  For finite samples with binary outcomes penalized logistic regression such as
ridge logistic regression (RR) has the potential of achieving smaller mean
squared errors (MSE) of coefficients and predictions than maximum likelihood
estimation. There is evidence, however, that RR is sensitive to small or sparse
data situations, yielding poor performance in individual datasets. In this
paper, we elaborate this issue further by performing a comprehensive simulation
study, investigating the performance of RR in comparison to Firth's correction
that has been shown to perform well in low-dimensional settings. Performance of
RR strongly depends on the choice of complexity parameter that is usually tuned
by minimizing some measure of the out-of-sample prediction error or information
criterion. Alternatively, it may be determined according to prior assumptions
about true effects. As shown in our simulation and illustrated by a data
example, values optimized in small or sparse datasets are negatively correlated
with optimal values and suffer from substantial variability which translates
into large MSE of coefficients and large variability of calibration slopes. In
contrast, if the degree of shrinkage is pre-specified, accurate coefficients
and predictions can be obtained even in non-ideal settings such as encountered
in the context of rare outcomes or sparse predictors.
