Updated 11-25-02

Other references for Latent class models
McCutcheon, A. (1987) Latent Class Analysis, Sage Series #64.

Simpsons Paradox

Prevalence of Insanity in the Foreign Born Population Relative to the Native Born Population: Massachusetts, 1854

	Insane	Not Insane
foreign-born	625	229375
native-born	2007	892669

odds ratio = 1.2 (1.11, 1.32)

This statistic was used to make claims that foreigners were more likely to be insane.

BUT

What about if we adjust for Status: i.e. is the person in the ``Pauper Class'' or the ``Independent Class''

	Pauper Class
	Insane	Not Insane
foreign-born	182	9090
native-born	250	12513

	Independent
	Insane	Not Insane
foreign-born	442	220285
native-born	1757	880156

Odds ratio within Pauper Class = 1.00
Odds ratio within Independent Class = 1.00

Thus if you adjust for the status or class of the person we see that, in fact, there is no relationship between being foreign and being insane than native born persons

Latent Class Models

• Search for underlying cause of correlation
• Conditioned on the underlying classes, the individuals response to the questions will be independent
  • From our example, conditioning on status, the relationship between nativity and insanity is independent
• What about if we don't or can't observe this variable which should be conditioned on...i.e. it is latent
  • Like in exploratory factor analysis we can examine a model that hypothesizes k underlying latent classes and see how well it fits the data
  • Also like in exploratory factor analysis we need at least three observed variables otherwise we can't identify the model.
  • We will be able to estimate something like ``factor loadings'', i.e. an estimate of how each variable loads on a particular class

Chalk talk introducing the notation used in Heinen to represent the different kinds of probabilities and the latent class model

HANDOUT
Example from BK. Macready and Dayton (1977) "The use of probabilistic models in the assessment of mastery". 4 questions where each one is coded as a binary outcome.

HANDOUT
Example from BK. Menezes and Bartholomew (1996) "New developments in latent structure analysis applied to social attitudes", JRSS-A, 159, 213-224. 10 questions where each one is coded as a binary outcome.

HANDOUT
Example from Heinen page 46. Felling, Peters, and Schreuder (1987) "Religion in Dutch society 85: Documentation of a national survey on religious and secular attitudes in 1985" Amsterdam: Steinmetz Archive. 5 questions about women's liberation. Each one coded as a binary outcome.