scipy.stats.contingency.odds_ratio(table, *, kind='conditional')[source]#

Compute the odds ratio for a 2x2 contingency table.

tablearray_like of ints

A 2x2 contingency table. Elements must be non-negative integers.

kindstr, optional

Which kind of odds ratio to compute, either the sample odds ratio (kind='sample') or the conditional odds ratio (kind='conditional'). Default is 'conditional'.

resultOddsRatioResult instance

The returned object has two computed attributes:

  • If kind is 'sample', this is sample (or unconditional) estimate, given by table[0, 0]*table[1, 1]/(table[0, 1]*table[1, 0]).

  • If kind is 'conditional', this is the conditional maximum likelihood estimate for the odds ratio. It is the noncentrality parameter of Fisher’s noncentral hypergeometric distribution with the same hypergeometric parameters as table and whose mean is table[0, 0].

The object has the method confidence_interval that computes the confidence interval of the odds ratio.


The conditional odds ratio was discussed by Fisher (see “Example 1” of [1]). Texts that cover the odds ratio include [2] and [3].

New in version 1.10.0.



R. A. Fisher (1935), The logic of inductive inference, Journal of the Royal Statistical Society, Vol. 98, No. 1, pp. 39-82.


Breslow NE, Day NE (1980). Statistical methods in cancer research. Volume I - The analysis of case-control studies. IARC Sci Publ. (32):5-338. PMID: 7216345. (See section 4.2.)


H. Sahai and A. Khurshid (1996), Statistics in Epidemiology: Methods, Techniques, and Applications, CRC Press LLC, Boca Raton, Florida.


In epidemiology, individuals are classified as “exposed” or “unexposed” to some factor or treatment. If the occurrence of some illness is under study, those who have the illness are often classifed as “cases”, and those without it are “noncases”. The counts of the occurrences of these classes gives a contingency table:

            exposed    unexposed
cases          a           b
noncases       c           d

The sample odds ratio may be written (a/c) / (b/d). a/c can be interpreted as the odds of a case occurring in the exposed group, and b/d as the odds of a case occurring in the unexposed group. The sample odds ratio is the ratio of these odds. If the odds ratio is greater than 1, it suggests that there is a positive association between being exposed and being a case.

Interchanging the rows or columns of the contingency table inverts the odds ratio, so it is import to understand the meaning of labels given to the rows and columns of the table when interpreting the odds ratio.

Consider a hypothetical example where it is hypothesized that exposure to a certain chemical is assocated with increased occurrence of a certain disease. Suppose we have the following table for a collection of 410 people:

          exposed   unexposed
cases         7         15
noncases     58        472

The question we ask is “Is exposure to the chemical associated with increased risk of the disease?”

Compute the odds ratio:

>>> from scipy.stats.contingency import odds_ratio
>>> res = odds_ratio([[7, 15], [58, 472]])
>>> res.statistic

For this sample, the odds of getting the disease for those who have been exposed to the chemical are almost 3.8 times that of those who have not been exposed.

We can compute the 95% confidence interval for the odds ratio:

>>> res.confidence_interval(confidence_level=0.95)
ConfidenceInterval(low=1.2514829132266785, high=10.363493716701269)

The 95% confidence interval for the conditional odds ratio is approximately (1.25, 10.4).