  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.
