A shared parameter mixture model for longitudinal income data with missing responses and zero rounding

Abstract

The analysis of longitudinal income data is often made challenging for several reasons. For example, in a national Australian survey on income over time, a non-negligible proportion of responses are missing, and it is believed the missingness mechanism is non-ignorable. Also, there are a large number of reported zero incomes, some of which may be true zeros (corresponding to individuals who legitimately do not earn an income), while some may be false zeros (corresponding to individuals choosing to round their income to zero). We propose a new shared parameter mixture (SPM) model for analysing semi-continuous longitudinal income data, which addresses the two challenges of income non-response and zero rounding. This is accomplished by jointly modelling an individual's underlying income together with the probability of missingness and rounding to zero, where both probabilities are permitted to vary in a smooth manner with their underlying non-zero income. Applying the SPM model to the Australian income survey reveals that on average, older female individuals and individuals with a long-term health condition are considerably less likely to earn an income, while income tended to be highest for male individuals on fixed-term/permanent job contracts between ages 50 and 60. Furthermore there is evidence of both zero rounding, and conditional on the assumed missingness mechanism, individuals with incomes at the higher and lower ends are more likely to not report their income.