Latent Class Analysis

Concept

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Theoretically, Latent Class Analysis (LCA) is a model-based statistical method for identifying unobserved (latent) subgroups within a heterogeneous population using patterns of responses on observed categorical variables. It assumes that population heterogeneity can be explained by membership of a finite number of mutually exclusive latent classes, with observed variables being conditionally independent once class membership is taken into account. LCA exists to classify individuals into homogeneous groups when the underlying structure cannot be observed directly.

Mathematically, LCA is formulated as a finite mixture model in which the probability of an observed response pattern is expressed as a weighted sum of class-specific response probabilities. Parameters are estimated by maximum likelihood, most commonly using the Expectation-Maximisation (EM) algorithm. Model selection is typically based on information criteria such as the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC), while posterior probabilities are used to assign individuals to latent classes.

In practice, LCA is implemented by specifying candidate models with varying numbers of latent classes and selecting the model that provides the best balance between statistical fit and interpretability. In health economics, LCA is used to identify patient subgroups, classify patterns of healthcare utilisation, segment treatment preferences, analyse health-related behaviours and explore heterogeneity in discrete choice experiments. The resulting class memberships can subsequently inform subgroup analyses, resource allocation and personalised healthcare strategies.

Purpose

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Used to identify unobserved population subgroups, classify individuals according to response patterns, investigate heterogeneity in healthcare populations and improve the interpretation of complex multivariable datasets in health economic research.

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Mathematical Formulae

Primary Formula

P(Y = y) = ????? �? ? ????? P(Y? = y? | C = k)

where:

• �? = probability of membership in latent class k
• K = number of latent classes
• J = number of observed variables
• C = latent class
• P(Y? = y? | C = k) = conditional response probability within class k

Supporting Formulae

Log-likelihood:

ln(L) = ?? ln[P(Y?)]

Posterior class probability:

P(C = k | Y = y) = [�? ? P(Y = y | C = k)] � P(Y = y)

Bayesian Information Criterion:

BIC = ?2ln(L) + p ln(n)

where:

• p = number of estimated parameters
• n = sample size

Related Mathematical Methods

• Finite Mixture Models
• Expectation-Maximisation Algorithm
• Maximum Likelihood Estimation
• Bayesian Information Criterion
• Akaike Information Criterion
• Latent Profile Analysis
• Growth Mixture Modelling

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Example

A health economist analyses responses from 2,000 patients on eight categorical indicators describing healthcare utilisation, medication adherence and health status.

Latent Class Analysis is fitted using models containing two to six latent classes.

The four-class model produces:

• Log-likelihood = ?6,482
• BIC = 13,145
• Average posterior class probabilities > 0.90

The identified classes represent:

• Low healthcare utilisation
• High chronic disease management
• Frequent emergency care users
• Complex multimorbidity patients

These latent classes are subsequently incorporated into a cost-effectiveness analysis to evaluate whether interventions provide different economic value across patient subgroups.

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Excel Implementation

Function	Example Formula	Health Economics Application
LN	=LN(B2)	Calculate log-likelihood components.
SUM	=SUM(C2:C1001)	Sum log-likelihood contributions across observations.
EXP	=EXP(A2)	Convert log probabilities to probabilities during calculations.
PRODUCT	=PRODUCT(D2:K2)	Calculate conditional response probabilities for an observation.
Solver	Maximise the total log-likelihood by varying model parameters.	Estimate latent class probabilities using maximum likelihood.

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VBA (Optional)

A VBA routine can automate repeated latent class model estimation across candidate numbers of classes and summarise model fit statistics for model selection.

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Sources

• Lazarsfeld PF, Henry NW. Latent Structure Analysis. Houghton Mifflin.
• McCutcheon AL. Latent Class Analysis. Sage Publications.
• Hagenaars JA, McCutcheon AL, editors. Applied Latent Class Analysis. Cambridge University Press.
• Collins LM, Lanza ST. Latent Class and Latent Transition Analysis. Wiley.
• Muth�n B, Muth�n LK. Mplus User's Guide.
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
• ISPOR Good Practice Reports.