Sensitivity measures of influence on the loading matrix in exploratory factor analysis

For the detection of influential observations on the loading matrix of the factor analysis model, we propose to use the infinitesimal version of two matrix coefficients, including Escoufier (1973)'s also discussed the application in factor analysis of some sensitivity measures used for similar purposes in principal component analysis.     

The internal norm approach to influence diagnostics

Cook and Weisberg (1982) describe the external and internal norm approaches to assessing the influence of a subset of data on least squares regression estimates. External norms base influence measurement on the repeated sampling theory of the assumed model, while internal norm measures judge the influence of a size-k subset relative to all size-k subsets within the given data. Although intuitively appealing, intemal norms have been largely ignored in favor of external norms due to computational considerations. The purpose of this article is to present the internal norm approach as a viable alternative to external norm influence measurement. In addition to discussing conceptual and computational issues, empirical evidence is provided to show that the internal norm interpretation of influence is different from that of its external counterparts. Finally, comparisons are drawn between external calibration and internal scaling for evaluating influence measure values.     

Local influence in binary regression models,and its correspondence with global influence

We develop local influence diagnostics for a general binary regression model,and apply these methods to case-weight perturbations in four examples. In addition, we illustrate the correspondence between case-deletion diagnostics and local case-weight perturbation slopes and curvatures. We demonstrate that local influence diagnostics can provide a more computationally efficient means for obtaining analogous information to that yielded by case-deletion diagnostics, which can be thought of as global influence perturbations. We also assess the global consistency of patterns of local influence using these data examples.     

Sensitivity analysis in maximum likelihood factor analysis

The present paper deals with sensitivity analysis in maximum likelihood factor analysis. To investigate the influence of a small change of data we derive theoretical influence functions I(x; LL^T ) and I(x; Δ) for a common variance matrix T= LL^T and a unique variance matrix Δ respectively. Numerical examples are shown to illustrate our procedure.     

Influence measure for the L1 regression

Because outliers and leverage observations unduly affect the least squares regression, the identification of influential observations is considered an important and integrai part of the analysis. However, very few techniques have been developed for the residual analysis and diagnostics for the minimum sum of absolute errors, L₁ regression. Although the L₁ regression is more resistant to the outliers than the least squares regression, it appears that outliers (leverage) in the predictor variables may affect it. In this paper, our objective is to develop an influence measure for the L₁ regression based on the likelihood displacement function. We illustrate the proposed influence measure with examples.     

The influence in comparing covariance matrices

The influence measure for the likelihood ratio test for comparing two covariance matrices is derived using the influence curve approach under the normality assumption. The influence measure for testing the equality of covariance matrices against the arbitrariness of them is partitioned into three influence measures: one for testing the equality of covariance matrices against the proportionality of them, another for testing the proportionality against the equality of correlations between them and the other for testing the equality of correlations against the arbitrariness. This partition implies that an observation can be influential in performing some tests among the four tests but not in performing the remaining tests. Thus the partition is more informative than considering the influence measure for the test of equality alone. Each influence measure is useful for detecting outliers in performing the corresponding likelihood ratio test.     

Multiple deletion measures and conditional influence in regression model

The joint effect of the deletion of the ith and jih cases is given by Gray and Ling (1984), they discussed the influence measures for influential subsets in linear regression analysis. The present paper is concerned with multiple sets of deletion measures in the linear regression model. In particular we are interested in the effects of the jointly and conditional influence analysis for the detection of two influential subsets.     

Bayesian analysis of outlier problems using divergence measures

A Bayesian approach is presented for detecting influential observations using general divergence measures on the posterior distributions. A sampling-based approach using a Gibbs or Metropolis-within-Gibbs method is used to compute the posterior divergence measures. Four specific measures are proposed, which convey the effects of a single observation or covariate on the posterior. The technique is applied to a generalized linear model with binary response data, an overdispersed model and a nonlinear model. An asymptotic approximation using Laplace method to obtain the posterior divergence is also briefly discussed.     

Simultaneous transformations and influence in regression

Influence of simultaneous transformations on the response and some of explanatory variables to the same level in the linear regression is considered. An approximate, but useful diagnostic procedure is developed for practical use. An example is then given for illustrations.     

Stepwise local influence in generalized autoregressive conditional heteroskedasticity models

Detection of outliers or influential observations is an important work in statistical modeling, especially for the correlated time series data. In this paper we propose a new procedure to detect patch of influential observations in the generalized autoregressive conditional heteroskedasticity (GARCH) model. Firstly we compare the performance of innovative perturbation scheme, additive perturbation scheme and data perturbation scheme in local influence analysis. We find that the innovative perturbation scheme give better result than other two schemes although this perturbation scheme may suffer from masking effects. Then we use the stepwise local influence method under innovative perturbation scheme to detect patch of influential observations and uncover the masking effects. The simulated studies show that the new technique can successfully detect a patch of influential observations or outliers under innovative perturbation scheme. The analysis based on simulation studies and two real data sets show that the stepwise local influence method under innovative perturbation scheme is efficient for detecting multiple influential observations and dealing with masking effects in the GARCH model.     

Local influence in seemingly unrelated regression model with ridge estimate

Local influence is a well-known method for identifying the influential observations in a dataset and commonly needed in a statistical analysis. In this paper, we study the local influence on the parameters of interest in the seemingly unrelated regression model with ridge estimation, when there exists collinearity among the explanatory variables. We examine two types of perturbation schemes to identify influential observations: the perturbation of variance and the perturbation of individual explanatory variables. Finally, the efficacy of our proposed method is illustrated by analyzing [13 A. Munnell, Why has productivity declined? Productivity and public investment, New Engl. Econ. Rev. (1990), pp. 3–22. [Google Scholar]] productivity dataset.     

Local Influence Analysis in AB–BA Crossover Designs

The aim of this article is to develop methodology for detecting influential observations in crossover models with random individual effects. Various case‐weighted perturbations are performed. We obtain the influence of the perturbations on each parameter estimator and on their dispersion matrices. The obtained results exhibit the possibility to obtain closed‐form expressions of the influence using the residuals in mixed linear models. Some graphical tools are also presented.     

Singly and Doubly Censored Current Status Data: Estimation, Asymptotics and Regression

In biostatistical applications interest often focuses on the estimation of the distribution of time between two consecutive events. If the initial event time is observed and the subsequent event time is only known to be larger or smaller than an observed point in time, then the data is described by the well-understood singly censored current status model, also known as interval censored data, case I. Jewell et al. (1994) extended this current status model by allowing the initial time to be unobserved, with its distribution over an observed interval [A, B] known; the data is referred to as doubly censored current status data. This model has applications in AIDS partner studies. If the initial time is known to be uniformly distribute d, the model reduces to a submodel of the current status model with the same asymptotic information bounds as in the current status model, but the distribution of interest is essentially the derivative of the distribution of interest in the current status model. As a consequence the non-parametric maximum likelihood estimator is inconsistent. Moreover, this submodel contains only smooth heavy tailed distributions for which no moments exist. In this paper, we discuss the connection between the singly censored current status model and the doubly censored current status model (for the uniform initial time) in detail and explain the difficulties in estimation which arise in the doubly censored case. We propose a regularized MLE corresponding with the current status model. We prove rate results, efficiency of smooth functionals of the regularized MLE, and present a generally applicable efficient method for estimation of regression parameters, which does not rely on the existence of moments. We also discuss extending these ideas to a non-uniform distribution for the initial time.     

Influence and sensitivity measures in correspondence analysis

Since correspondence analysis appears to be sensitive to outliers, it is important to be able to evaluate the sensitivity of the data on the results. This article deals with measuring the influence of rows and columns on the results obtained with correspondence analysis. To establish the influence of individuals on the analysis, we use the notion of influence curve and we propose a general criterion based on the mean square error to measure the sensitivity of the correspondence analysis and its robustness. A numerical example is presented to illustrate the notions developed in this article.     

On influence diagnostic in univariate elliptical linear regression models

We discuss in this paper the assessment of local influence in univariate elliptical linear regression models. This class includes all symmetric continuous distributions, such as normal, Student-t, Pearson VII, exponential power and logistic, among others. We derive the appropriate matrices for assessing the local influence on the parameter estimates and on predictions by considering as influence measures the likelihood displacement and a distance based on the Pearson residual. Two examples with real data are given for illustration.     

Assessing case influence on confidence intervals in nonlinear regression

Diagnostic techniques are proposed for assessing the influence of individual cases on confidence intervals in nonlinear regression. The technique proposed uses the method of profile t-plots applied to the case-deletion model. The effect of the geometry of the statistical model on the influence measures is assessed, and an algorithm for computing case-deleted confidence intervals is described. This algorithm provides a direct method for constructing a simple diagnostic measure based on the ratio of the lengths of confidence intervals. The generalization of these methods to multiresponse models is discussed.     

Estimative influence measures for the multivariate general linear model

Within the context of the multiviriate general linear model, and using a Bayesian formulation and Kullback-Leibler divergences this paper provides a framework and the resultant methods for the problem of detecting and characterizing influential subsets of observations when the goal is to estimate parameters. It is further indicated how these influence measures inherently depend upon one's exact estimative intent. The relationship to previous work on observations influential in estimation is discussed. The estimative influence measures obtained here are also compared with predictive influence functions previously obtained. Several examples are presented illustrating the methodology.     

Quadratic perturbation expansions of certain functions of eigenvalues and eigenvectors and their application to sensitivity analysis in multivariate methods

Tanaka (1988) lias derived the influence functions, which are equivalent to the perturbation expansions up to linear terms, of two functions of eigenvalues and eigenvectors of a real symmetric matrix, and applied them to principal component analysis. The present paper deals with the perturbation expansions up to quadratic terms of the same functions and discusses their application to sensitivity analysis in multivariate methods, in particular, principal component analysis and principal factor analysis. Numerical examples are given to show how the approximation improves with the quadratic terms.     

Identity for the NPMLE in Censored Data Models

We derive an identity for nonparametric maximum likelihood estimators (NPMLE) and regularized MLEs in censored data models which expresses the standardized maximum likelihood estimator in terms of the standardized empirical process. This identity provides an effective starting point in proving both consistency and efficiency of NPMLE and regularized MLE. The identity and corresponding method for proving efficiency is illustrated for the NPMLE in the univariate right-censored data model, the regularized MLE in the current status data model and for an implicit NPMLE based on a mixture of right-censored and current status data. Furthermore, a general algorithm for estimation of the limiting variance of the NPMLE is provided. This revised version was published online in July 2006 with corrections to the Cover Date.