Using a two-part mixed-effects model for understanding daily, individual-level media behavior

Abstract

This study supports a strategic analytics proposal, namely that there is conceptual and practical utility in applying a two-part mixed-effects model for understanding individual differences in daily media use. Individual-level daily diary measures of media use typically contain information about a person’s likeliness to use media, extent of usage, and variation in use across days that, taken together, can provide data for evaluating media behavior that is otherwise masked by using aggregate measures. The statistical framework developed and demonstrated here focuses on these three metrics. The approach, applied to daily diary measures of television use in a large, representative U.S. sample, yields results that add value when weighing media strategies centered on the twin tactics of reach and frequency. The implications for the proposed analytic strategy are discussed.

Appendices

Appendix 1

The lognormal and gamma distributions are part of a class of generalized gamma distributions. Depending on their parameters, both can be monotonically declining or bell-shaped and skewed to the right. The lognormal distribution can be characterized by a mean (µ_log(y)) and a variance ($\sigma_{\log (y)}^{2}$), both of which relate to the log of y: log(y). The probability density function (PDF) of the lognormal distribution for a random variable Y can be given as

$$f\left( {y|\mu_{\log \left( y \right)} ,\sigma_{\log \left( y \right)}^{2} } \right) = \frac{1}{{y_{{i\sqrt {2\sigma^{2} \pi } }} }}\exp \left\{ { - \frac{1}{2}\left( {\frac{{\left( {\log \left( {y_{i} } \right) - \mu_{\log \left( y \right)} } \right)}}{{\sigma^{2}_{\log \left( y \right)} }}} \right)} \right\}^{{}}.$$

Note that y > 0 and both µ and σ² > 0.

The gamma distribution can be characterized by two parameters, namely a shape (γ) and scale (θ) parameter. The PDF of the gamma distribution for a random variable Y with γ and θ parameters can be given as

$$f\left( {y|\gamma ,\theta } \right) = - \theta \log \left\{ \gamma \right\} - \log \left\{ {\varGamma \left( \theta \right)} \right\} + \left( {\theta - 1} \right)\log \left\{ y \right\} - y/\gamma,$$

where $\varGamma \left( \theta \right)$ is the standard gamma function of the scale parameter θ. Note that y > 0, γ > , and θ > 0. It is useful to relate the shape and scale parameters to the mean, variance, and standard deviation of y:

$$\mu_{Y} = \gamma \theta ,\;\sigma_{Y}^{2} = \gamma \theta^{2} ,\sigma_{Y}^{{}} = \sqrt {\gamma \theta^{2} },$$

(8)

respectively.

Appendix 2

Empirical Bayes estimates of the random effects based on a two-part mixed-effects model. Values are the random effects relating to the individual estimated logits (upper figure), log time spent (middle figure), and variance of log time spent (bottom figure).