2 Methods
2.1 Notation
In the nonlinear mixed effect multiple response model, an “elementary” design ξi for one individual i is defined by nisampling times. It is composed of several sub-designs such that
ξi=(ξi1,ξi2,K,ξiK), with ξik being the sub-design associated with the kth response,
k=1,K,K. ξik is defined by tik1,tik2,K,tiknik, the vector of the nik sampling times for the
()
observations of the k response, so that ni=∑nik.
th
K
k=1
For N individuals, we define a “population design” composed of the N allocated elementary
inserm-00371363, version 1 - 27 Mar 2009
designs ξi, i=1,K,N. A population design is therefore described by the N elementary designs for a total number n of observations such that n=∑ni:
i=1N
Ξ={ξ1,K,ξN}
(1)
Usually population designs are composed of a limited number Q of groups of individuals with identical design within each group. Each of these groups is defined by an elementary designξq,q=1,K,Q, which is composed, for the kthresponse, of nqk sampling times
(t
qk1
,tqk2,K,tqknqk to be performed in a number Nq of individuals. The population design can
)
then be written as follows:
Ξ=[ξ1,N1];[ξ2,N2];K; ξQ,NQ
{}
(2)
A nonlinear mixed effects multiple response model or a multiple response population model is defined as follows. The vector of observations Yi for the ith individual is defined as the vector of the K different responses:
TTT
Yi= y,y,K,yi1i1iK
T
(3)