PROPENSITY SCORE MODELS Michael Massoglia

Department of SociologyUniversity of Wisconsin Madison

General Overview The logic of propensity models Application based discussion of some of

the key features Emphasis on working understanding use of

models Brief formal presentation of the models Empirical example Questions and discussion

Please interrupt with questions and clarifications

My orientation Not an advocate nor a detractor

Try to understand the strengths and weakness The research is vastly expanding in this area

Focus on 1 statistics program -- 2 modules Used in published work

Level of talk Data is often problematic in social

science research Propensity models

One tool that can help with data limitations

Part I: Basic LogicStandard Regression Estimator Net of controls, the estimate is based

upon mean differences on some outcome between those who experienced the event or treatment – marriage, incarceration, job -- and is assumed to be an average effect generalizable to the entire population Under conditions in which

1) The treatment is random and the 2) Population is homogeneous (prior)

Often unlikely in the social sciences

Problems of Experiential Design Many social processes cannot be randomly

designed Incarceration Marriage Drug use Divorce

And the list goes on Data limitations

Cross sectional, few waves, retrospective data, measures change

Propensity models attempt to replicated experimental design with statistics

Propensity models Rooted in classic experimental design

Treatment group Exposed to some treatment

Control group Not exposed to treatment

Individuals are statistically randomization into groups Identical (net of covariates)Or differ in ways unrelated to outcomes

Treatment can be seen as random Ignorable treatment (conditional independence)

assumption

Counterfactuals PSM: Toward a consideration of

counterfactuals Some people receive treatment -- marriage,

incarceration, job. The counterfactual

“What would have happened to those who, in fact, did receive treatment, if they had not received treatment (or the converse)?”

Counterfactuals cannot observed, but we can create an estimate of them Rubin “The fundamental problem…”

At the heart of PSM

Part II: Application Based Discussion Propensity Score

Calculate the predicted probability of some treatment Assuming the treatment can be manipulated

Comparatively minor debate in literature We have predicted probability (for everything)

Predicted probability is based observed covariates

Once we know the predicted probability 1) Find people who experiences a treatment 2) Match to people who have same* predicted

probability, but did not experience treatment 3) Observe differences on some outcome

The process of Matching All based on matching a treated to a controlled

1 program 2 modules Nearest neighbor matching

1-1 match Kernel matching

Weights for distance Radius matching

0.01 around each treated Stratification matching

Breaks propensity scores into strata based on region of common support Great visual from Pop Center at PSU http://help.pop.psu.edu/help-by-statistical-method/propensity-

matching/Intro%20to%20P-score_Sp08.pdf/?searchterm=None

3 Key Compondents Range of common support

Existence Condition Balancing Property

Ignorable treatment assumption Observed Covariates

Reviewers pay attention ? More so than other methods

Important to keep in mind: Cross group models Not within person “fixed effects models”

Range of Common Support We use data only from region of common

support: Violates existence condition. Assumption of common support (1)

Participants Nonparticipants

Predicted Probability

Range of matched cases.

Balanced Among those with the same predicted

probability of treatment, those who get treated and not treated differ only on their error term in the propensity score equation. But this error term is approximately

independent of the X’s. Ignorable treatment assumption

The reality: The same given the covariates

Observed Covariates Propensity models based on observed

covariates Much like many other regression based

models Yet, reviewers pay particular attention

Models get additional attention PSM

Cannot: Fix out some variables Fixed effects models: Hard to measure time stable

traits Can: Assess the role of unobserved variables

with simulations

Part 3: Brief Formal PresentationPropensity score

More formally: The propensity score for subject i (i = 1, …,

N), is the conditional probability of being assigned to treatment Zi = 1 vs. control Zi = 0 given a vector xi of observed covariates:

where it is assumed that, given the X’s the Zi’s are independent

)|1(Pr)( iiii Ze xXx

Assumption(s)

Given the X’s the Zi’s are independent (given covariates)

Moves propensity scores to logic to that of an experiment Substantively means

Treatment status is independent of observed variables Treatment status occurs at random

Ignorable Treatment Assumption (2) Stable unit treatment value assumption. The potential

outcomes on one unit should be unaffected by the particular assignment of treatments to the other units Issues of independence

)|1(Pr)( iiii Ze xXx

Part 4: Empirical Example 3 part process

1)Assign propensity scores Create your matching equation Some programs do this at the same they estimate

treatment score My view is do them separately

Greater flexibility if you have pp scores independent of treatment effects

High, low, females, makes 2) Create matched sample

Average treatment effect 3) Tests of robustness

Add on to Stata Can be done in SAS, S-Plus R, MPLS, SPSS* Stata-

PSMATCH2: Stata module for propensity score matching, common support graphing, and covariate imbalance testing psmatch2.ado

PSCORE – same basic features More user “friendly” pscore.ado

.net search psmatch2 .net search pscore .ssc install psmatch2, replace

Moving into stata Estimation of average treatment effects

based on propensity scores (2002) The Stata Journal Vol.2, No.4, pp. 358-377.

Walk through the process Create propensity score

From observed covariates in the data Use different matching groups

Estimates Test the robustness of effect

Bias from unobservables

Two quick notes 1) tab mypscore Estimated | propensity | score | Freq. Percent Cum.------------+----------------------------------- .000416 | 1 0.02 0.02 .000446 | 1 0.02 0.04 .0004652 | 1 0.02 0.05 .0005133 | 1 0.02 0.07 .0005242 | 1 0.02 0.09 .0005407 | 1 0.02 0.11 .0005493 | 1 0.02 0.13 .0005666 | 3 0.05 0.18 .0005693 | 1 0.02 0.20 .0005729 | 1 0.02 0.22

2) Bad Matching Equation: Link back to PSU

3) Link : IU

Sensitivity Tests

gen delta delta is the difference in treatment effect between

treated and untreated rbounds delta, gamma (1 (0.1)2) gamma: log odds of differential assignment due to

unobserved heterogeneity Rosenbaum bounds takes the difference in the

response variable between treatment and control cases as delta, and examines how delta changes based on gamma LINK TO IU 2

A few concluding comments

Propensity models Dependent on data

As are all models Reviewers and editors seem to care more

Yet weakness appear similar traditional regression models

You can empirically test the role of unobservables with simulations Significant advancement

Thank you! A small window into propensity models

Regression, matched sample, use as covariates, as an instrument

Longitudinal data perfectly measured on all variables over time

Open to an argument preferences Fixed effects models And variants: Difference in differences

Do not live in such world Propensity models help us through imperfect data

Questions? (5) Preference an open discussion