Entropy of a fitted latent class model

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Description

Calculates the entropy of a cross-classification table produced as a density estimate using a latent class model.

Usage

1

Arguments

lc

A model object estimated using the poLCA function.

Details

Entropy is a measure of dispersion (or concentration) in a probability mass function. For multivariate categorical data it is calculated

H = -∑_c p_c log(p_c)

where p_c is the share of the probability in the cth cell of the cross-classification table. A fitted latent class model produces a smoothed density estimate of the underlying distribution of cell percentages in the multi-way table of the manifest variables. This function calculates the entropy of that estimated probability mass function.

Value

A number taking a minumum value of 0 (representing complete concentration of probability on one cell) and a maximum value equal to the logarithm of the total number of cells in the fitted cross-classfication table (representing complete dispersion, or equal probability for outcomes across every cell).

See Also

poLCA

Examples

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data(carcinoma)
f <- cbind(A,B,C,D,E,F,G)~1
lca2 <- poLCA(f,carcinoma,nclass=2) # log-likelihood: -317.2568
lca3 <- poLCA(f,carcinoma,nclass=3) # log-likelihood: -293.705
lca4 <- poLCA(f,carcinoma,nclass=4,nrep=10,maxiter=5000) # log-likelihood: -289.2858 

# Maximum entropy (if all cases equally dispersed)
log(prod(sapply(lca2$probs,ncol)))

# Sample entropy ("plug-in" estimator, or MLE)
p.hat <- lca2$predcell$observed/lca2$N
H.hat <- -sum(p.hat * log(p.hat))
H.hat   # 2.42

# Entropy of fitted latent class models
poLCA.entropy(lca2)
poLCA.entropy(lca3)
poLCA.entropy(lca4)