Lecture_10 [Modo de compatibilidad]

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Techniques of Statistical

Analysis I

Lect_10: Causality

Bruno Arpino

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Association is not causation!!!

“Alternative explanations”

Spurious correlation and controlling for third

variables

Outline

2

variables

Simpson’s paradox

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Association

means that two variables move

together. Causation

means that one of the

variables is causing the movements in the other.

Remember: the direction of causality in a

Causality

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regression model is set a-priori by the

researcher

Evidence of association is necessary but not

sufficient evidence of causation

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There is empirical evidence that X has a causal effect on

Y if:

1. X and Y are associated (

relationship condition

)

Three necessary conditions for causation

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2. X temporally/logically precede Y (

antecedent

condition

)

3. The observed association between X and Y cannot be

explained by factors not controlled for in the study

(

lack of alternative explanations

condition)

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You might find an association but this cannot be

interpreted as causal if:

A. You mixed up causes and effects (

reverse causality

)

Association does not imply causation

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B. X and Y are mutually related (

bidirectional causality

)

C. X and Y are associated only because some third factor(s)

is related to both variables (

spurious association

)

Check out this wikipedia entry:

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The more firemen fighting a fire (X), the bigger the fire is

observed to be (Y).

Therefore we conclude

firemen cause fire size

.

A. You mixed up causes and

effects (reverse causality):

Examples of reverse causality

fi re s iz e 6

effects (reverse causality):

the bigger is the fire, the more

firemen will be sent to the scene

You find a strong correlation between X and Y.

But it’s too early to draw causal conclusion!!!

Ask yourself: is X the cause of Y or is it the

other way

other way around? (Stata cannot answer!!!)

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It’s a common practice in primary and secondary schools to divide

students in working groups according to their abilities.

It’s easy to find data showing association between working groups (X)

and student performance (Y).

E.g, the graph shows results from

Examples of reverse causality (cont’d)

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E.g, the graph shows results from

an ANOVA: students in group 1

had bad average performance

compared to students in the other

two groups.

But… is it because students were in

group 1 that they had lower performance?

Or… it’s that student who needed more help (“““less smart”””) were

included in group 1?

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Does education (X) have a causal effect on fertility (Y)?

Probably yes but… if you are a pregnant teen maybe you drop out

of school! (Think hard about sample selection, timing of events

and possible anticipation effects)

This paper, for example, addresses the possible reverse effect in

Examples of reverse causality (cont’d)

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This paper, for example, addresses the possible reverse effect in

the education-fertility relationship

http://www.econ.uconn.edu/Seminar%20Series/AminBehrmanSchoolingFertility.pdf

R Related to fertility: big families are more likely

to have a van than small families.

Is it that having a van makes you more fertile?

http://www.youtube.com/watch?v=dk8B_c991kA

http://www.youtube.com/watch?v=xDZSxFLcMVg&feature=player_embedded You are

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The more eggs (X) we have

the more chickens (Y) we have

… or the other way around???

Examples of bidirectional relations

The long contested issue: which came first,

the chicken or the egg?

B. X and Y are mutually related

(

bidirectional causality

)

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… N. of children

Quality of marital relation

N. of Children …

… N. of children

Economic wellbeing

N. of Children …

… Childcare choices

Mothers’ labor force participation

Childcare choices…

… Grandparenting

Cognitive Skills

Grandparenting

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Data show that the higher is the presence of storks (X) in a county the higher is the birth rate (Y)

Therefore we conclude that storks bring babies and cause the increase of the fertility rate

How to reduce fertility rates? Simple! Kill storks!

Examples of spurious correlation

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Problem considered, for example, by Kronmal (1993), JRSSA

Another long contested issue:

do storks bring babies?

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You found a strong positive association between Birth Rate and Stork-Rate. But it’s too early to draw causal conclusion!!!

Ask yourself: is there an alternative explanation to the observed relationship other than a causal effect? Are there common factors?

Think, for example, that in more rural areas both fertility and presence of stork are higher. What happens if we split the sample of counties by rurality?

Examples of spurious correlation (cont’d)

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Urban areas

Rural areas

Regression lines in:

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The relationship between the birth rate and the presence of storks disappear as soon as we hold “rurality” constant.

The relationship we found for the whole sample was due to the fact that both the birth rate and the presence of storks are higher in rural areas and lower in urban areas!!!

The common cause “rurality” falsified the picture!

Examples of spurious correlation (cont’d)

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Note that in the scatterplot each value of the stork presence is associated with many values of birth rate!

We will see that multiple regression is a way to control for confounding effects. Another way is splitting the sample at each value of the confounder (as done here)

If you continue to ignore rurality

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STORKS (X)

FERTILITY (Y)

RURALITY (C)

Confounders

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Rurality is a confounder. Confounders can reduce, reverse, cancel relationships.

The effect of storks represented by the black line was (entirely) confused with the effect of rurality!

In the storks example, X and Y are marginally dependent but conditionally (on C) independent!!!

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“Sleeping with one's shoes on is strongly associated with waking up with a headache”.

Therefore, sleeping with one's shoes on causes headache. But… how much did you drink yesterday?

Common cause: going to bed drunk

“As ice cream sales increase, the rate of drowning deaths increases sharply”

Therefore, eating ice cream causes drowning.

Examples of spurious correlation (cont’d)

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Therefore, eating ice cream causes drowning.

But… when is more likely that people buy ice creams and swim? Common cause: common (seasonal) trend (season of the year)

The common trend problem is very common! It happens every time we have time series data on variables that “naturally” move together.

Ice cream sells

Drownings

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(serious) Example of spurious correlation

In their paper “Driving Status and Risk of Entry Into Long-Term Care in Older Adults” appeared in 2006 in the American Journal of Public Health, Ellen E. Freeman and colleagues study whether not driving increases the probability of entering long-term care (LTC) institutions.

The authors consider that the association between driving or not with being in

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The authors consider that the association between driving or not with being in LCT or not might be spurious because of the elderly health status. They use a multivariate model where health status is controlled for.

The authors find that even after controlling for health (and other factors) not driving increases the likelihood of entry into LTC.

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Related to the role of third variable (confounders).

An association is reversed (change sign or disappear) when the whole sample is split into groups.

It seems that Edward H. Simpson not Homer deserves the merit of having explained it.

Simpson’s paradox

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( From Wikipedia) One of the best known real life examples of Simpson's paradox occurred when the University of California, Berkeley was sued for bias against women who had applied for admission to graduate schools there. The admission figures for the fall of 1973 showed that men

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However when examining the individual departments, it was found that no department was significantly biased against women. The data from the six largest departments is listed below.

Simpson’s paradox (cont’d)

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Bickel et al. ("Sex Bias in Graduate Admissions: Data From Berkeley“; Science 187 (4175): 398–404) concluded that women tended to apply to competitive departments with low rates of admission even among

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Is the weather today good? So, TOSA_I increases the likelihood of having good weather!

Do you have headache after the lecture? It’s just a coincidence!

With a decrease in the number of pirates, there has been an increase

in global warming over the

Causality and Coincidences

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in global warming over the same period.

Therefore, global warming is caused by a lack of pirates.

(This example is used satirically by the parody religion

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It’s a formal framework to think about causal effects (and draw causal inference)

Counterfactual refers to what would have happened if, contrary to fact, the value of X had been something other than what it actually was.

The counterfactual approach to causal

inference

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Example:

Does eating donuts cause Homer being overweight?

Should then the Government increase taxes on donuts?

Fact: Homer eats donuts (X= 1); Y₁ = 38 (Y = Body Mass Index)

Counterfactual: what would have been Homer’s BMI if he has not been eating donuts? Y₀ = ???

The causal effect of eating donuts for Homer is defined as a comparison of the two potential

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Let X be an indicator variable which assumes the value 1 for treated units (donuts) and 0 for untreated, or controls (no donuts).

Each unit, i, has two potential outcomes: Y_1i if X_i=1 and Y_0i if X_i= 0.

But only one potential outcome is observed at a given point in time for each unit (the fundamental problem of causal inference; Holland, 1986).

The fundamental problem of causal inference

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unit (the fundamental problem of causal inference; Holland, 1986).

The observed BMI for Homer is Y₁ = 38; but Y₀ is unobserved

How to solve the fundamental problem and estimate causal effects?

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Find a substitute for Homer’s unobservable counterfactual state.

VS

How to solve the fundamental problem

Donuts No donuts

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Finding a credible counterfactual substitute is the crux of all sound causal inference.

Is it Flanders a good match for Homer? Maybe not… think about third variables (Homer and Flanders are very different in many respects… NOT ONLY wrt eating donuts or not!!!).

?

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Overall lifestyle (eating too much, drinking alcohol, not doing sport) could be associated with eating donuts and BMI. If this is the case the

comparison is not fair for donuts.

VS

How to solve the fundam. problem (cont’d)

Healthy Lifestyle (C)

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The influence diagram is an easy and intuitive way to assess direction of bias caused by not taking into account for confounders. In this case the bias would be positive (“-” * “-” = “+”) because healthy lifestyle reduces both the quantity of donuts consumed and BMI.

Therefore the simple comparison Homer’s BMI (= 38) – Flanders’s BMI (= 25) = 13 would overestimate the effect due ONLY to donuts.

I.e., “true donuts effect” < 13

Donuts (X)

BMI (Y)

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What we do in practice is to focus on average causal effects (people eating donuts vs people not eating donuts) and not individual ones (Homer

Simpson vs Ned Flanders).

We need to collect (or find) data on a sample of people with different levels of X (eating and not eating donuts) who are comparable with respect to third variables (similar overall lifestyle).

How to solve the fundam. problem (cont’d)

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third variables (similar overall lifestyle).

Ideally we would compare people eating donuts (X=1) and others not eating donuts (X=0) with exactly the same lifestyle (comparison group).

Simply, it’s all about finding a good comparison group!

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To estimate the effect of a variable (e.g., education) on another (e.g., political behavior, religiosity, fertility intention…) we could in principle randomly assign people to different groups (e.g., force some children to stop studying, others to go ahead, etc.).

Random assignment guaranties that the treatment groups only differ by the

Randomized experiments vs Observational

Studies

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Random assignment guaranties that the treatment groups only differ by the values of education and not other factors (we remove by design

confounding effects). That’s why random experiments are considered the gold standard for causal inference.

Randomized experiments are very rare in social sciences because of ethical and practical reasons. (For examples see next slide)

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“Moving to Opportunity” (MTO) demonstration sponsored by the U.S. Department of Housing and Urban Development, a housing mobility experiment in five cities in which eligible ghetto residents are randomly assigned to receive (or not) various forms of assistance to relocate.

Research findings using MTO data are summarized by Del Conte, A., and Kling, J. (2001), “A Synthesis of MTO Research on Self Sufficiency, Safety

Examples of randomized experiments in

Social Sciences

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Kling, J. (2001), “A Synthesis of MTO Research on Self Sufficiency, Safety and Health, and Behavior and Delinquency,” Poverty Research News, 5, 3– 6.

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X can be considered a causal variable if it can be manipulated (Holland, 1986). I.e, somebody can (at least in principle) change its values.

E.g., education is a causal variable because people (and/or their parents) can decide how many years they want to study. Similarly, n. of children, place of residence, party identification, etc.

No causation without manipulation

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E.g., taxation, family allowances, quality of public transportation, election law are causal variables. Who decides is the Government (or the

Parliament) in this case.

For gender, race and other immutable characteristics it does not make sense to talk about causal effects. Unless we refer to the consequence of how they are perceived (Greiner and Donald B. Rubin, 2011; The Review of Economics and Statistics). One can hypothesize interventions that might change the decider’s