  The objective of disease mapping is to model data aggregated at the areal
level. In some contexts, however, (e.g. residential histories, general
practitioner catchment areas) when data is arising from a variety of sources,
not necessarily at the same spatial scale, it is possible to specify spatial
random effects, or covariate effects, at the areal level, by using a multiple
membership principle (MM) (Petrof et al. 2020, Gramatica et al. 2021). A
weighted average of conditional autoregressive (CAR) spatial random effects
embeds spatial information for a spatially-misaligned outcome and estimate
relative risk for both frameworks (areas and memberships). In this paper we
investigate the theoretical underpinnings of these application of the multiple
membership principle to the CAR prior, in particular with regard to
parameterisation, properness and identifiability. We carry out simulations
involving different numbers of memberships as compared to number of areas and
assess impact of this on estimating parameters of interest. Both analytical and
simulation study results show under which conditions parameters of interest are
identifiable, so that we can offer actionable recommendations to practitioners.
Finally, we present the results of an application of the multiple membership
model to diabetes prevalence data in South London, together with strategic
implications for public health considerations
