Geometric Mean Type of Proportional Reduction in Variation Measure for Two-Way Contingency Tables

Consider $R\times C$ contingency table with nominal categories of the explanatory variable $X$ and the response variable $Y$. Let $p_{ij}$ denote the probability that an observation will fall in the $i$th row and $j$th column of the table ($i=1,\ldots,R;j=1,\ldots,C)$. In addition, $p_{i\cdot}$ and $p_{\cdot j}$ are denoted as $p_{i\cdot}=\sum_{l=1}^{C}p_{il}$, $p_{\cdot j}=\sum_{k=1}^{R}p_{kj}$. The conventional PRV measure has the form

where $V(Y)$ is a measure of variation for the marginal distribution of $Y$, and $E[V(Y|X)]$ is the expectation for the conditional variation of $Y$ given the distribution of $X$ (see, Agresti,, 2003). $\Phi$ is using the weighted arithmetic mean of $V(Y|X=i)$, i.e, $\sum_{i=1}^{R}p_{i\cdot}V(Y|X=i)$. By changing the variation measure, various PRV measures can be expressed, such as uncertainty coefficient $U$ for the variation measure $V(Y)=-\sum_{j=1}^{C}p_{\cdot j}\log p_{\cdot j}$ called Shannon entropy and concentration coefficient $\tau$ for the variation measure $V(Y)=1-\sum_{j=1}^{C}p_{\cdot j}^{2}$ called Gini concentration (see, Agresti,, 2003). Tomizawa et al., (1997) proposed a generalized PRV measure $T^{(\lambda)}$ that includes $U$ and $\tau$ by using $V(Y)=\left(1-\sum_{j=1}^{C}p_{\cdot j}^{\lambda+1}\right)/\lambda$ as the variation measure which is Patil and Taillie, (1982) diversity index of degree $\lambda$ for the marginal distribution $p_{\cdot j}$. Furthermore, Momozaki et al., (2022) proposed an extension of generalized PRV (eGPRV) measure that includes $U$, $\tau$, and $T^{(\lambda)}$:

The variation measure used in the eGPRV measure $\Phi_{f}$ are $V(Y)=-\sum_{j=1}^{C}f(p_{\cdot j})$.

We propose a new PRV measure by using the weighted geometric mean of $V(Y|X=i)$ that aims to sensitively capture the association of each explanatory variable to the response variable. Assume that $p_{\cdot j}>0$ and $V(Y|X=i)$ is a real number greater than or equal to 0 ($i=1,\ldots,R;~{}j=1,\ldots,C$). We propose a geometric mean type of PRV (geoPRV) measure for $R\times C$ contingency tables defined as

where $V(Y)$ is a measure of variation for the marginal distribution of $Y$. The geoPRV measure can use the same variation as the conventional PRV measure, for example,

where the variation measure $V(Y)=-\sum_{j=1}^{C}f(p_{\cdot j})$. In addition, the following theorem for $\Phi_{Gf}$ holds.

A property of the geoPRV measure is that the larger the value of $\Phi_{G}$, the stronger the association between the response variable $Y$ and the explanatory variable $X$. In other words, the larger the value of $\Phi_{G}$, the more accurately you can predict the $Y$ category if you know the $X$ category than if you do not. In contrast, if the value of $\Phi_{G}$ is 0, the $Y$ category is not affected by the $X$ category at all.

Consider the artificial data in Table 1. These are data to clearly show the difference in characteristics between conventional PRV measures and the geoPRV measure. Table 1c shows the case where the explanatory variable in the first row has a complete association structure with the response variable in the third column. On the other hand, Table 1a and Table 1b show the case where the explanatory variable in the first row has a weak or slightly strong association structure to the response variable, respectively.

The values of $\Phi_{f}^{(\lambda)}$ and $\Phi_{Gf}^{(\lambda)}$ are provided in Table 2a and Table 2b, respectively. For instance, Table 2a shows that when Table 1c is parsed the measure $\Phi_{f}^{(\lambda)}=0.2628,~{}0.1990,~{}0.1784$ for each $\lambda$ and does not capture the complete association structure of the first row. In contrast, $\Phi_{Gf}^{(\lambda)}=1$ in all $\lambda$, allowing us to identify the local complete association structure. Similarly, consider the results of the $\Phi_{Gf}^{(\lambda)}$ and $\Phi_{f}^{(\lambda)}$ in any $\lambda$ from Table 1a to Table 1c. As can be seen from these results, the simulation also shows that $\Phi_{Gf}^{(\lambda)}$ changes significantly by capturing partially related structures compared to $\Phi_{f}^{(\lambda)}$.

Consider the artificial data in Table 3. These data are intended to examine the value of the geoPRV measure $\Phi_{Gf}$ as the association of the entire contingency table changes. Therefore, we obtained data suitable for the survey by converting the bivariate normal distribution with means $\mu_{1}=\mu_{2}=0$ and variances $\sigma^{2}_{1}=\sigma^{2}_{2}=1$, in which the correlation coefficient was changed from $0$ to $1$ by $0.2$, into the $4\times 4$ contingency tables with equal-interval frequency. From Theorem 2 and the properties of the PRV measures, when the absolute values of the correlation coefficients are the same, i.e., when the rows of the contingency table are simply swapped, the values are equal, so the results for the negative correlation coefficient case are omitted.

Table 4 shows the value of $\Phi_{f}^{(\lambda)}$ and $\Phi_{Gf}^{(\lambda)}$ for each value of $\rho$, respectively. We observe that the values of $\Phi_{f}^{(\lambda)}$ and $\Phi_{Gf}^{(\lambda)}$ increase as the absolute value of the $\rho$ increases. Besides, $\rho=0$ if and only if the measures show that it is independent of the table, and $\rho=1.0$ if and only if the measures confirm that there is a structure of all (or partially) complete association. Also, if there is a relationship only to the entire contingency table, the values of $\Phi_{Gf}^{(\lambda)}$ are found to be larger than the values of $\Phi_{f}^{(\lambda)}$ by Theorem 1, but the differences are small.

Consider the case where the PRV measure is adapted to the data in Table 5, a survey of cannabis use among students conducted at the University of Ioannina (Greece) in 1995 and published in Marselos et al., (1997). The students’ frequency of alcohol consumption is measured on a four-level scale ranging from at most once per month up to more frequently than twice per week while their trial of cannabis through a three-level variable (never tried–tried once or twice–more often). We can see the partial bias of the frequency for the first and second rows in the data.

The estimates of $\Phi_{f}^{(\lambda)}$ and $\Phi_{Gf}^{(\lambda)}$ are provided in Table 6a and Table 6b, respectively. For instance, when $\lambda=1$, the measure $\widehat{\Phi}_{f}^{(1)}=0.1034$ for Table 6a, and $\widehat{\Phi}_{Gf}^{(1)}=0.2992$ for Table 6b. $\widehat{\Phi}_{f}^{(1)}$ shows that the average condition variation of trying cannabis is $10.34\%$ smaller than the marginal variation, and similarly $\widehat{\Phi}_{Gf}^{(1)}$ shows that the average condition variation of trying cannabis is $29.92\%$ smaller. Based on the results of these values, the following can be interpreted from Table 5:

• (1)

  There is a strong association overall between alcohol consumption and cannabis use experience associated.

• (2)

  There are fairly strong associations between some alcohol consumption and cannabis use experience.

These interpretations seem to be intuitive when looking at Table 5. However, by analyzing using the measures, we have been able to present an objective interpretation numerically and to show how strongly associated structures are in the contingency table.

By analyzing multiple contingency tables using the measures, it is possible to numerically determine how much difference there are between the associations of the contingency tables. Therefore, consider the data in Table 7 are taken from Hashimoto, (1999). These data describe the cross-classifications of the father’s and son’s occupational status categories in Japan which were examined in 1975 and 1985. In addition, we can consider the father’s states as an explanatory variable and the son’s states as an response variable, since the father’s occupational status categories seem to have an influence on the son’s. The analysis of Table 7 aims to show what differences there are in the associations of occupational status categories for fathers and sons in 1975 and 1985.

Table 8 and Table 9 give the estimates of $\Phi_{g}^{(\omega)}$ and $\Phi_{Gg}^{(\omega)}$, respectively. Comparing the estimates for each $\omega$ in Table 8 and Table 9, we can see that the values for both measures are almost the same. In addition, comparing Table 8a and Table 8b, the estimate is slightly larger in Table 8b, so it can be assumed that Table 8b is more related, but there is little difference because all the confidence intervals are covered. When we also compare 9a and Table 9b, we can see that 9b is larger because the estimate is slightly larger in 9b. However, we can see that the confidence interval does not cover at $\omega=0.9$. From the results of these values, the following can be interpreted for Table 7a and Table 7b:

• (1)

  The occupational status categories of fathers and sons in 1975 and 1985 both have weak associations overall, further indicating that individual explanatory variables do not have remarkably associations.

• (2)

  Although the association of Table 7b is slightly larger than Table 7a, the results of the confidence intervals indicate that there is no statistical difference.

• (3)

  The partial association in Table 7b is slightly larger than Table 7a, and the results of confidence intervals indicate that there may be a statistical difference.

When there are statistical differences from the results of some confidence intervals, as in (3), it is affected by differences in the characteristics of variation associated with changing the tuning parameters. In this case, it is difficult to give an interpretation by referring to variation because there was no difference in the variation in the special cases (e.g., $\omega=0$). However, when there are differences in variation in special cases, further interpretation can be given by focusing on the characteristics.