Assessing Estimability of Latent Class Models Using a Bayesian

Estimation Approach

Elizabeth S. Garrett

Scott L. Zeger

Johns Hopkins University

Departments of Oncology and Biostatistics

ENARMarch 26, 2001Charlotte, NC

Motivation: We would like to investigate how many classes of depression exist using a latent class model applied to an epidemiologic sample of mental health symptoms.

Issue: Latent class models require a “large” sample size to be estimable.

Questions:

Is our sample size large enough?

How many classes can we fit “reliably” given the amount of data that we have?

Latent Class Model Overview(binary indicator case)

• There are M classes of depression. m represents the proportion of individuals in the population in class m (m = 1,…,M).

• Each individual is a member of one of the M classes, but we do not know which. The latent class of individual i is denoted by i (i 1,…M).

• Symptom prevalences vary by class. The prevalence for symptom j in class m is denoted by pmj.

• Given class membership, the symptoms are independent.

Likelihood Function

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For individual i,

Two ScenariosA latent class model with M classes may be only

weakly estimable if

(1) M classes are too many (over-parameterization). “True” number of classes is < M.

(2) There are “truly” M classes, but there is not enough data to identify all of the classes. Better to use fewer classes than to use the M class model with weakly estimated parameters.

Estimation of LC Models

• How large is large enough?

• Depends on– true class sizes

– prevalence of items/symptoms

– measurement error in items/symptoms

Example 1: Three class model• N = 500.

• True class sizes are 30%, 40%, 30%.

• Symptom prevalences range from 30% to 80%.

Example 2: Three class model• N = 500.

• True class sizes are 80%, 18%, 2%.

• Symptom prevalences range from 3% to 10%.Same sample size, but are

both estimable?

Distinction and Definitions

• Statistical identifiability refers to the “difficulty of distinguishing among two or more explanations of the same empirical phenomena.”1

• Not so much a concept that stems from limited data, but the ability to distinguish between explanations even if we had unlimited data.

• Classic example:– X is random variable such that E(X) = 1 - 2

– observations of X allow us to identify 1 - 2

– but, we cannot identify 1 or 2 regardless of the amount of information we collect on X.

– Infinitely many values of 1 and 2 can give rise to the same value of X

1 Franklin Fisher, “Statistical Identifiability”, International Encyclopedia of Statistics, ed. Kruskal and Tanur, 1977.

“Identifiability” in Latent Class Models

• Latent class issues with identifiability are “twofold”.

• Still have the statistical identifiability to contend with.– Local identifiability issues have been well-defined– Goodman, 1974, Biometrika– McHugh, 1956, Biometrika– Bandeen-Roche et al., 1997, JASA

• To distinguish issues stemming from limited data from classical identifiability we use alternative terminology: estimability.

• We say that a model is estimable in this context if there is enough data to uniquely estimate all parameters with some degree of certainty.

Local IdentifiabilityDefine p0 to be a vector of

parameters which defines the LC model.

If,

for all y and p in the neighborhood of p0, then we say that L is locally identifiable.

Maximum likelihood estimation is concerned with local identifiability

L y p L y p p p( ; ) ( ; )0 0 NOT locally identified!

Parameter Estimate

p0 p1

L(Y

|p)

Weak Identifiability / Weak Estimability• Bayesian concept

• Assume we partition parameter vector = (1, 2).

• If , then we say that 2 is not estimable.

• The data (Y) provides no information about 2 given 1.

• If most of the information we have about 2 is supplied by its prior distribution then:

• In words, if the prior distribution of 2 is approximately the same as its posterior, then we say that 2 is only weakly estimable or weakly identifiable.

P Y P( | , ) ( | ) 2 1 2 1

P Y P( | ) ( ) 2 2

Estimation Approaches

Maximum Likelihood Approach:Find estimates of p, , and that are most consistent with

the data that we observe conditional on number of classes, M.

Often used: EM algorithm (iterative fitting procedure)

Bayesian Approach:

Quantify beliefs about p, , and before and after observing data.

Often used: Gibbs sampler, MCMC algorithm

Bayesian Model Review• Every parameter, , in a Bayesian model has a prior

distribution associated with it:• The resulting posterior distribution is a combination of the

the prior distribution and the the likelihood function:

If there is a lot of information in the data, then the likelihood will

provide most of the information that determines the posterior.

If there is not a lot of information in the data, then the prior will provide most of the information that determines the posterior

P ( )

P Y P L Y( | ) ( ) ( | )

Estimability Assessment ApproachCompare posterior distribution to the prior distribution

– visual inspection: picture tells us how much information is coming from the prior and how much from the likelihood.

– quantify similarity/difference by a statistic (e.g. percent overlap). Bayes factor is also related to this idea.

p

De

nsi

ty

p p

Simulated Data Examples

Two class model

Class 1 Class 2 0.60 0.40

p1 0.75 0.25

p2 0.75 0.25

p3 0.25 0.75

p4 0.25 0.75

p5 0.25 0.75

N 500

LCED for 2 class data

Simulated Data Examples

Three class modelClass 1 Class 2 Class 3

0.30 0.40 0.30p1 0.10 0.75 0.90p2 0.10 0.50 0.90p3 0.10 0.50 0.90p4 0.10 0.25 0.90p5 0.10 0.25 0.90N 1000

LCED for 3 class data

Latent Class Analysis in Mental Health Research

Author Disorder Year EpidemiologicStudy

N # of classes

Eaton et al depression 1989 Yes 3198 3

Kendler et al. depression 1996 Yes 2058 7

Sullivan et al. depression 1998 No 2836 6

Parker et al. depression 1998 No 185 4

Kendler et al. psychoses 1998 No 343 6

Bucholz et al. alcoholism 1996 No 2551 4

Sullivan et al. bulimia nervosa 1998 No 3794 4

Sham et al. schizophrenia 1996 No 447 3

Eaves at al. conduct disorder 1993 No 676 4

Neuman et al. ADHD 1999 No 3493 6

Epidemiologic Catchment Area (ECA) Study

• Goal: To obtain epidemiologic sample on mental health data.

• Population: Community dwelling individuals over age 18 in five sites in the US.

• Instrument: Diagnostic Interview Schedule (DIS) includes 17 depression symptoms.

• Our sample:– Data from 1981, Baltimore site only

– Full information on depression symptoms as defined in the DSM-III on 2938 individuals.

Depression Symptom GroupsGroup Symptoms Prevalence

1 Loss of appetiteWeight lossWeight gain

0.11

2 InsomniaHypersomnia

0.14

3 Slow movementrestlessness

0.07

4 Disinterest in sex 0.04

5 Reduced energy/fatigue 0.09

6 Guilty/sinful 0.04

7 Reduced concentrationSlow thoughts

0.06

8 Thoughts of deathWant to dieSuicidal thoughtsSuicide attempt

0.12

Dysphoria 0.06

LCED for ECA data

Conclusions from ECA example

• The four class model is not an appropriate model for this data.

• This could be due to (1) Four class model is an overparameterization

(2) More than three classes are needed to describe depression, but the data set is too small to estimate more classes.

Aside: Looking at 2 statisticsMay say just compare 2 statistics from a maximum likelihood

estimation procedure

BUT! 2 relies on assumption that most patterns are relatively prevalent

which is not generally true.

• May see significant differences due to summing up very small differences over large number of samples.

• M and M -1 class models are not really “nested” in interpretation

Not a valid statistic for comparing LC models.

See Bayes Factor and BIC: these are better statistics for comparing LC models.

Extension: Quantifying Estimability• Calculate percent overlap between posterior and prior.

• Estimate “estimabilility index” for each class where there are K symptoms:

• Characteristics:

– 0 < < 1

– Larger numbers indicate weak estimability

• Also see: Bayes Factor

mk

K

K % o v erlap

1