March 23, 2021

Confounding

Outline

  1. Example
  2. Confounding
    • definition
    • sources of bias
    • graphs
    • when do we get confounding?

Example:

Karaivanov et al (2020), economists at SFU, investigate:

Have indoor mask mandates reduced COVID cases, on average?

Example:

They compare COVID cases in Ontario Public Health Units (PHU) with and without mask mandates

  • Correlation of mask mandate and COVID cases

Example:

poll

Example:

Why doesn’t correlation imply causation?

Confounding

Confounding

Correlation suffer from two sources of error:

random error: we observe patterns in \(X\) (independent variable) and \(Y\) (dependent variable) by chance, when there is in fact no relationship.

bias (confounding): the observed pattern between \(X\) and \(Y\) is not the true causal relationship between \(X\) and \(Y\).

Confounding

  • What is confounding?
  • Why does it happen?
    • What circumstances make it happen?
  • What is the direction of the bias it produces?

Confounding

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

  • This is not a chance correlation.
  • Two ways to explain why this happens (different explanations, but two sides of the same coin)

Confounding

Explanation 1:

Confounding happens when the cases we observe with different levels of \(X\) have different (factual and counterfactual) potential outcomes of \(Y\).

Example: Imagine we want to know whether mask mandate causes PHU to have lower COVID cases. Because of FPCI, we use correlation:

  • compare PHU 1 (with mask mandate) to PHU 2 (without)

Confounding

In correlation, assume that PHU 1 without mask mandate (counterfactual) is the same as PHU 2 without mask mandate (factual)

PHU Mask Mandate No Mandate
1 \(\mathrm{COVID \ Cases_{PHU \ 1}(Mandate)}\) \(\color{red}{\mathrm{COVID \ Cases_{PHU \ 1}(No \ Mandate)}}\)
\(\Downarrow{=}\) \(\Uparrow{=}\)
2 \(\color{red}{\mathrm{COVID \ Cases_{PHU \ 2}(Mandate)}}\) \(\mathrm{COVID \ Cases_{PHU \ 2}(No \ Mandate)}\)

Confounding

If this substitution is wrong: PHU 1 and 2 have different factual/counterfactual COVID caseloads, correlation is biased.

PHU Mask Mandate No Mandate
1 \(\boxed{\mathrm{COVID \ Cases_{PHU \ 1}(Mandate)}}\) \(\color{red}{\mathrm{COVID \ Cases_{PHU \ 1}(No \ Mandate)}}\)
\(\Downarrow{\neq}\) \(\Uparrow{\neq}\)
2 \(\color{red}{\mathrm{COVID \ Cases_{PHU \ 2}(Mandate)}}\) \(\boxed{\mathrm{COVID \ Cases_{PHU \ 2}(No \ Mandate)}}\)

Confounding

Why do these two PHUs have different potential outcomes?

  • There other differences besides mask mandate…

Explanation 2:

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

This can be understood visually

Causal Graphs

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. “Mask Mandate”, “COVID Cases”)

the arrows convey the direction of causality

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

Causal Graphs

For example

PHU 1 (that had a mask mandate) may have a larger population of university educated adults than PHU 2 (no mask mandate).

  • More educated residents might be more likely to work from home \(\xrightarrow{}\) mask mandate affects fewer people \(\xrightarrow{}\) more likely to implement.
  • More educated residents \(\xrightarrow{}\) work from home \(\xrightarrow{}\) fewer contacts \(\xrightarrow{}\) lower COVID cases

Causal Graphs

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 \(Y\)
    • a chain of two or more arrows that follows arrows backwards out of \(X\), changes direction once and follows arrows toward \(Y\): \(X \leftarrow W \leftarrow Z \rightarrow Y\)

Causal Graphs: Confounding

In reality, we don’t really know the variables and paths on these causal graphs.

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

Activity

In groups…

Imagine you look at the correlation between mask mandates and COVID Cases…

given the correlation you observe, propose a causal graph that would imply that the correlation is the result of confounding.

Share

Confounding:

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 have confounding, some which do not.

Additional Variables: Patterns

  • antecedent variables
    • sometimes confounding
    • sometimes no confounding
  • intervening variables
    • no confounding
  • reverse causality
    • yes, confounding.

Antecedent Variables

antecedent variable: a variable that affects \(X\)

  • e.g. in this path, \(W \xrightarrow{} X \xrightarrow{} Y\), \(W\) is an antecedent variable.

  • antecedent variables (\(W\)) do not produce confounding if the only causal path from \(W\) to \(Y\) passes through \(X\).
  • antecedent variables do produce confounding if there is another causal path from \(W\) to \(Y\) that does NOT include \(X\).

Antecedent Variable: Confounding?

  • No. No “backdoor” path.

Antecedent Variable: Confounding?

  • No. No “backdoor” path.

Antecedent Variable: Confounding?

  • Yes. Mandate \(\xleftarrow{}\) Positives \(\xrightarrow{}\)Stay Home\(\xrightarrow{}\)COVID Cases

Antecedent Variable: Confounding?

  • No. No “backdoor” path; apparent “backdoor” changes directions more than once.

Intervening Variables

intervening variable: a variable that affects \(Y\) and is affected by \(X\).

  • e.g. in this path, \(X \xrightarrow{} M \xrightarrow{} Y\), \(M\) is an intervening variable.

  • intervening variables (\(M\)) do not produce confounding because they are on the causal path from \(X\) to \(Y\). They do not produce backdoor path.

Intervening Variable

  • No backdoor path. Mask mandate affects COVID through mask wearing.

Reverse Causality

reverse causality describes the situation where the dependent variable \(Y\) actually causes the independent variable \(X\).

So while we use the correlation to describe the effect of \(X\) on \(Y\): \(X \to Y\), the correlation in fact is the result of the effect of \(Y\) on \(X\): \(Y \to X\).

This is a special case of bias or 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

Conclusion

Confounding

  1. bias: observed correlation \(\neq\) true casual relationship
  2. Why?
    • cases with different values of \(X\), different in other ways
    • if “third” variable affects \(X\), affects \(Y\) \(\to\) confounding.
  3. Causal graphs help us diagnose possible sources of confounding.

Next: Direction/Size of bias; turning correlation into causation.