Latent Variable Models:
Latent variable models are a broad subclass of latent structure models . They postulate some relationship between the statistical properties of observable variables (or “manifest variables”, or “indicators”) and latent variables. A special kind of statistical analysis corresponds to each kind of the latent variable models.
According to Bartholomew and Knott , the latent variable models (and corresponding areas of statistical analysis) can be categorized according to the types of the manifest and latent variables:
|Latent profile analysis
|Latent trait analysis
|Latent class analysis
A central assumption in these models is the local independence postulate.
In latent variable models the distribution of continuous variables is often assumed to be normal, distribution of categorical variables – binomial or multinomial.
Latent variable models are covered in statistics.com´s online course Introduction to Structural Equation Modeling.
 Bartholomew, D.J., and Knott, M. (1999). Latent Variable Models and Factor Analysis. London: Arnold.