brglmControl {brglm2}

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,
  check_aliasing = TRUE,
  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,
  check_aliasing = TRUE,
  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

epsilon

positive convergence tolerance epsilon. Default is 1e-06.

maxit

integer giving the maximal number of iterations allowed. Default is 100.

check_aliasing

logical indicating where a QR decomposition of the model matrix should be used to check for aliasing. Default is TRUE. See Details.

trace

logical indicating if output should be produced for each iteration. Default is FALSE.

type

the type of fitting method to be used. The options are "AS_mean" (mean-bias reducing adjusted scores), "AS_median" (median-bias reducing adjusted scores), "AS_mixed" (bias reduction using mixed score adjustments; default), "correction" (asymptotic bias correction), "MPL_Jeffreys" (maximum penalized likelihood with powers of the Jeffreys prior as penalty) and "ML" (maximum likelihood).

transformation

the transformation of the dispersion to be estimated. Default is "identity". See Details.

slowit

a positive real used as a multiplier for the stepsize. The smaller it is the smaller the steps are. Default is 1.

response_adjustment

a (small) positive constant or a vector of such. Default is NULL. See Details.

max_step_factor

the maximum number of step halving steps to consider. Default is 12.

a

power of the Jeffreys prior penalty. See Details.

...

further arguments passed to brglmControl(). Currently ignored in the output.

Details

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

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

When check_aliasing = TRUE (default), a QR decomposition of the model matrix is computed to check for aliasing. If the model matrix is known to be of full rank, then check_aliasing = FALSE avoids the extra computational overhead of an additional QR decomposition, which can be substantial for large model matrices. However, setting check_aliasing = FALSE tells brglmFit() that the model matrix is full rank, and hard to trace back errors will result if it is rank deficient.

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 response_adjustment 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, 2020, 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 \beta and the dispersion parameter \phi. 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 (2021) 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 ⁠[aut, cre]⁠ [e-mail]

References

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

Kosmidis I, Kenne Pagui E C, Sartori N (2020). Mean and median bias reduction in generalized linear models. Statistics and Computing, 30, 43-59. doi:10.1007/s11222-019-09860-6.

Firth D (1993). Bias reduction of maximum likelihood estimates. Biometrika, 80, 27-38. doi:10.2307/2336755.

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")