AA.MultS {Surrogate}

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.

See Also

Single.Trial.RE.AA

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)