 # Likelihood ratio is a ratio of odds (but not the usual odds ratio)

There seems to be an artificial distinction between odds ratios and diagnostic likelihood ratios

In the calculation of post test probability by means of pre test probability and likelihood ratio, we first convert to pretest odds to establish post test odds

Pretest odds * LR = post test odds

Algebra says LR is a ratio of odds.

But it is not discussed this way anywhere. Am I missing something fundamental?

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Tony Lachenbruch had a paper years ago that I wish I could find, where he pointed out that the odds ratio is the product of the likelihood ratio + and the likelihood ratio -.

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oh my.

This is because the pre-test odds is not the “baseline” odds!

It is the ‘prior’ odds, i.e. before test information is considered at all. This odds estimate comes from a model that does not have a parameter to represent the test.

The post-test odds is the posterior odds, i.e., after test information is known. It comes from the “posterior” model that has an extra parameter to represent the test.

Said another way, pre-test odds does not mean odds given a negative test. It means odds before you even know the test result.

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Well put. The odds ratio is the effect of going from “knowing the test negative” to “knowing it’s positive” whereas the likelihood ratio + is the effect of going from an unknown state to knowing the test is +. “Unknown state” is not always well defined so I stick to logistic regression for the diagnostic problem.

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Good discussion. Here is the Lachenbruch reference.

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This article discusses what is sometimes called the DOR - diagnostic odds ratio - in more detail.

The DOR reflects the odds of positivity in disease versus the odds of positivity in the non-diseased or, alternatively, the odds of disease in test positives versus the odds of disease in test negatives.

Like all other single indicators of test performance, the DOR is less useful for clinical decision making, or for indicating the informativeness of a single result from a quantitative test.

Glas AS, Lijmer JG, Prins MH, Bonsel GJ, Bossuyt PM.
The diagnostic odds ratio: a single indicator of test performance.
J Clin Epidemiol. 2003 Nov;56(11):1129-35.
doi: 10.1016/s0895-4356(03)00177-x.

(we were not aware of the Lachenbruch letter at the time…)

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Patrick it is fantastic to have you join the site. I look forward to continuing with more useful exchanges to come!

Happy to learn - and contribute, where possible - Frank!

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