Causal Inference - What If?

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# Causal Inference - What If?
### MAJ
### Mar 14, 2021

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## Recap ...

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$$
`\begin{aligned}
A, Y
\end{aligned}`
$$
]
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+ **Treatment and outcome**
random variables for treatment and outcome
]

.pull-left[
$$
`\begin{aligned}
Y^{a = 1}, Y^{a = 0}  
\end{aligned}`
$$
]
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+ **Counterfactual/Potential outcome**;
outcome that would have been observed under treatment value `$a$` - one outcome is realised, assumes no *interference*
]

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$$
`\begin{aligned}
Y^{a = 1} \ne Y^{a = 0}
\end{aligned}`
$$
]
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+ **Causal effect for individual** implied but not identifiable
]

.pull-left[
$$
`\begin{aligned}
Y = Y^A
\end{aligned}`
$$
]
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+ **Consistency**; if `$A_i = a$` then `$Y$` equal to potential outcome `$Y_i^a = Y_i^A = Y_i$` when observed
]

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## Recap ...

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$$
`\begin{aligned}
A = a
\end{aligned}`
$$
]
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+ **SUTVA**;
stable-unit-treatment-value - one version of treatment 
]

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$$
`\begin{aligned}
E(Y^{a = 1}) \ne E(Y^{a = 0})
\end{aligned}`
$$
]
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+ **ACE**;
avg causal effects for specified outcome of interest and treatments for a well defined population
]

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$$
`\begin{aligned}
P(Y=1|A=1) &= P(Y=1|A=0)
\end{aligned}`
$$
]
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+ **Independence**;
`$A$` does not predict `$Y$`, `$Y \perp A$` and `$A \perp Y$`
]

.pull-left[
$$
`\begin{aligned}
&E[Y|A=1] \text{ vs. } E[Y|A=0] \\
&E[Y^{a=1}] \text{ vs. } E[Y^{a=0}] 
\end{aligned}`
$$
]
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+ **Association vs Causation**;
difference in risk in population determined by actual treatment vs difference in risk under two different treatment values
]

+ Under what conditions can real world data be used for causal inference?

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## Randomized experiments

Randomized experiments address the main components of any causal question:

+ Does an action,

+ affect an outcome,

+ in a specific population?

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### Randomisation

+ Interested in potential outcomes `$Y^{a=1}$` and `$Y^{a=0}$`

+ Only see `$Y$` under the treatment `$A$` that participant received

+ In other words, a missing data problem

+ Randomisation ensures missing values occurred by chance

+ Expected to produce **exchangeability**

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### Exchangeability

+ Counterfactual outcome and actual treatment are independent; `$Y^a \perp A$` for all `$a$`

+ What?

+ Doesn't matter which group is assigned to treatment and which group is assigned to control

+ Counterfactual outcome `$Y^a$` is a fixed characteristic (like genetic makeup) existing before treatment was randomly assigned

+ `$Y^a$` encodes the outcome when assigned to `$a$`

+ If `$A$` is randomised, then is independent of your fixed counterfactual `$Y^a$`

+ `$\implies$` association equivalent to causation

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### Exchangeability

A randomised experiment **implies exchangeability**, `$Y^a \perp A$`, but if there is a treatment effect, then `$Y \perp A$` **does not hold** because treatment is assoc with observed outcome.

+ `$Y^a \perp A$` and `$Y \perp A$` are different things

+ `$Y^a \perp A$` - independence between **counterfactual** and observed treatment

+ `$Y \perp A$` - independence between the **observed outcome** and the observed treatment

+ Equal risk under no treatment (and treatment) in the treated and untreated groups implies exchangeable

$$
`\begin{aligned}
P(Y^{a} = 1 | A=1) = P(Y^{a} = 1 | A=0) \quad \forall a \implies \text{exchangeable}
\end{aligned}`
$$

+ Infeasible to test IRL because no counterfactual available to us

+ Assumption gives us equivalence of association and causality

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### Conditional randomisation

+ Previously considered **marginally randomised** experiment

+ Stratified randomisation `$\implies$` **conditionally randomised** experiments

+ Conditionally randomisation leads to conditional exchangeability `$Y^a \perp A | L$`

+ Observed risks are equal to the counterfactual risks within `$L$`

+ OK, so how do we estimate the causal effects now?

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### Standarisation

+ Average causal effect derived from weighting the within `$L$` effect estimates

+ Marginal counterfactual risk `$P(Y^a = 1)$` is the weighted average of the stratum-specific risks `$P(Y^a = 1|L=0)$` and `$P(Y^a = 1|L=1)$` with weights equal to proportion of individuals in population with `$L=0$` and `$L=1$`

$$
`\begin{aligned}
P(Y^a = 1) = \sum_l P(Y^a = 1|L = l)P(L=l)
\end{aligned}`
$$

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### Inverse probability weighting

+ Mathematically equivalent to standardisation

+ Based on the observed data:

.pull-left[
$$
`\begin{aligned}
&P(L=0) = 0.4 \\
&P(A=0|L=0)=0.5 \\
&P(Y=0|A=0, L=0) = 0.75 \quad \text {etc.}
\end{aligned}`
$$
]
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<img src="data:image/png;base64,#fig/tree1.png" width="100%" />
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+ Used to create the causal risk ratio

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### Inverse probability weighting

+ Under conditional exchangeability, simulate counterfactuals

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<img src="data:image/png;base64,#fig/tree2a.png" width="90%" />
Population with all untreated (10 die) gives `$P(Y^{a=0}=1)$` 
]

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<img src="data:image/png;base64,#fig/tree2b.png" width="90%" />
Population with all treated (10 die) gives `$P(Y^{a=1}=1)$`
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+ Causal risk ratio: `$P(Y^{a=1}=1)/P(Y^{a=0}=1) = 1$`

+ Weights arise from `$1/f(A|L)$`; one over the conditional probability of treatment

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### Summary

+ Randomisation allows us to assume exchangeability

+ Exchangeability allows us to infer the causal effects

+ Potential outcome independent of treatment

+ Stratified randomisation leads to conditional exchangeability

+ Standardization uses the probability of the strata and the
conditional probability of outcome given treatment and strata membership

+ IP weighting uses the conditional probability of treatment given the covariate

+ The end