Package 'detectseparation'

Generic method for checking for infinite estimates

Description

Generic method for checking for infinite estimates

Usage

check_infinite_estimates(object, ...)

checkInfiniteEstimates(object, ...)

Arguments

object	a fitted model object (e.g. the result of a glm call).
...	other options to be passed to the method.

See Also

check_infinite_estimates.glm

---

A simple diagnostic of whether the maximum likelihood estimates are infinite

Description

A simple diagnostic of whether the maximum likelihood estimates are infinite

Usage

## S3 method for class 'glm'
check_infinite_estimates(object, nsteps = 20, ...)

Arguments

object	the result of a glm call.
nsteps	starting from maxit = 1, the GLM is refitted for maxit = 2, maxit = 3, ..., maxit = nsteps. Default value is 30.
...	currently not used.

Details

check_infinite_estimates() attempts to identify the occurrence of infinite estimates in GLMs with binomial responses by successively refitting the model. At each iteration the maximum number of allowed IWLS iterations is fixed starting from 1 to nsteps (by setting control = glm.control(maxit = j), where j takes values 1, ..., nsteps in glm). For each value of maxit, the estimated asymptotic standard errors are divided to the corresponding ones from control = glm.control(maxit = 1). Then, based on the results in Lesaffre & Albert (1989), if the sequence of ratios in any column of the resultant matrix diverges, then complete or quasi-complete separation occurs and the maximum likelihood estimate for the corresponding parameter has value minus or plus infinity.

check_infinite_estimates() can also be used to identify the occurrence of infinite estimates in baseline category logit models for nominal responses (see brmultinom() from the brglm2 R package), and adjacent category logit models for ordinal responses (see bracl() from the brglm2 R package).

Value

An object of class inf_check that has a plot method.

A matrix inheriting from class inf_check, with nsteps rows and p columns, where p is the number of model parameters. A plot method is provided for inf_check objects for the easy inspection of the ratios of the standard errors.

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.

References

Lesaffre, E., & Albert, A. (1989). Partial Separation in Logistic Discrimination. *Journal of the Royal Statistical Society. Series B (Methodological)*, **51**, 109-116

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

See Also

multinom, detect_separation, brmultinom, bracl

Examples

# endometrial data from Heinze \& Schemper (2002) (see ?endometrial)
data("endometrial", package = "detectseparation")
endometrial_ml <- glm(HG ~ NV + PI + EH, data = endometrial,
                      family = binomial("probit"))
# clearly the maximum likelihood estimate for the coefficient of
# NV is infinite
(estimates <- check_infinite_estimates(endometrial_ml))
plot(estimates)



# Aligator data (Agresti, 2002, Table~7.1)
if (requireNamespace("brglm2", quietly = TRUE)) {
    data("alligators", package = "brglm2")
    all_ml <- brglm2::brmultinom(foodchoice ~ size + lake , weights = round(freq/3),
                         data = alligators, type = "ML", ref = 1)
    # Clearly some estimated standard errors diverge as the number of
    # Fisher scoring iterations increases
    plot(check_infinite_estimates(all_ml))
    # Bias reduction the brglm2 R packages can be used to get finite estimates
    all_br <- brglm2::brmultinom(foodchoice ~ size + lake , weights = round(freq/3),
                         data = alligators, ref = 1)
    plot(check_infinite_estimates(all_br))
}

---

Detect Infinite Estimates

Description

Method for glm that detects infinite components in the maximum likelihood estimates of generalized linear models with binomial responses.

Usage

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

detectInfiniteEstimates(
  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

For binomial-response generalized linear models with "log" link, separated data allocations do not necessarily lead to infinite maximum likelihood estimates. For this reason, for models with the "log" link detect_infinite_estimates() relies on an alternative linear optimization model developed in Schwendinger et al. (2021), and for all the other supported links it relies on the linear programming methods developed in Konis (2007). See detect_separation() for definitions and details.

detect_infinite_estimates() is a wrapper to the functions separator_ROI(), separator_lpSolveAPI() (a modified version of the separator() function from the **safeBinaryRegression** R package), and dielb_ROI().

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

detectInfiniteEstimates() is an alias for detect_infinite_estimates().

Author(s)

Ioannis Kosmidis [aut, cre] [email protected], Florian Schwendinger [aut] [email protected], Dirk Schumacher [aut] [email protected], Kjell Konis [ctb] [email protected]

References

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(3), 310–313. https://www.jstor.org/stable/2984941

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

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_separation, check_infinite_estimates, brglm_fit

Examples

# The classical example given in Silvapulle (1981) can be utilized
# to show that for the Log-Binomial model there exist data allocations
# which are separated but produce finite estimates.
data("silvapulle1981", package = "detectseparation")

# Since the data is separated the MLE does not exist for the logit link.
glm(y ~ ghqs, data = silvapulle1981, family = binomial(),
    method = "detect_infinite_estimates")

# However, for the log link all components of the MLE are finite.
glm(y ~ ghqs, data = silvapulle1981, family = binomial("log"),
    method = "detect_infinite_estimates")
glm(y ~ ghqs, data = silvapulle1981, family = binomial("log"), start = c(-1, 0))

---

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] [email protected], Dirk Schumacher [aut] [email protected], Florian Schwendinger [aut] [email protected], Kjell Konis [ctb] [email protected]

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)
}

---

Auxiliary function for the glm interface when method is detect_separation.

Description

Typically only used internally by detect_separation but may be used to construct a control argument.

Usage

detect_separation_control(
  implementation = c("ROI", "lpSolveAPI"),
  solver = "lpsolve",
  linear_program = c("primal", "dual"),
  purpose = c("find", "test"),
  tolerance = 1e-04,
  solver_control = list()
)

detectSeparationControl(
  implementation = c("ROI", "lpSolveAPI"),
  solver = "lpsolve",
  linear_program = c("primal", "dual"),
  purpose = c("find", "test"),
  tolerance = 1e-04,
  solver_control = list()
)

Arguments

implementation	should the implementation using ROI or the implementation using lpSolveAPI be used? Default is ROI.
solver	should the linear program be solved using the "lpsolve" (using the ROI.plugin.lpsolve package; default) or another solver? Alternative solvers are "glpk", "cbc", "clp", "cplex", "ecos", "gurobi", "scs", "symphony". If ROI.plugin.[solver] is not installed then the user will be prompted to install it before continuing.
linear_program	should detect_separation solve the "primal" (default) or "dual" linear program for separation detection? Only relevant if implementation = "lpSolveAPI".
purpose	should detect_separation simply "test" for separation or also "find" (default) which parameters are infinite? Only relevant if implementation = "lpSolveAPI".
tolerance	maximum absolute variable value from the linear program, before separation is declared. Default is 1e-04.
solver_control	a list with additional control parameters for the "solver". This is solver specific, so consult the corresponding documentation. Default is list() unless solver is "alabama" when the default is list(start = rep(0, p)), where p is the number of parameters.

Value

A list with the supplied linear_program, solver, solver_control, purpose, tolerance, implementation, and the matched separator function (according to the value of implementation).

---

detectseparation: Methods for Detecting and Checking for Separation and Infinite Maximum Likelihood Estimates

Description

detectseparation provides pre-fit and post-fit methods for the detection of separation and of infinite maximum likelihood estimates in binomial response generalized linear models.

Details

The key methods are detect_separation and check_infinite_estimates.

See Also

detect_separation, check_infinite_estimates

---

Histology grade and risk factors for 79 cases of endometrial cancer

Description

Histology grade and risk factors for 79 cases of endometrial cancer

Usage

endometrial

Format

A data frame with 79 rows and 4 variables:

NV

neovasculization with coding 0 for absent and 1 for present

PI

pulsality index of arteria uterina

EH

endometrium height

HG

histology grade with coding 0 for low grade and 1 for high grade

Source

The packaged data set was downloaded in .dat format from https://users.stat.ufl.edu/~aa/glm/data/. The latter link provides the data sets used in Agresti (2015).

The endometrial data set was first analyzed in Heinze and Schemper (2002), and was originally provided by Dr E. Asseryanis from the Medical University of Vienna.

References

Agresti, A. (2015). *Foundations of Linear and Generalized Linear Models*. Wiley Series in Probability and Statistics. Wiley

Heinze, G., & Schemper, M. (2002). A Solution to the Problem of Separation in Logistic Regression. *Statistics in Medicine*, **21**, 2409–2419

See Also

brglm_fit

---

Habitat preferences of lizards

Description

The lizards data frame has 23 rows and 6 columns. Variables grahami and opalinus are counts of two lizard species at two different perch heights, two different perch diameters, in sun and in shade, at three times of day.

Usage

lizards

Format

An object of class data.frame with 23 rows and 6 columns.

Details

• grahami. count of grahami lizards

• opalinus. count of opalinus lizards

• height. a factor with levels <5ft, >=5ft

• diameter. a factor with levels <=2in, >2in

• light. a factor with levels sunny, shady

• time. a factor with levels early, midday, late

Source

McCullagh, P. and Nelder, J. A. (1989) _Generalized Linear Models_ (2nd Edition). London: Chapman and Hall.

Originally from

Schoener, T. W. (1970) Nonsynchronous spatial overlap of lizards in patchy habitats. _Ecology_ *51*, 408-418.

See Also

brglm_fit

---

Separation Example Presented in Silvapulle (1981)

Description

Separation example presented in Silvapulle (1981).

Usage

silvapulle1981

Format

A data frame with 35 rows and 2 variables:

y

a factor with the levels case and none-case, giving the outcome of a standardized psychiatric interview

ghqs

an integer giving the general health questionnaire score.

References

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(3), 310–313. https://www.jstor.org/stable/2984941

---