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AppendixConsider a continuous variable Sj and the linear model
where z_{S}j is a (general) q—dimensional vector of covariates (including the intercept), n_{S} is a corresponding q—dimensional vector of coefficients, and ?j ~ N(0, a^{2}). The estimating equations for n_{S} can then be expressed as follows (Molenberghs and Verbeke, 2005):
Define It can be shown (Molenberghs and Verbeke, 2005) that n^{1/2}(n_{S} — n_{S}) has, asymptotically, a normal distribution with mean 0 and variancecovariance matrix that can be estimated by
For the failuretime variable Tj, consider the proportional hazard model (7.2) with coefficients n_{T}. Consider now a bivariate random variable (Tj, Sj). Let models (7.2) and (7.10) be marginal models for Tj and Sj, respectively. The estimating equations for the coefficient vector (nT, nS У are These equations can be seen as arising under “independence working assumptions” (Liang and Zeger, 1986). Denote the solution to (7.12) by (Пт, nS) • Let with W_{T},j(n_{T}) was defined in (6.25). Define A_{ST}(n_{T}, n_{S}) to be a block diagonal matrix with A_{T}(n_{T}), defined in (6.26), and A_{s}(n_{S}) on the diagonal and zeros elsewhere. Then it follows (see Appendix 6.5) that (nT, nSУ is asymptotically normal with mean (nT, nS У and the variancecovariance matrix that can be estimated by
In particular, the covariance between n_{T} and n_{S} can be estimated by 
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