  The conditionality principle $C$ plays a key role in attempts to characterize
the concept of statistical evidence. The standard version of $C$ considers a
model and a derived conditional model, formed by conditioning on an ancillary
statistic for the model, together with the data, to be equivalent with respect
to their statistical evidence content. This equivalence is considered to hold
for any ancillary statistic for the model but creates two problems. First,
there can be more than one maximal ancillary in a given context and this leads
to $C$ not being an equivalence relation and, as such, calls into question
whether $C$ is a proper characterization of statistical evidence. Second, a
statistic $A$ can change from ancillary to informative (in its marginal
distribution) when another ancillary $B$ changes, from having one known
distribution $P_{B},$ to having another known distribution $Q_{B}.$ This means
that the stability of ancillarity differs across ancillary statistics and
raises the issue of when a statistic can be said to be truly ancillary. It is
therefore natural, and practically important, to limit conditioning to the set
of ancillaries whose distribution is irrelevant to the ancillary status of any
other ancillary statistic. This results in a family of ancillaries for which
there is a unique maximal member. This also gives a new principle for
inference, the stable conditionality principle, that satisfies the criteria
required for any principle whose aim is to characterize statistical evidence.
