Non-normal simultaneous regression models for customer linkage analysis

Abstract

Simultaneous systems of equations with non-normal errors are developed to study the relationship between customer and employee satisfaction. Customers interact with many employees, and employees serve many customers, such that a one-to-one mapping between customers and employees is not possible. Analysis proceeds by relating, or linking, distribution percentiles among variables. Such analysis is commonly encountered in marketing when data are from independently collected samples. We demonstrate our model in the context of retail banking, where drivers of customer and employee satisfaction are shown to be percentile-dependent.

Appendix

1.1 Estimation algorithms

Estimation proceeds by recursively generating draws from the full conditional distribution of all model parameters. The likelihood of the data can be factored as:

$$\pi {\left( {\left. {y_{A} ,y_{B} } \right|\alpha ,\beta ,\theta _{A} ,\theta _{B} } \right)} = \pi {\left( {\left. {\varepsilon _{{y_{A} }} ,\varepsilon _{{y_{B} }} } \right|\alpha ,\beta ,\theta _{A} ,\theta _{B} } \right)} \times {\left| {J_{{{\left( {\varepsilon _{{_{{yA}} }} ,\varepsilon _{{_{{yB}} }} } \right)} \to {\left( {y_{A} ,y_{B} } \right)}}} } \right|}$$

where the first factor corresponds to Eq. 1 and the second factor corresponds to Eq. 2. θ _A and θ _B represent other model parameters specific to the assumed distributions of the error terms. All model parameters are estimated using a random-walk Metropolis–Hastings algorithm where the likelihood contribution is of the form:

$${\left[ {\alpha \left| {else} \right.} \right]} \propto {\prod\limits_{j = 1}^J \pi }{\left( {\varepsilon _{{y_{{A,j}} }} ,\varepsilon _{{y_{{B,j}} }} \left| {\alpha ,\beta ,\theta _{A} ,\theta _{B} } \right.} \right)} \times {\left| {J_{{{\left( {\varepsilon _{{_{{yA,j}} }} ,\varepsilon _{{_{{yB,j}} }} } \right)} \to {\left( {y_{{_{{A,j}} }} ,y_{{_{{B,j}} }} } \right)}}} } \right|}$$

where

$$J_{{{\left( {\varepsilon _{{_{{yA,j}} }} ,\varepsilon _{{_{{yB,j}} }} } \right)} \to {\left( {y_{{A,j}} ,y_{{B,j}} } \right)}}} = {\left| {\frac{{\partial \varepsilon _{j} }}{{\partial y_{j} }}} \right|} = 1 - \alpha _{1} \beta _{1} $$

1.2 Model 1: Asymmetric Laplace

$$\begin{array}{*{20}c} {\varepsilon _{yA} \sim {\text{AL}}\left( {0,\sigma _{yA} ,p_{yA} } \right)} \\ {\varepsilon _{yB} \sim {\text{AL}}\left( {0,\sigma _{yB} ,p_{yB} } \right)} \\ {f_p \left( {\left. y \right|\mu ,\sigma ,p} \right) = \frac{{p\left( {1 - p} \right)}}{\sigma }\exp \left\{ { - \rho _p \left( {\frac{{y - \mu }}{\sigma }} \right)} \right\}} \\ \end{array} $$

Prior distributions for all slope coefficients were specified as normal with mean zero and covariance equal to 100I. Inverted chi-square priors with 3 degrees of freedom and prior sum of squares equal to .01 were used for the scale parameters (σ _yA , σ _yB ). Priors for p _yA and $p_{y_B }$ were specified as uniform(0,1).

1.3 Model 2: Skewed t

$$\begin{array}{*{20}c} {\varepsilon _{y_A } \sim {\text{skewt}}\left( {0,\gamma _{y_A } ,\sigma _{y_A } ,\nu _{y_A } } \right)} \\ {\varepsilon _{y_B } \sim {\text{skewt}}\left( {0,\gamma _{y_B } ,\sigma _{y_B } ,\nu _{y_B } } \right)} \\ {p\left( {\left. {y_i } \right|\beta ,\sigma ,\nu ,\gamma } \right) = \frac{2}{{\gamma + \gamma ^{ - 1} }}\frac{{\Gamma \left( {\frac{{\nu + 1}}{2}} \right)}}{{\Gamma \left( {\frac{\nu }{2}} \right)\left( {\pi \nu } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}\sigma ^{ - 1} \times \left[ {1 + \frac{{\left( {y_i - x_i^\prime \beta } \right)^2 }}{{\nu \sigma ^2 }}\left\{ {\frac{1}{{\gamma ^2 }}I_{\left[ {0,\infty } \right)} \left( {y_i - x_i^\prime \beta } \right) + \gamma ^2 I_{\left( { - \infty ,0} \right)} \left( {y_i - x_i^\prime \beta } \right)} \right\}} \right]^{{{ - \left( {\nu + 1} \right)} \mathord{\left/ {\vphantom {{ - \left( {\nu + 1} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}} } \\ \end{array} $$

Prior distributions for all slope coefficients were specified as normal with mean zero and covariance equal to 100I. Inverted chi-square priors with 3 degrees of freedom and prior sum of squares equal to 0.01 were used for the scale parameters ($\sigma _{y_A } ,\sigma _{y_B }$). Priors for $\gamma _{y_A }$ and $\gamma _{y_B }$ were specified as inverted chi-square with 3 degrees of freedom and prior sum of squares equal to 0.1. A uniform(0,150) prior was used for $\nu _{y_A }$ and $\nu _{y_B }$.

1.4 Model 3: Mixture of multivariate normals

$$\left( {\begin{array}{*{20}c} {\varepsilon _{y_A } } \\ {\varepsilon _{y_B } } \\ \end{array} } \right) \sim \sum\limits_k {\phi _k N\left( {\mu _k ,\Sigma _k } \right)} $$

• 1. Generate z|μ _k , Σ _k , φ

  $$\begin{array}{*{20}l} {z_i \sim {\text{Multinomial}}\left( {\pi _i } \right)} \hfill \\ {\pi _i = \phi _k \frac{{\varphi \left( {\left. {\varepsilon _{y_A } ,\varepsilon _{y_B } } \right|\mu _k ,\Sigma _k } \right)}}{{\sum\limits_k {\varphi \left( {\left. {\varepsilon _{y_A } ,\varepsilon _{y_B } } \right|\mu _k ,\Sigma _k } \right)} }}} \hfill \\ \end{array} $$

Where ϕ(·) is the multivariate normal density and z _i is a latent indicator variable that assigns each observation in sample to one of the K mixture components.

• 2. φ|z

  $$\begin{array}{*{20}l} {\phi \sim {\text{Dirichlet}}\left( {\widetilde\alpha } \right)} \hfill \\ {\widetilde\alpha = n_k + \alpha _k } \hfill \\ {n_k = \sum\limits_{i = 1}^n {I\left( {z_i = k} \right)} } \hfill \\ \end{array} $$

Where α _k is a prior value that is set equal to 1 for all k.

• 3. μ _k , Σ _k |z, α, β

Conditional on the latent indicator variable z _i each observation can be assigned to one of the K mixture components. Inference for μ _k , Σ _k can then proceeds through the use of the standard multivariate regression:

$$\begin{array}{*{20}l} {\Sigma _k \sim {\text{IW}}\left( {\nu _0 + n_k ,V_0 + S_k } \right)} \hfill \\ {\mu _k \sim N\left( {\widetilde\mu ,\Sigma _k \otimes \left( {X_k^\prime X_k + A_\mu } \right)^{ - 1} } \right)} \hfill \\ \end{array} $$

Where: X _k  = ι _k , the unit vector with length equal to the number of observations assigned to component k and:

$$\widetilde\mu _k = {\text{vec}}\left( {\widetilde{\rm M}_k } \right),\;\widetilde{\rm M}_k = \left( {X_k^\prime X_k + A_\mu } \right)^{ - 1} \left( {X_k^\prime X_k \widehat{\rm M}_k + A_\mu \overline {\rm M} } \right),$$

$${\text{S}}_k = \left( {Y_k - X_k \tilde M_k } \right)^\prime \left( {Y_k - X_k \tilde M_k } \right) + \left( {\tilde M_k - \overline M } \right)^\prime A_\mu \left( {\tilde M_k - \overline M } \right),$$

and $\hat M = \left( {X_k^\prime X_k } \right)^{ - 1} \left( {X_k^\prime Y_k } \right)$

Prior parameters were specified as follows: ν ₀ = 3, $V_0 = \left[ {\begin{array}{*{20}c} {0.1}&0 \\ 0&{0.1} \\ \end{array} } \right]$, A = 0.01, $\overline M = \left[ {\begin{array}{*{20}c} 0&0 \\ \end{array} } \right]$

• 4. α, β|μ _k , Σ _k , z

α and β can be drawn using a standard random walk Metropolis-Hastings algorithm where the likelihood for the model can be computed as follows:

$$\prod\limits_i {\prod\limits_k {\varphi \left( {\left. {\varepsilon _{i,n_k }^{y_A } ,\varepsilon _{i,n_k }^{y_B } } \right|\mu _k ,\Sigma _k } \right)} } \left| {J_{\varepsilon \to y} } \right|$$

Prior distributions for both α and β were specified as multivariate normal with mean 0 and covariance equal to 100I. To improve mixing of the Markov chain we found it useful to draw the regression coefficients in two distinct blocks.