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Introduction to Latent Class Modeling using Latent GOLD

Copyright © 2012 by Statistical Innovations Inc.

ASSIGNED READING MATERIALS

1

SESSION 3

SESSION 3

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Assigned Reading: Latent Class: C3

11

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15

Basic and Advanced1

Jeroen K. Vermunt and Jay Magidson

Statistical Innovations Inc.

(617) 489-4490

http://www.statisticalinnovations.com

1

Excerpts from:

Excerpts from:

16

Technical Guide for Latent GOLD 5.0:

This document should be cited as “J.K. Vermunt and J. Magidson(2013). Technical Guide for Latent GOLD 5.0: Basic and Advanced. BelmontMassachusetts:StatisticalInnovations Inc.”

sum of the parameters of the three other categories. This guarantees thatthe parameters sum to zero since

∑3p=1 βp −

∑3p=1 βp = 0.

Instead of using effect coding, it is also possible to use dummy coding.Depending on whether one uses the first or the last category as referencecategory, the design matrix will look like this

category 1category 2category 3category 4

0 0 01 0 00 1 00 0 1

,

or thiscategory 1category 2category 3category 4

1 0 00 1 00 0 10 0 0

.

Whereas in effect coding the category-specific effects should be interpreted interms of deviation from the average, in dummy coding their interpretation isin terms of difference from the reference category. Note that the parameterfor the reference category is omitted, which implies that it is equated to 0.

2.5 Known-Class Indicator

Sometimes, one has a priori information – for instance, from an externalsource – on the class membership of some individuals. For example, in a four-class situation, one may know that case 5 belongs to latent class 2 and case11 to latent class 3. Similarly, one may have a priori information on whichclass cases do not belong to. For example, again in a four-class situation,one may know that case 19 does not belong to latent class 2 and that case 41does not belong to latent classes 3 or 4. In Latent GOLD, there is an option– called “Known Class” – for indicating to which latent classes cases do notbelong to.

Let τ i be a vector of 0-1 variables containing the “Known Class” infor-mation for case i, where τix = 0 if it is known that case i does not belong toclass x, and τix = 1 otherwise. The vector τ i modifies the general probabilitystructure defined in equation (1) as follows:

f(yi|zi, τ i) =K∑

x=1

τix P (x|zi) f(yi|x, zi) .

Assigned Reading: Technical G uide: Sec. 2.5

17

As a result of this modification, the posterior probability of belonging to classx will be equal to 0 if τix = 0.

The known-class option has three important applications.

1. It can be used to estimate models with training cases; that is, casesfor which class membership has been determined using a gold standardmethod. Depending on how this training information is obtained, themissing data mechanism will be MCAR (Missing Completely At Ran-dom, where the known-class group is a random sample from all cases),MAR (Missing At Random, where the known-class group is a randomsample given observed responses and covariate values), or NMAR (NotMissing At Random, where the known-class group is a non-randomsample and thus may depend on class membership itself). MAR oc-curs, for example, in clinical applications in which cases with more thana certain number of symptoms are subjected to further examination toobtain a perfect classification (diagnosis). NMAR may, for example,occur if training cases that do not belong to the original sample underinvestigation are added to the data file.

Both in the MAR and MCAR situation, parameter estimates will be un-biased. In the NMAR situation, however, unbiased estimation requiresthat separate class sizes are estimated for training and non-trainingcases (McLachlan and Peel, 2000). This can easily be accomplishedby expanding the model of interest with a dichotomous covariate thattakes on the value 0 for training cases and 1 for non-training cases.

2. Another application is specifying models with a partially missing dis-crete variable that affects one or more response variables. An importantexample is the complier average causal effect (CACE) model proposedby Imbens and Rubin (1997), which can be used to determine the effectof a treatment conditional on compliance with the treatment. Compli-ance is, however, only observed in the treatment group, and is missingin the control group. In Latent GOLD, this CACE model can be speci-fied as a LC Regression model, in which class membership (compliance)is known for the treatment group, and which a treatment effect is spec-ified only for the compliance class.

3. The known-class indicator can also be used to specify multiple-group LCmodels. Suppose we have a three-class model and two groups, say males

3 Latent Class Cluster Models

nominal covariate.should not only be used as the known-class indicator, but also as afemales to classes 4–6. To get the correct output, the grouping variablethere are six latent classes, were males may belong to classes 1–3 andand females. A multiple-group LC model is obtained by indicating that

18

proposed using random effects in LC Regression models for ranking data,and Muthen (2004) and Vermunt (2006) proposed including random effectsin LC growth models.

It has been observed that the solution of a LC Regression analysis may bestrongly affected by heterogeneity in the intercept. In rating-based conjointstudies, for example, it is almost always the case that respondents differ withrespect to the way they use the, say, 7-point rating scale: some respondentstend to give higher ratings than others, irrespective of the characteristics ofthe rated products. A LC Regression model captures this response hetero-geneity phenomenon via Classes with different intercepts. However, mostlikely, the analyst is looking for a relatively small number of latent classesthat differ in more meaningful ways with respect to predictor effects on theratings. By including a random intercept in the LC Regression model, forexample,

ηm,x,zit,F1i= βxm0 +

Q∑q=1

βx.q · y∗m · zpreditq + λx01 · y∗m · F1i,

it is much more likely that one will succeed in finding such meaningful Classes(segments). The random intercept, which may have a different effect in eachlatent class, will filter out (most of) the “artificial” variation in the intercept.

Another interesting application of random effects within latent classesoccurs in the context of LC growth modeling (Muthen, 2004; Vermunt, 2006).Suppose we have a model for a binary response variable measured at Toccasions in which zpred

it1 equals time and zpredit2 time squared. A LC growth

model with a random intercept and a random slope for the linear time effectwould be of the form:

ηx,zit,Fi= βx0 + βx1 · zpred

it1 + βx2 · zpredit2 + λ01 · F1i + λ02 · F2i + λ12 · F2i · zpred

it1 ,

where we assume that the β parameters are Class dependent and λ parame-ters Class independent. Similarly, LC growth models can be formulated fordependent variables of other scale types.

10 Multilevel LC Model

10.1 Model Components and Estimation Issues

To be able to explain the multilevel LC model implemented in Latent GOLD,we have to introduce and clarify some terminology. Higher-level observa-

j given all exogenous variable information in group j,jg

j j• for f(y |z , z ), which is the marginal density of all responses in group

These four equations show that a multilevel LC model is a model

jgg

jih ji jif(y |x, z ,F , x ,F )h=1

∏Hjgg

ji ji jif(y |x, z ,F , x ,F ) =

where

j jg gg g

ji ji ji ji ji jif(F ) P (x|z , x ,F ) f(y |x, z ,F , x ,F ) dF .jiF

∫x=1

∑Kjgg

ji jif(y |z , x ,F ) =

(29); that is,jgg

ji jii, f(y |z , x ,F ) has a structure similar to the one described in equationAssuming that the model of interest may also contain CFactors, for each case

i=1

∏jI

j jg gg g

j j ji jif(y |z , x ,F ) = f(y |z , x ,F ).

where

j jg g

j j j jg g g gf(F ) P (x |z ) f(y |z , x ,F ) dF , (33)

jgF

∫gx =1

∑gK

jg

j jf(y |z , z ) =

model equalsThe most general Latent GOLD probability structure for a multilevel LC

g gβ , and λ .

gj jgg gvariates) are denoted by x , F , and z , and group-level parameters by γ ,

group-level continuous factors (GCFactors), and group-level covariates (GCo-tities will be referred to using a superscript g: Group-level classes (GClasses),j. Rather than expanding the notation with new symbols, group-level quan-

jresponses of case i in group j, and with y the responses of all cases in groupji(at replication) t of case i belonging to group j, with y the full vector of

jitnumber of cases in group j. With y we denote the response on indicatorjThe index j is used to refer to a particular group and I to denote the

the multiple responses of an individual at the various time points.be the multiple time points within individuals and replications (or indicators)be individuals, for example, in longitudinal applications. “Cases” would thenvariable. It should, however, be noted that higher-level observations can also

records of cases belonging to the same group are connected by the Group IDtions will be referred to as groups and lower-level observations as cases. The

Assigned Reading: Technical G uide: Sec. 10

19

• containing GClasses (xg) and/or (at most three mutually independent)GCFactors (Fg

j ),

• containing GCovariates zgj affecting the group classes xg,

• assuming that the Ij observations for the cases belonging to group jare independent of one another given the GClasses and GCFactors,

• allowing the GClasses and GCFactors to affect the case-level latentclasses x and/or the responses yji.

GCFactors enter in exactly the same manner in the linear predictors forthe various types of response variables as case-level CFactors. We will refer totheir coefficients as λt,g

d (Cluster and DFactor) and λgxqd (Regression), where

we add a subscript m when needed. GCFactors can also be used in themodel for the latent classes. These terms are similar to those for nominal(Cluster and Regression) or ordinal (DFactor) dependent variables. We willdenote a GCFactor effect on the latent classes as λ0,g

xrd, 0 ≤ 1 ≤ R, where thesuperscript 0 refers to the model for the latent classes.

GClasses enter in the linear predictors of the models for the indica-tors as βt,g

xg and in the one of the model for the dependent variable asβg

x0,xg +∑Q

q=1 βgxq,xg ·zpred

jitq . Inclusion of GClasses in the model for the Clusters,DFactors, or Classes implies that the γ parameters become GClass depen-dent; that is ηx|zji,x

g = γxg ,x0 +∑R

r=1 γxg ,xr · zcovjir . Note that this is similar to

a LC Regression analysis, where xg now plays the role of x, and x the role ofa nominal or ordinal y variable.

The remaining linear predictor is the one appearing in the multinomiallogistic regression model for the GClasses. It has the form ηxg |zg

i= γg

xg ,0 +∑Rg

r=1 γgxg ,r ·zg,cov

jr . This linear predictor is similar to the one for the Clusters orClasses (in a standard LC model), showing that GCovariates may be allowedto affect GClasses in the same way that covariates may affect Classes.

Below we will describe the most relevant special cases of this very generallatent variable model,43 most of which were described in Vermunt (2002b,2003, 2004, and 2005) and Vermunt and Magidson (2005b). We then devotemore attention to the expressions for the exact forms of the various linearpredictors in models with GClasses, GCFactors, and GCovariates.

43In fact, the multilevel LC model implemented in Latent GOLD is so general that manypossibilities remain unexplored as of this date. It is up to Latent GOLD Advanced usersto further explore its possibilities.

20

Model restrictions Both in Cluster and in DFactor models, one canequate the λ’s are across indicators of the same type for selected CFactors(“Equal Effects”) and/or fix some of the λ’s to zero. The same applies tothe β’s corresponding to the GClasses.

In Regression, one can use the parameter constraints “Class Indepen-dent”, “No Effect”, and “Merge Effects”, implying equal λ’s (β’s) amongall Classes, zero λ’s (β’s) in selected Classes, and equal λ’s (β’s) in selectedClasses.

ML (PM) estimation and technical settings Similar to what was dis-cussed in the context of CFactors, with GCFactors, the marginal densityf(yj|zj) described in equation (33) is approximated using Gauss-Hermitequadrature. With three GCFactors and B quadrature nodes per dimension,the approximate density equals

f(yj|zj, zgj ) ≈

Kg∑xg=1

B∑b1=1

B∑b2=1

B∑b3=1

P (xg|zgj ) f(yj|zj, x

g, F gb1

, F gb2

, F gb3

) P gb1

P gb2

P gb3

.

ML (PM) estimates are found by a combination of the upward-downwardvariant of the EM algorithm developed by Vermunt (2003, 2004) and Newton-Raphson with analytic first-order derivatives.44

The only new technical setting in multilevel LC models is the same one asin models with CFactors; that is, the number of quadrature nodes to be usedin the numerical integration concerning the GCFactors. As already explainedin the context of models with CFactors, the default value is 10, the minimum2, and the maximum 50.

10.2 Application Types

10.2.1 Two-level LC or FM model

The original multilevel LC model described by Vermunt (2003) and Vermuntand Magidson (2005b) was meant as a tool for multiple-group LC analysisin situations in which the number of groups is large. The basic idea wasto formulate a model in which the latent class distribution (class sizes) isallowed to differ between groups by using a random-effects approach rather

44Numeric second-order derivatives are computed using the analytical first-order deriva-tives.

21

than by estimating a separate set of class sizes for each group – as is done ina traditional multiple-group analysis.

When adopting a nonparametric random-effects approach (using GClasses),one obtains the following multilevel LC model:

f(yj) =Kg∑

xg=1

P (xg)

Ij∏i=1

K∑x=1

P (x|xg)T∏

t=1

f(yjit|x)

,

in which the linear predictor in the logistic model for P (x|xg) equals ηx|xg =γxg ,x0. Here, we are in fact assuming that the intercept of the model for thelatent classes differs across GClasses.

When adopting a parametric random-effects approach (GCFactors), oneobtains

f(yj) =∫ ∞

−∞f(F g

1j)

Ij∏i=1

K∑x=1

P (x|F g1j)

T∏t=1

f(yjit|x)

dF g1j,

where the linear term in the model for P (x|F g1j) equals ηx|F g

1j= γx0+λ0,g

x01 ·Fg1j.

Note that this specification is the same as in a random-intercept model fora nominal dependent variable.

Vermunt (2005) expanded the above parametric approach with covariatesand random slopes, yielding a standard random-effects multinomial logisticregression model, but now for a latent categorical outcome variable. Withcovariates and multiple random effects, we obtain

f(yj|zj) =∫Fg

j

f(Fgj )

Ij∏i=1

K∑x=1

P (x|zji,Fgj )

T∏t=1

f(yjit|x)

dFgj ,

where the linear predictor for x equals

ηx|zji,Fgj

= γx0 +R∑

r=1

γxr · zcovjir +

Dg∑d=1

λ0,gx0d · F

gdj +

Dg∑d=1

R∑r=1

λ0,gxrd · F

gdj · zcov

jir ,

Whereas in the Cluster and Regression Modules, this is a random-effectsmultinomial logistic regression model, in the DFactor Module, we use arandom-effects ordinal logistic regression model for each of the discrete ordi-nal factors.

22

Also when adopting a nonparametric random-effects approach, one mayinclude covariates in the multilevel LC model; that is,

ηx|zji,xg = γxg ,x0 +

R∑r=1

γxg ,xr · zcovjir .

This yields a model for the latent classes in which the intercept and thecovariate effects may differ across GClasses. In fact, we have a kind of LCRegression structure in which the latent classes serve as a nominal dependentvariable and the GClasses as latent classes.

An important extension of the above nonparametric multilevel LC modelsis the possibility to regress the GClasses on group-level covariates. This partof the model has the same form as the multinomial logistic regression modelfor the Clusters or Classes in a standard LC or FM model.

In the Cluster and DFactor Modules, it is possible to allow GCFactorsand/or GClasses to have direct effects on the indicators. As suggested byVermunt (2003), this is a way to deal with item bias. Below, we will discussvarious other applications of this option.

10.2.2 LC (FM) regression models for three-level data

Another application type of the Latent GOLD multilevel LC option is three-level regression modeling (Vermunt, 2004). This application type concernsthe Regression Module.

A three-level LC (FM) Regression model would be of the form

f(yj|zj) =Kg∑

xg=1

P (xg)

Ij∏i=1

K∑x=1

P (x)Ti∏

t=1

f(yjit|x, zpredjit , xg)

.

Suppose we have a LC Regression model for a binary outcome variable. Thesimplest linear predictor in a model that includes GClasses would then be

ηzjit,x,xg = βx0 +Q∑

q=1

βxq · zpredjitq + βg

0,xg ,

which is a model in which (only) the intercept is affected by the GClasses. Amore extended model is obtained by assuming that also the predictor effectsvary across GClasses; that is,

ηzjit,x,xg = βx0 +Q∑

q=1

βxq · zpredjitq + βg

0,xg +Q∑

q=1

βgq,xg · zpred

jitq .

23

for the first GClass, or are equal to zero for the last GClass.

the dependent variable either sum to zero across GClasses, are equal to zero

0 ≤ q ≤ Q and 1 ≤ x ≤ K. In other words, the parameters in the model forgxqKxq1gxq,x

g g gβ = 0, β = 0, or β = 0, forgx =1

gK∑the most general model, this isstraints have to be imposed on the parameters involving the GClasses. InIt should be noted that in each of the above three models, identifying con-

gxq,xgx0,xjitq jitqpred g g pred

q=1 q=1

∑ ∑Q Q

gjitz ,x,x x0 xqη = β + β · z + β + β · z .

teractions. Such a model is defined as

gto be Class dependent, which implies including Classes-GClasses (x-x ) in-The most extended specification is obtained if all the effects are assumed

dictors that change values across cases to depend on the GClasses.change values across replications to be Class dependent and effects of pre-In practice, it seems to be most natural to allow effects of predictors that

and/or discrete random effects (GClasses).This variation can be modelled using continuous random effects (GCFactors)cept and possibly also the time slope is allowed to vary across individuals.a LC growth model: class membership depends on time, where the inter-regression model for the (time-specific) latent classes will have the form ofof one of the other types of Latent GOLD models. The multinomial logisticmodel for the time points may have the form of a LC Cluster model, but alsoand the index j for the cases (time points are nested within cases). The LCcation of such a model would involve using the index i for the time pointschange over time could be described using a (LC) growth model. Specifi-to build a time-specific latent classification, while the pattern of (latent)sponse variables) for each time point. The multiple responses could be usedSuppose one has a longitudinal data set containing multiple indicators (re-

10.2.4 LC growth models for multiple indicators

24

USER’S GUIDE

Jeroen K. Vermunt

& Jay Magidson

LATENT GOLD® 4.0

Statistical Innovations

Thinking outside the brackets!TM25

ClassPred Tab: Restricting Cases Known (Not) to Belong to a Certain Class orClasses

With this option one can specify that one or more specific cases can belong to a certain class or certain classesonly. To use this feature, select a variable from the list box in the ClassPred Tab to be used as the Known ClassIndicator and click Known Class. The variable moves to the Known Class Indicator Box and the Assignment Tablebecomes active. For each category of the Known Class indicator you then specify to which classes the cases withthat category code may belong (or not belong) using the Assignment Table. For example, Figure 5-12 illustratesa 4-Cluster model (4 columns) where the variable 'classind' is used as the Known Class Indicator. Cases for which'classind=1' are allowed to be in cluster 1 only; those for which 'classind=2' are allowed to be in cluster 2 only; allother cases ('classind=3') may be assigned to any of the 4 clusters.

Figure 5-12. ClassPred Tab for Cluster Model with a Known Class Indicator

This option is useful if you have a priori class membership information for some cases (pre-assigned or pre-clas-sified cases) or if membership to certain classes is very implausible for some combinations of observed scores.

KNOWN CLASS - CLASS INDICATOR

In applications where a subset of the cases are known with certainty not to belong to a particular class, or particular classes, you can take advantage of this information to restrict their posterior membership probability to0 for one or more classes and hence classify these cases into one of the remaining class(es) with a total probability = 1. This feature allows more control over the segment definitions to ensure that the resulting classes

CHAPTER 5. BASIC STEPS FOR MODEL DEVELOPMENT

26

are most meaningful. Common applications include:

1) using new data to refine old segmentation models while maintaining the segment classifications ofthe original sample

2) archetypal analysis - define class membership a priori based on extreme response patterns thatreflect theoretical "archetypes"

3) partial classification -- high cost (or other factors) may preclude all but a small sample of cases frombeing classified with certainty. These cases can be assigned to their respective classes with 100%certainty, and the remaining would be classified by the LC model in the usual way

4) certain cases may be known to be "type 1 OR type 2" (e.g., 'clinically depressed' or 'troubled'). Byexcluding such cases from being in say class 3 = 'healthy', such cases can be pre-assigned to bein class 1 or 2, while additional cases may be freely classified into any class

5) post-hoc refinement of class assignment where modal assignment for certain cases is judged to beimplausible based on the desired interpretation of the classes.

In addition, this option may also be used to specify multiple group models by including the group variable as botha Known Class Indicator and as an active covariate. For further details of this, see section 2.5 of Technical Guide.

Note: The Known Class option is not available in cluster and regression if the Range option hasbeen used in the Variables Tab.

For DFactor models, this option applies only to levels of DFactor 1.

To select known classes (clusters/classes/DFactor1 levels):

Select one variable from those appearing in Variables List Box(located in the upper left-hand portion of the ClassPred Tab).Variables appearing here are those that have not been previouslyselected as Indicators or Covariates.

Click Known Class to move that variable to the Known Class Boxand the class assignment window beneath the Known Class Boxbecomes active.

A separate row appears for each category/code/value taken on bythe known class indicator

A separate column appears for each class.

Click on the appropriate boxes to select or deselect the possibleassignment of the categories to certain classes.

A checkmark off means that the posterior membership probability is restricted to zero for that class for cases inthat category of the known class indicator.

By default, the checks are assigned as follows:

For a K-class model, a category with a code of K on the Known Class Indicator is assigned to only class

LATENT GOLD® 4.0 USER'S GUIDE

27

K. Categories coded less than 1, greater than K or missing are assigned to all classes (i.e., no restric-tions). Missing values are not shown in the table.

Note: For the example in Figure 5-12 above, all cases are coded either 1, 2 or 3 on the variable'classind' (i.e., no missing values). Those coded 'classind=1' and 'classind=2' are main-tained at their default specifications on the table, while the default specification for casescoded 'classind=3' was changed (from 'cluster 3 only' to 'any cluster' -- all 4 cluster columnschecked). This specification would be obtained by default if those coded '3' on the classindvariable were instead coded as 'missing'. In this situation, the table would differ from thatshown in Figure 5-12, in that the 3rd row of the table would not appear, since that categorywould be coded 'missing'.

For further information, see section 2.5 of the Technical Guide.

CHAPTER 5. BASIC STEPS FOR MODEL DEVELOPMENT

28