  A major challenge in turbulence research is to understand from first
principles the origin of anomalous scaling of the velocity fluctuations in
high-Reynolds-number turbulent flows. One important idea was proposed by
Kolmogorov [J. Fluid Mech. {\bf 13}, 82 (1962)], which attributes the anomaly
to the variations of the locally averaged energy dissipation rate. Kraichnan
later pointed out [J. Fluid Mech. {\bf 62}, 305 (1973)] that the locally
averaged energy dissipation rate is not an inertial-range quantity and a proper
inertial-range quantity would be the local energy transfer rate. As a result,
Kraichnan's idea attributes the anomaly to the variations of the local energy
transfer rate. These ideas, generally known as refined similarity hypotheses,
can also be extended to study the anomalous scaling of fluctuations of an
active scalar, like the temperature in turbulent convection. In this paper, we
examine the validity of these refined similarity hypotheses and their
extensions to an active scalar in shell models of turbulence. We find that
Kraichnan's refined similarity hypothesis and its extension are valid.
