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Abstract
Inference about a scalar parameter of interest is a core statistical task that has attracted immense research in statistics. The Wald statistic is a prime candidate for the task, on the grounds of the asymptotic validity of the standard normal approximation to its finite-sample distribution, simplicity and low computational cost. It is well known, though, that this normal approximation can be inadequate, especially when the sample size is small or moderate relative to the number of parameters. A novel, algebraic adjustment to the Wald statistic is proposed, delivering significant improvements in inferential performance with only small implementation and computational overhead, predominantly due to additional matrix multiplications. The Wald statistic is viewed as an estimate of a transformation of the model parameters and is appropriately adjusted, using either maximum likelihood or reduced-bias estimators, bringing its expectation asymptotically closer to zero. The location adjustment depends on the expected information, an approximation to the bias of the estimator, and the derivatives of the transformation, which are all either readily available or easily obtainable in standard software for a wealth of models. An algorithm for the implementation of the location-adjusted Wald statistics in general models is provided, as well as a bootstrap scheme for the further scale correction of the location-adjusted statistic. Ample analytical and numerical evidence is presented for the adoption of the location-adjusted statistic in prominent modelling settings, including inference about log-odds and binomial proportions, logistic regression in the presence of nuisance parameters, beta regression, and gamma regression. The location-adjusted Wald statistics are used for the construction of significance maps for the analysis of multiple sclerosis lesions from MRI data.
Highlights
Testing hypotheses and constructing confidence intervals for scalar parameters are key statistical tasks that are usually carried out relying on large-sample results about likelihood-based quantities
Another recent example is in Kosmidis and Firth (2010), who illustrate that the finite-sample bias of the maximum likelihood (ML) estimator of the precision parameter in beta regression models results in excessively narrow Wald-type confidence intervals (CIs) and anti-conservative Wald tests, and propose the use of a reduced-bias (RB) estimator to alleviate those issues
The LA Wald statistics are successfully used within the mass univariate probit regression framework in Ge et al (2014, § 4.1) for the construction of brain significance maps to visualize the strength of association of patient characteristics to the occurrence of multiple sclerosis lesions from MRI data
Summary
Testing hypotheses and constructing confidence intervals for scalar parameters are key statistical tasks that are usually carried out relying on large-sample results about likelihood-based quantities. One prominent example is the use of a moment-based estimator for the dispersion parameter in generalized linear models with unknown dispersion, as is recommended in McCullagh and Nelder (1989, § 8.3) and implemented in the summary.glm function of the stats R package (R Core Team, 2018) Another recent example is in Kosmidis and Firth (2010), who illustrate that the finite-sample bias of the ML estimator of the precision parameter in beta regression models results in excessively narrow Wald-type confidence intervals (CIs) and anti-conservative Wald tests, and propose the use of a reduced-bias (RB) estimator to alleviate those issues.
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