Marginal versus conditional versus 'structural source' models: a rationale for an alternative to log-linear methods for capture-recapture estimates.

Log-linear models for capture-recapture type data are widely used for estimating sizes of populations. Log-linear methods model conditional interactions between the sources. Often, however, the marginal associations are more appropriate and easier for the practitioner to conceptualize. Analyses here of previously published data on cases of spina bifida in upstate New York are used to show how the assumption that sources are conditionally independent can give biased estimates if in fact the sources are marginally independent. A plausible model for the structural sources of interactions between the sources of information about spina bifida cases is developed which implies marginal independence of two of the sources rather than conditional independence. Estimates of the population total based on marginal independence are derived and give larger estimates of the population total than those derived based upon conditional dependence. When investigators can in fact model the likely underlying relationships of the sources in the population, we suggest considering modelling the potential interdependencies of the sources, which we term 'structural source modeling'.