  Likelihood-based methods of statistical inference provide a useful general
methodology that is appealing, as a straightforward asymptotic theory can be
applied for their implementation. It is important to assess the relationships
between different likelihood-based inferential procedures in terms of accuracy
and adherence to key principles of statistical inference, in particular those
relating to conditioning on relevant ancillary statistics. An analysis is given
of the stability properties of a general class of likelihood-based statistics,
including those derived from forms of adjusted profile likelihood, and
comparisons are made between inferences derived from different statistics. In
particular, we derive a set of sufficient conditions for agreement to
$O_{p}(n^{-1})$, in terms of the sample size $n$, of inferences, specifically
$p$-values, derived from different asymptotically standard normal pivots. Our
analysis includes inference problems concerning a scalar or vector interest
parameter, in the presence of a nuisance parameter.
