Potential Outcome Framework

There are at least three different school of thoughts regarding causality: 1) granger causality; 2) Rubin’s potential outcome framework and 3) Pearl’s causality. Each of these thoughts have their own pros and cons. I will quickly discuss about the Rubin’s potential outcome framework and show the fundamental problem of causal inference. For an example, say a researcher wants to study impact of certain treatment (say performance enhancing drug say \(T\)) on some outcomes (say capacity to solve riddle quizzes say \(Y\)).

He randomly sampled the total population and divided the total sample population \(N\) to the control group of \({{N}_{co}}\) and treatment group \({{N}_{Tr}}\) such that \({{N}_{co}}+{{N}_{Tr}}=N\).

Say the treatment group are exposed to the treatment i.e. \(T=1\) for intervening with performance enhancing drug and control group is not intervened or given a placebo i.e. \(T=0\).

After this experiment, researcher takes test of all the individual and records numbers of minutes to solve the questions.

The average outcome of treated group is \(E\left[ Y\left( 1 \right)|T=1 \right]\) and average outcome of control group is given as \(E\left[ Y\left( 0 \right)|T=0 \right]\). Both averages can be estimated easily as the data are observed. The simple difference of means \(SDoM\) between average outcome of treated and average outcome of control group given as:

\[SDoM=E\left[ Y\left( 1 \right)|T=1 \right]-E\left[ Y\left( 0 \right)|T=0 \right]\]

The potential outcome framework, however, look at this problem and ask a simple question i.e. “What is the counterfactual?”. In another word, what would be the outcome of those who are treated if they were not been treated i.e. \(E\left[ Y\left( 0 \right)|T=1 \right]=??\). Similarly, what would be the outcome of control if they had been treated i.e. \(E\left[ Y\left( 1 \right)|T=0 \right]=??\). Then, the average treatment effect is given as:

\[ATE=\left\{ E\left[ Y\left( 1 \right)|T=1 \right]-E\left[ Y\left( 0 \right)|T=1 \right] \right\}-\left\{ E\left[ Y\left( 0 \right)|T=0 \right]-E\left[ Y\left( 1 \right)|T=0 \right] \right\}\]

Where,

\(E\left[ Y\left( 1 \right)|T=1 \right]\) represents, given treated, the average outcome of treated group (observed in data). \(E\left[ Y\left( 0 \right)|T=1 \right]\) represents, given treated, the average outcome of treated group if they were controlled (unobserved). \(E\left[ Y\left( 0 \right)|T=0 \right]\) represents, given controlled, the average outcome of controlled group (observed in data). \(E\left[ Y\left( 1 \right)|T=0 \right]\) represents, given controlled, the average outcome of controlled group if they were treated (unobserved).

We can also correct this estimate with sampling weights as:

\[ATE=\lambda \left\{ E\left[ Y\left( 1 \right)|T=1 \right]-E\left[ Y\left( 0 \right)|T=1 \right] \right\}-\left( 1-\lambda \right)\left\{ E\left[ Y\left( 0 \right)|T=0 \right]-E\left[ Y\left( 1 \right)|T=0 \right] \right\}\]

Where, \(\lambda =\frac{{{N}_{Tr}}}{N}\) or proportion of treated group and \(1-\lambda =1-\frac{{{N}_{Tr}}}{N}=\frac{{{N}_{co}}}{N}\) proportion of control group.

If we closely look at estimate of \(ATE\) we find that each counterfactual of treated and control group is missing data problem, this is the fundamental problem of causal inference.

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