Odds Ratios and Probability Ratios

Odds ratios are my favorite way to describe the strength of the relationship between two dichotomous variables.  One could use probability ratios, but there are problems with probability ratios, which I illustrate here.

Suppose that we randomly assign 200 ill patients to two groups.  One group is treated with a modern antibiotic.  The other group gets a homeopathic preparation.  At the end of two weeks we blindly determine whether or not there has been a substantial remission of symptoms of the illness.  The data are presented in the table below.

 Remission of Symptoms Treatment No Yes Antibiotic 10 90 100 Homeopathic 60 40 100 70 130 200

Odds of Success

For the group receiving the antibiotic, the odds of success (remission of symptoms) are 90/10 = 9.  For the group receiving the homeopathic preparation the odds of success are 40/60 = 2/3 or about .67.  The odds ratio reflecting the strength of the relationship is 9 divided by 2/3 = 13.5.  That is, the odds of success for those treated with the antibiotic are 13.5 times higher than for those treated with the homeopathic preparation.

Odds of Failure

For the group receiving the homeopathic preparation, the odds of failure (no remission of symptoms) are 60/40 = 1.5.  For those receiving the antibiotic the odds of failure are 10/90 = 1/9.  The odds ratio is 1.5 divided by 1/9 = 13.5.  Notice that this ratio is the same as that obtained when we used odds of success, as, IMHO, it should be.

Now let us see what happens if we use probability ratios.

Probability of Success

For the group receiving the antibiotic, the probability of success is 90/100 = .9.  For the homeopathic group the probability of success is 40/100 = .4.  The ratio of these probabilities is .9/.4 = 2.25.  The probability of success for the antibiotic group is 2.25 times that of the homeopathic group.

Probability of Failure

For the group  receiving the homeopathic preparation the probability of failure is 60/100 = .6.  For the antibiotic group it is 10/100 = .10.  The probability ratio is .6/.1 = 6.  The probability of failure for the homeopathic group is six times that for the antibiotic group.

The Problem

With probability ratios the value you get to describe the strength of the relationship when you compare (A given B) to (A given not B) is not the same as what you get when you compare (not A given B) to (not A given not B).  This is, IMHO, a big problem.  There is no such problem with odds ratios.

Another Example

According to a report provided by Medscape, among the general population in the US, the incidence of narcissistic personality disorder is 0.5%.  Among members of the US Military it is 20%.

Probability of NPD.

The probability that a randomly selected member of the military will have NPD is 20%, the probability that a randomly selected member of the general population will have NPD is 0.5%.  This yields a probability ratio of .20/.005 = 40.

Probability of NOT NPD.

The probability that a randomly selected member of the general population will not have NPD is .995.  The probability that a randomly selected member of the military will not have NPD is .80.  This yields a probability ratio of .995/.8 = 1.24.

Odds of NPD

For members of the military, .2/.8 = .25.  For members of the general population, .005/.995 = .005.  The odds ratio is (.2/.8) / (.005/.995) = 49.75.

Odds of NOT NPD.

For members of the military, .8/.2 = 4.  For members of the general population, .995/.005 = 199.  The odds ratio is (.995/.005) / (.8/.2) = 49.75.