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## Exact logistic regression

Stata's exlogistic fits exact logistic regression models and provides more reliable statistical inference with small-sample datasets. The dependent variable can be Bernoulli (0 or 1) or binomial (the number of successes in n trials). Exact joint hypothesis tests can be performed, and predictions with exact confidence intervals can be obtained.

### Example

Stata’s exact logistic regression provides better coverage in small samples than does standard logistic regression.

It also provides parameter estimates and confidence intervals where standard asymptotic methods cannot.

Such cases include small-data problems with binary regressors for which the outcome is 1 whenever the regressor is 1.

In the example below, every treated patient exhibits a positive response. Standard logistic regression cannot estimate the treatment effect. Stata’s exlogistic can:

. exlogistic response treatment gender hypertension

Enumerating sample-space combinations:
Observation 1:  Enumerations =          2
Observation 2:  Enumerations =          3
(omitted)
Observation 56: Enumerations =       6329
Observation 57: Enumerations =       3528
note: CMLE estimate for treatment is +inf; computing MUE.

Exact logistic regression                          Number of obs =       57
Model score   = 14.56624
Pr >= score   =   0.0013

response   Odds ratio       Suff.  2*Pr(Suff.)     [95% conf. interval]

treatment     21.66646*        20        0.0003     3.302679       +inf
gender     1.303983         22        0.9448     .2871768    6.012212
hypertens~n     2.431485         25        0.3072     .5447126    11.66149

(*) median unbiased estimates (MUE)


Parameter estimates, standard errors, and CIs are calculated on the basis of permutation without recourse to asymptotic assumptions and results.

Not only is the treatment effect estimated, but tests of significance and reported confidence intervals are based on exact methods.

Stata also includes exact Poisson regression for count data.