Latent Class Analysis vs. Cluster Analysis - differences in inferences? What are the differences in inferences that can be made from a latent class analysis (LCA) versus a cluster analysis?  Is it correct that a LCA assumes an underlying latent variable that gives rise to the classes, whereas the cluster analysis is an empirical description of correlated attributes from a clustering algorithm?  It seems that in the social sciences, the LCA has gained popularity and is considered methodologically superior given that it has a formal chi-square significance test, which the cluster analysis does not.  
It would be great if examples could be offered in the form of, "LCA would be appropriate for this (but not cluster analysis), and cluster analysis would be appropriate for this (but not latent class analysis).
Thanks!
Brian
 A: The difference is Latent Class Analysis would use hidden data (which is usually patterns of association in the features) to determine probabilities for features in the class. Then inferences can be made using  maximum likelihood to separate items into classes based on their features.
Cluster analysis plots the features and uses algorithms such as nearest neighbors, density, or hierarchy to determine which classes an item belongs to.
Basically LCA inference can be thought of as "what is the most similar patterns using probability" and Cluster analysis would be "what is the closest thing using distance".
A: A latent class model (or latent profile, or more generally, a finite mixture model) can be thought of as a probablistic model for clustering (or unsupervised classification). The goal is generally the same - to identify homogenous groups within a larger population. I think the main differences between latent class models and algorithmic approaches to clustering are that the former obviously lends itself to more theoretical speculation about the nature of the clustering; and because the latent class model is probablistic, it gives additional alternatives for assessing model fit via likelihood statistics, and better captures/retains uncertainty in the classification. 
You might find some useful tidbits in this thread, as well as this answer on a related post by chl. 
There are also parallels (on a conceptual level) with this question about PCA vs factor analysis, and this one too. 
A: Latent Class Analysis is in fact an Finite Mixture Model (see here). The main difference between FMM and other clustering algorithms is that FMM's offer you a "model-based clustering" approach that derives clusters using a probabilistic model that describes distribution of your data. So instead of finding clusters with some arbitrary chosen distance measure, you use a model that describes distribution of your data and based on this model you assess probabilities that certain cases are members of certain latent classes. So you could say that it is a top-down approach (you start with describing distribution of your data) while other clustering algorithms are rather bottom-up approaches (you find similarities between cases).
Because you use a statistical model for your data model selection and assessing goodness of fit are possible - contrary to clustering. Also, if you assume that there is some process or "latent structure" that underlies structure of your data then FMM's seem to be a appropriate choice since they enable you to model the latent structure behind your data (rather then just looking for similarities).
Other difference is that FMM's are more flexible than clustering. Clustering algorithms just do clustering, while there are FMM- and LCA-based models that


*

*enable you to do confirmatory, between-groups analysis,

*combine Item Response Theory (and other) models with LCA,

*include covariates to predict individuals' latent class membership,

*and/or even within-cluster regression models in latent-class regression,

*enable you to model changes over time in structure of your data etc.


For more examples see:

Hagenaars J.A. & McCutcheon, A.L. (2009). Applied Latent Class
  Analysis. Cambridge University Press.

and the documentation of flexmix and poLCA packages in R, including the following papers:

Linzer, D. A., & Lewis, J. B. (2011). poLCA: An R package for
  polytomous variable latent class analysis. Journal of Statistical
  Software, 42(10), 1-29.
Leisch, F. (2004). Flexmix: A general framework for finite mixture
  models and latent glass regression in R. Journal of Statistical
  Software, 11(8), 1-18.
Grün, B., & Leisch, F. (2008). FlexMix version 2: finite mixtures with
  concomitant variables and varying and constant parameters. Journal of
  Statistical Software, 28(4), 1-35.

