1. Correlation: Recap

• definition
• attributes
• problems

2. Problems with Correlation

• Random correlation
• Bias in correlation (confounding)
• Today is a very good day to ask questions

We “solve” the FPCI by comparing factual outcomes in different cases that have different exposures to the “cause”

• \(\mathrm{Case \ A}\) is exposed to a “cause”
• \(\mathrm{Case \ B}\) is not exposed to a “cause”

This is correlation: the degree of association/relationship between the observed values of \(X\) (the independent variable) and \(Y\) (the dependent variable)

• To infer causality, we assume that \(\mathrm{Case \ B}\) is the same as the counterfactual \(\color{red}{\mathrm{Case \ A}}\)

Correlations have

• direction:
  • positive: implies that as \(X\) increases, \(Y\) increases
  • negative: \(X\) increases, \(Y\) decreases
• strength (has nothing to do size of effect):
  • strong: \(X\) and \(Y\) almost always move together (near \(1,-1\))
  • weak: \(X\) and \(Y\) do not move together very much (near \(0\))
• magnitude:
  • this is the how much \(Y\) changes with \(X\).
  • The larger the effect of \(X\) on \(Y\), the steeper the slope

Two types of problems

• random association: correlations between \(X\) and \(Y\) occur by chance and do not reflect \(X\) causing \(Y\).

• bias (spurious correlation, confounding): \(X\) and \(Y\) are correlated but the correlation does not result from a causal relationship between those variables

What is the problem?

If correlation can occur by chance:

• we can observe a positive correlation between \(X\) and \(Y\) and conclude that \(X\) causes \(Y\), even if \(X\) has no effect on \(Y\)
• failure of weak severity

What is the solution?

1. Obtain probability of observing this correlation by chance:
   • strength of correlation
   • size of the sample (\(N\))
   • assume we know the random process generating our observations
2. \(p\)-values: probability of this correlation by chance
   • Given same \(N\), stronger correlation has lower \(p\)
   • Given same strength, correlation with more \(N\) has lower \(p\)
   • probability we incorrectly infer a real relationship (due to chance), if used correctly

\(p\) hacking:

What is the problem with looking at many correlations and reporting only those that are “significant”? (\(p < 0.05\))

• We play a game where you win if you roll “20” on a 20-sided die (probability is 1 in 20 (\(0.05\))
• What is the probability you win if you roll once?
• What is the probability you win if you roll twenty times?

• 64%! Looking at lots of correlations, you are likely to find at some “unlikely” correlations

Practice

Mueller and Schwarz (2023) investigate:

During the period from 2015 to the end of 2017, Trump posted more than 300 messages that can be classified as “Anti-Muslim”.

Did Trump’s tweeting of anti-Muslim messages on social media increase anti-Muslim hate crimes?

We can’t observe the US in the absence of Trump tweeting against Muslims, so authors use correlation…

Trump’s Twitter and Hate Crimes

\(X\) (Independent Variable): Trump’s Twitter gained attention after he announced run for President (2015-2017)

Trump’s Twitter and Hate Crimes

\(Y\) (Dependent variable): When Trump gained prominence, anti-Muslim hate crimes increased

As Trump’s Tweeting against Muslims reached more people (change in observed \(X\)), anti-Muslim hate crimes increased (change in \(Y\))

In groups, discuss:

Is this correlation be convincing that Trump’s tweets caused anti-Muslim hate crimes?

Why or why not?

In this case, correlation “solves” FPCI by plugging in these factual potential outcomes

Why doesn’t correlation imply causation?

confounding is when there is a systematic observed correlation between \(X\) and \(Y\) that does NOT reflect the causal effect of \(X\) on \(Y\).

• This is not a chance correlation: if we looked at more data, relationship would persist
• Two ways to explain why this happens (different explanations, but two sides of the same coin)

Mueller and Schwarz look at the correlation of Trump’s Twitter activity and Hate Crimes over time:

When Trump tweeted more (and had more followers) (2015-2017), hate crimes were higher than when Trump tweeted less (and had fewer followers) (2010-2014).

• What do we have to assume in order for this correlation to imply causation? (HINT: Think about the potential outcomes we just looked at)

In order for Mueller and Schwarz’s correlation to imply causation, need to assume that:

\(\color{red}{\mathrm{AntiMuslim \ Hate \ Crime_{USA \ 2015-17}(No \ Trump \ Tweets)}}\) \(=\) \(\mathrm{AntiMuslim \ Hate \ Crime_{USA \ 2010-14}(No \ Trump \ Tweets)}\)

If assumption is wrong…

and Anti-Muslim hate crimes in 2015-2017 would have been different from 2010-2014 even without Trump’s Tweets…

…this comparison leads to confounding.

Confounding

In correlation, Mueller and Schwarz assume that US (2015-17) without Trump tweets (counterfactual) is the same as US (2010-14) without Trump tweets (factual)

Confounding

If this substitution is wrong: USA in 2010-14 vs USA 2015-17 have different potential outcomes of hate crime, correlation is biased.

Maybe \(\mathrm{Hate \ Crime_{USA \ 2015-17}(Trump \ Tweets)} = \\ \color{red}{\mathrm{Hate \ Crime_{USA \ 2015-17}(No \ Tweets)}}\): hate crimes would have gone up ANYWAY

Why does confounding happen?

Explanation 1:

Confounding happens when cases that experience different levels of \(X\) have systematically different (factual and counterfactual) potential outcomes of \(Y\).

Cases with different levels of cause \(X\)

• systematically different in their factual/counter-factual values of \(Y\).

Does social media use cause increased polarization?

Individual data on social media use and affective polarization from ANES 2024

Why does confounding happen?

Explanation 1:

Confounding happens when cases that experience different levels of \(X\) have systematically different (factual and counterfactual) potential outcomes of \(Y\).

What does that look like in the context of social media consumption and polarization?

::BOARD::

Why does confounding happen?

Explanation 1:

Confounding happens when cases that experience different levels of \(X\) have systematically different (factual and counterfactual) potential outcomes of \(Y\).

Cases that actually receive cause \(X\)…

1. have a different baseline outcome (different \(Y\) absent \(X\)): selection bias
2. respond differently to cause \(X\): heterogeneity bias

… than cases not actually receiving \(X\)

Explanation 2:

Confounding occurs when there are other differences between cases (call them variables, e.g. \(W\), etc.) that causally affect \(X\) and \(Y\).

The easiest way to understand this is visually.

Causal graphs represent a model of the true causal relationships between variables.

the nodes or dots correspond to variables

• can be labeled with generic names for independent/dependent variables (\(X\), \(Y\)) or meaningful names (e.g. “Trump Tweets”, “Hate Crimes”)

the arrows convey the direction of the flow of causality

• \(X \rightarrow Y\) means that \(X\) causes changes in \(Y\)
• \(X \leftarrow W\) means that \(W\) causes changes in \(X\)

Arrows alone do not indicate whether \(X\), e.g., increases or decreases \(Y\).

For example

Did Trump anti-Muslim tweets cause hate crimes?

• Islamist terrorist attacks \(\xrightarrow{}\) Trump anti-Muslim tweets
• Islamist terrorist attacks \(\xrightarrow{}\) News Coverage
• News coverage \(\xrightarrow{}\) anti-Muslim attitudes \(\xrightarrow{}\) Hate crimes

Did Trump anti-Muslim tweets cause hate crimes?

Maybe…

In a causal graph, there is confounding of correlation of \(X\) and \(Y\) if…

1. any variable \(W\) has a causal path (of any length) toward \(X\) and \(Y\)
2. (equivalently) there is backdoor path or non-causal path from \(X\) to \(Y\)
   • a chain of two or more arrows that follows arrows backwards from of \(X\), changes direction once and follows arrows toward \(Y\): \(X \leftarrow W \leftarrow Z \rightarrow Y\)

Does social media use cause increased polarization?

In groups: what other variables might affect social media use? polarization?

We don’t know the True causal graph (if we did, we wouldn’t need work so hard to evaluate causal claims)

Instead, these causal graphs help us think through possible scenarios that might produce bias/confounding of the correlation between \(X\) and \(Y\).

These examples illustrate the possibility that if causal graphs include variables in addition to the independent and dependent variables, there is a risk of confounding or bias.

Do all additional variables produce confounding?

No… We will discuss three different patterns of variables: some of which produce confounding, some which do not.

The most serious threat to empirical evidence of causality is confounding:

• We can’t observe causal effects of \(X\) for individual cases (FPCI)
• Using correlation may lead us astray if other factors affect \(X\) and \(Y\)
• (theories about) Confounding can be diagnosed visually