A brand choice model for TV advertising management using single-source data

Abstract

A brand choice model for TV advertising management using single-source data is proposed. The model replaces household-specific advertising exposure, which is often used as a covariate in a brand choice model, with gross rating points (GRP), a managerial control variable for advertising. In particular, given daily GRP, a probabilistic model of advertising exposure for heterogeneous customers is integrated into a brand choice model with advertising threshold under a Bayesian framework. Through hierarchical modeling, demographic information on panels provides managerial insights into advertising planning.

Appendices

Appendix 1: hierarchical structure of brand choice module

We set the hierarchical regression models for the household-specific parameters composing the brand choice module, as follows:

$$ {r_h} = {Z_h}\zeta + {\psi_h},\,{\psi_h}{{\sim }}N\left( {0,\nu } \right), $$

$$ \rho_{{jh}}^{*} = \ln \left( {\frac{{{\rho_{{jh}}}}}{{1 - {\rho_{{jh}}}}}} \right) = {Z_h}{\eta_j} + {\omega_{{hj}}},\,{\omega_{{jh}}}{{\sim }}N\left( {0,{\xi_j}} \right)\left( {0 \leqslant {\rho_{{jh}}} < 1} \right), $$

$$ \beta_h^{{(1)}} = {Z_h}{\gamma^{{(1)}}} + \delta_h^{{(1)}},\,\delta_h^{{(1)}}{{\sim }}N\left( {0,{\Lambda^{{(1)}}}} \right), $$

$$ \left\{ {{\alpha_h},\beta_h^{{(2)}}} \right\} = {Z_h}{\gamma^{{(2)}}} + \delta_h^{{(2)}},\,\delta_h^{{(2)}}{{\sim }}N\left( {0,{\Lambda^{{(2)}}}} \right) $$

where we assume that their error terms, ${\psi_h},{\omega_{{jh}}},\,\delta_h^{{(1)}}$ and $\delta_h^{{(2)}}$follow normal distributions.

Appendix 2: MCMC algorithm for hierarchical modeling of advertising exposure module

First of all, we have the likelihood function for the mean parameter of the number of times for advertising exposure based on Poisson distribution as

$$ L\left( {\left\{ {{\lambda_{{jh}}}} \right\};\left\{ {{F_{{jhw}}}} \right\}} \right) = \prod\limits_{{w = 1}}^W {\prod\limits_{{h = 1}}^H {\prod\limits_{{j = 1}}^J {\exp \left( { - {\lambda_{{jh}}}{\text{GR}}{{\text{P}}_{{jw}}}} \right)\frac{{{{\left( {{\lambda_{{jh}}}{\text{GR}}{{\text{P}}_{{jw}}}} \right)}^{{{F_{{jhw}}}}}}}}{{{F_{{jhw}}}!}}} } } $$

We assume that prior distribution for π(δ _j , κ _j ) in hierarchical model (7) as ${\delta_j} \sim N({\bar{\delta }_j},{\bar{\Sigma }_{{{\delta_j}}}});\,\bar{\delta } = 0,{\bar{\Sigma }_{{{\delta_j}}}} = 0.01{I_d}$ and ${\kappa_j}^{{ - 1}}\sim{\text{Gamma}}({\bar{\kappa }_j},{\bar{\Upsilon }_{{{\kappa_j}}}});{\bar{\kappa }_j} = g,{\bar{\Upsilon }_{{{\kappa_j}}}} = {\bar{\kappa }_j}$. Then the posterior density of exposure rate parameter $g\left( {\left\{ {{\lambda_{{jh}}}} \right\}\left| {\left\{ {{F_{{jhw}}}} \right\}} \right.} \right)$ is evaluated from the joint posterior density

$$ g\left( {\left\{ {{\lambda_{{jh}}}} \right\},\left\{ {{\delta_j}} \right\},\left\{ {{\kappa_j}} \right\}|\left\{ {{F_{{jhw}}}} \right\}} \right) \propto L\left( {\left\{ {{\lambda_{{jh}}}} \right\};\left\{ {{F_{{jhw}}}} \right\}} \right)P\left( {{\lambda_{{jh}}}|{\delta_j},{\kappa_j}} \right)\pi \left( {{\delta_j},{\kappa_j}} \right). $$

To get this joint posterior density, we employ the sampling procedure for MCMC from conditional posterior distributions:

• 1. “λ _jh |δ _j , κ _j ” $\left[ {{\lambda_{{jh}}} = {Z_h}{\delta_j} + {\omega_{{jh}}},{\omega_{{jh}}} \sim N\left( {0,{\kappa_j}} \right)} \right]$

  We use Metropolis–Hastings with a random walk algorithm $\lambda_{{jh}}^{{(k)}} = \lambda_{{jh}}^{{(k - 1)}} + {\tilde{\omega }_{{jh}}},\,{\tilde{\omega }_{{jh}}} \sim N(0,0.01),$ and $\lambda_{{jh}}^{{(0)}} = \sum\nolimits_{{w = 1}}^W {{F_{{jhw}}}} /\sum\nolimits_{{w = 1}}^W {{\text{GR}}{{\text{P}}_{{jw}}}},$ with acceptance probability α

  $$ \begin{gathered} \alpha \left( {\lambda_{{jh}}^{{(k)}},\lambda_{{jh}}^{{(k - 1)}}\left| {{\delta_j},{\kappa_j},{Z_h},{F_{{jhw}}},{\text{GR}}{{\text{P}}_{{jw}}}} \right.} \right) = \hfill \\ \min \left[ {\frac{{\exp \left( { - \frac{1}{2}{{\left( {\lambda_{{jh}}^{{(k)}} - {Z_h}{\delta_j}} \right)}^{\prime }}{\kappa_j}^{{ - 1}}\left( {\lambda_{{jh}}^{{(k)}} - {Z_h}{\delta_j}} \right)} \right)\prod\limits_{{w = 1}}^W {\exp \left( { - \exp \left( {\lambda_{{jh}}^{{(k)}}} \right){\text{GR}}{{\text{P}}_{{jw}}}} \right)\left( {\frac{{{{\left( {\exp \left( {\lambda_{{jh}}^{{(k)}}} \right){\text{GR}}{{\text{P}}_{{jw}}}} \right)}^{{{F_{{jhw}}}}}}}}{{{F_{{jhw}}}!}}} \right)} }}{{\exp \left( { - \frac{1}{2}{{\left( {\lambda_{{jh}}^{{\left( {k - 1} \right)}} - {Z_h}{\delta_j}} \right)}^{\prime }}{\kappa_j}^{{ - 1}}\left( {\lambda_{{jh}}^{{\left( {k - 1} \right)}} - {Z_h}{\delta_j}} \right)} \right)\prod\limits_{{w = 1}}^W {\exp \left( { - \exp \left( {\lambda_{{jh}}^{{(k - 1)}}} \right){\text{GR}}{{\text{P}}_{{jw}}}} \right)\left( {\frac{{{{\left( {\exp \left( {\lambda_{{jh}}^{{(k - 1)}}} \right){\text{GR}}{{\text{P}}_{{jw}}}} \right)}^{{{F_{{jhw}}}}}}}}{{{F_{{jhw}}}!}}} \right)} }},1} \right]. \hfill \\ \end{gathered} $$

• 2. “δ _j |λ _jh , κ _j ”

  $$ \begin{gathered} {\delta_j}{{\sim }}N\left( {{\delta_j}^{*},{\Sigma_{{{\delta^{*}}}}}} \right); \hfill \\ {\delta_j}^{*} = {\left( {{Z_h}\prime{Z_h} + {{\bar{\Sigma }}_{{{\delta_j}}}}^{{ - 1}}} \right)^{{ - 1}}}\left( {{Z_h}\prime{Z_h}{{\hat{\delta }}_j} + {{\bar{\Sigma }}_{{{\delta_j}}}}^{{ - 1}}\bar{\delta }} \right),{\Sigma_{{{\delta_{{_j}}}^{*}}}} = {\left( {{Z_h}\prime{Z_h} + {{\bar{\Sigma }}_{{{\delta_j}}}}^{{ - 1}}} \right)^{{ - 1}}},{{\hat{\delta }}_j} = {({Z_h}\prime{Z_h})^{{ - 1}}}{Z_h}\lambda_{{jh}} \hfill \\ \end{gathered} $$

• 3. “κ _j |λ _jh ,δ _j ”

  $$ \kappa_j^{{ - 1}} \sim Gamma\left( {{{\bar{\kappa }}_j} + H,\mathop{{\left( {\bar{\Upsilon }_{\kappa }^{{ - 1}} + R} \right)}}\nolimits^{{ - 1}} } \right),\,R = \sum\limits_{{h = 1}}^H \,\mathop{{\left( {{\lambda_{{jh}}} - {Z_h}{\delta_j}} \right)}}\nolimits^{\prime} \left( {{\lambda_{{jh}}} - {Z_h}{\delta_j}} \right) $$