# Should one derive risk difference from the odds ratio?

**URL:** <https://discourse.datamethods.org/t/should-one-derive-risk-difference-from-the-odds-ratio/4403>\
**Category:** bayes\
**Tags:** absolute-risk\
**Created:** [May 13, 2021, 11:49pm UTC](https://discourse.datamethods.org/t/should-one-derive-risk-difference-from-the-odds-ratio/4403 "2021-05-13T23:49:41Z")\
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**Author:** ![R\_cubed](https://discourse.datamethods.org/user_avatar/discourse.datamethods.org/r_cubed/32/1518_2.png) [@R\_cubed](https://discourse.datamethods.org/u/R_cubed)\
**Post date:** [May 28, 2023, 3:47pm UTC](https://discourse.datamethods.org/t/should-one-derive-risk-difference-from-the-odds-ratio/4403/594 "2023-05-28T15:47:14Z")

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From the paper:

> … the RR when looked at from the diagnostic testing angle, is now understood to be a likelihood ratio **and therefore a ratio of two odds** not … simply considered as a ratio of two conditional probabilities…

I have no idea why you are converting proportions to odds ratios here. This flies in the face of decades of mathematical work on the statistical properties of estimators and experiments.

In an experimental design context, the only thing we have control over is the error probabilities, which are more informative than likelihoods.

After the data are collected, an observed proportion can be useful to estimate a future probability.

Pawitan, paraphrasing Fisher wrote:

> whenever possible to get exact results, we should base inference on probability statements, otherwise they should be based on the likelihood.

**Pawitan, Y. (2001).** In all likelihood: statistical modelling and inference using likelihood. Oxford University Press. p. 15

The Bayes Factor has a frequentist interpretation of \frac{1- \bar{\beta}}{\alpha} , which is a **ratio of probabilities.**

[https://www.sciencedirect.com/science/article/pii/S002224961600002X](https://www.sciencedirect.com/science/article/pii/S002224961600002X)

Diagnostic testing is merely a specific case of the broader hypothesis testing problem, where:

Type I error = 1 - spec  
Type II error = 1 - sens  
Power = sens = 1 - \beta

> **[Sensitivity and specificity](https://en.wikipedia.org/wiki/Sensitivity_and_specificity)**
>
> In medicine and statistics, sensitivity and specificity mathematically describe the accuracy of a test that reports the presence or absence of a medical condition. If individuals who have the condition are considered "positive" and those who do not are considered "negative", then sensitivity is a measure of how well a test can identify true positives and specificity is a measure of how well a test can identify true negatives:
> If the true status of the condition cannot be known, sensitivity and...

Information is formalized as a (pseudo) distance metric of 2 probability distributions.

> **[Kullback–Leibler divergence](https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence)**
>
> In mathematical statistics, the Kullback–Leibler (KL) divergence (also called relative entropy and I-divergence), denoted 
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> {\\displaystyle D\_{\\text{KL}}(P\\parallel Q)}
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> , is a type of statistical distance: a measure of how much a model probability distribution Q is different from a true probability distribution P. Mathematically, it is defi A simple...

> [@Relating S-values to other measures of statistical information](https://discourse.datamethods.org/t/relating-s-values-to-other-measures-of-statistical-information/4293/5):
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> The Bayes factor bound is the most optimistic assessment of the information from the experiment that distinguishes between the 2 hypotheses. But before calculating the posterior odds via that relationship to the p-value, we need to rank experiments in terms of their error rates. Berger, Bayarri, and Selke make the case that Bayes Factors can be interpreted in the frequentist sense in terms of relative errors, giving \frac{1 - \bar \beta}{\alpha} – the ratio of expected power over type I error. …

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