= Using algebra to show why the maximum likelihood estimates are undefined when a response group only occurs in certain predictor groups =

We consider a binary outcome (positive/negative) consisting of groups with $$n_text{1}$$ positives and $$n_text{2}$$ 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 negative outcome when x=1. This circumstance is known as ''complete separation''. The log-likelihood may be written as

ln L = $$n_text{1}$$ ln p + $$n_text{2}$$ ln (1-p)

In a binary logistic regression

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

ln L = $$n_text{1}$$(a+bx) - $$n_text{1}$$ ln(1 + $$e^text{a+bx}$$) + $$n_text{2}$$ - $$n_text{2}$$ ln(1+$$e^text{a+bx}$$)

$$\frac{d}{db} = n_text{1}x - n_text{1}x \frac{e^text{a+bx}}{1+e^text{a+bx}} - n_text{2}x \frac{e^text{a+bx}}{1+e^text{a+bx}}$$

Since all the x=0 scores are in the 'positive' group (which we denote as group 1) and all the x=1 scores are in the 'negative' group (which we denote as group 2) we have

$$\frac{d}{db} = n_text{2}\frac{ e^text{a+b}}{1+e^text{a+b}}$$. 

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

This argument can be extended to continuous predictors where an outcome group only occurs for values below or above certain thresholds of the predictor. 

When the maximum likelihood estimates are undefined associated diagnostics such as -2 log likelihoods and R-squares may be outputted as 0 or 1.