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 is 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 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 (2020) 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.