Biometrical Journal 47 (2005) 5, 691--706 DOI: 10.1002/bimj.200310138
Local Influence in Comparative Calibration Models Under Elliptical t-Distributions Manuel Galea; 1 , Heleno Bolfarine2 , and Filidor Vilca3 1 2
3
Departamento de Estadstica, Universidad de Valparaso, Casilla 5030, Valparaso, Chile Departamento de Estatstica, Universidade de So Paulo, Caixa Postal 66281, CEP 05315970, So Paulo, Brasil Departamento de Estatstica, Universidade Estadual de Campinas, Caixa Postal 6065, CEP 13083970, Campinas, So Paulo, Brasil
Received 4 December 2003, revised 30 September 2004, accepted 6 December 2004
Summary In this paper we consider applications of local influence (Cook, 1986) to evaluate small perturbations in the model or data set in the context of structural comparative calibration (Bolfarine and Galea, 1995) assuming that the measurements obtained follow a multivariate elliptical distribution. Different perturbation schemes are investigated and an application is considered to a real data set, using the elliptical tdistribution.
Key words: Local influence; Comparative calibration; Elliptical distribution.
1 Introduction The main object of this paper is the study of local influence and diagnostics in comparative calibration models designed to compare the efficiency of several measuring devices (or instruments) when measuring the same unknown quantity x in a common group of individuals or experimental units. It is assumed that the observed measurements follow a multivariate elliptical distribution. Moreover, the comparative calibration model can be seen as a special case of the general multivariate measurement error model (Fuller, 1987). Comparing measuring devices which varies in pricing, fastness and other features, such as efficiency, has been of growing interest in several areas like engineering, medicine, psychology and agriculture. Grubbs (1948, 1973) reports data on an experiment designed for comparing three cronometers and Barnett (1969) reports on the comparison of four combinations of two instruments and two operators for measuring vital capacity. Several other examples in the medical area are reported in the literature specially in Kelly (1984, 1985), Chipkevitch et al. (1996) and Lu et al. (1997). Examples in agriculture are considered in Fuller (1987) and in psychology and education in Dunn (1992). Outliers and detection of influential observations is an important step in the analysis of a data set. There are several ways of evaluating the influence of perturbations in the data set and in the model given the parameter estimates. Important reviews can be found in the books by Cook and Weisberg (1982) and Chatterjee and Hadi (1988) and in the paper by Cook (1986). On the other hand, there are just a few works in the literature for diagnostic and influence of observations in models with measurement errors. Kelly (1984) considered a diagnostic procedure in the structural linear model based on the influence function. Tanaka et al. (1991) also consider the influence function introduced by Hampel for
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evaluating the influence of observations in the analysis of covariance structures. Recently, Zhao, and Lee (1998) define leverage of one observation and Cook’s distance in a simultaneous equation model. Rather than eliminating cases, the approach proposed by Cook (1986) is a general method for evaluating, under the maximum likelihood estimators, the influence of small perturbations in the model or data set. Additional results on local influence and applications in linear regression and mixed models can be found in Bechman et al. (1987), Lawrance (1988), Thomas and Cook (1990), Tsai and Wu (1992), Paula (1993), Galea et al. (1997) and Lesaffre and Verbeke (1998). Zhao and Lee (1998) and Kwan and Fung (1998) apply the local influence approach for factor analysis and simultaneous equations under the normality assumption. Recently, Galea et al. (2002a) apply the local influence method in functional and structural comparative calibration models also under the normal distribution assumption. However, no application of local influence has been considered for comparative calibration under elliptical models. Thus, the main object of this paper is to apply the approach of local influence to elliptical measurement error models. As typically considered in the literature, the relevance of using the t-distribution is related to its capability of downweighting influent observations. See, for example, Lange et al. (1989). Several perturbation schemes are considered such as case perturbation and response perturbation. We also investigate a decomposition of the Mahalanobis distance of the model in two terms, one in the direction of the unobserved true values of the characteristic being measured and the other corresponding to the model error. In Section 2 the elliptical structural comparative calibration model is considered and in Section 3 the main concepts of local influence are revised. In Section 4 model curvatures are considered for different perturbation schemes and in Section 5 an illustration of the methodology is presented for a real data set. Finally in the Section 6 it presents some concluding remark.
2 Elliptical Comparative Calibration Models Suppose that we have at our disposal p 2 instruments for measuring a characteristic of interest x in a group of n experimental units. Let xi be the true (unknown) value in unit i and yij the measured value obtained with instrument j in unit i, i ¼ 1; . . . ; n and j ¼ 1; . . . ; p. A model typically considered in the literature (see Jaech, 1964; Cochran, 1968; Barnett, 1969; Williams, 1969; Shyr and Gleser, 1986) for such situation is given by yij ¼ aj þ bj xi þ eij. The measurement errors eij are assumed to be mutually independent, and also independent of the (random) true values x1 ; . . . ; xn . Further, for each j, the random variables e1j ; . . . ; enj are assumed to have a common normal distribution with mean 0 and variance fj , j ¼ 1; . . . ; p. Finally, we assume that x1 ; . . . ; xn are a random sample from a normal distribution with mean mx and variance fx . It is easily seen that the parameters of this model are not identifiable (Shyr and Gleser, 1986) consequently restrictions must be placed on the parameters to identify them. Barnett (1969) consider that there is a reference instrument (instrument 1) which measures without bias (additive or multiplicative) the quantity of interest and consequently assumed that a1 ¼ 0 and b1 ¼ 1. Theobald and Mallison (1978) adopt the usual factor analytic constraints mx ¼ 0 and fx ¼ 1. Using such constraint is equivalent to changing the definition of the quantity to be measðxi m Þ ured from xi to the standardized value zi ¼ pffiffiffiffi x . More recently; Lu et al. (1997) consider mx and fx fx known to identify the model. For diagnostic purposes, any parameterization yields the same results using local influence. In this paper we use the parametrization of Theobald and Mallison (1978), which presents computational advantages. Thus, the model, in matrix notation, is Y i ¼ m þ lzi þ ei ¼ m þ LUi ;
ð2:1Þ
T
T
where m ¼ ðm1 ; . . . ; mp Þ , l ¼ ðl1 ; . . . ; lp Þ are p 1 vectors, L ¼ ðl; I p Þ is a p ðp þ 1Þ matrix, Y i ¼ ðyi1 ; . . . ; yip ÞT and ei ¼ ðei1 ; . . . ; eip ÞT are p 1 random vectors U i ¼ ðzi ; eTi ÞT is of dimension ðp þ 1Þ 1 and I p denotes the identity matrix of dimension p, i ¼ 1; . . . ; n and the random vectors www.biometrical-journal.de
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U 1 ; . . . ; U n are iid Npþ1 ð0; SÞ, where 1 0 S¼ 0 DðfÞ with DðfÞ ¼ diag ðf1 ; . . . ; fp Þ. Note that m ¼ ð0; a2 ; . . . ; ap ÞT þ mx ð1; b2 ; . . . ; bp ÞT
ð2:2Þ pffiffiffiffi and l ¼ fx
ð1; b2 ; . . . ; bp ÞT for the model proposed in Barnett (1969) and m ¼ ða1 ; . . . ; ap ÞT þ mx ðb1 ; . . . ; bp ÞT pffiffiffiffi and l ¼ fx ðb1 ; . . . ; bp ÞT corresponding to the model considered in Lu et al. (1997). Hence, by working with the general parametrization in (2.1) the models considered in Barnett (1969), in Theobald and Mallison (1978) and Lu et al. (1997) are all contemplated. Though model (2.1) is useful in many areas (Fuller, 1987; Dunn, 1992), it suffers from the same lack of robustness against outlying/influential observations as other statistical models based on the normal distribution. The above model, with DðfÞ ¼ fI p , has been applied to a data set reported in Barnett (1969) corresponding to measurements by four instruments of the vital capacity of the human lung in a group of 72 patients. A study of local influence for detecting the effect of small perturbations of the model or data on the maximum likelihood estimators for that data set is presented in Galea et al. (2002a). According to that study, some observations seem to exert moderate to strong influence on the maximum likelihood estimators. Observations that come out as the most influential ones are 23, 30 and 67. The main object of this paper is to consider an extension of that study by analyzing the effect of the observations on the t-distribution, specially to detect the tradeoff between the number of influential observations and the degrees of freedom parameter of the t-distribution. We hope also with that study to be able to find the most adequate degrees of freedom parameter so that outliers are well accommodated. A vast statistical literature exists on robust modelling methods, with some authors concentrating more on methods for outliers (influential observation) identification (Barnett and Lewis, 1995; Cook, 1986) and others on methods for outliers accommodation (Huber, 1981; Hampel et al., 1986) We follow here the robust statistical modeling approach described by Lange et al. (1989) and consider a version of model (2.1) in which the normal distributions for the zi and ei are replaced by elliptical t-distributions. Recently, several authors have considered the multivariate t-distribution as an alternative to the normal model because it can naturally accommodate outliers present in the data. Thus, the t-model provides a robust procedure for analysing data sets which may present outliers. Rubin (1983) obtained maximum likelihood estimators for the parameters of the multivariate t-model by using the EM-algorithm; Little (1988) extends the results in Rubin (1983) for the case of incomplete data sets, that is, data sets with missing data. Sutradhar and Ali (1986) consider maximum likelihood estimation in the multivariate t-regression model. Lange et al. (1989) discuss the use of the t-distribution in regression and in problems related to multivariate analysis. More recently, Sutradhar (1993) has considered a score test aiming at testing if the covariance matrix is equal to some specified covariance matrix (diagonal, for example), using the t-distribution; Bolfarine and Arellano (1994) introduce t-functional and t-structural measurement error models and Bolfarine and Galea (1996) use the t-distribution in structural comparative calibration models. The t-distribution incorporates an additional parameter n, namely the degrees of freedom parameter, which allows adjusting for the kurthosis of the distribution. In this work we suppose that the degrees of freedom n is known. Fernandez and Steel (1999), alert on problems with the estimation of n and they notice that the log-likelihood function can be unbounded and that indeed it corresponds to an nonregular estimation problem. See also Taylor and Verbyla (2004). Due to this, it is suggested (see Lange et al., 1989) to estimate q ¼ ðmT ; lT ; mT ÞT considering a set of acceptable values for n and to choose the one that maximizes the log-likelihood function. Specifically we replace the multivariate normal distribution in (2.1) with the multivariate elliptical t-distribution and obtain the elliptical t-structural comparative calibration model: i ¼ 1; 2; . . . ; n Y i ¼ m þ lzi þ ei ; zi ð2:3Þ Ui ¼ ; 1 i n; are iid tpþ1 ð0; S; nÞ ; ei www.biometrical-journal.de
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where n represents the t-distribution degrees of freedom. It follows from (2.3) that Y 1 ; . . . ; Y n are iid tp ðm; V; nÞ, with density given by: 1
y 2 IRp ; ð2:4Þ fY ðyÞ ¼ jVj2 gððy mÞT V 1 ðy mÞÞ ; 12ðnþpÞ p þ n n d , kðp; nÞ ¼ G where gðdÞ ¼ kðp; nÞ 1 þ =ðnpÞp=2 G , d ¼ ðy mÞT V 1 ðy mÞ and n 2 2 V ¼ llT þ DðfÞ is the scale matrix. Thus, the main object of this paper is to consider the approach of local influence for the elliptical t-structural comparative calibration model given in (2.3). To estimate q we used the EM algorithm, see Lange et al. (1989) and Bolfarine and Galea (1996).
3 Local Influence Let lðqÞ denote the log-likelihood function from the postulated model (here q ¼ ðmT ; lT ; mT ÞT ) and let w be a q 1 perturbation vector restricted to some open subset of Rq . The perturbations are made in the likelihood function such that it takes the form lðq j w). Denoting the vector of no perturbation by w0 , we assume lðq j w0 Þ ¼ l(q). To asses the influence of the perturbations on the maximum likelihood estimate of q, one may consider the likelihood displacement ^Þ lðq ^w Þ ; LDðwÞ ¼ 2½lðq ^ denotes the maximum likelihood estimators under the models lðqjwÞ and lðqÞ, respec^w and q where q tively. In some situations, it may be of interest to assess the influence on a subset q1 of q ¼ ðqT1 ; qT2 ÞT . For example, one may have interest on q1 ¼ l or q1 ¼ f. In such situations, the likelihood displacement can be defined as ^2 ðq ^1w ÞÞ; ^Þ lðq ^1w ; q LDðwÞ ¼ 2½lðq ^w ¼ ðq ^T ; q ^T T ^ ^ ^1w can be obtained from q where q 1w 2w Þ and q2 ðq1w Þ is the maximum likelihood estimate of q2 for q1w fixed in the perturbed model. The idea of local influence (Cook, 1986) is concerned in characterizing the behavior of LDðwÞ at w0 . The procedure consists of selecting a unit direction l, jj l jj ¼ 1, and then to consider the plot of LDðw0 þ alÞ against a with a 2 R. This plot is called lifted line. Notice that since LDðw0 Þ ¼ 0, LDðw0 þ alÞ has a local minimum at a ¼ 0. Each lifted line can be characterized by considering the normal curvature Cl ðqÞ around a ¼ 0. The suggestion is to consider the direction lmax corresponding to the largest curvature Cl max ðqÞ. The index plot of lmax may reveal those observations that under small perturbations exert notable influence on LDðwÞ. Cook (1986) showed that the normal curvature at the direction l takes the form Cl ðqÞ ¼ 2jlT DT L1 Dl j;
ð3:1Þ
where -- L is the observed Fisher information matrix for the postulated model ðw ¼ w0 Þ and D is the p* q matrix with elements Dij ¼
@ 2 lðqjwÞ ; @qi @wj
^ and w ¼ w0 ; i ¼ l; . . . ; p* and j ¼ 1; . . . ; q; p* ¼ 3p. Therefore, the maximization evaluated at q ¼ q of (3.1) is equivalent to finding the largest absolute eigenvalue Cl max of the matrix B ¼ DT L1 D, and lmax is the corresponding eigenvector. For the subset q1 , the curvature at the direction l is given by Cl ðq1 Þ ¼ 2j lT DT ðL1 B22 ÞDl j ; www.biometrical-journal.de
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where
B22 ¼
0 0
0 L1 22
695
;
and L22 is obtained from the partition of L according to the partition of q. The eigenvector lmax corresponds to the largest absolute eigenvalue of the matrix B ¼ DT ðL1 B22 Þ D. Another important direction, according to Escobar and Meeker (1992) (see also Verbeke and Molenberghs, 2000) is l ¼ ein , which corresponds to the i-th position, where there is a one. In that case, the normal curvature, called the total local influence of individual i, is given by Ci ¼ 2jeTin Bein j ¼ 2jbii j, where bii is the i-th element diagonal of B, i ¼ 1; . . . ; n. Verbeke and Molenberghs (2000) propose to q P Ci =q. We use lmax consider the i-th observation as influential if Ci is larger than the cutoff value 2 i¼1 and Ci as diagnostics for local influence. Fung and Kwan (1997) presented an interesting discussion on the application of the local influence for other influence measures than the likelihood displacement. They show that an influence measure, namely T^w, is scale invariant if G_ ¼ @ T^w =@wjw¼w0 ¼ 0. When this derivative is nonzero the ordering among the components of lmax is not necessarily preserved under changes in the scale. In particular, ^w Þ=@wj for the likelihood displacement, G_ ¼ @lðq w¼w0 ¼ 0. This property also follows, for instance, for the influence measures proposed in Thomas and Cook (1990) and Paula (1993). But this property is not shared by other influence measures, as pointed out by Fung and Kwan (1997).
4 Curvature Derivation for Elliptical Comparative Calibration Models In this Section we derive the observed information matrix and the D matrix for different schemes of perturbations for an arbitrary elliptical distribution, that is, for any generator density g : R 7! ½0; 1Þ such that f01 up1 gðu2 Þ du < 1. Perturbation of cases weights is considered to detect cases (observations) with large contribution to the likelihood function and which can exert great influence on the maximum likelihood estimators. Perturbation of the observations is used to detect observations (responses) that exert great influence on the maximum likelihood estimators. On the other hand, degrees of freedom perturbation can be useful to evaluate sensitivity of the maximum likelihood estimators a small modifications in this parameter, which controls the distribution kurtosis and thus, the downweighting of influential observations.
4.1
The observed information matrix
From (2.4) we have that the log-likelihood function is given by: lðqÞ ¼
n P
‘i ðqÞ;
ð4:1Þ
i¼1
where ‘i ðqÞ ¼ 12 logjV j þ log ðgðdi ÞÞ and di ¼ di ðqÞ ¼ ðyi mÞT V 1 ðyi mÞ, i ¼ 1; 2; . . . ; n and V is as in (2.4). The matrix of second derivatives with respect to q is given by: 0 1 Lmm Lml Lmf 2 @ lðqÞ Lll Llf A L¼ ð4:2Þ ¼@ @q @qT q¼q^ L ff
^ is the maximum likelihood estimator of q. The elements of this matrix are given in the where q appendix. www.biometrical-journal.de
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4.2
Perturbation of cases weights
We considered the vector of weights w ¼ ðw1 ; . . . ; wn ÞT , for weighting the contribution of each case in the log-likelihood. Thus the perturbed log-likelihood is: lðq j wÞ ¼
n P
wi ‘i ðqÞ ;
ð4:3Þ
i¼1
where ‘i ðqÞ is defined in (4.1). The vector of no perturbation is w0 ¼ ð1; . . . ; 1ÞT ¼ 1. Let Dg ¼ ðDg1 ; Dg2 ; . . . ; Dgn Þ the submatrix of D in (3.1), associated to the parameter g. That is, Dgi is the i-th column of Dg ; i ¼ 1; . . . ; n and g ¼ m; l; f. Consequently, using (4.1), (4.3) and calculus of vector derivatives (Nel, 1980), we have, after some computations that: Dmi ¼ 2Wg ðdi Þ V 1 Xi ;
ð4:4Þ
Dli ¼ c1 ½I p þ 2Wg ðdi Þ ðD1 ðfÞ X i X Ti c1 ci1 I p Þ D1 ðfÞl ;
ð4:5Þ
Dfi ¼ 12 D1 ðfÞ ½1 c1 D1 ðfÞ DðlÞ l þ Wg ðdi Þ D2 ðfÞ ½DðXi Þ Xi þ 2c1 ci2 DðlÞ X i c2 ci1 DðlÞ l ;
ð4:6Þ
k 1 where DðaÞ ¼ Diag ða1 ; . . . ; ap Þ, for a 2 Rp, Dk ðaÞ ¼ Diag ðak 1 ; . . . ; ap Þ; k ¼ 1; 2; 3, M ¼ D ðfÞ 0 g ðdi Þ ; ci1 ¼ X Ti MX i and ci2 ¼ X Ti D1 ðfÞ l, llT D1 ðfÞ; c ¼ 1 þ lT D1 ðfÞ l, X i ¼ Y i m, Wg ðdi Þ ¼ gðdi Þ i ¼ 1; . . . ; n. Expressions (4.4)--(4.6) are evaluated at the maximum likelihood estimators to yield the 3p n matrix D in (3.1) which takes the form
D ¼ ðDTm ; DTl ; DTf ÞT :
4.3
ð4:7Þ
Perturbation of the observations
In this Section, the measured values obtained with the instruments are perturbed. With this scheme we want to detect measures of the instruments that it can be exert a potential influence on the maximum likelihood estimators. Let Y i ðwi Þ the perturbation in observation Y i , where wi ¼ ðwi1 ; . . . ; wip ÞT ; i ¼ 1; . . . ; n. Some situations of interest in this case are: (a) Simultaneous perturbations of the measurements of the p instruments: Y i þ wi ; additive perturbation Y i ðwi Þ ¼ Y i wi ; multiplicative perturbation where denotes Hadamard product. (b) Perturbing the measurements from one instrument. Suppose that it is of interest perturbing the measurements from one specific instrument, say, k; k ¼ 1; . . . ; p. In this case Y i þ wi ek ; additive perturbation Y i ðwi Þ ¼ Y i 1p ðwik Þ; multiplicative perturbation where ek is the k-th unit vector of Rp and 1p ðwik Þ ¼ ð1; . . . ; 1; wik ; 1; . . . ; 1ÞT , is of dimension p. Note that in the above perturbation schemes there exists wi0 such that Y i ðwi0 Þ ¼ Y i , for example, in (a) wi0 ¼ ð0; . . . ; 0ÞT in the additive case and wi0 ¼ ð1; . . . ; 1ÞT in the multiplicative case. Let W ¼ ðw1 ; . . . ; wn Þ a p n matrix, whose columns are wi ; i ¼ 1; . . . ; n. Denote w ¼ Vec ðWÞ ¼ ðwT1 ; . . . ; wTn ÞT . Thus, the perturbed log-likelihood function is given by: lðq j wÞ ¼
n P
‘i ðq j wi Þ ;
ð4:8Þ
i¼1
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1 ‘njVj þ ‘ngðdi ðwi ÞÞ, with di ðwi Þ ¼ ðY i ðwi Þ mÞT V 1 ðY i ðwi Þ mÞ and Y i ðwi Þ 2 as defined in (a) or (b), i ¼ 1; . . . ; n. Differentiating Lðq j wÞ with respect to w and q and proceeding similarly as in the case-weight perturbation situation, it follows that:
where ‘i ðq j wi Þ ¼
Case (a): Let Dagi ðDm gi Þ the i-th submatrix of dimension p p, of Dg with respect to the additive (multiplicative) perturbation scheme, g ¼ m; l; f and i ¼ 1; . . . ; n. Thus, Dmi ¼ 2ðWg0 ðdi Þ dim XTi Wg ðdi Þ I p Þ V 1 ;
ð4:9Þ
Dli ¼ 2Wg0 ðdi Þ dil XTi V 1 þ 2c1 Wg ðdi Þ ð2V 1 X i lT D1 ðfÞ X i lT þ ci2 I p Þ D1 ðfÞ;
ð4:10Þ
Dfi ¼ 2Wg0 ðdi Þ dif XTi V 1 2Wg ðdi Þ ðD1 ðfÞ DðXi Þ V 1 þ c2 D2 ðfÞ ðDðlÞ lX Ti M c1 ci2 D2 ðfÞ DðlÞÞ;
i ¼ 1; . . . ; n;
ð4:11Þ
@di ; g ¼ m; l; f are given in the appendix. @g With respect to the multiplicative scheme, it follows that
where dig ¼
a Dm gi ¼ Dgi DðY i Þ;
g ¼ m; l; f;
i ¼ 1; . . . ; n
ð4:12Þ
The above expressions are all evaluated at the maximum likelihood estimators. Case (b): Without loss of generality, we can take k ¼ 1. To obtain the matrix D in the additive scheme, we can multiply Dagi ; g ¼ m; l; f given in (4.11) from the right by e1 ¼ ð1; 0; . . . ; 0ÞT 2 Rp. Analogously, in the multiplicative scheme, it is sufficient multiplying Dm mi , given in (4.12), from the right by the same expression. Expressions for Wg0 ðdÞ and Wg ðdÞ are obtained in Galea et al. (1997) for some distributions in the elliptical family. 4.4
Perturbation of the degrees of freedom
In this Section the following perturbed model is considered: Y i tp ðm; V; n0 hðwi ÞÞ;
ð4:13Þ
with the Y i being independent, i ¼ 1; . . . ; n, where h is positive and differentiable and further, there exists w0i such that hðw0i Þ ¼ 1. The purpose in this scheme of perturbation is to evaluate of sensitivity of the maximum likelihood estimators to small modifications of n0 . Under the perturbed model, the log-likelihood function is given by n P
lðq j wÞ ¼
‘i ðq j wi Þ ;
ð4:14Þ
i¼1
1 1 di ‘njVj ðni þ pÞ ‘n 1 þ , where ni ¼ n0 hðwi Þ; kðni ; pÞ as in 2 2 ni (2.3) with n replaced by ni and di as defined in (4.1), i ¼ 1; . . . ; n. According to our notation, w ¼ ðw1 ; . . . ; wn ÞT . Hence, following similar procedures as considered in Section 4.3, it follows that:
where ‘i ðq j wi Þ ¼ ‘nkðni ; pÞ
D ¼ ðDq1 ; . . . ; Dqn Þ; where
ð4:15Þ
1 @di 0 Wg ðdi Þ þ Wg ðdi Þ ; Dqi ¼ n0 h ðw0i Þ n0 þ p @q 0
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^. The funci ¼ 1; . . . ; n. Expression (4.15) should be evaluated at the maximum likelihood estimate q wi tion h can be defined as in Escobar and Meeker (1992), namely, hðwi Þ ¼ a , with a > 0 and hn i 0 ; an0 . For example, we wi 2 ½1; 1; i ¼ 1; . . . ; n. Thus, ni ¼ n0 hðwi Þ takes values in the interval a can take a ¼ 2 and h0 ðw0i Þ ¼ ‘n 2, i ¼ 1; . . . ; n. If wi ¼ w for all i ¼ 1; . . . ; n; hðwÞ is a scalar type ^w Þ around w ¼ w0 it follows function. In this case, considering a Taylor expansion of order 2 of lðq that LD1 ðwÞ ffi DT ðLÞ1 Dðw w0 Þ2 ; where D ¼ n0 h0 ðw0 Þ
ð4:16Þ
Wg ðdi Þ @di ; Wg0 ðdi Þ þ n0 þ p @q i¼1 n P
which is of dimension 3p 1. Note that lim Cl ðqÞ ¼ 0; for all l; jj l jj ¼ 1; meaning that for large n0 n0 !1 (close to normality) there are no directions of local influence, which is reasonable since the normal model is independent of n0 . Thus in the application presented below, we consider only the case of small n0 .
5 Application In this section we analyze a real data set given in Barnett (1969). Two instruments used for measuring the vital capacity of the human lung and operated by skilled and unskilled operators were compared on a common group of 72 patients. We will focus on the parameter set q and perturbation of cases. All computations were performed in S-Plus. ^Þ for checking the fit of the t-modLange et al. (1989) proposed the Mahalanobis-like distance di ðq els and for identifying outliers. In this case, the outliers may occur either at the level of the measurement error ei, called e-outliers (Pinheiro et al., 2001), or at the level of the unobserved random variable zi , called z-outliers. In the first case, some unusual within-individual values may be observed, whereas in the second case unusual individuals are observed. In effect, following to Pinheiro et al. (2001), we can write; ^Þ ¼ e^T D1 ðf ^Þ þ di2 ðq ^Þ ; ^ Þ e^i þ ð^zi Þ2 ¼ di1 ðq di ðq i
ð5:1Þ
^Þ, i ¼ 1; 2; . . . ; n. Note that, under normality, ^^ say, where e^i ¼ Y i m l^ zi and ^ zi ¼ Eðzi j Y i ; q ^Þ and di1 ðq ^Þ are expected to be close to p Eðdi ðqÞÞ ¼ p, Eðdi1 ðqÞÞ ¼ p and Eðz2i Þ ¼ 1. Therefore, di ðq ^ and di2 ðqÞ close to 1 under normal model, and can be used as diagnostics statistics for identifying outliers. Figure 1 presents these diagnostic statistics for Barnett’s data. Patients 25 and 67 present ^Þ and di1 ðq ^Þ, suggesting outlying observations at the within-patients level. The large values of di ðq ^ di2 ðqÞ plot gives some indication that patients 23 and 30 are possible z-outliers. The log-likelihood function was used for the choice the degrees of freedom in the t-model. The summary of results is in Table 1. Figure 2 displays the transformed distance plots, see Lange et al. (1989), for the normal and t-models with n0 ¼ 11, t11 , for Barnett’s data. As seen, from Table 1 and Figure 2, the t11 model seems to present a good fit. On the other hand, Figure 3(a) gives the index plot of lmax for the perturbation of degrees of freedom parameter in the t11 model. For n0 ¼ 11, the maximum likelihood estimators are stable with respect to small perturbations in the degrees of freedom parameter, with a little influence of the patients 4, 23 and 30. Figure 3(b) shows plots of the likelihood displacement LDðw0 þ almax Þ versus a for perturbation of case weights for normal and t11 -models. It is worth noticing that the LD curve for the t11 -model has a smaller change than the curve for the normal model. Thus, for the present data set n0 ¼ 11 seems to be a adequate value of parameter degrees of freedom. www.biometrical-journal.de
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Summary of model fitting.
Table 1 t-Model
^Þ lðq
n¼1 n¼4 n ¼ 10 n ¼ 11 n ¼ 13 n ¼ 14 n ¼ 50 n ¼ 100 n ! 1, Normal Model
2086.32 2065.48 2063.18 2063.16 2063.17 2063.19 2063.90 2064.16 2064.50
Figures 4 and 5 present graphics of local influence for the perturbation of case weights for the normal and t11 models. As expected for the t11 -model the influence of the observations is less than the normal model. However, as n increases (close to the normal model), some observations (4, 23, 30) show significant influence on the maximum likelihood estimators, as was also verified in Galea et al. (2002a) for the normal model with DðfÞ ¼ fI p . This shows that the t-model with small degrees of freedom can be very useful for accommodating influent observations present in the data sets, which is not the case with normal models. Similar results were also obtained in Galea et al. (2002b) in the structural errors-in-variables models using a t-distribution. Next we compare local influence and case deletion diagnostics. Following Zhao and Lee (1998), Cook’s distance can be defined by ^ðiÞ q ^ÞT ðLÞ ðq ^ðiÞ q ^Þ ðq ; ð5:2Þ 3p ^ðiÞ denotes the parameter estimates without case i. Figures 6 gives the index plot i ¼ 1; . . . ; n, where q of Di , i ¼ 1; . . . ; n for the normal and t11 -model. Once again cases 4, 23 and 30 are prominent in the normal model. Whereas, for the t11 -model there are no global influential observations on the maximum likelihood estimators. Similar results are observed in Figure 7 with the Likelihood Displacement, ^Þ lðq ^ðiÞ ÞÞ. The Tables 2 and 3 present the maximum likelihood estimate for f , j ¼ 1; 2; 3; 4 Dvi ¼ 2ðlðq j Di ¼
15
15
(b)
(a)
di1 di2
10
di
di1, di2
10
5
5
0
0
10
20
30
40
50
60
70
0
0
10
Figure 1
20
30
40
50
60
70
Index
Index
^Þ and (b) di1 ðq ^Þ and di2 ðq ^Þ. Index plot Mahalanobis distance for Normal Model, (a) di ðq www.biometrical-journal.de
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2.5 2
1.5
1.5
1
Transformed Distance
Transformed Distance
(a) 2
0.5 0 −0.5 −1 −1.5
(b)
1 0.5 0 −0.5 −1 −1.5
−2
−2
−2.5 −2.5
−2.5 −2.5
−2
−1.5
−1
Figure 2
−0.5 0 0.5 1 Expected Normal Deviate
1.5
2
2.5
−1.5
−1
−0.5 0 0.5 1 Expected Normal Deviate
1.5
2
0.8
1.6
0.7
1.4
0.6
1.2 LD(ω(a))
0.9
1.8
0.5
1
0.4
0.8
0.3
0.6
0.2
0.4
0.1
0.2
10
20
2.5
t(11) Normal
(b)
(a)
0 0
2
Plots of Transformed Distances: (a) Normal Model and (b) t11 -model.
1
| lmax |
−2
30
40 Index
50
60
0 −1
70
−0.5
0 a
0.5
1
Figure 3 (a) Index plot of lmax for perturbation of degrees of freedom for t11-model, (b) Plots of the likelihood displacement LDðw(a)) versus a for directions lmax for normal and t11 -models.
1
1 (b)
(a)
0.9
0.8
0.8
0.7
0.7
0.6
0.6 | lmax |
| lmax |
0.9
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0 0
10
Figure 4
20
30
40 Index
50
60
70
0 0
10
20
30
40 Index
50
60
70
Index plot of lmax for perturbation of case weights: (a) Normal Model (b) t11 -model.
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2.5
2.5 (b)
(a)
2
1.5
1.5
C
Ci
i
2
1
1
0.5
0.5
0 0
10
30
40 Index
50
60
0 0
70
10
20
30
40 Index
50
60
70
Index plot of Ci for perturbation of case weights: (a) Normal Model and (b) t11 -model.
Figure 5
0.15
20
0.15
(a)
(b)
0.1
Di
D
i
0.1
0.05
0.05
0 0
10
20
30
40 Index
50
60
0 0
70
2
1.8
1.4
1.2
1.2 Dvi
1.6
1.4
Dvi
30
40 Index
50
60
70
2
(a)
1.6
1
0.8
0.6
0.6
0.4
0.4
0.2
0.2
10
20
30
Figure 7
40 Index
50
60
70
(b)
1
0.8
0 0
20
Index plot of Cook’s distance: (a) Normal Model (b) t11 -model.
Figure 6
1.8
10
0 0
10
20
30
40 Index
50
60
70
Likelihood Displacement: (a) Normal Model (b) t11 -model. www.biometrical-journal.de
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Table 2
^ for Normal Model. Relative changes and f jðIÞ
I
RC1
RC2
RC3
RC4
^ f 1ðIÞ
^ f 2ðIÞ
^ f 3ðIÞ
^ f 4ðIÞ
-4 23 30 4, 23 4, 30 23, 30 4, 23, 30
-15.47 00.51 01.00 15.78 14.35 01.52 14.73
-18.65 12.07 06.25 32.10 24.74 20.54 40.51
-13.18 03.82 00.58 18.73 13.74 04.73 19.87
-09.11 00.83 01.03 09.19 07.77 02.53 07.18
50248.08 42473.37 50506.14 50748.51 42319.27 43038.17 51013.88 42848.85
19150.75 15578.36 16839.93 17953.94 13002.31 14413.46 15218.03 11392.93
29235.73 33088.05 30351.09 29403.87 34712.63 33251.30 30618.27 35045.66
38843.19 42381.64 38520.20 38444.84 42414.05 41861.40 37860.65 41631.59
^ for t11-model. Relative changes and f jðIÞ
Table 3 I
RC1
RC2
RC3
RC4
^ f 1ðIÞ
^ f 2ðIÞ
^ f 3ðIÞ
^ f 4ðIÞ
-4 23 30 4, 23 4, 30 23, 30 4, 23, 30
-14.06 00.01 00.85 14.23 12.97 00.73 13.21
-18.57 11.32 06.64 30.11 24.89 18.38 36.70
-10.76 02.57 00.10 13.97 10.73 02.60 13.84
-08.27 01.08 01.55 07.61 06.41 02.92 05.43
39765.62 34173.03 39760.98 40103.12 34106.26 34608.52 40054.35 34511.39
14328.55 11667.04 12706.22 13377.50 10014.23 10762.88 11694.87 09070.12
25770.52 28543.76 26431.80 25796.92 29370.56 28535.04 26439.36 29336.49
33306.86 36062.75 32947.23 32790.85 35840.89 35443.40 32332.58 35114.94
and its relative changes, if it is not considered the three most influential observations, according the global/local influence method, using the normal and t distributions. The relative change is defined as: ðf ^ ^ j fjðIÞ Þ ^ RCj ¼ 100%, where f jðIÞ denotes the maximum likelihood estimate for fj after obser^ f j vations belonging to the index set I are removed. As was expected the largest variations for the maximum likelihood estimate for fj, j ¼ 1; 2; 3; 4 are observed under the normal model. This variation ^ ) under the normal model when observations 4, 23 and 30 are reaches a maximum of 40.51% (in f 2 removed. However, under the t11 -model the maximum variation is of 36.70% when the same observations are removed. Thus is, under the t11 -model, the maximum likelihood estimators are robust in the sense that discrepant observations (outliers) are downweighted.
6 Final Conclusions In this paper we discuss parameter estimation and model diagnostics for the comparative calibration model under elliptical distributions. The model considered includes several parametrizations previously considered in the literature under normality. Closed form expressions are obtained for the observed information matrix and for the D matrix under several perturbation schemes within the elliptical family of distributions. The special cases of the normal and t model are discussed in detail. The t-distribution provides an extension of the normal model with heavier tails than the normal distribution being thus useful for adjusting statistical models with larger than normal tails. The degrees of freedom of the t-distribution can be used for adjusting the distribution kurtosis and providing more robust procedures than the ones that follow by using the normal distribution with moderate additional computational effort. We consider three perturbation schemes. One, denominated case weighting (case-
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weight perturbation), is intended to evaluate whether the contribution of observations with different weights affects the maximum likelihood estimator of q. Perhaps, this is the method most commonly used to evaluate the influence of a small modification of the model. The perturbation scheme where we consider measurements perturbation, can be used to analyse the sensibility of the maximum likelihood estimators when individual or simultaneous measurements performed with the instruments are affected by small perturbations. This scheme is analogous to the response perturbation used in linear regression (Cook, 1986). Finally, we consider perturbation of the degrees of freedom parameter, n. Since n is assumed known, this type of perturbation allows studying the sensibility of the maximum likelihood estimators of q with increasing degrees of freedom. In the application we used global and local influence for the normal and t-model with eleven degrees of freedom. Other values were also tried but with similar results. The empirical study provides new evidence on the robustness aspects of the maximum likelihood estimators for the t-distribution with small degrees of freedom, as also shown by Lange et al. (1989) in regression and multivariate analysis. The study also shows the necessity of using (developing) diagnostic techniques in models with tails heavier than the normal.
Appendix: Computing the observed information matrix in the elliptical structural model In this appendix the observed information matrix is obtained for the elliptical structural model. From (4.1), it follows that @‘i ðqÞ 1 @‘njVj ¼ þ Wg ðdi Þ dig ; ðA:1Þ @g 2 @g @di ; g ¼ m; l; f and di ¼ XTi V 1 X i ; Xi ¼ Y i m; i ¼ 1; . . . ; n: Further, using results in @g Nel (1980) related to vector derivatives it follows that,
with dig ¼
@‘njVj @‘njVj ¼ 0; ¼ 2c1 D1 ðfÞ l ; @m @l @‘njVj ¼ D1 ðfÞ 1 c1 D2 ðfÞ DðlÞ l ; @f
ðA:2Þ
dim ¼ 2V 1 Xi ;
ðA:3Þ
dil ¼ 2c1 D1 ðfÞ Xi X Ti D1 ðfÞ l þ 2c2 ci1 D1 ðfÞ l ; 2
1
ðA:4Þ
2
dif ¼ D ðfÞ DðXi Þ Xi þ 2c ci2 D ðfÞ DðlÞX i c2 ci1 D2 ðfÞ DðlÞ l;
ðA:5Þ
i ¼ 1; 2; . . . ; n :
From (A.1) it follows that the observed, per element, information matrix is given by 3 2 2 @ ‘i @ 2 ‘i @ 2 ‘i 6 @m @mT @m @lT @m @fT 7 7 6 7 6 6 @ 2 ‘i @ 2 ‘i 7 Li ¼ Li ðq j Y i Þ ¼ 6 7; 6 @l @lT @l @fT 7 7 6 4 @ 2 ‘i 5 @f @fT
ðA:6Þ
i ¼ 1; . . . ; n, where @ 2 ‘i 1 @ 2 ‘njVj ¼ þ Wg0 ðdi Þ dig ditT þ Wg ðdi Þ digtT ; 2 @g @tT @g @tT
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with digtT ¼
@ 2 di ; i ¼ 1; 2; . . . ; n and g; t ¼ m; l; f, where @g @tT
@ 2 ‘njVj @ 2 ‘njVj @ 2 ‘njVj ¼ ¼ ¼ 0; @m @mT @m @fT @m @lT
ðA:8Þ
@ 2 ‘njVj ¼ 2c1 ðV 1 c1 MÞ ; @l @lT @ 2 ‘njVj ¼ 2c1 V 1 D1 ðfÞ DðlÞ ; @l @fT
ðA:9Þ ðA:10Þ
@ 2 ‘njVj ¼ D2 ðfÞ þ 2c1 D3 ðfÞ D2 ðlÞ @f @fT c2 D1 ðfÞ DðlÞ MDðlÞ D1 ðfÞ ;
ðA:11Þ
dimmT ¼ 2V 1 ; 1
ðA:12Þ 1
dimlT ¼ 2c ðD ðfÞ
lXTi D1 ðfÞ
þ ci1 ðV
1
1
c MÞÞ ;
ðA:13Þ
dimfT ¼ 2D2 ðfÞ DðXi Þ 2c1 D1 ðfÞ lXiT DðlÞ D2 ðfÞ 2c1 ci2 D2 ðfÞ DðlÞ þ 2c2 MXi lT DðlÞ D2 ðfÞ ;
ðA:14Þ
dillT ¼ 2c1 D1 ðfÞ X i X Ti D1 ðfÞ þ 4c2 D1 ðfÞ X i X Ti M þ 2c2 ci1 ðV 1 3c1 MÞ þ 4c2 MX i X Ti D1 ðfÞ;
ðA:15Þ
dilfT ¼ 2c1 D1 ðfÞ X i X Ti DðlÞ D2 ðfÞ 2c2 D1 ðfÞ X i X Ti MD1 ðfÞ DðlÞ þ 2c1 ci2 D2 ðfÞ DðX i Þ 2c2 ci1 D2 ðfÞ DðlÞ þ 4c3 ci1 MD1 ðfÞ DðlÞ 4c2 MXi XTi DðlÞ D2 ðfÞ;
ðA:16Þ
diffT ¼ 2D3 ðfÞ D2 ðX i Þ 4c1 ci2 D3 ðfÞ DðlÞ DðX i Þ 2c1 D2 ðfÞ DðlÞ Xi XTi DðlÞ D2 ðfÞ þ 2c2 D2 ðfÞ DðlÞ Xi XTi MD1 ðfÞ DðlÞ þ 2c2 ci1 D3 ðfÞ D2 ðlÞ 2c3 ci1 D1 ðfÞ DðlÞ MDðlÞ D1 ðfÞ þ 2c2 D1 ðfÞ DðlÞ MXi XTi DðlÞ D2 ðfÞ;
ðA:17Þ
i ¼ 1; 2; . . . ; n. Thus, the complete observed information matrix is Lob ðq j Y Þ ¼
n P
Li ðq j Y i Þ. Evaluat-
i¼1
^ it follows that Lob ðq ^ j YÞ ¼ L given in (4.2). ing the observed information matrix at q Acknowledgements The authors acknowledges the partial financial support from CNPq and Fapesp, Brasil and Projects Fondecyt 1000424 and 1030588, Chile. The authors also are grateful to the referees whose comments and suggestions were valuable to improve the exposition of the paper.
References Barnett, V. D. (1969). Simultaneous pairwise linear structural relationships. Biometrics 25, 129--142. Barnett, V. and Lewis, T. (1995). Outliers in statistical data. 3rd Edition. John Wiley, New York.
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Bechman, R., Nachtsheim, C., and Cook, R. (1987). Diagnostics for mixed-model analysis of variance. Technometrics 29, 413--426. Bolfarine, H. and Arellano-Valle, R. B. (1994). Robust modeling in measurement error models using the Student-t distributions. Brazilian Journal of Probability and Statistics 8, 67--84. Bolfarine, H. and Galea, M. (1995). Structural comparative calibration using the EM algorithm. Journal of Applied Statistics 22, 277--292. Bolfarine, H. and Galea-Rojas, M. (1996). One structural comparative calibration under a t-models. Computational Statistics 11, 63--85. Chatterjee, S. and Hadi, A. S. (1988). Sensivity Analysis in Linear Regression. John Wiley, New York. Cheng, C. and van Ness, J. (1999). Statistical Regression With Measurement Error. Arnold, London. Chipkevitch, E., Nishimura, R., Tu, D., and Galea-Rojas, M. (1996). Clinical measurement of testicular volume in adolescents: Comparison of the reliability of 5 methods. Journal of Urology 156, 2050--2053. Cochran, W. G. (1968). Errors of measurement in statistics. Technometrics 10, 637--666. Cook, R. D. (1986). Assessment of local influence. Journal of the Royal Statistical Society B 48, 133--169. Cook, R. D. and Weisberg, S. (1982). Residuals and Influence in Regression. Chapman and Hall, London. Dunn, G. (1992). Design and Analysis of Reliability: The statistical evaluation of measurement errors. Edward Arnold, New York. Escobar, E. and Meeker, W. (1992). Assessing influence in regression analysis with censored data. Biometrics 48, 507--528. Ferna´ndez, C. and Steel, M. (1999). Multivariate student t regression models: pitfalls and inference. Biometrika 86, 153--167. Fuller, W. A. (1987). Measurement error models. John Wiley, New York. Fung, W. and Kwan, C. (1997). A note on local influence based on normal curvature. Journal of the Royal Statistical Society B 59, 839--843. Galea, M., Paula, G. A., and Bolfarine, H. (1997). Local influence in elliptical linear regression models. The Statistician 46, 71--79. Galea, M., Bolfarine, H., and de Castro, M. (2002a). Local influence in comparative calibration models. Biometrical Journal 44, 59--81. Galea, M., Bolfarine, H., and Vilca, F. (2002b). Influence diagnostics for structural errors-in-variables model under the student-t distribution. Journal of Applied Statistics 29, 1191--1204. Grubbs, F. E. (1948). On estimating precision of measuring instruments and product variability. Journal of the American Statistical Association 43, 243--264. Grubbs, F. E. (1973). Errors of measurement, precision, accuracy and the statistical comparison of measuring instruments. Technometrics 15, 53--66. Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J., and Stahel, W. A. (1986). Robust Statistics. John Wiley, New York. Huber, P. J. (1981). Robust Statistics. John Wiley, New York. Jaech, J. L. (1964). A program to estimate measurement errors in nondestructive avaluation of fuel element quality. Technometrics 6, 293--300. Kelly, G. (1984). The influence function in the errors in variables problem. The Annals of Statistics 12, 87--100. Kelly, G. (1985). Use of the structural equations model in assessing the reliability of a new measurement technique. Applied Statistics 34, 258--263. Kwan, C. and Fung, W. (1998). Assessing local influence for specific restricted likelihood: Application to factor analysis. Psychometrika 63, 35--46. Lange, K. L., Little, R. J., and Taylor, J. (1989). Robust statistical modelling using the t-distribution. Journal of the American Statistical Association 84, 881--896. Lawrance, A. J. (1988). Regression transformation diagnostic using local influence. Journal of the American Statistical Association 83, 1067--1072. Lesaffre, E. and Verbeke, G. (1998). Local influence in linear mixed models. Biometrics 54, 570--583. Little, R. J. (1988). Robust estimation of the mean and covariance matrix from data with missing values. Applied Statistics 37, 23--38. Lu, Y., Ye, K., Mathur, A., Hui, S., Fuerst, T., and Genant, H. (1997). Comparative calibration without a gold standard. Statistics in Medicine 16, 1889--1905. Nel, D. G. (1980). On matrix differentiation in Statistics. South African Statistical Journal 14, 137--193. Paula, G. A. (1993). Assessing local influence in restricted regressions models. Computational Statistics and Data Analysis 16, 63--79.
www.biometrical-journal.de
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Pinheiro, J., Liu, Ch., and Wu, Y. (2001). Efficient algorithms for robust estimation in linear mixed.effects models using the multivariate t-distribution. Journal of Computational and Graphical Statistics 10, 249--276. Rubin, D. B. (1983). Iteratively reweighted least squares. Encyclopedia of the Statistical Sciences 4, 272--275. Shyr, J. and Gleser, L. (1986). Inference about comparative precision in linear structural relationships. Journal of Statistical Planning and Inference 14, 339--358. Sutradhar, B. C. (1993). Score tests for the covariance matrix of the elliptical t-distribution. Journal of Multivariate Analysis 46, 1--12. Sutradhar, B. C. and Ali, M. M. (1986). Estimation of the parameter of a regression model with a multivariate terror variable. Communications in Statistics -- Theory and Methods 15, 429--450. Tanaka, Y., Watadani, S., and Moon, S. (1991). Influence in covariance structure analysis with an application to confirmatory factor analysis. Commutations in Statistics-Theory and Methods 20, 3805--3821. Taylor, J. and Verbyla, A. (2004). Joint modelling of location and scale parameters of the t distribution. Statistical Modelling 4, 91--112. Theobald, C. M. and Mallison, J. R. (1978). Comparative callibration, linear structural relationship and congeneric measurements. Biometrics 34, 39--45. Thomas, W. and Cook, R. D. (1990). Assessing influence on predictions from generalized linear models. Technometrics 32, 59--65. Tsai, C. L. and Wu, X. (1992). Assessing local influence in linear regression models with first-order autoregressive or heteroscedastic error structure. Statistics and Probability Letters 14, 247--252. Verbeke, G. and Molenberghs, G. (2000). Linear mixed models for longitudinal data. Springer, New York. Williams E. J. (1969). Regression methods in calibration problems. Bulletin of the International Statistical Institute 43, 17--28. Zhao, Y. and Lee, A. (1998). Influence diagnostics for simultaneous equations models. Australian and New Zealand Journal Statistics 40, 345--357.
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