Odds Ratio

An odds ratio is the division of two odds. An odds is formed by two numbers representing two different events (most often event vs. nonevent), such as the odds of winning versus losing, or the odds of voting for the Democratic candidate versus voting for the Republican candidate (or a non-Democratic candidate). As such, it is different from a rate, which gives the number of events occurring in a population standardized to a base, such as per hundred or per thousand. Odds ratios are useful statistical tools in categorical data analysis for studying association or concordance in the data.

G. Udny Yule was notably the first statistician to apply odds ratios to measure the association in contingency tables. Odds ratios and log-odds ratios are common tools for log-linear modeling. To demonstrate the application of odds ratios, let us examine in Table 1 some classic data presented by M. Greenwood and G. U. Yule in their classic 1915 study of the effect of antityphoid inoculations.

This is a typical 2 × 2 table with the columns recording the two outcomes and the rows representing the two categories in the exposure variable. More generally, the columns may contain any response or dependent variable and the rows, any explanatory or independent variable; indeed, the rows and columns can simply contain two variables whose association is under investigation. The cells can be indicated by A,B,C, and D (see Table 2).

There are three nonredundant ways of forming odds ratios (OR) in a 2 × 2 table:

For studying association between two categorical variables, OR3 typically is used. This odds ratio is also known as the cross-product ratio because

where hereafter we ignore the subscript because it is the only odds ratio we examine. Returning to the data on antityphoid inoculations, we find the odds ratio or cross-product ratio to be

The ratio suggests that there is an association between inoculations and typhoid occurrences. The odds of having typhoid prevented are close to three times greater for those who had received inoculations than for those who had not. Statistics like these may give some evidence to government agencies for policy making such as implementing vaccination programs.

Note that Yule's Q is simply a function of the odds ratio that has been normed to fall between −1 and +1:

Because the Q statistic is close to 0.5 (with zero indicating no relationship), it suggests some moderate positive association between inoculations and typhoid prevention.

Often, it is convenient to work with (natural-) log-odds ratios, which have two useful properties. Log-odds ratios have zero as the value for indicating no [Page 760]relation between the two variables, whereas the value for odds ratios is one. Zero for no relationship may be intuitively appealing for some researchers. One also can compute asymptotic standard errors for log-odds

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