In [Required
assumptions](https://developers.google.com/meridian/docs/causal-inference/required-assumptions), it was
stated that the 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 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).

## Meridian's causal graph

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.
![Causal DAG with lagged treatment effect](https://developers.google.com/static/meridian/images/lagged-treatment-effect.png)

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\\)) satisfy the backdoor criterion.

The main conclusions are that for each time period, the MMM regression should
condition on:

1. Paid and organic media from the current time period and all preceding time periods, up to an assumed maximum lag duration.
2. Non-media treatment variables of the current time period only.
3. 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.

Meridian can also support a causal graph with lagged controls, as discussed in
[When lagged controls are needed](https://developers.google.com/meridian/docs/advanced-modeling/control-variables#when_lagged_controls_are_needed).

## Meridian's full-funnel causal graph

To resolve the mediator-confounder dilemma, Meridian supports a full-funnel
causal graph using a two-stage approach (see [Full-Funnel
MMM](https://developers.google.com/meridian/docs/advanced-modeling/full-funnel)).

### Single time period

![The full-funnel causal DAG (truncated)](https://developers.google.com/static/meridian/images/full-funnel-truncated.png)

In the diagram, we see the full-funnel causal graph within one time point.

Treatment variables are split into two nodes:

- \\(B\\) denotes Brand marketing, which is marketing expected to influence the brand equity variable \\(G\\)
- \\(O\\) denotes other marketing, which is marketing that may be influenced by the brand equity variable \\(G\\).

Practically, it is helpful to identify which marketing variables are considered
Brand marketing by asking: "Does this marketing variable have a causal effect on
the brand equity variable?" If the answer is yes, it should be included in
\\(B\\). If the answer is no, it should be included in \\(O\\). Recall that an
arrow in a DAG represents a *potential* causal relationship. It does not require
that the causal relationship truly exists. Therefore, each marketing
variable in the \\(O\\) node does not necessarily need to be causally affected
by the brand equity variable. For this reason, it is helpful to first
determine the variables that should be classified as Brand marketing and then
classify the remaining variables as Other marketing.

To give a concrete example, imagine we have **branded GQV (bGQV)** as a
brand equity variable, categorizing it in the \\(G\\) node. A video ad channel
is expected to drive brand equity and increase bGQV. We would consider the
video ad channel a Brand marketing variable, categorizing it in the \\(B\\)
node. On the other hand, a lower-funnel, performance-driven marketing channel
is likely to be driven by brand equity. Thus, we would consider this Other
marketing, categorizing it in the \\(O\\) node. For more on this example, see
the [Meridian Full-Funnel](https://developers.google.com/meridian/notebook/meridian-full-funnel)
walkthrough.

Control variables are also split into three nodes:

- \\(V\\) denotes variables that confound with both Brand marketing and the brand equity variable \\(G\\).
- \\(Z\\) denotes variables that confound with marketing (whether Brand or Other) and the KPI \\(K\\).
- \\(W\\) denotes variables that confound with brand equity and the KPI \\(K\\).

Each of \\(V\\), \\(Z\\), and \\(W\\) are confounders for a causal effect we
estimate in the full-funnel model: \\(V\\) are confounders for the causal effect
Brand marketing has on the brand equity variable, \\(Z\\) are confounders for
the causal effect marketing has on the KPI, and \\(W\\) are confounders for the
causal effect brand equity has on the KPI. Just as in a single-stage MMM, it is
important to consider confounders for all estimated causal effects.

### Multiple time periods with lagged effects

In the following diagram, multiple time periods are incorporated into the graph,
with the number following each node's name denoting the time period. This adds
a diagram for the *following lagged effects assumed in Meridian's full-funnel
causal graph*:

- Brand marketing \\(B\\) on the KPI \\(K\\)
- Other marketing \\(O\\) on the KPI \\(K\\)
- Brand equity \\(G\\) on the KPI \\(K\\)
- Brand marketing \\(B\\) on Brand equity \\(G\\)

![The full-funnel causal DAG](https://developers.google.com/static/meridian/images/full-funnel.png)