Statistical modeling under partial identification: Distinguishing three types of identification regions in regression analysis with interval data

Abstract

One of the most promising applications of the methodology of imprecise probabilities in statistics is the reliable analysis of interval data (or more generally coarsened data). As soon as one refrains from making strong, often unjustified assumptions on the coarsening process, statistical models are naturally only partially identified and set-valued parameter estimators (identification regions) have to be derived. In this paper we consider linear regression analysis under interval data in the dependent variable. While in the traditional case of neglected imprecision different understandings of regression modeling lead to the same parameter estimators, we now have to distinguish between two different types of identification regions, called (Sharp) Marrow Region (SMR) and (Sharp) Collection Region (SCR) here. In addition, we propose the Set-loss Region (SR) as a compromise between SMR and SCR based on a set-domained loss function. We elaborate and discuss some fundamental properties of these regions and then illustrate the methodology in detail by an example, where the influence of different covariates on wine quality, measured by a coarse rating scale, is investigated. We also compare the different identification regions to classical estimates from a naive analysis and from common interval censorship modeling.

Reliable Inference in Categorical Regression Analysis for Non‐randomly Coarsened Observations

“…While in categorical cases showing large state spaces, one can focus on the most important coarse categories to avoid the explosion of the number of coarse categories, the problem has to be approached totally differently for continuous variables (cf., e.g. Schollmeyer & Augustin, ). Because most questionnaires involve questions showing a manageable number of categorical answers (cf., e.g.…”

“…Model assumptions and the inherent model uncertainty in case of partially identified models have already been studied in literature (cf., e.g. Ponomareva & Tamer, ; Schollmeyer & Augustin, ). In particular, in this case—unlike in the identified case—the interpretation of a linear model (in the sense of Freedman () who distinguishes between a structural or a descriptive view) is crucial and can lead to different results.…”

“…The impact of model assumptions in the context of incomplete data is an important aspect of study in literature (cf., e.g. Hüllermeier, ; Sánchez & Couso, ; Schollmeyer & Augustin, ). We study in particular the effect of the regression model assumptions on the estimated coarsening parameters.…”

Statistics with Imprecise Probabilities—A Short Survey

“…[56]); see also [8,Sect. 7.8.2], who try to characterize and unify these approaches by the concept of cautious data completion, and the concept of collection regions in [60].…”

On the testability of coarsening assumptions: A hypothesis test for subgroup independence

“…Here, we use the cautious approach developed in [20], which refers to the latent likelihood and is -just as e.g. [17] (in the context of misclassification) and [25] -strongly influenced by the methodology of partial identification (cf. [16]).…”