ABSTRACT
Exploratory latent class analysis (ELCA) (Clogg, 1981; Goodman, 1974; Haberman, 1988; Vermunt, 1997) is used to group responses xij of persons i = 1,…, N to items j = 1,…,J into classes q = 1,…,Q such that persons with similar responses are assigned to the same class. In this chapter we restrict ourselves to dichotomous data xijє{0,1}. Each class q is characterized by J class specific probabilities π qj , indicating the probability of the response ‘1’ on item j in class and a weight ωq , indicating the unconditional probability that a person’s latent class membership τ equals q. Let X = [x 1,…, x N ], θ = [ω, π 1,…,π Q ] , π q = [π q1,…,π qJ ], x i = [x i1,…,xij ] and ω = [ω 1 ,…, ωQ ] . The density of the data given the parameters of ELCA is then given by https://www.w3.org/1998/Math/MathML"> P ( X| θ ) = ∏ i = 1 N P ( x i | θ ) = ∏ i = 1 N [ ∑ q = 1 Q P ( x i , τ = q | θ ) ] = ∏ i = 1 N [ ∑ q = 1 Q ω q ∏ j = 1 J π q j x i j ( 1 − π q j ) ( 1 − x i j ) ] . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781410612021/8fa18355-d1a3-4748-9061-9351c0b528f0/content/math_26_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
