Abstract
SummaryFor the estimation of cumulative link models for ordinal data, the bias reducing adjusted score equations of Firth in 1993 are obtained, whose solution ensures an estimator with smaller asymptotic bias than the maximum likelihood estimator. Their form suggests a parameter-dependent adjustment of the multinomial counts, which in turn suggests the solution of the adjusted score equations through iterated maximum likelihood fits on adjusted counts, greatly facilitating implementation. Like the maximum likelihood estimator, the reduced bias estimator is found to respect the invariance properties that make cumulative link models a good choice for the analysis of categorical data. Its additional finiteness and optimal frequentist properties, along with the adequate behaviour of related asymptotic inferential procedures, make the reduced bias estimator attractive as a default choice for practical applications. Furthermore, the estimator proposed enjoys certain shrinkage properties that are defensible from an experimental point of view relating to the nature of ordinal data.
Highlights
In many models with categorical responses the maximum likelihood estimates can be on the boundary of the parameter space with positive probability
While there is no ambiguity in reporting an estimate on the boundary of the parameter space, as is for example an infinite estimate for the parameters of a logistic regression model, estimates on the boundary can (i) cause numerical instabilities to fitting procedures, (ii) lead to misleading output when estimation is based on iterative procedures with a stopping criterion, and more importantly, (iii) cause havoc to asymptotic inferential procedures, and especially to the ones that depend on estimates of the standard error of the estimators
The above expression agrees with the results in Kosmidis and Firth (2009, §4.3), where it is shown that for generalized linear models reduction of bias via adjusted score functions is equivalent to replacing the actual count yr with the parameter-dependent adjusted count yr + grhr/(2wr) (r = 1, . . . , n)
Summary
In many models with categorical responses the maximum likelihood estimates can be on the boundary of the parameter space with positive probability. Several simulation studies on well-used models for discrete responses have demonstrated that bias reduction via the adjustment of the log-likelihood derivatives (Firth, 1993) offers a solution to the problems relating to boundary estimates; see, for example, Mehrabi and Matthews (1995) for the estimation of simple complementary log-log models, Heinze and Schemper (2002) and Bull et al (2002); Kosmidis and Firth (2011) for binomial and multinomial logistic regression, respectively, and Kosmidis (2009) for binomial-response generalized linear models. The exposition of the methodology is accompanied by a parallel discussion of the corresponding implications in the application of the models through examples with artificial and real data
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