Inferential Issues Research Articles

Calibrating the Degrees of Freedom for Automatic Data Smoothing and Effective Curve Checking

Curve fitting and curve checking based on the local polynomial regression technique are commonly used data-analytic methods in statistics. This article examines, in nonparametric settings, both the asymptotic expressions and empirical formulas for degrees of freedom (DF), a notion introduced by Hastie and Tibshirani, of linear smoothers. The asymptotic results give useful insights into the nonparametric modeling complexity. Meanwhile, by substituting the exact DFs by the empirical formula, an empirical version of the generalized cross-validation (EGCV) is obtained. An automatic bandwidth selection method based on minimizing EGCV is proposed for conducting local smoothing. This procedure preserves full benefits of the ordinary and generalized cross-validation, but offers a substantial reduction in computational burden. Furthermore, the EGCV-minimizing bandwidth can be extended in a very simple manner to fit multivariate models, such as the varying-coefficient models. Applications of calibrating DFs to important inferential issues, such as assessing the validity of useful model assumptions and measuring the significance of predictor variables based on the generalized likelihood ratio statistics are also discussed. Simulation studies are presented to illustrate the performance of the proposed procedures in a range of statistical problems.

Bayesian inference for a generalized population attributable fraction: the impact of early vitamin A levels on chronic lung disease in very low birthweight infants.

In this paper, the population attributable fraction is studied using the potential responses framework of Rubin's causal model. This framework facilitates definition of a general measure of population attributable effect which can accommodate many-valued and multivariate exposures as well as many-valued responses. Inferential issues are considered from the Bayesian perspective. Finite population inference is emphasized with inference in the case of a fully observed population given particular attention. The key inferential issue concerns computation of the posterior distribution of unobserved potential responses, given observed responses, exposures and covariates. A dependency on model parameters about which observed data are uninformative is highlighted and this reflects the unobservable nature of causal effects. In an application to a small cohort study of respiratory problems in very low birthweight infants, posterior inferences were found to be insensitive to assumptions concerning the joint distribution of potential response variables but sensitive to the assumption of weak ignorability, a weaker form of the more familiar assumption of no confounding by omitted covariates. In a model-based set-up, the weak ignorability assumption is identified with setting a model parameter to zero, and consequently uncertainty concerning this assumption can, in principle, be handled via the prior distribution for the model parameters.

Minimax Entropy Principle and Its Application to Texture Modeling

This article proposes a general theory and methodology, called the minimax entropy principle, for building statistical models for images (or signals) in a variety of applications. This principle consists of two parts. The first is the maximum entropy principle for feature binding (or fusion): for a given set of observed feature statistics, a distribution can be built to bind these feature statistics together by maximizing the entropy over all distributions that reproduce them. The second part is the minimum entropy principle for feature selection: among all plausible sets of feature statistics, we choose the set whose maximum entropy distribution has the minimum entropy. Computational and inferential issues in both parts are addressed; in particular, a feature pursuit procedure is proposed for approximately selecting the optimal set of features. The minimax entropy principle is then corrected by considering the sample variation in the observed feature statistics, and an information criterion for feature pursuit is derived. The minimax entropy principle is applied to texture modeling, where a novel Markov random field (MRF) model, called FRAME (filter, random field, and minimax entropy), is derived, and encouraging results are obtained in experiments on a variety of texture images. The relationship between our theory and the mechanisms of neural computation is also discussed.

The Econometrics of the ‘Market Model’: Cointegration, Error Correction and Exogeneity

This paper investigates the market model from the perspective of modern time series econometrics, focusing upon the assumptions and restrictions that are made in order to arrive at the returns regression that is the market model. The fact that stock prices and indices are typically non-stationary leads naturally to the topic of cointegration, and the implications of this for the market model are developed. Exogeneity within cointegrated systems is considered, whilst testing, estimation and inferential issues are also discussed. Three examples are provided to illustrate the proposed methodology for modelling the relationship between a stock price and a market index.

The Econometrics of the ‘Market Model’: Cointegration, Error Correction and Exogeneity

This paper investigates the market model from the perspective of modern time series econometrics, focusing upon the assumptions and restrictions that are made in order to arrive at the returns regression that is the market model. The fact that stock prices and indices are typically non-stationary leads naturally to the topic of cointegration, and the implications of this for the market model are developed. Exogeneity within cointegrated systems is considered, whilst testing, estimation and inferential issues are also discussed. Three examples are provided to illustrate the proposed methodology for modelling the relationship between a stock price and a market index.

The role of probability-based inference in an intelligent tutoring system

Probability-based inference in complex networks of interdependent variables is an active topic in statistical research, spurred by such diverse applications as forecasting, pedigree analysis, troubleshooting, and medical diagnosis. This paper concerns the role of Bayesian inference networks for updating student models in intelligent tutoring systems (ITSs). Basic concepts of the approach are briefly reviewed, but the emphasis is on the considerations that arise when one attempts to operationalize the abstract framework of probability-based reasoning in a practical ITS context. The discussion revolves around HYDRIVE, an ITS for learning to troubleshoot an aircraft hydraulics system. HYDRIVE supports generalized claims about aspects of student proficiency through probabilitybased combination of rule-based evaluations of specific actions. The paper highlights the interplay among inferential issues, the psychology of learning in the domain, and the instructional approach upon which the ITS is based.

THE ROLE OF PROBABILITY‐BASED INFERENCE IN AN INTELLIGENT TUTORING SYSTEM

ABSTRACTProbability‐based inference in complex networks of interdependent variables is an active topic in statistical research, spurred by such diverse applications as forecasting, pedigree analysis, troubleshooting, and medical diagnosis. This paper concerns the role of Bayesian inference networks for updating student models in intelligent tutoring systems (ITSs). Basic concepts of the approach are briefly reviewed, but the emphasis is on the considerations that arise when one attempts to operationalize the abstract framework of probability‐based reasoning in a practical ITS context. The discussion revolves around HYDRIVE, an ITS for learning to troubleshoot an aircraft hydraulics system. HYDRIVE supports generalized claims about aspects of student proficiency through probability‐based combination of rule‐based evaluations of specific actions. The paper highlights the interplay among inferential issues, the psychology of learning in the domain, and the instructional approach upon which the ITS is based.

Assessing Drug and Alcohol Treatment Needs of General and Special Populations: Conceptual, Empirical and Inferential Issues

“Needs assessments” have become increasingly important in guiding resource allocations to publicly funded drug and alcohol treatment programs. Although technical progress in conducting needs assessments in recent years has occurred, virtually all approaches used to identify treatment needs of general and special populations suffer from conceptual, empirical, and inferential problems. This article clarifies the more important problems and offers recommendations about how to improve needs assessments.

Distribution theory and inference for polynomial-normal densities

This paper considers a class of densities formed by taking the product of nonnegative polynomials and normal densities. These densities provide a rich class of distributions that can be used in modelling when faced with non-normal characteristics such as skewness and multimodality. In this paper we address inferential and computational issues arising in the practical implementation of this parametric family in the context of the linear model. Exact results are recorded for the conditional analysis of location-scale models and an importance sampling algorithm is developed for the implementation of a conditional analysis for the general linear model when using polynomial-normal distributions for the error.

A stochastic model for neuronal bursting

A new stochastic model for bursting of neuronal firing is proposed. It is based on stochastic diffusion and related to the first passage time problem. However, the model is not of renewal type. Its form and parameters are physiologically interpretable. Parametric and non-parametric inferential issues are discussed.

EVALUATING CLASSIFICATIONS OF JOB BEHAVIOR: A CONSTRUCT VALIDATION OF THE ABILITY REQUIREMENT SCALES

This article discusses the major inferential issues arising in the development of behavioral classification systems. Subsequently, we discuss the implications of these inferential issues for evaluating the construct validity of systems designed to assess the requirements of human task performance. Fleishman's (1975b, 1982) ability requirement taxonomy and its associated job analysis system, the Manual for the Ability Requirement Scales (MARS), were then evaluated with respect to these criteria. In particular, a variety of criteria relevant to internal and external validity was reviewed. It was found that the ability requirement taxonomy and the associated measurement system provides a meaningful description of job activities with respect to these criteria. It was argued that application of these construct validity principles might contribute much to our understanding of human performance.

A mixed integer programming approach for accommodating outliers with the minimax absolute error criterion

Minimax absolute error (MMAE) is a data fitting criterion which has several desirable features. However at the same time, this criterion is especially sensitive to the influence of outliers. In this paper we discuss a minimax problem which can be solved by a mixed integer programming model, which is paired with the MMAE criterion to provide simultaneous determination of candidate outliers along with model parameter estimation. We limit discussion to the case of one dependent variable per observation and focus attention on data fitting rather than inferential issues.

Towards a process paradigm in psychophysiology

This article discusses a theory and method for classifying psychophysiological phenomena as processes prior to considering them as experimental responses. The initial aim then is to construe experimental units as continuous subject-centred processes whose beginning and end depends on the individuals' actions. Boundaries of such processes are systematically defined by first identifying the process as a dimension. Activities are then classified relative to this dimension. Process units are specified by considering two classes of activity which form opposite ends of the dimension of process. The two classes are equivalent in all respects save along the one dimension. These two classes are termed ‘pro’- and ‘null’-equivalence and are the process methods version of familiar H 1 and H 0 of hypothetico-deductive procedures. Proportional presence or absence of a parameter value during pro- and null-equivalence of a process is the basis for inferring that a particular measure does or does not typify the process. Actual parameter values of a measure can then form credible limits by which to classify other processes as equivalent or not. These limits can indicate for example whether two experimental effects are dealing with the same or a different process. There are some difficulties in applying this process method, but the general conclusion is that classifying psychophysiological responses at the outset in terms of process clarifies technical and inferential issues in experimentation.

Monotone Regression Splines in Action

Piecewise polynomials or splines extend the advantages of polynomials to include greater flexibility, local effects of parameter changes and the possibility of imposing useful constraints on estimated functions. Among these constraints is monotonicity, which can be an important property in many curve estimation problems. This paper shows the virtues of monotone splines through a number of statistical applications, including response variable transformation in nonlinear regression, transformation of variables in multiple regression, principal components and canonical correlation, and the use of monotone splines to model a dose-response function and to perform item analysis. Computational and inferential issues are discussed and illustrated.