Log Odds Ratio (Logit) Measure

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

• AKA: Logit Transformation.
• Context:
  • range: Log-Odds Score.
  • It can convert an odds ratio to a value on the real number line, allowing for easier interpretation and statistical analysis.
  • It can be used to express the log odds of an outcome given a set of predictor variables in logistic regression.
  • It can be referenced by a Likelihood Ratio Test for statistical inference on odds ratios.
  • It can be used to compare the relative risk of an outcome between two groups, while adjusting for confounding variables.
  • It can be approximately normally distributed, making it suitable for various statistical tests and confidence interval calculations.
  • It can be symmetric around zero, with positive values indicating increased odds and negative values indicating decreased odds.
  • It can be used to express the strength of association between categorical variables in contingency tables.
  • Range: $(-\infty, +\infty)$, where 0 indicates equal odds, positive values indicate higher odds, and negative values indicate lower odds.
  • ...
• Example(s):
  • If the outcome probability is $p = 1/10$, then $\operatorname{logit}(1/10) = \log_e(\frac{1/10}{9/10}) = -2.1972$.
  • In a study comparing the effectiveness of two treatments, a log odds ratio of 1.5 indicates that the odds of a successful outcome are $e^{1.5} \approx 4.48$ times higher for the first treatment compared to the second.
  • A logistic regression model predicting the probability of a disease based on age might have a coefficient of 0.05 for the age variable, meaning that for each year increase in age, the log odds of having the disease increase by 0.05.
  • ...
• Counter-Example(s):
  • Risk Ratio, which measures the ratio of probabilities rather than odds.
  • Absolute Risk Reduction, which measures the absolute difference in risk between two groups.
  • Log Likelihood Ratio, which compares the goodness of fit of two competing statistical models.
• See: Logistic Regression, Generalized Linear Mixed Model, Odds Ratio, Contingency Table, Confounding Variable, Statistical Inference, Confidence Interval, Association Measure.

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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 :[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.

2008

• (Brown, Wang, et al., 2008) ⇒ C. Hendricks Brown, Wei Wang, Sheppard G Kellam, Bengt O. Muthén, Hanno Petras, Peter Toyinbo, Jeanne Poduska, Nicholas Ialongo, Peter A Wyman, Patricia Chamberlain, and The Prevention Science and Methodology Group. (2008). “Methods for Testing Theory and Evaluating Impact in Randomized Field Trials: Intent-to-treat Analyses for Integrating the Perspectives of Person, Place, and Time.” In: Drug and Alcohol Dependence Journal, 95. doi:10.1016/j.drugalcdep.2007.11.013
  • QUOTE: In the generalized linear mixed effects model analysis presented in Table 7 for males, which adjusts for classroom variation and baseline aggressive, disruptive behavior ratings from first-grade teachers, the reduction in lifetime drug abuse/dependence disorders for GBG compared with internal GBG controls was large and significant, with a log odds ratio (OR) of 0.999 (p = 0.035)

2005

• (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]

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.

2000b

• 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.

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