estimating the dose-response function through the glm approach
DESCRIPTION
ESTIMATING THE DOSE-RESPONSE FUNCTION THROUGH THE GLM APPROACH Barbara Guardabascio , Marco Ventura Italian National Institute of Statistics 7 th June 2013, Potsdam. Outline of the talk. Motivations;. literature references;. our contribution to the topic;. - PowerPoint PPT PresentationTRANSCRIPT
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ESTIMATING THE DOSE-RESPONSE FUNCTION THROUGH THE GLM APPROACH
Barbara Guardabascio, Marco Ventura
Italian National Institute of Statistics
7th June 2013, Potsdam
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Outline of the talk
Motivations;
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literature references;
our contribution to the topic;
the econometrics of the dose-response;
how to implement the dose-response;
our programs;
applications.
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Motivations
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Main question:
how effective are public policy programs with continuous treatment exposure?
Fundamental problem:
treated individuals are self-selected and not randomly.
Treatment is not randomly assigned
(possible) solution:
estimating a dose-response function
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Motivations
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E[y
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0 2 4 6 8 10Treatment level
Dose Response Low bound
Upper bound
Confidence Bounds at .95 % levelDose response function = Linear prediction
Dose Response Function
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E[y
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-E[y
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Treatment Effect Low bound
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Confidence Bounds at .95 % levelDose response function = Linear prediction
Treatment Effect Function
What is a dose-response function?
It is a relationship between treatment and an outcome variable e.g.: birth weight, employment, bank debt, etc
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Motivations
How can we estimate a dose-response function?
It can be estimated by using the Generalized Propensity Score (GPS)
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Literature references
1. Propensity Score for binary treatments:
Rosenbaum and Rubin (1983), (1984)
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3. Generalized Propensity Score for continuous treatments:
Hirano and Imbens, 2004; Imai and Van Dyk (2004)
2. for categorical treatment variables:
Imbens (2000), Lechner (2001)
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Our contribution
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Ad hoc programs have been provided to STATA users (Bia and Mattei, 2008), but …
… these programs contemplate only Normal distribution of the treatment variable
(gpscore.ado and doseresponse.ado)
We provide new programs to accommodate other distributions, not Normal.
(gpscore2.ado and doseresponse2.ado)
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The econometrics of the dose-response
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{Yi(t)} set of potential outcomes for
Where is the set of potential treatments over [t0, t1]
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The econometrics of the dose-response
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N individuals, i=1 … N
Xi vector of pre-treatment covariates;
Ti level of treatment delivered;
Yi (Ti) outcome corresponding to the treatment Ti
Let us suppose to have
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The econometrics of the dose-response
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Hirano-Imbens define the GPS as the conditional density of the actual treatment given the covariates
)()( tYEt i
We want the average dose response function
)|( XTrR
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The econometrics of the dose-response
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Within strata with the same r(t,x) the probability that T=t does not depend on X
),(|}{1 xtrtTX
Balancing property:
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The econometrics of the dose-response
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This means that the GPS can be used to eliminate any bias associated with differences in the covariates and …
tXTtY |)(
If weak unconfoundedness holds we have
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rRtTYE
rXtrtYErt
,|
),(|)(),(
The dose-response function can be computed as:
The econometrics of the dose-response
),(,)( XtrtEt
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1. Regress Ti on Xi and
The dose-respone can be implemented in 3 steps:
FIRST STEP:
take the conditional distribution of the treatment giventhe covariates Ti| Xi
How to implement the GPS
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2i
' ,D ~ |)( XXTf ii
Where f(.) is a suitable transformation of T (link) D is a distribution of the exponential family
β parameters to be estimated
σ conditional SE of T|X
How to implement the GPS
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GPS
2' ˆ,ˆ,ˆ iii XTDR
1a. Test the balancing property
How to implement the GPS
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Model the conditional expectation of E[Yi| Ti, Ri ] as a function of Ti and Ri
SECOND STEP:
iiiiii
iii
RTRRTT
RTYErt
52
432
210
,|),(
How to implement the GPS
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Estimate the dose-response function by averaging the estimated conditionl expectation over the GPS at each level of the treatment we are interested in
THIRD STEP:
N
iiXtrtN
t ),(ˆ,ˆ1
)(
How to implement the GPS
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Where is the novelty?
in the FIRST STEP
Instead of a ML we use a GLM
exponential distribution (family)
combined with a link function
How to implement the GPS
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our programs
Link\Distr Normal Inv. Normal
Binomial Poisson Neg. Binomial
Gamma
Identity X X X X X X
Log X X X X X X
Logit X
Probit X
Cloglog X
Power X X X X X X
Opower X
Nbin X
Loglog X
Logc X
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We have written two programs:
doserepsonse2.ado;
estimates the dose-response function and graphs the result.
It carries out step 1 – 2 – 3 of the previous slides by running other 2 programs
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our programs
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gpscore2.ado:
evaluates the gpscore under 6 different distributional assumptions
step 1 of the previous slides
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doseresponse_model.ado:
Carries out step 2 of the previous slides
our programs
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doseresponse2 varlist , outcome(varname) t(varname) family(string) link(string) gpscore(newvarname) predict(newvarname) sigma(newvarname) cutpoints(varname) nq_gps(#) index(string) dose_response(newvarlist)
Optionst_transf(transformation) normal_test(test) normal_level(#) test_varlist(varlist) test(type) flag(#) cmd(regression_cmd) reg_type_t(string) reg_type_gps(string) interaction(#) t_points(vector) npoints(#) delta(#) bootstrap(string) filename(filename) boot_reps(#) analysis(string) analysis_leve(#) graph(filename) flag_b(#) opt_nb(string) opt_b(varname) detail
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our programs
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gpscore2 varlist , t(varname) family(string) link(string) gpscore(newvarname) predict(newvarname) sigma(newvarname) cutpoints(varname) index(string) nq_gps(#)
Options
t_transf(transformation) normal_test(test) normal_level(#) test_varlist(varlist) test(type) flag_b(#) opt_nb(string) opt_b(varname) detail
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our programs
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Application
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Data set by Imbens, Rubin and Sacerdote (2001);
The winners of a lottery in Massachussets:amount of the prize (treatment) Ti
earnings 6 years after winning (outcome) Yi
age, gender, education, # of tickets bought, working status, earnings before winning up to 6 Xi
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Application: flogit
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Fractional data: flogit model.
Treatment: prize/max(prize)
outcome: earnings after 6 year
family(binomial) link(logit)
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Application: flogit
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Dose Response Low bound
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Confidence Bounds at .95 % levelDose response function = Linear prediction
Dose Response Function
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E[y
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Treatment Effect Low bound
Upper bound
Confidence Bounds at .95 % levelDose response function = Linear prediction
Treatment Effect Function
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Application: count data
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Count data: Poisson model.
Treatment: years of college+ high school
outcome: earnings after 6 year
family(poisson) link(log)
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Application: count data
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Dose Response Low bound
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Confidence Bounds at .95 % levelDose response function = Linear prediction
Dose Response Function
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Treatment Effect Low bound
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Confidence Bounds at .95 % levelDose response function = Linear prediction
Treatment Effect Function
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Application: gamma distribution
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Gamma distribution:
Treatment: age
outcome: earnings after 6 year
family(gamma) link(log)
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Application: gamma distribution
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Dose Response Low bound
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Confidence Bounds at .95 % levelDose response function = Linear prediction
Dose Response Function
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Upper bound
Confidence Bounds at .95 % levelDose response function = Linear prediction
Treatment Effect Function