The complete separation occurs when all subjects are completely classified into each response group according to the values of the explanatory variable(s). Consider the following data set (Table 1).²⁾

The estimated odds ratio (OR) using X=0 as the reference is infinite [OR=(a/b)/(c/d)]=(5/0)/(0/5)=25/0]. Note that if either (b) or (c) is equal to zero, then the OR is undefined. We also obtain an infinite value of standard error for ORSEOR=OR1a+1b+1c+1d=25/0×15+10+10+15=infinite×infinite. Also note that if any one of the four cell frequencies is equal to zero, then SE (OR) is undefined. Gart and Zweifel³⁾ suggested improved estimates of OR and SE (OR) which are calculated after adding 0.5 to each cell. According to their suggestion, we obtain OR'=(5.5/0.5)/(0.5/5.5)=121 and SE (OR')=252.76.

We can obtain the OR and its 95% confidence limits for Table 1 using the following commands in PROC FREQ procedure (here, we do not consider the small sample size problem).

DATA CS; INPUT X Y Z; CARDS;
1 1 5
1 0 0
0 1 0
0 0 5
;
PROC FREQ DATA=CS; TABLES X*Y/ALL; WEIGHT Z; RUN;

<OUTPUT>
Odds ratio = 121.000, 95% confidence limits = 2.0169 ~
7259.1773

Other options, RELRISK or MEASURES in TABLES statement cannot calculate OR and SE (OR) in this case.

We can also obtain the OR and its 95% confidence limits for Table 1 using the following commands in PROC LOGISTIC procedure.

PROC LOGISTIC DESCENDING DATA=CS; MODEL Y=X/
FIRTH; WEIGHT Z; RUN;

<OUTPUT>
Odds ratio = 120.976, 95% confidence limits = 1.364 ~ >999.999

We can see that both procedures give the same ORs, but somewhat different 95% confidence limits. However if we do not use the FIRTH option in MODEL statement, then values of OR and upper 95% confidence limit are infinites.