  Different relationships between the H$\alpha$ and Ca II chromospheric
emissions have been reported in solar-type stars. In particular, the
time-series of emissions in these two lines are clearly anti-correlated for a
few percent of the stars, contrary to what is observed on the Sun. Our
objective is to characterise these relationships in more detail using
complementary criteria, and to constrain the properties of filaments and plages
that are necessary to explain the observations. We analysed the average level
and variability of the H$\alpha$ and Ca II emission for 441 F-G-K stars, paying
particular attention to their (anti-)correlations on both short and long
timescales. We also computed synthetic H$\alpha$ and Ca II time-series for
different assumptions of plage and filament properties and compared them with
the observations. We were not able to find plage properties that, alone, are
sufficient to reproduce the observations at all timescales simultaneously, even
when allowing different H$\alpha$ and Ca II emission relationships for
different stars. We also specified the complex and surprising relationship
between the average activity levels of both indexes, in particular for
low-activity stars.We conclude that plages alone are unlikely to explain the
observed variety of relationships between Ca II and H$\alpha$ emission, and
that the presence of other phenomena like filaments may help to reconcile the
models with observations.
