East Carolina University

Department of Psychology

**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**.

Contact Information for the Webmaster,

Dr. Karl L. Wuensch

This page most recently revised on
24. October, 2009.