1.3 Left truncation

The concept of left truncation, or delayed entry, is well illustrated by the data set oldmort that was discussed in detail in Example 1.1. Please note the difference compared to left censoring. Unfortunately, you may still see articles where these two concepts are confused.

It is illustrative to think of the construction of the data set oldmort as a statistical follow-up study, starting on January 1, 1860. At that day, all persons present in the parish and 60 years of age or above, are included in the study. It is decided that the study will end at 31 December 1879, that is, the study period (follow-up time) is 20 years. The interesting event in this study is death. This means that the start event is the sixtieth anniversary of birth and the final event is death. Due to the calendar time constraints (and migration), all individuals will not be observed to die (especially those who live long), and moreover, some individuals will enter the study after the “starting” event, the sixtieth anniversary. A person who enter late, say he is 65 on January 1, 1860, had not been included had he died at age 63 (say). Therefore, in the analysis, we must condition on the fact that he was alive at 65. Another way of saying this is to say that this observation is left truncated at age 65.

People being too young at the start date will be included from the day they reach 60, if that happens before the closing date, December 31, 1879. They are not left truncated, but will have a higher and higher probability of being right censored, the later they enter the study.