### Mischievous Odds Ratios

#### Posted by plosmedicine on 30 Mar 2009 at 23:50 GMT

Author: 'William' 'Steinsmith'
Position: MD
Institution: none
E-mail: bbhaywood@aol.com
Submitted Date: February 03, 2006
Published Date: February 7, 2006
This comment was originally posted as a “Reader Response” on the publication date indicated above. All Reader Responses are now available as comments.

Reitsma, et al, have explored -- in a population of patients anticoagulated with coumarin congeners -- the connection between the presence of mutant alleles of a single gene and the risk of hemorrhage(1).

Using as their denominator the odds for bleeding in a non-mutant patient, and using as their numerators the odds for patients with each of the two mutant alleles, the authors propose the resulting odds ratios as surrogates for the relative risk of hemorrhage.

It should be noted, however, that the conflation of an odds ratio with a relative risk is not generally justified (2,3). The relative risk is the ratio of two probabilities (p2:p1), whereas the corresponding odds ratio is (p1:1-p1)/(p2:1-p2). Equating these two ratios requires that p1=p2, i.e., that the risk ratio be unity.

In the Reitsma paper, none of the eight presented odds ratios in Table 2 turns out identical with the corresponding calculated risk ratio, and the most discordant pair of values diverge by a factor of about 1.4, i.e., the odds ratio of 2.6 corresponding to a relative risk of 1.9.

Mischievous conflation of odds ratios with probability ratios is widespread in the literature dealing with laboratory testing, with the odds ratio (confusingly termed the "likelihood ratio") typically presented as surrogate for the corresponding ratio of probabilities.

The power of a positive laboratory test to enhance the likelihood of disease presence in a given patient (properly termed the "positive probability-based likelihood ratio") is the ratio of two probabilities: the probability that the positive-tested patient is truly diseased (termed the "positive predictive value"} divided by the probability of disease in the pre-test population (termed the "disease prevalence")

Expressed explicitly in terms of the sub-categories of the test population, the positive predictive value is the ratio represented by (True Positives):(True Positives + False Positives), and the prevalence is the ratio represented by (True Positives + False Negatives):(True Positives + False Negatives + True Negatives + False Positives).

The calculus is easily adapted to computation of the probability-based likelihood ratio for the absence of disease in a given patient. In this case, the post-negative-test probability of disease absence (termed the "negative predictive value") is the ratio represented by (True Negatives):(True Negatives + False Negatives), and the pre-test probability is 1 minus the disease prevalence. The negative probability-based likelihood ratio is then the ratio represented by the post-test probability divided by the pre-test probability.

A more descriptive term for the probability-based likelihood ratio would be the "probability magnifying power," since it leads to the expanded probability of the presence (or absence) of disease yielded by a positive (or negative) test result.

-William Steinsmith, MD

(1) Reitsma PH, et al, Coumarin Sensitivity and Bleeding Risk. PLos Med. 2005. 2(10): e312

(2) Van den Ende J, et al, The Trouble with Likelihood Ratios. The Lancet. 2005. 366: 548

(3) Wessler AM & Bailey KR, A Critique on Contemporary Reporting of Likelihood Ratios in Test Power Analysis. Mayo Clin Proc. 2004. 79(10): 1317-1318, 1341

No competing interests declared.