where yik, k=1,K,K, is the vector of observations for the kth response. Each of these responses is associated with a known function fk which defines the nonlinear structural model. The K functions fk can be grouped in a vector of multiple response models F, such as:
T
F(θi,ξi)= f1θi,ξi1
()
T
,f2θi,ξi2
()
T
,K,fKθi,ξiK
()
T
(4)
where θi is the vector of all the individual parameters needed for all the response models in individual i. The vector of individual parameters θi depends on β, the p-vector of the fixed effects parameters and on bi the vector of the p random effects for individual i. The relation
inserm-00371363, version 1 - 27 Mar 2009
between θi and (β,bi) is modelled by a functiong,θi=g(β,bi), which is usually additive, so that θi=β+bi, or exponential so that θi=βexp(bi). It is assumed that bi～N(0, )with defined as a p×p-diagonal matrix, for which, each diagonal elementωr2,r=1,K,p,
represents the variance of the rthcomponent of the vector bi. The statistical model is thus given by:
Yi=F(g(β,bi),ξi)+εi
(5)
where εi is the vector composed of the K vectors of residual errors εik, k=1,K,K, associated with the K responses. We also suppose εik～N(0,Σik) with Σik a nik×nik-diagonal matrix such that
Σik(β,bi,σinterk,σslopek,ξik)=diagσinterk+σslopekfk(g(β,bi),ξik)
where σinterk and σslopek qualify the model for the variance of the residual error of the kthresponse. The case σslopek=0 returns a homoscedastic error model, whereas the case
()
2
(6)
σinterk=0 returns a constant coefficient of variation error model. The general case where the
two parameters differ from 0 is called a combined error model. We then note
Σi(β,bi,σinter,σslope,ξi) the variance of εi, over the K responses, such that Σi is a ni×ni-
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