AA.MultS: Compute the multiple-surrogate adjusted association
In Surrogate: Evaluation of Surrogate Endpoints in Clinical Trials

AA.MultS

R Documentation

Compute the multiple-surrogate adjusted association

Description

The function AA.MultS computes the multiple-surrogate adjusted correlation. This is a generalisation of the adjusted association proposed by Buyse & Molenberghs (1998) (see Single.Trial.RE.AA) to the setting where there are multiple endpoints. See Details below.

Usage

AA.MultS(Sigma_gamma, N, Alpha=0.05)

Arguments

`Sigma_gamma`	The variance covariance matrix of the residuals of regression models in which the true endpoint (`T`) is regressed on the treatment (`Z`), the first surrogate (`S1`) is regressed on `Z`, ..., and the `k`-th surrogate (`Sk`) is regressed on `Z`. See Details below.
`N`	The sample size (needed to compute a CI around the multiple adjusted association; `\gamma_M`)
`Alpha`	The `\alpha`-level that is used to determine the confidence interval around `\gamma_M`. Default `0.05`.

Details

The multiple-surrogate adjusted association (\gamma_M) is obtained by regressing T, S1, S2, ..., Sk on the treatment (Z):

T_{j}=\mu_{T}+\beta Z_{j}+\varepsilon_{Tj},

S1_{j}=\mu_{S1}+\alpha_{1}Z_{j}+\varepsilon_{S1j},

\ldots,

Sk_{j}=\mu_{Sk}+\alpha_{k}Z_{j}+\varepsilon_{Skj},

where the error terms have a joint zero-mean normal distribution with variance-covariance matrix:

{\boldsymbol{\Sigma}=\left(\begin{array}{cc} \sigma_{TT} & \Sigma_{\boldsymbol{S}T}\\ \Sigma^{'}_{\boldsymbol{S}T} & \Sigma_{\boldsymbol{SS}} \\ \end{array}\right).}

The multiple adjusted association is then computed as

\gamma_M = \sqrt(\frac{\left(\Sigma^{'}_{ST} \Sigma^{-1}_{SS} \Sigma_{ST}\right)}{\sigma_{TT}})

Value

An object of class AA.MultS with components,

`Gamma.Delta`	An object of class `data.frame` that contains the multiple-surrogate adjusted association (i.e., `\gamma_M`), its standard error, and its confidence interval (based on the Fisher-Z transformation procedure).
`Corr.Gamma.Delta`	An object of class `data.frame` that contains the bias-corrected multiple-surrogate adjusted association (i.e., corrected `\gamma_M`), its standard error, and its confidence interval (based on the Fisher-Z transformation procedure).
`Sigma_gamma`	The variance covariance matrix of the residuals of regression models in which `T` is regressed on `Z`, `S1` is regressed on `Z`, ..., and `Sk` is regressed on `Z`.
`N`	The sample size (used to compute a CI around the multiple adjusted association; `\gamma_M`)
`Alpha`	The `\alpha`-level that is used to determine the confidence interval around `\gamma_M`.

Author(s)

Wim Van der Elst, Ariel Alonso, & Geert Molenberghs

References

Buyse, M., & Molenberghs, G. (1998). The validation of surrogate endpoints in randomized experiments. Biometrics, 54, 1014-1029.

Van der Elst, W., Alonso, A. A., & Molenberghs, G. (2017). A causal inference-based approach to evaluate surrogacy using multiple surrogates.

Examples

data(ARMD.MultS)

# Regress T on Z, S1 on Z, ..., Sk on Z 
# (to compute the covariance matrix of the residuals)
Res_T <- residuals(lm(Diff52~Treat, data=ARMD.MultS))
Res_S1 <- residuals(lm(Diff4~Treat, data=ARMD.MultS))
Res_S2 <- residuals(lm(Diff12~Treat, data=ARMD.MultS))
Res_S3 <- residuals(lm(Diff24~Treat, data=ARMD.MultS))
Residuals <- cbind(Res_T, Res_S1, Res_S2, Res_S3)

# Make covariance matrix of residuals, Sigma_gamma
Sigma_gamma <- cov(Residuals)

# Conduct analysis
Result <- AA.MultS(Sigma_gamma = Sigma_gamma, N = 188, Alpha = .05)

# Explore results
summary(Result)

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