Comparing estimation approaches for the illness-death model under left truncation and right censoring

Left-truncated data arise when lifetimes are observed only if they are larger than independent truncation times. For example, in a cross-sectional sampling, only individuals who live long enough to be present on the sampling day are observed. There are several ways to perform statistical inference under this setting. One can do the following: (i) use an unconditional approach, (ii) condition on the value of the truncation variable, or (iii) condition on all the history up to the time of truncation. The latter two approaches are equivalent when analyzing univariate survival outcomes but differ under the multi-state framework. In this paper, we consider the illness-death model and compare between the three estimation approaches in a parametric regression framework. We show that approach (ii) is more efficient than the standard approach (iii), although it requires more computational effort. Approach (i) is the most efficient approach, but it requires knowledge on the distribution of the truncation variable and hence is less robust. The methods are compared using a theoretical example and simulations and are applied to intensive care units data collected in a cross-sectional design, where the illness state corresponds to a bloodstream infection.