logbin fits a generalised linear model (GLM) with a
binomial error distribution and log link
function. Predictors are assumed to be continuous, unless
they are of class factor, or are character or
logical (in which case they are converted to
factors). Specifying a predictor as monotonic using
the mono argument means that for continuous terms,
the associated coefficient will be restricted to be
non-negative, and for categorical terms, the coefficients
will be non-decreasing in the order of the factor
levels. This allows semi-parametric monotonic regression
functions, in the form of unsmoothed step-functions. For
smooth regression functions see logbin.smooth.
As well as allowing monotonicity constraints, the function
is useful when a standard GLM routine, such as
glm, fails to converge with a log-link
binomial model. For convenience in comparing convergence on
the same model, logbin can be used
as a wrapper function to glm and glm2
through the method argument.
If glm does achieve successful convergence,
and logbin converges to an interior point, then the two
results will be identical. However, as illustrated in one of
the examples below, glm may still experience convergence
problems even when logbin converges to an interior point.
Note that if logbin converges to a boundary point, then it
may differ slightly from glm even if glm successfully
converges, because of differences in the definition of the parameter
space. logbin produces valid fitted values for covariate
values within the Cartesian product of the observed range of covariate
values, whereas glm produces valid fitted values just
for the observed covariate combinations (assuming it successfully
converges). This issue is only relevant when logbin
converges to a boundary point. The adaptive barrier approach defines
the parameter space in the same way as glm, so the
same comments apply when comparing its results to those from
method = "cem" or "em".
The main computational method is an EM-type algorithm which accommodates
the parameter contraints in the model and is more stable than iteratively
reweighted least squares. This is done in one of two ways,
depending on the choice of the method argument.
method = "cem" implements a CEM algorithm (Marschner, 2014),
in which a collection of restricted parameter spaces is defined
that covers the full parameter space, and an EM algorithm is applied within each
restricted parameter space in order to find a collection of
restricted maxima of the log-likelihood function, from
which can be obtained the global maximum over the full
parameter space. See Marschner and Gillett (2012) for further
details.
method = "em" implements a single EM algorithm
on an overparameterised model, and the MLE of this model
is transformed back to the original parameter space.
Acceleration of the EM algorithm in either case can be
achieved through the methods of the turboem
package, specified through the accelerate argument. However,
note that these methods do not have the guaranteed convergence of
the standard EM algorithm, particularly when the MLE is on the
boundary of its (possibly constrained) parameter space.
Alternatively, an adaptive barrier method can be used by specifying
method = "ab", which maximises the likelihood subject to
constraints on the fitted values.