In
statistics, separation is a phenomenon associated with models for
dichotomous
A dichotomy is a partition of a whole (or a set) into two parts (subsets). In other words, this couple of parts must be
* jointly exhaustive: everything must belong to one part or the other, and
* mutually exclusive: nothing can belong simult ...
or categorical outcomes, including
logistic and
probit regression
In statistics, a probit model is a type of regression where the dependent variable can take only two values, for example married or not married. The word is a portmanteau, coming from ''probability'' + ''unit''. The purpose of the model is to e ...
. Separation occurs if the predictor (or a
linear combination of some subset of the predictors) is associated with only one outcome value when the predictor range is split at a certain value.
The phenomenon
For example, if the predictor ''X'' is continuous, and the outcome ''y'' = 1 for all observed ''x'' > 2. If the outcome values are perfectly determined by the predictor (e.g., ''y'' = 0 when ''x'' ≤ 2) then the condition "complete separation" is said to occur. If instead there is some overlap (e.g., ''y'' = 0 when ''x'' < 2, but ''y'' has observed values of 0 and 1 when ''x'' = 2) then "quasi-complete separation" occurs. A
2 × 2 table with an empty (zero) cell is an example of quasi-complete separation.
The problem
This observed form of the data is important because it sometimes causes problems with the estimation of regression coefficients. For example,
maximum likelihood (ML) estimation relies on maximization of the likelihood function, where e.g. in case of a
logistic regression
In statistics, the logistic model (or logit model) is a statistical model that models the probability of an event taking place by having the log-odds for the event be a linear combination of one or more independent variables. In regression a ...
with completely separated data the maximum appears at the parameter space's margin, leading to "infinite" estimates, and, along with that, to problems with providing sensible
standard errors. Statistical software will often output an arbitrarily large parameter estimate with a very large standard error.
Possible remedies
An approach to "fix" problems with
ML estimation is the use of
regularization (or "
continuity corrections").
In particular, in case of a logistic regression problem, the use of ''exact logistic regression'' or ''Firth logistic regression'', a bias-reduction method based on a penalized likelihood, may be an option.
Alternatively, one may avoid the problems associated with likelihood maximization by switching to a
Bayesian
Thomas Bayes (/beɪz/; c. 1701 – 1761) was an English statistician, philosopher, and Presbyterian minister.
Bayesian () refers either to a range of concepts and approaches that relate to statistical methods based on Bayes' theorem, or a followe ...
approach to inference. Within a Bayesian framework, the pathologies arising from likelihood maximization are avoided by the use of
integration rather than
maximization, as well as by the use of sensible
prior probability distributions.
References
Further reading
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External links
Logistic regression using Firth's bias reduction: a solution to the problem of separation in logistic regression
{{DEFAULTSORT:Separation (Statistics)
Logistic regression