A powerful nonparametric test of the effect of dementia duration on mortality

A continuous-time multi-state history is semi-Markovian, if an intensity to migrate from one state into a second, depends on the duration in the first state. Such duration can be formalised as marker, entering the intensity process of the transition counts. We derive the intensity process, prove its predictability and the martingale property of the residual to the integrated intensity. In particular, we verify the usual conditions for the respective filtration. As a consequence, according to Nielsen and Linton (1995), a kernel estimator of the transition intensity, including the duration dependence, converges point-wise at a slow rate, compared to the fast rate of the Markovian kernel estimator, i.e. when ignoring duration dependence. By using the inequality of the two rates, we follow Gozalo (1993) and show that the (properly penalised) maximal difference of the two kernel estimators on a random grid of points is asymptotically $ \chi <^>2_1 $ chi 12-distributed. As compared to the $ \chi <^>2 $ chi 2 goodness-of-fit test on a fixed grid, with less power, the estimator of the variance needs to be shown to converge uniformly. The data example is a sample of 130,000 German women, observed over a period of 10 years, and we model the intensity to die with dementia, potentially dependent on the disease duration. As usual, the models under the null and the alternative hypothesis need to be enlarged to allow for independent right-censoring. We find a significant effect of dementia duration, nearly independent of the bandwidth. A Monte Carlo simulation confirms level and high power of the test with the penalised maximal difference, under the conditions of our application. It also reminds that hyperparameter selection is sensitive to the application.