Log Odds Ratio (Logit) Measure

AKA: Logit Transformation.
Context:
- range: Log-Odds Score.
- It can convert between Risk Ratio and Odds Ratio. (?)
- It can be referenced by a Likelihood Ratio Test (for statistical inference).
Example(s):
- If outcome probabity is [math]\displaystyle{ p = 1/10 }[/math], then [math]\displaystyle{ logit(1/10) = log_e(\frac{1/10}{9/10}) = -2.1972 }[/math].
- …
Counter-Example(s):
- Log Likelihood Ratio.
See: Logistic Regression, Linear Mixed-Effects Model.

References

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 :[math]\displaystyle{ {\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})}.\, }[/math] ...
  ... 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.

(Ramani et al., 2005) ⇒ A. K. Ramani, Razvan C. Bunescu, Raymond Mooney and E. M. Marcotte. (2005). “Consolidating the Set of Known Human Protein-Protein Interactions in Preparation for Large-Scale Mapping of the Human Interactome." Genome Biology, volume 6, number 5, r40.
- QUOTE: … For both benchmarks, the scoring scheme for measuring interaction set accuracy is in the form of a log odds ratio of gene pairs either sharing annotations or physically interacting. To evaluate a dataset, we calculate a log likelihood ratio (LLR) as: [math]\displaystyle{ LLR = ln (\frac{P(D \vert I)} {P(D \vert \sim I)}) }[/math]

(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.

(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.

http://www.sjsu.edu/faculty/gerstman/EpiInfo/bin-mult.htm
- QUOTE: We then make the logit transformation to outcome p, where the logit is the natural log of the odds: [math]\displaystyle{ logit = log_e(\frac{p}{1-p}) }[/math] For example, if the probability of an outcome, p = .1, then the logit = loge(.1 / .9) = -2.1972.