Draw your assumptions before your conclusions

• Causal diagrams
• Directed acyclic graphs (DAGs)

DAGs

Arah, O. A. 2008. “The Role of Causal Reasoning in Understanding Simpson’s Paradox, Lord’s Paradox, and the Suppression Effect: Covariate Selection in the Analysis of Observational Studies.” Emerg Themes Epidemiol 5 (February): 5. https://doi.org/10.1186/1742-7622-5-5

DAGs

Arah, O. A. 2008. “The Role of Causal Reasoning in Understanding Simpson’s Paradox, Lord’s Paradox, and the Suppression Effect: Covariate Selection in the Analysis of Observational Studies.” Emerg Themes Epidemiol 5 (February): 5. https://doi.org/10.1186/1742-7622-5-5

When we observe a statistical association between two variables

1) Two variables are a cause and the consequence

2) Two variables have a common cause (a confounder)

• A and Y are associated even though A does not cause Y
• Example: L - smoking, A - carrying a lighter, Y - lung cancer

When we observe a statistical association between two variables

3) Two variables share a common consequence or a child of a common consequence, which was conditioned on

• E and D are associated even though E does not cause D (or vice versa)

Hernán, Miguel A., Sonia Hernández-Díaz, and James M. Robins. 2004. “A Structural Approach to Selection Bias.” Epidemiology 15 (5): 615–25

Associatioin vs Causation

• "Association, unlike causation, is a symmetric relationship between two variables; thus, when present, association flows between two variables regardless of the direction of the causal arrows"

• Association is not causation

Marginal vs conditional probabilities

• Conditional
  • The probability that an event Y occurs, given we know some other event A has occurred, is the conditional probability of Y given A
  • $Pr\left(Y|A\right)$
• Marginal(unconditional)
  • $Pr\left(Y\right)$
  • Remembering standardization/adjustment formula
    • $Pr\left(Y^a\right)={\Sigma }_{l}Pr(Y=1|A=a,L=l)\ast Pr(L=l)$
    • Probability of PO $Y^a$ marginal over confounder $L$

Causal Inference in Statistics: A Primer. Judea Pearl, Madelyn Glymour, Nicholas P. Jewell

• $A\perp \perp Y|B$
• $A\perp /\perp /Y$

Conditioning

• $A\perp \perp Y|L$

• $A\perp /\perp /Y|L$

Biasing path

For the association between A and Y

• A --> [L] <--Y

Backdoor path

• A <-- L --> Y

Exchangeability, positivity, consistency

inference is never assumption-free

Systematic bias

• Infinite sample size won't help eliminating systematic bias
• The magnitude of the effect is off

• Structural definition of bias

  • Common causes --> confounding

  

  • Conditioning on common effects --> selection bias (collider stratification bias)

  

  

  • Information (measurement) error --> later in the book

Effect modification structure

• Causal vs surrogate efect modifiers

• Structure of the association of the effect modifier with the outcome matters

Interaction in DAGs

• Augmented DAGs

Right. Personally, I’ve found it useful to augment the DAG for heterogeneity as in these from my SER 2015 conference presentation: pic.twitter.com/8QJ26BTx6Y

— Onyi Arah🎄waiting for vaccine🌍 MD, DSc, PhD (@oacarah) September 5, 2019