This chapter will discuss how Lexis surfaces can help support thinking about, through and beyond questions of identifying APC effects in population data. I argue for the routine use of Lexis surfaces when working with population data as part of a broader research workflow. Lexis surfaces of observed data allow for geometric features in the data to be noticed, which can help the researcher either in choosing between existing statistical model specifications, or in developing new model specifications. Lexis surfaces can then be used later in the research workflow to show model-predicted surfaces, allowing researchers to identify whether the proposed model structure is an appropriate representation of key data features; this use later in the research workflow is complemented by the additional use of Lexis surfaces to show the residuals between model-predicted and observed data, allowing structural misspecification of the data to be more readily recognised.

The uses of Lexis surfaces, as applied to both observed data and model predictions and residuals, is illustrated by an extended case study. This case study shows how male death rates attributable to suicide, alcohol and drug-related deaths varied by age and year in Scotland in the most deprived fifth of areas since 1975. Lexis surfaces of these three causes of death indicate that they fit into one of two broad geometric features: the ‘truncated triangle’, for drug-related deaths and suicides; and the ‘band and ellipse’, for alcohol-related deaths. Neither of these geometric forms conform in a straightforward way to linear and distinct APC effects. These two geometric forms are operationalised into two novel statistical model structures, which are fitted to all three datasets and compared against a range of more generic model specifications. The two new model structures, which would not have been developed without first looking at the data as a Lexis surface, outperform the majority of these more generic models using standard penalised model fit metrics when applied to the datasets they were designed for, demonstrating they correctly identify key features in the underlying population structure of the phenomena. The structural features of the two model specifications are substantively meaningful and help highlight similarities and differences in the likely aetiology of the three causes of death.