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Exploring Heterogeneneity of Treatment Effects usingFinite Mixture Models
Partha Deb
Hunter College and the Graduate Center, CUNYNBER
August 2013
Partha Deb (Hunter College) FMM Aug 2013 1 / 37
Introduction: Heterogeneity of treatment effects
There are many intuitive reasons for expecting that treatment effectsare not constant
In most pragmatic trials in which experimental conditions are not tightlycontrolled
actual treatment is heterogeneous across sites or geographiesintensity of treatment (dose) variescompliance to treatment varies by individual or group characteristicseffects of treatment varies by biological characteristics
Partha Deb (Hunter College) FMM Aug 2013 2 / 37
Introduction: Heterogeneity of treatment effects
Heterogeneities in each of these dimensions lead to heterogeneity oftreatment effects (HTE).
When treatment effects are heterogeneous, the typically small benefitsfound in studies can be misleading because small average effects mayreflect a mixture of substantial benefits for some, little benefit for many,and even possibly harmful for a few
Heterogeneous treatment effects can be categorized into three groups(Gadbury and Iyer, 2000; Gadbury, Iyer, and Albert, 2004; Shen, et al.2013)
Partha Deb (Hunter College) FMM Aug 2013 3 / 37
Introduction: Heterogeneity of treatment effects
Let Y1 and Y0 be the potential outcomes under the intervention and thecontrol
Benefit is defined as Y1 − Y0 > 0
Harm occurs when Y1 − Y0 < 0
Treatment benefit rate TBR = Pr(Y1 − Y0 > 0)
Treatment harm rate THR = Pr(Y1 − Y0 < 0)
For the fraction 1− TBR − THR of the population, Y1 − Y0 = 0
Partha Deb (Hunter College) FMM Aug 2013 4 / 37
Introduction: Heterogeneity of treatment effects
Let Y1 and Y0 be the potential outcomes under the intervention and thecontrol
Benefit is defined as Y1 − Y0 > 0
Harm occurs when Y1 − Y0 < 0
Treatment benefit rate TBR = Pr(Y1 − Y0 > 0)
Treatment harm rate THR = Pr(Y1 − Y0 < 0)
For the fraction 1− TBR − THR of the population, Y1 − Y0 = 0
Partha Deb (Hunter College) FMM Aug 2013 4 / 37
Introduction: Heterogeneity of treatment effects
TBR and THR may be characterized by a small set of observedcovariates in some instances
TBR and THR are more likely to be determined by complexconfigurations of observed and unobserved covariates
Partha Deb (Hunter College) FMM Aug 2013 5 / 37
Introduction: Finite mixture models
When data has been drawn from a finite number of distinct populationsbut in which the sub-populations are not identified
A finite mixture model allows one to identify and estimate theparameters of interest for each sub-population in the data, not just ofthe overall mixed population
Partha Deb (Hunter College) FMM Aug 2013 6 / 37
Introduction: A graphical view
Mixture of normal densities
Partha Deb (Hunter College) FMM Aug 2013 7 / 37
Introduction: A graphical view
Mixture of normal densities
Partha Deb (Hunter College) FMM Aug 2013 8 / 37
Introduction: A graphical view
Mixture of normal densities
Partha Deb (Hunter College) FMM Aug 2013 9 / 37
Introduction: Published applications
Effects of job loss on BMI and alcohol consumption – Deb, Gallo,Ayyagari, Fletcher and Sindelar, Journal of Health Economics 2011
Effect of prenatal care on birth weight – Conway and Deb, Journal ofHealth Economics 2005
Price elasticities of medical care use – Deb and Trivedi, Journal ofApplied Econometrics 1997; Deb and Trivedi, Journal of HealthEconomics 2002
Partha Deb (Hunter College) FMM Aug 2013 10 / 37
Outline of talk
Introduction
Model
Mixture densityPropertiesComplications
Example: color of wine
Application: healthcare expenditures in the RAND Health InsuranceExperiment
Partha Deb (Hunter College) FMM Aug 2013 11 / 37
Outline of talk
Introduction
Model
Mixture densityPropertiesComplications
Example: color of wine
Application: healthcare expenditures in the RAND Health InsuranceExperiment
Partha Deb (Hunter College) FMM Aug 2013 11 / 37
Outline of talk
Introduction
Model
Mixture densityPropertiesComplications
Example: color of wine
Application: healthcare expenditures in the RAND Health InsuranceExperiment
Partha Deb (Hunter College) FMM Aug 2013 11 / 37
Model: Common mixture component densities
The density function for a C -component finite mixture is
f (y |x; θ1, θ2, ..., θC ; π1, π2, ..., πC ) =C
∑j=1
πj fj (y |x; θj )
where 0 < πj < 1, and ∑Cj=1 πj = 1
Normal (Gaussian)GammaPoissonNegative BinomialStudent-t
Partha Deb (Hunter College) FMM Aug 2013 12 / 37
Model: Basic properties
The conditional mean of the outcome in a finite mixture model is alinear combination of component means:
E(yi |xi ) =C
∑j=1
πjλij where λij = Ej (yi |xi )
The marginal effect of a covariate in a finite mixture model is a linearcombination of component marginal effects:
∂Ej (yi |xi )∂xi
=∂λij
∂xi−→ within component
∂E(yi |xi )∂xi
=C
∑j=1
πj∂λij
∂xi−→ overall
Partha Deb (Hunter College) FMM Aug 2013 13 / 37
Model: Basic properties
The conditional mean of the outcome in a finite mixture model is alinear combination of component means:
E(yi |xi ) =C
∑j=1
πjλij where λij = Ej (yi |xi )
The marginal effect of a covariate in a finite mixture model is a linearcombination of component marginal effects:
∂Ej (yi |xi )∂xi
=∂λij
∂xi−→ within component
∂E(yi |xi )∂xi
=C
∑j=1
πj∂λij
∂xi−→ overall
Partha Deb (Hunter College) FMM Aug 2013 13 / 37
Model: Basic properties
Prior probability that observation yi belongs to component c is oftenspecified as a constant
Pr[yi ∈ population c |xi , Θ] = πc
c = 1, 2, ..C
It cannot be used to classify individual observations into types
Posterior probability that observation yi belongs to component c :
Pr[yi ∈ population c |xi , yi ;Θ] =πc fc(yi |xi , Θc)
∑Cj=1 πj fj (yi |xi , Θj )
c = 1, 2, ..C
It can be used to classify individual observations into types
Partha Deb (Hunter College) FMM Aug 2013 14 / 37
Model: Basic properties
Prior probability that observation yi belongs to component c is oftenspecified as a constant
Pr[yi ∈ population c |xi , Θ] = πc
c = 1, 2, ..C
It cannot be used to classify individual observations into types
Posterior probability that observation yi belongs to component c :
Pr[yi ∈ population c |xi , yi ;Θ] =πc fc(yi |xi , Θc)
∑Cj=1 πj fj (yi |xi , Θj )
c = 1, 2, ..C
It can be used to classify individual observations into types
Partha Deb (Hunter College) FMM Aug 2013 14 / 37
Model: Estimation challenges
The number of components has to be specified - we usually have littletheoretical guidance
Even if prior theory suggests a particular number of components we maynot be able to reliably distinguish between some of the components
In some cases additional components may simply reflect the presence ofoutliers in the data
Likelihood function may have multiple local maxima
Partha Deb (Hunter College) FMM Aug 2013 15 / 37
Example: Color of wine
Results of a chemical analysis of wines grown in the same region in Italybut derived from three different cultivars (grape variety)
Data characteristicsCultivar Freq. % of total Color intensity (mean)
1 59 33.15 5.5282 71 39.89 3.0863 48 26.97 7.396
Total 178 100 5.058
Partha Deb (Hunter College) FMM Aug 2013 16 / 37
Example: Color of wine
0.0
5.1
.15
.2D
ensi
ty
0 5 10 15Color
kernel = epanechnikov, bandwidth = 0.7076
Wine Color Density
Partha Deb (Hunter College) FMM Aug 2013 17 / 37
Example: Color of wine
Suppress information on class (cultivar)
Estimate a Finite mixture of Normals with 3 components
Use estimates of posterior probabilities to assign observations into oneof 3 classes
Partha Deb (Hunter College) FMM Aug 2013 18 / 37
Example: Color of wine
Estimates from finite mixture of normals with 3 componentsParameter component 1 component 2 component 3
Constant 4.929 2.803 7.548(0.334) (0.244) (0.936)
π 0.365 0.323 0.312(0.176) (0.107) (0.117)
Data characteristicsCultivar Freq. % of total Color (mean)
1 59 33.15 5.5282 71 39.89 3.0863 48 26.97 7.396
Total 178 100 5.058
Partha Deb (Hunter College) FMM Aug 2013 19 / 37
Example: Color of wine
Predicted CultivarCultivar 1 2 3 Total
1 No. 42 3 14 59% 71.2 5.1 23.7 100.0
2 No. 15 56 0 71% 21.1 78.9 0.0 100.0
3 No. 19 0 29 48% 39.6 0.0 60.4 100.0
Partha Deb (Hunter College) FMM Aug 2013 20 / 37
Application: Medical Care Use
Heterogeneity of insurance effects on healthcare expenditure
Data from the Rand Health Insurance Experiment (RHIE)
Conducted by the RAND Corporation from 1974 to 1982
Individuals were randomized into insurance plans
Widely regarded as the basis of the most reliable estimates of priceelasticities
Partha Deb (Hunter College) FMM Aug 2013 21 / 37
Application: Medical Care Use
Heterogeneity of insurance effects on healthcare expenditure
Data from the Rand Health Insurance Experiment (RHIE)
Conducted by the RAND Corporation from 1974 to 1982
Individuals were randomized into insurance plans
Widely regarded as the basis of the most reliable estimates of priceelasticities
Partha Deb (Hunter College) FMM Aug 2013 21 / 37
Application: Medical Care Use
Data collected from about 8,000 enrollees in 2,823 families from sixsites across the country
Each family was enrolled in one of fourteen different insurance plans foreither three or five years
The FFS plans ranged from free care to 95% coinsurance
Data from all 5 years of the experiment
Number of observations: 20,186
Partha Deb (Hunter College) FMM Aug 2013 22 / 37
Application: Medical Care Use0
.000
5.0
01.0
015
.002
Den
sity
0 10000 20000 30000 40000$
Mean expenditure = $ 220
Density of positive expenditures
0.0
05.0
1.0
15D
ensi
ty0 100 200 300
$Mean expenditure = $ 220
Density of positive expenditures < $300
Partha Deb (Hunter College) FMM Aug 2013 23 / 37
Application: Medical Care Use
0.1
.2.3
Den
sity
0 5 10ln($)
Mean expenditure = $ 220
Density of log positive expenditures
Partha Deb (Hunter College) FMM Aug 2013 24 / 37
Application: Medical Care Use
Explanatory variablesIDP 1 if individual deductible plan, 0 otherwiseLC ln(coinsurance+1), 0≤coinsurance≤100LPI f(annual participation incentive payment)FMDE f(maximum dollar expenditure)LINC ln(family income)LFAM ln(family size)EDUCDEC education of the household head in yearsPHYSLIM 1 if the person has a physical limitationNDISEASE number of chronic diseasesPHINDEX index of health (larger is worse)
AGE, FEMALE, CHILD, BLACK
Partha Deb (Hunter College) FMM Aug 2013 25 / 37
Application: Medical Care Use
Estimate a 2-component finite mixture of gamma densities
Highlight differences in treatment effects by component (class)Highlight differences in distributions of expenditures by classExplore sources / correlates of class differences
Estimate a 3-component finite mixture of gamma densities
Partha Deb (Hunter College) FMM Aug 2013 26 / 37
Application: Medical Care Use
−40
−30
−20
−10
010
% e
ffect
c=1
log(coinsurance)
c=2
c=1
plan has deductible
c=2
% Effect 95 % confidence interval
π1 = 0.75 , π2 = 0.25
Parameter estimates2−component gamma finite mixture model
Partha Deb (Hunter College) FMM Aug 2013 27 / 37
Application: Medical Care Use
020
040
060
080
0
π1 = 0.75 , π2 = 0.25
Average E(y|x) by component2−component gamma finite mixture model
E(y|x,c=1) E(y|x,c=2)
Partha Deb (Hunter College) FMM Aug 2013 28 / 37
Application: Medical Care Use
0.0
05.0
1.0
15.0
2
0 100 200 300x
y|x f(y|x,c=1) f(y|x,c=2)
π1 = 0.75 , π2 = 0.25
Distributions of y|x and f(y|x) by component2−component gamma finite mixture model
Partha Deb (Hunter College) FMM Aug 2013 29 / 37
Application: Medical Care Use
05
1015
0 .2 .4 .6 .8 1x
p(c=1|y,x) p(c=2|y,x)
π1 = 0.75 , π2 = 0.25
Distributions of posterior probabilities by component2−component gamma finite mixture model
Partha Deb (Hunter College) FMM Aug 2013 30 / 37
Application: Medical Care Use
logcidp
fmde
lpi
xage
black
femaleeducdec
linc
lfam
physlm
phindex
hlthf
hlthp
c=1c=2
π1 = 0.75π2 = 0.25
Values of covariates are standardizedAn observation is classified into class c if p(c|y,x)>0.7)
Average covariate values by component assignment2−component gamma finite mixture model
Partha Deb (Hunter College) FMM Aug 2013 31 / 37
Application: Medical Care Use
−40
−20
020
40%
effe
ct
c=1
log(coinsurance)
c=2
c=3
c=1
plan has deductible
c=2
c=3
% Effect 95 % confidence interval
π1 = 0.71 , π2 = 0.25 , π3 = 0.04
Parameter estimates3−component gamma finite mixture model
Partha Deb (Hunter College) FMM Aug 2013 32 / 37
Application: Medical Care Use
050
01,
000
1,50
02,
000
π1 = 0.71 , π2 = 0.25 , π3 = 0.04
Average E(y|x) by component3−component gamma finite mixture model
E(y|x,c=1) E(y|x,c=2) E(y|x,c=3)
Partha Deb (Hunter College) FMM Aug 2013 33 / 37
Application: Medical Care Use
0.0
05.0
1.0
15.0
2
0 100 200 300x
y|x f(y|x,c=1) f(y|x,c=2) f(y|x,c=3)
π1 = 0.71 , π2 = 0.25 , π3 = 0.04
Distributions of y|x and f(y|x) by component3−component gamma finite mixture model
Partha Deb (Hunter College) FMM Aug 2013 34 / 37
Application: Medical Care Use
020
4060
8010
0
0 .2 .4 .6 .8 1x
p(c=1|y,x) p(c=2|y,x) p(c=3|y,x)
π1 = 0.71 , π2 = 0.25 , π3 = 0.04
Distributions of posterior probabilities by component3−component gamma finite mixture model
Partha Deb (Hunter College) FMM Aug 2013 35 / 37
Application: Medical Care Use
logc
idp
fmde
lpi
xage
black
female
educdec
linc
lfam
physlm
phindex
hlthf
hlthp
c=1c=2c=3
π1 = 0.71π2 = 0.25π3 = 0.04
Values of covariates are standardizedAn observation is classified into class c if p(c|y,x)>0.7)
Average covariate values by component assignment3−component gamma finite mixture model
Partha Deb (Hunter College) FMM Aug 2013 36 / 37
Conclusions
Heterogeneity in treatment effects can be important to identify andexplore
When treatment effects are heterogeneous, the typically small benefitsfound in studies can be misleading
Finite mixture models are a useful way to model heterogeneoustreatment effects
FMM can uncover otherwise hidden heterogeneity
As described, FMM can be applied when outcomes are continuous ordiscrete
But not for binary or “severely” limited outcomes
Extension to collection of binary outcomes is referred to as the Grade OfMembership model
Partha Deb (Hunter College) FMM Aug 2013 37 / 37
Conclusions
Heterogeneity in treatment effects can be important to identify andexplore
When treatment effects are heterogeneous, the typically small benefitsfound in studies can be misleading
Finite mixture models are a useful way to model heterogeneoustreatment effects
FMM can uncover otherwise hidden heterogeneity
As described, FMM can be applied when outcomes are continuous ordiscrete
But not for binary or “severely” limited outcomes
Extension to collection of binary outcomes is referred to as the Grade OfMembership model
Partha Deb (Hunter College) FMM Aug 2013 37 / 37
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