Mean Shift Outlier Model Research Articles

Simultaneous variable selection and outlier identification in linear regression using the mean-shift outlier model

We provide a method for simultaneous variable selection and outlier identification using the mean-shift outlier model. The procedure consists of two steps: the first step is to identify potential outliers, and the second step is to perform all possible subset regressions for the mean-shift outlier model containing the potential outliers identified in step 1. This procedure is helpful for model selection while simultaneously considering outlier identification, and can be used to identify multiple outliers. In addition, we can evaluate the impact on the regression model of simultaneous omission of variables and interesting observations. In an example, we provide detailed output from the R system, and compare the results with those using posterior model probabilities as proposed by Hoeting et al. [Comput. Stat. Data Anal. 22 (1996), pp. 252–270] for simultaneous variable selection and outlier identification.

Detection of outliers in multilevel models

This paper studies outlier detection for multilevel models. Approximate formulae for outlier detection in estimating both fixed and random parameters under the mean-shift outlier model are derived, and a test for multiple outliers is proposed. These results can be used to detect outlier units at any levels. Detection of outlier units related to random parts is also studied. Analysis of an example shows that the proposed method is effective in identifying outliers in multilevel models.

Diagnostics analysis for log-Birnbaum–Saunders regression models

In this paper, several diagnostics measures are proposed based on case-deletion model for log-Birnbaum–Saunders regression models (LBSRM), which might be a necessary supplement of the recent work presented by Galea et al. [2004. Influence diagnostics in log-Birnbaum–Saunders regression models. J. Appl. Statist. 31, 1049–1064] who studied the influence diagnostics for LBSRM mainly based on the local influence analysis. It is shown that the case-deletion model is equivalent to the mean-shift outlier model in LBSRM and an outlier test is presented based on mean-shift outlier model. Furthermore, we investigate a test of homogeneity for shape parameter in LBSRM, which is a problem mentioned by both Rieck and Nedelman [1991. A log-linear model for the Birnbaum–Saunders distribution. Technometrics 33, 51–60] and Galea et al. [2004. Influence diagnostics in log-Birnbaum–Saunders regression models. J. Appl. Statist. 31, 1049–1064]. We obtain the likelihood ratio and score statistics for such test. Finally, a numerical example is given to illustrate our methodology and the properties of likelihood ratio and score statistics are investigated through Monte Carlo simulations.

Influence diagnostics in nonlinear reproductive dispersion mixed models

The class of nonlinear reproductive dispersion mixed models (NRDMMs) is an extension of nonlinear reproductive dispersion models and generalized linear mixed models. This paper discusses the influence analysis of the model based on Laplace approximation. The equivalence of case-deletion models and mean-shift outlier models in NRDMMs is investigated, and some diagnostic measures are proposed via the case-deletion method. We also investigate the assessment of local influence of various perturbation schemes. The proposed method is illustrated with an example.

On influence assessment for LAD regression

Least absolute deviations (LAD) regression, i.e. L 1 regression, is more resistant to the outliers in the response variable than the least-squares regression, but is relatively sensitive to outlying observations in explanatory variables. However, some but few attentions have been contributed to the influence assessment for LAD regression, especially for LAD nonlinear regression. In this paper, we propose several diagnostic measures, which can be used for LAD regression models. The quasi-likelihood distance based on the L 1 objective function, Cook distance based on the elliptical norm and some other diagnostic measures are introduced for LAD regression, and two examples are given to illustrate the use of these measures. The diagnostic models for LAD regression are also investigated. It is proved that the estimators of the case deletion model (CDM) and the mean shift outlier model (MSOM) are equal in linear and nonlinear LAD regression models.

INFLUENCE ANALYSIS ON EXPONENTIAL NONLINEAR MODELS WITH RANDOM EFFECTS

This paper presents a unified diagnostic method for exponential nonlinear models with random effects based upon the joint likelihood given by Robinson in 1991. The authors show that the case deletion model is equivalent to mean shift outlier model. From this point of view, several diagnostic measures, such as Cook distance, score statistics are derived. The local influence measure of Cook is also presented. Numerical example illustrates that our method is available.

Influence Diagnostics and Outlier Tests for Semiparametric Mixed Models

SummarySemiparametric mixed models are useful in biometric and econometric applications, especially for longitudinal data. Maximum penalized likelihood estimators (MPLEs) have been shown to work well by Zhang and co-workers for both linear coefficients and nonparametric functions. This paper considers the role of influence diagnostics in the MPLE by extending the case deletion and subject deletion analysis of linear models to accommodate the inclusion of a nonparametric component. We focus on influence measures for the fixed effects and provide formulae that are analogous to those for simpler models and readily computable with the MPLE algorithm. We also establish an equivalence between the case or subject deletion model and a mean shift outlier model from which we derive tests for outliers. The influence diagnostics proposed are illustrated through a longitudinal hormone study on progesterone and a simulated example.

Influence analysis in nonlinear models with random effects

In this paper, a unified diagnostic method for the nonlinear models with random effects based upon the joint likelihood given by Robinson in 1991 is presented. It is shown that the case deletion model is equivalent to the mean shift outlier model. From this point of view, several diagnostic measures, such as Cook distance, score statistics are derived. The local influence measure of Cook is also presented. A numerical example illustrates that the method is available.

Influence Analysis for Linear Measurement Error Models

In this paper, we present a unified diagnostic method for linear measurement error models based upon the corrected likelihood of Nakamura (1990, Biometrika, 77, 127–137). Both global influence and local influence are discussed. The case-deletion model and mean-shift outlier model are considered, and they are shown to be approximately equivalent. Several diagnostic measures are derived and discussed. It is found that they can be written in terms of the residual and leverage measure. Some existing results are improved. Numerical example illustrates that our method is useful for diagnosing influential observations.

Collapsibility hypotheses and diagnostic bounds in regression analysis

Deleted-case diagnostic statistics in regression analysis are based on changes in estimates due to deleting one or more cases. Bounds on these statistics, suggested in the literature for identifying influential cases, are widely used. In a linear regression model for Y in terms of X and Z, the model is “collapsible” with respect to Z if the Y−X relation is unchanged by deleting Z from the model. Deleted-case diagnostic statistics can be viewed as test statistics for collapsibility hypotheses in the mean shift outlier model. It follows that, for any given case, all deleted-case statistics test the same hypothesis, hence all have the same p-value, while the bounds correspond to different levels of significance among the several statistics. Furthermore, the bound for any particular deleted-case statistic gives widely varying levels of significance over the cases in the data set.

Influence diagnostics in nonlinear reproductive dispersion models

We extend some diagnostic methods studied by Pregibon (1981, Ann. Statist. 9, 705–724), Cook and Weisberg (1982, Residuals and Influence in Regression, Chapman & Hall, New York), and Wei (1998, Exponential Family Nonlinear Models, Springer, Singapore) to nonlinear reproductive dispersion models (NRDM, Jorgensen, 1997, The Theory of Dispersion Models, Chapman & Hall, London). The equivalency of case-deletion models and mean-shift outlier models in NRDM is investigated. Some of the diagnostic measures are illustrated with an example.

The mean-shift outlier model in general weighted regression and its applications

Consider the general weighted linear regression model y= Xβ+ ε , where E( ε)=0, Cov( ε)= Vσ 2, σ 2 is an unknown positive scalar, and V is a symmetric positive-definite matrix not necessary diagonal. Two models, the mean-shift outlier model and the case-deletion model, can be employed to develop multiple case-deletion diagnostics for the linear model. The multiple case-deletion diagnostics are obtained via the mean-shift outlier model in this article and are shown to be equivalent to the deletion diagnostics via the case deletion model obtained by Preisser and Qaqish (1996, Biometrika, 83, 551–562). In addition, computing the multiple case-deletion diagnostics obtained via the mean-shift outlier model is faster than computing the one based on the more commonly used case-deletion model in some situations. Applications of the multiple deletion diagnostics developed from the mean-shift outlier model are also given for regression analysis with the likelihood function available and regression analysis based on generalized estimating equations. These applications include survival models and the generalized estimating equations of Liang and Zeger (1986, Biometrika, 73, 13–22). Several numerical experiments as well as a real example are given as illustrations.

Influence analysis on linear models with random effects

In this paper, a unified diagnostic method for linear models with random effects based upon the joint likelihood given by Robinson (in 1991) is presented. The case deletion model is equivalent to mean shift outlier model, as well as case weights model. From this point of view, several new diagnostic measures, such as Cook distance, WK diagnostics are derived. Some previous results are improved. Numerical examples illustrate the method is available.

Assessment of Model Adequacy for Markov Regression Time Series Models

In Markov regression models with time series data, we apply asymptomatic results to obtain the quasi-score, quasi-Wald, and quasi-likelihood ratio tests for assessing model adequacy. Based on limited simulation studies, we show that these three test statistics, particularly the quasi-score test, perform reasonably well in small samples. In addition, we apply these tests to the mean-shift outlier model to examine outliers. The usefulness of these tests is demonstrated via the analysis of three practical examples.

On statistical models for regression diagnostics

In regression diagnostics, the case deletion model (CDM) and the mean shift outlier model (MSOM) are commonly used in practice. In this paper we show that the estimates of CDM and MSOM are equal in a wide class of statistical models, which include LSE, MLE, Bayesian estimate andM-estimate in linear and nonlinear regression models; MLE in generalized linear models and exponential family nonlinear models; MLEs of transformation parameters of explanatory variables in a Box-Cox regression models and so on. Furthermore, we study some models, in which, the estimates are not exactly equal but are approximately equal for CDM and MSOM.

An index sampling algorithm for the bayesian analysis of a class of model selection problems

In this paper, we present an application of the Gibbs sampler yielding an algorithm which facilitates the Bayesian analysis of a class of data partitioning or model selection problems. The resulting index sampling algorithm is applied to a univariate mean-shift outlier model for the purpose of illustration.

Measures of error with outliers in regression

This paper deals with a formal identification of outliers in regression based on tests of hypotheses. The hypothesis is not the standard one but is based on performance criteria that relates to the coefficient estimation and predictive capabilities of the model. The cri-teria include the trace of the mean square error matrix on the coefficients and integrated mean square error of prediction. Both the mean shift outlier model and the variance in-flation model are discussed.