Fisher信息矩阵用于非线性混合效应的多重效应模型：用于的药代动(2)

model of a drug and its metabolite. However, the accuracy of the development of MF by linearization for multiple responses has not yet been evaluated. Even if the same linearization as in the single response is used, computation can become more complicated for multiple responses. Indeed, some parameters can be included in several responses and the information on those parameters is therefore obtained from each of those response profiles. This is usual in the PKPD context where PD response depends on the PK parameters. Moreover, as noted previously, use of the linearization around the expectation of the random effects appears to be inadequate for joint estimation of multiple response models [6, 7]. The appropriateness of its use in the context of design evaluation is thus also questionable and should be investigated. The objective of this study was therefore to evaluate the first order approximation to compute the Fisher information matrix in NLMEM with multiple responses. To do this, we considered a PKPD simulation example associated with a population design. Then, we compared the predicted standard errors (SE), computed from the approximated expression of MF to those given by the evaluation of MF without linearization obtained by stochastic approximation using the SAEM algorithm of MONOLIX. We also performed another evaluation by comparison of those predicted SE to the empirical ones, obtained by estimation on simulated datasets using three different estimation algorithms: FO and FOCE (with NONMEM); SAEM (with MONOLIX). Based on those simulations, we also compared the performance of those three estimation methods in the same simultaneous analysis of this PKPD model.

In Section 2, we introduce the notations, describe the PKPD example and present the methodology used to evaluate MF and to compare the estimation methods. Section 3 describes the results of the evaluation and the comparison. Discussion of the results is provided in Section 4. The development of MF for multiple responses is given in detail in the Appendix.

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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

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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)

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

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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-

diagonal matrix composed of each diagonal element of Σik with k=1,K,K. σslope and σinter are two vectors of the K components σinterk and σslopek, k=1,K,K, respectively. Finally, conditionally on the v …… 此处隐藏：5893字，全部文档内容请下载后查看。喜欢就下载吧 ……