Economics 105: Statistics

Economics 105: Statistics• GH 19 not due Thur• RAP assignment … datasets to look at

• Find the “codebook” or “survey instrument” and look at the questions they asked

Brief Introduction to Research Design

Design Notation

Internal Validity

Experimental Design

Design Notation• Observations or measures are indicated with an “O”• Treatments or programs with an “X”• Groups are shown by the number of rows• Assignment to group is by “R,N,C”

– Random assignment to groups– Nonequivalent assignment to groups– Cutoff assignment to groups

• Time

Design Notation Example

R O1,2 X O1,2

R O1,2 O1,2

Os indicate differentwaves of

measurement.

Vertical alignmentof Os shows that

pretest and posttestare measured at same time.

X is the treatment.There are twolines, one foreach group.

R indicates the groups

are randomly assigned.

Subscriptsindicate

subsets ofmeasures.

Types of DesignsRandom assignment?

Control group or multiple measures?

No

Yes

Yes

Randomized(true experiment)

Quasi-experiment

No

Nonexperiment

Non-Experimental Designs

X O

O X O

X O O

Post-test only (case study)

Single-group, pre-test, post-test

Two-group, post-test only(static group comparison)

Experimental Designs

R O1 X O1,2

R O1 O1,2

R X O1,2

R O1,2

• Pretest-Posttest Randomized Experiment Design

• If continuous measures, use t-test

• If categorical outcome, use chi-squared test

• Posttest only Randomized Experiment Design

• Less common due to lack of pretest

• Probabilistic equivalence between groups

Experimental DesignsR O X O

R O O

R X O

R O

• Advantages• Information is available on the effect of treatment

(independent variable), the effect of pretesting alone, possible interaction of pretesting & treatment, and the effectiveness of randomization

• Disadvantages• Costly and more complex to implement

Solomon Four-Group Design

Establishing Cause and Effect

Single-Group Threats

Multiple-Group Threats

“Social” Interaction Threats

• Internal validity is the approximate truth about inferences regarding cause-effect relationships.

Internal Validity

Threats to Internal ValidityR X

OR

OHistory

MaturationTesting

InstrumentationMortality

Regression to the meanSelection

Selection-historySelection- maturation

Selection- testingSelection- instrumentation

Selection- mortality*Selection- regressionDiffusion or imitation*

Compensatory equalization*Compensatory rivalry*

Resentful demoralization*

Single-Group

Multiple-Group

Social Interaction

Single-Group Threatsto Internal Validity

Administerprogram

Measureoutcomes

X O

Two designs:

Administerprogram

Measureoutcomes

X O

Measurebaseline

O

Post-test only a single group

What is a “single-group” threat?

• Diabetes educational program for newly diagnosed adolescents in a clinic

• Pre-post, single group design• Measures (O) are paper-pencil, standardized

tests of diabetes knowledge (e.g. disease characteristics, management strategies)

Example

• Any other event that occurs between pretest and posttest

• For example, adolescents learn about diabetes by watching The Health Channel

Program Posttest

X O

Pretest

O

History Threat

• Normal growth between pretest and posttest.• They would have learned these concepts anyway,

even without program.

Program Posttest

X O

Pretest

O

Maturation Threat

• The effect on the posttest of taking the pretest• May have “primed” the kids or they may have

learned from the test, not the program• Can only occur in a pre-post design

Program Posttest

X O

Pretest

O

Testing Threat

• Any change in the test from pretest and posttest• So outcome changes could be due to different

forms of the test, not due to program• May do this to control for “testing” threat, but

may introduce “instrumentation” threat

Program Posttest

X O

Pretest

O

Instrumentation Threat

• Nonrandom dropout between pretest and posttest• For example, kids “challenged” out of program by

parents or clinicians• Attrition

Program Posttest

X O

Pretest

O

Mortality Threat

• Group is a nonrandom subgroup of population.• For example, mostly low literacy kids will appear

to improve because of regression to the mean.• Example: height

Program Posttest

X O

Pretest

O

Regression Threat

When you select a sample from

the low end of a distribution ...

the group will do better on a

subsequent measure.

The group mean on the first measure

appears to “regress toward the mean” of

the population.

Selectedgroup’smean

Overallmean

Regression to the mean

Overallmean

Regression to the Meanpre-test scores ~ N

post-test scores ~ N & assuming no effect of treatment pgm

Regression to the Mean

Regression to the MeanSir Francis Galton (1822 – 1911)903 adult children & their 250 parents

Regression to the Mean

• How to Reduce the effects of RTM (Barnett, et al., International Journal of Epidemiology, 2005)

1. When designing the study, randomly assign subjects to treatment and control (placebo) groups. Then effects of RTM on responses should be same across groups.

2. Select subjects based on multiple measurements

• RTM increases with larger variance (see graphs) so subjects can be selected using the average of 2 or more baseline measurements.

Multiple-Group Threats to Internal Validity

• When you move from single to multiple group research the big concern is whether the groups are comparable.

• Usually this has to do with how you assign units (e.g., persons) to the groups (or select them into groups).

• We call this issue selection or selection bias.

The Central Issue

Administerprogram

Measureoutcomes

Measurebaseline

Alternativeexplanations

Alternativeexplanations

X OO

OODo not

administerprogram

Measureoutcomes

Measurebaseline

The Multiple Group Case

• Diabetes education for adolescents

• Pre-post comparison group design

• Measures (O) are standardized tests of diabetes knowledge

Example

• Any other event that occurs between pretest and posttest that the groups experience differently.

• For example, kids in one group pick up more diabetes concepts because they watch a special show on Oprah related to diabetes.

X OO

OO

Selection-History Threat

• Differential rates of normal growth between pretest and posttest for the groups.

• They are learning at different rates, even without program.

X OO

OO

Selection-Maturation Threat

• Differential effect on the posttest of taking the pretest.

• The test may have “primed” the kids differently in each group or they may have learned differentially from the test, not the program.

X OO

OO

Selection-Testing Threat

• Any differential change in the test used for each group from pretest and posttest

• For example, change due to different forms of test being given differentially to each group, not due to program

X OO

OO

Selection-Instrumentation Threat

• Differential nonrandom dropout between pretest and posttest.

• For example, kids drop out of the study at different rates for each group.

• Differential attrition

X OO

OO

Selection-Mortality Threat

• Different rates of regression to the mean because groups differ in extremity.

• For example, program kids are disproportionately lower scorers and consequently have greater regression to the mean.

X OO

OO

Selection-Regression Threat

“Social Interaction” Threats to Internal Validity

• All are related to social pressures in the research context, which can lead to posttest differences that are not directly caused by the treatment itself.

• Most of these can be minimized by isolating the two groups from each other, but this leads to other problems (for example, hard to randomly assign and then isolate, or may reduce generalizability).

What Are “Social” Threats?

• Diffusion or imitation of Treatment• Compensatory Equalization of Treatment• Compensatory Rivalry• Resentful Demoralization

What Are “Social” Threats?

What is a Clinical Trial?• “A prospective study comparing the effect and

value of intervention(s) against a control in human beings.”

• Prospective means “over time”; vs. retrospective• It is attempting to change the natural course of a

disease• It is NOT a study of people who are on drug X

versus people who are not

• http://www.clinicaltrials.gov/info/resources

Example: Job Corps• What is Job Corps? http://jobcorps.doleta.gov/

• January 5, 2006 Thursday Late Edition – Final

SECTION: Section C; Column 1; Business/Financial Desk; ECONOMIC SCENE; Pg. 3

HEADLINE: New (and Sometimes Conflicting) Data on the Value to Society of the Job Corps

BYLINE: By Alan B. Krueger.

Alan B. Krueger is the Bendheim professor of economics and public affairs at Princeton University. His Web site is www.krueger.princeton.edu.

He delivered the 2005 Cornelson Lecture in the Department of Economics here at Davidson (that’s the big econ lecture each year).

Example: Job Corps• Quotations from “New (and Sometimes Conflicting) Data on the Value

to Society of the Job Corps” by Alan B. Krueger.

• Since 1993, Mathematica Policy Research Inc. has evaluated the performance of the Job Corps for the Department of Labor.

• Its evaluation is based on one of the most rigorous research designs ever used for a government program. From late 1994 to December 1995, some 9,409 applicants to the Job Corps were randomly selected to be admitted to the program and another 6,000 were randomly selected for a control group that was excluded from the Job Corps.

• Those admitted to the program had a lower crime rate, higher literacy scores and higher earnings than the control group.

RCT for Credit Card Offers

Source: Agarwal, et al. (2010), Journal of Money, Credit & Banking, 42 (4)

A1: 0% APR for first 8 months & 9.99% on balance transfers, then 9.99% on purchases

A2: 0% APR for first 12 months, & 9.99% on balance transfers, then 9.99% on purchases

A3: 0% APR for first 8 months & 8.99% on balance transfers, then 8.99% on purchases

RCT for Education in India

Source: Banerjee, et al. (2007), Quarterly Journal of Economics

RCT for Education in India

RCT for the Effect of High Rewards on Performance

Source: Ariely, Gneezy, Loewenstein, and Mazar (2009), Review of Economic Studies

RCT for the Effect of High Rewards on Performance

Random assignment !

Recommended Reading

Amazon link Amazon link

Amazon link

Introduction to Regression Analysis• Correlation analysis only measures the strength of

the association (linear relationship) between two variables … not necessarily a causal relationship

• Regression analysis is used to:– Predict the value of a dependent variable based on the

value of at least one independent variable– Explain the impact of changes in an independent variable

on the dependent variable

• Dependent variable: the variable we wish to predict or explain variation in ... outcome variable, Y.

• Independent variables: the variables used to explain variation in Y ... covariates, explanatory variables, r.h.s. vars, X-variables

Types of Relationships

Y

X

Y

X

Y

Y

X

X

Linear relationships Curvilinear relationships

Types of Relationships

Y

X

Y

X

Y

Y

X

X

Strong relationships Weak relationships

(continued)

Types of Relationships

Y

X

Y

X

No relationship

(continued)

Deterministic Linear Models• Theoretical Model:

– b0 and b1 are constant terms

• b0 is the intercept

• b1 is the slope

– Xi is a predictor of Yia

bb0

Xi

Yi

(continued)

Pop Random Error for this Xi value

Y

X

Observed Value of Y for Xi

Xi

Pop Slope = β1

Pop Intercept = β0

εi

Stochastic Simple Linear Population Regression Model

Gauss-Markov Assumptions• (1) Zero conditional mean

– Idiosyncratic, “white noise”– Measurement error on Y– Omitted relevant explanatory variables … why?

• (2)– Homoskedastic errors

• (3) – No serial correlation among errors (autocorrelation)

Y

X

E[Y|X] = 0+ 1X

Gauss-Markov Assumptions(4)

– Linear in the parameters + error– Variation in Y is caused by , the error (as well as X)– Not

(5) Random sample of data• are i.i.d.

• (Ancillary) errors are normally distributed

Stochastic Linear Models• Assumptions so far imply• • • Need to estimate population intercept & slope• Take a sample of data & obtain the sample regression line

•

The sample regression line equation provides an estimate of the population regression line

Sample Regression Equation (Prediction Line)

Estimate of the regression

intercept

Estimate of the regression slope

Estimated (or predicted) Y value for observation i

Value of X for observation i

The individual random error terms ei have a mean of zero

Other notation:

chosen in samplenot chosen in sample

estimated error for X3

(residual)

Y

X

Observed Value of Y for X3

Predicted Value of

Y for X3

X3

ε3

Sample Regression Equation

e3

Sample Regression Equation• Residual, ei, is the prediction error

• Positive errors• Negative errors

Y

X