Unit 2: Correlation and Causality

© Judith D. Singer, Harvard Graduate School of Education Unit #2/Slide #1

Unit 2: Correlation and Causality

© Judith D. Singer, Harvard Graduate School of Education Unit #2/Slide #2

The S-030 roadmap: Where’s this unit in the big picture?

Unit 2:Correlation

and causality

Unit 3:Inference for the regression model

Unit 4:Regression assumptions:

Evaluating their tenability

Unit 5:Transformations

to achieve linearity

Unit 6:The basics of

multiple regression

Unit 7:Statistical control in

depth:Correlation and

collinearityUnit 10:

Interaction and quadratic effects

Unit 8:Categorical predictors I:

Dichotomies

Unit 9:Categorical predictors II:

Polychotomies

Unit 11:Regression modeling

in practice

Unit 1:Introduction to

simple linear regression

Building a solid

foundation

Mastering the

subtleties

Adding additional predictors

Generalizing to other types of

predictors and effects

Pulling it all

together

© Judith D. Singer, Harvard Graduate School of Education Unit #2/Slide #3

In this unit, we’re going to learn about…

• Developing a heuristic understanding of the correlation coefficient (r)– Understanding correlation as regression on standardized variables– The relationship between r and R2—how large is large?

• From correlation to causality– Randomized experiments: The “gold standard” for establishing causality– What can you do when randomized experiments aren’t possible?– When might an observed correlation not indicate a causal relationship? – Spurious correlation, confounding, Simpson’s paradox, reciprocal

causation and ecological correlation– Conditions for establishing causality

© Judith D. Singer, Harvard Graduate School of Education Unit #2/Slide #4

RQ: Is there a link between TV exposure and attention deficit problems?

Read the Pediatrics article Listen to the NPR Interview

© Judith D. Singer, Harvard Graduate School of Education Unit #2/Slide #5

Developing a measure of “co-relation”: Meet Karl Pearson(27 March 1857 – 27 April 1936)

Heredity: relationships between siblings and spouses (Pearson & Lee, 1903, On the laws of inheritance in

man, Biometrika)

• Pearson, Galton’s advisee and the first Galton Professor of Eugenics at University College, London (shown here with Galton at right)

• Fun fact: Born Carl (with a C) changed his name to Karl (with a K) after Karl Marx

• Developed, or named, many of the basic tools of modern statistics, including standard deviation, 2 goodness of fit, and correlationLearn more about Karl Pearson

Pearson’s “problems” to solve: (1) Neither variable is an “outcome” or a “predictor”(2) The measure of correlation should be

dimensionless, (eg., applicable for inches or feet, digit span or stature)

His solution: Re-express (transform) both variables on new “standard” scales that essentially eliminate the particular metrics of the original scales

© Judith D. Singer, Harvard Graduate School of Education Unit #2/Slide #6

Transformation and standardization: Re-expressing a variable’s scale

ID OwnIQ SOwnIQ FostIQ SFostIQ

1 68 -1.9985 63 -2.3080 2 71 -1.7943 76 -1.4535 . . .25 95 -0.1606 96 -0.138926 96 -0.0925 93 -0.3361. . .52 129 2.1539 117 1.2415 53 131 2.2900 132 2.2274

VarsdVarVarSVar

Standardization: A particular transformation that

yields a new variable with mean = 0 and sd = 1

Transformation:Any re-expression of a variable’s

scale

8.132

FC

CentigradeFarenheit

Standardization...Forces the sample mean of the new

variable to be 0 and its sd to be 1The new values measure an individual’s

distance from the sample mean in sd unitsDoesn’t change anyone’s relative rankDoesn’t create a normally distributed variable

69.1436.97

OwnIQSOwnIQ

21.1511.98

FostIQSFostIQ

Mean- 2sd

Mean- 1sd Mean Mean

+ 1sdMean+ 2sd

Mean

Mean- 1sdMean- 2sd

Mean+ 2sdMean+ 1sd

Mean = 98.11sd = 15.21

Mean = 97.36sd = 14.69

© Judith D. Singer, Harvard Graduate School of Education Unit #2/Slide #7

^ SOwnIQSFostIQ 8767.00000.0

Using a regression on standardized variables to understand correlation

Does 0.8767 seem familiar?

Standardized regression line goes

precisely through (0,0)

At average X (SOwnIQ=0), we predict average Y (SFostIQ=0)

Slope of the standardized

regression line assesses the estimated

difference in FostIQ(measured in standard

deviation units)per standard deviation

in OwnIQ

Pearson product-moment coefficient, r

© Judith D. Singer, Harvard Graduate School of Education Unit #2/Slide #8

How do we interpret r?

© Judith D. Singer, Harvard Graduate School of Education Unit #2/Slide #9

Plots to help develop your intuition for interpreting r

Cool interactive applet for learning more about correlation

© Judith D. Singer, Harvard Graduate School of Education Unit #2/Slide #10

Understanding the relationship between r and R2 (and their use as measures of “effect size”)

1-R2R2r

96%4%.2

99%1%.1Not uncommon in social sciences, but when r < .2, you have very

little explanatory power (R2 < 4%)

75%25%.5

84%16%.4

91%9%.3Covers most “statistically significant” correlations in social sciences, but even when r = .5, you’re only explaining 25% of the

variance in Y64%36%.6

51%49%.7Rare in the social sciences and even when r = .7, you’re still

explaining less than ½ the variance in Y

36%64%.8

19%81%.9

10%90%.95

Extremely rare in the social sciences, unless you have

aggregate data or a coding problem(!)

Small: r=.10

Medium: r=.30

Large: r=.50

Cohen’s guidelines

Another way of thinking about r is

as a measure of effect size

© Judith D. Singer, Harvard Graduate School of Education Unit #2/Slide #11

From correlation to causality

Identified mechanism You have a sound theory to explain how a change in the predictor produces a change in the outcome

No plausible alternative explanationThere’s no rival predictor that can explain away the

observed correlation

Responsiveness You demonstrate that a change in the predictor

actually produces a change in the outcome

ConsistencyYou find the same result in other populations, with different characteristics,

at different times

“I interpreted…Galton…to mean that there was a category broader

than causation, namely correlation…and that this new

conception of correlation brought psychology, anthropology,

medicine, and sociology in large parts into the field of mathematical treatment. It was Galton who first freed me from the prejudice that sound mathematics could only be applied to natural phenomenon

under the category of causation. Here for the first time was a possibility, I will not say a

certainty, of reaching knowledge—as valid as physical knowledge was then thought to be—in the

field of living forms and above all in the field of human conduct.”

Karl Pearson, 1889

What do we really mean when we say: “Associated with”“Related to”“Explained by”“Varies with”“Covaries with”

Counterfactual reasoning provides a powerful lens for thinking about these

questionsYou’d like to know what outcome values these

individuals would have had if they had received a “different treatment”

(ie, if they had different predictor values)?

Four criteria for establishing causality

Highest priorities for design and analysis

and often the hardest to establish

© Judith D. Singer, Harvard Graduate School of Education Unit #2/Slide #12

Why randomized experiments are the “gold standard”

Four important attributes of randomized experiments

The researcher actively intervened

in the system, actually changing X (the treatment) and seeing what

happens to Y

Any difference found in Y must be

due to the changing of X (the

treatment) because there’s no

other plausible explanation

Because of random assignment, groups are

guaranteed to be initially equivalent, on average, on all observable (and unobservable) characteristics

4.82(4.02)

5.00(3.50)

October(pretest)

BTLArthur

5.69(3.87)

7.04(3.56)

February (midterm)

7.16(3.75)

9.13(4.35)

May/June (posttest)

(n=51)(n=57)

RQ: Can narrative skills be ‘taught’ via TV to English Language Learners?

Narrative development in bilingual kindergarteners: Can Arthur help? Yuuko Uchikoshi (2005)

Developmental Psychology

The control group provides the ideal counterfactual—our best estimate

of what the treatment group

would have looked like if it didn’t

receive the treatment

© Judith D. Singer, Harvard Graduate School of Education Unit #2/Slide #13

There will always be studies where researchers have the burden of proof

Ethics: Morally, there are some treatments to which you can’t expose peopleDoes radiation cause cancer?

Time: Practically, some information is better than no informationDoes quality child care cause better life outcomes?Availability of data: With so much data, shouldn’t we analyze it?

How might you try to establish responsiveness? The key question is:

How are predictor values assigned?

•They’re not: they’re immutable characteristics of people

•Participants choose them•Researchers assign them (but not

randomly)•Outside forces inadvertently

change them•External raters assign them using

a ranking criterion (e.g., identifying those above a cut-score)How might you eliminate alternative explanations? The key question is:

Can the findings be explained away?

•Can you establish that the groups were equivalent initially?

•Can you isolate that portion of the variation in X that’s exogenous (not subject to selection bias) ?

•Can you rule out other explanations for the observed association?

“You can’t fix by analysis what you bungled by design…”

Light, Singer and Willett (1990)

When participants (or even researchers) choose, the conclusions are weaker because they’re subject to selection bias

Natural experiments

Regression discontinuitydesigns

Matching (especially propensity score matching is very popular now)

Instrumental variables

Feasibility: Logistically, there are some treatments to which you can’t assign peopleDoes education cause increased income?

Many would argue that these can’t be “causes”

Let’s think about how you might go about doing this

© Judith D. Singer, Harvard Graduate School of Education Unit #2/Slide #14

Non-experimental data: Might the correlation be causal?

CorrelationX Y X Y

Causation

But, just because we haven’t done an experiment doesn’t mean the correlation isn’t causal

X Y Sample Tobacco Industry Statements– “[the causes of diseases] are complex,

and the mechanism of causation, as well as the possible role of any cigarette smoke constituent in causation, have not been scientifically established”

– [At] least two standards for establishing causation exist. An epidemiological standard of causation, based primarily on statistical evidence, … [and] the more rigorous traditional scientific standard…[which] requires, among other things… well-designed and conducted … experiments.”

US Committee on Gov’t Reform “When forced to take legally binding positions, the

tobacco industry still does not accept scientific consensus …

that…cigarettes cause disease in smokers [and] that

environmental tobacco smoke causes disease in nonsmokers.

Read the Waxman (2002) report Tobacco industry statements in the Department of Justice Lawsuit

Does Imply ?

© Judith D. Singer, Harvard Graduate School of Education Unit #2/Slide #15

Even experiments aren’t foolproof: The MRFIT trial

Heart Attack Study Finds Men Heeding Health Advice BetterA federally financed study of 12,866 men -- half exhorted to improve their health habits and half getting only "usual care" from their doctors--has produced an unexpected result: Both groups had the same rate of heart attacks, but it was only one-fourth the rate of the general population of the same age.

What happened [is that] almost all Americans were reading and hearing advice to smoke less, eat fewer fats and lower their cholesterol level and blood pressure. Exhorted or not, most of the men in the study and their doctors apparently got the same message, and did even better than the average American.

17 September 1982

Find Article on LexisNexis

© Judith D. Singer, Harvard Graduate School of Education Unit #2/Slide #16

Spurious Correlation: Common Response to a Third Variable

There’s a 3rd variable, Z, which causes changes in

X and may—or may not—also

cause change in Y

X Y

Z

SES is often the “3rd variable”It is easy to prove that the wearing of tall hats

and the carrying of umbrellas enlarges the chest, prolongs life, and confers comparative immunity from disease… A university degree, a daily bath,

the owning of thirty pairs of trousers, a knowledge of Wagner’s music, a pew in church, anything, in short, that implies more means and better nurture…can be statistically palmed off as a magic spell conferring all sorts of privileges…The mathematician whose correlations would fill a Newton with admiration, may, in collecting and

accepting data and drawing conclusions from them, fall into quite crude errors by just such

popular oversights --George Bernard Shaw (1906)

Pauperism Poorhouses

Yule (1899) An investigation into the causes of changes in pauperism in England

Pigou (1899)

Yule’s footnote 25“Strictly speaking, for

‘due to’ read ‘associated with.’ ”

?

But not all spurious correlations are nonsense

Polio Incidence

Soft DrinkConsumption

© Judith D. Singer, Harvard Graduate School of Education Unit #2/Slide #17

Confounding: A “confusion of effects”:A third variable may (or may not) explain away (or reduce)

the correlation

CocaineRace

X Y

Z

?

There’s a 3rd variable, Z, which is correlated with X and

which causes changes in Y, but we don’t know if this

explains away the correlation between X & Y

 Crack cocaine study faulted on race factor

A study carried out four years ago has created the false perception that crack cocaine smoking is more common among blacks and Hispanics than among white Americans, say scientists who reanalyzed the findings in a new report.

The 1988 National Household Survey on Drug Abuse said that rates of crack use among blacks and Hispanics were twice as high as among whites. But the study failed to take into account social factors such as where the users lived and how easily the drug could be obtained, according to researchers writing in yesterday's issue of the Journal of the American Medical Association. The authors, from Johns Hopkins University, said that when adjusted for those factors, the study found equivalent use of crack among blacks, Hispanics and whites.

"Researchers have the responsibility to go beyond the reporting of racial and ethnic differences" because such findings "are often presented as if a person's race has intrinsic explanatory power," the authors wrote.

25 Feb 1993

Find Article on LexisNexis

© Judith D. Singer, Harvard Graduate School of Education Unit #2/Slide #18

Simpson’s paradox: A third variable may reverse(!) the correlation

Some confounders don’t just ‘explain

away’ the association, they reveal a reversal in the direction of the

effect

X Y

Z

+

Sex bias in graduate admissions:UC Berkeley (1973)

N apps % acceptedMen 8442 44%

Women 4321 34%

-

Learn more about Simpson’s paradox

AdmissionMale +

-

r = -0.56

© Judith D. Singer, Harvard Graduate School of Education Unit #2/Slide #19

Reciprocal causation: Do happy mothers make happy babies? Or it is the other way around?

X may cause Y or Y may cause

X—with the data we have, we just

can’t tell

X Y?

?

Cross-sectional observational studies are particularly susceptible to questions of

reciprocal causation• Special education placements and reading

scores: the more segregated the placement, the lower reading scores

• Motherhood and suicide risk: married women with children are at lower risk of suicide than unmarried women; the more children the lower the risk

• Depression and smoking: teens who are daily smokers are more likely to be seriously depressed

© Judith D. Singer, Harvard Graduate School of Education Unit #2/Slide #20

Ecological correlation: Aggregate and individual correlations may differ

Aggregate data describe aggregate relationships, not individual level relationships

Xaggregate

Yaggregate

?-.526-.619

.77348 states

.9469 regions

Robinson, W.S. (1950). Ecological Correlations and the Behavior of Individuals.

American Sociological Review 15: 351–357.

.118.20397,272 individuals

Foreign BornBlack

Unit of analysis

Correlations with Illiteracy

(Rural, low foreign born, but lots of

illiterates)

(Urban, lots of foreign born, but also lots of very literate folks)

© Judith D. Singer, Harvard Graduate School of Education Unit #2/Slide #21

Where to go to learn more about establishing causalityIn recent years, there has been an explosion of interest in the conditions necessary for establishing

causal inferences. Different disciplines use different standards and approaches, and there is much to learn by reading broadly. Here are some sources that I find particularly interesting and insightful:

General overviews (warning: some of these are very

technical)• Shadish

, WR, Cook, TF and Campbell, ST (2002). Experimental and Quasi-Experimental Designs for Generalized Causal Inference, Boston, MA: Houghton Mifflin.

• Holland PW (1986) Statistics and causal inference (with discussion). Journal of the American Statistical Association, 81, 945-970. [Link to search results for article and discussion.]

• Freedman, DA (2004). Graphical models for causation and the identification problem. Evaluation Review, 28(4), 267-293.

• Rubin, DB (2005) Causal inference using potential outcomes: Design, Modeling, Decisions. Journal of the American Statistical Association, 100, 322-331.

Discussions focused on sociology• Winship

, C & Morgan, S (1999). The estimation of causal effects from observational data. Annual Review of Sociology, 25, 659-707.

• Morgan, SL & Winship, C (2007). Counterfactuals and Causal Inference: Methods and Principles for Social Research. NY: Cambridge U Press.

Discussions focused on education• Shavelson, RJ & Towne, L eds (2002)

Scientific Research in Education. Washington, DC: National Academy Press.

• Angrist, JD (2004). American education research changes tack. Oxford Review of Economic Policy, 20(2), 198-212

• Gamse, BC & Singer, JD (2005) Lessons from the Red Sox Playbook. Harvard Education Letter, 21(2), 7-8

• Cook, TD (2001) Sciencephobia: Why education researchers reject randomized experiments. Education Next, Fall, 63-68

Discussions focused on psychology• Rutter M (2007) Proceeding from Observed

Correlation to Causal Inference: The Use of Natural Experiments, Perspectives on Psychological Science, 2(4) 377-395.

Discussions focused on epidemiology• Rothman, KJ & Greenland, S (2005). Causation

and causal inference in epidemiology, American Journal of Public Health, Supplement 1, 95(1), S144-S150

• Maldonado, G & Greenland, S (2002). Estimating causal effects. International Journal of Epidemiology, 31, 422-429.

• Krieger, N (1994). Epidemiology and the web of causation: Has anyone seen the spider? Social Science and Medicine. 39(7) 887-903.

© Judith D. Singer, Harvard Graduate School of Education Unit #2/Slide #22

What’s the big takeaway from this unit?

• Correlation coefficients are nifty tools when used correctly– Having a scale free measure of association is a powerful concept; you

can develop your intuition about the meaning of correlations and that intuition will carry across all types of variables

– The size of a correlation tells you about the strength of a relationship, not its magnitude. For the magnitude, you need the slope

• Correlation Causality– Randomized experiments are the gold standard for establishing

causality, but even with them there can be limits to your inferences– There are many reasons why you might find a correlation between an

outcome and a predictor; learn how to think about alternative explanations and evaluate whether they’re likely to hold in any given context

– When analyzing data, consider the steps involved in going from correlation to causality and decide how far your inferences can go

• There are many other issues involved in moving from correlation to causality– But before being able to tackle these more technical treatments, you need to

know much more about the basic regression approach– We offer entire classes, A-164: Program evaluation and S-290:

Quantitative methods for improving causal inference

© Judith D. Singer, Harvard Graduate School of Education Unit #2/Slide #23

options nodate nocenter nonumber;

title1 "Unit 2: IQs of Cyril Burt's identical twins";footnote1 "Program: Unit 2--Burt analysis.sas";

*-----------------------------------------------------*Be sure to update the infile reference to the file's location on your computer*-----------------------------------------------------*;

*---------------------------------------------------- *Input Burt data and name variables in dataset*-----------------------------------------------------*;

data one; infile 'm:\datasets\Burt.txt'; input ID 1-2 OwnIQ 4-6 FostIQ 8-10;

*-----------------------------------------------------*Estimate bivariate correlation between owniq & fostiq (Pearson correlation coefficient)*-----------------------------------------------------*;

proc corr data=one; var owniq fostiq;

run;

quit;

Appendix 1: Annotated PC SAS code for Unit 2, Burt data

Don’t forget the semicolon at the end of every statement;

Don’t forget to specify the location of the raw data, and check that you are indicating the appropriate drive

proc corr estimates bivariate correlations between variables you specify. Its var statement syntax is var1 var2 var3 … varn (note that it has neither an * (like proc gplot) or an = (like proc reg)

© Judith D. Singer, Harvard Graduate School of Education Unit #2/Slide #24

Appendix 2: My $0.02 on the need for randomized trials in education

© Judith D. Singer, Harvard Graduate School of Education Unit #2/Slide #25

Glossary terms included in Unit 2

• Aggregate data• Causality• Confounder• Correlation• Reciprocal causation• Spurious• Standardization• Transformation