Estimating SEM with Ordinal Data and Scale Usage Heterogeneity

In the social sciences, it is often useful to introduce latent variables and use structural equation modeling to quantify relations among observable and latent variables. Data gathered through surveys is frequently ordinal in nature, as Likert-scale type questions are used in most questionnaires. Although a large number of publications shows that treating ordinal data as metric can lead to wrong results, this practice is still common. Moreover, respondents treat the rating scales differently. There may be "yea"-sayers whose responses are biased in the positive direction (or the opposite). Some respondents may also use some part of the scale only. This problem can be explicitly addressed by modeling heterogeneity scale usage. This note shows how to estimate SEM with ordinal data and scale usage heterogeneity