Valentino Dardanoni

Inference for Lorenz curve orderings

In this paper we consider the issue of performing statistical inference for Lorenz curve orderings. This involves testing for an ordered relationship in a multivariate context and making comparisons among more than two population distributions. Our approach is to frame the hypotheses of interest as sets of linear inequality constraints on the vector of Lorenz curve ordinates, and apply order-restricted statistical inference to derive test statistics and their sampling distributions. We go on to relate our results to others which have appeared in recent literature, and use Monte Carlo analysis to highlight their respective properties and comparative performances. Finally, we discuss in general terms the issue and problems of framing hypotheses, and testing them, in the context of the study of income inequality, and suggest ways in which the distributional analyst could best proceed, illustrating with empirical examples.

A unified approach to likelihood inference on stochastic orderings in a nonparametric context

Abstract For data in a two-way contingency table with ordered margins, we consider various hypotheses of stochastic orders among the conditional distributions considered by rows and show that each is equivalent to requiring that an invertible transformation of the vectors of conditional row probabilities satisfies an appropriate set of linear inequalities. This leads to the construction of a general algorithm for maximum likelihood estimation under multinomial sampling and provides a simple framework for deriving the asymptotic distribution of log-likelihood ratio tests. The usual stochastic ordering and the so called uniform and likelihood ratio orderings are considered as special cases. In particular, for each of these three orderings we determine the transformation required to apply the estimation algorithm; we then consider testing the hypothesis that the rows are identically distributed against the alternative that they are stochastically ordered, as well as testing each stochastic order against an u...

Mobility comparisons: does using different measures matter?

In this paper we review alternative measure of intergenerational mobility, emphasizing the distinction between absolute, relative and ordinal mobility. We then compare the performance of various mobility indices using real data. From Treiman and Ganzeboom (1990) dataset we compare the degree of occupational and educational intergenerational (father-son) mobility in 16 countries in a single year (comprised between 1968 and 1982). From three Bank of Italy surveys (1993-1995-1998) we obtain a comparable measure of social prestige and we show that intergenerational mobility in Italy across regions or age cohort exhibits different trends according to different indicators. We suggest that ordinal relative and absolute measures provide divergent indications whenever we compare mobility data with markedly different marginal distributions.

What’s so special about Euclidean distance?

In this paper, we provide an application-oriented characterization of a class of distance functions monotonically related to the Euclidean distance in terms of some general properties of distance functions between real-valued vectors. Our analysis hinges upon two fundamental properties of distance functions that we call “value-sensitivity” and “order-sensitivity”. We show how these two general properties, combined with natural monotonicity considerations, lead to characterization results that single out several versions of Euclidean distance from the wide class of separable distance functions. We then discuss and motivate our results in two different and apparently unrelated application areas—mobility measurement and spatial voting theory—and propose our characterization as a test for deciding whether Euclidean distance (or some suitable variant) should be used in your favourite application context.

The Measurement of Rank Mobility

In this paper we investigate the problem of measuring social mobility when the social status of individuals is given by their rank. In order to sensibly represent the rank mobility of subgroups within a given society, we address the problem in terms of partial permutation matrices which include standard (“global”) matrices as a special case. We first provide a characterization of a partial ordering on partial matrices which, in the standard case of global matrices, coincides with the well-known “concordance” ordering. We then provide a characterization of an index of rank mobility based on partial matrices and show that, in the standard case of comparing two global matrices, it is equivalent to Spearman’s index.

Positive Quadrant Dependence and Marginal Modeling in Two-Way Tables With Ordered Margins

For a collection of two-way tables, where subjects are cross-classified according to the same pair of ordinal categorical variables conditionally on the value of one or more discrete explanatory variables, we propose a general approach to likelihood inference that combines marginal modeling with fitting and testing of inequality constraints such as those implied by the assumption that one marginal distribution is stochastically larger than the other, positive dependence and stronger positive dependence. The approach is based on parameterizing bivariate conditional distributions with global logits and global log-odds ratios, and we provide a general framework for handling models defined by equality and inequality constraints on these parameters. In this way, such models as marginal homogeneity, proportional odds among row or columns margins, and Plackett distribution may be treated together with various models defined by inequality constraints on the same parameters, such as, for instance, those implied by the positive quadrant dependence. For this class of models, we define a Fisher scoring algorithm for computing maximum likelihood estimates and derive the asymptotic distribution of the likelihood ratio tests that turn out to be of the chi-bar squared type when inequalities are involved. When the main interest is on positive dependence, we derive tight bounds on the asymptotic distribution of these statistics that are independent of the marginal logits. These also may be removed by conditioning on one or both observed margins in each table, and we describe in detail how the approach may be adapted to these sampling schemes. Three applications to real datasets are discussed.

Fibrosis staging in chronic hepatitis C: analysis of discordance between transient elastography and liver biopsy

Summary.  In chronic hepatitis C, transient elastography (TE) accurately identifies cirrhosis, but its ability to assess significant fibrosis (Metavir ≥ F2) is variable. Constitutional and liver disease‐related factors may influence TE and here we examined the variables associated with differences. Three hundred consecutive hepatitis C virus (HCV)‐RNA positive patients had biochemical tests, TE and a biopsy performed on the same day. The Dale model was used to identify the variables associated with discordance between biopsy and elastography results. In 97 patients (34.2%), TE and histological assessment were discordant. Seventy‐six of 286 (26.6%) had stage ≥F2 and TE < 7.1 kPa (false negative); 21 of 286 (7.3%) had stage

Income distribution dynamics: monotone Markov chains make light work

This paper considers some aspects of the dynamics of income distributions by employing a simple Markov chain model of income mobility. The main motivation of the paper is to introduce the techniques of “monotone” Markov chains to this field. The transition matrix of a discrete Markov chain is called monotone if each row stochastically dominates the row above it. It will be shown that by embedding the dynamics of the income distribution in a monotone Markov chain, a number of interesting results may be obtained in a straightforward and intuitive fashion.

Incentive and selection effects of Medigap insurance on inpatient care.

The Medicare program, which provides insurance coverage to the elderly in the United States, does not protect them fully against high out-of-pocket costs. For this reason private supplementary insurance, named Medigap, has been available to cover Medicare gaps. This paper studies how Medigap affects the utilization of inpatient care, separating the incentive and selection effects of supplementary insurance. For this purpose, we use two alternative estimation methods: a standard recursive bivariate probit and a discrete multivariate finite mixture model. We find that estimated incentive effects are modest and quite similar across models. There seems to be very significant selection, with the presence of both adversely and advantageously selected individuals, stemming from the multidimensional nature of residual heterogeneity.

GRADING ACROSS SCHOOLS

Abstract This paper reports some facts about grading standards across a varied sample of 16 countries participating in the 2003 OCSE-PISA Survey. Our main finding is that in all countries except Ireland and the USA there is conspicuous heterogeneity in standards across schools (Table 3, Figures 1 & 2). In most of the countries where heterogeneity is present a grading-on-a-curve practice emerges, with grading standards increasing with average competence of the schools students (Table 4, Figures 3 & 4). Where this phenomenon is more pronounced, it may be related to existence of a tracking (as opposed to comprehensive) school system (Table 5, Figure 5).