The definitions described in the previous sections can be extended for channels with reach and frequency data. The potential outcomes can be written more generally as \( \overset \sim Y_{g,t}^{ \left( \left\{ x_{g,t,m}^{(\ast)} \right\}, \left\{ r_{g,t,n}^{(\ast)} \right\}, \left\{ f_{g,t,n}^{(\ast)} \right\} \right) } \)

The incremental KPI of the \(q^{th}\) channel with reach and frequency data is defined as:

Where \(r^{(0)}_{g,t,n}\) denotes the observed reach values for all channels except channel \(q\), which is set to zero everywhere. More specifically:

- \(r^{(0,q)}_{g,t,q} = 0\ \forall\ g,t\)
- \(r^{(0,q)}_{g,t,n} = r_{g,t,n}\ \forall\ g,t,n \neq q\)

Note that the frequency counterfactual values don't matter when the reach is zero; the incremental KPI should be zero regardless. These are arbitrarily set to historical values in this definition.

The ROI of the \(q^{th}\) channel with reach and frequency data is defined as:

\[ROI^{(rf)}_q = \dfrac{IncrementalSales^{(rf)}_q}{Cost^{(rf)}_q}\]

Where \(Cost^{(rf)}_q=\sum\limits_{g,t} \overset \sim r_{g,t,q}\).

To define response curves, note that there are many ways to scale spend for channels with reach and frequency data. For any given spend level, there are any number of reach and frequency combinations that can result in that spend level. Meridian focuses primarily on two types of response curve:

A

*reach*response curve where reach is scaled, holding frequency constant at the historical values for each geo and time period.A

*frequency*response curve where frequency is scaled, holding reach constant at the historical values for each geo and time period.

The reach response curve is defined as the following function:

Where \(r_{g,t,n}^{(\omega,q)}\) denotes the observed reach values for all channels except channel \(q\), which is scaled by \(\omega\) everywhere. More specifically:

- \(r_{g,t,q}^{(\omega,q)}=\omega \cdot r_{g,t,q}\ \forall\ g,t\)
- \(r_{g,t,n}^{(\omega,q)}=r_{g,t,n}\ \forall\ g,t,n \neq q\)

The frequency response curve is defined as the following function:

Where \(f_{g,t,n}^{(\omega,q)}\) denotes the observed frequency values for all channels except channel \(q\), which is scaled by \(\omega\) everywhere. More specifically:

- \(f_{g,t,q}^{(\omega,q)}=\omega \cdot f_{g,t,q}\ \forall\ g,t\)
- \(f_{g,t,n}^{(\omega,q)}=f_{g,t,n}\ \forall\ g,t,n \neq q\)

Note that for \(\omega < \dfrac{1}{min_{g,t} f_{g,t}}\), the counterfactual frequency \(\omega \cdot f_{g,t,q}\) will be less than one for some combinations of \(g,t\). Although it is not possible to have average frequency values below one, Meridian's model specification allows incremental KPI to be estimated for such implausible values. Be careful when interpreting response curves for such small values of \(\omega\).