Influence diagnostics for the Grubbs's model

In this paper we consider applications of local influence (Cook, 1986) to evaluate small perturbations in the model or in data sets of several measuring devices, assuming Grubbs's model. Different perturbation schemes are investigated and an application is considered to two real data sets.     

The Conway–Maxwell–Poisson-generalized gamma regression model with long-term survivors

In this paper, we proposed a flexible cure rate survival model by assuming the number of competing causes of the event of interest following the Conway–Maxwell distribution and the time for the event to follow the generalized gamma distribution. This distribution can be used to model survival data when the hazard rate function is increasing, decreasing, bathtub and unimodal-shaped including some distributions commonly used in lifetime analysis as particular cases. Some appropriate matrices are derived in order to evaluate local influence on the estimates of the parameters by considering different perturbations, and some global influence measurements are also investigated. Finally, data set from the medical area is analysed.     

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.     

Bayesian local influence for survival models

The aim of this paper is to develop a Bayesian local influence method (Zhu et al. 2009, submitted) for assessing minor perturbations to the prior, the sampling distribution, and individual observations in survival analysis. We introduce a perturbation model to characterize simultaneous (or individual) perturbations to the data, the prior distribution, and the sampling distribution. We construct a Bayesian perturbation manifold to the perturbation model and calculate its associated geometric quantities including the metric tensor to characterize the intrinsic structure of the perturbation model (or perturbation scheme). We develop local influence measures based on several objective functions to quantify the degree of various perturbations to statistical models. We carry out several simulation studies and analyze two real data sets to illustrate our Bayesian local influence method in detecting influential observations, and for characterizing the sensitivity to the prior distribution and hazard function.     

On local influence for elliptical linear models

The local influence method plays an important role in regression diagnostics and sensitivity analysis. To implement it, we need the Delta matrix for the underlying scheme of perturbations, in addition to the observed information matrix under the postulated model. Galea, Paula and Bolfarine (1997) has recently given the observed information matrix and the Delta matrix for a scheme of scale perturbations and has assessed of local influence for elliptical linear regression models. In the present paper, we consider the same elliptical linear regression models. We study the schemes of scale, predictor and response perturbations, and obtain their corresponding Delta matrices, respectively. To illustrate the methodology for assessment of local influence for these schemes and the implementation of the obtained results, we give an example.     

Local influence diagnostics for incomplete overdispersed longitudinal counts

We develop local influence diagnostics to detect influential subjects when generalized linear mixed models are fitted to incomplete longitudinal overdispersed count data. The focus is on the influence stemming from the dropout model specification. In particular, the effect of small perturbations around an MAR specification are examined. The method is applied to data from a longitudinal clinical trial in epileptic patients. The effect on models allowing for overdispersion is contrasted with that on models that do not.     

Local Influence Analysis for The Ridge Regression Under Stochastic Linear Restrictions

In this article, we assess the local influence for the ridge regression of linear models with stochastic linear restrictions in the spirit of Cook by using the log-likelihood of the stochastic restricted ridge regression estimator. The diagnostics under the perturbations of constant variance, responses and individual explanatory variables are derived. We also assess the local influence of the stochastic restricted ridge regression estimator under the approach suggested by Billor and Loynes. At the end, a numerical example on the Longley data is given to illustrate the theoretic results.     

Model influence functions based on mixtures

Influence functions are considered as diagnostics for model departures in parametric Bayesian inference. A baseline model density is expressed as a mixture; then the mixing distribution is perturbed. This is designed to engender perturbations which are plausible a priori. The influence of perturbations is measured for both Bayes estimates and their associated posterior expected losses. To assess the plausibility of perturbations a posteriori, an additional influence function is constructed for the Bayes factor comparing the perturbed and baseline models. The effect of perturbation on various estimands is incorporated in the analysis.     

Testing interaction in some predator–prey populations

This paper deals with a testing problem for each of the interaction parameters of the Lotka–Volterra ordinary differential equations system~(ODE). In short, when the rates of birth and death are fixed, we would like to test if each interaction parameter is higher or lower than a fixed reference rate. We choose a statistical model where the actual population sizes are modelled as random perturbations of the solutions to this ODE. By assuming that the random perturbations follow correlated Ornstein–Uhlenbeck processes, we propose the uniformly most powerful test concerning each interaction parameter of the ODE and, we establish the asymptotic properties of the test. Further, we illustrate the suggested test on the Canadian mink–muskrat data set. This research has received the financial support from Natural Sciences and Engineering Research Council of Canada and Institut des Sciences Mathématiques.     

Diagnostics in elliptical regression models with stochastic restrictions applied to econometrics

We propose an influence diagnostic methodology for linear regression models with stochastic restrictions and errors following elliptically contoured distributions. We study how a perturbation may impact on the mixed estimation procedure of parameters in the model. Normal curvatures and slopes for assessing influence under usual schemes are derived, including perturbations of case-weight, response variable, and explanatory variable. Simulations are conducted to evaluate the performance of the proposed methodology. An example with real-world economy data is presented as an illustration.     

Latent variable model for mixed correlated power series and ordinal longitudinal responses with non ignorable missing values

We propose a joint model based on a latent variable for analyzing mixed power series and ordinal longitudinal data with and without missing values. A bivariate probit regression model is used for the missing mechanisms. Random effects are used to take into account the correlation between longitudinal responses. A full likelihood-based approach is used to yield maximum-likelihood estimates of the model parameters. Our model is applied to a medical data set, obtained from an observational study on women where the correlated responses are the ordinal response of osteoporosis of the spine and the power series response of the number of joint damages. Sensitivity analysis is also performed to study the influence of small perturbations of the parameters of the missing mechanisms and overdispersion of the model on likelihood displacement.     

Spatio-temporal functional regression on paleoecological data

There is much interest in predicting the impact of global warming on the genetic diversity of natural populations and the influence of climate on biodiversity is an important ecological question. Since Holocene, we face many climate perturbations and the geographical ranges of plant taxa have changed substantially. Actual genetic diversity of plant is a result of these processes and a first step to study the impact of future climate change is to understand the important features of reconstructed climate variables such as temperature or precipitation for the last 15,000 years on actual genetic diversity of forest. We model the relationship between genetic diversity in the European beech (Fagus sylvatica) forests and curves of temperature and precipitation reconstructed from pollen databases. Our model links the genetic measure to the climate curves. We adapt classical functional linear model to take into account interactions between climate variables as a bilinear form. Since the data are georeferenced, our extensions also account for the spatial dependence among the observations. The practical issues of these methodological extensions are discussed.     

Sensitivity analysis for ranked data

Sensitivity analysis is to study the influence of a small change in the input data on the output of the analysis. Han and Huh (1995) developed a quantification method for the ranked data. However, the question of stability in the analysis of ranked data has not been considered. Here, we propose a method of sensitivity analysis for ranked data. Our aim is to evaluate perturbations by using a graphical approach suggested by Han and Huh (1995). It extends the results obtained by Tanaka (1984) and Huh (1989) for the sensitivity analysis in Hayashi’s third method of quantification and those by Huh and Park (1990) for the principal component reduction of the case influence derivatives in regression. A numerical example is provided to explain how to conduct sensitivity analysis based on the proposed approach.     

Influential data points in predictive logistic models

The influence of individual points in an ordinal logistic model is considered when the aim is to determine their effects on the predictive probability in a Bayesian predictive approach. Our concern is to study the effects produced when the data are slightly perturbed, in particular by observing how these perturbations will affect the predictive probabilities and consequently the classification of future cases. We consider the extent of the change in the predictive distribution when an individual point is omitted (deleted) from the sample by use of a divergence measure suggested by Johnson (1985) as a measure of discrepancy between the full data and the data with the case deleted. The methodology is illustrated on some data used in Titterington et al. (1981).     

Influence diagnostics in Gaussian spatial linear models

Spatial linear models have been applied in numerous fields such as agriculture, geoscience and environmental sciences, among many others. Spatial dependence structure modelling, using a geostatistical approach, is an indispensable tool to estimate the parameters that define this structure. However, this estimation may be greatly affected by the presence of atypical observations in the sampled data. The purpose of this paper is to use diagnostic techniques to assess the sensitivity of the maximum-likelihood estimators, covariance functions and linear predictor to small perturbations in the data and/or the spatial linear model assumptions. The methodology is illustrated with two real data sets. The results allowed us to conclude that the presence of atypical values in the sample data have a strong influence on thematic maps, changing the spatial dependence structure.     

The influence function of penalized regression estimators

To perform regression analysis in high dimensions, lasso or ridge estimation are a common choice. However, it has been shown that these methods are not robust to outliers. Therefore, alternatives as penalized M-estimation or the sparse least trimmed squares (LTS) estimator have been proposed. The robustness of these regression methods can be measured with the influence function. It quantifies the effect of infinitesimal perturbations in the data. Furthermore, it can be used to compute the asymptotic variance and the mean-squared error (MSE). In this paper we compute the influence function, the asymptotic variance and the MSE for penalized M-estimators and the sparse LTS estimator. The asymptotic biasedness of the estimators make the calculations non-standard. We show that only M-estimators with a loss function with a bounded derivative are robust against regression outliers. In particular, the lasso has an unbounded influence function.     

Influence Diagnostics for Simultaneous Equations Models

This paper presents influence diagnostics for simultaneous equations models. It proposes residuals, leverage and other influence measures. A missing data method is adopted to minimize the masking effect due to case deletions. The assessment of local influence is also considered. The paper shows how to evaluate the effects that perturbations to the endogenous variables, predetermined variables and case weights may have on the parameter estimates. The diagnostics are illustrated with two examples.     

Bayesian local influence analysis of skew-normal spatial dynamic panel data models

The existing studies on spatial dynamic panel data model (SDPDM) mainly focus on the normality assumption of response variables and random effects. This assumption may be inappropriate in some applications. This paper proposes a new SDPDM by assuming that response variables and random effects follow the multivariate skew-normal distribution. A Markov chain Monte Carlo algorithm is developed to evaluate Bayesian estimates of unknown parameters and random effects in skew-normal SDPDM by combining the Gibbs sampler and the Metropolis–Hastings algorithm. A Bayesian local influence analysis method is developed to simultaneously assess the effect of minor perturbations to the data, priors and sampling distributions. Simulation studies are conducted to investigate the finite-sample performance of the proposed methodologies. An example is illustrated by the proposed methodologies.     

Influence measure in maximum likelihood estimate for models of lifetime data

We use the local influence approach to develop influence measures for identifying observations that strike a disproportionate effect on the maximum likelihood estimate of parameters in models for lifetime data. The proposed method for developing influence measures can be applied to a wide variety of models and we use the exponential model to illustrate the details. In particular, we show that the proposed measure is equivalent to the martingale residual under the exponential model.