Nonparametric Estimators of Dose-Response Functions
Nonparametric Estimators of
Dose-Response Functions
Michela Bia, Alessandra Mattei, Alfonso Flores-Lagunes,
Carlos Flores
CEPS/INSTEAD, Luxembourg
Department of Statistics "G. Parenti", University of Florence
State University of New York - Binghamton
University of Miami
IRVAPP
June, 7th 2012
Nonparametric Estimators of Dose-Response Functions
Motivation
The evaluation process in the field of economics, sociology, law and
many other settings relies on the implementation of
non-experimental techniques.
In many empirical studies, the units under analysis may receive
different treatment levels, in which case the estimation of
Dose-response Function is needed (Flores et al., 2012; Bia and
Mattei, 2012; Hirano and Imbens, 2004).
A key assumption (uncounfoundedness) is made, in order to adjust
for systematic differences between groups receiving different levels
of the treatment in a set of pre-treatment variables.
Nonparametric Estimators of Dose-Response Functions
Motivation
The evaluation process in the field of economics, sociology, law and
many other settings relies on the implementation of
non-experimental techniques.
In many empirical studies, the units under analysis may receive
different treatment levels, in which case the estimation of
Dose-response Function is needed (Flores et al., 2012; Bia and
Mattei, 2012; Hirano and Imbens, 2004).
A key assumption (uncounfoundedness) is made, in order to adjust
for systematic differences between groups receiving different levels
of the treatment in a set of pre-treatment variables.
Nonparametric Estimators of Dose-Response Functions
Motivation
The evaluation process in the field of economics, sociology, law and
many other settings relies on the implementation of
non-experimental techniques.
In many empirical studies, the units under analysis may receive
different treatment levels, in which case the estimation of
Dose-response Function is needed (Flores et al., 2012; Bia and
Mattei, 2012; Hirano and Imbens, 2004).
A key assumption (uncounfoundedness) is made, in order to adjust
for systematic differences between groups receiving different levels
of the treatment in a set of pre-treatment variables.
Nonparametric Estimators of Dose-Response Functions
Motivation
The evaluation process in the field of economics, sociology, law and
many other settings relies on the implementation of
non-experimental techniques.
In many empirical studies, the units under analysis may receive
different treatment levels, in which case the estimation of
Dose-response Function is needed (Flores et al., 2012; Bia and
Mattei, 2012; Hirano and Imbens, 2004).
A key assumption (uncounfoundedness) is made, in order to adjust
for systematic differences between groups receiving different levels
of the treatment in a set of pre-treatment variables.
Nonparametric Estimators of Dose-Response Functions
Reference Literature
Propensity score-based methods have been widely employed in the
last decades for the evaluation of causal effects, but usually confined
to the binary treatment case (Rosenbaum and Rubin, 1983).
Over the last years, several extensions of propensity score
techniques have been proposed and developed to deal with arbitrary
treatment regimes:
I
Imbens (2000), Lechner (2001) in the case of categorical
treatment variables;
I
Joffe and Rosenbaum (1999) for ordinal treatment variables;
I
Hirano and Imbens (2004), Imai and Van Dyk (2004) and
Flores et al. (2012) for continuous treatment variables.
Nonparametric Estimators of Dose-Response Functions
Reference Literature
Propensity score-based methods have been widely employed in the
last decades for the evaluation of causal effects, but usually confined
to the binary treatment case (Rosenbaum and Rubin, 1983).
Over the last years, several extensions of propensity score
techniques have been proposed and developed to deal with arbitrary
treatment regimes:
I
Imbens (2000), Lechner (2001) in the case of categorical
treatment variables;
I
Joffe and Rosenbaum (1999) for ordinal treatment variables;
I
Hirano and Imbens (2004), Imai and Van Dyk (2004) and
Flores et al. (2012) for continuous treatment variables.
Nonparametric Estimators of Dose-Response Functions
Reference Literature
Propensity score-based methods have been widely employed in the
last decades for the evaluation of causal effects, but usually confined
to the binary treatment case (Rosenbaum and Rubin, 1983).
Over the last years, several extensions of propensity score
techniques have been proposed and developed to deal with arbitrary
treatment regimes:
I
Imbens (2000), Lechner (2001) in the case of categorical
treatment variables;
I
Joffe and Rosenbaum (1999) for ordinal treatment variables;
I
Hirano and Imbens (2004), Imai and Van Dyk (2004) and
Flores et al. (2012) for continuous treatment variables.
Nonparametric Estimators of Dose-Response Functions
Reference Literature
Propensity score-based methods have been widely employed in the
last decades for the evaluation of causal effects, but usually confined
to the binary treatment case (Rosenbaum and Rubin, 1983).
Over the last years, several extensions of propensity score
techniques have been proposed and developed to deal with arbitrary
treatment regimes:
I
Imbens (2000), Lechner (2001) in the case of categorical
treatment variables;
I
Joffe and Rosenbaum (1999) for ordinal treatment variables;
I
Hirano and Imbens (2004), Imai and Van Dyk (2004) and
Flores et al. (2012) for continuous treatment variables.
Nonparametric Estimators of Dose-Response Functions
Reference Literature
Propensity score-based methods have been widely employed in the
last decades for the evaluation of causal effects, but usually confined
to the binary treatment case (Rosenbaum and Rubin, 1983).
Over the last years, several extensions of propensity score
techniques have been proposed and developed to deal with arbitrary
treatment regimes:
I
Imbens (2000), Lechner (2001) in the case of categorical
treatment variables;
I
Joffe and Rosenbaum (1999) for ordinal treatment variables;
I
Hirano and Imbens (2004), Imai and Van Dyk (2004) and
Flores et al. (2012) for continuous treatment variables.
Nonparametric Estimators of Dose-Response Functions
Reference Literature
Propensity score-based methods have been widely employed in the
last decades for the evaluation of causal effects, but usually confined
to the binary treatment case (Rosenbaum and Rubin, 1983).
Over the last years, several extensions of propensity score
techniques have been proposed and developed to deal with arbitrary
treatment regimes:
I
Imbens (2000), Lechner (2001) in the case of categorical
treatment variables;
I
Joffe and Rosenbaum (1999) for ordinal treatment variables;
I
Hirano and Imbens (2004), Imai and Van Dyk (2004) and
Flores et al. (2012) for continuous treatment variables.
Nonparametric Estimators of Dose-Response Functions
Our contribution to the existing literature
I
Hirano and Imbens (2004) address estimation of DRF by
introducing the concept of the Generalized Propensity Score
(GPS) and implement their approach in a parametric setting.
I
Flores at al. (2012) use the GPS within a weighting and
introduce a Inverse Weighted (IW) kernel estimator for
semiparametrically estimating the DRF.
We propose two new semiparametric estimators of DRF based
on spline methods, developing a set of Stata programs to
I
I
I
I
I
i) estimate the GPS by using generalized linear models
ii) common support condition
iii) test the balancing property of the estimated generalized
propensity score
iv) estimate the DRF using either penalized spline estimators
or IW kernel estimator.
Nonparametric Estimators of Dose-Response Functions
Our contribution to the existing literature
I
Hirano and Imbens (2004) address estimation of DRF by
introducing the concept of the Generalized Propensity Score
(GPS) and implement their approach in a parametric setting.
I
Flores at al. (2012) use the GPS within a weighting and
introduce a Inverse Weighted (IW) kernel estimator for
semiparametrically estimating the DRF.
We propose two new semiparametric estimators of DRF based
on spline methods, developing a set of Stata programs to
I
I
I
I
I
i) estimate the GPS by using generalized linear models
ii) common support condition
iii) test the balancing property of the estimated generalized
propensity score
iv) estimate the DRF using either penalized spline estimators
or IW kernel estimator.
Nonparametric Estimators of Dose-Response Functions
Our contribution to the existing literature
I
Hirano and Imbens (2004) address estimation of DRF by
introducing the concept of the Generalized Propensity Score
(GPS) and implement their approach in a parametric setting.
I
Flores at al. (2012) use the GPS within a weighting and
introduce a Inverse Weighted (IW) kernel estimator for
semiparametrically estimating the DRF.
We propose two new semiparametric estimators of DRF based
on spline methods, developing a set of Stata programs to
I
I
I
I
I
i) estimate the GPS by using generalized linear models
ii) common support condition
iii) test the balancing property of the estimated generalized
propensity score
iv) estimate the DRF using either penalized spline estimators
or IW kernel estimator.
Nonparametric Estimators of Dose-Response Functions
Our contribution to the existing literature
I
Hirano and Imbens (2004) address estimation of DRF by
introducing the concept of the Generalized Propensity Score
(GPS) and implement their approach in a parametric setting.
I
Flores at al. (2012) use the GPS within a weighting and
introduce a Inverse Weighted (IW) kernel estimator for
semiparametrically estimating the DRF.
We propose two new semiparametric estimators of DRF based
on spline methods, developing a set of Stata programs to
I
I
I
I
I
i) estimate the GPS by using generalized linear models
ii) common support condition
iii) test the balancing property of the estimated generalized
propensity score
iv) estimate the DRF using either penalized spline estimators
or IW kernel estimator.
Nonparametric Estimators of Dose-Response Functions
Our contribution to the existing literature
I
Hirano and Imbens (2004) address estimation of DRF by
introducing the concept of the Generalized Propensity Score
(GPS) and implement their approach in a parametric setting.
I
Flores at al. (2012) use the GPS within a weighting and
introduce a Inverse Weighted (IW) kernel estimator for
semiparametrically estimating the DRF.
We propose two new semiparametric estimators of DRF based
on spline methods, developing a set of Stata programs to
I
I
I
I
I
i) estimate the GPS by using generalized linear models
ii) common support condition
iii) test the balancing property of the estimated generalized
propensity score
iv) estimate the DRF using either penalized spline estimators
or IW kernel estimator.
Nonparametric Estimators of Dose-Response Functions
Our contribution to the existing literature
I
Hirano and Imbens (2004) address estimation of DRF by
introducing the concept of the Generalized Propensity Score
(GPS) and implement their approach in a parametric setting.
I
Flores at al. (2012) use the GPS within a weighting and
introduce a Inverse Weighted (IW) kernel estimator for
semiparametrically estimating the DRF.
We propose two new semiparametric estimators of DRF based
on spline methods, developing a set of Stata programs to
I
I
I
I
I
i) estimate the GPS by using generalized linear models
ii) common support condition
iii) test the balancing property of the estimated generalized
propensity score
iv) estimate the DRF using either penalized spline estimators
or IW kernel estimator.
Nonparametric Estimators of Dose-Response Functions
Our contribution to the existing literature
I
Hirano and Imbens (2004) address estimation of DRF by
introducing the concept of the Generalized Propensity Score
(GPS) and implement their approach in a parametric setting.
I
Flores at al. (2012) use the GPS within a weighting and
introduce a Inverse Weighted (IW) kernel estimator for
semiparametrically estimating the DRF.
We propose two new semiparametric estimators of DRF based
on spline methods, developing a set of Stata programs to
I
I
I
I
I
i) estimate the GPS by using generalized linear models
ii) common support condition
iii) test the balancing property of the estimated generalized
propensity score
iv) estimate the DRF using either penalized spline estimators
or IW kernel estimator.
Nonparametric Estimators of Dose-Response Functions
Application (1/2)
We use a data set collected by Imbens, Rubin and Sacerdote
(2001).
The winners of the Megabucks lottery in Massachusetts in the
mid-1980’s represent the reference population.
We implement our programs to semi-parametrically estimate the
average potential post-winning labor earnings for each amount of
the lottery prize (DRF).
Nonparametric Estimators of Dose-Response Functions
Application (1/2)
We use a data set collected by Imbens, Rubin and Sacerdote
(2001).
The winners of the Megabucks lottery in Massachusetts in the
mid-1980’s represent the reference population.
We implement our programs to semi-parametrically estimate the
average potential post-winning labor earnings for each amount of
the lottery prize (DRF).
Nonparametric Estimators of Dose-Response Functions
Application (1/2)
We use a data set collected by Imbens, Rubin and Sacerdote
(2001).
The winners of the Megabucks lottery in Massachusetts in the
mid-1980’s represent the reference population.
We implement our programs to semi-parametrically estimate the
average potential post-winning labor earnings for each amount of
the lottery prize (DRF).
Nonparametric Estimators of Dose-Response Functions
Application (2/2)
The assignment of the prize is random, but unit and item
nonresponse led to a selected sample where the amount of the prize
received is no longer independent of background characteristics.
As in the binary treatment case, the extent to which the bias
generated by unobservable confounding factors is reduced heavily
depends on the richness of the pre-treatment variables available in
the empirical study.
Nonparametric Estimators of Dose-Response Functions
Application (2/2)
The assignment of the prize is random, but unit and item
nonresponse led to a selected sample where the amount of the prize
received is no longer independent of background characteristics.
As in the binary treatment case, the extent to which the bias
generated by unobservable confounding factors is reduced heavily
depends on the richness of the pre-treatment variables available in
the empirical study.
Nonparametric Estimators of Dose-Response Functions
Estimation Strategy
[ outline ]
Estimation Strategy
The Stata program
Illustrative Example
On the agenda
Nonparametric Estimators of Dose-Response Functions
Estimation Strategy
Basic Framework (1/2)
We refer to the potential outcome approach to causal inference
(Rubin, 1974, 1978).
We estimate a continuous DRF that relates each value of the dose,
i.e., lottery prize level, to the post-winning labor earnings.
Formally, consider a set of N individuals, and denote each of them
by subscript i: i = 1, . . . , N. For each individual i, we observe a
vector of pre-treatment covariates, Xi , the received lottery prize
level, Ti , and the correspondent value of the outcome for this
treatment level, Yi = Yi (Ti ).
Nonparametric Estimators of Dose-Response Functions
Estimation Strategy
Basic Framework (1/2)
We refer to the potential outcome approach to causal inference
(Rubin, 1974, 1978).
We estimate a continuous DRF that relates each value of the dose,
i.e., lottery prize level, to the post-winning labor earnings.
Formally, consider a set of N individuals, and denote each of them
by subscript i: i = 1, . . . , N. For each individual i, we observe a
vector of pre-treatment covariates, Xi , the received lottery prize
level, Ti , and the correspondent value of the outcome for this
treatment level, Yi = Yi (Ti ).
Nonparametric Estimators of Dose-Response Functions
Estimation Strategy
Basic Framework (1/2)
We refer to the potential outcome approach to causal inference
(Rubin, 1974, 1978).
We estimate a continuous DRF that relates each value of the dose,
i.e., lottery prize level, to the post-winning labor earnings.
Formally, consider a set of N individuals, and denote each of them
by subscript i: i = 1, . . . , N. For each individual i, we observe a
vector of pre-treatment covariates, Xi , the received lottery prize
level, Ti , and the correspondent value of the outcome for this
treatment level, Yi = Yi (Ti ).
Nonparametric Estimators of Dose-Response Functions
Estimation Strategy
Basic Framework (1/2)
We refer to the potential outcome approach to causal inference
(Rubin, 1974, 1978).
We estimate a continuous DRF that relates each value of the dose,
i.e., lottery prize level, to the post-winning labor earnings.
Formally, consider a set of N individuals, and denote each of them
by subscript i: i = 1, . . . , N. For each individual i, we observe a
vector of pre-treatment covariates, Xi , the received lottery prize
level, Ti , and the correspondent value of the outcome for this
treatment level, Yi = Yi (Ti ).
Nonparametric Estimators of Dose-Response Functions
Estimation Strategy
Basic Framework 2/2
Average dose-response function µ(t) = E [Yi (t)]
Unconfoundedness assumption Yi (t) ⊥ Ti |Xi for all t ∈ T
Generalized Propensity Score r (T , X ) = fT |X (T |x)
The GPS is like a balancing score (e.g., Rosenbaum and Rubin,
1983)...within strata with the same value of r (t, X ), the probability
that T = t does not depend on the value of X .
Nonparametric Estimators of Dose-Response Functions
Estimation Strategy
Basic Framework 2/2
Average dose-response function µ(t) = E [Yi (t)]
Unconfoundedness assumption Yi (t) ⊥ Ti |Xi for all t ∈ T
Generalized Propensity Score r (T , X ) = fT |X (T |x)
The GPS is like a balancing score (e.g., Rosenbaum and Rubin,
1983)...within strata with the same value of r (t, X ), the probability
that T = t does not depend on the value of X .
Nonparametric Estimators of Dose-Response Functions
Estimation Strategy
Basic Framework 2/2
Average dose-response function µ(t) = E [Yi (t)]
Unconfoundedness assumption Yi (t) ⊥ Ti |Xi for all t ∈ T
Generalized Propensity Score r (T , X ) = fT |X (T |x)
The GPS is like a balancing score (e.g., Rosenbaum and Rubin,
1983)...within strata with the same value of r (t, X ), the probability
that T = t does not depend on the value of X .
Nonparametric Estimators of Dose-Response Functions
Estimation Strategy
Basic Framework 2/2
Average dose-response function µ(t) = E [Yi (t)]
Unconfoundedness assumption Yi (t) ⊥ Ti |Xi for all t ∈ T
Generalized Propensity Score r (T , X ) = fT |X (T |x)
The GPS is like a balancing score (e.g., Rosenbaum and Rubin,
1983)...within strata with the same value of r (t, X ), the probability
that T = t does not depend on the value of X .
Nonparametric Estimators of Dose-Response Functions
Estimation Strategy
Basic Framework 2/2
Average dose-response function µ(t) = E [Yi (t)]
Unconfoundedness assumption Yi (t) ⊥ Ti |Xi for all t ∈ T
Generalized Propensity Score r (T , X ) = fT |X (T |x)
The GPS is like a balancing score (e.g., Rosenbaum and Rubin,
1983)...within strata with the same value of r (t, X ), the probability
that T = t does not depend on the value of X .
Nonparametric Estimators of Dose-Response Functions
Estimation Strategy
Treatment Effect Estimation: Example
Normal distribution of the treatment given the covariates
f (T )|X ∼ N b0 + b1 X , σ 2
Estimated Generalized Propensity Score
R̂i = √
1
2πσ̂ 2
exp(−
1
(f (Ti ) − b̂0 − b̂1 Xi )2 )
2σ̂ 2
Nonparametric Estimators of Dose-Response Functions
Estimation Strategy
Treatment Effect Estimation: Example
Normal distribution of the treatment given the covariates
f (T )|X ∼ N b0 + b1 X , σ 2
Estimated Generalized Propensity Score
R̂i = √
1
2πσ̂ 2
exp(−
1
(f (Ti ) − b̂0 − b̂1 Xi )2 )
2σ̂ 2
Nonparametric Estimators of Dose-Response Functions
Estimation Strategy
Parametric Approach
Estimation through a quadratic approximation:
E (Yi |Ti , R̂i ) = α0 + α1 Ti + α2 Ti2 + α3 R̂i + α4 Rˆi2 + α5 Ti R̂i
Average potential outcome at treatment level t, given αˆ0 ,, αˆ2 ...αˆ5
P
E\
[Y (t)] = N1 ni=1 (αˆ0 + αˆ1 t + αˆ2 t 2 + αˆ3 r̂ (t, Xi ) + ... + αˆ5 tr̂ (t, Xi ))
Nonparametric Estimators of Dose-Response Functions
Estimation Strategy
Parametric Approach
Estimation through a quadratic approximation:
E (Yi |Ti , R̂i ) = α0 + α1 Ti + α2 Ti2 + α3 R̂i + α4 Rˆi2 + α5 Ti R̂i
Average potential outcome at treatment level t, given αˆ0 ,, αˆ2 ...αˆ5
P
E\
[Y (t)] = N1 ni=1 (αˆ0 + αˆ1 t + αˆ2 t 2 + αˆ3 r̂ (t, Xi ) + ... + αˆ5 tr̂ (t, Xi ))
Nonparametric Estimators of Dose-Response Functions
Estimation Strategy
Flexible specification of the Generalized Propensity Score
We estimate the conditional distribution of the treatment given the
covariates under 4 different distributional assumptions of the
treatment conditional on the covariates:
I
i) Gaussian Distribution,
I
ii) Inverse Gaussian Distribution
I
iii) Gamma Distribution
I
iv) Beta Distribution
Nonparametric Estimators of Dose-Response Functions
Estimation Strategy
Flexible specification of the Generalized Propensity Score
We estimate the conditional distribution of the treatment given the
covariates under 4 different distributional assumptions of the
treatment conditional on the covariates:
I
i) Gaussian Distribution,
I
ii) Inverse Gaussian Distribution
I
iii) Gamma Distribution
I
iv) Beta Distribution
Nonparametric Estimators of Dose-Response Functions
Estimation Strategy
Flexible specification of the Generalized Propensity Score
We estimate the conditional distribution of the treatment given the
covariates under 4 different distributional assumptions of the
treatment conditional on the covariates:
I
i) Gaussian Distribution,
I
ii) Inverse Gaussian Distribution
I
iii) Gamma Distribution
I
iv) Beta Distribution
Nonparametric Estimators of Dose-Response Functions
Estimation Strategy
Flexible specification of the Generalized Propensity Score
We estimate the conditional distribution of the treatment given the
covariates under 4 different distributional assumptions of the
treatment conditional on the covariates:
I
i) Gaussian Distribution,
I
ii) Inverse Gaussian Distribution
I
iii) Gamma Distribution
I
iv) Beta Distribution
Nonparametric Estimators of Dose-Response Functions
Estimation Strategy
Flexible specification of the Generalized Propensity Score
We estimate the conditional distribution of the treatment given the
covariates under 4 different distributional assumptions of the
treatment conditional on the covariates:
I
i) Gaussian Distribution,
I
ii) Inverse Gaussian Distribution
I
iii) Gamma Distribution
I
iv) Beta Distribution
Nonparametric Estimators of Dose-Response Functions
Estimation Strategy
Nonparametric Approach: Spline Estimators (1/4)
Following Ruppert et al (2003), we use bivariate basis function.
The additive model for truncated lines is:
t
bi +
E (Yi |Ti , R̂i ) = a0 +at Ti +ar R
K
X
k=1
r
ukt (Ti −kkt )+ +
K
X
bi −k r )+
ukr (R
k
k=1
where k1t < . . . < kKt t and k1 < . . . < kKr r are K t and K r distinct
bi .
knots in the support of T and the support ofthe estimated GPS, R
Nonparametric Estimators of Dose-Response Functions
Estimation Strategy
Nonparametric Approach: Spline Estimators (2/4)
We also allowed for interactions in the model by adding the
following product terms:
bi + λTi R
bi +
E (Yi |Ti , R̂i ) = a0 + at Ti + ar R
t
K
X
r
ukt (Ti
−
kkt )+
k=1
+
K
X
bi − k r )+
ukr (R
k
k=1
t
r
+
K
X
bi (Ti
vkr R
k=1
Kt X
Kr
X
k=1
−
kkt )+
+
K
X
b i − k r )+ +
vkt Ti (R
k
k=1
tr
t
r
b
vkk
0 (Ti − kk )+ (Ri − kk 0 )+
k 0 =1
The model is obtained from the basis functions
1, Ti t, (Ti − k1t ) . . . , (Ti − kKt t )
b t , (R
bi − k r ) . . . , (R
bi − k t r )
1, R
1
i
K
(1)
Nonparametric Estimators of Dose-Response Functions
Estimation Strategy
Nonparametric Approach: Spline Estimators (3/4)
An alternative to tensor product splines is given by the so-called
radial basis functions, which are basis functions of the form
C (k(t, r )0 − (k, k 0 )0 k) for some univariate function C . Here we
consider the following function
C
t ! t 2
t t
t
t
k
k
k
−
=
−
log −
r
r
kr r
kr kr where k · k is the Euclidean norm.
Nonparametric Estimators of Dose-Response Functions
Estimation Strategy
Nonparametric Approach: Spline Estimators (4/4)
We assume that
K
X
bi +λTi R̂i +
E (Yi |Ti , R̂i ) = a0 +at Ti +ar R
uk C
k=1
t !
Ti
kk −
R̂i
kkr (2)
where u1 , · · · , uk are random variables with mean 0, and
− 12 0
− 12
2
)(Ω
variance-covariance
matrix
Cov
(u)
=
σ
(Ω
u
k
k ) , with
"
t t !#
k
kk 0 k
−
.
Ωk = C kr
kr0 k
bt
R
i
k
1≤k,k 0 ≤K
Let
=b
r (t, Xi ) denote the estimated score at a specific
treatment level, t. Given the estimated parameters of the additive,
tensor or radial regression functions, the average potential outcome
bt.
at treatment level t is estimated averaging as over R
i
Nonparametric Estimators of Dose-Response Functions
Estimation Strategy
Nonparametric approach: Inverse-Weighting
Average potential outcome estimated by I-W Kernel Estimator
D0 (t)S2 (t) − D1 (t)S1 (t)
E\
[Y (t)] =
S0 (t)S2 (t) − S12 (t)
Sj (t) =
N
X
k̃hX (Ti − t)(Ti − t)j
i=1
Dj (t) =
N
X
k̃hX (Ti − t)(Ti − t)j Yi
i=1
k̃hX (Ti − t) =
Kh (Ti − t)
Rˆt
i
Nonparametric Estimators of Dose-Response Functions
Estimation Strategy
Nonparametric approach: Inverse-Weighting
Average potential outcome estimated by I-W Kernel Estimator
D0 (t)S2 (t) − D1 (t)S1 (t)
E\
[Y (t)] =
S0 (t)S2 (t) − S12 (t)
Sj (t) =
N
X
k̃hX (Ti − t)(Ti − t)j
i=1
Dj (t) =
N
X
k̃hX (Ti − t)(Ti − t)j Yi
i=1
k̃hX (Ti − t) =
Kh (Ti − t)
Rˆt
i
Nonparametric Estimators of Dose-Response Functions
Estimation Strategy
Nonparametric approach: Inverse-Weighting
Average potential outcome estimated by I-W Kernel Estimator
D0 (t)S2 (t) − D1 (t)S1 (t)
E\
[Y (t)] =
S0 (t)S2 (t) − S12 (t)
Sj (t) =
N
X
k̃hX (Ti − t)(Ti − t)j
i=1
Dj (t) =
N
X
k̃hX (Ti − t)(Ti − t)j Yi
i=1
k̃hX (Ti − t) =
Kh (Ti − t)
Rˆt
i
Nonparametric Estimators of Dose-Response Functions
Estimation Strategy
Nonparametric approach: Inverse-Weighting
Average potential outcome estimated by I-W Kernel Estimator
D0 (t)S2 (t) − D1 (t)S1 (t)
E\
[Y (t)] =
S0 (t)S2 (t) − S12 (t)
Sj (t) =
N
X
k̃hX (Ti − t)(Ti − t)j
i=1
Dj (t) =
N
X
k̃hX (Ti − t)(Ti − t)j Yi
i=1
k̃hX (Ti − t) =
Kh (Ti − t)
Rˆt
i
Nonparametric Estimators of Dose-Response Functions
The Stata program
[ outline ]
Estimation Strategy
The Stata program
Illustrative Example
On the agenda
Nonparametric Estimators of Dose-Response Functions
The Stata program
The syntax
DRF varlist , outcome(varname) treatment(varname)
gpscore(newvar) cutpoints(varname) index(string) nq_gps(num)
method(type) family(familyname) link(linkname) vce(string)
common(num) test(type) tpoints(vector) detail delta(num)
options for IW Kernel
bandwidth()
options for mtspline
degree1(num) degree2(num) nknots1(num) nknots2(num)
knots1(numlist) knots2(numlist) additive
options for radialpspline
nknots(num) knots(numlist) standardized
options for both mtspline and radialpspline
estopts(string)
Nonparametric Estimators of Dose-Response Functions
Illustrative Example
[ outline ]
Estimation Strategy
The Stata program
Illustrative Example
On the agenda
Nonparametric Estimators of Dose-Response Functions
Illustrative Example
The application on the Lottery Sample (1/2)
The sample we use in this analysis is the “winners” sample of 237
individuals who won a major prize in the lottery. The outcome of
interest is “year6” (earnings six years after winning the lottery), and
the treatment is “prize”, the prize amount. Control variables are
age, gender, years of high school, years of college, winning year,
number of tickets bought, working status after the winning, and
earnings s years before winning the lottery, s = 1, 2, . . . , 6.
Nonparametric Estimators of Dose-Response Functions
Illustrative Example
The application on the Lottery Sample (2/2)
DRF agew ownhs owncoll male tixbot workthen yearm1 yearm2
yearm3 yearm4, outcome(year6) treatment(prize) gpscore(gps)
method(radialpspline) family(gaussian) link(log) numoverlap(3)
nknots(4) tpoints(tp) flag(1) cutpoints(cut) index(p50) nq_gps(5)
detail delta(1)
DRF agew ownhs owncoll male tixbot workthen yearm1 yearm2
yearm3 yearm4, outcome(year6) treatment(prize) gpscore(gps)
method(iwkernel) family(gaussian) link(log) numoverlap(3)
tpoints(tp) flag(1) cutpoints(cut) index(p50) nq_gps(5) detail
delta(1)
Nonparametric Estimators of Dose-Response Functions
Illustrative Example
Results
Dose-response function and marginal treatment effect estimation
Penalized Spline Method
15
14
13
12
11
Dose−response function
16
15
10
14
13
12
11
10
9
10
20
30
40
50
60
Treatment
70
80
90
100
10
20
I−Weighting Kernel Method
30
40
50
60
Treatment
70
80
90
0
−.2
−.3
70
80
90
100
60
80
100
Radial Spline Method
Derivative
0
−.1
−.1
−.2
−.3
50
60
Treatment
40
.1
−.2
40
20
Treatment
0
Derivative
0
30
11
Penalized Spline Method
−.1
20
12
100
.1
10
13
9
0
.1
0
14
10
9
0
Derivative
Radial Spline Method
16
15
Dose−response function
Dose−response function
I−Weighting Kernel Method
16
−.3
0
10
20
30
40
50
60
Treatment
70
80
90
100
0
10
20
30
40
50
60
Treatment
70
80
90
100
Nonparametric Estimators of Dose-Response Functions
Illustrative Example
Bootstrap
bootstrap "DRF agew ownhs owncoll male tixbot workthen yearm1
yearm2 yearm3 yearm4, outcome(year6) treatment(prize)
gpscore(gps) method(radialpspline) family(gaussian) link(log)
numoverlap(3) nknots(4) tpoints(tp) flag(1) cutpoints(cut)
index(p50) nq_gps(5) detail delta(1)" _b, reps (50)
bootstrap "DRF agew ownhs owncoll male tixbot workthen yearm1
yearm2 yearm3 yearm4, outcome(year6) treatment(prize)
gpscore(gps) method(iwkernel) family(gaussian) link(log)
numoverlap(3) tpoints(tp) flag(1) cutpoints(cut) index(p50)
nq_gps(5) detail delta(1)" _b, reps (50)
Nonparametric Estimators of Dose-Response Functions
Illustrative Example
Bootstrap results
95% confidence intervals:
Dose-response function and marginal treatment effect estimation
Penalized Spline Method
30
30
30
25
20
15
10
5
Dose−response function
35
25
20
15
10
5
25
20
15
10
5
0
0
0
−5
−5
−5
−10
−10
−10
0
10
20
30
40
50
60
Treatment
70
80
90
100
0
10
20
I−Weighting Kernel Method
30
40
50
60
Treatment
70
80
90
100
0
Penalized Spline Method
.6
.4
.4
.2
−.2
−.4
−.6
−.6
−.8
−.8
−.4
−.6
−.8
−1
−1
−1
−1.2
−1.2
20
30
40
50
60
Treatment
70
80
90
100
100
−.2
−1.2
10
80
0
Derivative
−.4
60
.2
0
Derivative
0
−.2
40
Radial Spline Method
.6
.4
0
20
Treatment
.6
.2
Derivative
Radial Spline Method
35
Dose−response function
Dose−response function
I−Weighting Kernel Method
35
0
10
20
30
40
50
60
Treatment
70
80
90
100
0
10
20
30
40
50
60
Treatment
70
80
90
100
Nonparametric Estimators of Dose-Response Functions
On the agenda
[ outline ]
Estimation Strategy
The Stata program
Illustrative Example
On the agenda
Nonparametric Estimators of Dose-Response Functions
On the agenda
Possible extension
I
Non parametric specification of the generalized propensity
score
I
Basic assumption (what about relaxing CIA condition for
example?)
Nonparametric Estimators of Dose-Response Functions
On the agenda
Possible extension
I
Non parametric specification of the generalized propensity
score
I
Basic assumption (what about relaxing CIA condition for
example?)
Nonparametric Estimators of Dose-Response Functions
On the agenda
Possible extension
I
Non parametric specification of the generalized propensity
score
I
Basic assumption (what about relaxing CIA condition for
example?)
Nonparametric Estimators of Dose-Response Functions
On the agenda
Possible extension
I
Non parametric specification of the generalized propensity
score
I
Basic assumption (what about relaxing CIA condition for
example?)
Nonparametric Estimators of Dose-Response Functions
On the agenda
References
Bia, M. and Mattei, A. A Stata package for the estimation of the Dose
Response Function through adjustment for the generalized propensity
score. The Stata Journal, 8(3), 354 - 373, (2008).
Bia, M. and Mattei, A. Assessing the effect of the amount of financial
aids to Piedmont firms using the generalized propensity score. Statistical
Methods and Applications, forthcoming, (2012).
Flores, C. A., Flores-Lagunes, A., Gonzalez, A. and Neumann, T.C.
Estimating the effects of length of exposure to a training program: The
case of Job Corps. The Review of Economics and Statistics, 3, (2012).
Joffe, M. M. and Rosenbaum, P. R. Invited commentary: Propensity
scores. American Journal of Epidemiology, 150, 327-333, (1999).
Nonparametric Estimators of Dose-Response Functions
On the agenda
References
Hirano, K. and Imbens, G. The Propensity score with continuous
treatment. In Applied Bayesian Modelling and Causal Inference from
Missing Data Perspectives (eds A. Gelman and X.L. Meng), Wiley,(2004).
Imbens, G. W. The Role of the Propensity Score in Estimating
Dose-Response Functions. Biometrika, 87, 706-710, (2000).
Imai, K. and Van Dyk, D. A. Causal treatment with general treatment
regimes: Generalizing the Propensity Score. Journal of the American
Statistical Association, 99(467), 854-866, (2004).
Lechner, M. Identification and Estimation of Causal Effects of Multiple
Treatments under the Conditional Independence Assumption. In
Econometric Evaluations of Active Labor Market Policies in Europe (eds
M. Lechner and F. Pfeiffer), Heidelberg, Physica, (2001).