Interaction¶

This chapter considers a system that consists multiple treatment variables.

From Joint Intervention to Interaction¶

Intervention is to manually set the value of treatment variable ignoring the original distribution of the variable. Intervention on multiple variables are called joint intervention.

Let the treatment variable be $A$ and $B$. If the causal effect (measured in risk difference) of treatment $A$ is different with different values of $B$, then $B$ has an additive interaction on $A$. Notice that $B$ has an interaction on $A$ is equivalent to $A$ has an interaction on $B$.

\[ P(Y^{a = 1, b = 1} = 1) - P(Y^{a = 0, b = 1} = 1) \not = P(Y^{a = 1, b = 0} = 1) - P(Y^{a = 0, b = 0} = 1) \]

Superadditive and subadditive

Rewrite the above inequality as

\[ \begin{aligned} P(Y^{a = 1, b = 1} = 1) - P(Y^{a = 0, b = 0} = 1) &\not = [P(Y^{a = 1, b = 1} = 1) - P(Y^{a = 0, b = 1} = 1)] \\ &+ [P(Y^{a = 1, b = 0} = 1) - P(Y^{a = 0, b = 0} = 1)] \end{aligned} \]

If the $\not =$ can be replaced by $>$ or $<$, then the interaction is called superadditive or subadditive, respectively.

Similarly, if the causal effect (measured in risk ratio) of treatment $A$ is different with different values of $B$, or vice versa, then there is a multiplicative interaction between $A$ and $B$. If the $\not =$ can be replaced by $>$ or $<$, then the interaction is called supermultiplicative or submultiplicative, respectively.

\[ \frac{P(Y^{a = 1, b = 1} = 1)}{P(Y^{a = 0, b = 0} = 1)} \not = \frac{P(Y^{a = 1, b = 0} = 1)}{P(Y^{a = 0, b = 0} = 1)} \times \frac{P(Y^{a = 1, b = 1} = 1)}{P(Y^{a = 0, b = 1} = 1)} \]

Idenfication of Interactions¶

To identify the interaction between $A$ and $B$, we need make sure that consistency, exchangeability, positivity for $A$ is satisfied for all values of both $A$ and $B$.

Notice the difference between effect modification and interaction. If $B$ is randomly assigned, then $B$ interacts with $A$ is equivalent to that $B$ serves as an effect modifier for $A$.

Effect modification v.s. interaction

In an interaction, the two variables are of equal status, we can intervene on either or both of them. In effect modification, one variable is the treatment and the other is the modifier, we can only intervene on the treatment variable.

For treatment variables, we need consistency, exchangeability, positivity. But such conditions are not required for modifier variables.

Response Type¶

For each individual, the potential outcome pattern for receiving different levels of treatment is called response type.

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For binary outcome and one binary treatment, there are $2^{2^1} = 4$ response types. For binary outcome and two binary treatments, there are $2^{2 ^ 2} = 16$ response types.

Monotonicity

For a response type, when treatment variables except $A$ are fixed, if $Y^{a = 1} \geq Y^{a = 0}$, then the causal effect of $A$ on $Y$ is monotonic.

Sufficient Cause Interaction¶

Consider the mechanism of yielding certain outcome by controlling one binary treatment variable $A$. There are three types of mechanisms to inevitably yield the outcome.

With treatment $A$.
Without treatment $A$.
$A$ is irrelevant to the outcome.

Since the outcome is stochastic, there exists some unknown factors $U_1, U_2, U_0$ that leads to the outcome, respectively. For example, we can say $\{A = 1, U_1 = 1\}, \{A = 0, U_2 = 1\}$ or $\{U_3 = 1\}$ are sufficient-component causes for the outcome.

For models with $k$ binary treatment variables, there are $3^k$ possible mechanisms. If any individual $U_i = 1$ corresponds to multiple variables exists, then there exists sufficient cause interaction among these variables.

Let $k = 2$ and the treatment variables be $A, B$, respectively. The sufficient cause interaction is synergistic if $A = 1, B = 1$ is a sufficient-component cause for the outcome. The sufficient cause interaction is antagonistic if $A = 1, B = 0$ or $A = 0, B = 1$ are sufficient-component causes for the outcome.

When monotonicity holds, some sufficient causes are not possible.

Summary¶

Sufficient-component-cause framework and the counterfactual framework focus on different interaction questions:

Sufficient-component-cause framework focuses on the causal mechanism (reason) of the outcome.
Counterfactual framework focuses on the causal effect of the treatment.