Survival analysis of left truncated income protection insurance data

One of the main characteristics of Income Protection Insurance (IPI) claim duration data, which has not been considered in the actuarial literature on the topic, is left-truncation. Claimants that are observed are those whose sickness durations are longer than the deferred periods specified in the policies, and hence left-truncation exists in these data. This paper investigates a series of conditional mixture models when applying survival analysis to model sickness durations of IPI claimants, and examines the consequence of treating the IPI data with lengthy deferred periods as complete data and therefore ignoring the left-truncation by fitting the corresponding unconditional distributions. It also quantifies the extent of the bias in the resulting parameter estimates when ignoring the left-truncation in the data. Using the UK Continuous Mortality Investigation (CMI) sickness duration data, some well-fitting survival model results are estimated. It is demonstrated that ignoring the left-truncation in certain IPI data can lead to substantially different statistical estimates. We therefore suggest taking left-truncation into account by fitting conditional mixture distributions to IPI data. Furthermore, the best fitting model is extended by introducing a number of covariates into the conditional part to do regression analysis.