Professor Eric Beh

Ecological inference attempts to draw conclusions concerning individual-level relationships using data in the form of aggregates for groups in the population. The groups are often geographically defined. A fundamental statistical issue is how much information aggregate data contain concerning the relationships and parameters that we are trying to estimate. The information affects the standard errors of estimates as well as the power of any tests of hypothesis. It also affects the ability to tell, from the aggregate data, which different models under consideration are supported by the data. In this chapter likelihood-based methods are considered. We show in general how aggregation affects the information matrix associated with the maximum likelihood estimates compared with the case when individual-level data are available. Hypothesis testing using aggregate data is also considered. We apply this general approach to ecological inference in the case of several 2 by 2 tables and show how the information is affected by aggregation. Tests of the hypothesis that the parameters are constant across the groups are developed using aggregate data. We also consider how the addition of a small number of individual-level data obtained from a sample, ignoring the groups, increases the information concerning the parameters. The theory is illustrated through an example.

For more than 20 years, variants of correspondence analysis have arisen that accommodate for the structure of ordered categorical variables using orthogonal polynomials. When the visual display from this analysis is the biplot, projections linking the origin to the standard coordinate of each category is a common feature. In the case when a column variable, say, consists of ordered categories, the biplot can be constructed so that their standard coordinate is determined using orthogonal polynomials which require a set of a priori scores that reflect the ordered structure of the categories. When the first two polynomials are used to construct the biplot they produce a configuration of standard coordinates that appear to be parabolic in shape. This article verifies the exact nature of this parabolic relationship and examines the various features of this configuration of points. Particular emphasis is given to the focus, vertex, intercepts and directrix of this relationship and we also briefly examine the impact of choosing different a priori scores on these features. The R function, parabola.exe(), used to perform these calculates is included as supplementary material to this article. Supplementary files for this article are available online.

Research on taste-shape correspondences has focused on one-to-one taste/shape matching tasks. However, foods and drinks tend to involve multiple shapes (or shape attributes) and tastes that co-occur at different moments of our eating experiences. In the present research, we assessed whether food concepts involving more than one taste (e.g., ¿sweet and sour¿) would be associated with pairs of round and/or angular shapes. The participants matched pairs comprising angular and round shapes with ¿sweet and sour¿ food concepts more often than with other single taste and taste combination concepts, in a manner that is broadly consistent with studies involving one-to-one taste/shape matches. These results were observed both when the participants were presented with the shape pairs alone (Experiment 1) or along with a product's packaging (Experiment 2). We conclude by presenting possible explanations for the results obtained, as well as directions for future research.

The core of the paper consists of the treatment of two special decompositions for correspondence analysis of two-way ordered contingency tables: the bivariate moment decomposition and the hybrid decomposition, both using orthogonal polynomials rather than the commonly used singular vectors. To this end, we will detail and explain the basic characteristics of a particular set of orthogonal polynomials, called Emerson polynomials. It is shown that such polynomials, when used as bases for the row and/or column spaces, can enhance the interpretations via linear, quadratic and higher-order moments of the ordered categories. To aid such interpretations, we propose a new type of graphical display¿the polynomial biplot.

This paper presents a comprehensive statistical evaluation of the conductivity performance of PEDOT/PSS films generated at various spin coating speeds (1000 to 5000 rpm) by correspondence analysis (CA), regression and ANOVA models. Poly(3,4-ethylenedioxythiophene) (PEDOT) is a highly conducting insoluble polymer which, when doped with poly(styrene sulfonic acid) (PSS) in water, results in a dispersion with good film forming properties. Our interest in PEDOT/PSS is in its application as an interfacial layer and gate electrode in organic photovoltaic and organic field effect transistor devices, respectively. The generation of a homogeneous spun-cast layer of PEDOT/PSS over the entire substrate for these intended applications is a function of the spin-coating speed which, in turn, affects the thickness of the film and, consequently, its conductivity measured by the four point probe. From correspondence analysis, we find that only the 1000 rpm spin-coating speed is strongly associated with high thickness and high resistivity (i.e. low conductivity), although further analysis of speeds 2000 - 5000 rpm shows that the spin-coating speed is still associated with the thickness of the films, and subsequently, their conductivities, such that the thinnest and most conductive films result from the highest spin-coating speed (5000 rpm). We also observed the phenomenon of the horseshoe effect from the CA displays and discussed the reasons for not resorting to detrending solutions for dealing with its presence. From multiple and nonlinear regression models, spin-coating speeds 1000 and 2000 rpm were found to produce films of less precise conductivity values as reflected in the absolute value of mean of residuals of nonlinear regression models while more precise and optimal conductivity values are obtained when the films are generated between 3000 and 5000 rpm. The ANOVA models showed that the means of conductivity of films obtained at 1000 rpm is statistically significantly different to those from higher spin coating speeds and that there is no significant difference between the conductivity of films generated using spin coating speeds 2000 and 3000 rpm and between spin coating speeds 4000 and 5000 rpm. Further, the conductivity of films from spin coating speeds 4000-5000 rpm are higher and more precise than those obtained at 2000-3000 rpm. Based on the results of this study, we have standardised conductivity measurements of PEDOT/PSS films using the four-point probe by spin-coating at 4000-5000 rpm in order to obtain optimal and reproducible conductivity values.

Identification of differentially expressed genes (DEG) in transcriptomic analyses is one of the important tasks to find out significantly activated/deactivated pathways. Outliers and/or the missing values are commonly observed in microarray data; however, most available statistical methods did not deal with these issues and, therefore, their analytical results were frequently skewed and deteriorated. Here, we developed a novel technique robust against outliers and missing values: a dimension reduction procedure based on robust singular value decomposition (RSVD). The RSVD was evaluated by two numerical experiments: artificially prepared and nonsmall cell lung cancer data (gene expression data). Four conventional techniques, such as Student¿s t-test, SAM, Bayesian Robust Inference for Differential Gene Expression (BRIDGE) and Linear models for microarray data (Limma), were also performed. We evaluated the area under curve (AUC) form receiver operating characteristic curves of these five methods using two experiments with 50 different conditions. The AUC values of our methods showed significantly (p<0.05; Mann-Whitney test) higher than those of the other methods in both experiments. We believe our proposed technique is helpful for the identification of biologically meaningful genes that change in noisy microarray data.

This paper presents the R package CAvariants (Lombardo and Beh, 2017). The package performs six variants of correspondence analysis on a two-way contingency table. The main function that shares the same name as the package - CAvariants - allows the user to choose (via a series of input parameters) from six different correspondence analysis procedures. These include the classical approach to (symmetrical) correspondence analysis, singly ordered correspondence analysis, doubly ordered correspondence analysis, non symmetrical correspondence analysis, singly ordered non symmetrical correspondence analysis and doubly ordered non symmetrical correspondence analysis. The code provides the flexibility for constructing either a classical correspondence plot or a biplot graphical display. It also allows the user to consider other important features that allow to assess the reliability of the graphical representations, such as the inclusion of algebraically derived elliptical confidence regions. This paper provides R functions that elaborates more fully on the code presented in Beh and Lombardo (2014).

The analysis of the association between the two dichotomous variables of a 2 × 2 table arises as an important statistical issue in a number of diverse settings, such as in biomedical, medical, epidemiological, pharmaceutical or environmental research. When only the aggregate (or marginal) information is available, the analyst may determine the likely strength of the association between the variables. In this paper, we propose a new measure, called aggregate prediction index, that assesses the likely statistical significance of the association between the rows and columns of a 2 × 2 table where one variable is treated as a predictor variable and the other is treated as a response variable. Further insight into the predictor¿s potential strength can be visually obtained by performing an asymmetric version of correspondence analysis and considering a biplot display of the two variables - this issue shall also be explored in light of the new index.

Since its inception in the 1950s the odds ratio has become one of the most simple and popular measures available for analysing the association between two dichotomous variables. Since the direction and magnitude of the association can be captured in such a simple measure, its impact has been felt throughout much of scientific research, in particular in epidemiology and clinical trials. Despite this, its applicability for analysing aggregate data has rarely been considered. In this paper we shall express a new measure of association (the aggregate association index, or AAI), in terms of the classic odds ratio. The advantage of doing so is that we are able to explore the use of the odds ratio in a context for which it was not originally intended, and that is for the analysis of a 2×2 table where only the aggregate data is known. © 2013 Published by Elsevier B.V. All rights reserved.

Log-linear modeling is a popular statistical tool for analysing a contingency table. This presentation focuses on an alternative approach to modeling ordinal categorical data. The technique, based on orthogonal polynomials, provides a much simpler method of model fitting than the conventional approach of maximum likelihood estimation, as it does not require iterative calculations nor the fitting and re-fitting to search for the best model. Another advantage is that quadratic and higher order effects can readily be included, in contrast to conventional log-linear models which incorporate linear terms only. The focus of the discussion is the application of the new parameter estimation technique to multi-way contingency tables with at least one ordered variable. This will also be done by considering singly and doubly ordered two-way contingency tables. It will be shown by example that the resulting parameter estimates are numerically similar to corresponding maximum likelihood estimates for ordinal log-linear models. Copyright© 2004, Lawrence Erlbaum Associates, Inc.

An alternative approach to classical correspondence analysis was developed in [3] and involves decomposing the matrix of Pearson contingencies of a contingency table using orthogonal polynomials rather than via singular value decomposition. It is especially useful in analysing contingency tables which are of an ordinal nature. This short paper demonstrates that the confidence circles of Lebart, Morineau and Warwick (1984) for the classical approach can be applied to ordinal correspondence analysis. The advantage of the circles in analysing a contingency table is that the researcher can graphically identify the row and column categories that contribute or not to the hypothesis of independence. Copyright© 2001, Lawrence Erlbaum Associates, Inc.