detect_separation {detectseparation}

Detect Separation

Description

Method for glm that tests for data separation and finds which parameters have infinite maximum likelihood estimates in generalized linear models with binomial responses

detect_separation() is a method for glm that tests for the occurrence of complete or quasi-complete separation in datasets for binomial response generalized linear models, and finds which of the parameters will have infinite maximum likelihood estimates. detect_separation() relies on the linear programming methods developed in Konis (2007).

Usage

detect_separation(
  x,
  y,
  weights = NULL,
  start = NULL,
  etastart = NULL,
  mustart = NULL,
  offset = NULL,
  family = gaussian(),
  control = list(),
  intercept = TRUE,
  singular.ok = TRUE
)

detectSeparation(
  x,
  y,
  weights = NULL,
  start = NULL,
  etastart = NULL,
  mustart = NULL,
  offset = NULL,
  family = gaussian(),
  control = list(),
  intercept = TRUE,
  singular.ok = TRUE
)

Arguments

x

x is a design matrix of dimension n * p.

y

y is a vector of observations of length n.

weights

an optional vector of ‘prior weights’ to be used in the fitting process. Should be NULL or a numeric vector.

start

currently not used.

etastart

currently not used.

mustart

currently not used.

offset

this can be used to specify an a priori known component to be included in the linear predictor during fitting. This should be NULL or a numeric vector of length equal to the number of cases. One or more offset terms can be included in the formula instead or as well, and if more than one is specified their sum is used. See model.offset.

family

a description of the error distribution and link function to be used in the model. For glm this can be a character string naming a family function, a family function or the result of a call to a family function. For glm.fit only the third option is supported. (See family for details of family functions.)

control

a list of parameters controlling separation detection. See detect_separation_control() for details.

intercept

logical. Should an intercept be included in the null model?

singular.ok

logical. If FALSE, a singular model is an error.

Details

Following the definitions in Albert and Anderson (1984), the data for a binomial-response generalized linear model with logistic link exhibit quasi-complete separation if there exists a non-zero parameter vector \beta such that X^0 \beta \le 0 and X^1 \beta \ge 0, where X^0 and X^1 are the matrices formed by the rows of the model matrix $X$ corresponding to zero and non-zero responses, respectively. The data exhibits complete separation if there exists a parameter vector \beta such that the aforementioned conditions are satisfied with strict inequalities. If there are no vectors \beta that can satisfy the conditions, then the data points are said to overlap.

If the inverse link function G(t) of a generalized linear model with binomial responses is such that \log G(t) and \log (1 - G(t)) are concave and the model has an intercept parameter, then overlap is a necessary and sufficient condition for the maximum likelihood estimates to be finite (see Silvapulle, 1981 for a proof). Such link functions are, for example, the logit, probit and complementary log-log.

detect_separation() determines whether or not the data exhibits (quasi-)complete separation. Then, if separation is detected and the link function G(t) is such that \log G(t) and \log (1 - G(t)) are concave, the maximum likelihood estimates has infinite components.

detect_separation() is a wrapper to the detect_infinite_estimates() method. Separation detection, as separation is defined above, takes place using the linear programming methods in Konis (2007) regardless of the link function. The output of those methods is also used to determine which estimates are infinite, unless the link is "log". In the latter case the linear programming methods in Schwendinger et al. (2021) are called to establish if and which estimates are infinite. If the link function is not one of '"logit"', '"log"', '"probit"', '"cauchit"', '"cloglog"' then a warning is issued.

The coefficients method extracts a vector of values for each of the model parameters under the following convention: 0 if the maximum likelihood estimate of the parameter is finite, and Inf or -Inf if the maximum likelihood estimate of the parameter if plus or minus infinity. This convention makes it easy to adjust the maximum likelihood estimates to their actual values by element-wise addition.

detect_separation() can be passed directly as a method to the glm function. See, examples.

detectSeparation() is an alias for detect_separation().

Value

A list that inherits from class detect_separation, glm and lm. A print method is provided for detect_separation objects.

Note

For the definition of complete and quasi-complete separation, see Albert and Anderson (1984). Kosmidis and Firth (2021) prove that the reduced-bias estimator that results by the penalization of the logistic regression log-likelihood by Jeffreys prior takes always finite values, even when some of the maximum likelihood estimates are infinite. The reduced-bias estimates can be computed using the brglm2 R package.

detect_separation was designed in 2017 by Ioannis Kosmidis for the **brglm2** R package, after correspondence with Kjell Konis, and a port of the separator function had been included in **brglm2** under the permission of Kjell Konis. In 2020, detect_separation and check_infinite_estimates were moved outside **brglm2** into the dedicated **detectseparation** package. Dirk Schumacher authored the separator_ROI function, which depends on the **ROI** R package and is now the default implementation used for detecting separation. In 2022, Florian Schwendinger authored the dielb_ROI function for detecting infinite estimates in log-binomial regression, and, with Ioannis Kosmidis, they refactored the codebase to properly accommodate for the support of log-binomial regression.

Author(s)

Ioannis Kosmidis [aut, cre] [e-mail], Dirk Schumacher [aut] [e-mail], Florian Schwendinger [aut] [e-mail], Kjell Konis [ctb] [e-mail]

References

Konis K. (2007). *Linear Programming Algorithms for Detecting Separated Data in Binary Logistic Regression Models*. DPhil. University of Oxford. https://ora.ox.ac.uk/objects/uuid:8f9ee0d0-d78e-4101-9ab4-f9cbceed2a2a

Konis K. (2013). safeBinaryRegression: Safe Binary Regression. R package version 0.1-3. https://CRAN.R-project.org/package=safeBinaryRegression

Kosmidis I. and Firth D. (2021). Jeffreys-prior penalty, finiteness and shrinkage in binomial-response generalized linear models. *Biometrika*, **108**, 71–82. doi:10.1093/biomet/asaa052

Silvapulle, M. J. (1981). On the Existence of Maximum Likelihood Estimators for the Binomial Response Models. *Journal of the Royal Statistical Society. Series B (Methodological)*, **43**, 310–313. https://www.jstor.org/stable/2984941

Schwendinger, F., Grün, B. & Hornik, K. (2021). A comparison of optimization solvers for log binomial regression including conic programming. *Computational Statistics*, **36**, 1721–1754. doi:10.1007/s00180-021-01084-5

See Also

glm.fit and glm, detect_infinite_estimates, check_infinite_estimates, brglm_fit

Examples

# endometrial data from Heinze \& Schemper (2002) (see ?endometrial)
data("endometrial", package = "detectseparation")
endometrial_sep <- glm(HG ~ NV + PI + EH, data = endometrial,
                       family = binomial("logit"),
                       method = "detect_separation")
endometrial_sep
# The maximum likelihood estimate for NV is infinite
summary(update(endometrial_sep, method = "glm.fit"))


# Example inspired by unpublished microeconometrics lecture notes by
# Achim Zeileis https://eeecon.uibk.ac.at/~zeileis/
# The maximum likelihood estimate of sourhernyes is infinite
if (requireNamespace("AER", quietly = TRUE)) {
    data("MurderRates", package = "AER")
    murder_sep <- glm(I(executions > 0) ~ time + income +
                      noncauc + lfp + southern, data = MurderRates,
                      family = binomial(), method = "detect_separation")
    murder_sep
    # which is also evident by the large estimated standard error for NV
    murder_glm <- update(murder_sep, method = "glm.fit")
    summary(murder_glm)
    # and is also revealed by the divergence of the NV column of the
    # result from the more computationally intensive check
    plot(check_infinite_estimates(murder_glm))
    # Mean bias reduction via adjusted scores results in finite estimates
    if (requireNamespace("brglm2", quietly = TRUE))
        update(murder_glm, method = brglm2::brglm_fit)
}