Parameter Estimation And Objective Definition In Mean Variance Media Scheduling 1981

April 1980

In a recent article [3], the use of mean-variance analysis was considered as an alternative approach for media scheduling. This note extends these results with particular emphasis on the estimation of model parameters and increasing solution efficacy. Issues such as (1) the testing for independence in the underlying probability structure, (2) estimation of exposure probabilities and the use of sampling variances, (3) the specification of an objective function, and (4) ex post evaluation of schedule frequency obtained are briefly discussed.

1. A test for independence of exposure probabilities

The mean-variance tbrmulation [3] uses a binomial probabihty structure in deriving the expressions for the expected kth gross audience segment realized, the expected kth net audience segment obtained, schedule frequency and their variances. Specifically, if p~(t) denotes the probability that an individual in audience segment k is exposed to the/th cumulative insertion in medium i at time t, it is shown that the distribution of the size of the kth gross audience segment for medium i, denoted d~(t), is binomial with success probability

$q_{ii}^{k}(t)$ , where

$q_{ij}^{k}(t) = (1 - p_{i1}^{k}(t)) \cdot (1 - p_{ij}^{k}(t)) \cdots (1 - p_{ij}^{k}(t)). \quad (1)$

Similarly, the size of the kth net audience segment obtained given/(i) cumulative purchases in media i = 1,2, …,I in p.eriod t, denoted d/~(t), is distributed binomially with success probability

$Q_{j(i)}^{k}(t) = \prod_{i=1}^{I} q_{ij}^{k}(t) .$ (2)

Implicit in these developments is the assumption that the kth audience segment is relatively homogeneous. While care is taken to create a substantial degree of internal homogeneity in the segmentation process, it is advisable in practice to test whether individuals within the same audience segment have an approximately equal chance of being exposed, i An appropriate test for this assumption would be to consider an hypothesis of equal proportionate exposure between different subsections of the same audience segment. Thus, if there is reason to believe that the ith medium audience in the kth market segment, D~(t), contains a subsection D~ k(t) for which the exposure probabilities p~(t), ] = 1, 2, …, J, are substantially different, a suitable test statistic is given by

$\frac{\hat{p}_{ij}^{k}(t) - \hat{p}_{ij}^{\prime k}(t)}{\sqrt{[\hat{p}(1-\hat{p}) \cdot (1/n+1/n')]}},$ (3)

where n and n’ are the respective sample sizes and p represents a pooled estimate of the proportion of individuals exposed in segment k. If the exposure probabili. ties are equal, (3), of course, will be normally distributed.

¹ Of course, the very reason for segmenting the total audience is to create segments with a high degree of internal homogeneity as compared to likeness among segments. For example, d~](t) might refer to the number of women aged 25 to 34 (k) exposed via a purchase of/insertions in Time magazine (i) in period t.

2. Estimation of exposure probabilities

The test proposed above assumes that the implementation of the model is based on estimates for the probability structure $p_{ii}^{k}(t)$ . While such estimates can sometimes be obtained, e.g. in test market situations, it is generally more efficient to estimate the joint probabilities $q_{ij}^{k}(t)$ directly from campaign monitoring data. 2 When the latter course is taken, the variance expression for expected gross audience derived in [3] no longer applies as this expression only reflects the likely error in the joint probabilities $q_{ij}^{k}(t)$ due to randomness of exposure through the $p_{ii}^{k}(t)$ probability structure. However, since we are primarily concerned with the reliability of exposure estimates, it seems reasonable to employ the mean-variance format to include other sources of variation, e.g. sampling error, in the model. Thus, if $q_{ij}^k(t)$ estimates are obtained directly, their variances may be used to augment or replace the expressions in [3]. In particular, if $\hat{q}_{ii}^{k}(t)$ is an unbiased estimate of $q_{ij}^{k}(t)$ , obtained in a random sample of size n, the expected size of the kth gross audience segment for medium i is given by

D_i^k(t) \cdot (1 - \hat{q}_{ii}^k(t)) \tag{4}

for j = 1, 2, …, J, and its variance by

[D_i^k(t)]^2 \cdot \hat{q}_{ij}^k(t) \cdot (1 - \hat{q}_{ij}^k(t))/n . \tag{5}

Also note that the above procedure does not require any assumptions regarding the nature of the underlying probability structure $p_{ij}^k(t)$ , since (4) and (5), and similar expressions for net audience segments derived in [3] only depend on the joint probabilities $q_{ij}^k(t)$ . Hence the distribution of $p_{ij}^k(t)$ and the associated independence assumptions need only be considered if frequency controls are included in the model (see Section 4).

3. Objective function specification

The model in [3] seeks to minimize the variance of gross audience segments obtained. It may be desirable to balance risk in terms of several sources of schedule variance, viz. variance of gross audience segments, net audience and frequency, in a single objective. This may be accomplished by recasting the net audience expressions derived in [3] in incremental form. Recall that

the expected size of the kth net audience segment and its variance given j(i) cumulative insertions in medium i, i = 1, 2, …, I, are given by

\overline{E}_{i(t)}^{k}(t) = D^{k}(t) \cdot (1 - Q_{i(t)}^{k}(t)) \tag{6}

and

$\overline{V}_{i(i)}^{k}(t) = D^{k}(t) \cdot (1 - Q_{i(i)}^{k}(t)) \cdot Q_{i(i)}^{k}(t) , \qquad (7)$

where $D^k(t)$ denotes the combined media audience in the kth market segment. Using (6), net audience constraints of the type $\overline{E}_{I(t)}^k(t) \ge N^k(t)$ may be derived as

$\sum_{i} \sum_{j} \ln(1 - p_{ij}^{k}(t)) x_{ij}(t) \ge \ln[(1 - N^{k}(t))/D^{k}(t)]$ (8)

using a logarithmic transformation similar to that employed by Charnes et al. [1]. ³ Unfortunately, the logarithmic transformation makes it difficult to obtain an explicit representation of the variance of the kth net audience segment. To resolve this problem, we consider the use of an incremental argument similar to that employed for gross audience statistics. Specifically, if $Q_{j(i)}^k(t)$ denotes the probability that an individual in segment k is not exposed at least once given j(i) cumulative insertions in medium i, i = 1, 2, …, I, the effect of an additional insertion in any of the media may be represented by a revision of this probability to $Q_{j'(i)}^k(t)$ . Without loss of generality, assume the additional insertion was made in medium i’. Then

$Q_{j'(i)}^{k}(t) = \left[ \prod_{i \neq i'} q_{ij}^{k}(t) \right] \cdot q_{i',j+1}^{k}(t)$

= Q_{i(i)}^{k}(t) \cdot \left[ 1 - p_{i'+1}^{k}(t) \right]. \tag{9}

Thus the incremental expected kth net audience obtained by the additional insertion in medium i’ is given by

$\Delta \overline{E}_{j(i)}^{k}(t) = D^{k}(t) \cdot [Q_{j(i)}^{k}(t) - Q_{j(i)}^{k}(t)]$

$= D^{k}(t) \cdot Q_{j(i)}^{k}(t) \cdot [1 - p_{i'j+1}^{k}(t)]$ (10)

with incremental variance 4

$\Delta \overline{V}_{j(i)}^{k}(t) = D^{k}(t) \cdot \left[ (Q_{j(i)}^{k}(t))^{2} \cdot (1 - (1 - p_{i'j+1}^{k}(t))^{2}) - Q_{j(i)}^{k}(t) \cdot p_{i'j+1}^{k}(t) \right]$ (11)

⁴ See footnote 4 in [3].

^<sup>2 If $\hat{q}_{ij}^k(t)$ , j = 1, 2, …, J are unbiased estimates of $q_{ij}^k(t)$ based on independent random samples, $p_{ij}^k(t)$ is estimated by $\hat{p}_{ij}^k(t) = (\hat{q}_{i,j-1}^k(t) - \hat{q}_{ij}^k(t))/\hat{q}_{i,j-1}^k(t)$ .

^<sup>3 As noted by Professor Cooper, expression (8) is related to the more general Khinchin-Kullback-Leibler statistic in information theory. For further details on this aspect, see e.g. Sakaguchi [5] or Charnes, Cooper and Learner [2].

yielding net audience restrictions of the form 4. Ex post evaluation of schedule frequency

$\sum_{i} \sum_{j} \Delta \overline{E}_{j(i)}^{k}(t) \cdot X_{ij}(t) \geqslant N^{k}(t)$ (12)

with variance

\sum_{i} \sum_{j} \Delta \overline{V}_{j(i)}^{k}(t) \cdot X_{ij}(t) . \tag{13}

The additional flexibility in modeling schedule variance gained by the above linearization is obtained at the expense of additional estimation, requirements. Also, of course, this procedure assumes independence between successive insertions and between the different media whereas the logarithmic form (8) does not require independence (see Charnes et al. [ 1 ], and [3 ] ). In effect, insertions in different media are treated in (9) as successive purchases directed at the same target audience.

The variance of schedule frequency is readily specified. Noting that the expected number of exposures for an individual in segment k given/cumulative insertions in medium i in period t is given by

$p_{i1}^{k}(t) + p_{i2}^{k}(t) + \dots + p_{ii}^{k}(t)$ , (14)

the underlying distribution is recognized as generalized binomial in which the probability of ‘success’ is different for each ‘trial’. Hence, the variance of exposure frequency may be written as (see [4])

\sum_{i} \sum_{j} p_{ij}^{k}(t) \cdot [1 - p_{ij}^{k}(t)] \cdot X_{ij}(t) . \tag{15}

Using (13) and (15), the model in [3] may be refined to minimize a weighted linear combination of schedule variances. The revised objective takes the form

Min $W_G \cdot \sum_{t} \sum_{k} \sum_{i} \sum_{j} \Delta V_{ij}^{k}(t) \cdot X_{ij}(t)$

$+ W_N \cdot \sum_{t} \sum_{k} \sum_{i} \sum_{j} \Delta \overline{V}_{j(i)}^{k}(t) \cdot X_{ij}(t)$
$+ W_F \cdot \sum_{t} \sum_{k} \sum_{i} \sum_{j} p_{ij}^{k}(t) \cdot [1 - p_{ij}^{k}(t)] \cdot X_{ij}(t)$ , (16)

where the v~eights WG, WN, and WF reflect the relative importance attached to the different sources of schedule variance by the media planner.

Charnes et al. [ 1 ] first expressed concern regarding the use of average frequency as the sole measure of frequency control based on the observation that average frequency is relatively insensitive to the form of the underlying probability distribution. That is, regardless of the degree of correlation between audience segments obtained with successive insertions the expected number of individuals exposed by the/th advertisement in medium i is D~(O” p~(t). However, this relative insensitivity does not apply in the case of the variance of exposure frequency, and hence, the form of the entire frequency distribution. For example, if audience segrr~.ents are highly correlated, the distribution of frequency may become bimodal. In the extreme case of perfectly correlated audiences, and constant exposure probabilities pi(t)= p(t), the distribution degenerates to a two-point form, where a fraction p~(t) of audience segment k has been exposed / times and the remainder not at all. The variance in that case may be shown to be (/2). p~(t). [1 - p~(t)],/ times as large as the variance resulting under binomial assumptions.

Thus, a mean-frequency or a mean-variance frequency approach must be used with caution, particularly if the validity of the binomial model is questioned. To alleviate this concern, the following ex post method of determining the entire distribution of frequencies obtained is useful. Noting that the probability of an individual being exposed r times is given by the coefficient of z r in the generating function of the generalized binomial probability distribution given by

$G(z) = [p_{ii}^{k}(t) \cdot z + (1 - p_{ii}^{k}(t))]$ (17)

the entire frequency distribution obtained may be computed by recursively evaluating these coefficients as

$a_0^1 = [1 - p_{i1}^k(t)], a_1^1 = p_{i1}^k(t)$ and

a_r^{j+1} = [1 - p_{ij}^k(t)] \cdot a_r^j + p_{ij}^k(t) \cdot a_{r-1}^i, \tag{19}

where r= 1,2, …,/+ 1 and/= 1,2, …,& The resulting vectors (a~, a{, …, all) define the required frequency distribution given ] cumulative insertions in medium i, and may be used as an ex post means for schedule assessment or, alternatively, to formulate frequency controls, s S For example, the results in (18) and (19) may be used to define frequency constraints controlling specific fractiles of the audience exposed r or more times, (see Charnes et ~l. [11).

References

• [ 1 i A. Charnes, W.W. Cooper, J.K. Devoe, D.B. Learner and W. Reinecke, A goal programming model for media planning, Management Sci. 15 (1968) 423-430.
• [2] A. Charnes, W.W. Cooper and D.B. Learner, Constrained information theoretic characterizations in consumer purchase behaviour, J. Operational Res. Soc. 29 (9) (1978) 833-842.
• [3] C.A. de Kluyver, Media scheduling by mean-variance analysis, European J. Operational Res. 5 (1980) 112-117.
• [4] W. Feller, An Introduction to Probability Theory and its Applications, Vol. I (Wiley, New York, i 950).
• [5 ! M. Sakaguchi, Some entropy results in brand selection beha. vior, Rep. Statist. Appl. Res. Un. Japan. Sci. Engrs. 23 (1) (1976) 14-24.