Influence measures on corrected score estimators in functional heteroscedastic measurement error models

Journal of Multivariate Analysis 114 (2013) 1–15

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Influence measures on corrected score estimators in functional heteroscedastic measurement error models Patricia Giménez a , Manuel Galea b,∗ a

Universidad Nacional de Mar del Plata, FCEyN, Funes 3350, CP 7600, Mar del Plata, Bs.As., Argentina

b

Pontificia Universidad Católica de Chile and Laboratorio de Análisis Estocástico, Chile

article

info

Article history: Received 4 August 2010 Available online 20 July 2012 AMS subject classifications: 62H12 62J05 62J20 Keywords: Corrected score estimators Local influence Appropriate perturbation Comparative calibration models

abstract This paper deals with the local influence assessment of the effects of minor perturbations of data on corrected score estimators in the functional heteroscedastic measurement error models with known variances. By extending to the context of measurement error models the differential-geometrical framework proposed by Zhu et al. [H.T. Zhu, J.G. Ibrahim, S. Lee, H. Zhang, Perturbation selection and influence measures in local influence analysis, The Annals of Statistics 35 (2007) 2565–2588], an n-dimensional Riemannian manifold is defined. The associated metric tensor is utilized for the selection of appropriate perturbation schemes. The Levi-Civita connection and first and second derivatives of the corrected score estimators are used to construct influence measures. Simple formulas are obtained under different perturbation schemes. A comparison with the slope and curvature based diagnostics defined from the surface of the corrected score estimators formed by perturbation is included. A real data application and a simulated example illustrate the performance of the proposed diagnostics. © 2012 Elsevier Inc. All rights reserved.

1. Introduction Influence diagnostics have become an important tool for statistical analysis since the seminal work of Cook [5], where a perturbation scheme is introduced into the postulated model through a perturbation vector, and a differential comparison of the parameter estimates before and after perturbation is considered to study local influence. The approach is based on the analysis of the graph of the likelihood displacement versus the perturbation vector. Objective functions other than the likelihood displacement have been also used for local influence analysis in [18,31], amongst others. The influence on the maximum likelihood estimate of any parameter in a regression model is presented by Wu and Luo [33]. One of the advantages of the local influence concept (see [19]) is that it assesses the effect of joint perturbation on the data cases more easily than the global influence analysis. Accordingly, usually in a local sense, the results are free from masking effects that present difficulties to individual case deletion. For more details on relationships between local influence and case deletion; see [19,30]. Recently, Zhu et al. [37] have developed a differential-geometrical framework of a perturbation model, called the perturbation manifold. This method extends Cook’s approach in several aspects. First, it is shown that the metric tensor of the perturbation manifold provides important information about selecting an appropriate perturbation of a model. Second, new influence measures are defined for smooth objective functions, that avoid the scale dependence of normal curvature for objective functions that have nonzero first derivative at the critical point [11]. This paper deals with the local influence assessment in functional heteroscedastic measurement error models (also known as comparative calibration models), which can be seen as a special case of the linear multivariate measurement

∗

Correspondence to: Departamento de Estadística, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Macul, Santiago, Chile. E-mail address: [email protected] (M. Galea).

0047-259X/$ – see front matter © 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.jmva.2012.07.002

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error models. Influence diagnostics for measurement error models have received attention in the literature. Most works derive influence functions or apply the local influence method of Cook [5], that is, the so-called first order approach, Wu and Luo [33]. Kelly [17] gave an influence function for the structural models. Fuller [10] defined the hat matrix using the estimated predictor variable values and Wellman and Gunst [32] proposed a one-step approximation to Cook’s distance. Zhao et al. [35] and Zhao and Lee [34] derived the influence functions for generalized linear and non-linear measurement error models. Lee and Zhao [21] derived some useful diagnostics based on the likelihood displacement functions for generalized linear measurement models. Zhong et al. [36] presented a unified diagnostic method for linear measurement error models based upon the corrected likelihood of Nakamura [23]. Galea-Rojas et al. [12] considered influence and diagnostic methods in homoscedastic comparative calibration models in functional and structural versions using Cook’s approach based on the likelihood displacement. Rasekh and Fieller [25] considered the construction and properties of influence functions in the context of functional measurement error models with replicated data. Functional heteroscedastic comparative calibration models with known variances are typically used in comparing two or more analytical methods at several concentration levels. Estimation and hypothesis testing for these models are considered in [26,27,13,8], among others. de Castro et al. [6] apply the corrected score approach. A local influence study on maximum likelihood estimators and test statistics using the first order approach based on the likelihood function is developed by de Castro et al. [7,9]. In this paper we consider the assessment of effects of minor perturbations of data on corrected score estimators [23,14] in the functional heteroscedastic comparative calibration model with known variances. We extend to the context of measurement error models the differential geometrical framework proposed by Zhu et al. [37], based on the observed log-likelihood function of the related statistical model, where covariates are exactly observed. Recently, the approach of Zhu et al. [37] for the assessment of local influence has been applied by Shi et al. [28] for generalized linear models with missing covariates, Chen et al. [3] for nonlinear structural equation models and Chen et al. [4] for generalized linear mixed models. Application of this methodology in measurement error models has not been considered in the literature, neither selection of appropriate perturbation schemes is addressed nor second-order influence measures based on the surface formed by perturbation of estimators of the parameters of interest are calculated. To obtain the density of the perturbed model to define the perturbation manifold for our model, we propose a methodology to handle with a corrected log-likelihood, which is not a true log-likelihood function and is independent of the incidental parameters. The perturbation manifold under different perturbation schemes is obtained. The associated metric tensor is utilized for checking appropriate choice of a perturbation vector. We show that misleading conclusions may be drawn about influential cases if our model is arbitrarily perturbed. First and second-order terms on a covariant version of the Taylor’s theorem, based on the Levi-Civita connection, are used to calculate influence measures for the corrected score estimator. We obtain analytical expressions for corrected score estimators of the parameters of interest and simple formulas for first and second order derivatives needed for the calculation of the different influence measures. On the other hand, following the second-order approach of Wu and Luo [33], we study the slope and curvature associated with the surface of the corrected score estimator (CSE) formed by perturbation. This is referred as the CSE surface. We show that both approaches differ for our model. The paper is organized as follows. Section 2 presents the functional heteroscedastic measurement error model and discusses estimation by using the corrected score approach. Section 3 reviews the approach to local influence based on the perturbation manifold and that based on the CSE surface. In Section 4 we examine the local influence under data perturbation schemes inherent to the comparative calibration models, which consider case weight perturbation, perturbation of the instrument of reference and perturbation of one of the alternative instruments. Geometrical quantities for calculating first and second-order influence measures are obtained. The technique is illustrated on a real data set and a simulated example in Section 5. Concluding remarks are made in Section 6. 2. Model and corrected score approach Assume that we have at our disposal p + 1 (p ≥ 1) measuring instruments to measure a common characteristic in a group of n subjects. Let yij be the observed (measured) value corresponding to the unknown zj associated with subject j, j = 1, . . . , n, and obtained by using instrument i, i = 1, . . . , p. Relating the variables yj = (y1j , . . . , ypj )T and zj , we consider the linear model yj = α + βzj + ej , xj = zj + uj ,

(1)

j = 1, . . . , n,

(2)

where ej = (e1j , . . . , epj ) is the p × 1-dimensional error vector and α = (α1 , . . . , αp ) and β = (β1 , . . . , βp ) are p × 1vectors associated with the additive and multiplicative bias of the p measuring devices, which are usually the quantities of main interest. Moreover, Eq. (2) considers that one of the instruments measures the unknown and unobserved quantity zj without bias. In the literature [1], this instrument is typically called the reference instrument. The others p instruments are the alternative instruments. It is also considered that the measurement errors are heteroscedastic, that is T

ej ∼ Np (0, D(λj ))

T

and uj ∼ N (0, σj2 ),

j = 1, . . . , n

T

(3)

P. Giménez, M. Galea / Journal of Multivariate Analysis 114 (2013) 1–15

3

where D(λj ) denotes a diagonal matrix with elements λj = (λ1j , . . . , λpj )T and ej and uj are mutually independent. The variances λj and σj2 are assumed known and greater than 0, j = 1, . . . , n. This is a common setup in areas such as Analytical Chemistry ([26] and [27]) when interest is focused in comparing two or more analytical methods at several concentration levels. If variances cannot be assumed as known, they can be estimated by replications [16]. In this paper, the functional model is considered, where the values zj , j = 1, . . . , n, are modeled as unknown constants. They are incidental parameters since their number increase with the sample size. So, under these assumptions the model formulated by Eqs. (1)–(3) is known as a heteroscedastic functional comparative calibration model and it can be seen as a particular case of a measurement error model. Consider θ = (αT , βT )T the structural parameter of interest, Z = {zj }j=1,...,n the incidental parameters, Y = {yj }j=1,...,n , and X = {xj }j=1,...,n . The density of the postulated model is given by p(Y, X; Z, θ) = p1 (Y; Z, θ)p2 (X; Z) =

n 

p1j (yj ; zj , θ)

j =1

n 

p2j (xj ; zj ),

j =1

where p1j (yj ; zj , θ) =



1 exp − (yj − α − βzj )T D(λj )−1 (yj − α − βzj ) (2π )p/2 |D(λj )|1/2 2 1

1

 (4)

is the density in the absence of measurement error or unobserved density and p2j (xj ; zj ) =



1

(2π σj2 )1/2

exp −

1 2σj2

 (xj − zj )2 ,

j = 1, . . . , n.

(5)

Let

ℓ(θ, Z; Y, X) = ℓ1 (θ, Z; Y) + ℓ2 (Z; X) = log p1 (Y; Z, θ) + log p2 (X; Z), the log-likelihood of the postulated model. We call ℓ1 (θ, Z; Y) the unobserved log-likelihood or log-likelihood in the absence of measurement error. Maximum likelihood inference under this model is considered in [7]. de Castro et al. [6] also apply corrected score approach [23] for parameter estimation. For the sake of completeness, the corrected score approach for estimation in this model is sketched in the following paragraphs. Nakamura [23], Stefanski [29] and Giménez and Bolfarine [14] consider the use of corrected score functions in measurement error models. Giménez and Bolfarine [15] compare the corrected score approach with several others producing consistent estimators in functional comparative calibration models under the assumption of a homoscedastic model with known variances ratios. The corrected score approach depends on the existence of a function U∗ (θ; Y, X), independent of the incidental parameters, called a corrected score function, satisfying E[U∗ (θ; Y, X)|Y, Z] = U(θ; Y, Z),

(6)

for all Y, Z and θ , where U(θ; Y, Z) = ∂ℓ1 (θ, Z; Y)/∂θ is the unobserved score function. It follows from (6) that E[U∗ (θ; Y, X)] = 0, so that inference based on U∗ yields, under appropriate regularity conditions [14], consistent and asymptotically normal estimators. It can be shown, under the model specified by (1)–(3), that U∗j (θ; yj , xj ) =



D(λj )−1 (yj − α − βxj ) . D(λj )−1 [(yj − α − βxj )xj + βσj2 ]



(7)

T

ˆ , βˆ )T , satisfying The estimator θˆ = (α T

n 

ˆ yj , xj ) = 0, U∗j (θ;

(8)

j=1

is known as the corrected score estimator. By solving (8), it follows that

βˆ i =

S˜xyi S˜xxi − σ˜ i2

and αˆ i = y˜ i − βˆ i x˜ i ,

i = 1, . . . , p,

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where x˜ i =

n 

γij xj ,

j =1

S˜xyi =

y˜ i =

n 

γij yij ,

σ˜ i2 =

j =1

n 

γij (xj − x˜ i )(yij − y˜ i ),

n 

γij σj2 ,

j=1

S˜xxi =

j =1

n 

γij (xj − x˜ i )2 ,

j =1

with 1

γij =

λij n  j =1

,

i = 1, . . . , p, j = 1, . . . , n.

(9)

1

λij

3. First and second-order approach to local influence on corrected score estimator We are interested on the assessment of effects of minor perturbations of data on the corrected score estimator of θ . Let ω = (ω1 , . . . ωn )T be a perturbation vector, ω ∈ Ω ⊂ Rn , which is introduced to perturb U∗ (θ; Y, X). Let U∗ (θ; Y, X, ω) denote the perturbed corrected score for a given ω ∈ Ω , and assume that there is a point ω0 ∈ Ω , representing no ˆ perturbation, such that U∗ (θ; Y, X, ω0 ) = U∗ (θ; Y, X) for all θ . The perturbed corrected score estimator θ(ω) solves U∗ (θ; Y, X, ω) = 0. ˆ Let η(ω) ˆ denote a particular component of the vector θ(ω) (that is, αˆ i (ω) or βˆ i (ω), i = 1, . . . , p). We study each ˆ component of θ(ω) separately because as pointed out by Wu and Luo [33], this approach is sometimes more informative that studying mixed effects, which, from different sources, may cancel out each other. Following the differential-geometrical framework proposed by Zhu et al. [37] we obtain the perturbation manifold for the model and the geometrical quantities associated for checking appropriate choice of a perturbation vector and calculating first and second-order influence measures. In addition to this approach we also consider the second-order approach proposed by Wu and Luo [33], which in this case is based on an assessment of the directions corresponding to relatively large local maximum curvatures of the corrected T score estimator (CSE) surface with euclidean coordinates (ω1 , . . . , ωn , η(ω)) ˆ . The first derivative of η(ω) ˆ with respect to ω is not zero and contains useful information on the local influence worthy of examination. But, as argued by Wu and Luo [33], the curvature based diagnostic can provide the information that the first-order diagnostic fails to provide. However, we have the limitation that normal curvature is not scale invariant when applied to objective functions at a point with nonzero first derivative [11]. 3.1. Influence measures based on the perturbation manifold To obtain the first and second-order influence measures proposed by Zhu et al. [37] we need to know how the perturbation ω, introduced to perturb U∗ (θ; Y, X), affects the postulated model. This implies to find the density of the perturbed model p(Y, X; Z, θ, ω) = p1 (Y; Z, θ, ω)p2 (X; Z, ω), including structural and incidental parameters from which we can obtain U∗ (θ; Y, X, ω), with p(Y, X; Z, θ, ω) dYdX = 1 and p(Y, X; Z, θ, ω0 ) = p(Y, X; Z, θ). To assess the local influence of a perturbation, we are primarily interested in the behavior of p(Y, X; Z, θ, ω) as a function of ω around ω0 . If Eω denotes the expectation with respect to p(Y, X; Z, θ, ω), then, according to (6), we argue that under the perturbed model we should have Eω [U∗ (θ; Y, X, ω)|Y, Z, ω] = U(θ; Y, Z, ω),

(10)

where U(θ; Y, Z, ω) = ∂ log p1 (Y; Z, θ, ω)/∂θ is the corresponding perturbed unobserved score. Under different perturbation schemes, we seek p(Y, X; Z, θ, ω) such that (10) is verified. The perturbed model p(Y, X; Z, θ, ω), characterized by a set of perturbations ω can be regarded as an n-dimensional manifold M , with ω as a coordinate system. Z and θ are assumed fixed at a given value. Considering η(ω) ˆ : Rn → R as the objective function and ω(t ) a geodesic curve on M , first and second-order terms from a Taylor expansion of η(ω( ˆ t )) are used to define influence measures. An expected Fisher information matrix with respect to the perturbation vector ω defines a metric in the manifold. The metric matrix is denoted by G(ω) = (gij (ω)), with gij (ω) = Eω



 ∂ ∂ ℓ(ω; Y, X, Z, θ) ℓ(ω; Y, X, Z, θ) , ∂ωi ∂ωj

where ℓ(ω; Y, X, Z, θ) = log p(Y, X; Z, θ, ω).

i, j = 1, . . . , n,

P. Giménez, M. Galea / Journal of Multivariate Analysis 114 (2013) 1–15

5

(M, G) defines a Riemannian manifold, called a statistical perturbation manifold. The metric tensor G(ω) = (gij (ω)) is the Fisher information matrix with respect to the perturbation vector ω. The element gjj (ω) indicates the amount of perturbation introduced by the j-th component of ω. The elements gij (ω), i ̸= j, represent the association between different components of ω. Based on these observations, Zhu et al. [37] defined an appropriate perturbation as that satisfying that G(ω0 ) = cIn , where c > 0. This condition avoids any redundant components of ω, determines orthogonality between the different components of ω and ensures that the amounts of perturbation introduced by the components of ω are uniform. Then, we can easily pinpoint the cause of a large effect, at least locally. ˜ , defined by In applications, although G(ω0 ) ̸= cIn , we can always choose a new perturbation vector ω ω˜ = ω0 + c −1/2 G(ω0 )1/2 (ω − ω0 ),

(11)

˜ evaluated at ω equals cIn . such that G(ω) dω(t ) Consider ω(t ) a geodesic in M , unique and defined in an interval containing 0 such that ω(0) = ω0 and dt |t =0 = h ∈ 0 Tω0 , where Tω0 is the tangent space at point ω of the manifold M . Murray and Rice [22] state a covariant version of the Taylor theorem, which has the advantage that each term in the expansion of η(ω( ˆ t )) is a tensor, thus invariant to reparametrization, unlike the standard Taylor’s series expansion. First and second derivatives of η(ω( ˆ t )) on M , at t = 0, can be used to construct influence measures [37]. The first-order (FI) influence measure in the direction h ∈ Tω0 is given by 0

FIη, ˆ h = FIη(ω ˆ 0 ),h =

hT ∇ηˆ ∇ηTˆ h hT Gh

,

(12)

∂ η(ω ˆ 0)

where ∇ηˆ = ∂ω and G = G(ω0 ). A second-order (SI) influence measure in the direction h ∈ Tω0 is defined as SIη, ˆ h = SIη(ω ˆ 0 ),h =

˜ ηˆ h hT H hT Gh

,

(13)

˜ ηˆ = H˜ η(ω ˜ η(ω) where H ˆ at ω = ω0 , with the (i, j)th element of H given by ˆ ˆ 0 ) , is the covariant Hessian of η(ω) [H˜ η(ω) ](i,j) = ˆ

 ∂ η(ω) ˆ ∂ 2 η(ω) ˆ − g s,r (ω)Γijs (ω) , ∂ωi ∂ωj ∂ωr s ,r

where g s,r (ω) is the (s, r )th element of G(ω)−1 and

Γijk (ω) =

 ∂ ∂ ∂ gjk (ω) + gik (ω) − gij (ω) 2 ∂ωi ∂ωj ∂ωk 1



is the Christoffel symbol for the Levi-Civita connection. ˜ in (11), which yields If ω is not an appropriate perturbation, we can use an appropriate ω FIη( 0 = ˜ h |ω=ω ˆ ω), ˜

hT G−1/2 ∇ηˆ ∇ηTˆ G−1/2 h hT h

(14)

and SIη( 0 = ˜ h |ω=ω ˆ ω), ˜

˜ ηˆ G−1/2 h hT G−1/2 H hT h

.

(15)

The associated directions h are used to assess first and second order local influence. To identify influential observations, in this paper we inspect the eigenvectors hmax corresponding to maximum absolute values of FI and SI, given by the ˜ ηˆ G−1/2 , respectively. To analyze the contribution of Ej , the basic largest absolute eigenvalues of G−1/2 ∇ηˆ ∇ηTˆ G−1/2 and G−1/2 H perturbation vector with its j-th entry equal to 1 and zeros elsewhere, we can use Mj = |hmax |j . Another choice for identifying influential observations is to consider the aggregate contribution of the basic perturbations vectors to all eigenvectors that are associated with all nonzero eigenvalues; see [24,38]. According to Zhu and Lee [38], if the contribution of all basic ¯ of all Mj . If a particular Mj is much larger than M, ¯ then the perturbation vectors is uniform then each is equal to the mean M ¯ j-th observation can be regarded as influential. M + 2 sd(M ), where sd(M ) denotes the standard deviation of M1 , . . . , Mq , can be used as a benchmark to determine the significance of contributions from an individual case. However, we may use ¯ see for instance, [20]. To quantify the order of the influential points, the definition of the k0 -order different functions of M, ¯ , k0 M ¯ ], the j-th case is called a k0 th-order influential case ([39] and [4]) can be used. If Mj is in the interval [(k0 − 1)M influential case in the whole data set.

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P. Giménez, M. Galea / Journal of Multivariate Analysis 114 (2013) 1–15

3.2. Influence measures based on the CSE surface Following [33], the corrected score estimator (CSE) of η under U∗ (θ; Y, X, ω), η(ω) ˆ , can be regarded as a surface with T euclidean coordinates (ω1 , . . . , ωn , η(ω)) ˆ , called the CSE surface. Results of the first-order approach are based on the value of the maximum slope of the CSE surface at ω = ω0 , which is in the direction of the vector ∇ηˆ . The second-order approach to local influence is based on the calculus of the normal curvature at ω = ω0 on the CSE surface in the direction of the vector h, which is given by Cη, ˆ h =

hT Hηˆ h

(1 + ∇ηˆ ∇ηˆ )1/2 hT (In + ∇ηˆ ∇ηTˆ )h T

,

(16)

∂ 2 η(ω ˆ 0)

ˆ and In is the n × n identity matrix. where Hηˆ = ∂ω∂ωT is the Hessian matrix of η(ω) A large local change occurs at ω = ω0 in the direction along which the normal curvature is maximized. The maximum absolute value of Cη, ˆ h and the corresponding directions are the eigenvalue–eigenvector solution of the equation |Hηˆ −λBηˆ | = 0, where Bηˆ = (1 + ∇ηTˆ ∇ηˆ )1/2 (In + ∇ηˆ ∇ηTˆ ). The unit direction vector of the maximum absolute value of Cη, ˆ h , hmax C, is usually used as a second-order diagnostic. Note that if ω is an appropriate perturbation and ∇ηˆ = 0, then cSIη, ˆ h = Cη, ˆ h. In this paper, we have ∇ηˆ ̸= 0, so, in general SIη, and C provide different results. Moreover, SIη, ˆ h η, ˆ h ˆ h is scale invariant while Cη, ˆ h is not scale invariant at a point with a nonzero first derivative [11]. 4. Influence measures in the comparative calibration model 4.1. Case weight perturbation The case weights are often the basis for the study of influence; deleting a case is identical to attaching a zero weight to that case. Let ω = (ω1 , . . . , ωn )T denote an n × 1 vector of case weights, then the perturbed corrected score function can be denoted as U∗ (θ; Y, X, ω) =

n 

ωj U∗j (θ; yj , xj ),

(17)

j =1

with U∗j (θ; yj , xj ) defined in (7) and where ω0 = (1, . . . , 1)T yields the non-perturbed corrected score. Solving U∗ (θ; Y, X, ω) = 0 produces the perturbed corrected score estimators:

βˆ i (ω) =

S˜xyi (ω)

αˆ i (ω) = y˜ i (ω) − βˆ i (ω)˜xi (ω),

and

S˜xxi (ω) − σ˜ i2 (ω)

i = 1, . . . , p,

where x˜ i (ω) =

n 

γij (ω)xj ,

y˜ i (ω) =

n 

j =1

S˜xyi (ω) =

γij (ω)yij ,

σ˜ i2 (ω) =

j =1

n 

n 

γij (ω)σj2 ,

j=1

γij (ω)(xj − x˜ i (ω))(yij − y˜ i (ω)),

S˜xxi (ω) =

j =1

n 

γij (ω)(xj − x˜ i (ω))2 ,

j =1

with

γij (ω) =

ωj λij n  j =1

ωj λij

,

i = 1, . . . , p, j = 1, . . . , n.

It can be easily seen that if the density of the perturbed model is given by p(Y, X; Z, θ, ω) = p1 (Y; Z, θ, ω)p2 (X; Z), where p1 (Y; Z, θ, ω) =

n

p1j (yj ; zj , θ, ωj ) =

j=1

p1j (yj ; zj , θ, ωj ), with

1

(2π )

1 p/2

|D(ωj λj )|1/2 −1

 ω j

exp −

2

(yj − α − βzj )T D(λj )−1 (yj − α − βzj )



P. Giménez, M. Galea / Journal of Multivariate Analysis 114 (2013) 1–15

and p2 (X; Z) = score given by

n

j =1

7

p2j (xj ; zj ) with p2j (xj ; zj ) given in (5), then (10) is verified with the corresponding perturbed unobserved

U(θ; Y, Z, ω) =

n 

ωj Uj (θ; yj , zj ).

j =1

So, under the perturbed model, yj ∼ Np (α + βzj , D(ωj−1 λj )) and xj ∼ N (zj , σj2 ). After some calculations, we find the metric tensor G(ω) = (gij (ω)) and the Levi-Civita connection, with gij (ω) =

p(p + 1) 2ωi2

δij and Γijk (ω) = −

p(p + 1) 2ωi3

δij δik ,

i, j, k = 1 . . . n.

p(p+1)

Thus G(ω0 ) = 2 In and the perturbation in (17) is an appropriate one. Direct calculations lead to the covariant Hessian matrix of η(ω) ˆ at ω = ω0

˜ ηˆ = Hηˆ + diag(∇ηˆ ), H where diag(∇ηˆ ) represents the diagonal matrix with the elements of the vector ∇ηˆ in the diagonal.

After some algebraic derivations we have that the vectors of first derivatives of βˆ i (ω) and αˆ i (ω), i = 1, . . . , p, at ω0 = (1, 1, . . . , 1)T , are given by

∇βˆ i =

1 ∂ βˆ i (ω0 ) = γ i ⊙ (dxi ⊙ dyi − βˆ i sxi ), ∂ω S˜xxi − σ˜ i2

∇αˆ i =

∂ αˆ i (ω0 ) = γ i ⊙ (dyi − βˆ i dxi ) − ∇βˆ i x˜ i , ∂ω

where dxi = (x1 − x˜ i , . . . , xn − x˜ i )T ,

dyi = (yi1 − y˜ i , . . . , yin − y˜ i )T ,

γ i = (γi1 , . . . , γin )T and sxi = dxi ⊙ dxi − σ 2 , i = 1, . . . , p, with σ 2 = (σ12 , . . . , σn2 )T and S˜xxi , S˜xyi , σ˜ i2 and γij as defined in (9) and ⊙ denoting the element-wise product. The matrices of second derivatives at ω0 are given by ∂ 2 βˆ i (ω0 ) 1 Hβˆ = γ i γ Ti ⊙ (Ai + ATi ), =− i ˜ ∂ω∂ωT (Sxxi − σ˜ i2 )2 Hαˆ i =

(18)

∂ 2 αˆ i (ω0 ) = −γ i γ Ti ⊙ (Bi + BTi ) − (Ci + CiT ) − Hβˆ i x˜ i , ∂ω∂ωT

with Ai = (S˜xxi − σ˜ i2 )(dyi − βˆ i dxi )dTxi + (dxi ⊙ dyi − βˆ i sxi )sTxi ,

Bi = (dyi − βˆ i dxi ) ⊗ 1Tn

and Ci = ∇βˆ (γ i ⊙ dxi )T , i = 1, . . . , p, where ⊗ denotes the Kronecker product between matrices. i First and second-order influence measures are obtained applying formulas (12), (13) and (16). 4.2. Perturbation of the reference instrument measurements In this section the measurements obtained with the reference instrument are modified considering an additive perturbation scheme, leading to xj (ω) = xj + ωj ,

j = 1, . . . , n.

(19)

Replacing xj (ω) in (7) we obtain the perturbed corrected score U (θ; Y, X, ω), where ω = (0, . . . , 0) yields the nonperturbed corrected score. The perturbed corrected score estimators are given by ∗

βˆ i (ω) =

S˜xyi + S˜wyi S˜xxi + S˜wwi + 2S˜wxi − σ˜ i2

0

T

and αˆ i (ω) = y˜ i − βˆ i (ω)(˜xi − w ˜ i ),

where S˜wyi =

n 

γij (ωj − ω˜ i )(yij − y˜ i ),

S˜wxi =

j =1

S˜wwi =

n  j =1

n 

γij (ωj − ω˜ i )(xj − x˜ i ),

j =1

γij (ωj − ω˜ i )2 and ω˜ i =

n  j =1

γij ωj ,

i = 1, . . . , p.

(20)

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Eq. (10) is verified considering the density of the perturbed model given by p(Y, X; Z, θ, ω) = p1 (Y; Z, θ, ω)p2 (X; Z), where p1 (Y; Z, θ, ω) = j=1 p1j (yj ; zj (ω), θ), with p1j given by (4), substituting zj by zj (ω) = zj + ωj and p2 (X; Z) = p2j (xj ; zj ), with p2j given by (5), and where the perturbed unobserved score is given by

n

U(θ; Y, Z, ω) =

n 

n

j=1

Uj (θ; yj , zj (ω)).

j=1

That is, perturbing the observed covariate xj as xj (ω) = xj + ωj on the corrected score is equivalent to perturbing the unobserved covariate zj as zj (ω) = zj + ωj on the unobserved score. After some calculations, we have that the metric tensor G(ω) = (gij (ω)) and the Levi-Civita connection are given by T

ˆ ij ˆ D(λi )−1 β)δ gij (ω) = (β

and Γijk (ω) = 0,

∀ ω ∈ Ω , i, j, k = 1 . . . n.

Thus T

T

ˆ D(λ1 )−1 β, ˆ . . . , βˆ D(λn )−1 β), ˆ G = G(ω0 ) = diag(β indicates that the amount of perturbation introduced by ωi depends inversely on the variances λi . ˜ ω 0 = In ω is not an appropriate perturbation but we can always choose a new perturbation ω˜ = G(ω0 )1/2 ω such that G(ω)| and T

ˆ 1/2 ωj , ˜ = xj + (βˆ D(λj )−1 β) xj (ω)

j = 1, . . . , n.

(21)

First derivatives of βˆ i (ω) and αˆ i (ω) at ω0 = (0, . . . , 0)T are given by

∇βˆ i =

∂ βˆ i (ω0 ) 1 = γ i ⊙ {dyi − 2βˆ i dxi }, ˜ ∂ω Sxxi − σ˜ i2

∇αˆ i =

∂ αˆ i (ω0 ) = −βˆ i γ i − ∇βˆ i x˜ i , ∂ω

i = 1, . . . , p.

Second derivatives are given by Hβˆ =

∂ 2 βˆ i (ω0 ) −2γ i γ Ti 2βˆ i ⊙ (Di + DTi ) + (γ i γ Ti − diag(γ i )), = 2 2 ˜ ˜ ∂ω∂ωT (Sxxi − σ˜ i ) Sxxi − σ˜ i2

Hαˆ i =

∂ 2 αˆ i (ω0 ) = −γ i ∇βTˆ − ∇βˆ i γ Ti − Hβˆ i x˜ i , i ∂ω∂ωT

i

where Di = (dyi − 2βˆ i dxi )dTxi , i = 1, . . . , p. Formulas (14)–(15) are applied for the calculation of influence measures based of the perturbation manifold. ˜ ηˆ = Hηˆ , because G is independent of ω. In this case H 4.3. Perturbation of measurements of one of the alternative instruments In this section we introduce an additive perturbation in the measurement obtained from one alternative instrument. Without loss of generality, the measurements from instrument i = 1 are chosen to be perturbed leading to y1j (ω) = y1j + ωj , yij (ω) = yij ,

i ̸= 1, j = 1, . . . , n.

(22)

Replacing yij (ω) in (7) we obtain the perturbed corrected score and the perturbed corrected score estimators of parameters α1 and β1 ,

βˆ 1 (ω) =

S˜xy1 + S˜ωx1 S˜xx1 − σ˜ 12

and αˆ 1 (ω) = y˜ 1 + ω ˜ 1 − βˆ 1 (ω)˜x1 ,

where S˜xy1 , S˜ωx1 , σ˜ 12 and ω ˜ 1 are defined in (20). In this case, Eq. (10) is verified when the density of the perturbed model is given by p(Y, X; Z, θ, ω) = p1 (Y; Z, θ, ω)p2 (X; Z),

P. Giménez, M. Galea / Journal of Multivariate Analysis 114 (2013) 1–15

9

Fig. 1. Individual points and their standard deviations together with the corrected score regression line.

where p1 (Y; Z, θ, ω) = j=1 p1j (yj (ω); zj , θ), with p1j given by (4) with yj (ω) = yj + ωj δ1 instead of yj , p2 (X; Z) = p2j (xj ; zj ), with p2j (xj ; zj ) given by (5) and the corresponding perturbed unobserved score given by

n

U(θ; Y, Z, ω) =

n 

n

j =1

Uj (θ; yj (ω), zj ).

j =1

The metric tensor G(ω) = (gij (ω)) and the Levi-Civita connection are given by 1 gij (ω) = λ− i1 δij

and

Γijk (ω) = 0,

∀ ω ∈ Ω , i, j, k = 1 . . . n.

Thus, the amount of perturbation introduced by ωi increases when λi1 decreases and ω is not an appropriate perturbation. 1 −1 ˜ = G(ω0 )1/2 ω with G(ω0 ) = diag(λ− ˜ is an appropriate perturbation with However, choosing ω 11 , . . . , λn1 ), we have that ω ˜ ω0 = In and G(ω)| −1/2

˜ = y1j + λj1 y1j (ω) ˜ = yij , yij (ω)

ωj ,

i ̸= 1, j = 1, . . . , n.

(23)

In this case, only first-order influence measures can be calculated, because second derivatives of η(ω) ˆ are zero. The direction of maximum slope on the CSE surface with perturbation (22) is given by ∇ηˆ and the direction corresponding on −1/2 maximum FIη, ∇ηˆ . ˆ h with perturbation (23) is given by G First derivatives are given by

∇βˆ 1 =

γ ⊙ dx1 ∂ βˆ 1 (ω0 ) = 1 , ∂ω S˜xx1 − σ˜ 12

∇αˆ 1 =

∂ αˆ 1 (ω0 ) = γ 1 − ∇βˆ 1 x˜ 1 . ∂ω

5. Illustrative examples 5.1. Real data set To illustrate the computation of influence measures, we reanalyze the data set considered in [26]. The data consist of thirty pairs of determinations of the arsenate ion in natural river water (µg/g) by two methods, a continuous selective reduction and atomic absorption spectrometry (the reference instrument) and a non-selective reduction, cold trapping and atomic emission spectrometry (the alternative instrument). Each determination is the mean of three replicates, from which the variances of the errors are also obtained. Corrected score estimators are αˆ = 0.0022 and βˆ = 1.056. Data, standard deviations as well as the corrected score regression line are shown in Fig. 1. For illustration we include only the computation of first and second-order influence measures on the corrected score estimator of the slope, βˆ . Influence measures on the estimator of the intercept perform similarly. We obtained that case weight perturbation (17) is an appropriate one, so results of the first-order approach based on FIβ, ˆ h are the same as those that are based on the CSE surface. The maximum absolute value of FIβ, ˆ h equals 0.021012, while the

10

P. Giménez, M. Galea / Journal of Multivariate Analysis 114 (2013) 1–15

a

b

c

¯ Fig. 2. Index plot of (a) |hmax FI|, (b) |hmax SI| and (c) |hmax C|, for the case weight perturbation (17), where the dot lines are the lines of kM.

corresponding unit direction vector hmax FI can be used for identifying influential observations. High-order influential cases are the cases with strong influence compared with the average of the values |hmax FI|j of all cases. Fig. 2(a) displays index plot of |hmax FI|. Samples 19 and 22 are showed as potentially influent, and they are 5th- and 9th-order influential cases, respectively. Second-order influence measures SIβ, ˆ h and Cβ, ˆ h provides different results, with maximum absolute values given by 0.15620 and 0.064555, respectively. Fig. 2(b) and (c) gives the index plot of the corresponding unit directions |hmax SI| and |hmax C|, respectively. SI provides only case 22 as possibly influential, and it is 15th-order influential, while by using normal curvature cases 19 and 22 are identified as 4th and 5th-order influential cases, respectively. However, using first and second-order measures together the outstanding samples are the same from both approaches. With regard to additive perturbations of measurement from the reference instrument, perturbation defined in (19) is not appropriate. We noted that the amount of perturbation introduced by ωj increases when the variance λj decreases, j = 1, . . . , n. By using the approach based on the CSE surface, maximum absolute values of the slope and Cβ, ˆ h with perturbation (19) are given by 0.34882 and 0.95263, respectively. Fig. 3(b) and (d) gives the index plot of the corresponding unit directions. Using the slope, cases 19 and 22 are 5th- and 11th-order influential cases, respectively. Curvature identifies ˜ is defined in cases 23 as 4th-order influential and cases 22 and 25 as 9th-order influential. An appropriate perturbation ω ˜ is the same. Maximum absolute values of FIβ, (21), so that the amount of perturbation introduced for each component of ω ˆ h and SIβ, ˆ h are given by 0.0056187 and 0.011024, respectively. Fig. 3(a) and (c) shows the index plot of the corresponding unit directions and reveal only case 19 as the most influential, but not so preponderant, being 3rd-order influential by using both first and second order influence measures. Cases 22, 23 and 25 do not stand out as influential using the perturbation scheme (21). Analyzing the data set we notice that observations 22, 23 and 25 present the smaller values on variances for both measurement methods. This is the reason why perturbation (19) gives more weight to that observations and they appear as possibly influential by using the approach based on CSE surface. We also calculate the slope and normal curvature based on the CSE surface under the appropriate perturbation (21) and obtained the same conclusions than using the approach based on the perturbation manifold. Perturbing the alternative instrument we have only first order influence measures because second derivatives are zero. Maximum absolute values of FIβ, ˆ h and the slope of the CSE surface are given by 0.0038019 and 0.33738, respectively. The usual additive perturbation in (22) is not an appropriate one. It is noticed that the amount of perturbation introduced for the component of ωj is bigger when the variance of the alternative instrument λj1 is smaller, j = 1, . . . , n. An appropriate ˜ is defined in (23). By using this scheme the index plot of the unit direction of maximum FIβ, perturbation ω ˆ h reveals that case 19 is the most influential (Fig. 4(a)) but not so preponderant, while the index plot of the direction of maximum slope of the CSE surface based on perturbation (22) reveals cases 19 and 22 as the most influential and they appear as 5th- and 15th-order influential cases, respectively (Fig. 4(b)). We notice that observation 22 in the data set presents the smallest value of variance of the alternative instrument and therefore it receives a larger perturbation when scheme (22) is used.

P. Giménez, M. Galea / Journal of Multivariate Analysis 114 (2013) 1–15

a

b

c

d

11

Fig. 3. Index plot of (a) |hmax FI| and (c) |hmax SI| for the appropriate additive perturbation of the reference instrument (21); (b) |hmax slope| and (d) |hmax C| ¯ for the additive perturbation of the reference instrument (19), where the dot lines are the lines of kM.

a

b

Fig. 4. Index plot of (a) |hmax FI| for the appropriate additive perturbation of the alternative instrument (23) and (b) |hmax slope| for the additive perturbation ¯ of the alternative instrument (22), where the dot lines are the lines of kM.

First and second order measures based on both perturbation manifold and CSE surface, reveal cases 19 and 22 as the most influential under the different perturbation schemes. So, we discard singly and jointly these points and reanalyze the data set to study the changes on parameter estimates and conclusions of Wald test. We consider Wald type statistic based on corrected score estimators for testing the hypothesis of interest in comparative calibration experiments that both measurement methods perform similarly, that is H0 : α = 0

and β = 1.

(24)

This can be achieved using the Wald statistic considered in [6], W = (θˆ − θ 0 )T Vˆ −1 (θˆ − θ 0 ),

ˆ T , θˆ 0 = (0, 1)T and Vˆ is the sandwich estimator of the asymptotic covariance matrix of the corrected score where θˆ = (α, ˆ β) ˆ estimator θ . Under (24), W is asymptotically distributed according to a chi-square distribution with 2 degrees of freedom. The results of the analysis are summarized in Table 1. We conclude that the elimination of case 19 does not reveal significant changes on parameter estimates, and hypothesis H0 is accepted. However, the null hypothesis is rejected when case 22 is eliminated. This result and the changes in the parameter estimates reveal that case 22 is the most influential. Also it is noticed that the standard error of βˆ decreases. Joint elimination of cases 19 and 22 also shows significant changes on parameter estimates and a reduction of the standard error

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P. Giménez, M. Galea / Journal of Multivariate Analysis 114 (2013) 1–15 Table 1 Case deletion for α, ˆ βˆ and Wald test. Data set

αˆ (se(α) ˆ )

ˆ ) βˆ (se(β)

W (p-value)

Full data (D)

0.0022 (0.0056)

1.0559 (0.1450)

0.6719 (0.7147)

D − {19}

0.0039 (0.0066)

1.1178 (0.1639)

3.8613 (0.1451)

D − {22}

0.2396 (0.1392)

0.8285 (0.0565)

9.4512 (0.0089)

D − {19, 22}

0.3723 (0.0687)

0.8401 (0.0427)

35.411 (0.0000)

Fig. 5. Individual points and their standard deviations together with the corrected score regression line for a simulated data set with two outliers added.

of βˆ . W increases from 0.6719 to 35.411 and H0 would be rejected. This brings to light the importance of few observations on a decision like the acceptance or rejection of a hypothesis. A quick look at the data set does not allow us to suspect of possibly influential cases. The calculation of influence measures in the heteroscedastic comparative calibration model shows that the reciprocal of variances, as a measure of precision of the measurement methods, play an important role on detecting influential observations. 5.2. Simulated data To further investigate the empirical performance of the proposal measures, we generate a data set from model (1)–(2), with one alternative instrument (p = 1). We fix n = 50, α = 0 and β = 1. The true covariate z was generated as a normal with mean 2 and variance 3. Measurement error standard deviations σj2 and λj are picked from uniform distributions (0.5, 2) and (0.5, 4) respectively. We examined a scenario with two outliers, one on the reference instrument (x) and the other on the alternative instrument (y), changing the values of xmax and ymax , respectively. We change x10 to x10 + 3 to add an outlier in x and y18 to y18 + 3 to add an outlier in y. The goal of this simulation is to show the importance of choosing appropriate perturbation schemes and comparing the approaches based on the perturbation manifold and CSE surface. Since the quantity of perturbation introduced by schemes (19) and (22) depends on the reciprocal of the values of variances λj , we also modify the variances of cases 10 and 18 to show the effect of the measurement error variances on the influence measures. We substitute λ10 and λ18 by the maximum and minimum values of the λj on the whole data set, respectively. Data, standard deviations, as well as the corrected score line for this contaminated data set are shown in Fig. 5. In Fig. 6, we present index plots of first influence measures for the corrected score estimator βˆ . Fig. 6(a)–(c) represent the unit direction |hmax FI| based on the perturbation manifold, considering appropriate perturbations of case weights (17), reference instrument (21) and alternative instrument (23), respectively. As a guideline, the dashed lines represent the value of mean +2 sd.

P. Giménez, M. Galea / Journal of Multivariate Analysis 114 (2013) 1–15

a

b

c

d

e

f

13

Fig. 6. Index plots of first order influence measures for βˆ under case weight perturbation schemes (17) ((a) and (d)), perturbation of the reference instrument (21) and (19) ((b) and (e), respectively) and perturbation of one of the alternative instrument (23) and (22) ((c) and (f) respectively).

These schemes detect both cases 10 and 18 as potentially influential. Fig. 6(d)–(f) represents the unit direction |hmax slope| based on the CSE surface under perturbations of case weights (17), reference instrument (19) and alternative instrument (22), respectively. This approach does not consider the use of the metric tensor to define appropriate perturbation and it was shown that the quantity of perturbation introduced depends on the reciprocal of the error variances of the alternative instruments for perturbation of the reference and alternative instruments. Case weight perturbation is an appropriate scheme and parts (a) and (d) of Fig. 6 are the same. Observation 18 is highlighted for case weights and perturbation of the alternative instruments ((d) and (f)) but neither observations 10 and 18 appear as influential for perturbation of the reference instrument. It is noticed that observation 18 has the minimum value of variance on the whole data set. So, it can be detected by perturbation of the alternative instrument but the same does not occur with observations 10, which has the maximum value of variance on the whole data set. Second order influence measures are used together for analyzing local influence. Fig. 7 presents index plots. Fig. 7(a) and (b) represents the unit directions |hmax SI| under perturbation of case weights (17) and the reference instrument (21) highlighting observations 18 and 10, respectively. In index plots of |hmax C | based on the normal curvature of the CSE surface ((c) and (d)) only observation 18 appears as influential. The case weight perturbation scheme is an appropriate one, although the influence measures SI and C in (a) and (d) are not identical. Analogously to the first order approach, observation 10 is not detected under perturbation of the reference instrument (19) (d), due to the high value of their variance. This example shows that misleading conclusions may be drawn about influential cases when the model is arbitrarily perturbed. Outliers with high values of variances could be not detected. A computer program in R Code for calculating first and second order influence measures on corrected score estimators is available as Supplementary material of the paper. 6. Conclusions By extending a differential-geometrical framework proposed by Zhu et al. [37], we have defined a Riemannian manifold, called the perturbation manifold, for the assessment of local influence on corrected score estimators in the functional heteroscedastic comparative calibration model with known variances.

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a

b

c

d

Fig. 7. Index plots of second order influence measures for βˆ under case weight perturbation schemes (17) ((a) and (c)) and perturbation of the reference instrument (21) and (19) ((b) and (d), respectively).

A perturbation vector is introduced on the corrected score function (which is independent of the incidental parameters) and a methodology to find the density of the corresponding perturbed model (including structural and incidental parameters) is proposed. This density, as a function of the perturbation vector, defines the manifold. The associated metric tensor is obtained as an expected Fisher information matrix with respect to the perturbation vector and it plays a critical role for selecting an appropriate perturbation scheme. The Levi-Civita connection is calculated and used to define first and second-order influence measures. Simple formulas for these measures are presented. First derivatives of corrected score estimators at the critical point are nonzero and contain useful information on the local influence worthy examination. A first-order diagnostic is not always powerful, then, first and second-order influence measures are used together. For comparison, we also calculated the slope and curvature associated with the surface of the corrected score estimator (CSE surface) formed by perturbation, however the normal curvature is not scale invariant. The second-order influence measure based on the perturbation manifold addresses this drawback. We examine local influence under three data perturbation schemes inherent to the comparative calibration model. It is shown that case weight perturbation is an appropriate one. Then, first-order influence measures by both approaches are identical but second-order influence measures differ. Nevertheless, the numerical application shows that using together first and second order measures reveal the same possible influential cases by both approaches. Usual additive perturbation of the instrument of reference and of the alternative instrument are not appropriate and we notice that the amount of perturbation introduced by each component of the perturbation vector depends on the values of the known variances. However, we can choose appropriate perturbation by a transformation such that the amount of perturbation introduced by each component was the same. The normal curvature of the CSE surface does not take into account the values of variances. So, in this case, the illustrative examples show that different conclusions are obtained by using the perturbation manifold or the CSE surface approach. Outliers with high values of variances could be not detected if the model is arbitrarily perturbed. The model studied can be seen as a special case of the functional linear multivariate measurement error model. The methodology presented can be applied to the assessment of local influence on corrected score estimators under different functional measurement error models in the same way. On the other hand, corrected score estimators are also consistent estimators for a structural version of the model. The theory and procedures developed in the present paper could be extended for a structural model with slight modifications. As the corrected score methodology does not take into account the distribution of the true covariate, perturbations introduced on the corrected score function do not affect the distribution assumed for the true covariate in the structural model. We see that for all perturbation schemes considered in this paper only the conditional distribution of the response given the true

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covariate depends on the perturbation introduced. Thus, under the hypothesis of a non-differential measurement error [2], the density of the perturbed model in the structural model could be calculated by integrating the density of the perturbed model obtained in this paper for the functional version, with respect to the distribution of probability assumed for the true covariate. Then, for obtaining first and second order influence measures for the structural model, it is necessary to recalculate only the metric tensor and the Levy-Civita connection. Acknowledgments The authors acknowledges the partial financial support from Projects Fondecyt 1070919 and 1110318, and Laboratorio de Análisis Estocástico Chile, PBCT-ACT13, and MCT/CNPq /No 011/2008, Prosul, Brazil. The authors are grateful to the Associate Editor and reviewers for their helpful comments and suggestions that greatly improved the article. Appendix. 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