Left-Truncated Health Insurance Claims Data: Theoretical Review and Empirical Application

From a quarter million people sampled 2004 from AOK health claims and followed until 2013, we investigate the effect of having a stroke on the probability of dementia onset for Germans born in the first half of the 20$^{th}$ century. People deceased before 2004 are randomly left-truncated. Filtrations, modelling the missing data, enable to circumvent the unknown number of truncated persons by using a conditional instead of the full likelihood. Dementia onset after 2013 is a conditionally fixed right-censoring event. For each observed health history, including eventually the two diseases and inevitably death, Jacod's formula yields the conditional likelihood contribution. Asymptotic normality of the estimated intensities is derived, relative to a sample size definition that includes the truncated people. Yet, the standard error is observable. 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 $[0.048,0.051]$. The width halves, when we extend the analysis to an age-inhomogeneous model.