Repeated differentiation of the observed and expected (unconditional a
nd conditional) likelihood yokes leads to a 'dictionary' for translati
on between observed and expected likelihood calculations. As a particu
lar application a family of combinations of cumulants that are stable
in the sense of McCullagh is derived. These combinations are of an ele
mental character, and the examples of stable combinations of cumulants
presented by McCullagh are obtainable as linear combinations of those
elements.