CLASH OF CAUSAL
INFERENCE TECHNIQUES
TERRENCE WILLETT AND NATHAN PELLEGRIN
RP GROUP CONFERENCE
KELLOGG WEST APRIL 2014
OUTCOMES
• Describe purpose of regression and propensity score
matching (PSM)
• Explain data requirements and basic procedure of
regression and PSM
• Compare and contrast regression and PSM
• Identify additional resources for further exploration
WHY CAUSAL INFERENCE
• If you need to use statistics, then you should design a
better experiment. –attributed to Rutherford
• Most education research is observational/correlational,
not experimental.
COMMON SCENARIO
•
Did participation in an activity, class or support result in higher outcomes
for students than would have happened had they not participated?
•
Students self-selected to participate and/or were recruited to
participate.
•
When participants are compared to non-participants, differences in
outcomes can be attributed to differences in background variables or
motivation.
•
•
Can we determine if the participation caused a change in outcomes?
No, but…
REGRESSION
𝑦 = 𝑎0 + 𝑎1 𝑥1 + 𝑎2 𝑥2 …+𝑎𝑛 𝑥𝑛 + 𝑒
• Classic correlational technique
• Covariates used in model to attempt to control for differences in
background variables or motivation
• Background variables can include measures of or proxies for skill
level, social capital, or socio-economic status
• Measures of self-motivation often unavailable
• Models are imperfect and generally must be combined with other
evidence to more completely describe the possible influence of an
intervention, program or strategy
PROPENSITY SCORE MATCHING
• One of several ways to create a matched comparison
group of non-participants intended to be similar to
participant group for a valid comparison
• Likelihood or other techniques used to create a score
indicating the likelihood that a particular non-participant
would have been a participant based on similarity to one
or more participants
THE COUNTERFACTUAL
(POTENTIAL OUTCOMES) FRAMEWORK
∆ = 𝑌𝑡 − 𝑌𝑐
This shows the causal effect of some event (participation,
treatment, intervention) on an individual.
𝑌 𝑡 is the potential outcome under treatment.
𝑌 𝑐 is the potential outcome under non-treatment (control,
counterfactual) and thus cannot be observed in our universe*.
*In light of recently published evidence for inflationary theories of the
cosmos, we note that we may be living in a multiverse (Alan Guth, 2014).
Counterfactual conditions may obtain in alternate universes.
THE POTENTIAL OUTCOMES MATRIX
Potential Outcome
Yt
Actual
Treatment
Status
(T)
1
0
= observable
= not observable
Yc
AVERAGE TREATMENT EFFECT (ATE)
•
Participants and non-participants differ systematically (w.r.t.
demographics, trajectories, risk profiles, self-selection, etc.)
•
Different people respond differently to treatment (differential
response)
•
These facts must be taken into account when
modeling/computing treatment effects.
•
This means all four cells of the matrix must be estimated in
order to obtain an average treatment effect !
•
How ?
SYMBOLIC DERIVATION OF
AVERAGE EFFECTS
ATE
1
∆ = 𝑌𝑖 𝑡 − 𝑌𝑖 𝑐
2
𝑌𝑖 = 𝑇𝑖 𝑌𝑖 𝑡 − (1 − 𝑇𝑖 )𝑌𝑖 𝑐
3
𝐸[∆] = 𝐸[𝑌𝑖 𝑡] − 𝐸[𝑌𝑖 𝑐]
4
𝐸𝑁 [𝑌𝑖 |𝑇𝑖 = 1] − 𝐸𝑁 [𝑌𝑖 |𝑇𝑖 = 0]
5
𝐸 ∆ = 𝜌𝐸 𝑌𝑖 𝑡 𝑇𝑖 = 1 + 1 − 𝜌 𝐸 𝑌𝑖 𝑡 𝑇𝑖 = 0
−{𝜌𝐸 𝑌𝑖 𝑐 𝑇𝑖 = 1 + 1 − 𝜌 𝐸 𝑌𝑖 𝑐 𝑇𝑖 = 0 }
ATT
6
𝐸 ∆ 𝑇𝑖 = 1 = 𝐸 𝑌𝑖 𝑡 𝑇𝑖 = 1 − 𝐸 𝑌𝑖 𝑐 𝑇𝑖 = 1
ATU
7
𝐸 ∆ 𝑇𝑖 = 0 = 𝐸 𝑌𝑖 𝑡 𝑇𝑖 = 0 − 𝐸 𝑌𝑖 𝑐 𝑇𝑖 = 0
CONDITIONAL INDEPENDENCE
(PERFECT STRATIFICATION)
(SELECTION ON OBSERVABLES)
• If our observations include information on every one of
the variables influencing likelihood of participation or
differential responses*, then it is possible to avoid
omitted variable bias and so achieve CI (PS, SOO). And
if we have CI, this is like randomized assignment ….
𝐸 𝑌 𝑡 𝑇 = 1, 𝑿 = 𝐸 𝑌 𝑡 𝑇 = 0, 𝑿
𝐸 𝑌 𝑐 𝑇 = 1, 𝑿 = 𝐸 𝑌 𝑐 𝑇 = 0, 𝑿
* How often do we encounter such datasets in institutional research?
ASSUMING CI…
Potential Outcome
Yt
Actual
Treatment
Status
(T)
Y(t)
1
0
Yc
Y(c)
= observable
= not observable
AVERAGE TREATMENT EFFECT (ATE)
• PSM offers a way to plug values into (5), (6) and (7) to
obtain unbiased estimates.
• However, if we have CI, then why not just something
like….
𝑌𝑖 = 𝛼 + 𝛾𝑇𝑖 + 𝜷′𝑿𝑖 + 𝜀𝑖 ?
(and many other regression techniques)
There is no firm and fast answer to this question. Decisions
are based on pragmatic considerations. However…..
EXAMPLE
PROS AND CONS
• Regressions can be “easier” to run but harder to explain to a general audience
• PSM can be more time consuming to conduct but easier to explain to a general
audience
• Regressions tend to perform better with large data sets while PSM tends to
perform better with few observations provided the non-participant group has
sufficient numbers of individuals with the key confounding variables
• Regressions have been used for many years and are well described
mathematically with broad consensus on proper error terms
• PSM is newer and there is not consensus on optimal matching procedures or
proper error terms
• Regression will use all cases with non-missing data while PSM may only a subset
of cases from the pool of non-participants
• All analytic methods suffer if key variables are not available
• Conclusions can often be the same with either method
HOW TO RUN PSM
•
•
Create data file (95% of effort)
Match participants and non-participants on a set of control variables to
create a comparison group with similar proportions on all characteristics
(i.e. comparison group would have a similar percent female, Hispanic,
low income, etc. as compared to the participant group)
•
This step is referred to as “balancing” and generally must be repeated several
times to obtain balance on all variables of interest either by adjusting matching
criteria or removing variables
•
Run comparative analyses, which can include simple t-tests, post-PSM
regressions, or other techniques
•
Major packages that conduct PSM include STATA, R, and SAS
•
•
STATA version 12 and older have psmatch2, v13 has teffects psmatch
Note SPSS/PASW does not do PSM directly but there is an R plugin for SPSS
• http://arxiv.org/ftp/arxiv/papers/1201/1201.6385.pdf
AN ALTERNATIVE STRATEGY TO DECIDING
THIS BATTLE
(WARNING: EMPIRICISTS APPROACHING)
ALTERNATIVE PERSPECTIVES
• Estimating program effects based on observational data can also be understood as…
•
•
•
•
Delimiting the error “built-into” inductive reasoning about causes (the problem of induction)
An “inverse problem” in the study of social dynamics.
A missing data problem
An optimization problem
• From these alternate perspectives there is a large menu of methods and extensions, including Euclidean,
Mahalanobis and Gower’s distance, nearest neighbor, cosine similarity, kernal functions, genetic algorithms,
imputation, dimensionality reduction, and other supervised/unsupervised learning algorithms
• A criterion that can be applied to regression, PSM and other methods is this: how do they perform at
predicting new (future) observations? (false positives, false negatives, correlated errors)
• In empirical investigations we do not answer this question just once using new replications or sampling
under varying conditions. It is applicable in any situation where predictions are made concerning new
(possibly future) observations. We update our models based on new evidence. And in this respect,
regression and PSM methods can both be used as tools of discovery; as ways to extend our understanding
of the processes producing patterns we find in sets of observations (and in streams of information,
generally).
• SO: CHOOSE THE METHOD/MODEL WHICH YIELDS THE SMALLEST PREDICTION ERROR.
• That may decide a battle in a particular setting (or occasion), but the war between methods will go on ….
“The inability to predict outliers implies the
inability to predict the course of history”
― Nassim Nicholas Taleb, The Black Swan: The
Impact of the Highly Improbable
“If you insist on strict proof (or strict disproof) in the
empirical sciences, you will never benefit from
experience, and never learn from it how wrong you
are.“
— Karl Raimund Popper, The Logic of Scientific
Discovery: Logik Der Forschung (2002), 28.
FURTHER READING
•
Angrist, J. D., & Pischke, J. (2008). Mostly Harmless Econometrics: An Empiricists
Companion
•
Morgan, S., Harding, D. (2006) Matching Estimators of Causal Effects: From
Stratification and Weighting to Practical Data Analysis Routines
•
Caliendo and Kopeinig. 2005. Practical Guide for PSM
•
www.caliendo.de/Papers/practical_revised_DP.pdf
•
Rosenbaum, P. R., & Rubin, D. B. (1983). The central role of the propensity score in
observational studies for causal effects. Biometrika, 70, 41-55.
•
Padgett, R.; Salisbury, M.; An, B.; & Pascarella, E. (2010). Required, Practical, or
Unnecessary? An Examination and Demonstration of Propensity Score Matching Using
Longitudinal Secondary Data. New Directions for Institutional Research – Assessment
Supplement (pp. 29-42). San Francisco, CA: Jossey-Bass.
•
Soledad Cepeda, M.; Boston, R.; Farrar, J., & Strom, B. (2003). Comparison of Logistic
Regression versus Propensity Score When the Number of Events Is Low and There Are
Multiple Confounders. American Journal of Epidemiology, 158, 280-287.
•
http://www.biostat.jhsph.edu/~estuart/propensityscoresoftware.html
THANK YOU
Terrence Willett
Director of Planning, Research, and Knowledge Systems
Cabrillo College
terrence@cabrillo.edu
Nathan Pellegrin
Data Processing Specialist
Peralta District
npellegrin@peralta.edu