  Including covariates in loglinear models of population registers improves
population size estimates for two reasons. First, it is possible to take
heterogeneity of inclusion probabilities over the levels of a covariate into
account; and second, it allows subdivision of the estimated population by the
levels of the covariates, giving insight into characteristics of individuals
that are not included in any of the registers. The issue of whether or not
marginalizing the full table of registers by covariates over one or more
covariates leaves the estimated population size estimate invariant is
intimately related to collapsibility of contingency tables [Biometrika 70
(1983) 567-578]. We show that, with information from two registers, population
size invariance is equivalent to the simultaneous collapsibility of each margin
consisting of one register and the covariates. We give a short path
characterization of the loglinear model which describes when marginalizing over
a covariate leads to different population size estimates. Covariates that are
collapsible are called passive, to distinguish them from covariates that are
not collapsible and are termed active. We make the case that it can be useful
to include passive covariates within the estimation model, because they allow a
finer description of the population in terms of these covariates. As an example
we discuss the estimation of the population size of people born in the Middle
East but residing in the Netherlands.
