= Estimation in cases of infinite maximum likelihood estimates =

We consider a binary outcome (positive/negative) consisting of groups with n1 positives and n2 negatives and the probability of a positive outcome equal to p. Suppose we have a binary group predictor where we only get a positive outcome when x=0 and a negatuive outcome when x=1. The log-likelihood may be written as

ln L = n1 ln p + n2 ln (1-p)

In a binary logistic regression

$$p = \frac{e(a+bx)}{1+ e(a+bx)}$$ where x is the group predictor taking values 0 and 1. 

ln L = n1(a+bx) - n1 ln(1 + e(a+bx)) + n2 - n2 ln(1+e(a+bx))

$$\frac{d}{db} = n1x - n1x \frac{e(a+b)}{1+e(a+b)} - n2x \frac{e(a+b)}{1+e(a+b)}$$

Since the positive outcome only occurs in one of the predictor groups we have that all the x=0 scores are in the 'positive' group so

$$\frac{d}{db} = \frac{n2 e(a+b)}{(1+e(a+b)}$$. 

$$\frac{d}{db}$$ =0 for maximum likelihood estimates and $$\frac{d}{db}$$ can only equal zero with infinite estimates of a and b.