A Novel Indicator to Correct for Individual Reported Heterogeneity. An Application to Self-Evaluation of Later-Life Depression

Program evaluations often investigate complex or multi-dimensional constructs, such as individual opinions or attitudes, by means of ratings. A different interpretation of the same question may affect cross-country comparability, leading to the Differential Item Functioning problem. Anchoring vignettes were introduced in the literature as a way to adjust self-evaluations from this interpersonal incomparability. In this paper, we first introduce a new nonparametric solution to analyse anchoring vignette data, recoding a variable based on a rating scale to a new corrected-variable that guarantees comparability in any cross-country analysis. Then, we exploit the flexibility of a mixture model introduced to account for uncertainty in the response process (the CUP model) to test if the proposed solution is effectively able to remove this reported heterogeneity. This solution is easy to construct and has important advantages compared with the original nonparametric solution adopted with anchoring vignette data. The novel indicator is applied to investigate self-reported depression in an old population. Data that will be analysed come from the second wave of the Survey of Health, Ageing and Retirement in Europe, collected in 2006/2007. Results highlight the need of correcting for reported heterogeneity comparing individual self-evaluations. Once interpersonal incomparability resulting from the different uses of response scales is removed from the self-assessments, some estimates are reversed in magnitude and signs with respect to the analysis of the collected data.

Generalized residuals and outlier detection for ordinal data with challenging data structures

Motivated by the analysis of rating data concerning perceived health status, a crucial variable in biomedical, economic and life insurance models, the paper deals with diagnostic procedures for identifying anomalous and/or influential observations in ordinal response models with challenging data structures. Deviations due to some respondents’ atypical behavior, outlying covariates and gross errors may affect the reliability of likelihood based inference, especially when non robust link functions are adopted. The present paper investigates and exploits the properties of the generalized residuals. They appear in the estimating equations of the regression coefficients and hold the remarkable characteristic of interacting with the covariates in the same fashion as the linear regression residuals. Identification of statistical units incoherent with the model can be achieved by the analysis of the residuals produced by maximum likelihood or robust M-estimation, while the inspection of the weights generated by M-estimation allows to identify influential data. Simple guidelines are proposed to this end, which disclose information on the data structure. The purpose is twofold: recognizing statistical units that deserve specific attention for their peculiar features, and being aware of the sensitivity of the fitted model to small changes in the sample. In the analysis of the self-perceived health status, extreme design points associated with incoherent responses produce highly influential observations. The diagnostic procedures identify the outliers and assess their influence.

The number of response categories in ordered response models

The choice of the number m of response categories is a crucial issue in categorization of a continuous response. The paper exploits the Proportional Odds Models' property which allows to generate ordinal responses with a different number of categories from the same underlying variable. It investigates the asymptotic efficiency of the estimators of the regression coefficients and the accuracy of the derived inferential procedures when m varies. The analysis is based on models with closed-form information matrices so that the asymptotic efficiency can be analytically evaluated without need of simulations. The paper proves that a finer categorization augments the information content of the data and consequently shows that the asymptotic efficiency and the power of the tests on the regression coefficients increase with m. The impact of the loss of information produced by merging categories on the efficiency of the estimators is also considered, highlighting its risks especially when performed in its extreme form of dichotomization. Furthermore, the appropriate value of m for various sample sizes is explored, pointing out that a large number of categories can offset the limited amount of information of a small sample by a better quality of the data. Finally, two case studies on the quality of life of chemotherapy patients and on the perception of pain, based on discretized continuous scales, illustrate the main findings of the paper.

Huber M-estimator for Cumulative Odds Model with Application to the Measurement of Students' Final Exam Grades

The Huber M-estimator is proposed in this study as a robust method for estimating the parameters of the cumulative odds model, which includes a logistic link function and polytomous explanatory variables. With the help of an intensive Monte Carlo simulation study carried out using the statistical software R, this study evaluates the performance of the maximum likelihood estimator (MLE) and the robust technique developed. Bias, RMSE, and the Lipsitz Statistic are used to measure comparisons. When conducting the simulation study, different sample sizes, contamination proportions, and error standard deviations are considered. Preliminary findings indicate that the M-estimator with Huber weight estimates produces the best results for parameter estimation and overall model fitting compared to the MLE. As an illustration, the procedure is applied to real-world data of students' final exam grades as measured by two different estimators.