conditional exact logistic regression

Conditional Logistic Regression:

Taking the stratification into account by "conditioning out" (and not estimating) the stratum-specific intercepts gives consistent and asymptotically normal MLEs for the slope coefficients. If your nuisance parameters are not just stratum-specific intercepts, you can perform an exact conditional logistic regression.

The likelihood function to be maximized feels really like that in proc phreg with a discrete option. Each item (for a stratum) in the likelihood function is the conditional probability of observing subject 1 to h who actually had an event conditioning on h event happened (for any h subjects) in the stratum. The concept of sufficient statistic is not explicitly. Conditional logistic regression is called with the strata statement. More examples are from here and here.

Exact Conditional Logistic Regression
Exact conditional inference is based on generating the conditional distribution for the sufficient statistics of the parameters of interest. This distribution is called the permutation or exact conditional distribution. One example from sas suggests that we move from conditional regression to exact conditional regression when "you believe the data set is too small or too sparse for the usual asymptotics to hold.":

• by "sparse", they probably refer to the problem of complete separation of quasi-complete separation, and the exact method is one way to handle this, especially when the firth option is only available in sas 9.1.3;
• by "usual asymptotics", they probably refer to the issue of  large-sample asymptotic normality when we do not have a large sample size compared to the number of parameters to estimate.