Asymptotic Characteristics of the Non-Iterative Estimates of the Linear-by-Linear Association Parameter for Ordinal Log-Linear Models

Abstract

Over the past decade, a series of procedures has been introduced to estimate, using a non-iterative method, the linear-by-linear association parameter of an ordinal log-linear model. This paper will examine the two key non-iteratively determined estimates of the parameter for the analysis of the association between the two categorical variables that form a contingency table; these are the log and the Beh-Davy non-iterative estimates, referred to simply as the LogNI and the BDNI estimates, respectively. Such an examination will focus on determining their asymptotic characteristics. To do so, a computational study was undertaken for tables of varying sizes to show that these two estimates are asymptotically unbiased. It is also shown that both estimates are asymptotically normally distributed. On the basis of the standard errors, their relative efficiency was established for the 13 commonly analysed contingency tables that appear throughout the literature.

1. Introduction

Log-linear models (LLMs) are one of the most used tools in the analysis of the association between the categorical variables that form a contingency table. Extensive discussions on their development and application can be found in, for example, [1,2,3]. LLMs have been a focal point of research in the statistical and allied disciplines since 1950s and stem from the work of [4], who considered the association between three categorical variables. Since then, many notable contributions have been made to this area of research including those of [5,6,7,8,9,10,11,12,13]—although [14] points out that the term log-linear model was first coined by [15] (p. 119). Much of this work focused on the study of variables consisting of nominally structured categories. However, there are many practical settings where the variables consist of ordered categories. In the past, an analysis of ordered variables would have been undertaken by treating them as nominal variables (this still is the case in some studies even now). Therefore, where a variable consists of ordered categories, it is important to incorporate such a feature into the development of statistical tools used to analyse the association. Such models are commonly referred to as ordinal log-linear models, or OLLMs. A historical account on the development of OLLMs can be found by considering, for example, [3] (pp. 386–395) and [16] (pp. 145–154). Independent discussions of the origins of OLLMs have also been given by [17] and [18]; both authors focused their attention on a two-way contingency table. The development of OLLMs for a singly ordered (column variable) table was discussed by [18], while [17] considered a doubly ordered table. More recent discussions have been made of the type of OLLMs we consider in this paper, which span a variety of different contexts. For example, the books of [1,3] (Section 10.4), [16] (Section 6.2), [19,20] (Section 3.1), (Section 7.1), [21] (p. 248), [22] (Chapter 4), and [23] (Section 6.1.1) all provide various levels of introductory discussion of OLLMs. One may also refer to [24,25,26,27,28,29,30,31,32,33,34,35] for applications and further insights.

An important aspect of LLMs and OLLMs is the estimation of the parameters from these models. In the case of OLLMs for the doubly ordered contingency table (as considered by [17]), the primary parameter of interest is that which reflects the linear-by-linear association between two ordered variables. Traditionally, the estimation of this parameter has involved using a variety of iterative procedures. The two most used procedures are Newton’s algorithm and iterative proportional fitting; one may refer, for example, to [36,37,38,39,40] for more on these algorithms. One may also consider [16] and [19], who provide a detailed account of these methods and have provided a list of computer programs, including GLIM, SPSS, and SAS, for performing the necessary calculations. Rather than considering an iterative algorithm for estimating the association parameter in OLLMs, one may adopt a more direct approach that is built on a non-iterative framework. One such approach is to consider the method originally proposed by [41] and further studied by [40] through empirical studies. The former authors propose one method for estimating the linear-by-linear parameter through non-iterative (direct) means, while the latter authors considered three further, and related, methods. The accuracy and reliability of the non-iterative estimation for the OLLM of a doubly ordered contingency table was investigated by [42,43], who provided a simulation study of these four methods. They demonstrated that using their non-iterative techniques provides exceptionally stable and reliable estimates of the linear-by-linear association parameter when compared with Newton’s method. The links between these non-iterative methods, using the Box-Cox transformation, were demonstrated by [44]. It is this non-iterative method of estimation which we shall focus our attention on in this paper.

With the foundational mathematical links between the non-iterative estimates of the linear-by-linear association parameter established, and their reliability and stability verified through simulation, this paper identifies the asymptotic characteristics of the two key non-iterative estimates studied [40,41,42,43,44]. We shall be paying particular attention to the bias of the estimators—a topic that was briefly examined by [45]. We also examine here the asymptotic distribution and relative efficiency of each of the two non-iterative estimates. To undertake such a study, this paper is divided into five further sections. In Section 2, a brief review of the non-iterative estimation for OLLMs is given, highlighting the two key estimates for which the asymptotic characteristics are considered. Section 3 includes the theoretical and mathematical development of the asymptotic properties and the asymptotic distributions of the two estimates. An evaluation of these properties, through empirical studies and by simulation, is discussed in Section 4. Section 5 provides a discussion of these results and some remarks on the future research in this area of study.

2. Non-Iterative Estimation of the Linear-by-Linear Association Parameter

For a two-way contingency table, N, with I ordered rows and J ordered columns, denotes the proportion of individuals/units in the (i, j)’th cell as ${\mathrm{p}}_{\mathrm{ij}}={\mathrm{n}}_{\mathrm{ij}}/\mathrm{n}$, where ${\mathrm{n}}_{\mathrm{ij}}$ is the $\left(\mathrm{i},\mathrm{j}\right)$’th cell value of N, for $\mathrm{i}=1, 2, \dots \mathrm{I}$ and $\mathrm{j}=1, 2, \dots , \mathrm{J}$. Therefore, ${{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{I}}{{\displaystyle \sum }}_{\mathrm{j}=1}^{\mathrm{J}}{\mathrm{p}}_{\mathrm{ij}}=1$. Denote ${\mathrm{p}}_{\mathrm{i}•}$ and ${\mathrm{p}}_{•\mathrm{j}}$ as the marginal proportion of the $\mathrm{i}$’th row and $\mathrm{j}$’th column categories, respectively, such that ${{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{I}}{\mathrm{p}}_{\mathrm{i}•}={{\displaystyle \sum }}_{\mathrm{j}=1}^{\mathrm{J}}{\mathrm{p}}_{•\mathrm{j}}=1$. Moreover, let ${\mathrm{m}}_{\mathrm{ij}}$ be the expected cell frequency of the $\left(\mathrm{i},\mathrm{j}\right)$’th cell; for example, when the row and column variables are independent, ${\mathrm{m}}_{\mathrm{ij}}={\mathrm{np}}_{\mathrm{i}•}{\mathrm{p}}_{•\mathrm{j}}$, although any appropriate model of association or correlation may be used to determine ${\mathrm{m}}_{\mathrm{ij}}={\mathrm{np}}_{\mathrm{ij}}$. One such model is the ordinal log-linear model—also known as the uniform association model—for the doubly ordered two-way contingency table, which was considered by [40] (and many of the references given above) and is defined as

$$\ln {\mathrm{m}}_{\mathrm{ij}}=\mathsf{\mu }+{\mathsf{\alpha }}_{\mathrm{i}}+{\mathsf{\beta }}_{\mathrm{j}}+\mathsf{\phi }\left({\mathrm{u}}_{\mathrm{i}}-\overline{\mathrm{u}}\right)\left({\mathrm{v}}_{\mathrm{j}}-\overline{\mathrm{v}}\right).$$

(1)

The parameters are as follows: $\mathsf{\phi }$ is our parameter of interest and reflects the association between the two categorical variables; $\mathsf{\mu }$ is the grand mean of the expected cell frequencies, while ${\mathsf{\alpha }}_{\mathrm{i}}$ and ${\mathsf{\beta }}_{\mathrm{j}}$ are the main effects of the $\mathrm{i}$’th row and $\mathrm{j}$’th column, respectively, such that ${{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{I}}{\mathsf{\alpha }}_{\mathrm{i}}=0$ and ${{\displaystyle \sum }}_{\mathrm{j}=1}^{\mathrm{J}}{\mathsf{\beta }}_{\mathrm{j}}=0$. With there being $\mathrm{I}+\mathrm{J}$ parameters in (1), iterative techniques are commonly used to estimate them. However, the last three can be treated as nuisance parameters [41] (p. 76) and estimated by

$$\hat{\mathsf{\mu }}=\ln \left(\mathrm{n}\right)+\frac{1}{\mathrm{I}}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{I}}\ln \left({\mathrm{p}}_{\mathrm{i}•}\right)+\frac{1}{\mathrm{J}}{\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{J}}\ln \left({\mathrm{p}}_{•\mathrm{j}}\right),$$

$${\hat{\mathsf{\alpha }}}_{\mathrm{i}}=\ln \left({\mathrm{p}}_{\mathrm{i}•}\right)-\frac{1}{\mathrm{I}}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{I}}\ln \left({\mathrm{p}}_{\mathrm{i}•}\right)$$

and

$${\hat{\mathsf{\beta }}}_{\mathrm{j}}=\ln \left({\mathrm{p}}_{•\mathrm{j}}\right)-\frac{1}{\mathrm{J}}{\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{J}}\ln \left({\mathrm{p}}_{•\mathrm{j}}\right).$$

so that there is only a single parameter that requires estimating, that of $\mathsf{\phi }$. This suggests that a direct, non-iterative, method of estimating $\mathsf{\phi }$ can be used, and we shall now turn our attention to how this can be conducted.

To reflect the structure of the ordinal variable, scores are assigned to each of the categories. For (1), ${\mathrm{u}}_{\mathrm{i}}$ and ${\mathrm{v}}_{\mathrm{j}}$ are the scores associated with the $\mathrm{i}$’th row and $\mathrm{j}$’th column and, depending on the nature of the order, are chosen a priori. For example, when the row and column variables consist of increasingly ordered categories then ${\mathrm{u}}_1<\dots <{\mathrm{u}}_{\mathrm{I}}$ and ${\mathrm{v}}_1<\dots <{\mathrm{v}}_{\mathrm{J}}$. The mean of the row scores is denoted by $\overline{\mathrm{u}}={{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{I}}{\mathrm{p}}_{\mathrm{i}•}{\mathrm{u}}_{\mathrm{i}}$, while $\overline{\mathrm{v}}={{\displaystyle \sum }}_{\mathrm{j}=1}^{\mathrm{J}}{\mathrm{p}}_{•\mathrm{j}}{\mathrm{v}}_{\mathrm{j}}$ is the mean of the column scores.

The parameter of interest in (1) is $\mathsf{\phi }$ and is the measure of the linear-by-linear association between the variables since

$$\ln \left(\frac{{\mathrm{m}}_{\mathrm{ij}}{\mathrm{m}}_{\mathrm{i}+1,\mathrm{j}+1}}{{\mathrm{m}}_{\mathrm{i},\mathrm{j}+1}{\mathrm{m}}_{\mathrm{i}+1,\mathrm{j}}}\right)=\mathsf{\phi }\left({\mathrm{u}}_{\mathrm{i}+1}-{\mathrm{u}}_{\mathrm{i}}\right)\left({\mathrm{v}}_{\mathrm{j}+1}-{\mathrm{v}}_{\mathrm{j}}\right).$$

Therefore, when the row and column scores are chosen such that ${\mathrm{u}}_{\mathrm{i}+1}-{\mathrm{u}}_{\mathrm{i}}={\mathrm{v}}_{\mathrm{j}+1}-{\mathrm{v}}_{\mathrm{j}}=1$, $\mathsf{\phi }$ is the common log-odds ratio of the contingency table.

Four non-iterative estimation methods for the parameter $\mathsf{\phi }$ were considered by [40], but we shall restrict our attention to examining the two key methods. The first is the estimator

$${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}=\frac{1}{{\mathsf{\sigma }}_{\mathrm{I}}^2{\mathsf{\sigma }}_{\mathrm{J}}^2}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{I}}{\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{J}}{\mathrm{p}}_{\mathrm{i}•}{\mathrm{p}}_{•\mathrm{j}}\left({\mathrm{u}}_{\mathrm{i}}-\overline{\mathrm{u}}\right)\left({\mathrm{v}}_{\mathrm{j}}-\overline{\mathrm{v}}\right)\ln \left(\frac{{\mathrm{p}}_{\mathrm{ij}}}{{\mathrm{p}}_{\mathrm{i}•}{\mathrm{p}}_{•\mathrm{j}}}\right)$$

(2)

which is referred to as the LogNI estimate due to the presence of the logarithm function in the non-iterative (NI) estimate. The second estimate, and one that was originally proposed by [41], is

$${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}=\frac{1}{{\mathsf{\sigma }}_{\mathrm{I}}^2{\mathsf{\sigma }}_{\mathrm{J}}^2}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{I}}{\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{J}}{\mathrm{p}}_{\mathrm{ij}}\left({\mathrm{u}}_{\mathrm{i}}-\overline{\mathrm{u}}\right)\left({\mathrm{v}}_{\mathrm{j}}-\overline{\mathrm{v}}\right)$$

(3)

and is referred to as the BDNI (Beh-Davy NI) estimate of $\mathsf{\phi }$. For (2) and (3), ${\mathsf{\sigma }}_{\mathrm{I}}^2={{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{I}}{\mathrm{p}}_{\mathrm{i}•}{\mathrm{u}}_{\mathrm{i}}^2-{\overline{\mathrm{u}}}_{\mathrm{I}}^2$ and ${\mathsf{\sigma }}_{\mathrm{J}}^2={{\displaystyle \sum }}_{\mathrm{j}=1}^{\mathrm{J}}{\mathrm{p}}_{•\mathrm{j}}{\mathrm{v}}_{\mathrm{j}}^2-{\overline{\mathrm{v}}}_{\mathrm{J}}^2$. The estimate (3) can be obtained by considering the first-order Taylor series expansion $\ln \left(\frac{{\mathrm{p}}_{\mathrm{ij}}}{{\mathrm{p}}_{\mathrm{i}•}{\mathrm{p}}_{•\mathrm{j}}}\right)\approx \frac{{\mathrm{p}}_{\mathrm{ij}}}{{\mathrm{p}}_{\mathrm{i}•}{\mathrm{p}}_{•\mathrm{j}}}-1$. Equations (2) and (3) are also linked through the Box-Cox transformation and [45] discussed the features of the estimation procedure using this transformation. An excellent overview of the various features of ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$ and ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$ for estimating $\mathsf{\phi }$ from the OLLM of (1) was provided by [46] (p. 221–223) and [47]. The method introduced by [48] was also discussed by [47] and is a method of non-iterative estimation based on weighted mean and Mantel–Haenszel type estimators. A link that exists between ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$ and ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$ and the correspondence analysis of a double ordered two-way contingency table [49,50,51,52,53] was demonstrated by [54]. In doing so, [54] analysed the relationships that exist between the predictors of drug-likeness used in drug discovery to help filter out small molecules (drugs) that may fail clinical trials; see also [55], who applied a non-iterative approach to ordinal log-linear models in the investigation of a log permeability index in drug discovery. Further practical and technical features of ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$ and ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$ not discussed here have been examined in the following conference papers by the current authors: [56,57]. The latter of these involved modelling gender-specific associations between temperament and character traits and symptoms of psychological distress, such as anxiety, psychoticism, and depression, to motivate the value of creating gender-specific treatments for psychological distress.

To evaluate the performance and compare the non-iterative estimates of $\mathsf{\phi }$ using (2) and (3)—and two additional Taylor-series approximations of (2)—with the estimates obtained using the iterative algorithm of Newton’s method, [42,43] performed a series of simulation studies. As part of their studies, strategies needed to be considered for dealing with the presence of random zero-cell frequencies since the logarithm function in (2) leads to non-existent solutions for such cases. There are a variety of different ways in which these zeros can be dealt with. See, for example, [3] (Section 10.6.5), [23] (p. 119) and [58,59,60,61] for a discussion of this issue. In the simulations below, we shall be replacing a zero-cell frequency with 0.05, as [42,43,44] did. Note that determining ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$ when random zero-cell frequencies are present does not suffer from this problem. However, the simulation studies of [42,43] suggest that the accuracy and reliability of ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$ begins to falter as the dimension of the two-way table increases. Therefore, the discussions made in the following section are limited to the comparison of ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$ and ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$.

3. Asymptotic Characteristics of ${\widehat{\mathsf{\phi }}}_{LogNI}$ and ${\widehat{\mathsf{\phi }}}_{BDNI}$

In this section, the mean and variance of the sampling distributions of ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$ and ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$ are determined. Our discussion of the asymptotic distributional properties of the estimates is made using the delta method

$$\sqrt{\mathrm{n}}\left[\mathrm{g}\left({\mathrm{T}}_{\mathrm{n}}\right)-\mathrm{g}\left(\mathsf{\theta }\right)\right]\stackrel{\mathrm{d}}{\to }\mathrm{N}\left(0,\hspace{0.33em}{\left({\mathrm{g}}^{\prime }\left(\mathsf{\theta }\right)\right)}^2\mathrm{V}\left({\mathrm{T}}_{\mathrm{n}}\right)\right)$$

(4)

where $\mathrm{g}\left({\mathrm{T}}_{\mathrm{n}}\right)$ is a function of the variable ${\mathrm{T}}_{\mathrm{n}}$ and $\mathrm{g}\left(\mathsf{\theta }\right)$ is a function of the expected value of ${\mathrm{T}}_{\mathrm{n}}$: see, for example, [61] (p. 240) and [3] (p. 73). We shall also be making use of the following restrictions on ${\mathrm{u}}_{\mathrm{i}}$ and ${\mathrm{v}}_{\mathrm{j}}$

$${\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{I}}{\mathrm{p}}_{\mathrm{i}•}\left({\mathrm{u}}_{\mathrm{i}}-\overline{\mathrm{u}}\right)\left({\mathrm{u}}_{\mathrm{i}}^{\prime }-\overline{\mathrm{u}}\right)=\left\{\begin{array}{c}{\mathsf{\sigma }}_{\mathrm{I}}^2,\hspace{1em}{\mathrm{u}}_{\mathrm{i}}={\mathrm{u}}_{\mathrm{i}}^{\prime }\\ 0,\hspace{1em}{ \mathrm{u}}_{\mathrm{i}}\ne {\mathrm{u}}_{\mathrm{i}}^{\prime }\end{array}\right.\hspace{1em}\hspace{1em}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{I}}{\mathrm{p}}_{•\mathrm{j}}\left({\mathrm{v}}_{\mathrm{j}}-\overline{\mathrm{v}}\right)\left({\mathrm{v}}_{\mathrm{j}}^{\prime }-\overline{\mathrm{v}}\right)=\left\{\begin{array}{c}{\mathsf{\sigma }}_{\mathrm{J}}^2,\hspace{1em}{\mathrm{v}}_{\mathrm{j}}={\mathrm{v}}_{\mathrm{j}}^{\prime }\\ 0,\hspace{1em}{ \mathrm{v}}_{\mathrm{j}}\ne {\mathrm{v}}_{\mathrm{j}}^{\prime }\end{array}\right.$$

(5)

in the derivation of the following results, which determine the mean and variance of ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$ and ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$. The bias of these estimates is also considered.

3.1. Unbiasedness of ${\widehat{\phi }}_{LogNI}$ and ${\widehat{\phi }}_{BDNI}$

Suppose we first consider the non-iterative estimate of $\mathsf{\phi }$ given by ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$. By substituting ${\mathrm{n}}_{\mathrm{ij}}={\mathrm{np}}_{\mathrm{ij}}$ into (2), the estimate can be alternatively expressed as

$$\begin{gathered} {\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}=\frac{1}{{\mathsf{\sigma }}_{\mathrm{I}}^2{\mathsf{\sigma }}_{\mathrm{J}}^2}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{I}}{\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{J}}{\mathrm{p}}_{\mathrm{i}•}{\mathrm{p}}_{•\mathrm{j}}\left({\mathrm{u}}_{\mathrm{i}}-\overline{\mathrm{u}}\right)\left({\mathrm{v}}_{\mathrm{j}}-\overline{\mathrm{v}}\right)\ln \left({\mathrm{n}}_{\mathrm{ij}}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\frac{1}{{\mathsf{\sigma }}_{\mathrm{I}}^2{\mathsf{\sigma }}_{\mathrm{J}}^2}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{I}}{\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{J}}{\mathrm{p}}_{\mathrm{i}•}{\mathrm{p}}_{•\mathrm{j}}\left({\mathrm{u}}_{\mathrm{i}}-\overline{\mathrm{u}}\right)\left({\mathrm{v}}_{\mathrm{j}}-\overline{\mathrm{v}}\right)\ln \left({\mathrm{np}}_{\mathrm{i}•}{\mathrm{p}}_{•\mathrm{j}}\right) \end{gathered}$$

(6)

Suppose that the cell frequencies, represented as elements of the vector $\underset{~}{\mathrm{n}}=\left\{{\mathrm{n}}_{\mathrm{ij}}; \forall \mathrm{i}, \forall \mathrm{j}\right\}$, are independent Poisson random variables. Then, the random variable on the right-hand side of (6) is $\ln \left({\mathrm{n}}_{\mathrm{ij}}\right)$, so that

$$\begin{gathered} \mathrm{E}\left({\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}\right)=\frac{1}{{\mathsf{\sigma }}_{\mathrm{I}}^2{\mathsf{\sigma }}_{\mathrm{J}}^2}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{I}}{\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{J}}{\mathrm{p}}_{\mathrm{i}•}{\mathrm{p}}_{•\mathrm{j}}\left({\mathrm{u}}_{\mathrm{i}}-\overline{\mathrm{u}}\right)\left({\mathrm{v}}_{\mathrm{j}}-\overline{\mathrm{v}}\right)\mathrm{E}\left(\ln \left({\mathrm{n}}_{\mathrm{ij}}\right)\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\frac{1}{{\mathsf{\sigma }}_{\mathrm{I}}^2{\mathsf{\sigma }}_{\mathrm{J}}^2}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{I}}{\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{J}}{\mathrm{p}}_{\mathrm{i}•}{\mathrm{p}}_{•\mathrm{j}}\left({\mathrm{u}}_{\mathrm{i}}-\overline{\mathrm{u}}\right)\left({\mathrm{v}}_{\mathrm{j}}-\overline{\mathrm{v}}\right)\ln \left({\mathrm{np}}_{\mathrm{i}•}{\mathrm{p}}_{•\mathrm{j}}\right). \end{gathered}$$

(7)

By using the delta method, such that

$$\left[\ln \left({\mathrm{n}}_{\mathrm{ij}}\right)-\ln \left({\mathrm{m}}_{\mathrm{ij}}\right)\right]\stackrel{\mathrm{d}}{\to }\mathrm{N}\left(0,\hspace{0.33em}\frac{1}{{\mathrm{nm}}_{\mathrm{ij}}}\right)$$

(8)

the expectation of ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$ can be expressed as

$$\begin{gathered} \mathrm{E}\left({\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}\right)=\frac{1}{{\mathsf{\sigma }}_{\mathrm{I}}^2{\mathsf{\sigma }}_{\mathrm{J}}^2}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{I}}{\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{J}}{\mathrm{p}}_{\mathrm{i}•}{\mathrm{p}}_{•\mathrm{j}}\left({\mathrm{u}}_{\mathrm{i}}-\overline{\mathrm{u}}\right)\left({\mathrm{v}}_{\mathrm{j}}-\overline{\mathrm{v}}\right)\ln \left({\mathrm{m}}_{\mathrm{ij}}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\frac{1}{{\mathsf{\sigma }}_{\mathrm{I}}^2{\mathsf{\sigma }}_{\mathrm{J}}^2}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{I}}{\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{J}}{\mathrm{p}}_{\mathrm{i}•}{\mathrm{p}}_{•\mathrm{j}}\left({\mathrm{u}}_{\mathrm{i}}-\overline{\mathrm{u}}\right)\left({\mathrm{v}}_{\mathrm{j}}-\overline{\mathrm{v}}\right)\ln \left({\mathrm{np}}_{\mathrm{i}•}{\mathrm{p}}_{•\mathrm{j}}\right) \end{gathered}$$

(9)

There are many ways to model the cell values of the contingency table. By considering an unsaturated (linear-by-linear) association model of the type described in [41], (Equation (19)), ${\mathrm{m}}_{\mathrm{ij}}$ is the expected cell frequency of the (i, j)’th cell where

$${\mathrm{m}}_{\mathrm{ij}}={\mathrm{np}}_{\mathrm{i}•}{\mathrm{p}}_{•\mathrm{j}}\left(1+\mathsf{\phi }\frac{\left({\mathrm{u}}_{\mathrm{i}}-\overline{\mathrm{u}}\right)}{{\mathsf{\sigma }}_{\mathrm{I}}}\frac{\left({\mathrm{v}}_{\mathrm{j}}-\overline{\mathrm{v}}\right)}{{\mathsf{\sigma }}_{\mathrm{J}}}\right).$$

(10)

The link between this model and the OLLM of (1) was also shown by [40]. Substituting (10) into (9) yields, after simplification,

$$\mathrm{E}\left({\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}\right)=\frac{1}{{\mathsf{\sigma }}_{\mathrm{I}}^2{\mathsf{\sigma }}_{\mathrm{J}}^2}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{I}}{\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{J}}{\mathrm{p}}_{\mathrm{i}•}{\mathrm{p}}_{•\mathrm{j}}\left({\mathrm{u}}_{\mathrm{i}}-\overline{\mathrm{u}}\right)\left({\mathrm{v}}_{\mathrm{j}}-\overline{\mathrm{v}}\right)\ln \left(1+\mathsf{\phi }\frac{\left({\mathrm{u}}_{\mathrm{i}}-\overline{\mathrm{u}}\right)}{{\mathsf{\sigma }}_{\mathrm{I}}}\frac{\left({\mathrm{v}}_{\mathrm{j}}-\overline{\mathrm{v}}\right)}{{\mathsf{\sigma }}_{\mathrm{J}}}\right).$$

(11)

By using the first-order Taylor series expansion of the logarithm

$$\ln \left(1+\mathsf{\phi }\frac{\left({\mathrm{u}}_{\mathrm{i}}-\overline{\mathrm{u}}\right)}{{\mathsf{\sigma }}_{\mathrm{I}}}\frac{\left({\mathrm{v}}_{\mathrm{j}}-\overline{\mathrm{v}}\right)}{{\mathsf{\sigma }}_{\mathrm{J}}}\right)\approx \mathsf{\phi }\frac{\left({\mathrm{u}}_{\mathrm{i}}-\overline{\mathrm{u}}\right)}{{\mathsf{\sigma }}_{\mathrm{I}}}\frac{\left({\mathrm{v}}_{\mathrm{j}}-\overline{\mathrm{v}}\right)}{{\mathsf{\sigma }}_{\mathrm{J}}}$$

and after a little algebra using (5), we obtain the asymptotic expectation of ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$

$$\mathrm{E}\left({\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}\right)=\mathsf{\phi }.$$

(12)

Equation (12) asserts that ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$ is (approximately) an unbiased estimator of the linear-by-linear association parameter $\mathsf{\phi }$ in the OLLM of (1).

We can similarly show that ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$ is also an unbiased estimate of $\mathsf{\phi }$. Following the same argument made above for ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$, the expected value for ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$ can be derived by taking the expectation of both sides of (3), yielding

$$\mathrm{E}\left({\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}\right)=\frac{1}{{\mathrm{n}\mathsf{\sigma }}_{\mathrm{I}}^2{\mathsf{\sigma }}_{\mathrm{J}}^2}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{I}}{\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{J}}{\mathrm{m}}_{\mathrm{ij}}\left({\mathrm{u}}_{\mathrm{i}}-\overline{\mathrm{u}}\right)\left({\mathrm{v}}_{\mathrm{j}}-\overline{\mathrm{v}}\right).$$

(13)

By substituting (10) into (13), and after some simplification, we obtain the expectation of the BDNI estimate

$$\mathrm{E}\left({\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}\right)=\frac{\mathsf{\phi }}{{\mathsf{\sigma }}_{\mathrm{I}}^2{\mathsf{\sigma }}_{\mathrm{J}}^2}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{I}}{\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{J}}{\mathrm{p}}_{\mathrm{i}•}{\mathrm{p}}_{•\mathrm{j}}{\left({\mathrm{u}}_{\mathrm{i}}-\overline{\mathrm{u}}\right)}^2{\left({\mathrm{v}}_{\mathrm{j}}-\overline{\mathrm{v}}\right)}^2.$$

By considering the restriction (5) imposed on ${\mathrm{u}}_{\mathrm{i}}$ and ${\mathrm{v}}_{\mathrm{j}}$, this expectation simplifies to

$$\mathrm{E}\left({\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}\right)=\mathsf{\phi }.$$

(14)

3.2. Variance of ${\widehat{\phi }}_{LogNI}$ and ${\widehat{\phi }}_{BDNI}$

We now turn our attention to the derivation of the variance of ${\widehat{\phi }}_{LogNI}$ and ${\widehat{\phi }}_{BDNI}$. Consider first the variance of ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$, which is expressed as

$$\begin{gathered} \mathrm{V}\left({\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}\right)=\mathrm{V}[\frac{1}{{\mathsf{\sigma }}_{\mathrm{I}}^2{\mathsf{\sigma }}_{\mathrm{J}}^2}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{I}}{\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{J}}{\mathrm{p}}_{\mathrm{i}•}{\mathrm{p}}_{•\mathrm{j}}\left({\mathrm{u}}_{\mathrm{i}}-\overline{\mathrm{u}}\right)\left({\mathrm{v}}_{\mathrm{j}}-\overline{\mathrm{v}}\right)\ln \left({\mathrm{n}}_{\mathrm{ij}}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\frac{1}{{\mathsf{\sigma }}_{\mathrm{I}}^2{\mathsf{\sigma }}_{\mathrm{J}}^2}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{I}}{\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{J}}{\mathrm{p}}_{\mathrm{i}•}{\mathrm{p}}_{•\mathrm{j}}\left({\mathrm{u}}_{\mathrm{i}}-\overline{\mathrm{u}}\right)\left({\mathrm{v}}_{\mathrm{j}}-\overline{\mathrm{v}}\right)\ln \left({\mathrm{np}}_{\mathrm{i}•}{\mathrm{p}}_{•\mathrm{j}}\right)]. \end{gathered}$$

One may note that the second term on the right-hand side is a constant. Therefore, by expanding the above variance expression, the cross-product terms are

$$\frac{1}{{\mathsf{\sigma }}_{\mathrm{I}}^2{\mathsf{\sigma }}_{\mathrm{J}}^2}{\displaystyle \sum }_{\mathrm{i}\ne \mathrm{k}}^{\mathrm{I}}{\displaystyle \sum }_{\mathrm{j}\ne \mathrm{l}}^{\mathrm{J}}{\mathrm{p}}_{\mathrm{i}•}{\mathrm{p}}_{•\mathrm{j}}{\mathrm{p}}_{\mathrm{k}•}{\mathrm{p}}_{•\mathrm{l}}\left({\mathrm{u}}_{\mathrm{i}}-\overline{\mathrm{u}}\right)\left({\mathrm{u}}_{\mathrm{k}}-\overline{\mathrm{u}}\right)\left({\mathrm{v}}_{\mathrm{j}}-\overline{\mathrm{v}}\right)\left({\mathrm{v}}_{\mathrm{l}}-\overline{\mathrm{v}}\right)\mathrm{Cov}\left(\ln \left({\mathrm{n}}_{\mathrm{ij}}\right),\hspace{0.33em}\ln \left({\mathrm{n}}_{\mathrm{kl}}\right)\right)$$

and are zero from (5). Thus, by using the delta method and simplifying, the variance of ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$ is

$$\mathrm{V}\left({\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}\right)=\frac{1}{{\mathrm{n}\mathsf{\sigma }}_{\mathrm{I}}^2{\mathsf{\sigma }}_{\mathrm{J}}^2}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{I}}{\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{J}}{\mathrm{p}}_{\mathrm{i}•}{\mathrm{p}}_{•\mathrm{j}}{\left({\mathrm{u}}_{\mathrm{i}}-\overline{\mathrm{u}}\right)}^2{\left({\mathrm{v}}_{\mathrm{j}}-\overline{\mathrm{v}}\right)}^2{\mathrm{m}}_{\mathrm{ij}}^{-1}.$$

(15)

By following a similar derivation, the variance of ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$ is

$$\mathrm{V}\left({\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}\right)=\frac{1}{{\mathrm{n}}^2{\mathsf{\sigma }}_{\mathrm{I}}^2{\mathsf{\sigma }}_{\mathrm{J}}^2}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{I}}{\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{J}}{\mathrm{m}}_{\mathrm{ij}}{\left({\mathrm{u}}_{\mathrm{i}}-\overline{\mathrm{u}}\right)}^2{\left({\mathrm{v}}_{\mathrm{j}}-\overline{\mathrm{v}}\right)}^2.$$

(16)

When there is complete independence between the two categorical variables, (15) and (16) simplify to zero.

3.3. Asymptotic Distribution of ${\widehat{\phi }}_{LogNI}$ and ${\widehat{\phi }}_{BDNI}$

As we discussed when determining the expectation and variance of the estimates of ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$ and ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$, here we shall also use the delta method to determine their asymptotic distribution under the assumption that the observed cell frequencies are independent Poisson random variables. For large samples, the delta method implies that the asymptotic distributions of ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$ and ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$ are (approximately) true normal distributions, as (17) and (18) show, respectively:

$$\left({\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}-\mathsf{\phi }\right)\hspace{0.33em}\hspace{0.33em}\stackrel{\mathrm{d}}{\to }\hspace{0.33em}\hspace{0.33em}\mathrm{N}\left(0,\hspace{0.33em}\frac{1}{{\mathrm{n}\mathsf{\sigma }}_{\mathrm{I}}^2{\mathsf{\sigma }}_{\mathrm{J}}^2}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{I}}{\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{J}}{\mathrm{p}}_{\mathrm{i}•}{\mathrm{p}}_{•\mathrm{j}}{\left({\mathrm{u}}_{\mathrm{i}}-\overline{\mathrm{u}}\right)}^2{\left({\mathrm{v}}_{\mathrm{j}}-\overline{\mathrm{v}}\right)}^2{\mathrm{m}}_{\mathrm{ij}}^{-1}\right)$$

(17)

and

$$\left({\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}-\mathsf{\phi }\right)\hspace{0.33em}\hspace{0.33em}\stackrel{\mathrm{d}}{\to }\hspace{0.33em}\hspace{0.33em}\mathrm{N}\left(0,\hspace{0.33em}\frac{1}{{\mathrm{n}}^2{\mathsf{\sigma }}_{\mathrm{I}}^2{\mathsf{\sigma }}_{\mathrm{J}}^2}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{I}}{\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{J}}{\mathrm{m}}_{\mathrm{ij}}{\left({\mathrm{u}}_{\mathrm{i}}-\overline{\mathrm{u}}\right)}^2{\left({\mathrm{v}}_{\mathrm{j}}-\overline{\mathrm{v}}\right)}^2\right)$$

(18)

By comparing the expressions for (15) and (16) with (17) and (18), it can be observed that for ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$, the asymptotic variance is smaller than the exact variance. This implies that as the sample size increases, the accuracy and reliability of ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$ improves.

4. Empirical and Computational Study

To study the asymptotic characteristics of ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$ and ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$, 10,000 contingency tables of varying sizes are randomly generated. We consider contingency tables ranging in size from 2 × 2 to 5 × 5 since in many practical studies involving two-way contingency tables, rarely are tables of a size greater than 5 × 5 considered. However, in such cases, the conclusions from our study here are equally applicable. To reflect the varying magnitude of the cell frequencies (which are assumed to be a Poisson random variable), these values are randomly generated using various values of the Poisson parameter λ. Here, we consider λ values of 1.5, 5, 20, and 50; a small value of λ reflects, on average, small cell frequencies, while large values of λ reflect, on average, large cell frequencies. For each combination of dimension and λ, the non-iterative estimates of $\phi$ are calculated, yielding ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$ and ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$. Therefore, a population of 10,000 ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$ values and a population of 10,000 ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$ values is obtained. The mean of the ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$ values and the ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$ values is denoted by ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}^{\mathrm{p}}$ and ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}^{\mathrm{p}}$, respectively; note that ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}^{\mathrm{p}}$ and ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}^{\mathrm{p}}$ are therefore the population means of the two non-iterative estimates. From each population of 10,000 non-iterative estimate values, samples of ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$ and ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$ are randomly selected; small samples of ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$ and ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$ are deemed to be those representing 0.1%, 1%, and 5% of the population. Similarly, moderate sized samples (15%) and large samples (30% and 50%) are also considered. For each sample size, the mean of the sampling distribution of the ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$ and ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$ values obtained are denoted by ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}^{\mathrm{s}}$ and ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}^{\mathrm{s}}$, respectively.

4.1. Unbiasedness of ${\widehat{\phi }}_{LogNI}$ and ${\widehat{\phi }}_{BDNI}$

Consider Table 1, which summarises the computational study undertaken to verify the unbiasedness of the LogNI estimate; see (12). It can be observed that for smaller values of $\mathsf{\lambda }$ and small samples, the difference between ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}^{\mathrm{p}}$ and ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}^{\mathrm{s}}$ is relatively large. This difference arises because small values of $\mathsf{\lambda }$ increase the chances of a zero-cell value arising by chance. Even with the addition of 0.05 to account for the presence of the logarithm function in (2), it means that such a value may be a relatively serious remedy; to overcome this remedy, one could substitute for the zero-cell frequency an even smaller non-negative value; for example, [58] (p. 13) suggested replacing a random zero-cell frequency with 0.00000001. However, as the sample size increases, even for smaller $\mathsf{\lambda }$, the population mean of the estimates obtained from (2), ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}^{\mathrm{p}}$, and the mean of the sampling distribution, ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}^{\mathrm{s}}$, are virtually identical. It is also to be noted that for a larger value of $\mathsf{\lambda }$, the difference between ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}^{\mathrm{p}}$ and ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}^{\mathrm{s}}$ is zero for large samples of ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$ and is minimal even for the small sample sizes. Similar conclusions can be reached concerning the bias of the estimates, ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$. The unbiasedness of this non-iterative parameter estimate is apparent from observing the virtually zero difference between ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}^{\mathrm{p}}$ and ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}^{\mathrm{s}}$ that is summarised in Table 2 for the varying contingency table sizes and Poisson parameter $\mathsf{\lambda }$.

Table 1. Property of unbiasedness of ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$ for varying dimensions, sample sizes, and λ s of simulated contingency tables.

		Size of Contingency Table
λ	n	2 × 2		3 × 3		4 × 4		5 × 5		2 × 3		2 × 4		2 × 5		3 × 4		3 × 5		4 × 5
		${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}^{\mathrm{p}}$	${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}^{\mathrm{s}}$	${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}^{\mathrm{p}}$	${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}^{\mathrm{s}}$	${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}^{\mathrm{p}}$	${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}^{\mathrm{s}}$	${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}^{\mathrm{p}}$	${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}^{\mathrm{s}}$	${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}^{\mathrm{p}}$	${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}^{\mathrm{s}}$	${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}^{\mathrm{p}}$	${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}^{\mathrm{s}}$	${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}^{\mathrm{p}}$	${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}^{\mathrm{s}}$	${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}^{\mathrm{p}}$	${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}^{\mathrm{s}}$	${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}^{\mathrm{p}}$	${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}^{\mathrm{s}}$	${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}^{\mathrm{p}}$	${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}^{\mathrm{s}}$
1.5	0.1%	−0.0017	−0.0018	0.0026	0.0043	0.0033	0.0013	0.0040	0.0054	−0.0091	−0.0134	0.0025	0.0022	0.0044	0.0019	0.0073	0.0080	0.0024	0.0023	−0.0024	−0.0070
	1%	−0.0014	−0.0011	−0.0008	−0.0001	0.0016	0.0011	−0.0058	−0.0069	0.0058	0.0061	−0.0010	−0.0010	0.0076	0.0083	−0.0040	−0.0049	−0.0075	−0.0078	0.0038	0.0041
	5%	0.0021	0.0024	−0.0016	−0.0014	−0.0042	−0.0034	−0.0039	−0.0041	0.0015	0.0011	0.0010	0.0011	−0.0029	−0.0024	−0.0042	−0.0042	−0.0065	−0.0063	−0.0032	−0.0033
	15%	−0.0002	−0.0002	0.0095	0.0093	0.0025	0.0021	0.0044	0.0044	0.0096	0.0099	−0.0028	−0.0027	0.0023	0.0024	0.0043	0.0047	0.0068	0.0071	0.0039	0.0044
	30%	−0.0015	−0.0015	−0.0066	−0.0066	−0.0030	−0.0030	−0.0027	−0.0027	0.0013	0.0013	−0.0007	−0.0007	−0.0042	−0.0042	0.0043	0.0043	−0.0009	−0.0009	−0.0031	−0.0031
	50%	0.0016	0.0016	−0.0046	−0.0046	0.0034	0.0034	0.0043	0.0043	0.0047	0.0047	−0.0016	−0.0016	−0.0058	−0.0058	0.0028	0.0028	0.0011	0.0011	−0.0001	−0.0001
5	0.1%	0.0002	0.0004	−0.0008	0.0021	0.0001	−0.0002	0.0007	0.0028	−0.0041	−0.0028	0.0022	−0.0011	0.0029	0.0028	−0.0029	−0.0032	−0.0018	−0.0017	0.0001	0.0018
	1%	−0.0007	−0.0004	−0.0007	−0.0012	0.0025	0.0026	0.0017	0.0015	0.0066	0.0074	−0.0018	−0.0019	0.0026	0.0024	0.0006	0.0007	0.0011	0.0014	−0.0010	−0.0008
	5%	0.0002	0.0003	−0.0029	−0.0029	0.0007	0.0009	0.0000	0.0000	−0.0014	−0.0018	−0.0004	−0.0007	−0.0012	−0.0013	0.0022	0.0024	−0.0013	−0.0011	0.0002	0.0002
	15%	−0.0002	−0.0002	0.0020	0.0020	−0.0015	−0.0016	−0.0013	−0.0012	0.0019	0.0017	0.0030	0.0032	−0.0015	−0.0013	−0.0007	−0.0007	−0.0028	−0.0029	0.0014	0.0012
	30%	−0.0002	−0.0002	−0.0027	−0.0027	−0.0007	−0.0007	0.0001	0.0001	0.0028	0.0028	−0.0015	−0.0015	0.0005	0.0005	−0.0017	−0.0017	−0.0003	−0.0003	−0.0034	−0.0034
	50%	−0.0026	−0.0026	−0.0005	−0.0007	0.0023	0.0023	0.0027	0.0027	0.0016	0.0016	−0.0003	−0.0003	−0.0009	−0.0009	0.0013	0.0013	0.0018	0.0018	0.0021	0.0021
20	0.1%	−0.0004	−0.0013	0.0011	−0.0005	−0.0002	−0.0001	0.0012	0.0017	−0.0002	−0.0009	−0.0010	−0.0001	−0.0003	−0.0003	−0.0002	0.0004	0.0010	0.0005	−0.0001	−0.0004
	1%	0.0007	0.0004	−0.0001	−0.0003	−0.0006	−0.0007	0.0003	0.0003	0.0006	0.0004	0.0006	0.0006	−0.0002	0.0002	−0.0001	−0.0001	0.0003	0.0003	0.0004	0.0004
	5%	0.0019	0.0021	−0.0008	−0.0007	0.0010	0.0010	0.0005	0.0004	−0.0009	−0.0011	−0.0013	−0.0015	0.0006	0.0008	0.0004	0.0004	−0.0003	−0.0003	−0.0008	−0.0008
	15%	0.0018	0.0018	0.0003	0.0003	−0.0009	−0.0010	0.0003	0.0003	0.0018	0.0018	0.0005	0.0006	−0.0010	−0.0010	0.0007	0.0006	0.0011	0.0011	−0.0007	−0.0007
	30%	−0.0014	−0.0014	0.0005	0.0005	0.0003	0.0003	−0.0001	−0.0001	−0.0004	−0.0004	0.0006	0.0006	0.0010	0.0010	0.0014	0.0014	0.0004	0.0004	−0.0004	−0.0004
	50%	−0.0013	−0.0013	−0.0024	−0.0024	−0.0004	−0.0004	−0.0002	−0.0002	0.0004	0.0004	0.0013	0.0013	−0.0006	−0.0006	0.0017	0.0017	−0.0013	−0.0013	0.0007	0.0007
50	0.1%	0.0003	0.0018	−0.0001	−0.0001	0.0004	0.0010	−0.0004	−0.0006	−0.0007	−0.0002	0.0004	0.0019	0.0004	0.0002	−0.0004	−0.0004	0.0003	0.0004	−0.0006	−0.0005
	1%	−0.0007	−0.0003	0.0004	0.0004	0.0001	0.0001	−0.0002	−0.0001	−0.0007	−0.0006	−0.0001	−0.0003	−0.0002	−0.0002	−0.0001	−0.0002	0.0005	0.0005	0.0002	0.0002
	5%	0.0005	0.0004	0.0013	0.0012	−0.0005	−0.0005	−0.0001	−0.0001	0.0003	0.0004	−0.0005	−0.0005	0.0012	0.0012	0.0006	0.0006	0.0002	0.0002	−0.0004	−0.0004
	15%	0.0004	0.0004	−0.0006	−0.0006	0.0001	0.0001	−0.0003	−0.0003	−0.0008	−0.0008	0.0008	0.0008	0.0004	0.0004	−0.0003	−0.0002	−0.0002	−0.0002	−0.0005	−0.0005
	30%	0.0009	0.0009	0.0003	0.0003	−0.0003	−0.0003	0.0004	0.0004	−0.0006	−0.0006	0.0002	0.0002	0.0006	0.0006	0.0000	0.0000	0.0001	0.0001	0.0000	0.0000
	50%	0.0016	0.0016	0.0005	0.0005	−0.0003	−0.0003	−0.0004	−0.0004	−0.0003	−0.0003	0.0004	0.0004	0.0001	0.0001	0.0003	0.0003	0.0001	0.0001	−0.0002	−0.0002

Table 2. Property of unbiasedness of ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$ for varying dimensions and λ s of simulated contingency tables.

		Size of Contingency Table
λ	n	2 × 2		3 × 3		4 × 4		5 × 5		2 × 3		2 × 4		2 × 5		3 × 4		3 × 5		4 × 5
		${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}^{\mathrm{p}}$	${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}^{\mathrm{s}}$	${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}^{\mathrm{p}}$	${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}^{\mathrm{s}}$	${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}^{\mathrm{p}}$	${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}^{\mathrm{s}}$	${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}^{\mathrm{p}}$	${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}^{\mathrm{s}}$	${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}^{\mathrm{p}}$	${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}^{\mathrm{s}}$	${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}^{\mathrm{p}}$	${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}^{\mathrm{s}}$	${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}^{\mathrm{p}}$	${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}^{\mathrm{s}}$	${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}^{\mathrm{p}}$	${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}^{\mathrm{s}}$	${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}^{\mathrm{p}}$	${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}^{\mathrm{s}}$	${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}^{\mathrm{p}}$	${\widehat{\mathsf{\phi }}}_{\mathrm{BDNI}}^{\mathrm{s}}$
1.5	0.1%	−0.0010	−0.0020	0.0016	0.0025	0.0019	0.0023	0.0018	0.0015	−0.0048	−0.0012	0.0016	0.0017	0.0036	0.0002	0.0043	0.0040	0.0014	0.0045	−0.0021	−0.0002
	1%	−0.0010	−0.0004	0.0000	−0.0001	0.0009	0.0020	−0.0035	−0.0031	0.0018	0.0031	−0.0006	−0.0007	0.0040	0.0042	−0.0008	−0.0009	−0.0045	−0.0051	0.0025	0.0035
	5%	0.0010	0.0010	−0.0012	−0.0011	−0.0024	−0.0024	−0.0001	−0.0001	0.0008	0.0011	0.0003	0.0001	−0.0012	−0.0011	−0.0031	−0.0031	−0.0031	−0.0033	−0.0030	−0.0031
	15%	−0.0003	−0.0003	0.0042	0.0043	0.0024	0.0023	0.0036	0.0036	0.0057	0.0053	−0.0017	−0.0017	0.0019	0.0023	0.0007	0.0007	0.0043	0.0043	0.0014	0.0013
	30%	−0.0011	−0.0012	−0.0038	−0.0038	−0.0023	−0.0023	−0.0020	−0.0020	0.0009	0.0009	−0.0001	−0.0001	−0.0025	−0.0025	0.0033	0.0033	−0.0009	−0.0009	−0.0012	−0.0012
	50%	0.0007	0.0007	−0.0037	−0.0037	0.0024	0.0024	0.0021	0.0021	0.0032	0.0032	−0.0016	−0.0016	−0.0030	−0.0030	0.0011	0.0011	0.0010	0.0010	0.0001	−0.0001
5	0.1%	0.0001	0.0011	−0.0014	−0.0002	0.0002	0.0021	0.0005	0.0001	−0.0035	−0.0025	0.0018	0.0011	0.0029	0.0016	−0.0020	−0.0006	−0.0011	−0.0009	0.0003	0.0012
	1%	−0.0007	−0.0007	−0.0007	−0.0005	0.0022	0.0020	0.0010	0.0017	0.0045	0.0047	−0.0015	−0.0025	0.0020	0.0022	0.0009	0.0012	0.0014	0.0015	−0.0008	−0.0010
	5%	0.0001	0.0002	−0.0022	−0.0022	0.0006	0.0009	−0.0001	−0.0002	−0.0010	−0.0012	−0.0006	−0.0006	−0.0005	−0.0007	0.0020	0.0020	−0.0009	−0.0009	0.0002	0.0001
	15%	−0.0001	−0.0001	0.0012	0.0012	−0.0010	−0.0010	−0.0005	−0.0005	0.0015	0.0017	0.0021	0.0021	−0.0011	−0.0013	0.0000	0.0000	−0.0027	−0.0028	0.0013	0.0013
	30%	−0.0002	−0.0002	−0.0027	−0.0027	−0.0009	−0.0009	0.0003	0.0003	0.0024	0.0024	−0.0011	−0.0011	−0.0003	−0.0003	−0.0008	−0.0008	−0.0001	−0.0001	−0.0023	−0.0023
	50%	−0.0018	−0.0018	−0.0014	−0.0014	0.0016	0.0016	0.0027	0.0027	0.0022	0.0022	−0.0001	−0.0001	−0.0006	−0.0006	0.0007	0.0007	0.0019	0.0019	0.0013	0.0013
20	0.1%	−0.0004	−0.0018	0.0011	0.0009	−0.0002	−0.0004	0.0011	0.0008	−0.0002	−0.0012	−0.0010	0.0000	−0.0003	−0.0003	−0.0002	0.0000	0.0009	0.0012	−0.0001	−0.0001
	1%	0.0007	0.0012	−0.0002	−0.0002	−0.0007	−0.0006	0.0004	0.0004	0.0005	0.0009	0.0005	0.0002	−0.0001	−0.0001	−0.0001	0.0003	0.0002	0.0003	0.0004	0.0004
	5%	0.0019	0.0020	−0.0007	−0.0006	0.0010	0.0010	0.0004	0.0004	−0.0009	−0.0009	−0.0012	−0.0012	0.0006	0.0004	0.0005	0.0004	−0.0003	−0.0003	−0.0008	−0.0007
	15%	0.0018	0.0018	0.0003	0.0003	−0.0009	−0.0009	0.0003	0.0003	0.0017	0.0017	0.0005	0.0005	−0.0010	−0.0009	0.0006	0.0006	0.0012	0.0012	−0.0007	−0.0007
	30%	−0.0014	−0.0014	0.0006	0.0006	0.0002	0.0002	−0.0001	−0.0001	−0.0004	−0.0004	0.0006	0.0006	0.0009	0.0009	0.0014	0.0014	0.0003	0.0003	−0.0004	−0.0004
	50%	−0.0014	−0.0014	−0.0023	−0.0023	−0.0004	−0.0004	−0.0002	−0.0002	0.0004	0.0004	0.0013	0.0013	−0.0005	−0.0005	0.0017	0.0017	−0.0012	−0.0012	0.0007	0.0007
50	0.1%	0.0003	−0.0012	−0.0001	−0.0003	0.0004	0.0006	−0.0004	−0.0003	−0.0007	−0.0003	0.0004	0.0004	0.0004	−0.0003	−0.0004	−0.0004	0.0003	0.0008	−0.0006	0.0001
	1%	−0.0007	−0.0008	0.0004	0.0003	0.0001	0.0001	−0.0002	−0.0001	−0.0007	−0.0007	−0.0001	−0.0002	−0.0002	0.0000	−0.0001	−0.0002	0.0004	0.0005	0.0002	0.0002
	5%	0.0004	0.0004	0.0012	0.0012	−0.0005	−0.0005	−0.0001	−0.0001	0.0003	0.0003	−0.0005	−0.0005	0.0012	0.0013	0.0006	0.0007	0.0002	0.0001	−0.0004	−0.0004
	15%	0.0004	0.0004	−0.0006	−0.0006	0.0001	0.0001	−0.0003	−0.0003	−0.0008	−0.0008	0.0008	0.0008	0.0004	0.0004	−0.0002	−0.0002	−0.0002	−0.0002	−0.0004	−0.0004
	30%	0.0009	0.0009	0.0003	0.0003	−0.0003	−0.0003	0.0004	0.0004	−0.0006	−0.0006	0.0002	0.0002	0.0006	0.0006	−0.0001	−0.0001	0.0001	0.0001	0.0000	0.0000
	50%	0.0016	0.0016	0.0005	0.0005	−0.0002	−0.0002	−0.0004	−0.0003	−0.0003	−0.0003	0.0004	0.0004	0.0001	0.0001	0.0003	0.0003	0.0001	0.0001	−0.0002	−0.0002

4.2. Asymptotic Variances of ${\widehat{\phi }}_{LogNI}$ and ${\widehat{\phi }}_{BDNI}$

The variance of the estimates ${\widehat{\phi }}_{LogNI}$ and ${\widehat{\phi }}_{BDNI}$, defined by (17) and (18), respectively, is summarised in Table 3. The table shows that the variance of ${\widehat{\phi }}_{BDNI}$ becomes larger as one considers smaller values of λ and small-sized tables (in particular, for those of size 2 × 2, 2 × 3, 2 × 4, and 2 × 5). This is because when λ has a smaller value (λ = 1.5, say), the cell values that are close to, or exactly, zero dominate the association structure between the variables of the contingency table. Its impact therefore leads to ${\widehat{\phi }}_{BDNI}$ being a poor performing estimate of $\phi$. However, for moderate and larger-sized contingency tables, the variance of ${\widehat{\phi }}_{BDNI}$ is consistently smaller than the variance of ${\widehat{\phi }}_{LogNI}$ for all values of λ and contingency table sizes. When the value of λ is large (λ = 50, say), the variances for ${\widehat{\phi }}_{LogNI}$ and ${\widehat{\phi }}_{BDNI}$ are virtually identical.

Table 3. Asymptotic variances of ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$ and ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$ for varying dimensions and λ s of simulated contingency tables.

		Size of Contingency Table
λ	n	2 × 2		3 × 3		4 × 4		5 × 5		2 × 3		2 × 4		2 × 5		3 × 4		3 × 5		4 × 5
		$\mathrm{V}\left({\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}\right)$	$\mathrm{V}\left({\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}\right)$	$\mathrm{V}\left({\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}\right)$	$\mathrm{V}\left({\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}\right)$	$\mathrm{V}\left({\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}\right)$	$\mathrm{V}\left({\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}\right)$	$\mathrm{V}\left({\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}\right)$	$\mathrm{V}\left({\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}\right)$	$\mathrm{V}\left({\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}\right)$	$\mathrm{V}\left({\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}\right)$	$\mathrm{V}\left({\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}\right)$	$\mathrm{V}\left({\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}\right)$	$\mathrm{V}\left({\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}\right)$	$\mathrm{V}\left({\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}\right)$	$\mathrm{V}\left({\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}\right)$	$\mathrm{V}\left({\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}\right)$	$\mathrm{V}\left({\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}\right)$	$\mathrm{V}\left({\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}\right)$	$\mathrm{V}\left({\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}\right)$	$\mathrm{V}\left({\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}\right)$
1.5	0.1%	1.20 × 109	1.31 × 10−1	9.82 × 10−3	1.35 × 10−2	2.15 × 10−3	2.50 × 10−3	7.92 × 10−4	8.76 × 10−4	1.23 × 103	3.34 × 10−2	2.84 × 10−1	1.47 × 10−2	1.05 × 10−2	7.51 × 10−3	4.11 × 10−3	5.00 × 10−3	2.45 × 10−3	2.87 × 10−3	1.28 × 10−3	1.45 × 10−3
	1%	1.21 × 109	1.30 × 10−1	1.12 × 10−2	1.16 × 10−2	2.09 × 10−3	2.41 × 10−3	7.96 × 10−4	8.79 × 10−4	1.24 × 103	3.55 × 10−2	1.64 × 10−1	1.43 × 10−2	6.39 × 10−3	7.51 × 10−3	4.10 × 10−3	5.10 × 10−3	2.44 × 10−3	2.83 × 10−3	1.29 × 10−3	1.45 × 10−3
	5%	1.23 × 109	1.32 × 10−1	8.63 × 10−3	1.23 × 10−2	2.11 × 10−3	2.44 × 10−3	7.92 × 10−4	8.71 × 10−4	1.34 × 103	3.27 × 10−2	2.07 × 100	1.45 × 10−2	6.34 × 10−3	7.44 × 10−3	4.27 × 10−3	5.16 × 10−3	2.43 × 10−3	2.83 × 10−3	1.29 × 10−3	1.46 × 10−3
	15%	9.77 × 108	1.37 × 10−1	8.62 × 10−3	1.49 × 10−2	2.09 × 10−3	2.43 × 10−3	7.97 × 10−4	8.80 × 10−4	1.25 × 100	3.50 × 10−2	1.37 × 10−1	1.45 × 10−2	6.36 × 10−3	7.67 × 10−3	4.14 × 10−3	4.99 × 10−3	2.43 × 10−3	2.85 × 10−3	1.29 × 10−3	1.45 × 10−3
	30%	5.50 × 1010	1.21 × 10−1	8.56 × 10−3	1.11 × 10−2	2.10 × 10−3	2.45 × 10−3	7.97 × 10−4	8.77 × 10−4	3.93 × 101	3.23 × 10−2	2.27 × 10−2	1.45 × 10−2	6.28 × 10−3	7.61 × 10−3	4.10 × 10−3	5.09 × 10−3	2.44 × 10−3	2.82 × 10−3	1.29 × 10−3	1.46 × 10−3
	50%	2.74 × 1010	1.31 × 10−1	8.47 × 10−3	2.29 × 10−2	2.11 × 10−3	2.47 × 10−3	7.94 × 10−4	8.76 × 10−4	1.06 × 102	3.43 × 10−2	1.54 × 10−1	1.46 × 10−2	6.39 × 10−3	7.53 × 10−3	4.08 × 10−3	4.96 × 10−3	2.44 × 10−3	2.86 × 10−3	1.29 × 10−3	1.46 × 10−3
5	0.1%	3.16 × 10−3	3.38 × 10−3	5.44 × 10−4	5.57 × 10−4	1.66 × 10−4	1.68 × 10−4	6.62 × 10−5	6.68 × 10−5	1.30 × 10−3	1.34 × 10−3	6.94 × 10−4	7.08 × 10−4	4.35 × 10−4	4.40 × 10−4	2.97 × 10−4	3.02 × 10−4	1.89 × 10−4	1.91 × 10−4	1.05 × 10−4	1.06 × 10−4
	1%	3.17 × 10−3	3.38 × 10−3	5.44 × 10−4	5.56 × 10−4	1.65 × 10−4	1.67 × 10−4	6.63 × 10−5	6.69 × 10−5	1.29 × 10−3	1.34 × 10−3	6.94 × 10−4	7.06 × 10−4	4.34 × 10−4	4.40 × 10−4	3.00 × 10−4	3.05 × 10−4	1.88 × 10−4	1.90 × 10−4	1.04 × 10−4	1.05 × 10−4
	5%	3.17 × 10−3	3.38 × 10−3	5.46 × 10−4	5.59 × 10−4	1.65 × 10−4	1.68 × 10−4	6.64 × 10−5	6.70 × 10−5	1.28 × 10−3	1.32 × 10−3	6.95 × 10−4	7.06 × 10−4	4.33 × 10−4	4.39 × 10−4	2.98 × 10−4	3.03 × 10−4	1.89 × 10−4	1.91 × 10−4	1.04 × 10−4	1.05 × 10−4
	15%	3.17 × 10−3	3.38 × 10−3	5.50 × 10−4	5.64 × 10−4	1.65 × 10−4	1.67 × 10−4	6.62 × 10−5	6.67 × 10−5	1.28 × 10−3	1.32 × 10−3	6.97 × 10−4	7.08 × 10−4	4.36 × 10−4	4.41 × 10−4	3.01 × 10−4	3.06 × 10−4	1.89 × 10−4	1.91 × 10−4	1.04 × 10−4	1.05 × 10−4
	30%	2.09 × 102	3.39 × 10−3	5.45 × 10−4	5.57 × 10−4	1.65 × 10−4	1.67 × 10−4	6.64 × 10−5	6.69 × 10−5	1.30 × 10−3	1.33 × 10−3	6.95 × 10−4	7.09 × 10−4	4.36 × 10−4	4.42 × 10−4	2.99 × 10−4	3.04 × 10−4	1.89 × 10−4	1.91 × 10−4	1.05 × 10−4	1.06 × 10−4
	50%	3.17 × 10−3	3.39 × 10−3	5.47 × 10−4	5.60 × 10−4	1.65 × 10−4	1.67 × 10−4	6.64 × 10−5	6.69 × 10−5	1.29 × 10−3	1.32 × 10−3	6.97 × 10−4	7.10 × 10−4	4.37 × 10−4	4.43 × 10−4	3.00 × 10−4	3.05 × 10−4	1.90 × 10−4	1.92 × 10−4	1.05 × 10−4	1.06 × 10−4
20	0.1%	1.64 × 10−4	1.34 × 10−3	3.16 × 10−5	3.16 × 10−5	9.91 × 10−6	9.91 × 10−6	4.03 × 10−6	4.03 × 10−6	7.18 × 10−5	7.18 × 10−5	4.01 × 10−5	4.01 × 10−5	2.55 × 10−5	2.55 × 10−5	1.77 × 10−5	1.77 × 10−5	1.13 × 10−5	1.13 × 10−5	6.32 × 10−6	6.33 × 10−6
	1%	1.64 × 10−4	1.64 × 10−4	3.16 × 10−5	3.16 × 10−5	9.89 × 10−6	9.90 × 10−6	4.04 × 10−6	4.04 × 10−6	7.16 × 10−5	7.17 × 10−5	4.00 × 10−5	4.01 × 10−5	2.55 × 10−5	2.55 × 10−5	1.76 × 10−5	1.76 × 10−5	1.12 × 10−5	1.13 × 10−5	6.31 × 10−6	6.31 × 10−6
	5%	1.64 × 10−4	1.64 × 10−4	3.16 × 10−5	3.16 × 10−5	9.88 × 10−6	9.89 × 10−6	4.04 × 10−6	4.04 × 10−6	7.18 × 10−5	7.18 × 10−5	3.99 × 10−5	3.99 × 10−5	2.56 × 10−5	2.56 × 10−5	1.77 × 10−5	1.77 × 10−5	1.13 × 10−5	1.13 × 10−5	6.32 × 10−6	6.32 × 10−6
	15%	1.64 × 10−4	1.64 × 10−4	3.16 × 10−5	3.16 × 10−5	9.90 × 10−6	9.90 × 10−6	4.03 × 10−6	4.03 × 10−6	7.19 × 10−5	7.19 × 10−5	4.00 × 10−5	4.00 × 10−5	2.55 × 10−5	2.55 × 10−5	1.76 × 10−5	1.77 × 10−5	1.13 × 10−5	1.13 × 10−5	6.32 × 10−6	6.32 × 10−6
	30%	1.64 × 10−4	1.64 × 10−4	3.17 × 10−5	3.17 × 10−5	9.89 × 10−6	9.89 × 10−6	4.03 × 10−6	4.03 × 10−6	7.18 × 10−5	7.18 × 10−5	4.00 × 10−5	4.00 × 10−5	2.55 × 10−5	2.55 × 10−5	1.77 × 10−5	1.77 × 10−5	1.13 × 10−5	1.13 × 10−5	6.32 × 10−6	6.32 × 10−6
	50%	1.64 × 10−4	1.64 × 10−4	3.15 × 10−5	3.15 × 10−5	9.89 × 10−6	9.89 × 10−6	4.03 × 10−6	4.03 × 10−6	7.18 × 10−5	7.18 × 10−5	4.01 × 10−5	4.01 × 10−5	2.55 × 10−5	2.55 × 10−5	1.77 × 10−5	1.77 × 10−5	1.13 × 10−5	1.13 × 10−5	6.32 × 10−6	6.32 × 10−6
50	0.1%	2.55 × 10−5	2.55 × 10−5	4.98 × 10−6	4.98 × 10−6	1.57 × 10−6	1.57 × 10−6	6.43 × 10−7	6.43 × 10−7	1.12 × 10−5	1.12 × 10−5	6.31 × 10−6	6.31 × 10−6	4.03 × 10−6	4.03 × 10−6	2.79 × 10−6	2.79 × 10−6	1.79 × 10−6	1.79 × 10−6	1.00 × 10−6	1.00 × 10−6
	1%	2.54 × 10−5	2.54 × 10−5	4.99 × 10−6	4.99 × 10−6	1.57 × 10−6	1.57 × 10−6	6.42 × 10−7	6.42 × 10−7	1.13 × 10−5	1.13 × 10−5	6.31 × 10−6	6.31 × 10−6	4.03 × 10−6	4.03 × 10−6	2.80 × 10−6	2.80 × 10−6	1.79 × 10−6	1.79 × 10−6	1.00 × 10−6	1.00 × 10−6
	5%	2.54 × 10−5	2.54 × 10−5	4.99 × 10−6	4.99 × 10−6	1.57 × 10−6	1.57 × 10−6	6.42 × 10−7	6.42 × 10−7	1.12 × 10−5	1.12 × 10−5	6.31 × 10−6	6.31 × 10−6	4.03 × 10−6	4.03 × 10−6	2.80 × 10−6	2.80 × 10−6	1.79 × 10−6	1.79 × 10−6	1.01 × 10−6	1.01 × 10−6
	15%	2.55 × 10−5	2.55 × 10−5	4.98 × 10−6	4.98 × 10−6	1.57 × 10−6	1.57 × 10−6	6.42 × 10−7	6.42 × 10−7	1.13 × 10−5	1.13 × 10−5	6.32 × 10−6	6.32 × 10−6	4.03 × 10−6	4.03 × 10−6	2.80 × 10−6	2.80 × 10−6	1.79 × 10−6	1.79 × 10−6	1.00 × 10−6	1.00 × 10−6
	30%	2.55 × 10−5	2.55 × 10−5	4.98 × 10−6	4.98 × 10−6	1.57 × 10−6	1.57 × 10−6	6.43 × 10−7	6.43 × 10−7	1.13 × 10−5	1.13 × 10−5	6.32 × 10−6	6.32 × 10−6	4.03 × 10−6	4.03 × 10−6	2.80 × 10−6	2.80 × 10−6	1.79 × 10−6	1.79 × 10−6	1.00 × 10−6	1.00 × 10−6
	50%	2.54 × 10−5	2.54 × 10−5	4.99 × 10−6	4.99 × 10−6	1.57 × 10−6	1.57 × 10−6	6.42 × 10−7	6.42 × 10−7	1.13 × 10−5	1.13 × 10−5	6.29 × 10−6	6.29 × 10−6	4.03 × 10−6	4.03 × 10−6	2.80 × 10−6	2.80 × 10−6	1.79 × 10−6	1.79 × 10−6	1.00 × 10−6	1.00 × 10−6

4.3. Relative Efficiency of ${\widehat{\phi }}_{LogNI}$ and ${\widehat{\phi }}_{BDNI}$

To compare the variances of the two non-iterative estimation procedures given by (2) and (3), we consider their relative efficiencies

$$\mathrm{RE}=\frac{\mathrm{V}\left({\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}\right)}{\mathrm{V}\left({\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}\right)}$$

(19)

which are summarised in Table 4 based on the variances given in Table 3. Table 4 shows that ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$ is more efficient than ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$ for estimating $\mathsf{\phi }$ for tables of moderate and larger size. It is worth noting that for moderate and larger values of λ, the relative efficiency of ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$ approaches 1. This indicates that for contingency tables with large cell values, the performance of both estimates is virtually the same. However, the same cannot be said for small λ values.

Table 4. Relative efficiencies, RE, of ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$ and ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$ for varying dimensions and λ s of simulated contingency tables.

Size of Contingency Table
λ	n	2 × 2	3 × 3	4 × 4	5 × 5	2 × 3	2 × 4	2 × 5	3 × 4	3 × 5	4 × 5
1.5	0.1%	9,189,606,121	0.72784956	0.85885459	0.9045322	36992.41	19.31567	1.401092	0.821198	0.854602	0.885681
	1%	9,240,148,509	0.96202313	0.86569879	0.9060769	34833.59	11.42992	0.851444	0.80374	0.860328	0.885121
	5%	9,334,020,605	0.69906125	0.86225639	0.9085593	40875.57	142.9842	0.853147	0.828006	0.860919	0.886632
	15%	7,134,124,262	0.57758461	0.86198772	0.9054768	35.83611	9.424074	0.82911	0.829665	0.853502	0.889904
	30%	4.5422 × 1011	0.77400485	0.85878719	0.9092136	1218.304	1.569196	0.825333	0.806418	0.865789	0.884679
	50%	2.0874 × 1011	0.3701973	0.85574967	0.9067557	3102.509	10.56086	0.847628	0.822509	0.851845	0.888767
5	0.1%	0.93398803	0.97807489	0.98786916	0.9914141	0.969994	0.980504	0.987917	0.983266	0.98752	0.989555
	1%	0.93907845	0.97746633	0.98698452	0.9912539	0.964727	0.981822	0.986971	0.98311	0.98834	0.989857
	5%	0.93732391	0.97821917	0.98765407	0.9915809	0.969324	0.983593	0.987118	0.985381	0.986763	0.989947
	15%	0.93685705	0.97556275	0.98876169	0.9919514	0.970251	0.984432	0.988278	0.983998	0.987323	0.990359
	30%	61704.3588	0.97781952	0.98772586	0.9914344	0.971497	0.98123	0.986997	0.983168	0.987851	0.990201
	50%	0.93517417	0.97673207	0.98746003	0.9917618	0.973975	0.980896	0.986541	0.983451	0.987356	0.989454
20	0.1%	0.12285176	0.99973734	0.99973702	0.999817	0.999622	0.999756	0.999787	0.999703	0.999767	0.999772
	1%	0.99955224	0.99965713	0.99974376	0.9998317	0.999762	0.999692	0.999785	0.999755	0.999729	0.999804
	5%	0.99960642	0.99967657	0.99975318	0.9998185	0.999751	0.999771	0.999771	0.999717	0.999769	0.999792
	15%	0.99975382	0.99962495	0.99976403	0.9998131	0.99971	0.999797	0.999802	0.999726	0.999762	0.999803
	30%	0.99986221	0.99966588	0.99976152	0.9998227	0.999695	0.999799	0.999779	0.999745	0.999769	0.999796
	50%	0.99994944	0.99961342	0.99974001	0.9998151	0.999769	0.999815	0.999785	0.9997	0.999785	0.999781
50	0.1%	0.99995092	0.99995684	0.99996116	0.9999754	0.999963	0.99997	0.999977	0.999957	0.999963	0.999965
	1%	0.99998192	0.99995771	0.99996247	0.9999727	0.999968	0.999961	0.99998	0.999957	0.99997	0.99997
	5%	0.99996342	0.99994206	0.99996434	0.9999717	0.999979	0.999965	0.999967	0.999954	0.999962	0.999966
	15%	0.99991797	0.99995585	0.99996816	0.9999745	0.99997	0.999972	0.999972	0.999954	0.999963	0.999967
	30%	0.99998194	0.99995479	0.99996308	0.9999712	0.99997	0.999959	0.999972	0.999958	0.99997	0.99997
	50%	0.99993474	0.99994946	0.99997071	0.9999754	0.999956	0.999978	0.999973	0.999971	0.999969	0.999969

4.4. Graphical Display for the Asymptotic Distributions of ${\widehat{\phi }}_{LogNI}$ and ${\widehat{\phi }}_{BDNI}$

The asymptotic distributions of ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$ and ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$, defined by (17) and (18), are graphically demonstrated by considering Figure 1a–h and Figure 2a–h respectively. The graphical displays are shown for a small (λ = 1.5) and large (λ = 50) Poisson parameter and for a 2 × 2 and 4 × 5 contingency table. The same populations of 10,000 ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$ and 10,000 ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$ values are again considered here, and the sampling distributions have been generated for 1000 samples ranging in size from n = 10 to n = 5000. The histograms depict the sampling distribution of ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$ and ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$. They show that, for a small and large Poisson parameter, and for small and moderately sized contingency tables, the distribution of each non-iterative estimate is asymptotically normally distributed.

Figure 1. Sampling distribution of ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$ for various λ and n.

Figure 2. Graphical display of asymptotic distribution of ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}.$.

4.5. An Empirical Study

To highlight the practical implications of the results described above, we shall consider the relative efficiency of the estimates ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$ and ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$ by studying 13 of the datasets analysed by [40] (Table 1). These data were selected for their common use in categorical data analysis. They have been given the same labels here as were given by [40]; therefore, further details on the context of the data and the key references in which they appear can be found in [40]. These datasets are summarised in Table 5 and vary in size. The smallest contingency table (Visitor) is of size 3 × 3 while the largest contingency table analysed (Occupation) is of size 8 × 8. The sample size also varies, from n = 132 (for Visitor) to n = 25,263 (for Swedish). All the zero-cell frequencies have been replaced by 0.05. Table 5 summarises the relative efficiency of the estimates, as follows. Defined by (19), for each dataset. From these results, the variance of ${\hat{\mathsf{\phi }}}_{\mathrm{BDNI}}$ is always less than, or equal to, the variance of ${\hat{\mathsf{\phi }}}_{\mathrm{LogNI}}$ for all the datasets. Note that for the datasets Aberdeen, Denmark, and Swedish, all of which have very large cell frequencies, the two variances are identical, resulting in a relative efficiency of 1. This empirically verifies our earlier findings, which state that as the Poisson parameter increases (leading to large randomly generated cell frequencies), the variance of both estimates is virtually identical.

Table 5. Standard errors and efficiency of ${\widehat{\phi }}_{BDNI}$ and ${\widehat{\phi }}_{LogNI}$ for 13 datasets.

Data	$\mathbf{n}$	$\mathbf{V}\left({\widehat{\mathsf{\phi }}}_{\mathbf{LogNI}}\right)$	$\mathbf{V}\left({\widehat{\mathsf{\phi }}}_{\mathbf{BDNI}}\right)$	Efficiency
Calcium (4 × 4)	136	7.9032 × 10−5	6.0528 × 10−5	0.76587
Dreams (5 × 4)	233	2.2753 × 10−5	2.0794 × 10−5	0.91389
Aberdeen (4 × 5)	22,361	2.5000 × 10−9	2.5000 × 10−9	1.00000
Caithness (4 × 5)	5387	4.8400 × 10−8	3.6100 × 10−8	0.74587
Denmark (4 × 5)	7025	1.9600 × 10−8	1.9600 × 10−8	1.00000
Swedish (4 × 5)	25,263	1.6000 × 10−9	1.6000 × 10−9	1.00000
Visitor (3 × 3)	132	7.5690 × 10−5	5.9598 × 10−5	0.78740
Socec (4 × 6)	1660	3.8440 × 10−7	3.6000 × 10−7	0.93652
Occupation (8 × 8)	3498	1.3690 × 10−7	1.2960 × 10−7	0.94668
Dumping (4 × 3)	417	5.8081 × 10−6	5.7600 × 10−6	0.99172
Ideology (3 × 3)	1083	1.0201 × 10−6	8.4640 × 10−7	0.82972
Status (3 × 3)	3497	3.6100 × 10−8	2.8900 × 10−8	0.80055
Opinion (4 × 4)	926	1.3225 × 10−6	1.2544 × 10−6	0.94851

5. Conclusions

This paper examined the issues of bias, variance, and sampling distribution properties and was concerned with two standard approaches for non-iteratively estimating the linear-by-linear parameter of an ordinal log-linear model. We have done so assuming a Poisson sampling scheme, but it can also be adapted for a multinomial sampling scheme; see, for example, [62] for a study of asymptotic results for multinomial models. These approaches to estimation are easy to determine mathematically and so are easier to compute than their iterative counterparts (including Newton’s method and iterative proportional fitting). We have confirmed that the two non-iterative estimation procedures lead to estimates that are unbiased and asymptotically normally distributed. We have also demonstrated empirically, and through simulation, that the variance of the BDNI estimate is less than the variance of the LogNI estimate. This therefore suggests that the BDNI estimate is a more efficient, and reliable, non-iterative estimate than the LogNI estimate. This seems contrary to the findings originally proposed by [40,42,43] and so requires further examination.

In this paper we have examined the simple case where the OLLM consists of the estimation of a single parameter. At present, the non-iterative estimation procedures have not been generalised to incorporate estimations for models with multiple association parameters. In fact, there is only a limited amount of literature available on the specification of such models and the interpretation of the parameters. Therefore, an investigation of this issue, and into the properties of the non-iterative estimation for such models, is a topic for future consideration.