SEM: combining binary & ordinal variables


New Member
In my research I surveyed participants after they had used a website. I have 28 items which related 8 different concepts. Which means I have 8 latent variables with 3-6 observed variables each. All items are measured with 7-point likert scales. Additionally I have a binary variable of whether people bought something or not. My sample size is about 300.

I want to see if the latent variables together can predict whether users buy something, a binary variable. I was planning to model this by creating a latent variable with only the binary variable as indicator and setting the factor loading to 1. The other latent variables would influence this variable. (Additional to some relations among themselves as well)

Now I read that binary variables are a problem for structural equation modeling (SEM) and not sure if the combination of ordinal and binary variables is even possible. I understand normality is an issue, however 7-point likert scales aren't normally distributed either and are often used in SEM research (in psychology). Could anyone say anything about the possibility of using binary and ordinal variables in combination?

Additionally if I have doubts about the direction of an arrow in a model, based on conflicting theories and observations, are there ways to use SEM to explore such possibilities? I understand additional experiments are likely required conclude such a thing with certainty.