Sampling in the Lexis Diagram
Jens Lund
The Royal Veterinary & Agricultural University
Denmark
e-mail: jlund@dina.kvl.dk
Abstract
The Lexis Diagram is a (time,age)-coordinate system. We
represent individuals by lines with slope 1 from the time of birth to the
(time,age) of death. A basic assumption is that, given the birthtimes,
the lifetimes of individuals are independent. Four different ways to
sample individuals will be considered: A synthetic cohort study is when
we observe the time and age of individuals that die in a time window. A
cross sectional study is when we observe the ages of individuals alive at
a specific time point. A time window is when we observe all ages of
individuals alive in a time window, and the time and age of individuals
that die in the time window. A follow up on a cross sectional study is the
same as a time window, except that we only observe individuals alive at
the beginning of the time window. It is shown that when we condition on
the time of birth, we must analyse the lifetimes in the conditional
distribution given that we observe any information about the individual.
This most often means left truncation, and right censoring or right
truncation of the lifetimes. This analysis does not depend on the
distribution of the births. If we assume, that the process of births is
Poisson, then it is shown by likelihood arguments that we can analyse the
lifetimes as independent, in the stationary recurrence time distribution
and length biased distribution known from renewal theory. The two methods
are compared.