Arguably, the question is not very precise. Rather than enumerating all measures of association for $2\times 2$ tables, I shall concentrate on the way such measures may be constructed and how to select the one that is most appropriate with respect to hypothesis or constraints relevant to a cross-classification.
The very first questions to ask are: what does the table reflect (concordance, agreement, association between two attributes, etc.), do you seek an overall measure of association or do you think one of the two variables plays a specific role (which would justify the search for an "oriented" association), do you consider either or both of the margins fixed (row and/or columns totals)? All of this impact on the method to choose and the way to interpret the results.
The $2\times 2$ case
Two-by-two tables are often treated separately from $I\times J$ tables because we often consider that variables play a symmetric role in this particular case. Obviously, this is not always the case: cross-classification of exposure and disease, as commonly found in epidemiological studies, is an example where both variables play a distinct role, which may lend to more than a simple interpretation in terms of association. Another one is $2\times 2$ tables constructed for studying the screening properties of a given diagnostic instrument. Although the odds-ratio (compared to, e.g. the relative risk) keeps its nice properties, we may be interested in predictive/negative positive values or specificity/sensibility, which means working with other quantities of interest. Hence, the need to specifiy whether the problem at hand implies two variables that are purely acting in a symmetrical way, or not, because it influences the way we interpret the results or derive a useful measure of association, agreement, or discrimination.
For the sake of clarity, I will consider that data (counts) are arranged in the following way:
Basically, measures of association for $2\times 2$ tables can be grouped in two classes: those relying on (a) (a function of) the cross-product ratio and those based on (b) the product-moment (Pearson) correlation, or a function thereof.
The cross-product ratio, mostly known as the odds-ratio, is simply $\alpha=p_{11}p_{22}/p_{12}p_{21}$. It is invariant under rows and columns interchange, and transformations of margins that preserves $\sum_{i,j}p_{ij}=1$. In epidemiology, we usually think of it as a measure of association where rows (or columns) are fixed: $p_{11}/p_{12}$ is then the odds of being in the first column (e.g., diseased) conditional on being in the first row (e.g., exposed), and likewise $p_{21}/p_{22}$ is the odds for the second row, or in other words
$$
\alpha=\frac{p_{11}/p_{12}}{p_{21}/p_{22}}.
$$
Yule's $Q=(\alpha-1)/(\alpha+1)$ fall into the former case, (a). Yule also proposed a measure of "colligation", $Y$, as $(\sqrt{\alpha}-1)/(\sqrt{\alpha}+1)$. Yule's $Q$ can be interpreted as the difference between conditional probabilities of like and unlike "orders" for two individuals chosen at random; it is identical to Goodman and Kruskal's $\gamma$ measure of association for $I\times J$ tables.
For (b), we can derive a correlation coefficient for a $2\times 2$ table by thinking of the table as a combination of each of two variables scores (taking value 0 and 1, for the first and second row/column, resp.). Then, the coefficient $\rho$ is defined as the covariance divided by the square root of the product of the variances:
$$
\rho=\frac{p_{22}-p_{2\cdot}p_{\cdot 2}}{\sqrt{p_{1\cdot}p_{2\cdot}p_{\cdot 1}p_{\cdot 2}}},
$$
which is equivalent to putting $p_{11}p_{22}-p_{21}p_{12}$ in the numerator. Plugging in the observed counts, Pearson's $r$ is the MLE of $\rho$ under a multinomial sampling model.
It is invariant under rows and columns interchange, and positive linear transformation.
It can be shown (Yule, 1912) that $\rho$ is identical to Yule's $Y$ if we standardize our table such that row and column margins sum to 1/2, i.e. $p_{11}^*=p_{22}^*=0.5\left(\sqrt{\alpha}/(\sqrt{\alpha}+1)\right)$ and $p_{12}^*=p_{21}^*=0.5\left(1/(\sqrt{\alpha}+1)\right)$. By doing this, we remove the information coming from the margins, such that $Y=2(p_{11}^*-p_{12}^*)$.
Correlation-based measures are connected to the usual Pearson's chi-square statistic, since
$$
\Phi^2=\sum_{i=1}^2\sum_{j=1}^2\frac{(p_{ij}-p_{i\cdot}p_{\cdot j})^2}{p_{i\cdot}p_{\cdot j}},
$$
that is,
$$
\Phi^2=\frac{(p_{11}p_{22}-p_{21}p_{12})^2}{p_{1\cdot}p_{2\cdot}p_{\cdot 1}p_{\cdot 2}}=\rho^2.
$$
In a $2\times 2$ table, we thus have $r^2=\chi^2/N$. Pearson also proposed to use $\sqrt{\rho^2/(1+\rho^2)}$ as a measure of association, and he coined it the coefficient of mean square contingency.
As to how to choose the correct measure (a vs. b), it clearly depends on whether we want to be sensitive to marginal totals (in this case, $\rho$ cannot take its full range of possible values in $[-1;1]$), and whether we consider that we observe a full association even if one of the four cells is zero (in this case, $\rho$ cannot take the value $+1$ or $-1$ if only one of the cells is zero, which is not the case of Yule's $Q$).
Of note, correlation-based measures are better if they are used in a correlation matrix (e.g., for factor analysis), because we cannot guarantee that a matrix composed of Yule's $Q$ coefficient will be positive definite.
The $I\times J$ case
Like for the $2\times 2$ case, we can derive measures of association based on different quantities. Measures based on chi-square include
- Pearson's $P$ coefficient based on $\Phi^2$ (see above), $\sqrt{\Phi^2/(\Phi^2+1)}$ (to overcome the fact that $\Phi^2$ no longer lies in $[0;1]$ when $I$ or $J>2$);
- Tschuprow's $T=\left(\Phi^2/\sqrt{(I-1)(J-1)}\right)^{1/2}$, which behaves better than $P$ in square tables (in that it can reach a maximum value of 1, for full or complete association);
- Cramer's $V$ is another derivation, and $V=\left(\Phi^2/\text{min}(I-1,J-1)\right)^{1/2}$ (we have $V\geq T$ for all $I,J>2$).
These measures are all measures of association where none of the variables plays a specific role. In case a $\chi^2$ test is significant, it is more interesting to look at how the expected counts depart from the observed counts (i.e. look at the Pearson residuals) in all $(i,j)$ cells, or to use something like a mosaic plot.
Goodman and Kruskall (1954) also proposed a predictive measure of association between rows and columns, or more specifically a measure of proportional reduction in error in predicting one column category when the row category is known as opposed to the case when the latter one is unknown. This is called $\lambda_{C|R}$ and its MLE is
$$
\hat\lambda_{C|R}=\frac{\sum_{i=1}^Ix_{im}-x_{\cdot m}}{N-x_{\cdot m}}
$$
where $x_{im}$ and $x_{\cdot m}$ stand for the maximum for the $i$th row and the column totals. This measure is interesting because it has a nicer interpretation than $\chi^2$-based measure, but it also has some drawbacks: when there is statistical independence, $\lambda_{C|R}$ is not necessarily zero, for instance.
A measure of the proportion of explained variance (derived from Gini's total variation) may be derived from the total sum of squares (SS) in an $I\times J$ table
$$
\text{TSS}=\frac{N}{2}-\frac{1}{2N}\sum_{i=1}^Ix_{i\cdot}^2,
$$
which can be partitioned as a within- and between-group SS. Of interest here is the variance explained by considering the different categories (BSS) divided by the total variance, TSS. Like in the ANOVA framework, we have BSS=TSS-WSS, where
$$
\text{WSS}=\frac{N}{2}-\frac{1}{2}\sum_{j=1}^J\frac{1}{x_{\cdot j}}\sum_{i=1}^Ix_{ij}^2,
$$
so that we can derive BSS/TSS as
$$
\hat\tau_{R|C}=\frac{\sum_j\frac{1}{x_{\cdot j}}\sum_i x_{ij}^2-\frac{1}{N}\sum_ix_{i\cdot}^2}{N-\frac{1}{N}\sum_ix_{i\cdot}^2}.
$$
This measure can be interpreted as "the relative decrease in the proportion of incorrect predictions when we go from predicting the row category based only on the row marginal probabilities to predicting the row category based on the conditional proportions $p_{ij}/p_{\cdot j}$" (Bishop et al., 2007, p. 391).
Finally, measures based on the cross-product ratios are also available, as well as measures of agreement for ordinal variables, but I realize now that I need to stop (and thank the reader who reached the end of this overview).
A thorough overview of measures of association may be found in Bishop et al. (2007), from which I grabbed most of the above discussion, and of course Agresti (2002), about which Laura Thompson made a complete R adaptation in his textbook R (and S-PLUS) Manual to Accompany Agresti's Categorical Data Analysis.
References
- Agresti, A. (2002). Categorical Data Analysis. Wiley. Companion website
- Bishop, Y.M., Fienberg, S.E., and Holland, P.W. (2007). Discrete Multivariate Analysis. Springer.
- Goodman, L.A. and Kruskall, W.H. (1954). Measures of association for cross-classification. JASA, 49, 732-764.
- Yule, G.U. (1912). On the methods of measuring association between two attributes. Journal of the Royal Society, 75, 579-642.
