Distributional equivalence and subcompositional coherence in the analysis of contingency tables, ratio-scale measurements and compositional data

We consider two fundamental properties in the analysis of two-way tables of positive data: the principle of distributional equivalence, one of the cornerstones of correspondence analysis of contingency tables, and the principle of subcompositional coherence, which forms the basis of compositional data analysis. For an analysis to be subcompositionally coherent, it suffices to analyse the ratios of the data values. The usual approach to dimension reduction in compositional data analysis is to perform principal component analysis on the logarithms of ratios, but this method does not obey the principle of distributional equivalence. We show that by introducing weights for the rows and columns, the method achieves this desirable property. This weighted log-ratio analysis is theoretically equivalent to spectral mapping , a multivariate method developed almost 30 years ago for displaying ratio-scale data from biological activity spectra. The close relationship between spectral mapping and correspondence analysis is also explained, as well as their connection with association modelling. The weighted log-ratio methodology is applied here to frequency data in linguistics and to chemical compositional data in archaeology.

Biplots of fuzzy coded data

A biplot, which is the multivariate generalization of the two-variable scatterplot, can be used to visualize the results of many multivariate techniques, especially those that are based on the singular value decomposition. We consider data sets consisting of continuous-scale measurements, their fuzzy coding and the biplots that visualize them, using a fuzzy version of multiple correspondence analysis. Of special interest is the way quality of fit of the biplot is measured, since it is well-known that regular (i.e., crisp) multiple correspondence analysis seriously under-estimates this measure. We show how the results of fuzzy multiple correspondence analysis can be defuzzified to obtain estimated values of the original data, and prove that this implies an orthogonal decomposition of variance. This permits a measure of fit to be calculated in the familiar form of a percentage of explained variance, which is directly comparable to the corresponding fit measure used in principal component analysis of the original data. The approach is motivated initially by its application to a simulated data set, showing how the fuzzy approach can lead to diagnosing nonlinear relationships, and finally it is applied to a real set of meteorological data.

Adaptive model selection using empirical complexities

Given $n$ independent replicates of a jointly distributed pair $(X,Y)\in {\cal R}^d \times {\cal R}$, we wish to select from a fixed sequence of model classes ${\cal F}_1, {\cal F}_2, \ldots$ a deterministic prediction rule $f: {\cal R}^d \to {\cal R}$ whose risk is small. We investigate the possibility of empirically assessingthe {\em complexity} of each model class, that is, the actual difficulty of the estimation problem within each class. The estimated complexities are in turn used to define an adaptive model selection procedure, which is based on complexity penalized empirical risk.The available data are divided into two parts. The first is used to form an empirical cover of each model class, and the second is used to select a candidate rule from each cover based on empirical risk. The covering radii are determined empirically to optimize a tight upper bound on the estimation error. An estimate is chosen from the list of candidates in order to minimize the sum of class complexity and empirical risk. A distinguishing feature of the approach is that the complexity of each model class is assessed empirically, based on the size of its empirical cover.Finite sample performance bounds are established for the estimates, and these bounds are applied to several non-parametric estimation problems. The estimates are shown to achieve a favorable tradeoff between approximation and estimation error, and to perform as well as if the distribution-dependent complexities of the model classes were known beforehand. In addition, it is shown that the estimate can be consistent,and even possess near optimal rates of convergence, when each model class has an infinite VC or pseudo dimension.For regression estimation with squared loss we modify our estimate to achieve a faster rate of convergence.

Solving heterogeneous-agent models by projection and perturbation

The paper proposes a numerical solution method for general equilibrium models with a continuum of heterogeneous agents, which combines elements of projection and of perturbation methods. The basic idea is to solve first for the stationary solutionof the model, without aggregate shocks but with fully specified idiosyncratic shocks. Afterwards one computes a first-order perturbation of the solution in the aggregate shocks. This approach allows to include a high-dimensional representation of the cross-sectional distribution in the state vector. The method is applied to a model of household saving with uninsurable income risk and liquidity constraints. The model includes not only productivity shocks, but also shocks to redistributive taxation, which cause substantial short-run variation in the cross-sectional distribution of wealth. If those shocks are operative, it is shown that a solution method based on very few statistics of the distribution is not suitable, while the proposed method can solve the model with high accuracy, at least for the case of small aggregate shocks. Techniques are discussed to reduce the dimension of the state space such that higher order perturbations are feasible.Matlab programs to solve the model can be downloaded.

Geographical deviations in foreign trade statistics: A study into European trade with Latin American Countries, 1925

We have analyzed the spatial accuracy of European foreign trade statistics compared to Latin American. We have also included USA s data because of the importance of this country in Latin American trade. We have developed a method for mapping discrepancies between exporters and importers, trying to isolate systematic spatial deviations. Although our results don t allow a unique explanation, they present some interesting clues to the distribution channels in the Latin American Continent as well as some spatial deviations for statistics in individual countries. Connecting our results with the literature specialized in the accuracy of foreign trade statistics; we can revisit Morgernstern (1963) as well as Federico and Tena (1991). Morgernstern had had a really pessimistic view on the reliability of this statistic source, but his main alert was focused on the trade balances, not in gross export or import values. Federico and Tena (1991) have demonstrated howaccuracy increases by aggregation, geographical and of product at the same time. But they still have a pessimistic view with relation to distribution questions, remarking that perhaps it will be more accurate to use import sources in this latest case. We have stated that the data set coming from foreign trade statistics for a sample in 1925, being it exporters or importers, it s a valuable tool for geography of trade patterns, although in some specific cases it needs some spatial adjustments.

Analysis of matched matrices

We consider the joint visualization of two matrices which have common rowsand columns, for example multivariate data observed at two time pointsor split accord-ing to a dichotomous variable. Methods of interest includeprincipal components analysis for interval-scaled data, or correspondenceanalysis for frequency data or ratio-scaled variables on commensuratescales. A simple result in matrix algebra shows that by setting up thematrices in a particular block format, matrix sum and difference componentscan be visualized. The case when we have more than two matrices is alsodiscussed and the methodology is applied to data from the InternationalSocial Survey Program.

Small-area estimation with spatial similarity

A class of composite estimators of small area quantities that exploit spatial (distancerelated)similarity is derived. It is based on a distribution-free model for the areas, but theestimators are aimed to have optimal design-based properties. Composition is applied alsoto estimate some of the global parameters on which the small area estimators depend.It is shown that the commonly adopted assumption of random effects is not necessaryfor exploiting the similarity of the districts (borrowing strength across the districts). Themethods are applied in the estimation of the mean household sizes and the proportions ofsingle-member households in the counties (comarcas) of Catalonia. The simplest version ofthe estimators is more efficient than the established alternatives, even though the extentof spatial similarity is quite modest.

Identifying the gap between need and intervention for alcohol use disorders in Europe.

AIMS: A literature review of existing research on the prevalence of alcohol use disorders (AUDs) and availability of alcohol interventions in Europe was conducted. The review also explored what is known about the gap between need and provision of alcohol interventions in Europe. METHODS: The review search strategy included: (i) descriptive studies of alcohol intervention systems in Europe; (ii) studies of alcohol service provision in Europe; and (iii) studies of prevalence of AUD and alcohol needs assessment in Europe. RESULTS: Europe has a relatively high level of alcohol consumption and the resulting disabilities are the highest in the world. Most research on implementation of alcohol interventions in Europe has been restricted to screening and brief interventions. Alcohol needs assessment methodology has been developed but has not been applied in comparative studies across countries in Europe. CONCLUSIONS: This review points to key gaps in knowledge related to alcohol interventions in Europe. There is a lack of comparative data on variations in alcohol treatment systems across European countries and there is also a lack of comparative data on the prevalence of alcohol use disorders across European countries and the relative gap between need and access to treatment. The forthcoming Alcohol Measures for Public Health Research Alliance (AMPHORA) research project work package on 'Early identification and treatment' aims to address these gaps.

Variance reduction methods for simulation of densities on Wiener space

We develop a general error analysis framework for the Monte Carlo simulationof densities for functionals in Wiener space. We also study variancereduction methods with the help of Malliavin derivatives. For this, wegive some general heuristic principles which are applied to diffusionprocesses. A comparison with kernel density estimates is made.

On the accuracy of Latin American trade statistics: A nonparametric test for 1925

This paper proposes a nonparametric test in order to establish the level of accuracy of theforeign trade statistics of 17 Latin American countries when contrasted with the trade statistics of the main partners in 1925. The Wilcoxon Matched-Pairs Ranks test is used to determine whether the differences between the data registered by exporters and importers are meaningful, and if so, whether the differences are systematic in any direction. The paper tests for the reliability of the data registered for two homogeneous products, petroleum and coal, both in volume and value. The conclusion of the several exercises performed is that we cannot accept the existence of statistically significant differences between the data provided by the exporters and the registered by the importing countries in most cases. The qualitative historiography of Latin American describes its foreign trade statistics as mostly unusable. Our quantitative results contest this view.

Improving small area estimation by combining surveys: new perspectives in regional statistics

A national survey designed for estimating a specific population quantity is sometimes used for estimation of this quantity also for a small area, such as a province. Budget constraints do not allow a greater sample size for the small area, and so other means of improving estimation have to be devised. We investigate such methods and assess them by a Monte Carlo study. We explore how a complementary survey can be exploited in small area estimation. We use the context of the Spanish Labour Force Survey (EPA) and the Barometer in Spain for our study.

Country effects in ISSP-1993 environmental data: Comparison of SEM approaches

Structural equation models (SEM) are commonly used to analyze the relationship between variables some of which may be latent, such as individual ``attitude'' to and ``behavior'' concerning specific issues. A number of difficulties arise when we want to compare a large number of groups, each with large sample size, and the manifest variables are distinctly non-normally distributed. Using an specific data set, we evaluate the appropriateness of the following alternative SEM approaches: multiple group versus MIMIC models, continuous versus ordinal variables estimation methods, and normal theory versus non-normal estimation methods. The approaches are applied to the ISSP-1993 Environmental data set, with the purpose of exploring variation in the mean level of variables of ``attitude'' to and ``behavior''concerning environmental issues and their mutual relationship across countries. Issues of both theoretical and practical relevance arise in the course of this application.

Tying up the loose ends in simple correspondence analysis

Although correspondence analysis is now widely available in statistical software packages and applied in a variety of contexts, notably the social and environmental sciences, there are still some misconceptions about this method as well as unresolved issues which remain controversial to this day. In this paper we hope to settle these matters, namely (i) the way CA measures variance in a two-way table and how to compare variances between tables of different sizes, (ii) the influence, or rather lack of influence, of outliers in the usual CA maps, (iii) the scaling issue and the biplot interpretation of maps,(iv) whether or not to rotate a solution, and (v) statistical significance of results.