Latent Class Analysis - Statistical Innovations

Latent Class Analysis

Foundation Entries

SAGE Research Methods Foundations

By: Jay Magidson, Jeroen K. Vermunt & John P. Madura

Published:2020

Length: 10,000 Words

DOI: http://dx.doi.org/10.4135/9781526421036

Methods: Latent Class Analysis

Online ISBN: 9781526421036

Disciplines: Anthropology, Business and Management, Criminology and Criminal Justice,

Communication and Media Studies, Counseling and Psychotherapy, Economics, Education, Geography,

Health, History, Marketing, Nursing, Political Science and International Relations, Psychology, Social

Policy and Public Policy, Social Work, Sociology, Science, Technology, Computer Science, Engineering,

Mathematics, Medicine

Access Date: September 10, 2020

Publishing Company: SAGE Publications Ltd

City: London

© 2020 SAGE Publications Ltd All Rights Reserved.

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Not for redistribution beyond Jay Magidson's online course in Latent Class Modeling

http://dx.doi.org/10.4135/9781526421036

Abstract

Latent class (LC) analysis is a widely used method for extracting meaningful groups (LCs) from data.

The basic concept was introduced by Paul Lazarsfeld in 1950 for building typologies (or clusters) from

dichotomous variables as part of his more general latent structure analysis. In 1974, Leo Goodman

operationalized and extended LC analysis using maximum likelihood methods, which resolved previous

implementation problems. After 1995, extensions to traditional LC modeling took place, and Jeroen K.

Vermunt and Jay Magidson defined the LC model more generally as any model where some parameters differ

across unobserved subgroups called LCs. LC modeling has since become a general multivariate modeling

approach for revealing latent segments based on any set of observed indicators, across a wide range of

applications. In particular, LC generalizes traditional cluster, factor, and item response theory analyses and

also generalizes various kinds of regression modeling where the parameters that differ for different classes

are the regression coefficients. This entry discusses traditional LC modeling, tools for determining the number

of classes, approaches for identifying meaningful classes, and advanced LC regression models for analyzing

ratings and choice data. Several other topics—such as dealing with covariates using one-step and three-step

approaches, multilevel LC models, latent Markov models, and LC growth models—are also briefly discussed.

Introduction

Latent class (LC) analysis has become one of the most widely used methods for extracting meaningful groups

(LCs) from data. The concept of LCs was originally introduced by Paul Lazarsfeld (1950) in building typologies

(or clusters) from dichotomous variables as part of his more general latent structure analysis (Lazarsfeld &

Henry, 1968). However, implementation problems were substantial and not resolved until 1974 when Leo

Goodman operationalized and extended LC analysis in a major breakthrough using maximum likelihood

methods (Goodman, 1974a, 1974b).

Goodman’s algorithm, now referred to as an implementation of the expectation-maximization (EM) algorithm,

was made publicly accessible by Clifford Clogg’s computer program maximum likelihood latent structure

analysis (MLLSA). During the period 1974–1995, MLLSA was the primary LC program for extracting

underlying LCs from a set of categorical response variables (categorical indicators) as a method for

understanding the underlying structure of those variables.

In this entry, Goodman’s LC model is referred to as the traditional LC model. Numerous extensions to

traditional LC modeling took place after 1995, prompting Jeroen K. Vermunt and Jay Magidson (2004a)

to redefine the LC model more generally as any model where some parameters differ across unobserved

subgroups called LCs (or clusters or segments).

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Page 2 of 37 Latent Class Analysis


LC as a Measurement Model

Table 1 shows how the traditional LC model differs from other (traditional) latent variable measurement

models according to the scale types of the observed and latent variables (categorical vs. continuous).

Table 1. Traditional latent variable measurement/classification models.

Observed Indicators

Latent Variable

Continuous Categorical

Continuous Factor analysis Latent profile analysis

Categorical Item response theory Latent class analysis

LC modeling has evolved during the 21st century to become a general multivariate modeling approach for

revealing meaningful latent segments based on any set of observed indicators, not just categorical indicators,

and across a wide range of applications. In particular, LC generalizes traditional cluster, factor, and item

response theory (IRT) analyses in many ways and also generalizes various kinds of regression modeling

(including discrete choice models) where the parameters that differ for different classes are the regression

coefficients.

LC can be performed with cross-sectional or longitudinal data with response variables that are categorical,

continuous, or counts (binomial and/or Poisson) or combinations of these and other scale types. LC

clustering, discrete factor (DFactor) analysis, and LC regression analyses are generally conducted with cross-

sectional data, while LC growth modeling and mixture latent Markov (LM) or latent transition models can

provide valuable insights for longitudinal or panel data.

As used here, the term LC model refers to any statistical model conceptualized using one or more categorical

latent variables. Thus, LC modeling includes both traditional LC analysis and latent profile analysis (see Table

1), DFactor models containing two or more categorical latent variables, and hybrid models containing both

categorical and continuous latent variables such as the random intercept LC regression model illustrated later

to analyze ratings.

Example of LC Cluster Analysis

As an example of a traditional LC cluster analysis, Table 2 provides results from a 3-class LC model estimated

on responses from the 1982 General Social Survey (GSS; see McCutcheon, 1987). Two of the variables

ascertain the respondent’s opinion regarding (Y1) the purpose of surveys and (Y2) how accurate they are,

and the others are evaluations made by the interviewer of (Y3) the respondent’s levels of understanding of

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the survey questions and (Y1) cooperation shown in answering the questions. Alan McCutcheon named the

classes “Ideals,” “Believers,” and “Skeptics.”

Table 2. Parameter estimates from a traditional 3-class latent class analysis.a

White Respondents (n = 1,202)

Class 1 Class 2 Class 3

Ideals Believers Skeptics

LC probabilities 0.62 0.20 0.18

Conditional probabilities

PURPOSE

Good 0.89 0.92 0.16

Depends 0.05 0.07 0.22

Waste 0.06 0.01 0.62

ACCURACY

Mostly true 0.61 0.65 0.04

Not true 0.39 0.35 0.96

UNDERSTAND

Good 1.00 0.32 0.75

Fair, poor 0.00 0.68 0.25

COOPERATE

Interested 0.95 0.69 0.64

Cooperative 0.05 0.26 0.26

Impatient/hostile 0.00 0.05 0.10

aData consist of four variables obtained from 1982 General Social Survey.

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Comparing these classes with respect to their conditional probabilities shows that they clearly differ from

each other. The “Ideal” class (Class 1), representing an estimated 62% of the population, is favorable toward

surveys, approximately 90% agreeing that “Surveys serve a good purpose” (PURPOSE = Good), and are

rated by the interviewer as showing a “Good understanding of these survey questions” (UNDERSTAND =

Good).

The “Believers” (Class 2), representing an estimated 20% of the population, is similar to the “Ideals” in being

favorable toward surveys but differ from the Ideals in that they showed only a fair or poor understanding of

the questions (UNDERSTAND = Fair, poor). In contrast to both of these classes, the remaining 18%, called

“Skeptics” (Class 3), mostly state that “surveys are a waste of time” (PURPOSE = Waste).

This entry begins with a formal introduction to Goodman’s approach to LC (Goodman, 1974a, 1974b),

followed by the most important extensions. These extensions include (a) the ability to analyze a large

number of indicators, where the indicators are not only categorical (nominal and ordinal) indicators but also

continuous, count, and other scale types or any combination of these; (b) models for longitudinal multivariate

data that allow respondents to change from one LC (latent state) to another over time (LM or latent transition

models); (c) LC regression (including growth and conjoint models) for a repeated univariate response, where

the parameters that differ over classes are regression coefficients; and (d) the inclusion of covariates in any

LC model.

In addition, this entry discusses two recent approaches that are especially useful when analyzing a large

number of indicators and/or a large number of covariates—LC Tree modeling and three-step LC modeling.

Some advanced LC models, such as multilevel LC models, random intercept regression models, and scale-

adjusted choice models, are also introduced. All approaches discussed here have been implemented in the

program Latent GOLD (see, e.g., Vermunt & Magidson, 2016).

Traditional LC Analysis

This section introduces the basic ideas of traditional LC modeling for categorical (nominal or ordinal) response

variables. Formally, each LC corresponds to one of K mutually exclusive and exhaustive categories that

comprise the underlying categorical latent variable, X. Each observation is a member of one and only one

latent (unobservable) class. In other words, the population of interest contains K subgroups, but we do not

know to which subgroup an individual belongs—that is, the number of classes, K, is unknown.

Basic Ideas

The fundamental idea of traditional LC modeling is to extract that latent variable X with the fewest number of

classes sufficient to explain all the associations among the J observed response variables (indicators) y1, y2,

…, yj. For example, with J = 4 indicators, the joint response distribution can be expressed in terms of the LCs

as

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(1)

where P(X = k) is the probability of belonging to LC k = 1, 2, …, K and P(y1, y2, y3, y4 | X = k) is the

probability of observing a particular response pattern given that one belongs to LC k. Equation 1 shows that

each LC or subgroup has its own (joint response) probability P(y1, y2, y3, y4 | X = k) and that the overall

probability P(y1, y2, y3, y4) for the total population is obtained as a weighted average of the conditional

probabilities using the class proportions, P(X = k), as weights. That is, the population consists of a mixture of

K unordered (nominal) LCs.

We denote the number of categories for the indicators by M1, M2, M3, and M4, respectively, implying that

1 ≤ y1 ≤ M1, 1 ≤ y2 ≤ M2, and so on. Thus, in total, there are M1 · M2 · M3 · M4 possible response

patterns, where P(y1, y2, y3, y4) denotes the probability of occurrence of a particular response pattern on

these variables. The traditional K-class LC model is a measurement model in the sense that the model

includes parameters that define the underlying discrete latent variable in terms of the indicators. Specifically,

the model parameters consist of class sizes, P(X = k), that sum to 1, and for each indicator the conditional

probabilities, P(yj | X = k), that define class k in terms of the expected response distribution expected on

indicator j. The extent to which the conditional probabilities for a given class differ from the other classes

serves to define that class. For example, LC 3 in Table 2 differs primarily from the other classes in that they

have a relatively high probability of stating “surveys are a waste of time” (PURPOSE = Waste) and are not

accurate (ACCURACY = Not True), and hence that class was named “Skeptics.”

The key assumption made for any latent variable model is that the latent variable or variables explain

all associations between the indicators, as implied by the assumption of local independence (Vermunt &

Magidson, 2004b). That is, the indicators are mutually independent of each other conditional on class

membership. This assumption supports the meaningfulness of the classes as being the underlying source of

the associations among the indicators.

Revisiting our four-indicator example, local independence is expressed by the following equation:

(2)

where P(y1 | X = k) denotes the probability of giving response y1 on the first indicator given that one belongs

to LC k, P(y2 | X = k) is the corresponding conditional response probability for the second indicator, and

so on. As can be seen, conditional on the class one belongs to, the probability of occurrence of any given

response pattern can be computed as the product of the separate indicator response probabilities, which is

the definition of (conditional) independence.

This computation in Equation 2 can be illustrated with the example in Table 2. The local independence

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assumption implies that for LC 1 (“Ideals”) the probability of choosing the first, second, first, and first response

category on the four indicators equals 0.89, 0.39, 1.0, and 0.95, respectively. Thus, the probability for the full

response pattern, P(1, 2, 1, 1 | X = 1), is equal to (0.89)(0.39)(1.0)(0.95) = 0.32.

Substituting Equation 2 into Equation 1 yields the standard form of the LC model where the basic probability

parameters are included in the right side of the equation:

(3)

Table 2 provides the parameter estimates obtained by maximizing the likelihood function under the traditional

3-class LC model.

Determining the Number of Classes

The primary model selection problem is deciding on the number of classes. This can be done formally using

global goodness-of-fit tests, information criteria, or local goodness-of-fit tests (bivariate residuals [BVRs]).

The traditional global test compares maximum likelihood estimates for frequency counts expected under the

model with the actual observed frequencies using the likelihood ratio or Pearson lack-of-fit statistic, (L2 or X2;

Goodman, 1974a). The associated p value assesses the extent to which the estimated counts are sufficiently

close to the observed counts so that any differences can be explained by sampling variability.

In the special case where the expected frequencies in Equation 4 reproduce each of the corresponding

observed cell counts perfectly, the model fit is perfect and both L2 and X2 equals zero. To the extent that

the value for L2 exceeds 0, the L2 quantifies the lack of model fit, representing the amount of association

(nonindependence) that remains unexplained by that model. Since, the L2 for the null model of independence

(model H0 in Table 3) measures the total amount of association in the data, an R2-like measure can assess

the percentage of the association explained by each LC model (see % reduction of L2 in Table 3).

Estimates for the expected frequencies, ^F, are obtained by multiplying both sides of Equation 3 by the sample

size N to obtain F = N * P, and then plugging in the estimates for the probability parameters in the right side

of Equation 3 to obtain estimates for P:

(4)

However, when the number of indicators becomes large, the data become sparse (containing many empty or

small cell counts) in which case L2 and X2 no longer follow a χ2 distribution. An alternative is to use bootstrap

p values which do not depend upon the χ2 distribution (Langeheine et al., 1996). However, since computation

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of the bootstrap p value is somewhat complex and can take considerable time to compute for large models, a

popular alternative is to use information criteria such as the Bayesian information criteria (BIC).

Traditional exploratory applications proceed by estimating successive LC models with K = 1, 2, … classes

and stopping when the hypothesis of local independence can no longer be rejected at (say) the .05 level. For

our example, Table 3 shows that the 1-class and 2-class models are rejected because they fail to fit the data

(p < .05) but the 3-class model does provide an adequate fit (p = .11), and hence the number of classes K =

3 is accepted as the simplest solution that explains all of the associations among the indicators.

To pinpoint where any two-way residual associations remain, a local goodness-of-fit statistic such as the BVR

can be used to determine where violations of local independence may occur.

Table 3. Results from LC model fit for the General Social Survey data.

Model Description BICLL L² Npar df p Value % Reduction in L²(H0)

H0 1-class 5,787.0 257.3 6 29 2.0 × 10-38 0.0%

H1C 2-class 5,658.9 79.5 13 22 2.0 × 10-8 69.1%

H2C 3-class 5,651.1 22.1 20 15 0.11 91.4%

H3C 4-class 5,685.3 6.6 27 8 0.58 97.4%

Some additional models

H1D 2-class + direct effect 5,606.1 12.6 15 20 0.89 99.8%

H2F Basic 2-DFactor 5,640.1 11.1 20 15 0.75 95.7%

In practice, LC analysis is often conducted with a relatively large number of indicators, in which case the p

value determined from L2 or X2 is not valid because those statistics no longer follow a χ2 distribution due to

the data being sparse. An indication that p values cannot be trusted is when p values from L2 and X2 differ

substantially (in the extreme 1 and 0). In this case, one can use either a bootstrap of L2 to estimate the p

value or use an information criterion based on the log-likelihood (LL), such as AICLL or BICLL, to determine

the number of classes. BICLL reported in Table 3 (with nonsparse data) agrees that the best fit is provided

with 3 classes (BICLL reaches its minimum with 3 classes before increasing with 4 classes).

A bootstrap estimate of the p value corresponding to the difference in log-likelihood values for any two nested

models can also assist in the determination of the number of classes. This test finds that the 4-class model

(H3C) provides a significant improvement over the 3-class model. While this result is contrary to the BIC

criteria which suggests that the 3-class model is best, Table 3 shows that BIC suggests that a restricted

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4-class model (model H2C in Table 3) is preferred over the 3-class model. Model H2C is discussed in the

section “DFactor Models.”

Bivariate Residuals

Modification indices, called BVRs, help pinpoint where violations of local independence occur. BVRs are

Lagrange-type χ2 statistics which test each pair of indicators for local independence (Oberski & Vermunt,

2013; Vermunt & Magidson, 2016). As a rule of thumb, a BVR that exceeds a value of four indicates that a

significant residual association remains unexplained by the classes.

Table 4. Bivariate residuals for LC models.

Indicator Pair

LC Model

1-class 2-class 3-class 2-class + direct effect 2 DFactor

PURPOSE ACCURACY 61.64 0.07 0.11 0.01 0.01

PURPOSE UNDERSTAND 0.53 0.74 0.09 0.10 0.03

PURPOSE COOPERATE 10.59 0.05 0.12 0.07 0.10

ACCURACY UNDERSTAND 0.26 1.10 0.00 0.02 0.02

ACCURACY COOPERATE 8.61 0.41 0.25 0.22 0.44

UNDERSTAND COOPERATE 43.37 32.33 2.41 0.00 0.19

Recall from Table 3 that the 1- and 2-class models were both rejected (p < .05), which means that they

violate the assumption of local independence. The first two columns of Table 4 identify the particular indicator

pairs for which the local independence assumption is violated for these models—the residual association

remains large as indicated by a large BVR. The magnitude of the BVRs is reduced as the number of classes

is increased from 1 to 2, but not until a 3rd class is added is the model fit judged to be adequate (and the

BVRs are all small).

The 1-class model, H0, assumes that respondents are homogeneous and is referred to as the null model

because it assumes all indicators are mutually independent. Each BVR for this model is equivalent to the

Pearson χ2 lack-of-fit statistic (X2) divided by the degrees of freedom. From Table 4 we see that this model

fails to fit the data because four of the six indicators have large BVRs indicating that significant associations

exist between those pairs.

For this model, the two small BVRs (0.26 and 0.53) correspond to the pairs (PURPOSE, UNDERSTAND)

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and (ACCURACY, UNDERSTAND) suggest that a respondent’s level of understanding of the question

(UNDERSTAND) is statistically independent of their opinion of surveys as reflected by the indicators

PURPOSE and ACCURACY. Thus, we might expect the data to reflect two distinct dimensions—one

assessing the respondents’ level of understanding, the other their attitude toward surveys (favorable vs.

unfavorable).

The 2-class model, H1C, is a one-dimensional model which assumes the existence of two homogeneous

classes, mutual independence existing within each class (local independence). For these data, the 2-class

model is rejected (p < .05). Table 4 pinpoints that the reason for the lack of fit is that a residual association

between UNDERSTAND and COOPERATE remains unexplained (i.e., the BVR for this pair remains large).

As indicated in the 3rd column of Table 4, this association requires a 3rd class to explain. By accounting for a

2nd dimension, the 3-class model, H2C, reduces the BVR from 32.33 to the acceptable value of 2.41.

It should be noted that K-class models may sometimes represent fewer than K − 1 dimensions. In particular,

a one-dimensional model can capture a unique ordering among the classes. For example, data from Leonard

Pearlin and Joyce Johnson (1977), reanalyzed by Magidson & Vermunt (2001) suggested that the three

classes extracted from five dichotomous indicators of depression, represent three ordered classes. Class 1

(“Healthy” group) exhibits the lowest probabilities of depressive symptoms (lack of enthusiasm, low energy,

sleeping problem, poor appetite, and feeling hopeless) during the past week. At the other extreme Class

3 (“Depressed” class) exhibits the highest probabilities for these symptoms, and Class 2 falls somewhere

between these extremes on all symptoms. We might interpret Class 2 as having a “bad week” but not

necessarily being depressed. The summary of these results can be found in Figure 1 (for additional details of

this analysis, see Magidson & Vermunt, 2001).

Figure 1. Plot of class-specific response probabilities for each of five symptoms of depression.

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Classification

Traditional clustering approaches, such as K-means and hierarchical clustering, focus primarily on

classification. For example, Leonard Kaufman and Peter J. Rousseeuw (1990) define cluster analysis as

“classification of similar objects into groups, where the number of groups, as well as their forms are unknown”

(p. 1). N. Kumar (2005) provides a similar definition:

Cluster analysis divides data into groups (clusters) that are meaningful, useful, or both. If meaningful

groups are the goal, then the clusters should capture the natural structure of the data. (p. 487)

In contrast, LC is model based, classification being accomplished as a rigorous second step following

parameter estimation. This is similar to traditional factor analysis, where factor scores are estimated as

a second step following estimation of the model parameters (factor loadings). Specifically, for any given

observed response pattern (y1,…, yj), the posterior probability of belonging to class k, P(X = k | yj,…, yj),

can be expressed in terms of the LC model probability parameters (the unconditional class size probability,

P(X = k), and the conditional probabilities associated with each of the J indicators for respondents in this

class, P(yj | X = k), j = 1, 2, …, J.

(6)

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This result is an application of Bayes’ theorem, the last equality resulting from the assumption of local

independence (recall Equation 2).

Table 5 provides the posterior probabilities obtained from the 3-class model H2C. Given their response pattern,

respondents are typically assigned to the LC for which their posterior probability is highest (bolded in Table

5), a classification rule called modal assignment.

Table 5. Posterior classification probabilities for 3-class latent class model.

PURPOSE ACCURACY UNDERSTAND COOPERATION Ideal Believer Skeptic

Good Mostly true Good Interested 0.92 0.08 0.00



Fair, poor Interested 0.02 0.97 0.00



Not true Good Interested 0.88 0.06 0.06






Depends Mostly true Good Interested 0.87 0.10 0.04


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Impatient/Hostile 0.00 0.16 0.84

Waste Mostly true Good Interested 0.88 0.02 0.10










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These posterior probabilities provide useful information about how good the classification is. For example,

from the first row of Table 5, we see that for persons categorized as (Good, Mostly true, Good, Interested)

to the four indicators, the expected accuracy that they belong to the Ideal class is quite high—92%. Only

8% of these respondents would be expected to be misclassified—they really belong to the Believers class.

Averaging these different accuracies over all respondents yields an overall expected accuracy of 87%.

Extracting Meaningful LCs

As one moves from an exploratory to a more confirmatory use of LC modeling the question “What do you want

the LCs to represent?” becomes relevant. As argued by Christian Hennig (2015), the idea of “truth” in cluster

analysis depends on the context and the clustering aims. Thus, for example, if the goal of the LC analysis is to

extract the classes that represent different attitudes toward surveys, the association between UNDERSTAND

and COOPERATE should be ignored when deriving the classes. There are several approaches for modifying

the traditional LC approach that can accomplish this goal.

Next, three such approaches, each with its own advantages, are described, but for the current example, all

yield essentially the same two substantively meaningful LCs—one favorable and the other unfavorable toward

surveys. The approaches are as follows:

1. Add a direct effect to the model (see “Adding Direct Effects” section). This is the simplest approach

when dealing with a small number of indicators.

2. Use a separate discrete latent factor for each dimension, the first corresponding to the desired

classes, the second a nuisance factor (see “DFactor Models” section). Directly analogous to

traditional factor analysis, the DFactor approach is especially useful when the number of indicators is

large, in both exploratory settings where the number of dimensions is unknown and in confirmatory

settings.

3. Use LC tree models which impose a hierarchical tree structure on the data (see “LC Tree Models”

section). This approach is similar to the use of DFactors but easier to implement. It is designed

primarily for exploratory applications with a large number of indicators.

4. Weight the indicators to downplay the relative importance of one or more “less important” variables

(Eagle & Magidson, 2019).

Adding Direct Effects

While increasing the number of classes is one way to achieve an adequate fit, at some point increasing the

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number of classes from say K to K + 1 may result in classes that are less as opposed to more meaningful. For

example, the previous 3-class model distinguishes between both respondents who have a more versus less

favorable view of surveys (Classes 1 and 2 vs. Class 3) as well as respondents who show a good versus not-

so-good understanding of the data (Classes 1 and 3 vs. Class 2). If the latter distinction were not of interest,

one could modify the model in either of two ways:

1. by removing the indicator UNDERSTAND or

2. by adding a direct effect between UNDERSTAND and COOPERATE (Model H1D)

Approach 2 adds a parameter directly to the model to account for the association between UNDERSTAND

and COOPERATE, allowing that association to be explained directly rather than requiring it to be explained by

the classes (see Hagenaars, 1988, for details). As shown in Table 3, modifying the 2-class model (Model H1C)

by (a) adding a 3rd class (Model H2C) or (b) adding a direct effect (Model H1D), both provide an acceptable fit

to the data.

DFactor Models

DFactor models are restricted LC cluster models where ordinality restrictions are imposed on each DFactor

(Magidson & Vermunt, 2001). Each DFactor may have 2 or more levels. For example, Table 4 shows that a

DFactor model with two dichotomous DFactors V = 1, 2 and W = 1, 2 corresponds to a 4-class cluster model.

Similar to traditional factor analysis, the basic DFactor model restricts the factors to be independent of each

other.

LC factor models were introduced originally by Goodman (1974b) for confirmatory applications and extended

by Magidson &Vermunt (2001) who developed it for use as a general exploratory alternative to traditional LC

cluster modeling. DFactor models are analogous to traditional factor analysis where each factor corresponds

to a distinct dimension.

The dimensionality of a K-class LC cluster model is K − 1, corresponding to the K − 1 distinct contrasts

that can be formed by comparing classes. The 1-class model, which assumes mutual independence among

the indicators, means that the population is completely homogeneous, so there can be no justification for

subdividing the respondents into two or more subgroups.

The 2-class model is a one-dimensional model. As such, the classes can be interpretable as being high

versus low (e.g., favorable vs. unfavorable) on that dimension. The 3-class model is a two-dimensional model

and hence the solution can be plotted in a two-dimensional space using a barycentric coordinate plot as

described by Magidson &Vermunt (2001; see Figure 2).

Figure 2. Barycentric coordinate display of results reported for model H2C.

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Later, another two-dimensional LC model consisting of two DFactors, where each DFactor corresponds to

one of the dimensions, will be introduced.

Table 6. Relationship between DFactors V and W and the joint DFactor X = (V, W).

W = 1 W = 2

V = 1 X = 1 X = 2

V = 2 X = 3 X = 4

While the basic 2-DFactor model, H2F, is a restricted 4-class model, Table 3 shows that it has the same

number of parameters as the 3-class model H2C (Npar = 20) but fits better. Table 7 displays the probability

parameters for the joint DFactor X = (V, W), while Table 8 provides the probability parameters for each

DFactor separately.

Table 7. Joint DFactor output for the Basic 2-DFactor Model H2F.

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DFactor1 (V) 1 1 2 2

DFactor2 (W) 1 2 1 2

Class size 0.43 0.34 0.13 0.10

Indicators

PURPOSE

good 0.97 0.61 0.90 0.21

depends 0.03 0.13 0.10 0.18

waste 0.00 0.26 0.00 0.62

ACCURACY

mostly true 0.81 0.24 0.60 0.10

not true 0.19 0.76 0.40 0.90

UNDERSTAND

good 0.91 0.97 0.29 0.57

fair/poor 0.09 0.03 0.71 0.43

COOPERATE

interested 0.95 0.92 0.58 0.42

cooperative 0.05 0.08 0.34 0.39

impatient/hostile 0.00 0.00 0.07 0.19

Table 8. Parameters estimates for the Basic 2-DFactor Model H1C.

DFactor1 (V) DFactor2 (W)

Indicators Level 1 Level 2 Level 1 Level 2

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DFactor1 (V) DFactor2 (W)

size 0.77 0.23 0.56 0.44

PURPOSE

good 0.82 0.59 0.96 0.52

depends 0.07 0.13 0.04 0.14

waste 0.11 0.27 0.00 0.34

ACCURACY

mostly true 0.56 0.38 0.76 0.21

not true 0.44 0.62 0.24 0.79

UNDERSTAND

good 0.94 0.41 0.77 0.88

fair/poor 0.06 0.59 0.23 0.12

COOPERATE

interested 0.94 0.51 0.86 0.81

cooperative 0.06 0.36 0.12 0.15

impatient/hostile 0.00 0.13 0.02 0.04

Table 8 makes clear that DFactor1 distinguishes primarily between those having a good (V = 1) versus fair/

poor (V = 2) understanding of the questions. Figure 3 provides the biplot (Magidson & Vermunt, 2001), which

confirms graphically that the variable UNDERSTAND loads primarily on DFactor 1 (horizontal dimension),

while the survey attitude indicators (PURPOSE and ACCURACY) load primarily on DFactor 2 (vertical

dimension).

Figure 3. Biplot of DFactor 1 and DFactor 2.

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This interpretation is further supported by the estimated loadings (Table 9) showing that the indicator

UNDERSTAND loads primarily on DFactor1 (loading = .57), while PURPOSE and ACCURACY load primarily

of DFactor 2 (for details on how these loadings are computed, see Magidson & Vermunt, 2003; Vermunt &

Magidson, 2004c).

Table 9. Loadings output.

Loadings

Indicator DFactor1 DFactor2 R²

PURPOSE 0.19 0.45 0.26

ACCURACY 0.15 0.55 0.33

UNDERSTAND 0.57 0.14 0.36

COOPERATE 0.42 0.07 0.19

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Comparing the LC cluster and DFactor approaches, in practice, the DFactor model almost always fits better

and the parameters are easier to interpret since each DFactor corresponds to a distinct dimension. In

addition, fitting a 1-DFactor model with three or more levels, can be used to test whether the resulting classes

can be ordered. Specifically, by default equidistant scores are assigned to each level so that for a DFactor

with three levels, Level 2 can be interpreted as being midway between Levels 1 and 3. More general DFactor

models with unequal spacing can also be specified using the scores keyword in the Latent GOLD® syntax

(see Vermunt & Magidson, 2013; for related work on scoring LCs and the relationship to IRT models, see

Clogg, 1988; Formann, 1992; Heinen, 1996; Uebersax, 1993; Vermunt, 2001).

On the other hand, classification is more complex for DFactor models with two or more DFactors in that

posterior membership probabilities are available for each DFactor separately as well as the joint DFactor.

However, as an interpretive tool, the additional complexity may be worthwhile as one of the DFactors may

represent the factor of interest, while the others correspond to nuisance factors that allow the meaningful

factor to be measured without bias (as an example of how the DFactor model has been proven useful to

adjust for response style behavior, see Moors, 2003).

LC Tree Models

Especially when there are many indicators, the number of LCs may become large and thus more difficult

to interpret. A remedy for this is to structure the latent variable using LC tree models, which imposes a

hierarchical tree structure on the latent variable X (Van den Bergh et al., 2017, 2018) LC tree modeling begins

by extracting a relatively small number of interpretable root classes. Next, to the extent to which a root class

is heterogeneous, this class is split into two subclasses, and this splitting process continues to trace out the

full tree until all the heterogeneity is explained. The terminal nodes of the full tree replace the K classes in a

standard LC cluster analysis.

Each of the relatively small number of root classes represents an overall theme in the data, which makes

for easier interpretation of the K terminal classes. Thus, these root classes might be referred to as theme

classes.

To demonstrate the easier interpretation provided by the LC tree structure, an 8-class solution obtained in

choice modeling is re-examined using the LC tree approach. The LC tree solution yields three theme classes

which reveal the primary differences in preferences, each of which contains some within-class variation. For

each of these theme classes, the within-class variation is then modeled in a meaningful manner by splitting it

into two subclasses, which reveal secondary differences. Figure 4 depicts the LC tree process which results

in six classes that are much easier to interpret than the original eight classes.

Figure 4. General latent class tree model structure.

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This application demonstrates the improvement in interpretation which can result from the LC tree model

when there are many indicators and many classes. It is presented in the “SALC Tree Models” section, where

an extended LC tree model, referred to as the scale-adjusted LC (SALC) tree model, is also introduced.

The LC tree model can also provide useful insights in smaller applications. For example, when applied

to the GSS data, it yields two theme classes at the first level, corresponding to those who are favorable

versus unfavorable toward surveys, as previously estimated with the 2-class model H1C. In order to obtain an

adequate fit which explains all the heterogeneity, the LC tree approach then splits both of these classes to

account for the differences in understanding. Figure 5 shows how the largest BVR is reduced to acceptable

levels after this second split.

Figure 5. LC tree model using four indicators from the 1982 General Social Survey data (see McCutcheon,

1987).

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Extensions of Traditional LC Modeling

The most important extensions to traditional LC modeling involve the ability to analyze continuous indicators

and other scale types in additional to categorical indicators, the inclusion of covariates with either a one-

step or three-step modeling approach, multilevel LC models which yield a classification at multiple levels, LM

models which allow respondents to change from one LC to another over time, and LC regression modeling,

including LC ratings-based and choice-based conjoint models which allow LCs to be defined based on

differences in meaningful regression coefficients.

Continuous Indicators, Counts, and Other Scale Types

While traditional LC modeling was limited to categorical indicators, LC modeling today may be conducted not

only with response variables that are categorical (nominal and/or ordinal) but also continuous and/or counts

or a combination of these (or other) scale types continuous. This extension is direct by replacing the right-

most terms in Equation 4 by the appropriate distribution for each indicator. For example, if y3 is a count, the

Poisson or binomial distribution is used in Equation 4 for P(y3 | X = k). If y3 were continuous, P(y3 | X = k)

corresponds to the normal density with associated class-specific mean and variance parameters (μ3k, σ3k2 ).

Finite mixture modeling, which involves un-mixing continuous variables (McLachlan & Peel, 2004), predates

traditional LC modeling. In fact, it was Karl Pearson who first brought attention to this type of statistical

application. In his first statistical publication, Pearson (1883) dealt with the approximation of a complicated

continuous density as a finite mixture of simpler densities. In a classic application, he showed that the

asymmetric nature of the forehead to body length distribution in crabs can be explained as a mixture of

two normal probability density functions with different means and different variances (Pearson, 1894). He

interpreted the results as providing evidence that this population was evolving into two new species.

Since inclusion of continuous indicators introduces variance parameters into LC models, this allows classes

to be revealed that differ not only in their means for one or more indicators but also in variance. As explained

in the next subsection, the ability to account for variance heterogeneity allows LC models to extract segments

that are more meaningful than those obtained from the K-means clustering algorithm. While χ2 model fit

statistics such as L2 and X2 are not available for continuous variables, the BICLL statistic can be used with

such models since its computation requires only knowledge of the likelihood function.

Relationship to K-Means

When all variables in a LC model are continuous, LC models can be compared directly to the popular

K-means approach to clustering, which has been shown to be equivalent to maximizing the classification

likelihood of a restricted mixture model (Vermunt, 2011). The K-means Euclidean distance criterion translates

into implicit assumptions of (a) local independence and (b) equal within-cluster variances. These assumptions

are often referred to as sphericity because the locus of points associated each of the K-clusters correspond

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to equal-sized spheres.

In comparison to LC, K-means has been shown to perform poorly in cluster recovery because the K-means

assumption that residual variances are equal within each cluster fails to hold true in practice (Magidson &

Vermunt, 2002a, 2002b; Vermunt, 2011).

As an example, Figure 6 shows how patients from three known groups (those with “overt diabetes,” those

with “chemical diabetes,” and “normal” individuals who do not have diabetes), compare with respect to three

continuous measurements—GLUCOSE, INSULIN, and SSPG (Reaven & Miller, 1999). In particular, the

scatterplot of GLUCOSE by INSULIN reveals that the variances for these variables are much larger in the

Overt diabetes group,

Figure 6. Matrix scatterplot of diabetes data set by clinical classification.

Three-class LC models that allow variances to differ across classes not only provide the best fit to these data

in terms of the BIC statistic but also have been shown to recover the three structural groups with much higher

accuracy than K-means. Using simulated data, Vermunt (2011) confirms the general superiority of LC over

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K-means in recovering true class membership in the commonly occurring situation where variances differ

across classes.

The scatterplot between (y1) GLUCOSE and (y2) INSULIN in Figure 6 shows that the Overt group has larger

variances than the other groups, and also that a positive correlation between these variables remains within

this group, violating the local independence assumption. One way to handle such violations without increasing

the number of classes is to include a direct effect in the model. This is handled by applying the bivariate

normal distribution to the variable pair, P(y1, y2 | X = k) as shown in Equation 7, which introduces a covariance

parameter along with the mean and variance parameters:

(7)

Covariate Extension: One-Step and Three-Step Approaches

An important extension of the LC model involves inclusion of covariates predicting class membership (Dayton

& Macready, 1988; Kamakura, Wedel, & Agrawal, 1994). Denoting a person i’s covariate vector by zi, this

extended LC model is defined as:

(8)

The main change compared to the basic LC model is that the class membership probabilities may now be

dependent on zi, whereas the conditional probability of the indicators, P(yij | X = k), remains unchanged.

Note that an important additional assumption is made, namely that the effect of the zi on the yi is fully

mediated by the LCs. It is possible to test this assumption using local fit measures (BVRs) similar to those

discussed earlier, as well as to relax it by allowing for direct effects, which implies replacing P(yij | X = k)by P(yij | X = k, zi) for one or more of the yij. Typically, P(X = k | zi) is modeled using a multinomial logistic

specification; that is,

(9)

where γ0k and γpk represents the intercept and the slope of predictor zip for LC k. For identification, we assume

parameters sum to 0 across classes (effect coding) or are equated to 0 for one class (dummy coding).

The simultaneous modeling of responses yi and covariates zi using this one-step approach may sometimes

be impractical, especially when the number of possibly relevant covariates is large. Moreover, in most

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applications, we wish to obtain a clustering that is not affected by the chosen covariates but only by the

selected indicator variables. Therefore, most researchers prefer using a three-step approach involving:

1. estimating the LC model without covariates,

2. obtaining the individuals’ class assignments using the posterior membership probabilities, and

3. investigating how the class assignments are related to covariates.

However, as shown by Annabel Bolck, Marcel Croon, and Jacques Hagenaars (2004), this three-step

approach yields downward biased estimates of the covariate effects. Based on the work of these authors,

Vermunt (2010) proposed a simple method to adjust for this bias (see also Bakk, Tekle, & Vermunt, 2013).

The adjustment is based on the following relationship between the class assignments wi and the true class

memberships:

(10)

Note that this again is a LC model but with wi as a single “response” variable. The adjustment proposed by

Vermunt (2010) therefore involves estimating a LC model with zi as concomitant variables and wi as the single

response variable, while fixing the P(wi | X = k) at the values computed using the parameter estimates from

the first step.

Multilevel LC Analysis

Vermunt (2003) proposed the multilevel LC model, which can be used when individuals (lower level units

such as students) belong to groups (higher level units such as schools), and when the number of groups is

too large to use the grouping variable as a series of dummy variables in a LC model with covariates. The

description of the multilevel LC model requires expansion of our notation. We refer to a particular group or

higher level unit as g and to the response vector of a group and an individual within a group as yg and ygi,

respectively. The number of individuals within a group is denoted by ng, the group-level LC variable by V, a

group-level LC by d, and the number of group-level LCs by D. The lower level part of the two-level LC model

has the following form:

(11)

which is the same as a standard LC model, except for the fact that the lower level LC proportions are allowed

to differ across higher level LCs V. As in the standard LC model, we assume local independence across the

J indicators. The higher level part of the model which connects the responses of the ng persons in group g

equals

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(12)

As can be seen, the main additional model assumptions are each group belongs to one of D group-level

LCs and that the individuals’ responses within a group are independent given the group’s class membership.

Combining the Equations (11) and (12) yields the full equation of a two-level LC model:

(13)

As in a standard LC class model, we can include covariates predicting the higher level and lower level class

memberships, either using a one-step or a three-step approach. Moreover, the assumption that the ygij are

independent of the group’s class membership given the individual’s class membership can be tested and

relaxed (Nagelkerke, Oberski, & Vermunt, 2016).

LM or Latent Transition Models

A LM model is a LC model for longitudinal data in which persons are allowed to switch between latent states

across measurement occasions. It is also referred to as latent transition model (Collins & Lanza, 2010),

hidden Markov model (MacDonald & Zucchini, 1997; Visser, 2011), Markov switching or regime-switching

model.

More generally, a mixture LM model utilizes both LCs and latent states to study different transition patterns,

from one latent state to another, that occur for different LCs. For example, a Mover-Stayer model is a 2-class

LM model where one class consists of “stayers” who always remain in the same state, while the other class

are “movers” who change from one state to another over time.

LC Regression or Conjoint Models

LC regression models differ from LC cluster models in that the parameters that differ across unobserved

subgroups are regression coefficients rather than conditional probabilities. However, unlike standard

regression, where regression coefficients are treated as fixed effects, LC regression is more like mixed

models which allows for heterogeneity in the regression coefficients between observations.

Typical LC regression applications often involve multiple replications or repeated univariate responses for

each case, where the replications correspond to time (one replication for each time point), or different

situations, such as ratings of different products. Applications of the former include LC growth modeling, where

observations are clustered or grouped based on the way they change over time. The latter includes ratings-

based and choice-based conjoint applications, where observations are generally clustered or segmented

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based on the attributes that drive their ratings (or choices).

To facilitate LC regression models that involve multiple replications per case, the data are organized as a

long file rather than the typical wide file format. For example, in a taste testing experiment, consumers were

asked to rate each of J = 15 cracker products on an M = 9-point liking scale (Popper et. al., 2004; Magidson &

Vermunt, 2006). Figure 7 shows the data as a wide file in which each of the 15 ratings are stored in separate

columns, and each record corresponds to a different consumer (e.g., ratings for consumer #1101 are provided

in the first record). This format is typical for a LC cluster analysis.

Figure 7. Example of wide file format.

In contrast, Figure 8 shows the same data as a long file in which the product ratings for a given consumer

are stored in a single response variable RATING, with separate records for each of the 15 products

for that consumer. The latter format allows for the regression of the dependent variable RATING to be

performed as a function of the single nominal dependent variable PRODUCT or as a function of separate

product attributes. To accommodate the latter regression, the restructured data also include four appearance

attributes (JAPP1-JAPP4), four flavor attributes (FLV1-FLV4), and four texture attributes (TEX1-TEX4) for

each cracker product (Figure 8). The values for these sensory attributes were obtained from food experts

(see Popper et. al., 2004).

For these data, regardless of whether a LC cluster analysis is performed, a LC regression with product as a

nominal predictor, or LC regression with the product attributes as predictors, the resulting class assignments

are similar. In all three cases, a 2-class model consists of respondents who tend to rate all crackers relatively

low (Class 1) and respondents who tend to rate all crackers relatively high (Class 2)—see Figure 9. This result

is typical when analyzing ratings data. While successful in capturing the strong response level difference,

this type of result is not useful for food manufacturers, who want to know which types of crackers appeal to

different consumer segments.

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Figure 8. Example of long file format for LC Cluster model.

Figure 9. Results from standard 2-class LC analysis.

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Next, some approaches that separate out confounding factors, resulting in more meaningful classes, are

described. In particular, the random intercept model is very useful when analyzing ratings data to remove the

response level confound, by capturing local dependencies in a structured manner (see “Random Intercept

Model for Analyzing Ratings” section), and the SALC model (see “SALC Tree Models for Analyzing Choices”

section) is useful in adjusting classes to remove the confounding effects of scale in choice models.

Regardless of whether ratings or choices are analyzed, the challenge in both cases is to extract classes that

are meaningful and free from the potential confounding effects of response level (for the analysis of ratings)

or scale effects (for the analysis of choices).

Random Intercept Model for Analyzing Ratings

In practice, when LC regression is used with ratings data, care must be taken to avoid regression intercept

heterogeneity from dominating the model, resulting in LC segments that differ primarily in their ratings

style—one class tends to rate all objects high while a second class tends to give lower ratings to all objects.

To deal with this problem, a random intercept can be introduced into the LC regression model to account for

this heterogeneity, allowing the LCs to capture the more meaningful heterogeneity related to differences in

the regression coefficients. Using the ordinal scale type based on the adjacent-category logit model for m

rating, the LC random intercept regression model (Magidson & Vermunt, 2006) for the rating of product t can

be expressed as:

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(14)

where the continuous latent variable F is used to model the random intercept, allowing the regression

coefficients βk1, βk2, …, βk15 to assess liking for each of the 15 cracker products relative to individual i’s overall

liking of crackers (as reflected by their intercept). This is similar to centering where the ratings for an individual

is measured relative to their average rating for all 15 crackers. Figure 9 plots the results obtained from the

2-class random intercept regression model. Compared to the results from the standard LC (Figure 10), the

resulting classes now differ in their relative preferences for one cracker over another and thus are more useful

to food manufacturers.

Figure 10. Results from 2-class LC random intercept regression analysis.

Results from the alternative 2-class random intercept regression where the sensory attributes Z1 − ZQ are

utilized as predictors instead of PRODUCT yields results very similar to that in Figure 9, providing evidence

that class differences in the liking ratings reflect different preferences with respect to the cracker attributes.

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The specification of this latter model is

(15)

SALC Tree Models for Analyzing Choices

By asking respondents to choose between two or more alternatives rather than rate each alternative, choice-

based conjoint avoids the response style problem inherent in ratings-based conjoint. However, LC choice

modeling has its own unique scaling problem that should be dealt with in order to avoid the confounding

effects of scale classes (Groothuis-Oudshoorn et al., 2018). By decomposing utilities into separate scale and

preference components, the SALC model (Magidson & Vermunt, 2007) was introduced as a potential solution

to this problem, allowing LCs to reflect differences in preferences rather than differences in scale.

Table 9 provides results from a standard LC best-worst choice model where each respondent selected their

most and least preferred principle to be used in health plan design (for details of these data, see Louviere &

Flynn, 2007).

The class-specific utility estimates presented in Table 9 yield different rankings of the 15 principles, the

highest estimate corresponding to the principle with the largest utility for that class. Because of the relatively

large number of classes and the potential scale confound, interpretation becomes somewhat difficult.

Table 9. Utilities estimates for the 8-class model.

Description Class 1 Class 2 Class 3 Class 4 Class 5 Class 6 Class 7 Class 8

Culture of reflective improvement −0.39 −1.77 −1.18 −0.40 −1.03 −2.51 −0.48 −0.15

A respectful, ethical system −0.07 −0.35 1.06 1.29 −0.56 0.19 1.08 0.40

Comprehensiveness −0.55 −0.59 0.74 −1.40 0.57 −0.12 −1.46 0.45

Equity −0.64 −0.61 0.12 −2.05 −0.95 2.18 −1.89 2.65

People and family centered 0.90 0.94 −0.17 2.59 −1.47 1.90 −0.58 −0.91

Promoting wellness and prevent 0.25 −0.68 0.52 1.55 2.15 −0.47 3.31 0.24

Providing for future generations 0.82 0.07 −0.36 1.24 −0.26 0.17 0.16 0.32

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Description Class 1 Class 2 Class 3 Class 4 Class 5 Class 6 Class 7 Class 8

Public voice community engage −0.46 −1.53 −0.38 −0.70 −2.74 −1.31 −0.15 −1.04

Quality and safety 0.06 2.12 0.86 2.01 2.47 3.26 0.72 0.46

Social and environ shape health 0.14 −1.27 −1.08 0.87 −1.01 −1.90 2.46 1.14

Responsible spending 0.27 1.44 0.65 −1.24 1.38 −0.27 −0.70 −1.42

Shared responsibility −0.50 −0.95 −0.82 −0.62 −0.03 −0.73 −1.26 −0.62

Taking the long-term view 0.43 0.03 −0.92 −0.63 0.24 −0.60 −0.02 0.45

Transparency and accountability −0.41 0.08 1.12 −0.03 −0.23 0.28 −0.99 −1.20

Value for the money 0.15 3.06 −0.15 −2.48 1.47 −0.08 −0.20 −0.76

Table 10 provides results obtained for the three root level classes obtained from a LC tree analysis of these

data. Separate results are presented based on the standard LC choice model that does not adjust for scale

and the SALC model.

Table 10. Standard estimates (utilities) vs. scale-adjusted estimates (preferences) for 3-class models.a

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aThere are six indicators that stand out in discriminating across the three Non Adjusted and Adjusted classes.

The first is the equity indicator (seen in blue); the second is a set of three indicators associated with families,

prevention and future generations (seen in yellow); and the last is the pair of indicators associated with value

for the money and responsible spending (seen in green).

Note that the estimates for Class 2 of the standard 3-class model have very low magnitude as compared to

the other classes (standard deviation of .31 compared to 1.08 and 0.96 for the other classes). This reflects

a scale confound which makes this class difficult to interpret. In contrast, the results obtained from the

SALC model show no evidence of this confound as the magnitudes of the three class-specific estimates

are comparable (as reflected by similar standard deviations). The SALC model shows clear preference

differences between the three classes. Class 1 tends to choose “value for the money” as their most preferred

principle, Class 2 tends toward principles that are “people and family centered” whereas Class 3 chooses

“equity” as most important.

Example of SALC Tree Modeling

Using the 3-class SALC model as the root classes of the tree, within-heterogeneity is found to be sufficient

to split each class into two subclasses, revealing what appears to be secondary preferences. This yields six

terminal classes (Table 11). For example, the first two classes primarily prefer “value for money” (relatively

high preference utilities of 2.39 and 1.11 for this principle), and “responsible spending” but differ on the relative

importance of “people and family centered.”

Table 11. Estimated utilities revealing primary and secondary preferences for scale-adjusted LC tree model

with three theme classes.

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Further Reading

Goodman, L. A. (2002). Latent class analysis: The empirical study of latent types, latent variables, and latent

structures. In J. A. Hagenaars & A. L. McCutcheon (Eds.), Applied latent class analysis (2–55). Cambridge,

England: Cambridge University Press.

Hagenaars, J. A., & McCutcheon, A. L. (2002). Applied latent class analysis. Cambridge, England:

Cambridge University Press.

Magidson, J., & Vermunt, J. K. (2004). Latent class models. In D. Kaplan (Ed.), The SAGE handbook of

quantitative methodology for the social sciences (Chapter 10, pp. 175–198). Thousand Oaks, CA: SAGE.

Magidson, J., & Madura. J. P. (2018). Development of an adaptive typing tool from MaxDiff response data.

Sawtooth Software Conference Proceedings, Orlando, FL.

Moors, G. (2010). Ranking the ratings: A latent-class regression model to control for overall agreement in

opinion research. International Journal of Public Opinion Research, 22, 93–119.

Vermunt, J. K., Tran, B., & Magidson, J. (2008). Latent class models in longitudinal research. In S. Menard

(Ed.), Handbook of longitudinal research: Design, measurement, and analysis (pp. 373–385). Burlington, MA:

Elsevier.

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