Treatment effects under conditional independence

Author: Josef Perktold

This notebook illustrates the basic usage of the new treatment effect functionality in statsmodels.

The main class is statsmodels.treatment.treatment_effects.TreatmentEffect.

This class estimates treatment effect and potential outcome using 5 different methods, ipw, ra, aipw, aipw-wls, ipw-ra. The last three methods require both a treatment or selection model and an outcome model. Standard errors and inference are based on the joint GMM representation of selection or treatment model, outcome model and effect functions. The approach for inference follows Stata, however Stata support a wider range of models. Estimation and inference are valid under conditional independence or ignorability.

The outcome model is currently limited to a linear model based on OLS. Treatment is currently restricted to binary treatment which can be either Logit or Probit.

The example follows Cattaneo.

[1]:
import os
import numpy as np
from numpy.testing import assert_allclose
import pandas as pd

from statsmodels.regression.linear_model import OLS
from statsmodels.discrete.discrete_model import Probit
from statsmodels.treatment.treatment_effects import (
    TreatmentEffect
    )

from statsmodels.treatment.tests.results import results_teffects as res_st

# Load data for example
cur_dir = os.path.abspath(os.path.dirname(res_st.__file__))
file_name = 'cataneo2.csv'
file_path = os.path.join(cur_dir, file_name)
dta_cat = pd.read_csv(file_path)

methods = ['ra', 'ipw', 'aipw', 'aipw_wls', 'ipw_ra']
methods_st = [
    ("ra", res_st.results_ra),
    ("ipw", res_st.results_ipw),
    ("aipw", res_st.results_aipw),
    ("aipw_wls", res_st.results_aipw_wls),
    ("ipw_ra", res_st.results_ipwra),
    ]

# allow wider display of data frames
pd.set_option('display.width', 500)
[2]:
dta_cat.head()
[2]:
bweight mmarried mhisp fhisp foreign alcohol deadkids mage medu fage ... prenatal birthmonth lbweight fbaby prenatal1 mbsmoke_ mmarried_ fbaby_ prenatal1_ mage2
0 3459 married 0 0 0 0 0 24 14 28 ... 1 12 0 No Yes 0 1 0 1 576.0
1 3260 notmarried 0 0 1 0 0 20 10 0 ... 1 7 0 No Yes 0 0 0 1 400.0
2 3572 married 0 0 1 0 0 22 9 30 ... 1 3 0 No Yes 0 1 0 1 484.0
3 2948 married 0 0 0 0 0 26 12 30 ... 1 1 0 No Yes 0 1 0 1 676.0
4 2410 married 0 0 0 0 0 20 12 21 ... 1 3 1 Yes Yes 0 1 1 1 400.0

5 rows × 28 columns

Create TreatmentEffect instance and compute ipw

The TreatmentEffect class requires - a OLS model instance for the outcome model, - a results instance of the selection model and - a treatment indicator variable.

In the following example we use Probit as the selection model. Using Logit is also supported.

[3]:
# treatment selection model
formula = 'mbsmoke_ ~ mmarried_ + mage + mage2 + fbaby_ + medu'
res_probit = Probit.from_formula(formula, dta_cat).fit()

# outcome model
formula_outcome = 'bweight ~ prenatal1_ + mmarried_ + mage + fbaby_'
mod = OLS.from_formula(formula_outcome, dta_cat)

# treatment indicator variable
tind = np.asarray(dta_cat['mbsmoke_'])

teff = TreatmentEffect(mod, tind, results_select=res_probit)
Optimization terminated successfully.
         Current function value: 0.439575
         Iterations 6

After creating the TreatmentEffect instance, we can call any of the 5 methods to compute potential outcomes, POM0, POM1, and average treatment effect, ATE. POM0 is the potential outcome for the no treatment group, POM1 is the potential outcome for the treatment group, treatment effect is POM1 - POM0.

For example teff.ipw() computes POM and ATE using inverse probability weighting. The probability of treatment is also commonly called the propensity score. The summary of the estimation includes standard errors and confidence interval for POM and ATE.

Standard errors and other inferential statistics are based on the Generalized Method of Moments (GMM) representation of the selection and outcome models and the moment conditions for the results statistic. Method ipw uses the selection model but not the outcome model. Method ra uses the outcome model but not the selection model. The doubly robust estimators aipw, aipw-wls, ipw-ra include both selection and outcome models, where at least one of those two has to be correctly specified to get consistent estimates of the treatment effect. The moment conditions for the target variables, POM0, POM1, and ATE are based on POM0 and ATE. The remaining POM1 is computed as a linear combination of POM0 and ATE.

The internal gmm results are attached to the treatment results as results_gmm.

By default the treatment effect methods computes average treatment effect, where average is take over the sample observations. Option effect_group can be used to compute either average treatment effect on the treated, ATT, using effect_group=1 or average treatment effect on the non-treated using effect_group=0.

[4]:
res = teff.ipw()
res
[4]:
<class 'statsmodels.treatment.treatment_effects.TreatmentEffectResults'>
                             Test for Constraints
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
ATE         -230.6891     25.817     -8.936      0.000    -281.289    -180.089
POM0        3403.4632      9.571    355.586      0.000    3384.704    3422.223
POM1        3172.7741     24.001    132.193      0.000    3125.733    3219.815
==============================================================================
[5]:
res.summary_frame()
[5]:
coef std err z P>|z| Conf. Int. Low Conf. Int. Upp.
ATE -230.689070 25.816758 -8.935633 4.048542e-19 -281.288985 -180.089154
POM0 3403.463163 9.571412 355.586324 0.000000e+00 3384.703540 3422.222785
POM1 3172.774093 24.001059 132.193085 0.000000e+00 3125.732881 3219.815305
[6]:
print(res.results_gmm.summary())
                               _IPWGMM Results
==============================================================================
Dep. Variable:                      y   Hansen J:                    3.988e-09
Model:                        _IPWGMM   Prob (Hansen J):                   nan
Method:                           GMM
Date:                Thu, 02 Feb 2023
Time:                        19:16:58
No. Observations:                4642
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
p 0         -230.6891     25.817     -8.936      0.000    -281.289    -180.089
p 1         3403.4632      9.571    355.586      0.000    3384.704    3422.223
p 2           -1.5583      0.461     -3.380      0.001      -2.462      -0.655
p 3           -0.6485      0.055    -11.711      0.000      -0.757      -0.540
p 4            0.1744      0.036      4.836      0.000       0.104       0.245
p 5           -0.0033      0.001     -4.921      0.000      -0.005      -0.002
p 6           -0.2176      0.050     -4.390      0.000      -0.315      -0.120
p 7           -0.0864      0.010     -8.630      0.000      -0.106      -0.067
==============================================================================

average treatment effect on the treated

see more below

[7]:
teff.ipw(effect_group=1)
[7]:
<class 'statsmodels.treatment.treatment_effects.TreatmentEffectResults'>
                             Test for Constraints
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
ATE         -225.1796     23.658     -9.518      0.000    -271.549    -178.811
POM0        3362.8393     14.198    236.855      0.000    3335.012    3390.667
POM1        3137.6597     19.071    164.526      0.000    3100.281    3175.038
==============================================================================

average treatment effect on the untreated

[8]:
teff.ipw(effect_group=0)
[8]:
<class 'statsmodels.treatment.treatment_effects.TreatmentEffectResults'>
                             Test for Constraints
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
ATE         -231.8782     27.699     -8.371      0.000    -286.168    -177.588
POM0        3412.9116      9.283    367.634      0.000    3394.716    3431.107
POM1        3181.0334     26.120    121.786      0.000    3129.840    3232.227
==============================================================================

Other methods to compute ATE work in the same or similar way as for ipw for example regression adjustment ra and double robust ipw_ra.

[9]:
res_ra = teff.ra()
res_ra
[9]:
<class 'statsmodels.treatment.treatment_effects.TreatmentEffectResults'>
                             Test for Constraints
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
ATE         -239.6392     23.824    -10.059      0.000    -286.333    -192.945
POM0        3403.2423      9.525    357.288      0.000    3384.573    3421.911
POM1        3163.6031     21.864    144.698      0.000    3120.751    3206.455
==============================================================================
[10]:
res_ra.summary_frame()
[10]:
coef std err z P>|z| Conf. Int. Low Conf. Int. Upp.
ATE -239.639211 23.824021 -10.058722 8.408247e-24 -286.333435 -192.944988
POM0 3403.242272 9.525207 357.288006 0.000000e+00 3384.573209 3421.911335
POM1 3163.603060 21.863509 144.697867 0.000000e+00 3120.751371 3206.454750
[11]:
ra2 = teff.ipw_ra(effect_group=1, return_results=True)
ra2.summary_frame()
[11]:
coef std err z P>|z| Conf. Int. Low Conf. Int. Upp.
ATE -223.545262 23.794008 -9.395023 5.720507e-21 -270.180660 -176.909864
POM0 3361.204984 14.465009 232.367989 0.000000e+00 3332.854088 3389.555880
POM1 3137.659722 19.070923 164.525844 0.000000e+00 3100.281400 3175.038045

All methods in TreatmentEffect

The following computes and prints ATE and POM for all methods. (We include the call to TreatmentEffect as a reminder.)

[12]:
teff = TreatmentEffect(mod, tind, results_select=res_probit)

for m in methods:
    res = getattr(teff, m)()
    print("\n", m)
    print(res.summary_frame())

 ra
             coef    std err           z         P>|z|  Conf. Int. Low  Conf. Int. Upp.
ATE   -239.639211  23.824021  -10.058722  8.408247e-24     -286.333435      -192.944988
POM0  3403.242272   9.525207  357.288006  0.000000e+00     3384.573209      3421.911335
POM1  3163.603060  21.863509  144.697867  0.000000e+00     3120.751371      3206.454750

 ipw
             coef    std err           z         P>|z|  Conf. Int. Low  Conf. Int. Upp.
ATE   -230.689070  25.816758   -8.935633  4.048542e-19     -281.288985      -180.089154
POM0  3403.463163   9.571412  355.586324  0.000000e+00     3384.703540      3422.222785
POM1  3172.774093  24.001059  132.193085  0.000000e+00     3125.732881      3219.815305

 aipw
             coef    std err           z         P>|z|  Conf. Int. Low  Conf. Int. Upp.
ATE   -230.989648  26.214445   -8.811541  1.234375e-18     -282.369017      -179.610280
POM0  3403.355674   9.568514  355.682783  0.000000e+00     3384.601731      3422.109616
POM1  3172.366025  24.427402  129.869153  0.000000e+00     3124.489197      3220.242854

 aipw_wls
             coef    std err           z         P>|z|  Conf. Int. Low  Conf. Int. Upp.
ATE   -227.195618  27.372036   -8.300282  1.038644e-16     -280.843822      -173.547414
POM0  3403.250651   9.596571  354.631943  0.000000e+00     3384.441717      3422.059585
POM1  3176.055033  25.654641  123.800406  0.000000e+00     3125.772860      3226.337206

 ipw_ra
             coef    std err           z         P>|z|  Conf. Int. Low  Conf. Int. Upp.
ATE   -229.967078  26.629411   -8.635830  5.830196e-18     -282.159765      -177.774391
POM0  3403.335639   9.571288  355.577620  0.000000e+00     3384.576260      3422.095018
POM1  3173.368561  24.871955  127.588224  0.000000e+00     3124.620425      3222.116697

Results in Stata

The results in statsmodels are very close to the results in Stata because both packages use the same approach.

[13]:
for m, st in methods_st:
    print("\n", m)
    res = pd.DataFrame(st.table[:2, :6], index = ["ATE", "POM0"], columns=st.table_colnames[:6])
    print(res)

 ra
                b         se           z        pvalue           ll           ul
ATE   -239.639211  23.824021  -10.058722  8.408247e-24  -286.333435  -192.944988
POM0  3403.242272   9.525207  357.288005  0.000000e+00  3384.573209  3421.911335

 ipw
                b         se           z        pvalue           ll           ul
ATE   -230.688638  25.815244   -8.936140  4.030006e-19  -281.285586  -180.091690
POM0  3403.462709   9.571369  355.587873  0.000000e+00  3384.703170  3422.222247

 aipw
                b         se           z        pvalue           ll           ul
ATE   -230.989201  26.210565   -8.812828  1.220276e-18  -282.360964  -179.617438
POM0  3403.355253   9.568472  355.684297  0.000000e+00  3384.601393  3422.109114

 aipw_wls
                b         se           z        pvalue           ll           ul
ATE   -227.195618  27.347936   -8.307597  9.765984e-17  -280.796587  -173.594649
POM0  3403.250651   9.596622  354.630065  0.000000e+00  3384.441618  3422.059684

 ipw_ra
                b         se           z        pvalue           ll           ul
ATE   -229.967078  26.626676   -8.636718  5.785117e-18  -282.154403  -177.779752
POM0  3403.335639   9.571260  355.578657  0.000000e+00  3384.576315  3422.094963

Treatment effects without inference

It is possible to compute POM and ATE without computing standard errors and inferential statistics. In this case the GMM model is not computed.

[14]:
for m in methods:
    print("\n", m)
    res = getattr(teff, m)(return_results=False)
    print(res)

 ra
(-239.63921146433495, 3403.2422719354863, 3163.6030604711514)

 ipw
(-230.6886377952614, 3403.462708684556, 3172.774070889295)

 aipw
(-230.98920111257803, 3403.355253173835, 3172.366052061257)

 aipw_wls
(-227.19561818674947, 3403.250650975787, 3176.0550327890373)

 ipw_ra
(-229.96707793513315, 3403.3356393074205, 3173.3685613722873)

Treatment effect on the treated

Treatment effects on subgroups are not available for aipw and aipw-wls.

effect_group choses the group for which treatement effect and potential outcomes are computed Options are “all” for sample average treatment effect, 1 for average treatment effect on the treated and 0 for average treatment effect on the untreated.

Note: The row labels in the pandas dataframe, POM and ATE, are the same even for treatment effect on subgroups.

[15]:
for m in methods:
    if m.startswith("aipw"):
        continue
    res = getattr(teff, m)(effect_group=1)
    print("\n", m)
    print(res.summary_frame())

 ra
             coef    std err           z         P>|z|  Conf. Int. Low  Conf. Int. Upp.
ATE   -223.301651  22.742195   -9.818826  9.342545e-23     -267.875534      -178.727767
POM0  3360.961373  12.757489  263.450069  0.000000e+00     3335.957154      3385.965592
POM1  3137.659722  19.070923  164.525844  0.000000e+00     3100.281400      3175.038045

 ipw
             coef    std err           z         P>|z|  Conf. Int. Low  Conf. Int. Upp.
ATE   -225.179608  23.658119   -9.518069  1.764269e-21     -271.548669      -178.810546
POM0  3362.839334  14.197866  236.855264  0.000000e+00     3335.012028      3390.666640
POM1  3137.659726  19.070923  164.525845  0.000000e+00     3100.281404      3175.038049

 ipw_ra
             coef    std err           z         P>|z|  Conf. Int. Low  Conf. Int. Upp.
ATE   -223.545262  23.794008   -9.395023  5.720507e-21     -270.180660      -176.909864
POM0  3361.204984  14.465009  232.367989  0.000000e+00     3332.854088      3389.555880
POM1  3137.659722  19.070923  164.525844  0.000000e+00     3100.281400      3175.038045

Treatment effect on the untreated

Similar to ATT, we can compute average treatment effect on the untreated by using option effect_group=0.

[16]:
for m in methods:
    if m.startswith("aipw"):
        # not available
        continue
    res = getattr(teff, m)(effect_group=0)
    print("\n", m)
    print(res.summary_frame())

 ra
             coef    std err           z         P>|z|  Conf. Int. Low  Conf. Int. Upp.
ATE   -243.375488  24.902030   -9.773319  1.465697e-22     -292.182569      -194.568406
POM0  3412.911593   9.283454  367.633805  0.000000e+00     3394.716358      3431.106829
POM1  3169.536106  23.128805  137.038471  0.000000e+00     3124.204480      3214.867731

 ipw
             coef    std err           z         P>|z|  Conf. Int. Low  Conf. Int. Upp.
ATE   -231.878176  27.699436   -8.371224  5.702294e-17     -286.168073      -177.588279
POM0  3412.911593   9.283454  367.633804  0.000000e+00     3394.716357      3431.106829
POM1  3181.033418  26.119760  121.786472  0.000000e+00     3129.839629      3232.227206

 ipw_ra
             coef    std err           z         P>|z|  Conf. Int. Low  Conf. Int. Upp.
ATE   -231.125972  28.813022   -8.021580  1.043933e-15     -287.598458      -174.653487
POM0  3412.911593   9.283454  367.633805  0.000000e+00     3394.716358      3431.106829
POM1  3181.785621  27.301318  116.543297  0.000000e+00     3128.276021      3235.295221

The docstring for the TreatmentEffect class and it’s methods can be obtained using help

help(teff)