Left-truncated health insurance claims data: theoretical review and empirical application

From the inventory of the health insurer AOK in 2004, we draw a sample of a quarter million people and follow each person's health claims continuously until 2013. Our aim is to estimate the effect of a stroke on the dementia onset probability for Germans born in the first half of the 20th century. People deceased before 2004 are randomly left-truncated, and especially their number is unknown. Filtrations, modelling the missing data, enable circumventing the unknown number of truncated persons by using a conditional likelihood. Dementia onset after 2013 is a fixed right-censoring event. For each observed health history, Jacod's formula yields its conditional likelihood contribution. Asymptotic normality of the estimated intensities is derived, related to a sample size definition including the number of truncated people. The standard error results from the asymptotic normality and is easily computable, despite the unknown sample size. The claims data reveal that after a stroke, with time measured in years, the intensity of dementia onset increases from 0.02 to 0.07. Using the independence of the two estimated intensities, a 95% confidence interval for their difference is [0.053, 0.057]. The effect halves when we extend the analysis to an age-inhomogeneous model, but does not change further when we additionally adjust for multi-morbidity.