Computational Statistics and Data Analysis 68 (2013) 52–65 Contents lists available at SciVerse ScienceDirect Computational Statistics and Data Analysis journal homepage: www.elsevier.com/locate/csda A reformulation of the aggregate association index using the odds ratio Eric J. Beh ∗ , Duy Tran, Irene L. Hudson School of Mathematical and Physical Sciences, University of Newcastle, Australia article info Article history: Received 29 August 2012 Received in revised form 6 June 2013 Accepted 6 June 2013 Available online 14 June 2013 Keywords: 2 × 2 contingency table Aggregate association index (AAI) Chi-squared statistic Fisher’s criminal twin data New Zealand voting data (1883–1919) abstract 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 vari- ables. 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. Crown Copyright © 2013 Published by Elsevier B.V. All rights reserved. 1. Introduction The odds ratio remains one of the most simple, influential and diversely used measures of association available to the analyst. Due to the widespread use of logistic regression, the odds ratio is widely used in many fields of medical and social science research. It is commonly used in survey research, in epidemiology (Rothman, 2002; Rothman et al., 2008) and to express the results of some clinical trials, such as in case-control studies (Miettinen, 1976). The odds ratio underpins the field of meta-analysis (Cheung et al., 2012). Meta analysis is a statistical method used to compare and combine effect sizes from a pool of relevant empirical studies. It is now a standard approach to synthesise research findings in many disciplines, including medical and healthcare research, and climate change research (Hudson, 2010) and increasingly in genome-wide studies (Nakaoka and Inoue, 2009; Kraft et al., 2009; Schurink et al., 2012) and drug discovery (Hudson et al., 2012). The odds ratio is often used as an alternative to the relative risk measure (Zhang and Yu, 1998; Montreuil et al., 2005; Schmidt and Kohlmann, 2008; Viera, 2008) in many applications where it is important to measure the strength, and direction, of the association between two dichotomous variables from a 2 × 2 table. Despite its popularity, using the odds ratio in cases where only the margins of the 2 × 2 table are available has rarely been considered. One exception to this is Plackett (1977) who showed that the margins do not provide enough information to make inferences about the cell values. There are a host of techniques that lie within the ecological inference literature that one may consider for inferring cell values, or some function of them; none of them, however, consider the odds ratio. For example, King’s (1997) groundbreaking parametric and non-parametric approaches may be considered. King (1997) also describes the ecological inference problem at length. Other strategies include Goodman’s (1953) ecological regression, Freedman et al.’s (1991) neighbourhood model, Chamber and Steel’s (2001) semi-parametric approach, Steel et al.’s (2004) homogeneous model and Wakefield’s (2004) Bayesian extension of this model. A comprehensive review of these ecological inference techniques, and their application to early New Zealand gender and voter turnout data was given by Hudson et al. (2010). Wakefield et al. (2011) further discuss ∗ Corresponding author. Tel.: +61 2 4921 5113. E-mail address: eric.beh@newcastle.edu.au (E.J. Beh). 0167-9473/$ – see front matter Crown Copyright © 2013 Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.csda.2013.06.009