March 25, 2021

Confounding

Outline

  1. Confounding
    • sources of bias
    • direction of bias
  2. Solutions to Confounding

Example:

Recent mass shooting in Boulder, CO has renewed calls in the United States to impose gun control legislation.

Does reducing the number of guns reduce firearms deaths?

Correlation: Guns and Gun Deaths

POLL

Confounding?

Confounding:

Confounding occurs when these other differences between cases (third variables, e.g. \(W\)) causally affect \(X\) and \(Y\).

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

  1. some variable \(W\) has causal paths toward \(X\) and \(Y\)
  2. (equivalently) there is backdoor path or non-causal path from \(X\) to \(Y\)

Confounding:

Third Variable? Key Attribute Confounding?
Antecedent Variables
(\(W\))
Yes \(W \to X\) If only causal path from \(W\) to \(Y\) contains \(X\): No
If a causal path from \(W\) to \(Y\) excludes \(X\): Yes
Intervening Variables
(\(M\))
Yes \(X \to M \to Y\) No
Reverse Causality No \(Y \to X\) Yes

Upward or Downward bias?

Confounding: Direction of Bias

Product of signs on causal path from \(W \to X\) and \(W \to Y\) gives us direction of bias created by confounding

\(W \xrightarrow{+} X\) \(W \xrightarrow{-} X\)
\(W \xrightarrow{+} Y\) \(Correlation(X,Y)\)
Biased (+)
\(Correlation(X,Y)\)
Biased (-)
\(W \xrightarrow{-} Y\) \(Correlation(X,Y)\)
Biased (-)
\(Correlation(X,Y)\)
Biased (+)

Downward bias

Confounding: Direction of Bias

Confounding: Direction of Bias

Confounding: Direction of Bias

Confounding: Direction of Bias

Confounding: Direction of Bias

Confounding: Direction of Bias

Confounding: Direction of Bias

Confounding: Direction of Bias

Fixing Confounding

POLL

Pandemic Misinformation

Story above not false, but misleading:

  • most widely shared story about the vaccine this year
  • \(.0018\%\) of US vaccine recipients have died
  • apprx. 8000 people die in the US die each day for other reasons.

Pandemic Misinformation

Pandemic Misinformation

What can be done to limit the negative effects of pandemic misinformation?

  • Does thinking about the accuracy of information make people less likely to share misinformation?

Pandemic Misinformation

What if we survey Facebook users:

  • look at previous Facebook history to see if they shared vaccine misinformation
  • ask them if they assess the accuracy of information before sharing links

Does a negative correlation imply causation?

  • What could we do to avoid confounding?

Pandemic Misinformation

Pennycook, et al (2020) run this experiment:

  • Show people pandemic-related stories that have been independently evaluated as false or true
  • “If you were to see the above on social media, how likely would you be to share it?”
  • Randomly assign some to assess the accuracy of non-pandemic news before they look at pandemic news.
  • People “nudged” to think about accuracy 3.9 ppt more likely to share true rather than false stories

Experiments

FPCI: We cannot know the causal effect of \(X\) on \(Y\) for a specific case.

Correlation of \(X\) and \(Y\) for different cases may suffer from confounding

Experiments are a solution

Allow us to treat correlation as an estimate (an inference about) the average causal effect of \(X\) on \(Y\).

  • We can’t know the causal effect for individual cases, but can get the average causal effect across all cases

Experiments

Experiments give us unbiased (no confounding) relationship between \(X\) and \(Y\), with assumptions:

  1. Random Assignment to “Treatment” and “Control”
  2. Exclusion Restriction (only one thing is changing: \(X\))


Technically, there are more assumptions, but not important for this class

Experiments

Randomization solves Confounding

  • Randomization balances cases with same potential outcomes in treatment and control
  • Randomization balances cases with similar values of confounding variable \(W\) in treatment and control (breaks the link \(W \to X\))

Cases are about the same on average: - cases in control are observable “counterfactuals” for cases in treatment - EXACTLY like with random sampling (to the board)

Experiments

Exclusion Restriction means we don’t add confounding

If in the COVID misinformation experiment, “Treatment” group …

  • Asked to assess the accuracy of information (\(X\))
  • Told that their social media shares were tracked by the government (\(Z\))

Two things are different between treatment and control group;

  • we don’t know which one does the work

Experiments

Experiments are a solution to confounding/FPCI

  • We can’t always use them
  • We make trade-offs by using experiments
  • What other options are there?