Assessing the Number of Components: Information Criteria Sample Clauses

Assessing the Number of Components: Information Criteria. When fitting a finite mixture model, the number of components, C, is typically as- sumed to be fixed and known. Unfortunately, in many applications, a priori informa- tion regarding the number of components, C, is not available. In latent class analysis, selecting the appropriate number of components relies on the fundamental assump- tion of local independence. The axiom of local independence for latent class models states that observed features are statistically independent within a given latent class. Thus, selecting the appropriate number of latent classes ideally leads to a model in which the underlying classes fully account for the population heterogeneity. Even in the cross-sectional context, there is currently no consensus regarding the best approach for selecting the number of components in a finite mixture model. With that said, there are two general approaches to address this issue. The first approach is based on information criteria, while the second involves hypothesis testing. Model selection based on information criteria is motivated by the Kullback-Leibler (KL) information [43]. Intuitively, the KL information is a measure of the difference between the proposed statistical model and the true distribution of the observed data. Using the notation of McLachlan and Peel [54], assume that the true density of the observed data is f (ω) and denote the estimated model being considered by f (ω; ψˆ). The KL information of f (ω) with respect to f (ω; ψˆ) is then: f (ω) ; f ω; ψ = { ( ˆ)} ∫  f (ω)  f )ˆ f (ω) log  ( dω ω; ψ = ∫ f (ω) logf (ω) dω − ∫ f (ω) logf (ω; ψˆ) dω ≥ 0. Since the first term above does not depend on the fitted model, estimation of the Kullback-Leibler information is based solely on the second term. Now, η (y; F ) = ∫ f (ω) logf (ω; ψˆ) dω = ∫ logf (ω; ψˆ) dF (ω) , n where F denotes the true cumulative distribution and y = (yT , . . . , yT )T is the ob- served data. Replacing the distribution function, F, with the empirical distribution function, Fˆn, yields the following estimate of η (y; F ): η (y; Fˆn) = ∑ 1
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