Computational Aspects
Separation: Categorical Variables
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.