Sampling bias in population studies - how to use the Lexis diagram (Q2711677)

Let \(\sigma_i\), \(i\in I\subseteq Z\), be birth-times of some individuals and \(X_i\) are their life-times. Let in \(R^2\) the horizontal coordinate be calendar time and the vertical one the age of the individual, so the individual's life is represented by a line from \((\sigma_i,0)\) to \((\sigma_i+X_i,X_i)\). The collection of all such life lines is called the Lexis diagram. The Lexis process is defined as the point process \((\sigma_i,X_i)_{i\in I}\).NEWLINENEWLINENEWLINEIn this paper connections of some sampling patterns (cohort studies, time window, cross sectional studies, synthetic cohort and follow-up on cross-sectional studies) and areas in the Lexis diagram are described. Using Poisson properties of the Lexis process the author derives likelihood functions which correspond to these sampling patterns. Generalizations to the case of different states of individuals (e.g., healthy - invalid - dead) are considered.