  One of the most important issues in the critical assessment of
spatio-temporal stochastic models for epidemics is the selection of the
transmission kernel used to represent the relationship between infectious
challenge and spatial separation of infected and susceptible hosts. As the
design of control strategies is often based on an assessment of the distance
over which transmission can realistically occur and estimation of this distance
is very sensitive to the choice of kernel function, it is important that models
used to inform control strategies can be scrutinised in the light of
observation in order to elicit possible evidence against the selected kernel
function. While a range of approaches to model criticism are in existence, the
field remains one in which the need for further research is recognised. In this
paper, building on earlier contributions by the authors, we introduce a new
approach to assessing the validity of spatial kernels - the latent likelihood
ratio tests - and compare its capacity to detect model misspecification with
that of tests based on the use of infection-link residuals. We demonstrate that
the new approach, which combines Bayesian and frequentist ideas by treating the
statistical decision maker as a complex entity, can be used to formulate tests
with greater power than infection-link residuals to detect kernel
misspecification particularly when the degree of misspecification is modest.
This new approach avoids the use of a fully Bayesian approach which may
introduce undesirable complications related to computational complexity and
prior sensitivity.
