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Latent class analysis (LCA)

Highlights
Use gsem's lclass() option to fit

  • Latent class models
  • Latent profile models
  • Path models with categorical latent variables
  • Multiple-group models with known groups

 

Categorical latent variables measured by

  • Binary items
  • Ordinal items
  • Continuous items
  • Count items
  • Categorical items
  • Fractional items
  • Survival items

 

Model-based method of classification

Goodness of fit: G2, AIC, BIC

Estimate probabilities, means, counts for items in each class

Estimate proportion of population in each class

Predict class membership

Multiple options for obtaining starting values

Robust and cluster–robust standard errors
Support for complex survey data

 

What's this about?

We believe that there are groups in our population and that individuals in these groups behave differently. But we don't have a variable that identifies the groups. The groups may be consumers with different buying preferences, adolescents with different patterns of behavior, or health status classifications. LCA lets us identify and understand these unobserved groups. It lets us know who is likely to be in a group and how that group's characteristics differ from other groups.

In latent class models, we use a latent variable that is categorical to represent the groups, and we refer to the groups as classes.

Latent class models contain two parts. One fits the probabilities of who belongs to which class. The other describes the relationship between the classes and the observed variables.

The LCA models that Stata can fit include the classic models:

  • probability of class membership
  • binary items

 

And extensions:

  • covariates determining the probability of class membership
  • items that are binary, ordinal, continous, or even any of the other types that Stata's gsem can fit
  • SEM path models that vary across latent classes

 

Let's see it work


Let's work with a classic model using an example of teen behavior (but on fictional data).

We have a set of observed variables that indicate whether adolescents have consumed alcohol (alcohol), have more than 10 unexcused absences from school (truant), have used a weapon in a fight (weapon), have engaged in vandalism (vandalism), and have stolen objects worth more than $25 (theft). We will use these items to fit a latent class model with three unobserved behavior classes. We type

. gsem (alcohol truant weapon theft vandalism <-), logit lclass(C 3)


We will not show the output of this command. If we had included predictors of the class probabilities or fit a latent profile model with continuous outcomes or fit a path model, the results would be more interesting. In this classic model, however, the reported coefficients are not very informative.

Instead, we will use the estat lcprob and estat lcmean commands to estimate statistics that we can interpret easily.

estat lcprob reports the probabilities of class membership.

. estat lcprob

Latent class marginal probabilities Number of obs = 10,000

                   Delta-method
       Margin      Std. Err.             [95% Conf. Interval]
C
1  .1631459   .0390465             .1001516   .2545543
2  .7979467   .0389126             .7110459   .8637217
3  .0389074   .016552               .0167174   .087918

These are the expected proportions of the population in each class.

estat lcmean reports the estimated mean for each item in each class.

. estat lcmean

Latent class marginal means Number of obs = 10,000

                               Delta-method
                        Margin   Std. Err.                [95% Conf. Interval]
1
   alcohol      .7453054   .055844               .6217856   .8389348
   truant        .3461541   .0511504              .2537076   .4518892
   weapon     .0928717  .0273732               .0513735     .162161
   theft           .0207514  .0341546              .0007855   .3635664
   vandalism  .2407638  .0519997              .1536777   .3564169
2
   alcohol      .3120356   .0150695              .2832886   .3423065
   truant        .0626883   .0076641              .0492432   .0794975
   weapon    .0089407   .0023358               .0053525   .0148983
   theft          .0123995   .002113                .0088731   .0173028
   vandalism .0471581   .005303                .0377877   .0587103
3
   alcohol      .7227077   .0346378              .6500293  .7852786
   truant        .4910226   .0426645              .4084191  .5741192
   weapon     .2985073   .0498659              .2106263  .4042766
   theft           .6199426  .18702                  .2560826  .8854454
   vandalism .5883386   .0735655              .4407238  .7216031

The results are the probabilities of alcohol, truant, etc., for each class. Our items are binary events. Had alcohol been the amount of alcohol consumed per day, estat lcmean would have reported average alcohol consumption for each class.

Let's summarize the results from estat lcprob and estat lcmean.

                 Class 1 Class 2 Class 3
Pr(Class)       0.16      0.80      0.04
Probability of
alcohol           0.75      0.31      0.72
truant             0.35      0.06      0.49
weapon          0.09      0.01      0.30
theft               0.02       0.01      0.62
vandalism      0.24       0.05      0.59

The table reveals that;

1) 16%, 80%, and 4% percent of our students are predicted to be in class 1, class 2, and class 3, respectively.
2) Class 2 is best behaved judging by the probabilities of alcohol, truant, ..., and vandalism.
3) Class 1 is the next best behaved.
4) Class 3 is the worst behaved.


We can use margins and marginsplot to visually compare the probabilities of participating in these activities across classes.

 

Extensions
We fit our classic LCA model by typing

. gsem (alcohol truant weapon theft vandalism <-), logit lclass(C 3)


If we believe class membership depends on parents' income, we can include it in the model for C by typing

. gsem (alcohol truant weapon theft vandalism <-, logit)
             (C <- income), lclass(C 3)

We moved logit inside the parentheses for the five behavior items. This means it applies only to those equations. We don't need to say that the model for C is multinomial logit; that is automatic.

We are not limited to logit models for our items. If the behavior items are instead continuous, we can type

. gsem (alcohol truant weapon theft vandalism <-, gaussian), lclass(C 3)


If they are ordinal, we can type

. gsem (alcohol truant weapon theft vandalism <-, ologit), lclass(C 3)


And if the behavior items are of differing types, we can even type

. gsem (alcohol <-, gaussian)
            (truant <-, poisson)
            (weapon <-, logit)
            (theft <-, ologit)
            (vandalism <-, logit),
            lclass(C 3)


Still, this just scratches the surface of what we can do with gsem's new latent class features. For instance, gsem fits path models such as

. gsem (y1 <- y2 x1 x2)
            (y2 <- y3 x1 x3)
            (y3 <- x2 x3 x4)


and we can allow them to vary across classes,

. gsem (y1 <- y2 x1 x2)
            (y2 <- y3 x1 x3)
            (y3 <- x2 x3 x4),
            lclass(C 2)

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