Structural equation models of change measurement.

We know from studies that variables like our salary or height are related to our health. These quantities can be seen directly or read off with a defined measurement, e.g. salary in euros or height in meters. However, many of the phenomena we are interested in are not directly visible, e.g. motivation to change behavior. Motivation cannot be measured in euros or in meters. 

Since there is no single tape measure for these latent variables, they must first be made "visible." This is done through measurable, observable indicators of the latent variable behind them, e.g. responses to items on a questionnaire measuring motivation. In a latent variable model, the latent variable is related to the observed indicators. Specifically, it is a multivariate regression model that describes the relationship between multiple observed dependent variables and one or more latent variables. In the model, the portion of the variance of the observed indicators that is not explained by the latent variable is the measurement error. The explicit consideration of measurement errors in latent variable models is not done in this form in usual analysis procedures and allows for 1) a better approximation of the true value of the construct being measured and 2) a measurement error-adjusted modeling of the relationships among the latent variables.

If you want to find out whether a physical activity promotion program actually motivates people to exercise more, it is helpful to be able to represent changes in physical activity over time in a mathematical model. Changes and developmental trajectories can be represented as latent variables in complex structural equation models. We use a variety of statistical methods for latent change modeling, including latent growth and change models. This allows modeling of both average change over time and intraindividual change at the latent trait level. In addition to modeling change processes without measurement error, these methods allow, among other things, 1) modeling complex trajectories of a variable that has been studied multiple times or parallel trajectories of multiple variables simultaneously, 2) examining determinants and consequences of change jointly in a single model, and 3) accounting for heterogeneity in development trajectories that cannot be attributed to observed variables. To represent this unobserved heterogeneity, we use mixture distribution models such as Growth Mixture Modeling. This involves dividing the population under study into a priori unknown subgroups of people with homogeneous trajectories of a variable over time ("trajectory types").