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Appendix

Consider a continuous variable Sj and the linear model

where zSj is a (general) q—dimensional vector of covariates (including the intercept), nS is a corresponding q—dimensional vector of coefficients, and ?j ~ N(0, a2). The estimating equations for nS can then be expressed as follows (Molenberghs and Verbeke, 2005):

Define

It can be shown (Molenberghs and Verbeke, 2005) that n1/2(nS — nS) has, asymptotically, a normal distribution with mean 0 and variance-covariance matrix that can be estimated by

For the failure-time variable Tj, consider the proportional hazard model (7.2) with coefficients nT.

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 WT,j(nT) was defined in (6.25). Define AST(nT, nS) to be a block- diagonal matrix with AT(nT), defined in (6.26), and As(nS) 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 variance-covariance matrix that can be estimated by

In particular, the covariance between nT and nS can be estimated by

 
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