials

Assess surrogacy in the information-theoretic causal-inference framework (Individual Causal Association, ICA) for a continuous surrogate and true endpoint measured repeatedly over time in a single-trial setting

Description

This function quantifies surrogacy under Galecki's model. It is a special case without subject-specific random effects. See Details below.

Usage

ICA_contcont_long_galecki(
  V0,
  V1,
  p,
  grid_V = seq(-1, 1, by = 0.1),
  return_candidates = FALSE,
  tol = 1e-12
)

Arguments

V0, V1	Treatment-specific residual 2 x 2 covariance matrices for T,S.
p	(numeric) number of time points.
grid_V	Sensitivity grid for unidentified residual correlations.
return_candidates	Logical. If TRUE, the retained admissible residual covariance completion candidates are returned.
tol	Numerical tolerance used when checking positive definiteness and validating ICA values.

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.

• Num.Pos.Def.V: An object of class numeric that contains the number of positive definite V matrices.

• R2_Lambda: A scalar or vector that contains the individual causal association R_{\Lambda}^2.

• V_candidates: retained residual covariance candidates, if requested.

Galecki's model:

The evolution of y_j over time is defined using a marginal model without subject-specific random effects.

T_{0j} = x_{T0j} \gamma_{T0} + \varepsilon_{T0j}

T_{1j} = x_{T1j} \gamma_{T1} + \varepsilon_{T1j}

S_{0j} = x_{S0j} \gamma_{S0} + \varepsilon_{S0j}

S_{1j} = x_{S1j} \gamma_{S1} + \varepsilon_{S1j}

It can be written as y=x \gamma + \varepsilon. It is assumed that \varepsilon \sim N(0, \Sigma), or, equivalently, y \sim N(x \gamma, \Sigma_y). Essentially, Galecki (1994) assumed that the covariance matrix is separable, i.e. it can be written as \Sigma_y = \Sigma = R \otimes V , where V is assumed to be unstructured and invariant across time, and R reflects a general correlation matrix.

Under Galecki's model, the vector of individual causal treatment effects \Delta is:

\Delta = \mu_{\Delta}+ \varepsilon_{\Delta}

and the marginal covariance matrix of the individual causal treatment effects is \Sigma_{\Delta} = \Sigma_u \otimes R.

Under Galecki's model, the ICA is given by:

R_{\Lambda}^2 = 1 - (1-\rho_{u}^2)^p

In this model \rho_u=\rho_{\Delta}=corr(\Delta T, \Delta S), which is the metric used in the cross-sectional setting.

\rho_u = \frac{ \sqrt{\sigma_{T0T0} \, \sigma_{S0S0}} \, \rho_{T0S0} + \sqrt{\sigma_{T1T1} \, \sigma_{S1S1}} \, \rho_{T1S1} - \sqrt{\sigma_{T1T1} \, \sigma_{S0S0}} \, \rho_{T1S0} - \sqrt{\sigma_{T0T0} \, \sigma_{S1S1}} \, \rho_{T0S1} }{ \sqrt{ (\sigma_{T0T0} + \sigma_{T1T1} - 2 \sqrt{\sigma_{T0T0} \, \sigma_{T1T1}} \, \rho_{T0T1}) (\sigma_{S0S0} + \sigma_{S1S1} - 2 \sqrt{\sigma_{S0S0} \, \sigma_{S1S1}} \, \rho_{S0S1}) } }

References

Deliorman, G., Pardo, M. C., Van der Elst W., Molenberghs G., & Alonso, A. (submitted in 2026). Assessing Surrogate Endpoints in Longitudinal Studies: An Information-Theoretic Causal Inference Approach.

Examples

V0 <- matrix(c(0.04513345, 0.04375355,
               0.04375355, 0.04866820),
              nrow = 2, byrow = TRUE,
              dimnames = list(c("T", "S"), c("T", "S")))


V1 <- matrix(c(0.05231045, 0.05177398,
               0.05177398, 0.05702684),
               nrow = 2, byrow = TRUE,
               dimnames = list(c("T", "S"), c("T", "S")))

ica_galecki<- ICA_contcont_long_galecki(V0 = V0,  V1 = V1,  p=5,
                          grid_V = seq(-1, 1, by = 0.5),
                          return_candidates = FALSE,
                          tol = 1e-12)

summary_longitudinal_ica(ica_galecki)

Surrogate documentation built on Oct. 16, 2026, 9:07 a.m.