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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 x2 x3

Enumerating sample-space combinations:
observation 1:   enumerations =          2
observation 2:   enumerations =          4
(omitted)
observation 57:  enumerations =       1659
observation 58:  enumerations =        900
note: CMLE estimate for treatment is +inf; computing MUE

Exact logistic regression                        Number of obs =         58
Model score   =   13.87402
Pr >= score   =     0.0020

response   Odds Ratio       Suff.  2*Pr(Suff.)     [95% Conf. Interval]

treatment     2.101107*         16      0.5277      .2616515       +Inf

x2     .5183876          13      0.7180      .0690557    4.336805

x3       .11389           7      0.0068      .0172465    .6023146

(*) 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.