Causal Inference: experimental and quasi-experimental methods

Draft

©G. Mason

(2005)

Scientific truth always goes through three stages. First, people say it conflicts with the Bible; next they say it has been discovered before; and lastly they say that they always believed it

Louis Agassiz, Swiss naturalist

We do not now a truth without knowing its cause

Aristotle, Nicomachean Ethics

Development of Western science is based on two great achievements: the invention of the formal logical system (Euclidean geometry) by the Greek philosophers, and the discovery of the possibility to find out causal relationships by systematic experiment (during the Renaissance)

Albert Einstein

Models of cause and effect 1

• Fishbone

Effect

A

B C

• Path Diagram (Regression sentence)

Y

X1

X2

XK

:

B1

B2

BK

Error

Models of cause and effect 2

Concept of causality

• Causality often implies inevitability, but the reality is that causal statements usually reflect degrees of uncertainty.

• Causality and probability are fundamentally connected because we want to :– Know the causes of an event– Measure the relative strength of these causes

Randomized experiments

• Classic experiment is the random, double-blind experiment (RDE):– subjects are selected randomly into a treatment

and control group– each subject received a code– an independent third party assigns codes

randomly to treatment and control group members.

– the treatment is not identifiable (i.e., the real and fake pill are identical.

– those administering the treatments and placebo have no knowledge of what subject receives.

Key benefits of the RDE

• randomization creates statistically equivalent groups

• in the absence of any interventions (the drug under tests), the incidence of disease is the same for both groups

• the groups are the “same” (statistically, except that one gets the drug and the other a placebo

• analysis can be done by difference of means tests or other basic techniques.

Limits of RDE

• In social science, randomized double blind experiments are often not feasible:– human subjects are unreliable (they move,

die or otherwise fail to participate in the full experiment).

– many see the administration of a placebo as withholding a treatment.

– social policy cannot be masked (creating a placebo is difficult).

Quasi-experimental designs

• Most policy testing in social sciences uses a quasi-experimental design.

• Two approaches exist– Multivariate (regression) models specify

dependent variable outcome, and include dummy variables to identify those in the program. Other covariates are included to control for the interventions.

– Matching: Program participants and non-participants serve as the basis for the treatment and control groups.

Four potential models for evaluating policy

1. Randomized control (RC)

2. Natural experiments (difference-in- difference, discontinuity)

3. Quasi-experimental methods– Heckman two step– Statistical Matching

4. Instrumental variables (treated separately)

Randomized control

Attempts to create a situation where

Cov (X’, ) = 0, or

E(T’, ) = E(T”, W), where W are the omitted variables that determine selection into treatment.

Natural experiments

• Create a “split” in the sample, where treated and untreated are classified by a variable that is not related to the the treatment.

• This split occur “naturally” where the program change occurs in one area/jurisdiction, not in others that are “closely similar.”

• Difference-in-differences (DID) methods are a common evaluation framework.

Difference in Differences

• The DID estimator uses the average before and after values for an outcome variable for the program and comparison group.

DID = [Yp (t=a) – Yp(t=b) ] – [Yc

(t=a) - Yc(t=b) ]

• Example:– Avg. earned income before - program group = $4500

– Avg. earned income after - program group = $6500

– Avg. earned income before - comparison group = $10,500

– Avg. earned income after - comparison group = $11,000

DID = [6500 - 4500] – [11,000 – 10,500] = $1,500

= net impact attributable to the program (treatment)

Net impact using DID

Yp (t=a)

t=b t=a

Yc (t=a)

Yc (t=b)

Yp (t=b)

a

b

c

d

e

bc = de

Y(income,

hours, etc.)

Time

Causal Inference – comparison in regression

Problem: Estimate effect of treatment (T) on observed outcome (Y), or estimate B in

Yi = B0 + B1Ti + i = Xi B + I (where X = [1, T]Assume

– dichotomous treatment variable: T=1 if treated, 0 otherwise– homogeneous treatment effect (B) (every i experiences

same effect) “average treatment effect”– Linear (no dose)– no covariates mediate the outcome

Bhat = Ybar(T=1) – YBar (T=0)

T = 1 (treatment) = 0 otherwise Y

i

B

The simple comparison group model

OLS assumption E (X’, ) = 0, or

E (X’, ) = X’ (Y- BOLSX)

which then creates the OLS estimator

BOLS = (X’X) –1 X’y

But, with omitted variables, the validity of OLS requires the omitted variables to be uncorrelated with T (the treatment). This is the essence of the self-selection problem.

No covariates is the key assumption

Selection and attrition

Random selectioninto program

Non-random selection into program(self-selection)

participant choice program manager choice (creaming)

Non-random selection out ofprogram (attrition)

participant choice program manager choice (creaming)

Random samples may be upset by self-selection and attrition (or both)

The original motivation for this procedure was to correct evaluation studies for sample distortion caused by self-selection.

Two steps:1. Estimate the probability of participation for each participant

and non-participant.

Yi = B0 + B1Xi1 + B2Xi2 + … BkXik + ei (Y=1 for a participant, for a non-participant).

2. For each participant/non-participant a unique probability of participation will be estimated. Call this λi Now, this is inserted into the outcome regression

Wi = B0 + B1Xi1 + B2Xi2 + … BkXik + Bl λi + ei, where Wi is the outcome for person i (wages, hours worked, etc.)

Heckman two step procedure (basic)

Matching

In social experiments, participants differ from non-participants because:– failure to hear of program– constraints on participation or completion– selection by staff

Matching participants and non-participants can be accomplished via– pair-wise– statistical

Matching Process

PARTICIPANTS

NON-PARTICIPANTS

MatchingProcess

Pairwise Statistical

PROGRAMGROUP

COMPARISONGROUP

Pair wise matching

• The theory will indicate those attributes that are likely to make a difference in the quasi-experiment.– For labour markets, gender, education and rural-

urban location are important– For health policy, age, rural-urban, and family

history might be important.

• The analyst starts with the first variable, and divides the participants and non-participants into two sets.

• Within the sets the samples are classified with respect to the second, variable and so on.

Non-Participants

Men

Participants

Men

Women

Women

Graduate

High School or Less

College

Graduate

High School or Less

College

Comparison

Program

GENDER EDUCATION

Graduate

High School or Less

College

Graduate

High School or Less

College

Pair wise Matching

Statistical Matching

• Matching is needed because we cannot randomly allocate clients to the program and comparison groups. Program benefits cannot be withheld.

• Logit model provides the estimate of the propensity to participate for participants and non-participants.

• The key idea is that we estimate that propensity to participate is based on observed attributes of the participants and non-participants.

• Participants are assigned a “Y”value of 1 and non-participants are assigned a “Y” value of 0.

• A logistic regression then estimates the propensity to participate.• Note that even though a non-participant actually did not participate the

model will assign a score between 0 and 1. Typically non-participants will have lower scores than participants, but there will be an overlap.

• The overlap is termed the region of common support.

Rationale for statistical matching

• Matching is needed because we cannot randomly allocate EI clients to the program and comparison groups. Part II benefits cannot be withheld.

• Logit model provides the estimate of the propensity to participate for participants and non-participants.

• The key idea is that we estimate that propensity to participate is based on observed attributes of the participants and non-participants.

• Participants are assigned a “Y”value of 1 and non-participants are assigned a “Y” value of 0.

• A logistic regression then estimates the propensity to participate.• Note that even though a non-participant actually did not participate the

model will assign a score between 0 and 1. Typically non-participants will have lower scores than participants, but there will be an overlap.

• The overlap is termed the region of common support.

Statistical matching simplified

0 1

Program

Comparison

X X X X X XX

X X X X X X

Each participant is matched to a "nearestneighbour" non-participant. Most non-

participants are not matched toparticipants and are discarded from the

sample survey and the analysis

Propensity to participate

X

The logit model

LPM Model Pi = E(Y = 1| Xi)

= B1 + B2X2 + B3X3 +..+BkXk

Logit Model Pi = E(Y = 1| Xi)

= 1/[1+ e – (BiXi)]

In Log Odds format

Li = ln(Pi/1-Pi) = Zi = BiXi

Region of Common Support

• Each participant has the value of 1 for P and each non-participant has the value 0.

• However, once the model is estimated, each participant and non-participant has a score between 0 and 1. Participants tend to have scores closer to 1 and non-participants are closer to 0.

• The distribution of scores can be graphed.

EI

Clie

nts

Participants

Non-participants

Matching variables age gender income prior interventions region time on EI ......

Statistical matching

Twin 1(Program)

Twin 2(Comparison)

Twin 10,000(Program)

Twin 10,000(Comparison)

:

Difference in pre and postprogram earnings, hours,

etc. regressed againstintervention dummy

variables, active/reachback, etc. for all

twinned pairs

Analysis

Statistical matching and structural modelling

Statistical Matching

Statistical Matching: The region of common support

0 1

Probability of participation(propensity score)

Participants

Non-participants

Overlap

Re

lati

ve

fre

qu

en

cy

Issues in Matching

• The matching is limited to the variables available in the administrative files.

• The balancing test compares the program and participant groups for each covariate using a t test for differences in means.

• Two key weaknesses are:– matching on the observed variables may not align the

program and comparison groups on the non-observed variables.

– The statistical quality of the match is very important Every additional variable that is introduced to the matching equation process, potentially improves the closeness of the match.

Application of the DID estimatorin a matching context• When combined with control, it measures the impact

of the observed differences between the two groups, which is participation in treatment (program)

• Cannot measure the net impact of different interventions unless these are added as covariates.

• This requires a 1 – 1 match between a program participant and a non-participant (i.e.., matched program and comparison groups)

DIDi-j = B1T1,i-j + B2T2,i-j + ..BkTk,i-j + ui-j