Tuesday, January 18, 2011

Example 8.21: latent class analysis

Latent class analysis is a technique used to classify observations based on patterns of categorical responses. Collins and Lanza's book,"Latent Class and Latent Transition Analysis," provides a readable introduction, while the UCLA ATS center has an online statistical computing seminar on the topic.

We consider an example analysis from the HELP dataset, where we wish to classify subjects based on their observed (manifest) status on the following variables: 1) on street or in shelter in past 180 days [homeless], 2) CESD score above 20, 3) received substance abuse treatment [satreat], or 4) linked to primary care [linkstatus]. We arbitrarily specify a three class solution.

Support for this method in SAS is available through the proc lca and proc lta add-on routines created and distributed by the Methodology Center at Penn State University. While it's customary in R to use researcher-written routines, it's less so for SAS; the machinery which allows independently written procs thus has the potential to mislead users. It bears explicitly stating that third-party procs probably don't have the same level of robustness or support as those distributed by SAS Institute.

The proc lca code assumes that the data exist in the dataset ds. The current coding of 0's and 1's needs to be changed to 1's and 2's.

data ds_0; set "c:\book\help.sas7bdat"; run;

data ds; set ds_0;
homeless = homeless+1;
cesdcut = (cesd > 20) + 1;
satreat = satreat+1;
linkstatus = linkstatus+1;

The call to the LCA procedure specifies the number of classes, the variables to include, the number of categories per variable, and information about the starting values and random starts. It's highly recommended to run a "large" number of random starts to ensure that the true maximum likelihood estimate is reached (the 20 we used is likely too few for more complex models).

proc lca data=ds;
title '3 class model';
nclass 3;
items homeless cesdcut satreat linkstatus;
categories 2 2 2 2;
seed 42;
nstarts 20;

The output begins with diagnostic information, and indicates that 40% of the seeds were associated with the best fitting model.

Data Summary, Model Information, and Fit Statistics (EM

Number of subjects in dataset: 431
Number of subjects in analysis: 431

Number of measurement items: 4
Response categories per item: 2 2 2 2
Number of groups in the data: 1
Number of latent classes: 3
Rho starting values were randomly generated (seed = 42).

No parameter restrictions were specified (freely estimated).

Seed selected for best fitted model: 1486228051
Percentage of seeds associated with best fitted model: 40.00%

The model converged in 3241 iterations.

Maximum number of iterations: 5000
Convergence method: maximum absolute deviation (MAD)
Convergence criterion: 0.000001000

A number of fit statistics are provided to help with model comparison (e.g. number of classes, constraints in more complex models).

Fit statistics:
Log-likelihood: -1032.48
G-squared: 1.22
AIC: 29.22
BIC: 86.15
CAIC: 100.15
Adjusted BIC: 41.72
Entropy R-sqd.: 0.94
Degrees of freedom: 1

The results indicate that 22% of subjects are in class 1, just 8% in class 2, and 70% in class 3.

Parameter Estimates
Gamma estimates (class membership probabilities):
Class: 1 2 3
0.2163 0.0785 0.7052

The next set of output describes the classes. The prevalence for each level of each variable is described for each class. The last response category is redundant (equal to 1 minus the sum of the other probabilities).

Rho estimates (item response probabilities):
Response category 1:
Class: 1 2 3
homeless : 0.2703 1.0000 0.5625
cesdcut : 0.1154 0.4214 0.1678
satreat : 0.0004 0.0000 1.0000
linkstatus : 0.6029 1.0000 0.5855

Response category 2:
Class: 1 2 3
homeless : 0.7297 0.0000 0.4375
cesdcut : 0.8846 0.5786 0.8322
satreat : 0.9996 1.0000 0.0000
linkstatus : 0.3971 0.0000 0.4145

Members of class 1 were primarily homeless subjects with a larger proportion of high scores on the CESD, with substance abuse treatment history, and 40% of whom linked to primary care. Class 2 (the smallest group) was comprised of non-homeless subjects with lower CESD scores, substance abuse treatment, but no linkage. Class 3 was 44% homeless, had high levels of CESD, did not report substance abuse treatment, and 41% linked to primary care.


We begin by reading in the data, Then we use the within() function (section 1.3.1) to generate a dataframe with the variables of interest.

ds = read.csv("http://www.math.smith.edu/r/data/help.csv")
ds = within(ds, (cesdcut = ifelse(cesd>20, 1, 0)))

The poLCA package supports estimation of latent class models in R. The poLCA() function, like proc lca, can incorporate polytomous categorical variables, but also like proc lca requires the variables to be coded starting with positive integers. We specify 10 repetitions (with random starting values).

res2 = poLCA(cbind(homeless=homeless+1,
cesdcut=cesdcut+1, satreat=satreat+1,
linkstatus=linkstatus+1) ~ 1,
maxiter=50000, nclass=3,
nrep=10, data=ds)

This generates the following output:

Model 1: llik = -1032.889 ... best llik = -1032.889
Model 2: llik = -1032.889 ... best llik = -1032.889
Model 3: llik = -1032.484 ... best llik = -1032.484
Model 4: llik = -1032.889 ... best llik = -1032.484
Model 5: llik = -1032.889 ... best llik = -1032.484
Model 6: llik = -1032.484 ... best llik = -1032.484
Model 7: llik = -1032.484 ... best llik = -1032.484
Model 8: llik = -1032.889 ... best llik = -1032.484
Model 9: llik = -1032.889 ... best llik = -1032.484
Model 10: llik = -1032.889 ... best llik = -1032.484
Conditional item response (column) probabilities,
by outcome variable, for each class (row)

Pr(1) Pr(2)
class 1: 0.2703 0.7297
class 2: 1.0000 0.0000
class 3: 0.5625 0.4375

Pr(1) Pr(2)
class 1: 0.1154 0.8846
class 2: 0.4213 0.5787
class 3: 0.1678 0.8322

Pr(1) Pr(2)
class 1: 0 1
class 2: 0 1
class 3: 1 0

Pr(1) Pr(2)
class 1: 0.6029 0.3971
class 2: 1.0000 0.0000
class 3: 0.5855 0.4145

Estimated class population shares
0.2162 0.0785 0.7053

Predicted class memberships (by modal posterior prob.)
0.181 0.1137 0.7053

Fit for 3 latent classes:
number of observations: 431
number of estimated parameters: 14
residual degrees of freedom: 1
maximum log-likelihood: -1032.484

AIC(3): 2092.967
BIC(3): 2149.893
G^2(3): 1.221830 (Likelihood ratio/deviance statistic)
X^2(3): 1.233247 (Chi-square goodness of fit)

The results are consistent with those found in proc lca. We note that, also similar to proc lca the global maximum likelihood estimates were reached 3 times out of 10-- this can be discerned by examination of the 10 model results. It's always a good idea to fit a large number of iterations to ensure that the global maximum likelihood estimates have been reached.


Anonymous said...

Is there a way to save class membership for each subject in poLCA ?

Nick Horton said...

The "poLCA()" function returns an object of class "poLCA": one of the entries is called "posterior". This consists of a matrix of posterior class membership probabilities, and should do the trick as a way of merging back class membership for each subject.

Anonymous said...

How I should perform the trick? for merging back?

Anonymous said...

is there a way to save class membership using Proc LCA for each subject?

Ken Kleinman said...

According to the FAQ for Proc LCA (http://methodology.psu.edu/ra/lcalta/faq):

"Can I obtain the predicted probability of membership in each latent class/status for each individual?
Yes. Posterior probabilities of class or status membership are available by using the OUTPOST option in PROC LCA and PROC LTA. See the user's guide for details on the syntax. If the goal is to assign individuals to a latent class based on their predicted probabilities and link class membership to outcomes, note that this approach does not incorporate the uncertainty of class membership into the analysis, thus biasing inference."

Sarah Anoke said...

This is in response to April's "Anonymous" comment, and references the poLCA manual: http://userwww.service.emory.edu/~dlinzer/poLCA/poLCA-manual-1-3-1.pdf.

As Nick mentioned, the "poLCA()" function returns an object of class "poLCA", a list of 28 elements. One of these elements is "posterior", "an N by R matrix containing each observation’s posterior class membership probabilities". N is defined as the “number of cases used in the model,” and R is the number of classes specified in the call to poLCA(). It is worth noting that poLCA has a logical argument, na.rm, that specifies how to deal with missing values – if TRUE (the default), then these cases are listwise deleted before estimating the model.

Another interesting element of a “poLCA” object is “predclass,” a vector with integer values, of length N, “of predicted class memberships.”

I fully recommend reading the poLCA manual for more details; there is some latent class model theory but everything after chapter 5 should be of practical interest.

Anna said...

I have a few questions about adding complication to a baseline model:

1) Adding covariates:
a. I do not see G2 anymore in the output. The only fit stat I get it Log-likelihoo. Any way to get all the fit stats?

b. When running a model with covariates (even with the OUTPAR option) one can obtain a different solution compared to the baseline model without covariates. It seems that in this case arguing about the effects of covariates cannot be done with reference to classes identified in a baseline model, but the meaning of classes should be discussed from the model with covariate, right?

c. Other programs (e.g Latent Gold) allow to use covariates as ‘inactive’ so that they do not affect the solution, but they are only used to describe class membership probabilities. Is this possible in PROC LCA? And how can one judge absolute model fir when df is high (btw, what is the threshold after which we can say that G2 doesn’t follow a chi-square distribution?)

2) Multi-group LCA
a. I thought that adding a variable to define multiple-groups ,imposing measurement invariance, would be mathematically equivalent to adding that variable as covariate. However, from my empirical tries, I obtain different results.

b. when running a model with multiple groups and imposing measurement invariance, class membership probabilities are given for each group separately. Is it possible to get in the output the class membership probabilities for the whole sample?

Isabel said...

Hello everybody,

I am conducting an LTA analysis and, in my case, the % of seeds in the best fitting model is 48%. I have seen that your example has 40% seeds. What is the minimum % needed for the model to be identified?
Thank you in advance :)

best wishes,

Jenni Miller said...


I am conducting an LCA and am wondering if how you code your binary indicators matters? For example 1=yes and 2=no verses 1=no and 2=yes.

Nick Horton said...

Nope, the coding doesn't matter. If you make the switch this will just cause all of your parameter estimates to flip sign. (I'd encourage you to try this on a minimally reproducible example.)


Anonymous said...

I wonder how I can compare the fit statistics when I have more than 25 variables. which one is more reliable (log liklihood, G-square, AIC, BIC, CAIC, Adjusted BIC or Entropy)?
Thanks by advance

Nick Horton said...

I've found Sonja Swanson's excellent paper to be helpful with those questions:


A Monte Carlo investigation of factors influencing latent class analysis: an application to eating disorder research.

Swanson SA1, Lindenberg K, Bauer S, Crosby RD.
Author information
Latent class analysis (LCA) has frequently been used to identify qualitatively distinct phenotypes of disordered eating. However, little consideration has been given to methodological factors that may influence the accuracy of these results.
Monte Carlo simulations were used to evaluate methodological factors that may influence the accuracy of LCA under scenarios similar to those seen in previous eating disorder research.
Under these scenarios, the aBIC provided the best overall performance as an information criterion, requiring sample sizes of 300 in both balanced and unbalanced structures to achieve accuracy proportions of at least 80%. The BIC and cAIC required larger samples to achieve comparable performance, while the AIC performed poorly universally in comparison. Accuracy generally was lower with unbalanced classes, fewer indicators, greater or nonrandom missing data, conditional independence assumption violations, and lower base rates of indicator endorsement.
These results provide critical information for interpreting previous LCA research and designing future classification studies.