# poLCA.entropy: Entropy of a fitted latent class model In poLCA: Polytomous variable Latent Class Analysis

## Description

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

## Usage

 `1` ``` poLCA.entropy(lc) ```

## 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).

`poLCA`

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18``` ```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) ```

### Example output

```Loading required package: scatterplot3d
Conditional item response (column) probabilities,
by outcome variable, for each class (row)

\$A
Pr(1)  Pr(2)
class 1:  0.0000 1.0000
class 2:  0.8835 0.1165

\$B
Pr(1)  Pr(2)
class 1:  0.0169 0.9831
class 2:  0.6456 0.3544

\$C
Pr(1)  Pr(2)
class 1:  0.2391 0.7609
class 2:  1.0000 0.0000

\$D
Pr(1)  Pr(2)
class 1:  0.4589 0.5411
class 2:  1.0000 0.0000

\$E
Pr(1)  Pr(2)
class 1:  0.0214 0.9786
class 2:  0.7771 0.2229

\$F
Pr(1)  Pr(2)
class 1:  0.5773 0.4227
class 2:  1.0000 0.0000

\$G
Pr(1)  Pr(2)
class 1:  0.0000 1.0000
class 2:  0.8835 0.1165

Estimated class population shares
0.5012 0.4988

Predicted class memberships (by modal posterior prob.)
0.5 0.5

=========================================================
Fit for 2 latent classes:
=========================================================
number of observations: 118
number of estimated parameters: 15
residual degrees of freedom: 103
maximum log-likelihood: -317.2568

AIC(2): 664.5137
BIC(2): 706.0739
G^2(2): 62.36543 (Likelihood ratio/deviance statistic)
X^2(2): 92.64814 (Chi-square goodness of fit)

Conditional item response (column) probabilities,
by outcome variable, for each class (row)

\$A
Pr(1)  Pr(2)
class 1:  0.9427 0.0573
class 2:  0.0000 1.0000
class 3:  0.4872 0.5128

\$B
Pr(1)  Pr(2)
class 1:  0.8621 0.1379
class 2:  0.0191 0.9809
class 3:  0.0000 1.0000

\$C
Pr(1)  Pr(2)
class 1:  1.0000 0.0000
class 2:  0.1425 0.8575
class 3:  1.0000 0.0000

\$D
Pr(1)  Pr(2)
class 1:  1.0000 0.0000
class 2:  0.4138 0.5862
class 3:  0.9424 0.0576

\$E
Pr(1)  Pr(2)
class 1:  0.9449 0.0551
class 2:  0.0000 1.0000
class 3:  0.2494 0.7506

\$F
Pr(1)  Pr(2)
class 1:  1.0000 0.0000
class 2:  0.5236 0.4764
class 3:  1.0000 0.0000

\$G
Pr(1)  Pr(2)
class 1:  1.0000 0.0000
class 2:  0.0000 1.0000
class 3:  0.3693 0.6307

Estimated class population shares
0.3736 0.4447 0.1817

Predicted class memberships (by modal posterior prob.)
0.3729 0.4322 0.1949

=========================================================
Fit for 3 latent classes:
=========================================================
number of observations: 118
number of estimated parameters: 23
residual degrees of freedom: 95
maximum log-likelihood: -293.705

AIC(3): 633.41
BIC(3): 697.1357
G^2(3): 15.26171 (Likelihood ratio/deviance statistic)
X^2(3): 20.50336 (Chi-square goodness of fit)

Model 1: llik = -292.493 ... best llik = -292.493
Model 2: llik = -291.9084 ... best llik = -291.9084
Model 3: llik = -289.7889 ... best llik = -289.7889
Model 4: llik = -289.7889 ... best llik = -289.7889
Model 5: llik = -289.2858 ... best llik = -289.2858
Model 6: llik = -289.7889 ... best llik = -289.2858
Model 7: llik = -289.7889 ... best llik = -289.2858
Model 8: llik = -289.2858 ... best llik = -289.2858
Model 9: llik = -289.7889 ... best llik = -289.2858
Model 10: llik = -289.2858 ... best llik = -289.2858
Conditional item response (column) probabilities,
by outcome variable, for each class (row)

\$A
Pr(1)  Pr(2)
class 1:  0.0000 1.0000
class 2:  0.0000 1.0000
class 3:  0.9422 0.0578
class 4:  0.4634 0.5366

\$B
Pr(1)  Pr(2)
class 1:  0.0905 0.9095
class 2:  0.0000 1.0000
class 3:  0.8584 0.1416
class 4:  0.0000 1.0000

\$C
Pr(1)  Pr(2)
class 1:  0.0186 0.9814
class 2:  0.1561 0.8439
class 3:  1.0000 0.0000
class 4:  1.0000 0.0000

\$D
Pr(1)  Pr(2)
class 1:  1.0000 0.0000
class 2:  0.2421 0.7579
class 3:  1.0000 0.0000
class 4:  0.9404 0.0596

\$E
Pr(1)  Pr(2)
class 1:  0.0000 1.0000
class 2:  0.0000 1.0000
class 3:  0.9443 0.0557
class 4:  0.2341 0.7659

\$F
Pr(1)  Pr(2)
class 1:  1.0000 0.0000
class 2:  0.3823 0.6177
class 3:  1.0000 0.0000
class 4:  1.0000 0.0000

\$G
Pr(1)  Pr(2)
class 1:  0.0000 1.0000
class 2:  0.0000 1.0000
class 3:  1.0000 0.0000
class 4:  0.3482 0.6518

Estimated class population shares
0.0936 0.343 0.3751 0.1882

Predicted class memberships (by modal posterior prob.)
0.1186 0.3136 0.3729 0.1949

=========================================================
Fit for 4 latent classes:
=========================================================
number of observations: 118
number of estimated parameters: 31
residual degrees of freedom: 87
maximum log-likelihood: -289.2858

AIC(4): 640.5717
BIC(4): 726.4629
G^2(4): 6.423452 (Likelihood ratio/deviance statistic)
X^2(4): 10.08438 (Chi-square goodness of fit)

[1] 4.85203
[1] 2.424357
[1] 2.693452
[1] 2.494442
[1] 2.458558
```

poLCA documentation built on May 29, 2017, 5:59 p.m.