March 28, 2018
If \(X \rightarrow Y\):
It is possible that \(X,Y\) are correlated, without causation
spurious correlation: when the observed correlation between \(X\) (independent variable) and \(Y\) (dependent variable) inaccurately reflects the true causal relationship \(X \rightarrow Y\)
\[Correlation_{True}(X,Y)  Correlation_{Observed}(X,Y) \neq 0\]
confounding variables are the source of spurious correlation:
All of these are examples of bias: true causal relationship of \(X,Y\) not observed due to confounding
designbased solutions:
Choose comparison so that we eliminate possible confounding variables
Compare cases in same place and time
Compare cases in same place and time with different exposure to the cause
Compare same case to itself over time
Compare same case to itself over time
Compare same case to itself over time against another case to itself
Compare same case to itself over time against another case to itself
Compare cases where cause/independent variable/\(X\) is (asif) randomly assigned by "nature"
Compare cases where cause/independent variable/\(X\) is (asif) randomly assigned by "nature"
How Bias Solved 
Which Bias Removed 
Assumes  Internal Validity 
External Validity 


Adjustment  Hold constant 
All measured confounding variables 
Condition all confounders 
Lowest  Highest 
Similar Cases  Hold constant 
Cases' shared confounding variables 
No diff. b/t cases 
Middle  Middle 
Same Case  Hold constant 
Case's unchanging confounding variables 
No confounding trends 
Middle  Middle 
Diff in Diff  Hold constant 
Case's unchanging variables Cases' shared trends 
Cases have parallel trends 
Higher  Lower 
Natural Experiment  Break \(W \rightarrow X\) link  All confounding variables  \(X\) asif random  Highest  Lowest 