William GreeneDepartment of EconomicsStern School of Business

New York University

Some Applications of Latent Class ModelingIn Health Economics

Outline

• Theory: Finite Mixture and Latent Class Models • Applications

o Obesityo Self Assessed Healtho Efficiency of Nursing Hommes

Latent Classes• A population contains a mixture of individuals

of different types (classes)• Common form of the data generating

mechanism within the classes• Observed outcome y is governed by the

common process F(y|x,j )

• Classes are distinguished by the parameters, j.

A Latent Class Hurdle NB2 Model

• Analysis of ECHP panel data (1994-2001)• Two class Latent Class Model

o Typical in health economics applications• Hurdle model for physician visits

o Poisson hurdle for participation and negative binomial intensity given participation

o Contrast to a negative binomial model

Density? Note significant mass below zero. Not a gamma or lognormal or any other familiar density.

How Finite Mixture Models Work

ˆ

1 y - 7.05737 1 y - 3.25966F(y) =.28547 +.71453

3.79628 3.79628 1.81941 1.81941

Find the ‘Best’ Fitting Mixture of Two Normal Densities

1000

2 i jji=1 j=1

j j

y -μ1 LogL = log π

σ σ

Maximum Likelihood Estimates

Class 1 Class 2

Estimate Std. Error Estimate Std. error

μ 7.05737 .77151 3.25966 .09824

σ 3.79628 .25395 1.81941 .10858

π .28547 .05953 .71453 .05953

Mixing probabilities .715 and .285

Approximation

Actual Distribution

A Practical Distinction• Finite Mixture (Discrete Mixture):

o Functional form strategyo Component densities have no meaning o Mixing probabilities have no meaningo There is no question of “class membership”o The number of classes is uninteresting – enough to get a good fit

• Latent Class:o Mixture of subpopulationso Component densities are believed to be definable “groups”

(Low Users and High Users in Bago d’Uva and Jones application)o The classification problem is interesting – who is in which class?o Posterior probabilities, P(class|y,x) have meaningo Question of the number of classes has content in the context of the analysis

Why Make the Distinction?

• Same estimation strategy• Same estimation results• Extending the latent class model

o Allows a rich, flexible model specification for behavioro The classes may be governed by different processes

Antecedents• Pearson’s 1894 study of crabs in Naples – finite mixture

of two normals – seeking evidence of two subspecies.• Some of the extensions I will note here have already

been (implicitly) employed in earlier literature. o Different underlying processeso Heterogeneous class probabilitieso Correlations of unobservables in class probabilities with

unobservables in structural (within class) models• One has not and is not widespread (yet)

o Cross class restrictions implied by the theory of the model

Switching Regressions• Mixture of normals with heterogeneous mean

o y ~ N(b0x0,02) if d=0,

y ~ N(b1x1,12) if d=1, P(d=1)=(cz).

o d is unobserved (latent switching). • Becomes a latent class model when regime 0 is

a demand function and regime 1 is a supply function, d=0 if excess supply

• The two regression equations may involve different variables – a true latent class model

Endogenous Switching (ca.1980)0 0 0

1 1 1

Regime 0: y

Regime 1: y

Regime Switch: d* = , d = 1[d* > 0]

Regime 0 governs if d = 0, Probability = 1- ( )

Regime 1 governs if d = 1, Probability = ( )

Endogenous

x

x

z

z

z

i i i

i i i

i

i

i

u

20 0

20 1

0 0 0 1 1

0

Switching: ~ 0 , ?

0 1

This is a latent class model with different processes in the two classes.

There is correlation between the unobservabl

i

i

i

N

es that govern the class

determination and the unobservables in the two regime equations.

Not identified. Regimes do not coexist.

Outcome Inflation Models• Lambert 1992, Technometrics. Quality control problem. Counting defects

per unit of time on the assembly line. • How to explain the zeros; is the process under control or not?• Two State Outcome: Prob(State=0)=R, Prob(State=1)=1-R

o State=0, Y=0 with certaintyo State=1, Y ~ some distribution support that includes 0, e.g., Poisson.o Prob(State 0|y>0) = 0o Prob(State 1|y=0) = (1-R)f(0)/[R + (1-R)f(0)]o R = Logistic probability

• “Nonstandard” latent class model• Recent users have extended this to “Outcome Inflated Models,” e.g., twos

inflation in models of fertility; inflated responses in health status.

Split Population Survival Models• Schmidt and Witte 1989 study of recidivism• F=1 for eventual failure, F=0 for never fail.

F is unobserved. P(F=1)=, P(F=0)=1- • C=1 for a recidivist, observed. Prob(F=1|C=1) = 1.• Density for time until failure actually occurs is × g(t|F=1).• Density for observed duration (possibly censored)

o P(C=0)=(1- ) + (G(T|F=1)) (Observation is censored)o Density given C=1 = g(t|F=1)o G=survival function, t=time of observation.

• Unobserved F implies a latent population split.• They added covariates to : i =logit(zi).• Different models apply to the two latent subpopulations.

Variations of Interest• Heterogeneous priors for the class probabilities

• Correlation of unobservables in class probabilities with unobservables in regime specific models

• Variations of model structure across classes

• Behavioral basis for the mixed models with implied restrictions

Applications• Obesity: Heterogeneous class probabilities,

generalized ordered choice; Endogenous class membership

• Self Assessed Health: Heterogeneous subpopulations; endogenous class membership

• Cost Efficiency of Nursing Homes: theoretical restrictions on underlying models

Modeling Obesity with a Latent Class Model

Mark HarrisDepartment of Economics, Curtin University

Bruce HollingsworthDepartment of Economics, Lancaster University

Pushkar MaitraDepartment of Economics, Monash University

William GreeneStern School of Business, New York University

300 Million People Worldwide. International Obesity Task Force: www.iotf.org

Costs of Obesity• In the US more people are obese than smoke or use

illegal drugs• Obesity is a major risk factor for non-communicable

diseases like heart problems and cancer• Obesity is also associated with:

o lower wages and productivity, and absenteeismo low self-esteem

• An economic problem. It is costly to society:o USA costs are around 4-8% of all annual health care

expenditure - US $100 billion o Canada, 5%; France, 1.5-2.5%; and New Zealand 2.5%

Measuring Obesity

• An individual’s weight given their height should lie within a certain rangeo Body Mass Index (BMI)o Weight (Kg)/height(Meters)2

• WHO guidelines: o Underweight BMI < 18.5o Normal 18.5 < BMI < 25o Overweight 25 < BMI < 30o Obese BMI > 30o Morbidly Obese BMI > 40

Two Latent Classes: Approximately Half of European Individuals

Modeling BMI Outcomes• Grossman-type health production function

Health Outcomes = f(inputs)• Existing literature assumes BMI is an ordinal, not cardinal,

representation of individuals.o Weight-related health statuso Do not assume a one-to-one relationship between BMI levels and

(weight-related) health status levels• Translate BMI values into an ordinal scale using WHO guidelines• Preserves underlying ordinal nature of the BMI index but recognizes

that individuals within a so-defined weight range are of an (approximately) equivalent (weight-related) health status level

Conversion to a Discrete Measure

• Measurement issues: Tendency to under-report BMIo women tend to under-estimate/report weight; o men over-report height.

• Using bands should alleviate this• Allows focus on discrete ‘at risk’ groups

A Censored Regression Model for BMI

Simple Regression Approach Based on Actual BMI:BMI* = x′ + , ~ N[0,2]

Interval Censored Regression ApproachWT = 0 if BMI* < 25 Normal

1 if 25 < BMI* < 30 Overweight 2 if BMI* > 30 Obese

Inadequate accommodation of heterogeneity Inflexible reliance on WHO classification Rigid measurement by the guidelines

An Ordered Probit ApproachA Latent Regression Model for “True BMI”

BMI* = x′ + , ~ N[0,σ2], σ2 = 1 “True BMI” = a proxy for weight is

unobservedObservation Mechanism for Weight Type

WT = 0 if BMI* < 0 Normal 1 if 0 < BMI* < Overweight

2 if < BMI* Obese

Heterogeneity in the BMI Ranges

• Boundaries are set by the WHO narrowly defined for all individuals

• Strictly defined WHO definitions may consequently push individuals into inappropriate categories

• We allow flexibility at the margins of these intervals

• Following Pudney and Shields (2000) therefore we consider Generalised Ordered Choice models - boundary parameters are now functions of observed personal characteristics

Generalized Ordered Probit Approach

A Latent Regression Model for True BMIBMIi* = ′xi + i , i ~ N[0,σ2], σ2 = 1

Observation Mechanism for Weight TypeWTi = 0 if BMIi* < 0 Normal

1 if 0 < BMIi* < i(wi) Overweight

2 if (wi) < BMIi* Obese

Latent Class Modeling• Several ‘types’ or ‘classes. Obesity be due to genetic reasons

(the FTO gene) or lifestyle factors

• Distinct sets of individuals may have differing reactions to various policy tools and/or characteristics

• The observer does not know from the data which class an individual is in.

• Suggests a latent class approach for health outcomes(Deb and Trivedi, 2002, and Bago d’Uva, 2005)

Latent Class Application

• Two class model (considering FTO gene):o More classes make class interpretations much

more difficulto Parametric models proliferate parameters

• Endogenous class membership: Two classes allow us to correlate the equations driving class membership and observed weight outcomes via unobservables.

• Theory for more than two classes not yet developed.

Heterogeneous Class Probabilities

• j = Prob(class=j) = governor of a detached natural process. Homogeneous.

• ij = Prob(class=j|zi,individual i)Now possibly a behavioral aspect of the process, no longer “detached” or “natural”

• Nagin and Land 1993, “Criminal Careers…

Endogeneity of Class Membership

,

,

Class Membership: C* = , C = 1[C* > 0] (Probit)

BMI|Class=0,1 BMI* = , BMI group = OP[BMI*, ( )]

10Endogeneity: ~ ,

10

Bivaria

z

x w

i i

c i c i c i

i c

c i c

u

uN

te Ordered Probit (one variable is binary).

Full information maximum likelihood.

Model Components• x: determines observed weight levels within classes For observed weight levels we use lifestyle factors such as

marital status and exercise levels• z: determines latent classes For latent class determination we use genetic proxies such as

age, gender and ethnicity: the things we can’t change• w: determines position of boundary parameters within classes For the boundary parameters we have: weight-training

intensity and age (BMI inappropriate for the aged?) pregnancy (small numbers and length of term unknown)

Data• US National Health Interview Survey (2005);

conducted by the National Center for Health Statistics

• Information on self-reported height and weight levels, BMI levels

• Demographic information• Split sample (30,000+) by gender

Outcome Probabilities• Class 0 dominated by normal and overweight probabilities ‘normal weight’ class• Class 1 dominated by probabilities at top end of the scale ‘non-normal weight’• Unobservables for weight class membership, negatively correlated with those

determining weight levels:

Normal Overweight Obese

Normal Overweight Obese

Class 0

Class 1

Classification (Latent Probit) Model

BMI Ordered Choice Model• Conditional on class membership, lifestyle factors• Marriage comfort factor only for normal class women• Both classes associated with income, education• Exercise effects similar in magnitude• Exercise intensity only important for ‘non-normal’ class:• Home ownership only important for .non-normal.class, and negative: result of

differing socieconomic status distributions across classes?

Effects of Aging on Weight Class

Effect of Education on Probabilities

Effect of Income on Probabilities

Inflated Responses in Self-Assessed Health

Mark HarrisDepartment of Economics, Curtin University

Bruce HollingsworthDepartment of Economics, Lancaster University

William GreeneStern School of Business, New York University

Introduction

• Health sector an important part of developed countries’ economies: E.g., Australia 9% of GDP

• To see if these resources are being effectively utilized, we need to fully understand the determinants of individuals’ health levels

• To this end much policy, and even more academic research, is based on measures of self-assessed health (SAH) from survey data

SAH vs. Objective Health MeasuresFavorable SAH categories seem artificially high. 60% of Australians are either overweight or obese (Dunstan et. al, 2001) 1 in 4 Australians has either diabetes or a condition of impaired glucose metabolism Over 50% of the population has elevated cholesterol Over 50% has at least 1 of the “deadly quartet” of health conditions (diabetes, obesity, high blood pressure, high cholestrol) Nearly 4 out of 5 Australians have 1 or more long term health conditions (National Health Survey, Australian Bureau of Statistics 2006) Australia ranked #1 in terms of obesity ratesSimilar results appear to appear for other countries

SAH vs. Objective Health

Our objectives

1. Are these SAH outcomes are “over-inflated”

2. And if so, why, and what kinds of people are doing the over-inflating/mis-reporting?

HILDA Data

The Household, Income and Labour Dynamics in Australia (HILDA) dataset:1. a longitudinal survey of households in Australia2. well tried and tested dataset3. contains a host of information on SAH and other health

measures, as well as numerous demographic variables

Self Assessed Health• “In general, would you say your health is: Excellent, Very

good, Good, Fair or Poor?"• Responses 1,2,3,4,5 (we will be using 0,1,2,3,4)• Typically ¾ of responses are “good” or “very good” health; in

our data (HILDA) we get 72%• Similar numbers for most developed countries• Does this truly represent the health of the nation?

Recent Literature - Heterogeneity• Carro (2012)

o Ordered SAH, “good,” “so so,” bad”o Two effects: Random effects (Mundlak) in latent

index function, fixed effects in threshold• Schurer and Jones(2011)

o Heterogeneity, panel data, o “Generalized ordered probit:” different slope

vectors for each outcome.

Kerkhofs and Lindeboom, Health Economics, 1995

• Subjective Health Measures and State Dependent Reporting Errors• Incentive to “misreport” depends on employment status: employed,

unemployed, retired, disabled• Ho = an objective, observed health indicator• H* = latent health = f1(Ho,X1)• Hs = reported health = f2(H*,X2,S)

o S = employment status, 4 observed categorieso Ordered choice, o Boundaries depend on S,X2; Heterogeneity is induced by incentives produced by

employment status

A Two Class Latent Class Model

True Reporter Misreporter

Reporter Type Model

*

= 1 if r* > 0 True reporter

0 if r* 0 Misreporter

r is unobserved

r r rr x

r

Y=4

Y=3

Y=2

Y=1

Y=0

Pr(true,y) = Pr(true) * Pr(y | true)

• Mis-reporters choose either good or very good• The response is determined by a probit model

* m m mm x

Y=3

Y=2

Observed Mixture of Two Classes

Pr( ) Pr( ) Pr( | ) Pr( ) Pr( | )y true y true misreporter y misreporter

Who are the Misreporters?

Priors and PosteriorsM=Misreporter, T=True reporter

Priors : Pr( ) ( ), Pr( ) ( )

Posteriors:

Noninflated outcomes 0, 1, 4

Pr( | 0,1,4) 0, Pr( | 0,1,4) ( )

Inflated outcomes 2, 3

Pr(

r r

r

M x T x

M y T y x

Pr( 2 | )Pr( )| 2)

Pr( 2 | )Pr( ) Pr( 2 | )Pr( )

y M MM y

y M M y T T

General Results

Poor Fair Good Very Good Excellent0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Sample

Predicted

Mis-Reporting

Whither the EM Algorithm?

• An Algorithm, not a modelo E step: Compute posterior probabilities, ij

o M step: In each class, estimate class specific parameters using a (class and individual) weighted log likelihood, using the posteriors as weights.

• Cannot impose cross class restrictions• Cannot model endogeneity

Latent Class Efficiency Studies• Battese and Coelli – growing in weather

“regimes” for Indonesian rice farmers

• Kumbhakar and Orea – cost structures for U.S. Banks

• Greene (Health Economics, 2005) – revisits WHO Year 2000 World Health Report

Studying Economic Efficiency in Health Care

• Hospital and Nursing Homeo Cost efficiencyo Role of quality (not studied today)

• Agency for Health Reseach and Quality (AHRQ)

Stochastic Frontier Analysis• logC = f(output, input prices, environment) + v + u• ε = v + u

o v = noise – the usual “disturbance”o u = inefficiency

• Frontier efficiency analysiso Estimate parameters of modelo Estimate u (to the extent we are able – we use E[u|ε])o Evaluate and compare observed firms in the sample

Nursing Home Costs

• 44 Swiss nursing homes, 13 years• Cost, Pk, Pl, output, two environmental variables• Estimate cost function• Estimate inefficiency

Estimated Cost Efficiency

Inefficiency?

• Not all agree with the presence (or identifiability) of “inefficiency” in market outcomes data.

• Variation around the common production structure may all be nonsystematic and not controlled by management

• Implication, no inefficiency: u = 0.

A Two Class Model

• Class 1: With Inefficiencyo logC = f(output, input prices, environment) + vv + uu

• Class 2: Without Inefficiencyo logC = f(output, input prices, environment) + vv o u = 0

• Implement with a single zero restriction in a constrained (same cost function) two class model

• Parameterization: λ = u /v = 0 in class 2.

LogL= 464 with a common frontier model, 527 with two classes

Conclusion

Latent class modeling provides a rich, flexible platform for behavioral model building.

Thank you.