Longitudinal and Time-Series Analysis

Abstract

Longitudinal analysis is concerned with studying the progression of the values of a variable over time for the members of a population. If time is defined as a categorical variable, longitudinal analysis is closely related to multivariate analysis, studying vectors of outcomes. When time is a continuous variable, longitudinal analysis studies the subjects' curves (trajectories), and random coefficient models are well suited for this purpose. We can associate each time point with a separate variable, in the spirit of the original definition of the term variable. Then longitudinal analysis is the study of collections of variables; in most applications the variables are strongly associated. Features of this association are frequently the targets of inference. 11.1 Introduction In longitudinal studies we work with populations and variables that are well defined at specified time points. For example, all those obliged to make annual declarations of income to the tax authority of a country are a population and the income of each of them is well defined by the rules of the authority in every year. Figure 11.1 summarises the income of a small random sample of taxpayers aged 25 in 1995 (year 0) in a country over the period of seven years, 1996-2002, coded as 1-7. The left-hand panel plots the values of income on the original scale and the right-hand panel on the logarithmic scale. The seven values of a subject are connected by straight lines. In the diagram, we can identify and describe certain features, such as the extent to which the lines criss-cross, how much they fan out over the seven years, and for how many subjects there are substantial changes from one year to the next. As an alternative, the values of the income may be described by a seven-variate distribution. The normal distribution would be convenient for this purpose because we are familiar with it. By applying the log-transformation to each component, the assumption of normality becomes much more palatable, so a