There is a description in "Latent Class Analysis" by McCutcheon AL (1987, p.7): "Lazarsfeld coined the term latent structure analysis to describe the use of mathematical models for characterizing latent variables in the analysis of attitudinal measures from survey research (Henry 1983). Lazarsfeld included factor analysis as the latent structure method for characterizing continuous latent variables (factors) based on continuous observed variables.

Latent class analysis, then, can be considered a qualitative data analog to factor analysis which enables researchers to empirically identify discrete latent variables from two or more discrete observed variables (Green 1951, 1952). Two additional techniques complete the array of latent structure methods: latent trait analysis enables the characterization of continuous latent variables from discrete observed variables, and latent profile analysis enables the characterization of discrete latent variables from continuous observed variables." Also in the book, latent class models are thought in the sense of probability instead of covariance (please let me know if I am wrong). I also don't know if latent trait analysis and latent profile analysis uses which approach (probability or covariance) or the two approaches are just equivalent mathematically (like the GLM for linear and logistic regression?). Thanks beforehand for the teaching. Simon.