A generalisation of the aggregate association index (AAI): incorporating a linear transformation of the cells of a 2 × 2 table

Abstract

The analysis of aggregate, or marginal, data for contingency tables is an increasingly important area of statistics, applied sciences and the social sciences. This is largely due to confidentiality issues arising from the imposition of government and corporate protection and data collection methods. The availability of only aggregate data makes it difficult to draw conclusions about the association between categorical variables at the individual level. For data analysts, this issue is of growing concern, especially for those dealing with the aggregate analysis of a single 2 × 2 table or stratified 2 × 2 tables and lies in the field of ecological inference. As an alternative to ecological inference techniques, one may consider the aggregate association index (AAI) to obtain valuable information about the magnitude and direction of the association between two categorical variables of a single 2 × 2 table or stratified 2 × 2 tables given only the marginal totals. Conventionally, the AAI has been examined by considering ${\mathrm{p}}_{11}$—the proportion of the sample that lies in the (1, 1)th cell of a given 2 × 2 table. However, the AAI can be expanded for other association indices. Therefore, a new generalisation of the original AAI is given here by reformulating and expanding the index so that it incorporates any linear transformation of ${\mathrm{p}}_{11}$. This study shall consider the consistency of the AAI under the transformation by examining four classic association indices, namely the independence ratio, Pearson’s ratio, standardised residual and adjusted standardised residual, although others may be incorporated into this general framework. We will show how these indices can be utilised to examine the strength and direction of association given only the marginal totals. Therefore, this work enhances our understanding of the AAI and establishes its links with common association indices.

1 Introduction

Data plays an important role in many disciplines from discovering new drugs to cure diseases such as cancer (e.g. Cancer Council Australia: cancercouncil.com.au, Cancer Research Institute: cancerresearch.org), for eliminating terrorism (e.g. National Security Agency—USA’s mass collection of phone data), studying consumer behaviour (e.g. Nielsen.com—a global information and measurement company with headquarters in New York, USA) or health/disease surveillance (e.g. Australian National Notifiable Diseases Surveillance System). One can also refer to Greenland (2003), Xun et al. (2010), Chau (2010), Valle and Clark (2013) for health/disease related studies. However, due to confidentiality issues and availability imposed by policies from governments and organisations, detailed data is usually not available for the public and data aggregation is often the only option. For more details about aggregate data, one can refer to, for example, King (1997).

This study will focus on the case of two dichotomous categorical variables that are cross-classified to form a 2 × 2 contingency table. However, we shall restrict our attention to when only aggregate, or marginal, totals are known. The issues arising from the analysis of aggregated categorical data (especially those from a 2 × 2 table) have attracted considerable attention in the statistical and allied disciplines. This attention was sparked by the comments of Fisher (1935, page 48) who said:

“Let us blot out the contents of the table, leaving only the marginal frequencies. If it be admitted that these marginal frequencies by themselves supply no information on the point at issue, namely, as to the proportionality of the frequencies in the body of the table we may recognize that we are concerned only with the relative probabilities of occurrence of the different ways in which the table can be filled in, subject to these marginal frequencies.”

The following decades saw considerable attention given to investigating Fisher’s concerns. For example, Yates (1984) agreed with Fisher’s comments, but stipulated that they are true “except in extreme cases and in repeated sampling”. Plackett (1977), Aitkin and Hinde (1984), and Barnard (1984) also confirmed Fisher’s observation. Goodman (1953, 1959) used accounting identify functions (ecological regressions) to express the relationships within multiple 2 × 2 tables and to estimate cell proportions given only marginal totals. However, the problems with his approach are twofold. Firstly, for G stratified 2 × 2 tables there are two unknown parameters (denote them as P_1g and P_2g for g = 1, 2,..., G) so that there are G equations with 2G unknowns leading to the indeterminacy problem. Secondly, to avoid this problem, Goodman’s technique set ${\text{P}}_{{1{\text{g}}}} = {\text{P}}_{1}$ and ${\text{P}}_{{2{\text{g}}}} = {\text{P}}_{2}$ which assumes that a homogeneous association exists for all G tables which is often not the case. Subsequent solutions that fall within the suite of Ecological Inference (EI) techniques, focus their attention on estimating P_1g and P_2g or some function of them (including the odds ratio) to analyse the association for multiple tables. The popularity and dissemination of methods and computational platforms for the analysis of aggregated categorical data increased significantly after King (1997) introduced his suite of EI techniques; these techniques are centred on P_1g and P_2g being modelled by a truncated bivariate normal distribution. Of particular note are the software packages EI (version 1.3-3, 2016), and EzI (King 2004; Benoit and King 2003), which perform statistical procedures, diagnostics and produce graphics for ecological inference. In R, one can also refer to the packages ecoreg (Jackson 2006), Zelig (Imai et al. 2008, 2009), eco (Imai et al. 2011), eiPack (Lau et al. 2007), and eiCompare (Collingwood et al. 2016). More recently, Knudson et al. (2021) discussed the PyEI Python package.

King (2004) introduced EI strategies that include using prior information to aid ecological inference and Markov Chain Monte Carlo methods to extend King’s ecological inference approach. Steel et al. (2004) introduced the homogeneous model, which overcame most of the assumptions in the linear regression approach proposed earlier by Goodman (1953) and addresses various assumptions of other EI techniques. Their solution to resolving the EI problem is to assume, for the g’th 2 × 2 table, that its (1, 1)’th cell is a binomial random variable with parameters being the marginal total of its first row and $\uppi _{{1{\text{g}}}} = {\text{P}}\left( {{\text{Column }}1|{\text{ Row }}1} \right)$ while the (2, 1)’th cell is also a binomial random variable where its parmeters are the marginal total of its second row and $\uppi _{{2{\text{g}}}} = {\text{P}}\left( {{\text{Column }}2|{\text{ Row }}1} \right)$. This alone is not enough to solve the EI problem so Steel et al. (2004) assume constancy of the second parameters so that $\uppi _{{1{\text{g}}}} =\uppi _{1}$ and $\uppi _{{2{\text{g}}}} =\uppi _{1}$. Current EI techniques are mainly applied to multiple tables and still suffer from there being major shortfalls in the assumptions required for their implementation. Hudson et al. (2010) compared those EI techniques just described (and more) in terms of the goodness of fit to the known true values. They concluded that “all EI methods make assumptions about the data to compensate for the loss of information due to aggregation” (p. 198). See also Hudson et al. (2005) for preliminary work carried out on their study. For more details about the EI framework, one can also refer to Chambers and Steel (2001), Salway and Wakefield (2004), King et al. (2004), Wakefield (2004), Glynn and Wakefield (2010) and Xun et al. (2010).

To overcome difficulties inherent in the EI assumptions, Beh (2008) proposed the aggregate association index (AAI). The novelty of this index is that, unlike EI, it assesses the statistical significance of the association between two dichotomous variables when only the marginal totals are known. The AAI is applicable to stratified 2 × 2 tables (Beh et al. 2014; Tran et al. 2018) and, unlike all EI techniques, to a single table (Tran et al. 2012b). The core notion of the AAI largely depends on a conditional probability P₁ which can be expressed as a simple linear transformation of p₁₁—the proportion of individuals/units classified into the (1, 1)th cell of a 2 × 2 table. While the AAI is sufficient in itself to determine the existence of an association and can be linked with the odds ratio (Beh et al. 2013), it still lacks formal links with other association indices for a 2 × 2 table—such as the independence ratio (Goodman 1996), Pearson’s ratio (Mirkin 2001), the standardised residual, and the adjusted standardised residual (Agresti 2013). These classic indices can be used to determine the strength of the association and are all expressible as a linear transformation of p₁₁. It must be pointed out that the AAI approaches the analysis of aggregate data for 2 × 2 tables in a very different way to EI. We therefore do not view AAI as being a strict member of the family of EI techniques but rather a mechanism for better understanding the association structure between two dichotomous variables where limited information is available.

As a result, the main objective of this study is to broaden our understanding of the AAI for a single 2 × 2 table through exploring its connection with the above classic association indices. We will show how these indices can still be utilised to examine the strength and direction of association given only the marginal totals. The paper is organised as follows: Sect. 2 defines the notation to be used in this paper and the AAI proposed by Beh (2008). Section 2 also introduces Fisher’s criminal twin data and the nightmare tendency data discussed by Steyn (2002) and Larsen and Marx (2012) which will be used as case studies. Section 3 develops the general linear transformation of p₁₁ and its mathematical link with the AAI. Section 4 focuses on the linear transformation of the independence ratio, Pearson’s ratio, standardised residual, and adjusted standardised residual using the AAI. Section 5 re-examines Fisher’s criminal twin case study to illustrate the results for a 2 × 2 table in terms of a full-data analysis and an aggregate-data analytic approach. Section 6 performs this analysis on the nightmare tendency data. Finally, Sect. 7 gives a discussion and highlights future research directions for the generalisation of the AAI.

2 The aggregate association index (AAI)

2.1 Case studies

We consider two case studies in this paper.

The first involves the following 2 × 2 table that was considered by Fisher (1935, p. 48) in a study carried out on criminal twins of the same sex. The 30 individuals in Table 1 were classified by twin type (Monozygotic, Dizygotic) and conviction status (Convicted, Not convicted). For example, of the 13 sets of monozygotic twins, 10 of them include a twin that is convicted of a crime, while of the 17 dizygotic sets of twins, only 2 of them have a twin that is convicted of a crime. The original source of this data can be found in Lange (1929) (in German) which was then translated into English by the author in 1931 (Lange 1931).

Table 1 Conviction of same-sex twins of criminals

With the joint cell values treated as being known, the Pearson chi-squared test of independence of Table 1 gives a test statistic of 13.032 with an associated p value of 0.0003, which shows a statistically significant association between the types of criminal twin and whether their sibling has been convicted of a crime. One may also incorporate Yates’ continuity correction to deal with the small cell frequencies and arrive at a different test statistic but the same conclusion.

Our second case study is based on data from Hersen (1971) who was interested in examining the link between the occurrence of nightmare dreams and a variety of personality characteristics from a sample of 352 hospitalised patients; see Table 2 which cross-classifies the gender of the patient and their tendency to have a nightmare. Table 2 is based on the 2 × 3 contingency table of Hersen (1971, Table 1) where his “Frequent” and “Sometimes” categories are aggregated to form the “Often” category. See also Steyn Jr (2002, Table 8) and Larsen and Marx (2012, Table 9.4.1) who studied the association between the two variables of Table 2.

Table 2 Nightmare tendency by gender

For Table 2, Pearson’s chi-squared statistic is 0.387 (without applying Yates’ continuity correction) and, with a p value of 0.534, shows that a statistically significant association does not exist between gender and nightmare tendency. However, as we shall show in Sect. 6, given only the marginal totals, such a conclusion is not as evident.

Therefore, the main objective for both data sets is to investigate any possible association between the two variables given only the marginal totals and to then examine how this association can be assessed. The following sections of this paper will thus focus on utilising marginal totals by considering the aggregate association index (AAI) and its extensions.

2.2 The AAI

Consider a single two-way contingency table where both variables are dichotomous in nature. Suppose that n individuals/units are classified into this table such that the number of classifications made in the (1, 1)th cell is denoted by ${\mathrm{n}}_{11}$. Let the i'th row marginal frequency be denoted by ${\mathrm{n}}_{\mathrm{i}\bullet }$, for i = 1, 2 and the jth column marginal frequency by ${\mathrm{n}}_{\bullet \mathrm{j}}$, for j = 1, 2. Also, denote the ith row and jth column marginal proportion by ${\mathrm{p}}_{\mathrm{i}\bullet }={\mathrm{n}}_{\mathrm{i}{\bullet }}/\mathrm{n}$ and ${\mathrm{p}}_{{\bullet }\mathrm{j}}={\mathrm{n}}_{{\bullet }\mathrm{j}}/\mathrm{n}$ respectively. The (i, j)th cell proportion is denoted by ${\mathrm{p}}_{\mathrm{ij}}={\mathrm{n}}_{\mathrm{ij}}/\mathrm{n}$. Table 3 provides a description of the notation used in this paper.

Table 3 Notation of a general 2 × 2 contingency table

Traditionally, Pearson’s (1904, eq (xxviii)) chi-squared test statistic can be calculated as

$${\mathrm{X}}^{2}=\mathrm{n}\frac{{\left({\mathrm{n}}_{11}{\mathrm{n}}_{22}-{\mathrm{n}}_{12}{\mathrm{n}}_{21}\right)}^{2}}{{\mathrm{n}}_{1{{ \bullet }}}{\mathrm{n}}_{2{ \bullet }}{\mathrm{n}}_{{ \bullet }1}{\mathrm{n}}_{{ \bullet }2}}.$$

(1)

Define ${\mathrm{P}}_{1}={\mathrm{n}}_{11}/{\mathrm{n}}_{1{\bullet }}$ to be the conditional probability of an individual/unit being classified into ‘Column 1’ given that they are classified in ‘Row 1’. Then the chi-squared test statistic (1) can be formulated in terms of ${\mathrm{P}}_{1}$ and the marginal totals by:

$${\mathrm{X}}^{2}\left({\mathrm{P}}_{1}|{\mathrm{p}}_{1{\bullet }},\hspace{0.33em}{\mathrm{p}}_{{\bullet }1}\right)=\mathrm{n}{\left(\frac{{\mathrm{P}}_{1}-{\mathrm{p}}_{{\bullet }1}}{{\mathrm{p}}_{2{\bullet }}}\right)}^{2}\left(\frac{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}}{{\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{{\bullet }2}}\right)\hspace{0.33em}.$$

(2)

When the cell values of Table 2 are unknown, it is not possible to calculate ${\mathrm{P}}_{1}$ and so (2) cannot be determined. However, the extremes of the permissible values of the (1,1)th cell frequency are well understood (Fréchet 1951; Duncan and Davis 1953) to lie within the interval

$${\mathrm{L}}_{{\mathrm{n}}_{11}}=\mathrm{max}\left(0,\hspace{0.33em}{\mathrm{n}}_{{\bullet }1}-{\mathrm{n}}_{2{\bullet }}\right)\le {\mathrm{n}}_{11}\le \mathrm{min}\left({\mathrm{n}}_{{\bullet }1},\hspace{0.33em}{\mathrm{n}}_{1{\bullet }}\right)={\mathrm{U}}_{{\mathrm{n}}_{11}}.$$

(3)

Therefore (3) leads to:

$${\mathrm{L}}_{{\mathrm{p}}_{11}}=\mathrm{max}\left(0,\hspace{0.33em}{\mathrm{p}}_{{\bullet }1}-{\mathrm{p}}_{2{\bullet }}\right)\le {\mathrm{p}}_{11}\le \mathrm{min}\left({\mathrm{p}}_{{\bullet }1},\hspace{0.33em}{\mathrm{p}}_{1{\bullet }}\right)={\mathrm{U}}_{{\mathrm{p}}_{11}}$$

(4)

and, from (4), the bounds for ${\mathrm{P}}_{1}$ are:

$${\mathrm{L}}_{{\mathrm{P}}_{1}}=\mathrm{max}\left(0,\hspace{0.33em}\frac{{\mathrm{p}}_{{\bullet }1}-{\mathrm{p}}_{2{\bullet }}}{{\mathrm{p}}_{1{\bullet }}}\right)\le {\mathrm{P}}_{1}\le \mathrm{min}\left(\frac{{\mathrm{p}}_{{\bullet }1}}{{\mathrm{p}}_{1{\bullet }}},\hspace{0.33em}1\right)={\mathrm{U}}_{{\mathrm{P}}_{1}}.$$

(5)

In addition, Beh (2010) showed that when only the marginal totals are available, and a test of the association is made at ${\alpha }$ level of significance, the bounds of ${\mathrm{P}}_{1}$ are

$$ {\text{L}}_{{\upalpha {\text{P}}_{1} }} = \max \left( {0,\;{\text{p}}_{ \bullet 1} - {\text{p}}_{2 \bullet } \sqrt {\frac{{\upchi _{\upalpha }^{2} }}{{\text{n}}}\frac{{{\text{p}}_{ \bullet 1} {\text{p}}_{ \bullet 2} }}{{{\text{p}}_{1 \bullet } {\text{p}}_{2 \bullet } }}} } \right) \le {\text{P}}_{1} \le \min \left( {1,\;{\text{p}}_{ \bullet 1} + {\text{p}}_{2 \bullet } \sqrt {\frac{{\upchi _{\upalpha }^{2} }}{{\text{n}}}\frac{{{\text{p}}_{ \bullet 1} {\text{p}}_{ \bullet 2} }}{{{\text{p}}_{1 \bullet } {\text{p}}_{2 \bullet } }}} } \right) = {\text{U}}_{{\upalpha {\text{P}}_{1} }} , $$

(6)

where $\upchi _{\upalpha }^{2}$ is the 1 − α percentile of the chi-squared distribution with 1 degree of  freedom.

With the bounds (5) and (6) of ${\text{P}}_{1}$, the ${\text{X}}^{2} \left( {{\text{P}}_{1} |{\text{p}}_{1 \bullet } ,\;{\text{p}}_{ \bullet 1} } \right)$ can be assessed against the critical value $\upchi _{\upalpha }^{2}$ to draw a conclusion about the association of a 2 × 2 table given only the marginal totals. Namely, rejecting the null hypothesis of complete independence (that is, ${\text{p}}_{11} = {\text{p}}_{1 \bullet } {\text{p}}_{ \bullet 1}$) between the two dichotomous variables requires that ${\text{X}}^{2} \left( {{\text{P}}_{1} |{\text{p}}_{1 \bullet } ,\;{\text{p}}_{ \bullet 1} } \right)$ be greater than the ${\upchi }_{{\upalpha }}^{2}$. Figure 1 illustrates how information about the association between the variables of a 2 × 2 table can be visually assessed using only the marginal totals. The shaded areas under the curve (referred to as the AAI curve) represent where a statistically significant association exists since ${\text{X}}^{2} \left( {{\text{P}}_{1} |{\text{p}}_{1 \bullet } ,\;{\text{p}}_{ \bullet 1} } \right)$ is greater than the $\upchi _{\upalpha }^{2}$ in these regions. If ${\text{L}}_{{\upalpha {\text{P}}_{1} }} < {\text{P}}_{1} < {\text{U}}_{{\upalpha {\text{P}}_{1} }}$ then, based only on the marginal totals, there is evidence that the row and column variables are independent at the α level of significance. However, if ${\text{L}}_{{{\text{P}}_{1} }} < {\text{P}}_{1} < {\text{L}}_{{\upalpha {\text{P}}_{1} }}$ or ${\text{U}}_{{\upalpha {\text{P}}_{1} }} < {\text{P}}_{1} < {\text{U}}_{{{\text{P}}_{1} }}$ then there is sufficient statistical evidence to suggest that the variables are significantly associated.

Fig. 1

A graphical interpretation of the aggregate association index (AAI)

As a result, Beh (2008) proposed the following index

$$ {\text{A}}_{\upalpha } \left( {{\text{P}}_{1} } \right) = 100\left( {1 - \frac{{\upchi _{\upalpha }^{2} \left[ {\left( {{\text{L}}_{{\upalpha {\text{P}}_{1} }} - {\text{L}}_{{{\text{P}}_{1} }} } \right) + \left( {{\text{U}}_{{{\text{P}}_{1} }} - {\text{U}}_{{\upalpha {\text{P}}_{1} }} } \right)} \right] + \mathop \smallint \nolimits_{{{\text{L}}_{{\upalpha {\text{P}}_{1} }} }}^{{{\text{U}}_{{\upalpha {\text{P}}_{1} }} }} {\text{X}}^{2} \left( {{\text{P}}_{1} |{\text{p}}_{1 \bullet } ,\;{\text{p}}_{ \bullet 1} } \right)\;{\text{dP}}_{1} }}{{\mathop \smallint \nolimits_{{{\text{L}}_{{{\text{P}}_{1} }} }}^{{{\text{U}}_{{{\text{P}}_{1} }} }} {\text{X}}^{2} \left( {{\text{P}}_{1} |{\text{p}}_{1 \bullet } ,\;{\text{p}}_{ \bullet 1} } \right)\;{\text{dP}}_{1} }}} \right). $$

(7)

Beh (2010) then provided an alternative, and equivalent, expression of (7):

$$ {\text{A}}_{\upalpha } \left( {{\text{P}}_{1} } \right) = 100\left( {1 - \frac{{\upchi _{\upalpha }^{2} \left[ {\left( {{\text{L}}_{{\upalpha {\text{P}}_{1} }} - {\text{L}}_{{{\text{P}}_{1} }} } \right) + \left( {{\text{U}}_{{{\text{P}}_{1} }} - {\text{U}}_{{{\alpha \mathrm{P}}_{1} }} } \right)} \right]}}{{{\text{k}}_{{{\text{P}}_{1} }} {\text{n}}\left[ {\left( {{\text{U}}_{{{\text{P}}_{1} }} - {\text{p}}_{ \bullet 1} } \right)^{3} - \left( {{\text{L}}_{{{\text{P}}_{1} }} - {\text{p}}_{ \bullet 1} } \right)^{3} } \right]}} - \frac{{\left( {{\text{U}}_{{\upalpha {\text{P}}_{1} }} - {\text{p}}_{ \bullet 1} } \right)^{3} - \left( {{\text{L}}_{{\upalpha {\text{P}}_{1} }} - {\text{p}}_{ \bullet 1} } \right)^{3} }}{{\left( {{\text{U}}_{{{\text{P}}_{1} }} - {\text{p}}_{ \bullet 1} } \right)^{3} - \left( {{\text{L}}_{{{\text{P}}_{1} }} - {\text{p}}_{ \bullet 1} } \right)^{3} }}} \right), $$

(8)

where

$$ {\text{k}}_{{{\text{P}}_{1} }} = \frac{1}{{3{\text{p}}_{2 \bullet }^{2} }}\left( {\frac{{{\text{p}}_{1 \bullet } {\text{p}}_{2 \bullet } }}{{{\text{p}}_{ \bullet 1} {\text{p}}_{ \bullet 2} }}} \right). $$

Here, ${\text{A}}_{\upalpha } \left( {{\text{P}}_{1} } \right)$ is referred to as the aggregate association index (AAI) and is formulated in terms of ${\text{P}}_{1}$ since the ${\text{X}}^{2} \left( {{\text{P}}_{1} |{\text{p}}_{1 \bullet } ,\;{\text{p}}_{ \bullet 1} } \right)$ is used to define the AAI curve in Fig. 1. This index quantifies, for a given level of significance α, and the marginal totals of a 2 × 2 table, how likely it is that a statistically significant association between two dichotomous variables exists. We can see then that the AAI is the ratio of the area defined by the statistically significant association region with respect to the total area under the AAI curve. Therefore, a value of ${\text{A}}_{\upalpha } \left( {{\text{P}}_{1} } \right)$ close to zero indicates that, given only the marginal totals, there is not enough evidence to suggest that a statistically significant association exists between the two variables at α. On the other hand, an AAI value close to 100 indicates that a statistically significant association between the variables is highly likely to exist.

3 The role of the linear transformation of p₁₁ in the AAI

3.1 The linear transformation of p₁₁

In Sect. 2 we discussed the original AAI of Beh (2008, 2010) in terms of ${\mathrm{P}}_{1}$ which is just a simple linear transformation of the ${\mathrm{p}}_{11}$. In addition, there are many other simple linear transformations of ${\mathrm{p}}_{11}$ that yield other popular association indices between two dichotomous variables. These include, but are not limited to, the independence ratio, Pearson’s ratio, the standardised residual, and the adjusted standardised residual; refer to Sect. 4 for more details.

Here we define the linear transformation of ${\mathrm{p}}_{11}$ as

$$\ell\left({\mathrm{p}}_{11}\right)={\mathrm{ap}}_{11}+\mathrm{b}.$$

(9)

The choice of index that is used to reflect the association between the row and column variables of a 2 × 2 contingency table can then be made by choosing the appropriate value of $\mathrm{a}$ and $\mathrm{b}$. We shall now turn our attention to determining the relationship between the linear transformation of ${\mathrm{p}}_{11}$ and the AAI.

3.2 The relationship between the linear transformation of p₁₁ and the AAI

3.2.1 The Bounds

From (3), the bounds of $\ell\left({\mathrm{p}}_{11}\right)$ are derived as follows:

$${\mathrm{L}}_{\ell}=\mathrm{a{\cdot}max}\left(0,\hspace{0.33em}{\mathrm{p}}_{{\bullet }1}-{\mathrm{p}}_{2{\bullet }}\right)+\mathrm{b}\le \ell\left({\mathrm{p}}_{11}\right)\le \mathrm{a{\cdot}min}\left({\mathrm{p}}_{{\bullet }1},\hspace{0.33em}{\mathrm{p}}_{1{\bullet }}\right)+\mathrm{b}={\mathrm{U}}_{\ell},$$

(10)

while the $100\left(1-{\alpha }\right)$% bounds of $\ell\left({\mathrm{p}}_{11}\right)$ are

$${\mathrm{L}}_{{\alpha }\ell}=\mathrm{max}\left(\mathrm{b},\hspace{0.33em}{\mathrm{ap}}_{1{\bullet }}\left({\mathrm{p}}_{{\bullet }1}-{\mathrm{p}}_{2{\bullet }}\sqrt{\frac{{\upchi }_{{\alpha }}^{2}}{\mathrm{n}}\frac{{\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{{\bullet }2}}{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}}}\right)+\mathrm{b}\right)\le \ell\left({\mathrm{p}}_{11}\right)\le \mathrm{min}\left({\mathrm{ap}}_{1{\bullet }}+\mathrm{b},\hspace{0.33em}{\mathrm{ap}}_{1{\bullet }}\left({\mathrm{p}}_{{\bullet }1}+{\mathrm{p}}_{2{\bullet }}\sqrt{\frac{{\upchi }_{{\alpha }}^{2}}{\mathrm{n}}\frac{{\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{{\bullet }2}}{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}}}\right)+\mathrm{b}\right)={\mathrm{U}}_{{\alpha }\ell} .$$

(11)

Following on from the discussions made concerning (5) and (6), we can conclude that when ${\mathrm{L}}_{{\alpha }\ell}<\ell\left({\mathrm{p}}_{11}\right)<{\mathrm{U}}_{{\alpha }\ell}$ then there is evidence that the row and column variables are independent at the ${\alpha }$ level of significance. However, if ${\mathrm{L}}_{{\alpha }\ell}<\ell\left({\mathrm{p}}_{11}\right)<{\mathrm{L}}_{\ell}$ or ${\mathrm{U}}_{{\alpha }\ell}<\ell\left({\mathrm{p}}_{11}\right)<{\mathrm{U}}_{\ell}$ then there is sufficient evidence to suggest that the variables in a 2 × 2 table are associated given only the marginal totals and an ${\alpha }$ level of significance.

We may also express the chi-squared statistic (2) in terms of $\ell\left({\mathrm{p}}_{11}\right)$. Doing so gives

$${\mathrm{X}}^{2}\left(\ell\left({\mathrm{p}}_{11}\right)|{\mathrm{p}}_{1{\bullet }},\hspace{0.33em}{\mathrm{p}}_{{\bullet }1}\right)=\mathrm{n}{\left(\frac{\ell\left({\mathrm{p}}_{11}\right)-\upmu \left({\mathrm{p}}_{11}\right)}{{\mathrm{ap}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}}\right)}^{2}\left(\frac{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}}{{\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{{\bullet }2}}\right)\hspace{0.33em},$$

(12)

where

$$\upmu \left({\mathrm{p}}_{11}\right)=\mathrm{E}\left(\ell\left({\mathrm{p}}_{11}\right)\right)=\mathrm{aE}\left({\mathrm{p}}_{11}\right)+\mathrm{b}={\mathrm{ap}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}+\mathrm{b}$$

(13)

is a linear function of the expectation of the (1, 1)th cell frequency under the null hypothesis of complete independence.

3.2.2 The curvature coefficient

Note that, irrespective of the choice of $\mathrm{a}\ne 0$ and $\mathrm{b}$, the chi-squared statistic is a quadratic function in terms of $\ell\left({\mathrm{p}}_{11}\right)$. Therefore, the turning point of the statistic occurs when

$$\frac{\partial {\mathrm{X}}^{2}\left(\ell\left({\mathrm{p}}_{11}\right)\right)}{\partial \ell\left({\mathrm{p}}_{11}\right)}=2\mathrm{n}\left(\frac{\ell\left({\mathrm{p}}_{11}\right)-\left({\mathrm{ap}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}+\mathrm{b}\right)}{{\left({\mathrm{ap}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}\right)}^{2}}\right)\left(\frac{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}}{{\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{{\bullet }2}}\right)\hspace{0.33em}=0$$

and is obtained when $\ell\left({\mathrm{p}}_{11}\right)={\mathrm{ap}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}+\mathrm{b}$. Similarly, the curvature of (12) is

$$\upgamma \left(\ell\left({\mathrm{p}}_{11}\right)\right)=\frac{{\partial }^{2}{\mathrm{X}}^{2}\left(\ell\left({\mathrm{p}}_{11}\right)\right)}{\partial {\ell}^{2}\left({\mathrm{p}}_{11}\right)}=\frac{2\mathrm{n}}{{\mathrm{a}}^{2}{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{{\bullet }2}}>0$$

(14)

so that (12) always has positive curvature, since $\mathrm{n}>0$ and ${\mathrm{a}}^{2}>0$; while this is apparent by observing that each term of (12) is positive, quantifying the curvature is useful for comparing AAI curves under different transformations. Therefore, the minimum value that the chi-squared statistic can take will coincide with $\ell\left({\mathrm{p}}_{11}\right)-\upmu \left({\mathrm{p}}_{11}\right)=0$ and the two local maximum values of the chi-squared statistic will lie at the bounds of $\ell\left({\mathrm{p}}_{11}\right)$ defined by (10). Thus, the coordinate of the turning point of the AAI curve is $\left(\upmu \left({\mathrm{p}}_{11}\right), 0\right)$. Furthermore, the curvature of ${\mathrm{X}}^{2}\left(\ell\left({\mathrm{p}}_{11}\right)|{\mathrm{p}}_{1{\bullet }},\hspace{0.33em}{\mathrm{p}}_{{\bullet }1}\right)$ is dependent on the chosen value of $\mathrm{a}$ in $\ell\left({\mathrm{p}}_{11}\right)$ and is independent of $\mathrm{b}$. Thus, we refer to “$\mathrm{a}$” as the curvature coefficient of the AAI curve. For each transformation, we shall be comparing the curvature of the AAI curve relative to the curve produced for ${\mathrm{p}}_{11}$. This will be quantified by

$$\Delta \left(\ell\left({\mathrm{p}}_{11}\right)\right)=\frac{\upgamma \left(\ell\left({\mathrm{p}}_{11}\right)\right)}{\upgamma \left({\mathrm{p}}_{11}\right)}$$

and termed the curvature index of the AAI curve for $\ell\left({\mathrm{p}}_{11}\right)$; here $\upgamma \left({\mathrm{p}}_{11}\right)$ is the curvature of the AAI curve when no linear transformation is applied to ${\mathrm{p}}_{11}$. This quantity $\upgamma \left({\mathrm{p}}_{11}\right)$ is defined by (21) below. For, example, a comparison of the curvature of the AAI curve for ${\mathrm{P}}_{1}$—see (7), or equivalently (8)—relative to that of ${\mathrm{p}}_{11}$ can be made by considering (2). In doing so, we obtain

$$\upgamma \left({\mathrm{P}}_{1}\right)=\frac{{\partial }^{2}{\mathrm{X}}^{2}\left({\mathrm{P}}_{1}\right)}{\partial {\mathrm{P}}_{1}^{2}}=\frac{2\mathrm{n}}{{\mathrm{p}}_{2{\bullet }}^{2}}\left(\frac{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}}{{\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{{\bullet }2}}\right)={\mathrm{p}}_{1{\bullet }}\upgamma \left({\mathrm{p}}_{11}\right)$$

so that

$$\Delta \left({\mathrm{P}}_{1}\right)=\frac{\upgamma \left({\mathrm{P}}_{1}\right)}{\upgamma \left({\mathrm{p}}_{11}\right)}={\mathrm{p}}_{1{\bullet }} .$$

Therefore, the curvature index of the AAI curve when expressed in terms of ${\mathrm{P}}_{1}$ is always less than the curvature of the AAI curve for ${\mathrm{p}}_{11}$. The exception to this is in the trivial case when ${\mathrm{p}}_{1{\bullet }}=1$ which we shall not be considering in our discussion.

3.2.3 The generalised AAI

By extending the definition of the AAI when it is expressed in terms of ${\mathrm{P}}_{1}$, one may consider the area that lies under the AAI curve defined by ${\mathrm{X}}^{2}\left(\ell\left({\mathrm{p}}_{11}\right)\right)$ but lies above the critical value ${\upchi }_{{\alpha }}^{2}$; such an area reflects how likely a statistically significant association exists between the variables. Thus, a new generalisation of the AAI may be considered for any simple linear transformation of ${\mathrm{p}}_{11}$. In this case, the generalised AAI can be expressed by

$${\mathrm{A}}_{{\alpha }}\left(\ell\left({\mathrm{p}}_{11}\right)\right)=100\left(1-\frac{{\upchi }_{{\alpha }}^{2}\left[\left({\mathrm{L}}_{{\alpha }\ell}-{\mathrm{L}}_{\ell}\right)+\left({\mathrm{U}}_{\ell}-{\mathrm{U}}_{{\alpha }\ell}\right)\right]+{\int }_{{\mathrm{L}}_{{\alpha }\ell}}^{{\mathrm{U}}_{{\alpha }\ell}}{\mathrm{X}}^{2}\left(\ell\left({\mathrm{p}}_{11}\right)|{\mathrm{p}}_{1{\bullet }},\hspace{0.33em}{\mathrm{p}}_{{\bullet }1}\right)\hspace{0.33em}\mathrm{d}\ell\left({\mathrm{p}}_{11}\right)}{{\int }_{{\mathrm{L}}_{\ell}}^{{\mathrm{U}}_{\ell}}{\mathrm{X}}^{2}\left(\ell\left({\mathrm{p}}_{11}\right)|{\mathrm{p}}_{1{\bullet }},\hspace{0.33em}{\mathrm{p}}_{{\bullet }1}\right)\hspace{0.33em}\mathrm{d}\ell\left({\mathrm{p}}_{11}\right)}\right).$$

(15)

Equation (15) may be simplified by evaluating the definite integrals. Doing so for x < y gives

$$\begin{aligned}&\int_{\mathrm{x}}^{\mathrm{y}}{\mathrm{X}}^{2}\left(\ell\left({\mathrm{p}}_{11}\right)|{\mathrm{p}}_{1{\bullet}},\hspace{0.33em}{\mathrm{p}}_{{\bullet}1}\right)\mathrm{d}\ell\left({\mathrm{p}}_{11}\right)\\&\quad =\frac{\mathrm{n}}{{\left({\mathrm{ap}}_{1{\bullet}}{\mathrm{p}}_{2{\bullet}}\right)}^{2}}\left(\frac{{\mathrm{p}}_{1{\bullet}}{\mathrm{p}}_{2{\bullet }}}{{\mathrm{p}}_{{\bullet}1}{\mathrm{p}}_{{\bullet }2}}\right)\int_{\mathrm{x}}^{\mathrm{y}}{\left(\ell\left({\mathrm{p}}_{11}\right)-\upmu \left({\mathrm{p}}_{11}\right)\right)}^{2}\mathrm{d}\ell\left({\mathrm{p}}_{11}\right)\\&\quad =\frac{\mathrm{n}}{3{\left({\mathrm{ap}}_{1{\bullet}}{\mathrm{p}}_{2{\bullet}}\right)}^{2}}\left(\frac{{\mathrm{p}}_{1{\bullet}}{\mathrm{p}}_{2{\bullet }}}{{\mathrm{p}}_{{\bullet}1}{\mathrm{p}}_{{\bullet }2}}\right)\left[{\left(\mathrm{y}-\upmu\left({\mathrm{p}}_{11}\right)\right)}^{3}-{\left(\mathrm{x}-\upmu\left({\mathrm{p}}_{11}\right)\right)}^{3}\right].\end{aligned}$$

Therefore, the generalised AAI_, may be expressed by

$$\begin{aligned}{\mathrm{A}}_{{\alpha }}\left(\ell\left({\mathrm{p}}_{11}\right)\right)&=100\left(1-\frac{{\upchi }_{{\alpha }}^{2}\left[\left({\mathrm{L}}_{{\alpha }\ell}-{\mathrm{L}}_{\ell}\right)+\left({\mathrm{U}}_{\ell}-{\mathrm{U}}_{{\alpha }\ell}\right)\right]}{{\mathrm{k}}_{\ell}\mathrm{n}\left[{\left({\mathrm{U}}_{\ell}-\upmu \left({\mathrm{p}}_{11}\right)\right)}^{3}-{\left({\mathrm{L}}_{\ell}-\upmu \left({\mathrm{p}}_{11}\right)\right)}^{3}\right]}\right. \\ & \quad\qquad \left. -\frac{{\left({\mathrm{U}}_{{\alpha }\ell}-\upmu \left({\mathrm{p}}_{11}\right)\right)}^{3}-{\left({\mathrm{L}}_{{\alpha }\ell}-\upmu \left({\mathrm{p}}_{11}\right)\right)}^{3}}{{\left({\mathrm{U}}_{\ell}-\upmu \left({\mathrm{p}}_{11}\right)\right)}^{3}-{\left({\mathrm{L}}_{\ell}-\upmu \left({\mathrm{p}}_{11}\right)\right)}^{3}}\right),\end{aligned}$$

(16)

where

$${\mathrm{k}}_{\ell}=\frac{1}{3{\mathrm{a}}^{2}\left({\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{{\bullet }2}\right)}.$$

(17)

From (15) and (16), it can be seen that the generalised AAI can be expressed in terms of any linear function of ${\mathrm{p}}_{11}$ and its interpretation remains the same as the original AAI. In addition, the value of ${\mathrm{A}}_{{\alpha }}\left(\ell\left({\mathrm{p}}_{11}\right)\right)$ remains constant regardless of which linear transformation is applied to ${\mathrm{p}}_{11}$; a proof of ${\mathrm{A}}_{{\alpha }}\left(\ell\left({\mathrm{p}}_{11}\right)\right)= {\mathrm{A}}_{{\alpha }}\left({\mathrm{P}}_{1}\right)$ is given in the Online Appendix. We shall refer to this feature as the homogeneity characteristic of the AAI. We now turn our attention to examining special cases of this generalised form of the AAI and its features.

4 Association indices in terms of the linear transformation of p₁₁

4.1 Classic association indices

In the case where the cell frequencies of a 2 × 2 table are known, there are many indices that can be considered for determing the association between the dichotomous variables. One may consider, for example, the wealth of measures discussed in Warrens (2008) and Abdesselam (2020a, b). Many of these measures can be included in the framework we describe below but, in addition to ${\mathrm{P}}_{1}$, we shall also focus our attention on the following:

• (a) The cell proportion: ${\mathrm{p}}_{11}$

• (b) Pearson’s contingency: ${\mathrm{P}}_{\mathrm{C}}={\mathrm{p}}_{11}-{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}$

• (c) Independence ratio: $\mathrm{B}={\mathrm{p}}_{11}/\left({\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}\right).$

• (d) Pearson’s ratio: $\mathrm{C}=\left({\mathrm{p}}_{11}-{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}\right)/\left({\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}\right).$

• (e) Standardised residual: $\mathrm{Z}=\left({\mathrm{p}}_{11}-{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}\right)/\sqrt{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}}$

• (f) Adjusted standardised residual: ${\mathrm{Z}}_{\mathrm{adj}}=\left({\mathrm{p}}_{11}-{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}\right)/\sqrt{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }2}}.$

From a historical perspective, Pearson (1904, page 5) also stated that:

“The total deviation of the whole classification system from independent probability must be some function of the [${\mathrm{p}}_{\mathrm{ij}}-{\mathrm{p}}_{\mathrm{i}{\bullet }}{\mathrm{p}}_{{\bullet }\mathrm{j}}$].”

and referred to ${\mathrm{p}}_{\mathrm{ij}}-{\mathrm{p}}_{\mathrm{i}{\bullet }}{\mathrm{p}}_{{\bullet }\mathrm{j}}$ as the contingency of the (i, j)th cell of the table. Pearson’s statement says that the information gained from obtaining the departure of the observed frequencies to their expected frequencies will help illustrate the nature of dependence or independence in a 2 × 2 table. Hence, Pearson’s ratio, the standardised residual and the adjusted standardised residual are all expressible in terms of the Pearson’s contingency ${\mathrm{P}}_{\mathrm{C}}={\mathrm{p}}_{11}-{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}$ and can be used to assess any symmetric association that exists between the variables. However, they are only applicable if the cell values of the 2 × 2 table are available. In the following section, we will show how the measures above can still be used to quantify the strength of the association in a 2 × 2 table given only the marginal totals.

4.2 The cell proportion $\left({\mathbf{p}}_{11}\right)$

Suppose we consider the untransformed case where the AAI is expressed solely in terms of ${\mathrm{p}}_{11}$. From (9), such a case arises when $\mathrm{a}=1$ and $\mathrm{b}=0$ and, using (10), $\ell\left({\mathrm{p}}_{11}\right)={\mathrm{p}}_{11}$ is bounded by

$${\mathrm{L}}_{{\mathrm{p}}_{11}}=\mathrm{max}\left(0,\hspace{0.33em}{\mathrm{p}}_{{\bullet }1}-{\mathrm{p}}_{2{\bullet }}\right)\le {\mathrm{p}}_{11}\le \mathrm{min}\left({\mathrm{p}}_{{\bullet }1},\hspace{0.33em}{\mathrm{p}}_{1{\bullet }}\right)={\mathrm{U}}_{{\mathrm{p}}_{11}},$$

(18)

and are akin to the Fréchet bounds of ${\mathrm{p}}_{11}$ (Fréchet 1951).

From (11), and based only on the marginal totals, the $100\left(1-{\alpha }\right)\%$ bounds of ${\mathrm{p}}_{11}$, are

$${\mathrm{L}}_{{\alpha }{\mathrm{p}}_{11}}=\mathrm{max}\left(0,\hspace{0.33em}{\mathrm{p}}_{1{\bullet }}\left({\mathrm{p}}_{{\bullet }1}-{\mathrm{p}}_{2{\bullet }}\sqrt{\frac{{\upchi }_{{\alpha }}^{2}}{\mathrm{n}}\frac{{\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{{\bullet }2}}{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}}}\right)\right)\le \, {\mathrm{p}}_{11}\le \mathrm{min}\left({\mathrm{p}}_{1{\bullet }},\hspace{0.33em}{\mathrm{p}}_{1{\bullet }}\left({\mathrm{p}}_{{\bullet }1}+{\mathrm{p}}_{2{\bullet }}\sqrt{\frac{{\upchi }_{{\alpha }}^{2}}{\mathrm{n}}\frac{{\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{{\bullet }2}}{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}}}\right)\right)={\mathrm{U}}_{{\alpha }{\mathrm{p}}_{11}}.$$

(19)

Hence, when ${\mathrm{L}}_{{\alpha }{\mathrm{p}}_{11}}<{\mathrm{p}}_{11}<{\mathrm{U}}_{{\alpha }{\mathrm{p}}_{11}}$ there is evidence that the row and column variables are independent at the ${\alpha }$ level of significance. However, if ${\mathrm{L}}_{{\mathrm{p}}_{11}}<{\mathrm{p}}_{11}<{\mathrm{L}}_{{\alpha }{\mathrm{p}}_{11}}$ or ${\mathrm{U}}_{{\alpha }{\mathrm{p}}_{11}}<{\mathrm{p}}_{11}<{\mathrm{U}}_{{\mathrm{p}}_{11}}$ then this constitutes evidence that the variables are associated at the ${\alpha }$ level of significance.

The chi-squared statistic (12) can therefore be expressed as a function of ${\mathrm{p}}_{11}$ such that

$${\mathrm{X}}^{2}\left({\mathrm{p}}_{11}|{\mathrm{p}}_{1{\bullet }},\hspace{0.33em}{\mathrm{p}}_{{\bullet }1}\right)=\mathrm{n}\frac{{\left({\mathrm{p}}_{11}-{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}\right)}^{2}}{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{{\bullet }2}}\hspace{0.33em},$$

(20)

This is one of the more common expressions of the chi-squared statistic for a 2 × 2 table proposed by Pearson (1904) and shows that there is complete independence between the two dichotomous variables when ${\mathrm{p}}_{11}={\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}$. From (14), the curvature of the ${\mathrm{X}}^{2}\left({\mathrm{p}}_{11}|{\mathrm{p}}_{1{\bullet }},\hspace{0.33em}{\mathrm{p}}_{{\bullet }1}\right)$ is:

$$\upgamma \left({\mathrm{p}}_{11}\right)=\frac{{\partial }^{2}{\mathrm{X}}^{2}\left({\mathrm{p}}_{11}\right)}{\partial {\left({\mathrm{p}}_{11}\right)}^{2}}=\frac{2\mathrm{n}}{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{{\bullet }2}}\hspace{0.33em}.$$

(21)

It can be seen from (21) that the curvature of the untransformed linear function of ${\mathrm{p}}_{11}$ depends only on the marginal totals of a 2 × 2 table and increases as $\mathrm{n}$ increases. This curvature index will be used for the various linear transformations of ${\mathrm{p}}_{11}$ described below for assessing how their curvature compares with (21).

From (18), (19), and (20), the AAI curve defined by the quadratic relationship between ${\mathrm{X}}^{2}\left({\mathrm{p}}_{11}|{\mathrm{p}}_{1{\bullet }},\hspace{0.33em}{\mathrm{p}}_{{\bullet }1}\right)$ and ${\mathrm{p}}_{11}$ can be graphically depicted in the same way that Fig. 1 shows the relationship between ${\mathrm{X}}^{2}\left({\mathrm{P}}_{1}|{\mathrm{p}}_{1{\bullet }},\hspace{0.33em}{\mathrm{p}}_{{\bullet }1}\right)$ and ${\mathrm{P}}_{1}$. As a result, the AAI can be expressed for ${\mathrm{p}}_{11}$ as

$$\begin{aligned}{\mathrm{A}}_{{\alpha }}\left({\mathrm{p}}_{11}\right)&=100\left(1-\frac{{\upchi }_{{\alpha }}^{2}\left[\left({\mathrm{L}}_{{\alpha }{\mathrm{p}}_{11}}-{\mathrm{L}}_{{\mathrm{p}}_{11}}\right)+\left({\mathrm{U}}_{{\mathrm{p}}_{11}}-{\mathrm{U}}_{{\alpha }{\mathrm{p}}_{11}}\right)\right]}{{\mathrm{k}}_{{\mathrm{p}}_{11}}\mathrm{n}\left[{\left({\mathrm{U}}_{{\mathrm{p}}_{11}}-{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}\right)}^{3}-{\left({\mathrm{L}}_{{\mathrm{p}}_{11}}-{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}\right)}^{3}\right]}\right.\\ & \left.\qquad \quad -\frac{{\left({\mathrm{U}}_{{\alpha }{\mathrm{p}}_{11}}-{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}\right)}^{3}-{\left({\mathrm{L}}_{{\alpha }{\mathrm{p}}_{11}}-{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}\right)}^{3}}{{\left({\mathrm{U}}_{{\mathrm{p}}_{11}}-{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}\right)}^{3}-{\left({\mathrm{L}}_{{\mathrm{p}}_{11}}-{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}\right)}^{3}}\right),\end{aligned}$$

where

$${\mathrm{k}}_{{\mathrm{p}}_{11}}=\frac{1}{3\left({\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{{\bullet }2}\right)} .$$

4.3 Pearson’s contingency (${\mathbf{P}}_{\mathbf{C}}$)

In his seminal paper, Pearson (1904) discussed the importance of the contingency in measuring association for a 2 × 2 table. Note that Pearson’s contingency, ${\mathrm{P}}_{\mathrm{C}}={\mathrm{p}}_{11}-{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1},$ is also a linear transformation of ${\mathrm{p}}_{11}$ with $\mathrm{a}=1$ and $\mathrm{b}=-{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}$. Therefore, with this specification, and by considering (10), the bounds of ${\mathrm{P}}_{\mathrm{C}}$ can be defined as

$${\mathrm{L}}_{{\mathrm{P}}_{\mathrm{C}}}=\mathrm{max}\left(-{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1},\hspace{0.33em}-{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }2}\right)\le {\mathrm{P}}_{\mathrm{C}}\le \mathrm{min}\left({\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{2{\bullet }},\hspace{0.33em}{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }2}\right)={\mathrm{U}}_{{\mathrm{P}}_{\mathrm{C}}}.$$

(22)

From (11), the $100\left(1-{\alpha }\right)\%$ bounds of ${\mathrm{P}}_{\mathrm{C}}$ based only on the marginal totals are

$${\mathrm{L}}_{{\alpha }{\mathrm{P}}_{\mathrm{C}}}=\mathrm{max}\left(-{\mathrm{p}}_{1{\bullet }.}{\mathrm{p}}_{{\bullet }1},\hspace{0.33em}-{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}\sqrt{\frac{{\upchi }_{{\alpha }}^{2}}{\mathrm{n}}\frac{{\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{{\bullet }2}}{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}}}\right)\le \, {\mathrm{P}}_{\mathrm{C}}\le \mathrm{min}\left({\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }2},\hspace{0.33em}{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}\sqrt{\frac{{\upchi }_{{\alpha }}^{2}}{\mathrm{n}}\frac{{\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{{\bullet }2}}{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}}}\right)={\mathrm{U}}_{{\alpha }{\mathrm{P}}_{\mathrm{C}}}.$$

(23)

From (22) and (23), we can conclude that if ${\mathrm{L}}_{{\alpha }{\mathrm{P}}_{\mathrm{C}}}<{\mathrm{P}}_{\mathrm{C}}<{\mathrm{U}}_{{\alpha }{\mathrm{P}}_{\mathrm{C}}}$ there is evidence that the row and column variables are independent at the ${\alpha }$ level of significance.

The chi-squared statistic (12) can be expressed as a function of ${\mathrm{P}}_{\mathrm{C}}$ and the marginal totals such that

$${\mathrm{X}}^{2}\left({\mathrm{P}}_{\mathrm{C}}|{\mathrm{p}}_{1{\bullet }},\hspace{0.33em}{\mathrm{p}}_{{\bullet }1}\right)=\mathrm{n}\frac{{\left({\mathrm{P}}_{\mathrm{C}}\right)}^{2}}{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{{\bullet }2}}\hspace{0.33em}.$$

(24)

Equation (24) is similar to the corresponding expression of the chi-squared statistic specified by (20). Equation (24) shows that, when the null hypothesis of complete independence is assumed, the expected value of ${\mathrm{P}}_{\mathrm{C}}$ is equal to 0. The AAI curve for ${\mathrm{X}}^{2}\left({\mathrm{P}}_{\mathrm{C}}|{\mathrm{p}}_{1{\bullet }},\hspace{0.33em}{\mathrm{p}}_{{\bullet }1}\right)$ can be graphically depicted in the similar way to that of Fig. 1 which shows the relationship between ${\mathrm{X}}^{2}\left({\mathrm{P}}_{1}|{\mathrm{p}}_{1{\bullet }},\hspace{0.33em}{\mathrm{p}}_{{\bullet }1}\right)$ and ${\mathrm{P}}_{1}$.

From (14), the curvature of ${\mathrm{X}}^{2}\left({\mathrm{P}}_{\mathrm{C}}|{\mathrm{p}}_{1{\bullet }},\hspace{0.33em}{\mathrm{p}}_{{\bullet }1}\right)$ can be obtained as:

$$\upgamma \left({\mathrm{P}}_{\mathrm{C}}\right)=\frac{{\partial }^{2}{\mathrm{X}}^{2}\left({\mathrm{P}}_{\mathrm{C}}\right)}{\partial {\left({\mathrm{P}}_{\mathrm{C}}\right)}^{2}}=\frac{2\mathrm{n}}{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{{\bullet }2}}\hspace{0.33em}=\upgamma \left({\mathrm{p}}_{11}\right).$$

(25)

so that the curvature index is $\Delta \left({\mathrm{P}}_{\mathrm{C}}\right)=1$. Despite the differences in the linear transformation bounds between the untransformed linear function of ${\mathrm{p}}_{11}$ (Sect. 4.2) and Pearson’s contingency ${\mathrm{P}}_{\mathrm{C}}$, (25) shows that the curvature of their corresponding AAI curves remains unchanged. This should also be apparent since the curvature coefficient “$\mathrm{a}$” is the same for ${\mathrm{p}}_{11}$ and ${\mathrm{P}}_{\mathrm{C}}$.

From (22), (23) and (24), the AAI is then expressed in terms of the Pearson’s contingency ${\mathrm{P}}_{\mathrm{C}}$ such that

$$\begin{aligned}{\mathrm{A}}_{{\alpha }}\left({\mathrm{P}}_{\mathrm{C}}\right)&=100\left(1-\frac{{\upchi }_{{\alpha }}^{2}\left[\left({\mathrm{L}}_{{\alpha }{\mathrm{P}}_{\mathrm{C}}}-{\mathrm{L}}_{{\mathrm{P}}_{\mathrm{C}}}\right)+\left({\mathrm{U}}_{{\mathrm{P}}_{\mathrm{C}}}-{\mathrm{U}}_{{\alpha }{\mathrm{P}}_{\mathrm{C}}}\right)\right]}{{\mathrm{k}}_{{\mathrm{P}}_{\mathrm{C}}}\mathrm{n}\left[{\left({\mathrm{U}}_{{\mathrm{P}}_{\mathrm{C}}}-{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}\right)}^{3}-{\left({\mathrm{L}}_{{\mathrm{P}}_{\mathrm{C}}}-{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}\right)}^{3}\right]}\right. \\ & \qquad \quad \left. - \, \frac{{\left({\mathrm{U}}_{{\alpha }{\mathrm{P}}_{\mathrm{C}}}-{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}\right)}^{3}-{\left({\mathrm{L}}_{{\alpha }{\mathrm{P}}_{\mathrm{C}}}-{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}\right)}^{3}}{{\left({\mathrm{U}}_{{\mathrm{P}}_{\mathrm{C}}}-{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}\right)}^{3}-{\left({\mathrm{L}}_{{\mathrm{P}}_{\mathrm{C}}}-{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}\right)}^{3}}\right),\end{aligned}$$

where

$${\mathrm{k}}_{{\mathrm{P}}_{\mathrm{C}}}=\frac{1}{3\left({\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{{\bullet }2}\right)}.$$

4.4 Independence ratio (B)

Define $\mathrm{B}={\mathrm{p}}_{11}/\left({\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}\right)$ as the independence ratio of the (1, 1)th cell of a 2 × 2 contingency table. This index has also been referred to as Pearson’s ratio (see, for example, Goodman 1996; Beh 2008) and as a contingency ratio (Greenacre 2009). Irrespective of how $\mathrm{B}$ is termed, it reflects the ratio of the cell proportion ${\mathrm{p}}_{11}$ to its expected value ${\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}$ (under the null hypothesis of complete independence). Hence a value of $\mathrm{B}$ close to 1 indicates likely independence of the variables of the table. In addition, $\mathrm{B}$ is a special case of $\ell\left({\mathrm{p}}_{11}\right)$ when $\mathrm{a}={\left({\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}\right)}^{-1}$ and $\mathrm{b}=0$. It can be shown that, for a 2 × 2 contingency table, the Pearson’s chi-squared statistic can thus be expressed in terms of B such that

$${\mathrm{X}}^{2}\left(\mathrm{B}|{\mathrm{p}}_{1{\bullet }},\hspace{0.33em}{\mathrm{p}}_{{\bullet }1}\right)=\mathrm{n}{\left(\mathrm{B}-1\right)}^{2}\left(\frac{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}}{{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }2}}\right).$$

(26)

One may note that, by (10), the Pearson’s ratio (B) is bounded by

$${\mathrm{L}}_{\mathrm{B}}=\mathrm{max}\left(0,\hspace{0.33em}\frac{{\mathrm{p}}_{{\bullet }1}-{\mathrm{p}}_{2{\bullet }}}{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}}\right)\le \mathrm{B}\le \mathrm{min}\left(\frac{1}{{\mathrm{p}}_{1{\bullet }}},\hspace{0.33em}\frac{1}{{\mathrm{p}}_{{\bullet }1}}\right)={\mathrm{U}}_{\mathrm{B}}.$$

(27)

When testing the association at the ${\alpha }$ level of significance, the Pearson’s ratio is bounded by

$${\mathrm{L}}_{{\alpha {\text B}}}=\mathrm{max}\left(0, 1-\sqrt{\frac{{\upchi }_{{\alpha }}^{2}}{\mathrm{n}}\frac{{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }2}}{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}}}\right)\le \mathrm{B}\le \mathrm{min}\left(\frac{1}{{\mathrm{p}}_{{\bullet }1}}, 1+\sqrt{\frac{{\upchi }_{{\alpha }}^{2}}{\mathrm{n}}\frac{{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }2}}{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}}}\right)={\mathrm{U}}_{\alpha \mathrm{B}}.$$

(28)

Therefore, we may conclude that when ${\mathrm{L}}_{\alpha \mathrm{B}}<\mathrm{B}<{\mathrm{U}}_{\alpha \mathrm{B}}$ there is evidence that the row and column variables are independent at the ${\alpha }$ level of significance. However, if ${\mathrm{L}}_{\mathrm{B}}<\mathrm{B}<{\mathrm{L}}_{\alpha \mathrm{B}}$ or ${\mathrm{U}}_{\alpha \mathrm{B}}<\mathrm{B}<{\mathrm{U}}_{\mathrm{B}}$ then there is evidence that the variables are associated.

The curvature of the AAI curve produced from ${\mathrm{X}}^{2}\left(\mathrm{B}|{\mathrm{p}}_{1{\bullet }},\hspace{0.33em}{\mathrm{p}}_{{\bullet }1}\right)$ is

$$\upgamma \left(\mathrm{B}\right)=\frac{{\partial }^{2}{\mathrm{X}}^{2}\left(\mathrm{B}\right)}{\partial {\left(\mathrm{B}\right)}^{2}}=2\mathrm{n}\frac{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}}{{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }2}}.$$

(29)

and can be written in terms of $\upgamma \left({\mathrm{p}}_{11}\right)$ such that:

$$\upgamma \left(\mathrm{B}\right)=2\mathrm{n}\frac{{\left({\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}\right)}^{2}}{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{{\bullet }2}}={\left({\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}\right)}^{2}\upgamma \left({\mathrm{p}}_{11}\right).$$

(30)

Therefore, the curvature index for the AAI curve when expressed in terms of the independence ratio is

$$\Delta \left(\mathrm{B}\right)=\frac{\upgamma \left(\mathrm{B}\right)}{\upgamma \left({\mathrm{p}}_{11}\right)}={\left({\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}\right)}^{2}.$$

(31)

Since this ratio is always less than 1, the curvature of the AAI curve in terms of the independence ratio, $\upgamma \left(\mathrm{B}\right)$, is always smaller than the curvature of the AAI curve in the untransformed case, $\upgamma \left({\mathrm{p}}_{11}\right)$.

The AAI curve defined from the quadratic relationship between ${\mathrm{X}}^{2}\left(\mathrm{B}|{\mathrm{p}}_{1{\bullet }},\hspace{0.33em}{\mathrm{p}}_{{\bullet }1}\right)$ and $\mathrm{B}$ may be graphically depicted in a way that is akin to Fig. 1. For such a relationship, the turning point of ${\mathrm{X}}^{2}\left(\mathrm{B}|{\mathrm{p}}_{1{\bullet }},\hspace{0.33em}{\mathrm{p}}_{{\bullet }1}\right)$ is located at $\mathrm{B}=1$ which coincides (as expected) with complete independence between the two dichotomous variables. Therefore, from (26), (27), (28), the AAI in terms of the independence ratio, $\mathrm{B}$, is calculated by

$${\mathrm{A}}_{{\alpha }}\left(\mathrm{B}\right)=100\left(1-\frac{{\upchi }_{{\alpha }}^{2}\left[\left({\mathrm{L}}_{\alpha \mathrm{B}}-{\mathrm{L}}_{\mathrm{B}}\right)+\left({\mathrm{U}}_{\mathrm{B}}-{\mathrm{U}}_{\alpha \mathrm{B}}\right)\right]}{{\mathrm{k}}_{\mathrm{B}}\mathrm{n}\left[{\left({\mathrm{U}}_{\mathrm{B}}-1\right)}^{3}-{\left({\mathrm{L}}_{\mathrm{B}}-1\right)}^{3}\right]}-\frac{{\left({\mathrm{U}}_{\alpha \mathrm{B}}-1\right)}^{3}-{\left({\mathrm{L}}_{\alpha \mathrm{B}}-1\right)}^{3}}{{\left({\mathrm{U}}_{\mathrm{B}}-1\right)}^{3}-{\left({\mathrm{L}}_{\mathrm{B}}-1\right)}^{3}}\right),$$

where

$${\mathrm{k}}_{\mathrm{B}}=\frac{1}{3}\left(\frac{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}}{{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }2}}\right).$$

4.5 Pearson’s ratio (C)

Rather than quantifying the independence ratio one may consider instead the index

$$\mathrm{C}=\frac{{\mathrm{p}}_{11}-{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}}{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}}=\frac{{\mathrm{p}}_{11}}{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}}-1 \;\left(=\mathrm{B}-1\right),$$

(32)

which is referred to as the Pearson’s ratio for the (1, 1)th cell of the 2 × 2 table.

For an IxJ contingency table, Mirkin (2001) described that Quetelet (1832) referred to indices related to $\mathrm{C}$ (but more generally expressed for the (i, j)th cell frequency) as a “degree of influence” between the categorical variables and, in doing so, referred to $\mathrm{C}$ as a Quetelet index. In the case of a 2 × 2 table, Pearson’s ratio ($\mathrm{C}$) can then be viewed as the “degree of influence” for the (1, 1)th cell value so that under complete independence the index is zero.

It can be seen that (32) is a special case of $\ell\left({\mathrm{p}}_{11}\right)$ when $\mathrm{a}={\left({\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}\right)}^{-1}$ and $\mathrm{b}=-1$. Therefore, $\mathrm{C}$ is bounded by

$${\mathrm{L}}_{\mathrm{C}}=\mathrm{max}\left(-1,\hspace{0.33em}-\frac{{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }2}}{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}}\right)\le \mathrm{C}\le \mathrm{min}\left(\frac{{\mathrm{p}}_{{\bullet }2}}{{\mathrm{p}}_{{\bullet }1}},\hspace{0.33em}\frac{{\mathrm{p}}_{2{\bullet }}}{{\mathrm{p}}_{1{\bullet }}}\right)={\mathrm{U}}_{\mathrm{C}},$$

(33)

while its $100\left(1-{\alpha }\right)\%$ bounds are

$${\mathrm{L}}_{\alpha \mathrm{C}}=\mathrm{max}\left(-1,-\sqrt{\frac{{\upchi }_{{\alpha }}^{2}}{\mathrm{n}}\frac{{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }2}}{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}}}\right)\le \mathrm{C}\le \mathrm{min}\left(\frac{{\mathrm{p}}_{{\bullet }2}}{{\mathrm{p}}_{{\bullet }1}},\hspace{0.33em}\sqrt{\frac{{\upchi }_{{\alpha }}^{2}}{\mathrm{n}}\frac{{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }2}}{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}}}\right)={\mathrm{U}}_{\alpha \mathrm{C}}.$$

(34)

From (12), the chi-squared statistic can then be expressed in terms of $\mathrm{C}$ and the marginal totals by

$${\mathrm{X}}^{2}\left(\mathrm{C}|{\mathrm{p}}_{1{\bullet }},\hspace{0.33em}{\mathrm{p}}_{{\bullet }1}\right)\hspace{0.33em}={\mathrm{nC}}^{2}\left(\frac{{\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{1{\bullet }}}{{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }2}}\right).$$

(35)

The curvature of the AAI curve produced using Pearson’s ratio (C) is defined as

$$\upgamma \left(\mathrm{C}\right)=\frac{{\partial }^{2}{\mathrm{X}}^{2}\left(\mathrm{C}\right)}{\partial {\mathrm{C}}^{2}}=2\mathrm{n}\frac{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}}{{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }2}}.$$

(36)

and this curve may be graphically depicted by plotting ${\mathrm{X}}^{2}\left(\mathrm{C}|{\mathrm{p}}_{1{\bullet }},\hspace{0.33em}{\mathrm{p}}_{{\bullet }1}\right)$ against $\mathrm{C}$ within the bounds given by (33). For such a relationship, the turning point of the AAI curve will be located at $\mathrm{C}=0$, which coincides with independence between the two dichotomous variables. Since the specification of $\mathrm{a}$ for Pearson’s ratio C is the same as the specification of $\mathrm{a}$ for the independence ratio B, the curvature formulations of these indices, (29) and (36), are identical and can be expressed in terms of $\upgamma \left({\mathrm{p}}_{11}\right)$ as:

$$\upgamma \left(\mathrm{C}\right)=\upgamma \left(\mathrm{B}\right)={\left({\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}\right)}^{2}\upgamma \left({\mathrm{p}}_{11}\right).$$

(37)

Also, since $\Delta \left(\mathrm{C}\right)={\left({\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}\right)}^{2}$, (37) is always less than 1, so that the curvature of the AAI curve using the independence ratio, $\upgamma \left(\mathrm{C}\right)$, is smaller than the curvature of the AAI curve in the untransformed case, $\upgamma \left({\mathrm{p}}_{11}\right)$.

From (33), (34), and (35), the AAI formulated in terms of Pearson’s contingency, C, is then

$${\mathrm{A}}_{{\alpha }}\left(\mathrm{C}\right)=100\left(1-\frac{{\upchi }_{{\alpha }}^{2}\left[\left({\mathrm{L}}_{\alpha \mathrm{C}}-{\mathrm{L}}_{\mathrm{C}}\right)+\left({\mathrm{U}}_{\mathrm{C}}-{\mathrm{U}}_{\alpha \mathrm{C}}\right)\right]}{{\mathrm{k}}_{\mathrm{C}}\mathrm{n}\left[{\mathrm{U}}_{\mathrm{C}}^{3}-{\mathrm{L}}_{\mathrm{C}}^{3}\right]}-\frac{{\mathrm{U}}_{\alpha \mathrm{C}}^{3}-{\mathrm{L}}_{\alpha \mathrm{C}}^{3}}{{\mathrm{U}}_{\mathrm{C}}^{3}-{\mathrm{L}}_{\mathrm{C}}^{3}}\right),$$

where

$${\mathrm{k}}_{\mathrm{C}}=\frac{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}}{3{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }2}}.$$

4.6 Standardised residual (Z)

In a manner similar to Pearson’s (1904) comment given in Sect. 4.1, Agresti (2013, p. 80) also noted that Pearson’s contingency ${\mathrm{p}}_{11}-{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}$ helps to reflect the nature of the dependence structure of the variables for the (1, 1)th cell. By assuming the (1, 1)th cell frequency follows a Poisson distribution, Agresti (2013, p. 80) discussed that the original Pearson’s contingency is not sufficient to quantify the strength of an association. Therefore, he (p. 80) discussed (in a more general setting) a more refined version of the Pearson’s contingency for the (1, 1)th cell of a 2 × 2 contingency table, although its history goes back further. This version is defined as

$$\mathrm{Z}=\frac{{\mathrm{p}}_{11}-{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}}{\sqrt{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}}}.$$

(38)

and is termed the standardised residual of the (1, 1)th cell. We can see that (38) is a special case of $\ell\left({\mathrm{p}}_{11}\right)$ where $\mathrm{a}={\left({\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}\right)}^{-1/2}$ and $\mathrm{b}=-{\left({\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}\right)}^{1/2}$ and under the null hypothesis of complete independence, $\mathrm{Z}=0$. From (10), $\mathrm{Z}$ is bounded by

$${\mathrm{L}}_{\mathrm{Z}}=\sqrt{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}}\mathrm{max}\left(-1,-\hspace{0.33em}\frac{{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }2}}{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}}\right)\le \mathrm{Z}\le \sqrt{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}}\mathrm{min}\left(\frac{{\mathrm{p}}_{2{\bullet }}}{{\mathrm{p}}_{1{\bullet }}},\hspace{0.33em}\frac{{\mathrm{p}}_{{\bullet }2}}{{\mathrm{p}}_{{\bullet }1}}\right)={\mathrm{U}}_{\mathrm{Z}},$$

(39)

while, from (11), the $100\left(1-\alpha \right)\%$ bounds of $\mathrm{Z}$ are

$${\mathrm{L}}_{\alpha \mathrm{Z}}={\mathrm{p}}_{2{\bullet }}\sqrt{\frac{{\mathrm{p}}_{1{\bullet }}}{{\mathrm{p}}_{{\bullet }1}}}\mathrm{max}\left(-\frac{{\mathrm{p}}_{{\bullet }1}}{{\mathrm{p}}_{2{\bullet }}},\hspace{0.33em}-\sqrt{\frac{{\upchi }_{{\alpha }}^{2}}{\mathrm{n}}\frac{{\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{{\bullet }2}}{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}}}\right)\le \mathrm{Z}\le \sqrt{\frac{{\mathrm{p}}_{1{\bullet }}}{{\mathrm{p}}_{{\bullet }1}}}\mathrm{min}\left({\mathrm{p}}_{{\bullet }2},\hspace{0.33em}{\mathrm{p}}_{2{\bullet }}\sqrt{\frac{{\upchi }_{{\alpha }}^{2}}{\mathrm{n}}\frac{{\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{{\bullet }2}}{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}}}\right)={\mathrm{U}}_{{\alpha Z}}.$$

(40)

From (12), the chi-squared statistic can then be expressed in terms of $\mathrm{Z}$ and marginal totals as:

$${\mathrm{X}}^{2}\left(\mathrm{Z}|{\mathrm{p}}_{1{\bullet }},\hspace{0.33em}{\mathrm{p}}_{{\bullet }1}\right)={\mathrm{nZ}}^{2}\left(\frac{1}{{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }2}}\right)\hspace{0.33em}.$$

(41)

so that the curvature of this quadratic relationship is

$$\upgamma \left(\mathrm{Z}\right)=\frac{{\partial }^{2}{\mathrm{X}}^{2}\left(\mathrm{Z}\right)}{\partial {\mathrm{Z}}^{2}}=\frac{2\mathrm{n}}{{\mathrm{p}}_{{\bullet }2}}\sqrt{\frac{{\mathrm{p}}_{1{\bullet }}}{{\mathrm{p}}_{{\bullet }1}}}.$$

(42)

Equation (42) can be expressed in terms of $\upgamma \left({\mathrm{p}}_{11}\right)$ by

$$\upgamma \left(\mathrm{Z}\right)=\left({\mathrm{p}}_{1{\bullet }}^{3/2}{\mathrm{p}}_{{\bullet }1}^{1/2}{\mathrm{p}}_{2{\bullet }}\right)\frac{2\mathrm{n}}{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{{\bullet }2}}=\left({\mathrm{p}}_{1{\bullet }}^{3/2}{\mathrm{p}}_{{\bullet }1}^{1/2}{\mathrm{p}}_{2{\bullet }}\right)\upgamma \left({\mathrm{p}}_{11}\right).$$

Here, the change in curvature from ${\mathrm{p}}_{11}$ to $\mathrm{Z}$ also depends only on the marginal totals while $\upgamma \left(\mathrm{Z}\right)$ is smaller than $\upgamma \left({\mathrm{p}}_{11}\right)$ since $\Delta \left(\mathrm{Z}\right)=\left({\mathrm{p}}_{1{\bullet }}^{3/2}{\mathrm{p}}_{{\bullet }1}^{1/2}{\mathrm{p}}_{2{\bullet }}\right)={\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}\sqrt{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}}$ is less than 1.

From (39), (40), and (41), the AAI may be expressed in terms of $\mathrm{Z}$ and its bounds such that

$${\mathrm{A}}_{{\alpha }}\left(\mathrm{Z}\right)=100\left(1-\frac{{\upchi }_{{\alpha }}^{2}\left[\left({\mathrm{L}}_{\alpha \mathrm{Z}}-{\mathrm{L}}_{\mathrm{Z}}\right)+\left({\mathrm{U}}_{\mathrm{Z}}-{\mathrm{U}}_{\alpha \mathrm{Z}}\right)\right]}{{\mathrm{k}}_{\mathrm{Z}}\mathrm{n}\left[{\mathrm{U}}_{\mathrm{Z}}^{3}-{\mathrm{L}}_{\mathrm{Z}}^{3}\right]}-\frac{{\mathrm{U}}_{\alpha \mathrm{Z}}^{3}-{\mathrm{L}}_{\alpha \mathrm{Z}}^{3}}{{\mathrm{U}}_{\mathrm{Z}}^{3}-{\mathrm{L}}_{\mathrm{Z}}^{3}}\right),$$

where

$${\mathrm{k}}_{\mathrm{Z}}=\frac{1}{3{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }2}}.$$

4.7 Adjusted standardised residual (${\mathbf{Z}}_{\mathbf{a}\mathbf{d}\mathbf{j}}$)

Haberman (1973) showed that $\mathrm{Z}$ is not a standard normal random variable and suggested that the standardised residual can be adjusted by dividing it by its standard error; see also Agresti (2013, p. 81). Hence, the adjusted standardised residual ${\mathrm{Z}}_{\mathrm{adj}}$ is defined as:

$${\mathrm{Z}}_{\mathrm{adj}}=\frac{{\mathrm{p}}_{11}-{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}}{\sqrt{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{{\bullet }2}}}$$

(43)

although it is also referred to under different names. For example, through the long history of (44) it is also referred to as Pearson’s (1904) product moment correlation coefficient for a 2 × 2 table while Agresti (2013, p. 81) refers to ${\mathrm{Z}}_{\mathrm{adj}}$ as the standardised residual. Under the null hypothesis of complete independence, ${\mathrm{Z}}_{\mathrm{adj}}$ is equal to zero.

We can see that (43) is a special case of $\ell\left({\mathrm{p}}_{11}\right)$ where $\mathrm{a}=1/\sqrt{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }2}}$ and $\mathrm{b}=-\sqrt{\left({\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}\right)/\left({\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }2}\right)}$. Therefore, from (10) and (11), ${\mathrm{Z}}_{\mathrm{adj}}$ is bounded by

$${\mathrm{L}}_{{\mathrm{Z}}_{\mathrm{adj}}}=\mathrm{max}\left(-\sqrt{\frac{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}}{{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }2}}},\hspace{0.33em}-\sqrt{\frac{{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }2}}{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}}}\right)\le {\mathrm{Z}}_{\mathrm{adj}}\le \mathrm{min}\left(\sqrt{\frac{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }2}}{{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }1}}},\hspace{0.33em}\sqrt{\frac{{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }1}}{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }2}}}\right)={\mathrm{U}}_{{\mathrm{Z}}_{\mathrm{adj}}},$$

(44)

while

$${\mathrm{L}}_{{\alpha }{\mathrm{Z}}_{\mathrm{adj}}}=\mathrm{max}\left(-\sqrt{\frac{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }1}}{{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }2}}},\hspace{0.33em}-\sqrt{\frac{{\upchi }_{{\alpha }}^{2}}{\mathrm{n}}}\right)\le {\mathrm{Z}}_{\mathrm{adj}}\le \mathrm{min}\left(\sqrt{\frac{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{{\bullet }2}}{{\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{2{\bullet }}}},\hspace{0.33em}\sqrt{\frac{{\upchi }_{{\alpha }}^{2}}{\mathrm{n}}}\right)={\mathrm{U}}_{{\alpha }{\mathrm{Z}}_{\mathrm{adj}}}.$$

(45)

From (12), the chi-squared statistic can thus be expressed in terms of ${\mathrm{Z}}_{\mathrm{adj}}$ and the marginal totals as

$${\mathrm{X}}^{2}\left({\mathrm{Z}}_{\mathrm{adj}}|{\mathrm{p}}_{1{\bullet }},\hspace{0.33em}{\mathrm{p}}_{{\bullet }1}\right)={\mathrm{nZ}}_{\mathrm{adj}}^{2}\hspace{0.33em},$$

(46)

so that the curvature of the associated AAI curve is

$$\upgamma \left({\mathrm{Z}}_{\mathrm{adj}}\right)=\frac{{\partial }^{2}{\mathrm{X}}^{2}\left({\mathrm{Z}}_{\mathrm{adj}}\right)}{\partial {\left({\mathrm{Z}}_{\mathrm{adj}}\right)}^{2}}=2\mathrm{n}\sqrt{\frac{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}}{{\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{{\bullet }2}}}.$$

(47)

Similar to the those transformations discussed above, (47) only depends on the marginal totals and can also be expressed in terms of $\upgamma \left({\mathrm{p}}_{11}\right)$ as:

$$\upgamma \left({\mathrm{Z}}_{\mathrm{adj}}\right)={\left({\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}\right)}^{3/2}{\left({\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{{\bullet }2}\right)}^{1/2}\left(\frac{2\mathrm{n}}{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{{\bullet }2}}\right)={\left({\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}\right)}^{3/2}{\left({\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{{\bullet }2}\right)}^{1/2}\upgamma \left({\mathrm{p}}_{11}\right).$$

(48)

From (48), one can see that the curvature $\upgamma \left({\mathrm{Z}}_{\mathrm{adj}}\right)$ is smaller than the curvature of the untransformed case $\upgamma \left({\mathrm{p}}_{11}\right)$ since $\Delta \left({\mathrm{Z}}_{\mathrm{adj}}\right)={\left({\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}\right)}^{3/2}{\left({\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{{\bullet }2}\right)}^{1/2}={\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}\sqrt{{\mathrm{p}}_{1{\bullet }}{\mathrm{p}}_{2{\bullet }}{\mathrm{p}}_{{\bullet }1}{\mathrm{p}}_{{\bullet }2}}$ is less than 1.

From (44), (45), and (46) the AAI may be expressed in terms of ${\mathrm{Z}}_{\mathrm{adj}}$ for a 2 × 2 table such that

$${\mathrm{A}}_{{\alpha }}\left({\mathrm{Z}}_{\mathrm{adj}}\right)=100\left(1-\frac{{\upchi }_{{\alpha }}^{2}\left[\left({\mathrm{L}}_{{\alpha }{\mathrm{Z}}_{\mathrm{adj}}}-{\mathrm{L}}_{{\mathrm{Z}}_{\mathrm{adj}}}\right)+\left({\mathrm{U}}_{{\mathrm{Z}}_{\mathrm{adj}}}-{\mathrm{U}}_{{\alpha }{\mathrm{Z}}_{\mathrm{adj}}}\right)\right]}{{\mathrm{k}}_{{\mathrm{Z}}_{\mathrm{adj}}}\mathrm{n}\left[{\mathrm{U}}_{{\mathrm{Z}}_{\mathrm{adj}}}^{3}-{\mathrm{L}}_{{\mathrm{Z}}_{\mathrm{adj}}}^{3}\right]}-\frac{{\mathrm{U}}_{{\alpha }{\mathrm{Z}}_{\mathrm{adj}}}^{3}-{\mathrm{L}}_{{\alpha }{\mathrm{Z}}_{\mathrm{adj}}}^{3}}{{\mathrm{U}}_{{\mathrm{Z}}_{\mathrm{adj}}}^{3}-{\mathrm{L}}_{{\mathrm{Z}}_{\mathrm{adj}}}^{3}}\right)$$

where

$${\mathrm{k}}_{{\mathrm{Z}}_{\mathrm{adj}}}=\frac{1}{3}.$$

Table 4 summaries the value of $\mathrm{a}$, $\mathrm{b}$, the curvature index (∆) and the location of the turning point (TP) – $\left(\upmu \left({\mathrm{p}}_{11}\right), 0\right)$ – for the quantities outlined in Sect. 4.1. Their value for the last four of these (to three decimal places) when applied to the twin data of Table 1 is given in Table 5.

Table 4 Summary of features for the seven linear transformations of p₁₁

Table 5 Value of the classic association indices in terms of the linear transformation $\ell \left( {{\text{p}}_{11} } \right)$ for Fisher’s twin data

5 Application: Fisher’s twin conviction data

5.1 Table 1 cells are known

As described in Sect. 2.1, applying Pearson’s chi-squared test of independence to Table 1 reveals that the association between the types of criminal twin and conviction status is statistically significantly associated at the 0.05 level of significance. Table 6 shows the value of the test statistic for four of the association indices described in Sect. 4.1 and their associated Monte-Carlo p value.

Table 6 Monte Carlo p values for the independence ratio, Pearson’s ratio, standardised residual, and adjusted standardised residual

There has been limited attention given to the distributional properties of the four measures of association in Table 6. However, their statistical significance can be assessed from their Monte Carlo p value which is summarised in the last column of Table 6. These p values were generated from 100,000 2 × 2 tables where each table has a sample size of 30 observations. The randomly generated 2 × 2 tables were created by assuming that each of the cell frequencies follows a Poisson distribution with an expected value calculated under the null hypothesis of complete independence. From Table 4, the association between conviction status and twin types is statistically significant and likely to be positive (given the value of the independence ratio is greater than 1 and the values of Pearson’s ratio, standardised residual, and adjusted standardised residual are all greater than 0). These findings are consistent with the Pearson’s chi-squared test of independence discussed in Sect. 2.1.

5.2 Aggregate data of Table 1

Suppose we now consider the case where only the marginal totals of Table 1 are known. The information about the nature of the association can be obtained with help from the AAI and the linear transformations of ${\mathrm{p}}_{11}$ discussed above. The AAI is 61.83 at the 5% level of significance. Therefore, 61.83% of contingency tables randomly generated with the marginal frequencies given by ${\mathrm{n}}_{1{\bullet }}=13$, ${\mathrm{n}}_{2{\bullet }}=17$, ${\mathrm{n}}_{{\bullet }1}=12$ and ${\mathrm{n}}_{{\bullet }2}=18$ will exhibit a statistically significant association between the two dichotomous variables. In other words, there is enough evidence in the margins of Table 1 to conclude that there is likely to be a statistically significant association between the twin types and conviction status. One may ask “how big is a big AAI value?”. Cheema et al. (2015) developed a strategy for answering this question using Monte Carlo simulation. This issue will be left for future discussion.

Figure 2 illustrates the AAI curve of ${\mathrm{X}}^{2}\left({\mathrm{P}}_{1}|{\mathrm{p}}_{1{\bullet }},\hspace{0.33em}{\mathrm{p}}_{{\bullet }1}\right)$ for Fisher’s data when only the marginal totals are assumed known; the horizontal dashed line is the critical value of the chi-squared statistic, 3.84, with ${\alpha }=0.05$ and 1 degree of freedom. It shows that the shaded region (which reflects a statistically significant association) dominates most of the area under the curve. By applying the linear transformation to each association index described above, the classic graphical representation of the AAI, given by Fig. 2, can be depicted where the horizontal axis is in terms of the magnitude of ${\mathrm{p}}_{11}$, ${\mathrm{P}}_{\mathrm{C}}$, $\mathrm{B}$, $\mathrm{C}$, $\mathrm{Z}$ or ${\mathrm{Z}}_{\mathrm{adj}}$. In particular, Fig. 3 shows as separate figures the AAI curve for these six association indices; note that they look identical to Fig. 2. While the AAI curve appears identical their difference lies in the scale of the horizontal axes. Therefore all seven AAI curves are simultaneously depicted in Fig. 4.

Fig. 2

AAI curve for the conditional proportion P₁ of Fisher’s twin data (Table 1)

Fig. 3

Individual AAI plots for six indices: Fisher’s twin data (α = 0.05)

Fig. 4

AAI curves of Fisher’s data in terms of ${\text{p}}_{11}$, ${\text{P}}_{1} , {\text{ P}}_{{\text{C}}}$, B, C, Z and Z_adj

Despite the different shape of the AAI curves in Fig. 4, the magnitude of the AAI across all different linear transformations is the same. Hence the conclusion concerning the statistically significant association between the twin types and conviction status remains unchanged when only the marginal totals are known.

In addition, by expressing the AAI in terms of the different indices, it is now possible to not only conclude if an association exists between the variables of a 2 × 2 table given only the marginal totals, but also to determine: (1) if an association is likely to be strong or weak; and (2) if it is likely to be a positive or negative association. For Table 1, these additional features can be seen by observing the area of the shaded regions in Fig. 4 relative to the total area under the AAI curve for each of the six indices. Note that the shaded region in Fig. 2 and Fig. 3 represent where a statistically significant association is detected between the two variables of Table 1. They graphically show for the six quantities outlined in Sect. 4.1, that the association (when only the marginal totals are assumed known) is likely to be a strong and positive since the right shaded area above the critical value of 3.84 is larger than the left shaded area. See Beh (2010, Sect. 4) for more details on this issue.

6 Application: nightmare tendency data

Consider now Table 2. While a chi-squared test of independence shows that there is no evidence of a statistically significant association existing between gender and nightmare tendency, this conclusion is based on the cell counts being known; see Sect. 2.1. We shall not go into as much detail with our analysis of the AAI for this data as we did in Sect. 5. We shall instead comment on the magnitude on its AAI and focus on the shape of the AAI curve for the seven simple measures of association that are described in Sect. 4.1.

Suppose now that we assume that only the marginal totals are know. Therefore, the only data we have available to us is how many hospitalised males and females that took part in the study and how many of the 352 participants tended to experience nightmares with “Often” or “Seldom” tendencies. In this case we can examine the association between the two variables using the AAI. Doing so, the AAI of Table 2, regardless of the linear transformation of ${\mathrm{p}}_{11}$ that is applied, is 94.12 at the 5% level of significance. Therefore, about 94% of contingency tables that are randomly generated with the same marginal totals as Table 2 will exhibit a statistically significant association between the two dichotomous variables. This apparent contradiction in the conclusion reached in Sect. 2.1 is because the observed value of ${\mathrm{n}}_{11}$ falls within a narrow interval where such conclusions will be reached. To explore this issue further, it is first important to note that the bounds of ${\mathrm{n}}_{11}$ are quite large; using (3), ${\mathrm{n}}_{11}$ is bounded by [0, 115]. If a test of association is performed at the 5% level of significance then, using (6), the upper and lower bounds of ${\mathrm{P}}_{1}$ that result in a non-significant association is 0.3799 and 0.5292. Multiplying these by ${\mathrm{n}}_{1{\bullet }}=115$, gives the an interval of of ${\mathrm{n}}_{11}\in \left(43.7, 60.9\right)$ where a statistically significant association conclusion is not reached. Since the observed value of ${\mathrm{n}}_{11}=55$ for Table 2 lies within this narrow interval, we reach the conclusions described in Sect. 2.1. If ${\mathrm{n}}_{11}$ were to lie outside of this interval—but still lie within the interval [0, 115]—then the conclusion will change.

An initial look of the magnitude of this association is made by visualising the AAI curve defined in terms of ${\mathrm{P}}_{1}$; Fig. 5 visually highlights the relationship between ${\mathrm{P}}_{1}$ and (2) for Table 2. It shows that the maximum value of Pearson’s chi-squared statistic occurs at the upper bound of (5) while the minimum chi-squared value is zero at independence; that is, when ${\mathrm{p}}_{{\bullet }1}=160/352=0.455$. It also shows the interval ${\mathrm{P}}_{1}\in \left(0.3799, 0.5292\right)$, derived in the previous paragraph, is where a statistically significant association is not observed between the variables. Figure 5 also shows that, when only the marginal totals are known, it is likely that the association between the two variables of Table 2 are quite strong, and (slightly) more likely to be positive than negative.

Fig. 5

AAI curve for the conditional proportion P₁ of the nightmare tendency data (Table 2)

Figure 6 shows the AAI curve for the six association indices ${\mathrm{p}}_{11}$, ${\mathrm{P}}_{\mathrm{C}}$, $\mathrm{B}$, $\mathrm{C}$, $\mathrm{Z}$ and ${\mathrm{Z}}_{\mathrm{adj}}$ plotted separately. Note that the shape of the six AAI curves appear identical to Fig. 5, although we need to keep in mind that the horizontal axis is not consistently scaled. In fact, the seven AAI curves that are depicted in Figs. 5 and 6 are of various shape and any difference between them is reflected in its curvature parameter, $\mathrm{a}$, and where the lowest part of the curve intercects the horizontal axis (corresponding to independence between the variables). Figure 7 simultaneously visualises all seven AAI curves and shows that AAI curve for ${\mathrm{P}}_{\mathrm{C}}$ and ${\mathrm{p}}_{11}$ are identical, verifying (25). It also shows that the curvature of AAI curve in terms of B, defined by (26). is smaller than the curvature of the AAI curve defined in terms of ${\mathrm{P}}_{1}$ and defined by (2), verifying (31). Figure 7 also shows that the AAI curve for C and B are identical, except for their intercept along the horizontal axis; this is because the two indices are related through C = B − 1.

Fig. 6

Individual AAI plots for six indices: nightmare tendency data (α = 0.05)

Fig. 7

A simultaneous view of the AAI curve of Table 2 where the index is expressed in terms of ${\text{p}}_{11}$, P₁, ${\text{P}}_{{\text{C}}}$, B, C, Z and Z_adj

7 Discussion

Given only the marginal totals of a single 2 × 2 table, this study has shown that the AAI can be expressed in terms of other classic association indices using a new and more general AAI formulation than was originally given by Beh (2008, 2010). We have explored the link between the AAI and the independence ratio, Pearson’s ratio, standardised residual and adjusted standardised residual through an investigation of the properties of the linear transformation of ${\mathrm{p}}_{11}$. It is also shown that the value of the AAI for a single 2 × 2 table is not dependent on the choice of association index considered so that is the homogeneity characteristic of the AAI is observed. We demonstrated such homogeneity analytically on Fisher’s data (Table 1) and the on the nightmare tendency data of Herson (1971) (Table 2). However, we note that the curvatures and vertices of the AAI curves change across the different indices due to the changing formulation of $\mathrm{a}$ and $\mathrm{b}$ in the linear transformation $\ell\left({\mathrm{p}}_{11}\right)$, as illustrated in Table 4.

The new generalisation of the AAI described here can likewise be extended to other association indices not discussed in this study, but which are nevertheless well-known in the statistical and allied literature. For example, Tan et al. (2004) discussed a suite of association indices for a 2 × 2 table such as the Kappa ($\upkappa$), Yule’s Q, Yule’s Y (Yule 1912), mutual information (M), J-Measure (J), Gini index (G), Collective strength (S), and the Klosgen (K) index. In addition, Janson and Vegelius (1981), Warrens (2008) and Abdesselam (2020a, b) also examined a large number of association indices for 2 × 2 tables and investigated general properties that these indices may satisfy. Earlier, Chung and Lee (2001) performed a simulation study to examine similarities among the following: the cosine coefficient, Jaccard coefficient, mutual information, Yule's Y, ${\upchi }^{2}$ statistic, and the likelihood ratio statistic. Many of these association indices involve a linear transformation of ${\mathrm{p}}_{11}$ ensuring that our methodology is also highly applicable to them. Given the wide range of literature that is now available on measures of association, the new generalisation of the AAI outlined in this paper can incorporate such measures for cases where only the marginal totals are known for investigation. Some of the above-mentioned indices, and the well-known odds ratio, involve a non-linear transformation of the cell proportion ${\mathrm{p}}_{11}$_. One can refer to Beh et al. (2013) for more details about the link between the AAI and the odds ratio. Link’s can also be made to generalise the above framework to cater for predictor/response variables. Such an association structure can be assessed using the Goodman-Kruskal tau index (Goodman and Kruskal 1954) and Lombardo and Beh (2016) showed how the AAI can be amended for such a structure. Therefore, linear transformations of ${\mathrm{p}}_{11}$ for predictor/response variables can also be considered as another path to adapt the approach outlined in this paper. Thus, this generalisation provides us with the opportunity to further extend the AAI to a non-linear functional framework, a topic for future work. Links between the AAI and log-linear models have been established by Cheema (2016, Chapter 6) and are particularly relevant to the closed-form parameter estimates available for ordinal categorical variables such as those described by Beh and Farver (2009) and Zafar et al. (2015).

Another direction of future research can involve determining an overall AAI curve for each association index. Such a strategy is appropriate for studies involving stratified aggregate data in the form of multiple 2 × 2 contingency tables. In this direction, the generalisation of the AAI discussed in this study will extend the work of Tran et al (2012a, 2018) and Beh et al. (2014) on gendered voting practices in early New Zealand (NZ) politics via the application of a generalised AAI for stratified 2 × 2 tables (Moore 2004, 2005; Keating 2015; Hudson et al. 2010). As such our work has clear practical implications to the political and social sciences (Greiner and Quinn 2009; Vasiljevic 2009; Glynn and Wakefield 2010; Enos and Lauderdale 2011; Puig and Ginebra 2014). Likewise for the growing areas of synthetic studies in general (Sutton et al. 2008; Cooper and Patall 2009), and specifically for drug discovery cheminformatics (Zafar et al. 2013) and importantly for climate change meta-analytic studies, as reviewed by Hudson (2011) within the framework of phenological climate change research (Hudson 2010).

Furthermore, Beh et al. (2015) introduced two approaches to minimise the impact that an increasing sample size of a 2 × 2 contingency table has on the AAI when analysing the association between the variables given only the aggregate data. Related to this issue is the assessment of the real magnitude of the AAI for a given sample size; see Cheema et al. (2015). These studies currently have not been incorporated with the linear transformation and therefore create an opportunity to further utilise the linear transformation in terms of the adjusted AAI. We leave such work for future consideration.

The development of ecological inference tools for RxC contingency tables has been available for quite some time. Important early contributions to this research include those of Brown and Payne (1986), Rosen et al. (2001) and Ferree (2004) while a more recent discussions have been made by Griener and Quinn (2009), Plescia and De Sio (2018) and Gnaldi et al. (2018). Therefore, extensions of the AAI to a single (and multiple) RxC table is an obvious avenue for future consideration. A preliminary strategy involves making use of components of the partition of the chi-squared statistic for nominal and ordered variables (Lancaster 1953; Beh and Davy 1998) and using the bounds of Dobra and Fienberg (2000, 2001) for ${\text{p}}_{{{\text{ij}}}}$ (for i = 1,..., R and j = 1,..., C) that are based soley on the marginal totals of the RxC contingency table. Note also that Pearson’s chi-squared statistic is just one member of a family of chi-squared statistics. Cressie and Read (1984) provided a general family of divergence statistics where special cases include Pearson’s statistic, the modified chi-squared statistic (Neyman 1949), the Freeman-Tukey statistic (Freeman and Tukey 1950), the log-likelihood ratio statistic (Wilks 1938) and the modified log-likelihood ratio statistic (Kullback 1959). Therefore, the approach described in this paper can be extended to these statistics by using the Cressie-Read family. A more general approach is to adapt the AAI so that it is in terms of Csiszár’s (1967) $\phi$-divergence statistic

$$ {\text{I}}_{\phi } = \mathop \sum \limits_{{{\text{i}} = 1}}^{2} \mathop \sum \limits_{{{\text{j}} = 1}}^{2} {\text{p}}_{{{\text{i}} \bullet }} {\text{p}}_{{ \bullet {\text{j}}}} \phi \left( {\frac{{{\text{p}}_{{{\text{ij}}}} }}{{{\text{p}}_{{{\text{i}} \bullet }} {\text{p}}_{{ \bullet {\text{j}}}} }}} \right) $$

where $\phi \left( {\text{x}} \right)$, for ${\text{x}} > 0$, is a convex function where $\phi \left( 1 \right) = \phi^{{{^{\prime}}\left( 1 \right)}} = 0$, $\phi^{{{^{\prime\prime}}\left( 1 \right)}} = 0$, $0 \times \phi \left( {{\text{p}}/0} \right) = {\text{p}}\mathop {\lim }\nolimits_{{{\text{x}} \to \infty }} \phi \left( {\text{x}} \right)/{\text{x}}$ and $0 \times \phi \left( {0/0} \right) = 0$; see, for example, Cressie and Pardo (2000) and Kateri (2018). The Cressie-Read family of divergence statistics is a special case of ${\text{I}}_{\phi }$ where $\phi \left( {\text{x}} \right) = \left( {{\updelta }\left( {{\updelta } + 1} \right)} \right)^{ - 1} {\text{x}}\left( {{\text{x}}^{{\updelta }} - 1} \right)$ for some value of ${\updelta } \in \left( { - \infty ,\infty } \right)$. Espendiller and Kateri (2016) also used Csiszár’s (1967) $\phi$-divergence statistic in tandem with the odds ratio of a 2 × 2 contingency table. Therefore, there is scope to provide a more general framework than given by Beh et al. (2013) which expressed the AAI in terms of the odds ratio. We shall leave any investigation of these generalisations for a later date.