Difference-in-Difference Evaluation Methods

Difference-in-Difference Evaluation Methods

MAAC 2015 Fall ConferenceTurf ValleyEllicott City, MarylandNovember 5, 2015

Todd Caldis, J.D., Ph.D.Senior EconomistCMS/OACT

• Evaluation problems arise from interventions that are perceived as likely to involve a desired effect on the outcome of an activity or process– An identifiable status quo, how things operated pre-

intervention– An identifiable intervention, departure from the

status quo: A new surgical procedure, a new drug therapy, a new insurance reimbursement policy, an adjustment to an existing social policy like the minimum wage

– In all cases a need to evaluate quantitatively the causal impact of the intervention

The Evaluation ProblemThe Evaluation Problem

3

• Snow’s 1855 study of cholera causation in London– In 1849 London serviced by 2 water companies

that both drew water from the Thames river in central London

– In 1852 one of the companies moved its water works upriver to an area freer of sewage

– Researchers showed that districts supplied by the upriver company post-intervention had sharply lower death rates from cholera relative to districts that continued to be supplied from the remaining downriver company

An Early Pioneering ExampleAn Early Pioneering Example

Evaluation Methods TodayEvaluation Methods Today

• Basic methods developed by econometricians, statisticians, and bio-statisticians

• Typically involve:– Collection of data for an intervention on a

pre and post basis– Specifying and estimating a regression

equation with coefficients intended to measure intervention effects

– Analysis of model results

5

Session GoalsSession Goals

• Specification of regression equations to measure intervention/treatment effects

– Simple Linear Regression and why it fails to get at the problem

– Fixed effects and dummy variable models that introduce important building blocks for an evaluation model

– True difference-in-difference regression models– Random assignment models

• Along the way will look at results of a couple estimated models

• Brief consideration of some methodological problems and possible fixes

6

or

• y is a vector of measured outcomes; x is matrix of measurements on explanatory variables for each member of the sample population; β

is a vector of coefficients to be estimated; ε is random error term• Estimated easily an efficiently by OLS by many statistical packages

with ability to analyze statistical significance of estimates and make forecasts

• Danger of omitted variable and confounding variable bias, in principle fixable

Simple Linear RegressionSimple Linear Regression

7

• d is a dummy or indicator variable, coded 0 or 1 (to indicate ‘M’ or ‘F’, ‘Union’ or ‘Non-Union’, etc.

• Allows modeling of effects of non-continuous, discrete variables; can include many such variables in a model

• Estimated coefficient on an indicator variable is interpreted as an ‘average effect’

• A building block to get where we want to go, but still not a model to measure intervention.

Dummy/Indicator Variables in Linear Regression

Dummy/Indicator Variables in Linear Regression

8

Why dummy variable alone generally can’t get at treatment effects?

Why dummy variable alone generally can’t get at treatment effects?

• A single dummy variable in isolation does not address two key factors critical to determining whether there is a policy effect or not:

1. The trend in the variable of interest prior to the policy intervention2. The trend in the variable of interest among those NOT subject to the policy intervention

9

• Estimates average invariant effects for each observational unit or each time period or both

• Requires repeated measurements on same observational units either within the same time period or in different time periods; panel or TSCS data sets

• Interpretation of subscript notation: i is person or observational and t is time period

• Coding of variables to estimate the fixed effects; LSDV estimation methods when not interested in the fixed effects themselves

Fixed Effects Models I: SpecificationFixed Effects Models I: Specification

10

Fixed Effects Models II: Their UsesFixed Effects Models II: Their Uses

• Allow researchers to account for unobserved average effects attributable to unobserved factors unique to each observational unit or unique to each time period

• Partial answer to ‘implicit’ omitted variable problem inherent in even the simplest linear regression model

• As Mundlak (1978) shows in a famous paper a partial fix for endogeneity bias when IV estimation is infeasible

• A building block for evaluation modeling, but still not an evaluation model. Why?

11

• Card and Krueger’s 1994 study of change in fast food employment after NJ raised its minimum wage to $5.05 from $4.25 in February, 1993

• Used the adjacent state of PA whose minimum wage remained unchanged as the control group

• Only computed means (regression unneeded) with no explanatory variables:

• Footnote: How Snow was able to implement his pioneering DD model

Difference-in Difference: A simple empirical example

Difference-in Difference: A simple empirical example

12

DD Simple Minimum Wage ModelDD Simple Minimum Wage Model

Variable NJ PA Difference: NJ - PA

Average FTE Employment per Restaurant after

21.03(

21.17()

-0.14(

Average FTE Employment per Restaurant before

20.44(

23.33(

-2.89()

Change in Average per Restaurant FTE

0.59(

-2.16()

2.76(

13

DD in Regression: The More Typical Situation

DD in Regression: The More Typical Situation

• Quite similar to model without explanatory variables

• Compute effects as before using coefficients, ignoring coefficients on explanatory variables

14

• Think back to the DD result of a 2.76 per restaurant increase in average FTE employment. Does it make sense?– In terms of economic theory no!– Omission of explanatory variables may be a factor– More importantly trends PA and NJ may not truly be

parallel universes, trends that are being differenced may not actually be the same across the 2 states (the deltas)

• Whenever construct a DD model the similarity of the comparison groups in every respect except treatment is key.

Picking Comparison GroupsPicking Comparison Groups

15

• Models we have examined are often called quasi-experiments because they are not true experiments, but an attempt to get at the effects we might ideally study with experimental methods, generally too costly and time consuming

• Experimental methods randomly assign study populations into treatment and control groups in order to assure that there are no systematic differences in the two groups other than the treatment– Treatment effects can then be estimated using what

are essentially simple dummy variable models

Sidenote: Random Assignment/ Experimental Models

Sidenote: Random Assignment/ Experimental Models

16

Evaluation of Effects of Medicare Payment Cuts on Hospital Performance

Large-Scale DD ExampleLarge-Scale DD Example

17

• Economic theory predicts that payment cuts should have ‘real world’ effects on quantity or quality of what firms provide to consumers

• Balanced Budget Act of 1997 imposed a variety of cuts in Medicare payment on acute hospitals

– DSH, Medical Education, Outliers– Likely to differentially effect hospitals depending on nature and scope of

their operations and their relative dependence on the Medicare program as a source of patients

– “Natural Experiment” for evaluating the effects of payment cuts on Medicare patients

• ACA mandates reduction of annual updates for most Medicare FFS payment systems below market basket– Theoretically the equivalent of an annual payment cut

The Evaluation Issue and Its RelevanceThe Evaluation Issue and Its Relevance

18

1. Develop a method for identifying ‘large-cut,’ ‘moderate-cut,’ and ‘small-cut’ hospitals.

---A key step because of issues of endogeneity bias---A story for another day

2. Use the hospital identifiers along with other relevant variables in a DD regression equation to predict patient outcomes such as mortality.

Evaluation Strategy(OACT Contractors Vivian Wu of USC and Yu-Chu Shen of the Naval Postgraduate School)

Evaluation Strategy(OACT Contractors Vivian Wu of USC and Yu-Chu Shen of the Naval Postgraduate School)

19

Illustrating the DD Evaluation‘Goal’(A ‘made-up’ example)

Illustrating the DD Evaluation‘Goal’(A ‘made-up’ example)

Changes in 30-day Mortality

Small-Cut Hospitals(Control Group)

Large-Cut Hospitals

(Treatment Group)

Difference-in-

Differences (DD)

 

1995-1997 (pre-BBA) -3% -3% -3-(-3)= -0%

2001-2005 (post-BBA) -7% -4% -4-(-7)= 3%

Difference-in-difference-in-differences (DDD)

(3-0)= +3%

20

The DD Regression Equation(Simplified)

The DD Regression Equation(Simplified)

21

Estimation Results IEstimation Results I

One-year mortality

Initial BBA period (1998-2000) AMI   CHF   Stroke  

moderate cut -0.19 [-0.71,0.34] 0.35 [-0.09,0.78] 0.48+ [-0.01,0.97]

large cut -0.04 [-0.64,0.55] 0.48+ [-0.02,0.99] 0.55+ [-0.04,1.14]

Post BBA period (2001-2005)

moderate cut 0.43 [-0.14,1.00] 0.43+ [-0.04,0.89] 0.31 [-0.19,0.81]

large cut 1.22** [0.54,1.91] 0.74** [0.20,1.28] 0.55+ [-0.05,1.15]Extended Post-BBA (2006-2009)

moderate cut 0.63+ [-0.01,1.27] 0.82** [0.25,1.38] 0.55+ [-0.05,1.14]

large cut 1.17** [0.38,1.95] 0.68* [0.01,1.35] 0.59+ [-0.10,1.28]

N (discharges)2,131,

0423,351,1

282,272,3

15

22

Estimation Results IIEstimation Results II

1-year mortality

Initial BBA period (1998-2000) Pneumonia

Hip Fracture

moderate cut 0.21[-0.24,0.65] 0.10

[-0.38,0.59]

large cut 0.56*[0.03,1.09] 0.04

[-0.52,0.59]

Post BBA period (2001-2005)

moderate cut 0.43+[-0.05,0.91] 0.39

[-0.09,0.87]

large cut 1.10**[0.52,1.67] 0.56+

[-0.02,1.15]

Extended Post-BBA (2006-2009)

moderate cut 0.43[-0.12,0.99] 0.56*

[0.02,1.10]

large cut 0.07[-0.59,0.74] 0.81*

[0.18,1.44]

N (discharges)3,388,68

81,688,16

5

23

Charted Results(An Example---Much more avaialable)

Charted Results(An Example---Much more avaialable)

24

Takeaways from Hospital Price Cut Study

Takeaways from Hospital Price Cut Study

• DD a versatile technique for uncovering ‘treatment’ effects buried in large data sets

• Regression coefficient estimates and their significance are the core of the story

• Presentational options exist for making DD effects easier to view

Where we have comeWhere we have come

• Why and how the logit regression model is interpreted as a probability model about discrete choices (logit as odds ratio)

• Where coefficient estimates come from, max of nonlinear likelihood function

• Kinds of analytics that can be done with logit models

• Looked simple example to ullustrate extension of the method to more than 2 choices in multinomial logit models

• Ready to learn more!

26

ReferencesReferences

Joshua D. Angrist and Jorn-Steffen Pischke, Mostly Harmless Econometrics, Princeton University Press, 2009.

A. Colin Cameron and Pravin K. Trivedi, Microeconometrics: Methods and Applications, Cambridge University Press, 2006.

Vivian Y. Wu and Yu-Chu Shen, “Reductions in Medicare Payments and Patient Outcomes,” Medical Care Vol. 51,

No. 11, pp. 970-977 (Nov., 2013)

Jeffrey M. Wooldridge, Econometric Analysis of Cross Section and Panel Data, The MIT Press, 2002.

27

Thank You to ColleaguesThank You to Colleagues

This presentation also drew upon an earlier presentation about DD in the work of CMS\OACT presented jointly with Andrea Sisko and Matt Rader of CMS\OACT in Spring, 2015.

Contact InformationContact Information

todd.caldis@cms.hhs.gov