# scipy.stats.contingency.odds_ratio

scipy.stats.contingency.odds_ratio(table, *, kind='conditional')

  Compute the odds ratio for a 2x2 contingency table.

  Parameters:
    table array_like of ints

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

    kind str, 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'`.

  Returns:
    result `OddsRatioResult` instance

      The returned object has two computed attributes:

      statisticfloat

        - 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.

  See also

  `scipy.stats.fisher_exact`
  `relative_risk`

  Notes

  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.

  References

  [1]

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

  [2]

  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.)

  [3]

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

  [4]

  Berger, Jeffrey S. et al. “Aspirin for the Primary Prevention of Cardiovascular Events in Women and Men: A Sex-Specific Meta-analysis of Randomized Controlled Trials.” JAMA, 295(3):306-313, DOI:10.1001/jama.295.3.306, 2006.

  Examples

  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.

  In [4], the use of aspirin to prevent cardiovascular events in women and men was investigated. The study notably concluded:

  > …aspirin therapy reduced the risk of a composite of cardiovascular events due to its effect on reducing the risk of ischemic stroke in women […]

  The article lists studies of various cardiovascular events. Let’s focus on the ischemic stoke in women.

  The following table summarizes the results of the experiment in which participants took aspirin or a placebo on a regular basis for several years. Cases of ischemic stroke were recorded:

                        Aspirin   Control/Placebo
      Ischemic stroke     176           230
      No stroke         21035         21018

  The question we ask is “Is there evidence that the aspirin reduces the risk of ischemic stroke?”

  Compute the odds ratio:

      >>> from scipy.stats.contingency import odds_ratio
      >>> res = odds_ratio([[176, 230], [21035, 21018]])
      >>> res.statistic
      0.7646037659999126

  For this sample, the odds of getting an ischemic stroke for those who have been taking aspirin are 0.76 times that of those who have received the placebo.

  To make statistical inferences about the population under study, we can compute the 95% confidence interval for the odds ratio:

      >>> res.confidence_interval(confidence_level=0.95)
      ConfidenceInterval(low=0.6241234078749812, high=0.9354102892100372)

  The 95% confidence interval for the conditional odds ratio is approximately (0.62, 0.94).

  The fact that the entire 95% confidence interval falls below 1 supports the authors’ conclusion that the aspirin was associated with a statistically significant reduction in ischemic stroke.