March 13, 2019
A research design has internal validity when the observed relationship between \(X\) and \(Y\) it finds is not a biased (systematically incorrect) estimate of the causal effect of \(X\) on \(Y\) (does not suffer from confounding).
Research design has external validity when the \(X\) and \(Y\) we examine match the causal theory and the cases we study match the set of cases/population the causal theory is supposed to describe
Recall what hypotheses say: If \(X\) causes \(Y\)
We should observe that as \(X\) changes, the potential outcomes of \(Y\) change.
But FPCI says we can only ever see one potential outcome per case.
We always examine the relationship between the observed \(X\) and the observed \(Y\)
To infer a causal relationship between \(X\) and \(Y\) based on relationship between the observed \(X\) and the observed \(Y\), one of the following must be true
All "solutions" to the FPCI make assumptions about the cases we compare that allow us to accept one of these two points (previous slide), so we can infer causality from how observed values of \(X\) and \(Y\) are related.
We want to know whether having fewer guns reduces suicide, accidents, and aggressive uses of guns.
Imagine the following experiment:
We collect data on legal gun owners. We randomly assign half of them to receive a letter from the government that offers to pay them money to hand over their guns (a gun buyback). The other half receives no letter. We then compare suicide rates, accident rates, and gun victimization rates among the friends and families of people who handed over their guns against people who did not hand over their guns. We observe that suicide, accident, and gun victimization rates were lower in the "give-up-guns" group. Can we infer that having fewer guns caused a reduction?
degree of association or relationship between the observed values taken by two variables (\(X\) and \(Y\))
But with some assumptions: correlation \(\to\) causation
To understand assumptions, need to know what problems arise
bias (spurious correlation, confounding): \(X\) and \(Y\) are correlated but the correlation does not result from causal relationship between those variables
random association: correlations between \(X\) and \(Y\) occur by chance and do not reflect
confounding occurs when some other variable \(W\) is causally linked to \(X\) (independent variable) and \(Y\) (dependent variable).
If we diagram causal links between variables using this notation: \(X \to Y\) implies \(X\) causes \(Y\), then…
confounding occurs when there is a path between \(X\) and \(Y\) that is non-causal (goes the "wrong way" on at least one arrow)
Do all third variables create confounding/bias?
\(W\) produces no confounding under the following conditions:
intervening variable: a variable through which \(X\) causes \(Y\)
\[X \rightarrow W \rightarrow Y\]
antecedent variable: a variable that affects \(Y\) only through \(X\)
\[Z \rightarrow X \rightarrow Y\]
\(Z\) is intervening variable