Auxiliary function for glm fitting using the brglmFit method.

Description

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

Usage

brglmControl(
  epsilon = 1e-06,
  maxit = 100,
  trace = FALSE,
  type = c("AS_mixed", "AS_mean", "AS_median", "correction", "MPL_Jeffreys", "ML"),
  transformation = "identity",
  slowit = 1,
  response_adjustment = NULL,
  max_step_factor = 12,
  a = 1/2
)

brglm_control(
  epsilon = 1e-06,
  maxit = 100,
  trace = FALSE,
  type = c("AS_mixed", "AS_mean", "AS_median", "correction", "MPL_Jeffreys", "ML"),
  transformation = "identity",
  slowit = 1,
  response_adjustment = NULL,
  max_step_factor = 12,
  a = 1/2
)

Arguments

Details

brglmControl provides default values and sanity checking for the various constants that control the iteration and generally the behaviour of brglmFit.

When trace is true, calls to cat produce the output for each iteration. Hence, options(digits = *) can be used to increase the precision.

transformation sets the transformation of the dispersion parameter for which the bias reduced estimates are computed. Can be one of "identity", "sqrt", "inverse", "log" and "inverseSqrt". Custom transformations are accommodated by supplying a list of two expressions (transformation and inverse transformation). See the examples for more details.

The value of response_adjustment is only relevant if brglmFit is called with start = NULL, and family is binomial or poisson. For those models, an initial maximum likelihood fit is obtained on adjusted data to provide starting values for the iteration in brglmFit. The value of response_adjustment governs how the data is adjusted. Specifically, if family is binomial, then the responses and totals are adjusted by and 2 * response_adjustment, respectively; if family is poisson, then the responses are adjusted by and response_adjustment. response_adjustment = NULL (default) is equivalent to setting it to "number of parameters"/"number of observations".

. . When type = "AS_mixed" (default), mean bias reduction is used for the regression parameters, and median bias reduction for the dispersion parameter, if that is not fixed. This adjustment has been developed based on equivariance arguments (see, Kosmidis et al, 2019, Section 4) in order to produce regression parameter estimates that are invariant to arbitrary contrasts, and estimates for the dispersion parameter that are invariant to arbitrary non-linear transformations. type = "AS_mixed" and type = "AS_mean" return the same results if brglmFit is called with family binomial or poisson (i.e. families with fixed dispersion).

When type = "MPL_Jeffreys", brglmFit will maximize the penalized log-likelihood

l(beta, phi) + a log det i(beta, phi)

where i(beta, phi) is the expected information matrix about the regression parameters β and the dispersion parameter φ. See, vignette("iteration", "brglm2") for more information. The argument $a$ controls the amount of penalization and its default value is a = 1/2, corresponding to maximum penalized likelihood using a Jeffreys-prior penalty. See, Kosmidis & Firth (2019) for proofs and discussion about the finiteness and shrinkage properties of the maximum penalized likelihood estimators for binomial-response generalized linear models.

The estimates from type = "AS_mean" and type = "MPL_Jeffreys" with a = 1/2 (default) are identical for Poisson log-linear models and logistic regression models, i.e. for binomial and Poisson regression models with canonical links. See, Firth (1993) for details.

brglm_control is an alias to brglmControl.

Value

a list with components named as the arguments, including symbolic expressions for the dispersion transformation (Trans) and its inverse (inverseTrans)

Author(s)

Ioannis Kosmidis ioannis.kosmidis@warwick.ac.uk

References

Kosmidis I, Kenne Pagui EC, Sartori N (2019). Mean and median bias reduction in generalized linear models. *arXiv e-prints*, arXiv:1804.04085. To appear in Statistics and Computing, <URL: https://arxiv.org/abs/1804.04085>.

Kosmidis I and Firth D (2019). Jeffreys-prior penalty, finiteness and shrinkage in binomial-response generalized linear models. *arXiv e-prints*, arXiv:1812.01938

#' Firth D (1993). Bias reduction of maximum likelihood estimates. Biometrika, **80**, 27-38

See Also

brglm_fit and glm.fit

Examples

data("coalition", package = "brglm2")
## The maximum likelihood fit with log link
coalitionML <- glm(duration ~ fract + numst2, family = Gamma, data = coalition)

## Bias reduced estimation of the dispersion parameter
coalitionBRi <- glm(duration ~ fract + numst2, family = Gamma, data = coalition,
                    method = "brglmFit")
coef(coalitionBRi, model = "dispersion")

## Bias reduced estimation of log(dispersion)
coalitionBRl <- glm(duration ~ fract + numst2, family = Gamma, data = coalition,
                    method = "brglmFit", transformation = "log")
coef(coalitionBRl, model = "dispersion")

## Just for illustration: Bias reduced estimation of dispersion^0.25
my_transformation <- list(expression(dispersion^0.25), expression(transformed_dispersion^4))
coalitionBRc <- update(coalitionBRi, transformation = my_transformation)
coef(coalitionBRc, model = "dispersion")