Conformal Normal Curvature Research Articles

Conformal normal curvature and detection of masked observations in multivariate null intercept measurement error models

Measurement errors occur very commonly in practice. After fitting the model, influence diagnostics is an important step in statistical data analysis. The most frequently used diagnostic method for measurement error models is the local influence. However, this methodology may fail to detect masked influential observations. To overcome this limitation, we propose the use of the conformal normal curvature with the forward search algorithm. The results are presented through easy to interpret plots considering different perturbation schemes. The proposed methodology is illustrated with three real data sets and one simulated data set, two of which have been previously analyzed in the literature. The third data set deals with the stability of the hygroscopic solid dosage in pharmaceutical processes to ensure the maintenance of product safety quality. In this application, the analytical mass balance is subject to measurement errors, which require attention in the modeling process and diagnostic analysis.

Local influence analysis of the 2PL IRT model for binary responses

Detecting influence is an important diagnostic stage in every statistical analysis. This paper examines the local influence of minor perturbations of the model and data for the two-parameter logistic item response model for binary data. By applying Cook's approach in the corresponding EM algorithm used to obtain the maximum likelihood estimate of the model parameters, the normal curvature and the conformal normal curvature - building blocks for obtaining the diagnostic measures – are computed. The methodology is illustrated with two real examples and an artificial one.

Sensitivity analysis of partially linear models with response missing at random

ABSTRACTThis article investigates case-deletion influence analysis via Cook’s distance and local influence analysis via conformal normal curvature for partially linear models with response missing at random. Local influence approach is developed to assess the sensitivity of parameter and nonparametric estimators to various perturbations such as case-weight, response variable, explanatory variable, and parameter perturbations on the basis of semiparametric estimating equations, which are constructed using the inverse probability weighted approach, rather than likelihood function. Residual and generalized leverage are also defined. Simulation studies and a dataset taken from the AIDS Clinical Trials are used to illustrate the proposed methods.

Local influence analysis of nonlinear structural equation models with nonignorable missing outcomes from reproductive dispersion models

Nonlinear structural equation models with nonignorable missing outcomes from reproductive dispersion models are proposed to identify the relationship between manifest variables and latent variables in modern educational, medical, social and psychological studies. The nonignorable missing mechanism is specified by a logistic regression model. An EM algorithm is developed to obtain the maximum likelihood estimates of the structural parameters and parameters in the logistic regression model. Assessment of local influence is investigated in nonlinear structural equation models with nonignorable missing outcomes from reproductive dispersion models on the basis of the conditional expectation of the complete-data log-likelihood function. Some local influence diagnostics are obtained via observations of missing data and latent variables that are generated by the Gibbs sampler and Metropolis–Hastings algorithm on the basis of the conformal normal curvature. A simulation study and a real example are used to illustrate the application of the proposed methodologies.

Local influence analysis of multivariate probit latent variable models

The multivariate probit model is very useful for analyzing correlated multivariate dichotomous data. Recently, this model has been generalized with a confirmatory factor analysis structure for accommodating more general covariance structure, and it is called the MPCFA model. The main purpose of this paper is to consider local influence analysis, which is a well-recognized important step of data analysis beyond the maximum likelihood estimation, of the MPCFA model. As the observed-data likelihood associated with the MPCFA model is intractable, the famous Cook's approach cannot be applied to achieve local influence measures. Hence, the local influence measures are developed via Zhu and Lee's [Local influence for incomplete data model, J. Roy. Statist. Soc. Ser. B 63 (2001) 111–126.] approach that is closely related to the EM algorithm. The diagnostic measures are derived from the conformal normal curvature of an appropriate function. The building blocks are computed via a sufficiently large random sample of the latent response strengths and latent variables that are generated by the Gibbs sampler. Some useful perturbation schemes are discussed. Results that are obtained from analyses of an artificial example and a real example are presented to illustrate the newly developed methodology.

Assessing local influence for nonlinear structural equation models with ignorable missing data

A method is proposed in this paper to assess the local influence of minor perturbations for a nonlinear structural equation model with missing data that are missing at random. The main idea is to apply Zhu and Lee's (J. Roy. Statist. Soc. Ser. B 63 (2001) 111) approach to the conditional expectation of the complete-data log-likelihood function in the corresponding EM algorithm for deriving the conformal normal curvature. Building blocks for achieving the diagnostic measures are computed via latent variables that are generated by the Gibbs sampler and Metropolis–Hastings algorithm. It is shown that the proposed methodology is feasible for a wide variety of perturbation schemes. To illustrate the methodology, results that are obtained from analyses of some artificial examples, a simulation study, and a real example are presented.

Local influence analysis of nonlinear structural equation models

By regarding the latent random vectors as hypothetical missing data and based on the conditional expectation of the complete-data log-likelihood function in the EM algorithm, we investigate assessment of local influence of various perturbation schemes in a nonlinear structural equation model. The basic building blocks of local influence analysis are computed via observations of the latent variables generated by the Metropolis-Hastings algorithm, while the diagnostic measures are obtained via the conformal normal curvature. Seven perturbation schemes, including some perturbation schemes on latent vectors, are investigated. The proposed procedure is illustrated by a simulation study and a real example.

Local Influence Analysis for Mixture of Structural Equation Models

Finite mixture of structural equation models is very useful to analyze data from heterogeneous populations. In this article, we present a feasible procedure to assess local influence of minor perturbations for identifying influential aspects on the maximum likelihood estimation of a finite mixture of structural equation models. A Monte Carlo EM algorithm which treats both latent variables and allocation variables as hypothetical missing data is implemented. The local influence measures are developed on the basis of a Q-displacement function at the E-step of the algorithm. The diagnostic measures are based on the conformal normal curvature that can be computed easily. Building blocks of the diagnostic measures are derived, and they are evaluated via observations simulated by the Gibbs sampler from the appropriate conditional distributions. A number of interesting and novel perturbations are considered. The methodology is illustrated with a real example.

On local influence analysis of full information item factor models

The full information item factor (FIIF) model is very useful for analyzing relations of dichotomous variables. In this article, we present a feasible procedure to assess local influence of minor perturbations for identifying influence aspects of the FIIF model. The development is based on a Q-displacement function which is closely related with the Monte Carlo EM algorithm in the ML estimation. In the E-step of this algorithm, the conditional expectations are approximated by sample means of observations simulated by the Gibbs sampler from the appropriate conditional distributions. It turns out that these observations can be utilized for computing the building blocks of the proposed diagnostic measures. The diagnoses are based on the conformal normal curvature that can be computed easily. A number of interesting perturbation schemes are considered. The methodology is illustrated with two real examples.

Theory & Methods: Detection of influential observations in nonlinear reproductive dispersion models

A robust case‐deletion diagnostic which could avoid masking is presented for nonlinear reproductive dispersion models (NRDM) (Jorgensen, 1997) that include generalized linear models and exponential family nonlinear models as special cases. Based on the second‐order approximation of log‐likelihood displacement and Poon & Poon’s (1999) conformal normal curvature, a measure of local influence ranging from 0 to 1 is constructed. An example illustrates application of the techniques.

LOCAL INFLUENCE IN NONLINEAR REPRODUCTIVE DISPERSION MODELS

Nonlinear reproductive dispersion models (NRDM, Jorgensen 1997) include a wider range of distributions and nonlinear models such as the possibility of correlated errors and nonlinear hypotheses dropping the exponential family assumption. Based on the generalized Cook distance and the conformal normal curvature of Poon & Poon (1999), local influence of minor perturbations on the data set is investigated for NRDM. Two examples are used to illustrate our results.

Conformal Normal Curvature and Assessment of Local Influence

Summary In 1986, R. D. Cook proposed differential geometry to assess local influence of minor perturbations of statistical models. We construct a conformally invariant curvature, the conformal normal curvature, for the same purpose. This curvature provides a measure of local influence ranging from 0 to 1, with objective bench-marks to judge largeness. We study various approaches to using the conformal normal curvature and the relationships between these approaches.