In Required assumptions, it was stated that conditional exchangeability assumption holds if you assume a causal graph that meets the backdoor criterion.
A causal graph shows the relationship between variables. Variables are grouped into collections (nodes), and an arrow between nodes indicates that a causal effect might exist in the direction of an arrow. An arrow does not necessarily indicate that a causal relationship exists between every pair of variables, but it does indicate that a causal relationship cannot exist in the reverse direction for any pair of variables.
The backdoor criterion (Pearl, J., 2009) states that given a causal diagram, a set of variables \(Z\) satisfies the backdoor criterion relative to a treatment variable \(X\) and the response variable \(Y\) if both of the following are true:
- No node in \(Z\) is a descendant of \(X\), and
- \(Z\) blocks every path between \(X\) and \(Y\) that contains an arrow into \(X\)
Marketing mix modeling (MMM) is used to estimate the causal effect of paid media, organic media, and non-media variables on a KPI (such as sales). So paid media, organic media, and non-media are the treatment variables (\(X\)) and the KPI is the response variable (\(Y\)). To estimate this causal effect from an MMM regression, the MMM must condition on a carefully selected set of control variables that meets the backdoor criterion. To paraphrase the backdoor criterion conditions:
- You must not control for any mediators. Mediators are variables that lie in the causal pathway between \(X\) and \(Y\).
- You must control for all confounders. Confounders are variables that have a causal effect on both \(X\) and \(Y\).
The MMM treatment variable is a collection of any combination of paid media, organic media, and non-media treatment variables indexed over both geo and time. It is is unwieldy to represent the entire treatment in a graph, so consider a simplified graph that represents only two time periods within a single geo. Geos are assumed to be independent, so the same graph can be used to represent any geo, and there are no arrows or relationships between geos. Two time periods are enough to describe the pattern of lagged treatment effects, which you can assume is repeated indefinitely into the future (or up to some maximum lag duration).
In the following diagram, \(T\) denotes paid media, organic media, and non-media treatment variables, \(C\) denotes controls, and \(K\) denotes the KPI. The number following each variable denotes the time period. Within each time period, assume that treatment affects sales, and that controls affect both treatment and sales. In the following diagram, \(T\) from a previous time period affects sales in the current time period ("lagged effect"). The Meridian regression model applies adstock to paid and organic media, but not to non-media treatments. This effectively assumes that non-media treatments don't have lagged effects. Including non-media treatments in node \(T\) is still valid because an arrow indicates that a causal effect may exist between any pair of variables in connected nodes. Including non-media treatments in node \(T\) makes the DAG presentation cleaner, and the DAG is still valid for determining which variables satisfy the backdoor criterion.
Consider the task of estimating the causal effect of treatment (\(T1\) and \(T2\)) on the KPI for time period 2 (\(K2\)). From the graph, you can see that the time 2 controls (\(C2\)) satisfies the backdoor criterion.
The main conclusions are that for each time period, the MMM regression should condition on:
- Paid and organic media from the current time period and all preceding time periods, up to an assumed maximum lag duration.
- Non-media treatment variables of the current time period only.
- Control variables of the current time period only.
A few noteworthy details to consider are:
- An arrow from \(C1\) to \(C2\) has no implication on which variables to include in the regression.
- An arrow from \(C1\) to \(K2\) requires the regression to include lagged control variables. In practice, it is best to avoid this if at all possible, as it could significantly increase the number of regression parameters.
- An arrow from \(T1\) to \(C2\) is problematic. In this case, \(C2\) is both a mediator and a confounder. A single MMM regression model cannot be used to recover the causal joint treatment effect.
- Adding the path \(T2 \leftarrow K1 \rightarrow K2\) is problematic for the same reason. In this case, \(K1\) acts as both a mediator and confounder.