Input data

Define the following index variables:

• \(g=1,\ldots,G\) indexes the geographical units.

• \(t=1,\ldots,T\) indexes the time units.

  For paid and organic media variables, data for time periods \(t<1\) can be included in the model input data to accurately model lagged effects in the earlier time periods. If data for \(t<1\) is not provided, it is assumed that there is no media execution prior to \(t=1\).

• \(i=1,\ldots,N_C\) indexes the control variables

• \(i=1,\ldots,N_N\) indexes the non-media treatments

• \(i=1,\ldots,N_M\) indexes the paid media channels without reach and frequency data

• \(i=1,\ldots, N_{OM}\) indexes the organic media channels without reach and frequency data

• \(i=1,\ldots,N_{RF}\) indexes the paid media channels with reach and frequency data

• \(i=1,\ldots, N_{ORF}\) indexes the organic media channels with reach and frequency data

Meridian requires two main data arrays as model inputs (KPI and paid media). Organic media and non-media treatments can be also provided as optional inputs if they are available. For paid and organic media channels with reach and frequency data available by geo and time period, the reach and frequency data can optionally be used instead of a single media metric. You also have the option to include controls that are confounding variables or strong predictors of the KPI. It is preferable to provide revenue data (if the KPI is not revenue) and media spend data (if the media unit is not spend), which allows units to be converted to a currency value for ROI calculations.

Data	Dimensions	Model Input: Raw Units	Model Input: Unit Value	Scaled Units (used in model equation)	Value/Cost
KPI	$$G \times T$$	$$\overset{\cdot \cdot}{y}_{g,t}$$	$$u^{[Y]}_{g,t}$$	$$y_{g,t} = L^{[Y]}_{g,t} (\overset{\cdot \cdot}{y}_{g,t})$$	$$\overset{\sim}y_{g,t} = u^{[Y]}_{g,t} \cdot \overset{\cdot \cdot}{y}_{g,t}$$
Controls	$$G \times T \times N_C$$	$$\overset{\cdot \cdot}{z}_{g,t,i}$$	$$\text{N/A}$$	$$z_{g,t,i} = L^{[C]}_{g,i}(\overset{\cdot \cdot}{z}_{g,t,i})$$	$$\text{N/A}$$
Media	$$G \times T \times N_M$$	$$\overset{\cdot \cdot}{x}^{[M]}_{g,t,i}$$	$$u^{[M]}_{g,t,i}$$	$$x^{[M]}_{g,t,i} = L^{[M]}_{g,i}(\overset{\cdot \cdot}{x}^{[M]}_{g,t,i})$$	$$\overset{\sim}x_{g,t,i}^{[M]} = u^{[M]}_{g,t,i}\cdot\overset{\cdot \cdot}{x}^{[M]}_{g,t,i}$$
Reach	$$G \times T \times N_{RF}$$	$$\overset{\cdot \cdot}{r}^{[RF]}_{g,t,i}$$	$$u^{[RF]}_{g,t,i}$$	$$r_{g,t,i} = L^{[RF]}_{g,i}(\overset{\cdot \cdot}{r}^{[RF]}_{g,t,i})$$	$$\overset{\sim}r^{[RF]}_{g,t,i} = u^{[RF]}_{g,t,i} \cdot \overset{\cdot \cdot}{r}^{[RF]}_{g,t,i} \cdot f^{[RF]}_{g,t,i}$$
Frequency	$$G \times T \times N_{RF}$$	$$f^{[RF]}_{g,t,i}$$		$$\text{N/A}$$
Organic Media	$$G \times T \times N_{OM}$$	$$\overset{\cdot \cdot}{x}^{[OM]}_{g,t,i}$$	$$u^{[OM]}_{g,t,i}$$	$$x^{[OM]}_{g,t,i} = L^{[OM]}_{g,i}(\overset{\cdot \cdot}{x}^{[OM]}_{g,t,i})$$	$$\overset{\sim}x^{[OM]}_{g,t,i} = u^{[OM]}_{g,t,i}\cdot\overset{\cdot \cdot}{x}^{[OM]}_{g,t,i}$$
Organic Reach	$$G \times T \times N_{ORF}$$	$$\overset{\cdot \cdot}{r}^{[ORF]}_{g,t,i}$$	$$u^{[ORF]}_{g,t,i}$$	$$r^{[ORF]}_{g,t,i} = L^{[ORF]}_{g,i}(\overset{\cdot \cdot}{r}^{[ORF]}_{g,t,i})$$	$$\overset{\sim}r^{[ORF]}_{g,t,i} = u^{[ORF]}_{g,t,i} \cdot \overset{\cdot \cdot}{r}^{[ORF]}_{g,t,i} \cdot f^{[ORF]}_{g,t,i}$$
Organic Frequency	$$G \times T \times N_{ORF}$$	$$f^{[ORF]}_{g,t,i}$$		$$\text{N/A}$$
Non-media Treatments	$$G \times T \times N_N$$	$$\overset{\cdot \cdot}{x}^{[N]}_{g,t,i}$$	$$\text{N/A}$$	$$x^{N}_{g,t,i} = L^{N}_{g,i}(\overset{\cdot \cdot}{x}^{N}_{g,t,i})$$	$$\text{N/A}$$

Unit scaling is handled internally by Meridian. Geo population scaling is necessary for hierarchical modeling to put all geos on a comparable scale. Other standardization is done so that standardized prior distributions can be used, without the need to consider the scale of each variable.

Define \(p_g\) to be the population size of each geo, which is another model input that must be specified by the user. The linear scaling functions are summarized as follows:

Scaling function: KPI units

KPI units are population scaled to put all geos on roughly the same scale. This way model parameters don't need to scale with population size.

After population scaling, the KPI is scaled to have mean zero and standard deviation one. Centering to mean zero makes it reasonable to choose a prior centered at zero for the intercept terms (knot_values and tau_g). Scaling to standard deviation one puts parameters on a scale such that reasonable default priors can be assigned.

Notation: \(L^{[Y]}_{g,t} (\cdot)\)

Description:

1. Divide by geo population.
2. Center and scale the geo-scaled values to have mean zero and standard deviation one.

Definition:

\(L^{[Y]}_{g,t} (q) = \dfrac{\dfrac{q}{p_g} - m^{[Y]}}{s^{[Y]}}\)

Where:

• \(y^\dagger_{g,t} = \dfrac{\overset {\cdot \cdot} y_{g,t}}{p_g}\)
• \(m^{[Y]} = \frac{1}{GT}\sum\limits_{g,t} y^\dagger_{g,t}\)
• \(s^{[Y]} = \sqrt{\frac{1}{GT-1} \sum\limits_{g,t} \left( y^\dagger_{g,t}-m^{[Y]} \right)^2}\)

Scaling function: Control variables

Control variables only need to be population scaled if the values roughly scale with population size. Meridian has geo-specific random effect coefficients (gamma_gc), but it is better to scale the variable rather than relying on the model fit to obtain coefficients that scale with population size.

Control variables are scaled to have mean zero and standard deviation one. Centering to mean zero makes it reasonable to choose a prior centered at zero for the intercept terms (knot_values and tau_g). Scaling to standard deviation one makes the puts the coefficient mean (gamma_c) on a scale such that a reasonable non-informative default prior can be assigned.

Notation: \(L^{[C]}_{g,i} (\cdot)\)

Description:

1. It might make sense to do population scaling for certain controls. This can be handled using the control_population_scaling_id argument. By default, no controls are population scaled.

2. Center and scale each control variable to have mean zero and standard deviation one.

Definition:

\(L^{[C]}_{g,i}(q) = \dfrac{\dfrac{q}{p^{I^{[C]}_i}_g} - m^{[C]}}{s^{[C]}}\)

Where:

• \(I_i^{[C]} = 1\) if control_population_scaling_id=True is used for the variable \(i;0\) otherwise.

  • \(z^{\dagger}_{g,t,i} = \dfrac{\overset {\cdot \cdot} z_{g,t,i}}{p_g^{I_i^{[C]}}}\)
  • \(m^{[C]} = \frac{1}{GT}\sum\limits_{g,t} z^{\dagger}_{g,t,i}\)
  • \(s^{[C]} = \sqrt{\frac{1}{GT-1} \sum\limits_{g,t} \left( z^{\dagger}_{g,t,i}-m^{[C]} \right)^2}\)

Scaling function: Media units

Media units are population scaled to put all geos on roughly the same scale. This way the half saturation parameters (ec_m) don't need to scale with population size.

Media units are then scaled by the median non-zero value for each channel. This is done to give the ec_m parameter a more intuitive interpretation so that an ec_m value of one implies that the half-saturation point occurs at the median non-zero media units per capita.

Notation: \(L^{[M]}_{g,i} (\cdot)\)

Description:

1. Divide by geo population.
2. For each media channel, scale the geo-scaled values by the median non-zero value.

Definition:

\(L^{[M]}_{g,i} (q) = \dfrac{q}{p_g d^{[M]}}\)

Where:

• \(x^{\dagger [M]}_{g,t,i} = \dfrac{\overset {\cdot \cdot} x_{g,t,i}^{[M]}}{p_g}\)
• \(d^{[M]} = \text{Median}\left( \left\{ x^{\dagger [M]}_{g,t,i}:x^{\dagger [M]}_{g,t,i} > 0 \right\}_{g,t} \right)\)

Scaling function: Reach

Reach is population scaled to put all geos on roughly the same scale. Meridian has geo-specific random effect coefficients (beta_grf), but it is better to scale the variable rather than relying on the model fit to obtain coefficients that scale with population size.

Reach scaled by the median non-zero value for each channel, which puts the coefficient mean parameter (beta_rf) on a standardized scale so that a reasonable default prior for beta_rf can be assigned (when ModelSpec.rf_prior_type is set to 'coefficient'.)

Notation: \(L^{[RF]}_{g,i} (\cdot)\)

Description:

The scaling function is the same as for media units.

Scaling function: Organic media units

The scaling and rationale are the same as for paid media units.

Notation: \(L^{[OM]}_{g,i} (\cdot)\)

Description:

1. Divide by geo population.
2. For each organic media channel, scale the geo-scaled values by the median non-zero value.

Definition:

\(L^{[OM]}_{g,i} (q) = \dfrac{q}{p_g d^{[OM]}}\)

Where:

• \(x^{\dagger [OM]}_{g,t,i} = \dfrac{\overset {\cdot \cdot} x_{g,t,i}^{[OM]}}{p_g}\)
• \(d^{[OM]} = \text{Median}\left( \left\{ x^{\dagger [OM]}_{g,t,i}:x^{\dagger [OM]}_{g,t,i} > 0 \right\}_{g,t} \right)\)

Scaling function: Organic reach

The scaling and rationale are the same as for paid media reach.

Notation: \(L^{[ORF]}_{g,i} (\cdot)\)

Description:

The scaling function is the same as for organic media units.

Scaling function: Non-media treatments

Non-media treatment variables only need to be population scaled if the values roughly scale with population size. Meridian has geo-specific random effect coefficients (gamma_gn), but it is better to scale the variable rather than relying on the model fit to obtain coefficients that scale with population size.

Non-media treatment variables are scaled to have mean zero and standard deviation one. Centering to mean zero makes it reasonable to choose a prior centered at zero for the intercept terms (knot_values and tau_g). Scaling to standard deviation one puts the coefficient mean parameter (gamma_n) on a standardized scale so that a reasonable default prior can be assigned (when ModelSpec.non_media_treatments_prior_type is set to 'coefficient'.)

Notation: \(L^{[N]}_{g,i} (\cdot)\)

Description:

1. It might make sense to do population scaling for certain non-media treatments. This can be handled using the non_media_population_scaling_id argument. By default, non-media treatments are not population scaled.

2. Center and scale each non-media treatment variable to have mean zero and standard deviation one.

Definition:

\(L^{[N]}_{g,i}(q) = \dfrac{\dfrac{q}{p^{I^{[N]}_i}_g} - m^{[N]}}{s^{[N]}}\)

Where:

• \(I_i^{[N]} = 1\) if non_media_population_scaling_id=True is used for the variable \(i;0\) otherwise.

  • \(X^{\dagger [N]}_{g,t,i} = \dfrac{\overset {\cdot \cdot} x_{g,t,i}}{p_g^{I_i^{[N]}}}\)
  • \(m^{[N]} = \frac{1}{GT}\sum\limits_{g,t} x^{\dagger [N]}_{g,t,i}\)
  • \(s^{[N]} = \sqrt{\frac{1}{GT-1} \sum\limits_{g,t} \left( x^{\dagger [N]}_{g,t,i}-m^{[N]} \right)^2}\)