# Log Odds Ratio (Logit) Measure

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A Log Odds Ratio (Logit) Measure is a metric based on the logarithm of an odds ratio.

## References

### 2018

• http://en.wikipedia.org/wiki/Odds_ratio#Statistical_inference
• QUOTE: Several approaches to statistical inference for odds ratios have been developed.

One approach to inference uses large sample approximations to the sampling distribution of the log odds ratio (the natural logarithm of the odds ratio). If we use the joint probability notation defined above, the population log odds ratio is :${\log\left(\frac{p_{11}p_{00}}{p_{01}p_{10}}\right) = \log(p_{11}) + \log(p_{00}\big) - \log(p_{10}) - \log(p_{01})}.\,$ ...

... An alternative approach to inference for odds ratios looks at the distribution of the data conditionally on the marginal frequencies of X and Y. An advantage of this approach is that the sampling distribution of the odds ratio can be expressed exactly.

### 2000

• (Bland & Altman, 2000) ⇒ J. Martin Bland, and Douglas G. Altman. (2000). “The Odds Ratio.” In: Bmj, 320(7247).
• QUOTE: The sample odds ratio is limited at the lower end, since it cannot be negative, but not at the upper end, and so has a skew distribution. The log odds ratio,2 however, can take any value and has an approximately Normal distribution. It also has the useful property that if we reverse the order of the categories for one of the variables, we simply reverse the sign of the log odds ratio: log(4.89)=1.59, log(0.204)=−1.59.

We can calculate a standard error for the log odds ratio and hence a confidence interval. The standard error of the log odds ratio is estimated simply by the square root of the sum of the reciprocals of the four frequencies.

### 2000

• (Hosmer & Lemeshow, 2000) ⇒ David W. Hosmer, and Stanley Lemeshow. (2000). “Applied Logistic Regression, 2nd Edition." Wiley. ISBN:0471356328
• QUOTE: ... In summary, the interpretation of the estimated coefficient for a continuous variable is similar to that of nominal scale variables: an estimated log odds ratio. The primary difference is that a meaningful change must be defined for the continuous variable. ...

... In the previous section in this chapter we discussed the interpretation of an estimated logistic regression coefficient in the case when there is a single variable in the fitted model.