A Multiple-Record Systems Estimation Method that Takes.docx

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1、Biometrics 60, 510 516 June 2004 A Multiple-Record Systems Estimation Method that Takes Observed and Unobserved Heterogeneity into Account Elena Stanghellini Dipartimento di Scienze Statistiche, Universita di Perugia, 06100 Perugia, Italy email: elena.stanghellinistat.unipg.it and Peter G. M. van de

2、r Heijden Department of Methodology and Statistics, Utrecht University, P.O. Box 80.140, 3508 TC Utrecht, The Netherlands email: p.vanderheijdenfss.uu.nl Summary. We present a model to estimate the size of an unknown population from a number of lists that applies when the assumptions of (a) homogene

3、ity of capture probabilities of individuals and (b) marginal independence of lists are violated. This situation typically occurs in epidemiological studies, where the heterogeneity of individuals is severe and researchers cannot control the independence between sources of ascertainment. We discuss t

4、he situation when categorical covariates are available and the interest is not only in the total undercount, but also in the undercount within each stratum resulting from the cross- classication of the covariates. We also present several techniques for determining condence intervals of the undercoun

5、t within each stratum using the prole log likelihood, thereby extending the work of Cormack (1992, Biometrics 48, 567576). Key words: Conditional independence model; Extended latent class model; Graphical models; Hierarchical log linear models; Identiability; Observed information matrix; Prole log l

6、ikelihood; Strata. 1. Introduction In epidemiology multiple-record systems estimation methods are employed to estimate the size of an unknown population. Traditionally this was done using a log linear model. If there are two lists of individuals, it is necessary to assume indepen- dence of the two p

7、robabilities to be included in the lists. If there are more than two lists, and if other external informa- tion is not available, then it is possible to allow for interac- tion between the lists. An overview of this approach can be found in Bishop, Fienberg, and Holland (1975). In this tradi- tional

8、 approach an assumption is that in each record system the inclusion probabilities are homogeneous over the individu- als, i.e., individuals have the same probability of being caught (Chao et al., 2001). Violation of this homogeneity assumption will result in dependence between lists (see, for exampl

9、e, In- ternational Working Group for Disease Monitoring and Fore- casting, 1995). Many strategies are proposed to go around this restrictive and often unrealistic assumption and the objective of this article is to extend one of these approaches. Originally Bishop et al. (1975) proposed to tackle pos

10、sible heterogeneity by stratifying the sample in such a way that in each stratum the homogeneity would be fullled. This option is possible if there is an observed categorical covariate that is closely related to the probability of being in each of the lists. Later Alho (1990) and Huggins (1991) gene

11、ralized this approach by proposing a model where the capture probabil- ities of each list are a function of observed covariates that can be categorical or continuous. These two strategies have in common that they allow for observed heterogeneity, i.e., the heterogeneity of capture probabilities can

12、be taken into account by observed covariates. Alternatively, models are proposed for situations where such covariates are not available. One approach makes use of log linear models with homogeneous two-factor interac- tions (see International Working Group for Disease Monitor- ing and Forecasting, 1

13、995); a second one makes use of latent class models, where it is assumed that the individuals can be classied in a small number of groups (the latent classes) with homogeneous capture probabilities and lists are inde- pendent conditional on the latent classes (see Agresti, 1994; Coull and Agresti, 1

14、999); a third makes use of Rasch mod- els, where it is assumed that the individuals dier on a con- tinuous scale (see Darroch et al., 1993; Coull and Agresti, 1999; Fienberg, Johnson, and Junker, 1999). These three modeling strategies are closely related due to the relation that exists between the R

15、asch model, quasi-symmetry mod- els, and latent class analysis (see Lindsey, Clogg, and Greco, 1991). 510 A Multiple-Record Systems Estimation Method 511 js h=1 hjs H H t j t1 More recently, Biggeri et al. (1999) account for local de- pendence between a pair of lists in a latent class model. This ap

16、proach is justied in epidemiological studies to estimate the size of a human population as there is unobserved het- erogeneity between individuals and lists are set up for various purposes, so that the independence between lists is not guar- anteed. A parallel approach, which makes use of the Rasch

17、Xhj s with the convention that SR is running fastest, and let E(X) = m. In dening the likelihood for our model we should take into account that U is not observed. Let t = 2R. The marginal entries Y = H X are also independent Poisson random variables, with expected values E(Y js) = js. We denote by Y

18、 the vector with the counts in the marginal table, stacked model, has been proposed by Bartolucci and Forcina (2001). in a way that Y = LX with L = 1 IJ t. It then follows Their model includes parameters for the associations between that E(Y ) = = Lm. Moreover, as the entries in the cells two lists,

19、 after conditioning on the unobserved heterogeneity and marginalizing with respect to the other lists. In this article we extend the approach of Biggeri et al. (1999) by including a covariate in the model. We use a hier- where s = 0, . , 0, are zero by construction, the rst entry of each stratum j i

20、s not observable. We denote with Y the vector of the observable data obtained from Y by removing the entries 1, t + 1, . , (J 1)t + 1. Note that Y = LX , archical log linear model with one categorical latent variable with L = 1 IJ (t1). We will refer to X as the complete representing the unobserved

21、heterogeneity. The model can be data vector and to Y as the incomplete data vector. represented via a conditional independence graph (Whittaker, 1990; Edwards, 2000) with one unobserved node. As we are dealing with nonstandard latent variable models, we discuss identiability by assessing the rank of

22、 the information ma- trix. We provide separate estimates for the undercount in each stratum as dened by the covariate and complement the esti- mates with condence intervals evaluated by using the prole Let n = nj be the vector of the observed counts in stra- tum j and N = Nj be the vector of the tot

23、al counts in stratum j. The model considered in this article can be writ- ten as log(m) = Z with Z a design matrix and the vector of parameters. The unknown parameters of the models are N and and their estimates should be derived by maximizing the log likelihood l(y | MY = N ; , N ), with M = IJ 1 ,

24、 log likelihood, thereby extending the work of Cormack (1992). When we compare modeling each stratum separately with simultaneous modeling over strata, the advantage of the latter is that it allows to make the model more parsimonious by re- where P (Y | MY = N ) Nj 1 p Nj nj p nj q ()yjs 1 stricting

25、 certain parameters to be equal over the strata and to = n test for these equalities. We reanalyze the data on prevalence j j ()j () nj ! js s:s =0 yjs! of diabetes in a town in northern Italy (Bruno et al., 1994) using a covariate related to the severity of the illness. The re- sults are compared w

26、ith other studies made on the same data set. In Section 2 we present the general framework of log linear models with one latent categorical variable in the capture recapture context, while in Section 3 we present the modeling strategy and address the issue of identiability. In Section 4 we describe

27、how condence intervals for the undercount of each stratum can be derived and in Section 5 we report our analyses on the prevalence of diabetes in a small area of northern Italy. In Section 6 we draw our conclusions. 2. Generalities Assume that there are R lists denoted by Sr , r = 1, . , R, with lev

28、els indexed by sr , where sr = 1 indicates being on list r and sr = 0 indicates not being on list r. We denote a categorical latent variable by U with levels indexed by h, h = 1, . , H. The latent variable U accounts for unobserved heterogeneity of the individuals by assuming that individuals with t

29、he same level of h have identical probabilities of inclu- sion but individuals with dierent h may have dierent prob- abilities of inclusion. Assume also that categorical covariates are available. As we are not interested in modeling the inter- action of the covariates, we will assume that we have a

30、unique covariate C with levels indexed by j, j = 1, . , J , obtained by the cross-classication of the levels of the covariates. The interest is in estimating the undercount for each level j of C. Let s = s1, . , sR denote a certain pattern of captures. Then let Xhj s refer to the counts with U = h,

31、C = j and a pattern of captures s. We assume Xhj s to be independent Poisson random variables. We denote with X the vector of in which pjs = js/ s js, p = s:s=0 pjs, and qjs = pjs/p . j j However, maximum likelihood estimates for and N are usually derived in two steps: (a) the maximum likelihood es-

32、 timate of is derived by maximizing the conditional log like- lihood l(y | MY = n; ) with M = IJ 1 ; (b) the vector of the conditional estimates of Nj is derived as the integer part of nj /p (see Bishop et al., 1975, p. 237). Although point estimates of the undercount are derived conditionally on th

33、e nj , condence intervals will be evaluated by both considering nj as xed or as random variables. 3. The Model As already noticed, when estimating the size of a human pop- ulation by means of incomplete lists we cannot always assume the lists to act independently, even after controlling for the ob-

34、served and unobserved heterogeneity. Furthermore, we do not exclude the covariate C to be conditionally associated with the lists, even after conditioning on all the other variables. As there is no clear ordering between the variables, we consider all variables on an equal footing and we use a log l

35、inear model with one unobserved variable. We restrict to the class of models that are hierarchical. These models can be represented with an independence graph, that is a diagram in which the nodes correspond to variables. An edge between two nodes is missing whenever the two corresponding variables

36、are independent given all the other variables. The conditioning set for this independence to hold can be reduced to its minimum by using the parallelism between conditional independence properties and the notion of separation (see Edwards, 2000). However, not all log linear models with one latent va

37、riable are locally identiable and the identication state should then be checked. Rothenberg (1971) shows that a way to assess j Biometrics, June 2004 512 r p H 1, 1, 1, j j local identiability is by the rank of the observed informa- tion matrix evaluated at the maximum of the likelihood (see also Ca

38、tchpole and Morgan, 1997, for a general discussion of local identiability). Goodman (1974) discusses local identi- ability of latent class models when the assumption of local independence between the observed variables holds. We provide a separate estimate of the unobserved count in each stratum obt

39、ained from the cross-classication of the covariates. We complement the estimates with condence in- tervals for the unobserved count in each stratum using both the prole log likelihood and the delta method. Maximum likelihood estimation of the model can be per- formed with the EM algorithm. The EM al

40、gorithm alternates the following E and M steps until convergence: E-step: The conditional expectation of the E(X | LX = y) is computed given the current best parameter estimates of . M-step: New parameter estimates are computed by tting a log linear model for the complete data X. Due to the close si

41、milarity with the latent class model we refrain from describing the algorithm in detail but refer to associated with the rst entry on stratum r and Yr = yr the vector of the random variables Y augmented by Vr . With v we denote the vector obtained from n by substituting the rth element with nr + vr

42、. Let r denote the value of that maximizes l(yr | Mr Yr = v; ) where Mr is a matrix obtained from M by inserting a column in position r(t 1). The col- umn to be inserted is eJr , that is the rth column of IJ . Let N (r ) be the vector with nr + vr as the rth element and the conditional estimate Nj (

43、r ) as the jth element, j = r. The unconditional Poisson prole log likelihood for vr is then obtained as PP (vr ) = 2l(y; ) l(y; r ). Analogously, the unconditional multinomial prole log like- lihood for vr is obtained as PM = 2ly | MY = N () ly | MY = N (r ). By a parallel argument of Cormack (1992

44、), Theorem 2, if vr = Nr () nr then r = . However, in general, Nj () = Nj (r ). We indicate with Devp(Yr ; r ) the deviance of the Poisson model on Yr with parameter r and with Devp (Y; ) the deviance of the Poisson model on Y with parameter . The unconditional prole log likelihood can be obtained a

45、s a function of their dierence. As an exam- ple, the unconditional multinomial prole log likelihood can be written as Agresti and Lang (1993). The observed information matrix can be obtained by dif- ferentiating twice l(y; ) with respect to . Alternatively we can use the results of Lang (1992) that

46、showed that, for PM (vr ) = Di + 2 log Nr ()! (nr + vr )! log Nr () nr ! vr ! r () Poisson log linear models, Louiss formula (Louis, 1982) for the observed information matrix can be reformulated in a sim- plied way. Let the vector of the complete data X be Poisson distributed and the vector Y of the

47、 observed data be Y = TX with T such that (i) each element is a 0 or a 1 and (ii) there + Nr () nr log1 p vr log1 p (r ) + log j =r Nj ()! Nj (r )! is at most one 1 per column. Then the observed information Nj () nj ! log + N () n matrix of the incomplete data I( | y = Tx ) can be derived as I( | y = Tx ) = Zvar(X; ) var(X | TX = y; )Z. In our Nj (r ) nj ! case, T = L and conditions (i) and (ii) are fullled. Evaluation of the rank of the observed information matrix shows that all log1 pj () Nj (r ) nj j () j ( ) + n log , p ( ) models considered in th