Where separation or quasi-complete separation of categorical variables occur, no unique maximum likelihood estimates exist.

TABLE 11.1

No separation.

Treatment

Placebo

Surrogate

Y

A ^ 0

В yt 0

N

C’yt 0

Dft 0

Treatment

Placebo

Surrogate

Y

A ft 0

0

N

0

1)^0

TABLE 11.2

Complete separation.

Separation: Binary Variables

Let us consider the case of two binary variables, for example, where a binary surrogate S is regressed on a binary treatment variable Z, as in (11.4). Complete and quasi-complete separation relate to the existence of empty cells in the cross-tabulation of S and Z. Table 11.1 shows no separation, as there are no empty cells. Table 11.2 gives an example of complete separation, when the binary variable Z perfectly predicts S. Table 11.3 illustrates quasi-complete separation, as one table cell is empty.

For complete or quasi-complete separation, the likelihood has no maximum, although it is bounded above by a number less than zero (Allison, 2008). For two binary variables we estimate the log-odds ratio ф as

Here, we can see that if a zero occurs in the denominator but not in the numerator, then ф = +то; if a zero occurs in the numerator but not in the denominator, then ф = —то. Both are limiting cases. If a zero value occurs in both, then ф is undefined.