ESTIMATING THE AUDIENCE OF CANADIAN TELEVISION ADVERTISING SCHEDULES*

Marshall D. Rice Faculty of Administrative Studies York University

Christopher Kubas Kubas Consultants, Toronto

For the first time. a television advertising medra model is developed and tested on Canadian data. This new model, which is based on Waring's Theorem, is found to produce accurate estimates of the reach and frequency distribution of prototypical advertising schedules. The results produced by the WARlNG model are found to be better than the popular BBD-L.D approach developed for United States television. Due to its accurate results, the WARING model represents a significant advancement in advertising media planning in CoMda.

INTRODUCTION

Advertisers and advertising agencies are increasingly using computerized models to assist in their media planning (Leckenby and Boyd, 1984; Leckenby and Kishi. 1982a). These models are most frequently used to estimate the reach and frequency distribution of advertising media schedules. The reach of an advertising schedule is defined as the unduplicated proportion of a population or target market that is exposed to the media vehicle or advertising message at least once during a designated period of time. The frequency distribution is a description of how many times (0, 1, 2, 3 and so forth up to the total number of insertions in the schedule) a certain number or proportion of people or households in the target market will see or hear the advertisements in which the vehicles will be placed. Accurate estimates of reach and the frequency distribution are essential. as the advertiser wants to expose the audience within a specific repetition range deemed optimally effective.

Considerable research has been conducted on the development of magazine and television exposure distribution models. The majority of this work has bcen conducted in the United States using data from Simmons

The Authors wish to thank Mr. Bob Harris, President of Harris Media Systems and Mr. Peter Jones, President of the Bureau of Broadcast Measurement for kindly providing the data used in this study.

Market Research Bureau (SMRB) (Boyd and Leckenby, 1984; Chandon, 1976; Headen, Klompmaker and Ted, 1976, 1977; Kishi, 1983; Leckenby and Rice, 1985; Leckenby and Kishi, 1982b; Rust and Klompmaker, 1981). Internationally, research has examined estimation methods for French magazines (Agostini. 1961; Hofmans, 1966). Australian print media (Fry, 1973). and British print media (Caffyn and Sagovsky, 1963; Metheringham, 1964). Only one published article has appeared on magazine exposure estimation methods in Canada (Bower, 1963). No published work has been conducted on the development and use of Canadian television media models.

It is apparent that many advertisers and agencies are using models that were developed and tested using United States data. However, the smaller size and restrictive structure of the Canadian television environment will likely cause different patterns of viewer exposure as compared to the U.S. In particular, audience accumulation and program duplication might be unique to Canadian television. These differences could cause models that were parameterized and tested on U.S. data to perform poorly in Canada.

This study addresses this issue in several ways. First, the paper presents the first empirical test of the beta binomial distribution limited data media model (BBD-LD) on Canadian data. The BBD-LD has bcen shown to be one of the most accurate models for estimating exposure to U.S. television schedules (Leckenby and Rice, 1985). Secondly, a new model based on Waring's theorem (Waring, 1792) is developed and t e s a on Canadian data for the first time in this manuscript.

DESCRIPTION OF MODELS

(1) Beta Binomial Distribution Limited Data Model (BBDLD)

The beta binomial distribution direct estimation model (BBD-DE) is one of the most popular methods of vehicle exposure estimation. The model suggested by Greene and Stock (1967) incorporates both the beta distribution used by Hyett (1958) and the binomial distribution which serves

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as the basis for the Sainsbury Normal method (Caffyn and Sagovsky, 1963). The beta distribution is used to estimate the probability that an individual is exposed to any one, any two, any three, up to any number of vehicles included in the schedule. These probabilities are thcn used in the binomial distribution to estimate the distribution of exposure; that is, the proportion of the target audience that is exposed to one and only one insertion, two and only two insertions and so on.

The calculation of the beta binomial distribution involves the estimation of the exposure parameter a (alpha) and the non- exposure parameter p (beta). In the case of U.S. magazines, audience duplication data is available from SMRB and the alpha and beta parameters can be calculated directly using the BBD-DE method. The calculation of the BBD-DE involves the estimation of the average schcdule reach assuming one insertion per vehicle (known as R1) and the average reach assuming two insertions per vehicle (known as R2). The steps involved in the estimation of the BBD-DE are as follows:

R1 = X n i A i / h i Average Reach of one insertion in each of the vehicles in the schedule

Average rcach of two insertions in each and every distinct vehicle (1)

where: m = number of vehicles in schedule ni = numbcr of insertions in vehicle i N = total number of insertions i n

schedule R2i = accumulation of vehicle i over two

insertions Rij = net audience of vehicles i and j Ai = audience of vehicle

The alpha (exposure parameter) and beta (non- exposure) parameter are thcn calculated from R 1 and R2 as follows:

P(x:a,P,N) =

(3) h (a(a+l)(u+2) ... a+x-I)(P)(P+I)(Pt2) ...(P+ N-x-I) (x3 (a+p)(atP+I)(atP+2) ...( a + P + N - l )

whcrc: a$ are the BBD parameters N = number of insertions in schedule x = exposurenumber

As audience duplication data are not available for U.S. television shows, Leckenby and Rice (1985) calculated the parameters of the BBD indirectly using the method of "means and zeros". This procedure fits the BBD to the mean of the distribution (equivalent to gross rating points (GRPs) and the proportion of the audience that is not exposed to any vehicle in the schedule.

In ordcr to calculate the BBD-LD by the means and zeros algorithm, schedule reach must first be estimated. If duplication data were available, reach could be estimated via a number of different methods (Agostini, 1961; Hofmans, 1966; Kwerel, 1969). In this study, BBD-LD reach is estimated by averaging an "upper" and "lower" bound on reach. Multivariate binomial reach was used to estimate the upper bound (Leckenby and Rice, 1985). Multivariate binomial rcach is as follows:

where: Ru = uppcrboundofreach ai = single insertion audience rating of

ni = number of insertions in vehicle i vehicle i (4)

Multivariate binomial reach exceeds true reach because th is formulation assumes random within-vehicle duplication. Within vehicle duplication refers to the number or percentage of people exposed to the same vehicle on two different occasions. The random duplication hypothesis states that vehicle exposure constitutes a set of mutually independent events such that exposure to a certain vehicle docs not modify the probability of exposure to the same vehicle again. Clearly, this assumption is unrealistic. If an individual is exposed to a vehicle once, the person will be more or less likely to be exposed to the same vehicle a second time depending, to a large extent. on whether the individual liked or disliked program content.

(2) This causes random within-vehicle duplication to consistently underestimate true within-vehicle duplication and, when used in the binomial model, leads to an over- cstimation of reach. This has been demonstrated empirically by Leckenby and Rice (1985) and Chandon

Single inscrtion binomial reach proved to be a rcliable lower bound. That is, this formulation was found to be consistently lower than the true reach. The lower bound is calculated similarly to the upper bound (equation 4) except that the numbcr of insertions per vehicle is limited to one.

Alpha parameter R 1 (R2-R 1 ) a = 2Ri-RpR12

Bcta parameter p=- a(1 -R I ) R1

The final BBD distribution is then calculated using the following expansion. ( 1976).

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This effectively resolves the issue of random duplication in terms of its effect on the over-estimation of tfue reach.

When the uppedlower bound reach is chosen, the means and zeros algorithm is then utilized. In this procedure, an initial value of the exposure parameter alpha is chosen and the non- exposure parameter beta is calculated as follows:

~ ( N - G R P s ) = GRPs

N = total number of insertions in schedule

(5)

Reach is then calculated from these alpha and beta values, and this is compared to the estimated reach from the upper and lower bounds. If the initial reach is over- estimated then the alpha value is lowered and a new beta is calculated. This iteration is repeated until an alpha and beta value are determined that can estimate the upper/lower bound reach accurately. A BBD is then fit to these values through expansion.

The BBD-LD method has been shown to be one of the most accurate models of exposure for United States network television schedules. In a large scale test of 200 tabulated SMRB schedules, Leckenby and Rice (1985) found that the BBD-LD produced an average error in reach (AER) of 3.23% and an average error in the distribution of 18.77%. These results were considerably more accurate than the BBD Indirect Estimation method or binomial model also tested in this article.

The only required data for this model is the rating of the television shows to be tested. Canadian rating data obtained from the Bureau of Broadcast Measurement were utilized.

WARING MODEL

At the core of exposure distribution methodology is the inclusion/exclusion principle in probability theory (Riorden, 1958; Takacs, 1967; Feller, 1968). The inclusion/exclusion principle takes the following form:

where: P i = probability of at least one exposure ( a h )

S1 = grossaudience (6) S2 = sum of the duplications S3 = sum of the triplications S~ . .SN= higher order tuplications

The probability of exactly i exposures is then given by Waring’s Theorem (Waring, 1792).

We have Seen that the calculation of the beta binomial distribution can be accomplished after R1 (the average

reach of 1 insertion in each of the vehicles in the schedule) and R2 (the average reach of two insertions in each vehicle) are estimated. R1 is known for any schedule because the program rating and number of insertions are known.

Now, R2 can be calculated using the following formula:

where: R2/N = (N-1) S 132

where: S 1 = gross rating points S2 = sum of the duplicated audience N = # of insertions in schedule

The total number of insertions (N) and gross rating points are known for any schedule. The sum of the duplications (S2), however, are unknown and must be estimated. The estimation can be accomplished through the application of Waring’s theorem on prototypical tabulated schedules.

Specifically, the use of Waring’s theorem involves the backwards computation of numbers SN, S N - ~ ....... Sm from a series probabilities Po Pi ......., pN using the following rules:

for m = N-1, N-2, ....., 0, compute Sm = Pm + C (-1) k-m-1 k Sk (9)

The probabilities (PN) are the exposure levels of prototypical schedules. The number of interest in these calculations is S2, the sum of the duplicated audience.

Using this algorithm, the S2 values were calculated for 50 prototypical schedules tabulated from the 1985 Bureau of Broadcast Measurement (BBM) data tapes. BBM conducts studies of the television exposure habits of Canadians by having respondents complete a diary of their viewing behavior. The 1985 October-November Toronto BBM tapes were used to provide the data for this study. Readers are directed to the BBM technical guide for detailed information on the sampling frame and audience measurement methodology (BBM, 1985).

The S2 values generated from the tabulations then served as the dependent variable in a regression analysis using the known schedule characteristics of number of programs, number of insertions and gross rating points as the independent variables. A power function was found to produce the best fit of the data (R2 = .92, p I .01). Complete results of the regression can be found in Table 1.

The regression equation was then used to predict the S2 of new schedules and the alpha and beta parameters were

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calculated. The BBD was then expanded using formula 3 to produce the complcte distribution.

TEST PROCEDURE AND RESULTS

To test the BBD-LD and WARING model, a random sample of 92 television schedules were selected and tabulated from the 1985 Toronto BBM report. These 92 tabulated schedules represent the true or actual television viewing patterns of residents in this market. Thus, the tabulated schedules can serve as a benchmark against which the estimates of the models can be compared. In other words, a comparison of the reach and exposure distribution estimates of the models versus the tabulated data gives an objective assessment of model accuracy.

The television shows used in these schedules represented a wide variety of program types (daytime, prime-time, early evening and combination dayparts). The number of insertions ranged from 3 to 20 and GFWs varied from a low of 2.7 to a high of 252. Clearly, the variety of program types and intensity levels are representative of actual schedules used by practitioners.

The models were evaluated in terms of their ability to predict both reach and exposure distribution of the 92 tabulated schedules. All error calculations are based upon the absolute differences between the actual (tabulated) frequencies and those estimated by the two models. These were 1) error in reach (ERi); 2) error in the distribution (EDi) and; (3) numbcr of over-estimations (more than 5% over-estimation of the observed value of reach), under- estimations, (more than 5% under- estimation of the observed reach) and close estimations (within f 5% of the observed reach). Table 2 provides definitions for these error calculations.

Table 3 shows the average error in reach (AER) and the average emor in the distribution (AED) for all 92 schedules. As indicated in this table, the WARlNG model produced the most accurate estimates of reach and of the exposure distribution. Specifically, the WARING model produced an AER of 5.54% while the BBD-LD produced an AER of 8.19%. A t-test indicated the difference between these estimates was significant (T = 2.06, p I .05), Similarly, the WARING model produced significantly more accurate estimates of the distribution. producing an AED of 19 I .71% versus the BBD-LD estimate of 25.51% (T= 1.68, p .lo).

Table 4 shows the number of over, under and even estimations within +_ 5% of the observed reach. The WARING model estimated reach within 5% accuracy in 67 (72.8%) of the schedules. The BBD-LD approach, on the other hand, was within the 5% range in only 45 (49%) of the schedules. Both models showed a systematic propensity to over-estimate reach. This bias toward over- estimalion has been found in previous studies of the BBD (Leckenby and Rice, 1985).

Table 5 shows the actual and estimated distributions produced by the WARING and BBD-LD models for a

selected schedule. This schedule is characterized by 5 programs and low GRPs. For this schedule, the WARING model produced very good estimates, producing a 1.65% error in reach (ERi) and a 5.91% error in the distribution (EDi). This was considerably more accurate than the BBD- LD which produced an ERi of 4.43% and a EDi of 11.65%.

CONCLUSION AND IMPLICATIONS

A new model, based on WARING's Theorem, was developed and tested on 92 tabulated BBM television schedules. The WARING model was found to produce very accurate estimates of the reach and exposure distribution of these schedules. The model produced a 5.54% average error in reach and an average error in the distribution of 19.71%. The previously published BBD-LD model was tested on Canadian data for the first time in this study. This model produced results that were less accurate than the WARING approach.

The major reason why the WARlNG model produced more accurate results than the BBD-LD is that the latter model is weaker mathematically. The BBD-LD averages an upper and lower bound to produce an estimate of reach. There is no good theoretical reason for doing this, and in fact there may be good reasons why this should not be done. One questionable feature, for example, is that the low reach estimate does not change, regardless of the number of insertions in each vehicle. Each vehicle could have one insertion or several, and the Same low estimate would be obtained. This will lead to bias and increased error in the BBD-LD. The WARING model does not estimate reach by averaging an upper and lower value and thus does not encounter this problem.

It is interesting to note that BBD-LD model produced more accurate estimates for United States television than Canadian television. This could have occurred because of the high penetration of cable television in the Toronto area. The abundance of cable viewing options in this market makes channel switching more prevalent. This can lead to complex viewing patterns and less accurate estimates.

The WARlNG model is the first television media modcl specifically developed for the Canadian television environment. Media planning for this medium should be enhanced by the use of this model. The WARlNG model is practical and can be programmed on a personal computer for use in many advertising planning situations.

Cet ankle a QC st5lectionnt par M. Ronald J. Burke. This article has been selected by Mr. Ronald I. Burke.

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Table 1 Table 2

WARlNG Regression Results Definitions of Error ~~~

Indmndent Variablg Coefficient Estimate (1) Error In Reach: ERi

Log (GRPs) Log (number of programs) Log(number of insertions) Constant

F of Regression = 285.04**

1.1157** .3941* .3533**

-6.6684

n = 50 schedules

P I .01 * P 5 .15

* *

(a) ERi = lOi - eit where: O i = Actual or true tabulated

reach of schedule i ei = estimated reach of schedule i

(b) Average Error In Reach: AER

where: k = total number of schedules Cloi - eilk

(2) Error in Distribution: EDi

(a) EDi = loij - eijl where: Oij = observed frequency at exposure level

j of schedule i estimated frequency at exposure leveI j of schedule i

- 1, .... n)

eij =

(b) Average Error In Distribution: AED

where: k = total number of schedules C(l0ij - eijl)/k

(3) Accuracv With f 5% Reach:

Close estimation: Under estimation: Over estimation:

if 1% - ei I< 5% if Oi - ei > -5% if Oi - ei > +5%

Table 3 Table 4

Average Errors in Reach and Distribution for WARlNG and BBD-LD Models

Number of Over, Under and Even Estimations Within f 5% of Observed (Actual) Reach

Average Error in Average Error in Over 5% Under 5% Within 5% w Reach (AER) Distribution (AED)

WARING 24 1 67 92 WARING 5.54% 19.71% BBD-LD 47 0 45 92 BBD-LD 8.19% 25.51%

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Table 5

Estimated and Actual vabulated) Distributions for WARING and BBD-LD Models

Schedule 19

Rating Jnsertions

Vehicles: CIII Thur/Fri 1:00-1:30 1.85 2 CIII Thur/Fri 1:30-2:OO 2.00 3 CFTO Tue/Wed 6:00-6:30 4.35 2 m Thur 7:00-7:30 1.70 2 m Thur 7:30-8:00 1.80 1

Exposure Level ACTUAL WARING BBD-LD

0 88.80 87.15 84.37 1 4.80 7.27 10.47 2 3.30 2.86 3.31 3 1.60 1.37 1.19 4 .30 .70 .44 5 .90 .35 .15 6 -20 .I7 .05 7 .oo .08 .01 8 .10 .03 .oo 9 .oo .01 .oo 10 .oo .oo .oo

ER 1.65 4.43 ED 5.91 11.65

REFERENCES

Agostini. J.M., "How to Estimate Unduplicated Audiences," Journal of Advertising Research, March, 1961, pp. 11-14.

Boyd, Marsha and John Leckenby, "The Hofmans Beta Binomial Distribution," Current Issues and Research in Advertising, Graduate School of Business, The University of Michigan, Ann Arbor. 1984.

Bower, John, "Net Audience of US. and Canadian Magazines: Seven Tests of Agostini's Formula." Journal of Advertising Research, 3 (March, 1963). pp. 13-20.

Bureau of Broadcast Measurement, Why, What and How: A guide to the Measurement of Television Audiences by BBM, Bureau of Broadcast Measurement, 1985.

Caffyn, J. and M. Sagovsky, "Net Audiences of British Newspapers: A Comparison of the Agostini and Sainsbury Methods." Journal of Advertising Research, Vol. 3, No. 1 (March, 1963). pp. 21-25.

Chandon, Jean-Louis, A Comparative Study of Media Exposure Distribution Models , unpublished Ph.D. Dissertation, Northwestern University, Evanston. ILL., 1976.

Feller, William, An lntroduction to Probability Theory and Its Applications, Vol. 1 3rd. ed. New York: John Wiley & Sons, Inc., 1968.

Fry, Christopher "The Concept of Casualness", Feedback, No. 2. April 1973.

Greene, Jerome and Steven J. Stock, "Advertising Reoch and Freguency in Magazines," New York: Marketmath and Readers Digest, 1967.

Headen, Robert S., Jay E. Klompmaker and Jesse E. Ted, "Tv Audience Exposure." Journal of Advertising Research, Vol. 16. No. 5 (December, 1976). pp. 49-52.

Headen, Robert S.. Jay E. Klompmaker and Jesse E. Ted, "Predicting Audience Exposure to Spot Tv Advertising Schedules," J O W M ~ of Marketing Research, Vol. 14 (February,

Hofmans. Pierre "Measuring the Cumulative Net Coverage of a Combination of Media." Journal of Marketing Research,

Hyett, G.P. "The Measurement of Readership," paper read to the statistics seminar, London School of Economics, February, 1958.

Kishi, Shizue, "Exposure Distribution Models in Advertising Media", unpublished Ph.D. dissertation, Department of Advertising, The University of Illinois at Urbana-Champaign, Urbana, Illinois, 1983.

Kwerel, Seymour M.. Estimating Unduplicated Audience and Exposure Distribution," Journal of Advertising Research, Vol. 9, No. 2 (June, 1969), pp. 46-53.

Leckenby, John D. and Marsha Boyd., "How Media Directors View Reach/Frequency Evaluation Standards," Journal of Advertising Research, Vol. 24, No. 5, (Oct/Nov. 1984) pp.

Leckenby, John D. and Shiwe Kishi. "How Media Directors View ReachFrequency Estimation," Journal of Advertising Reseorch, Vol. 23. No. 3. (June/July, 1982a), pp. 64-69.

Leckenby, John D. and Shizue Kishi, "Performance of Four Exposure Distribution Models," Journal of Advertising Research, Vol. 22. No. 3, ( A p r W a y 1982b), pp. 35-44.

Leckenby. John D. and Marshall D. Rice, "A Beta Binomial Network Tv Exposure Model Using Limited Data." Journal of Advertising. Vol. 14, No. 3, 1985, pp. 25-31.

Metheringham. Richard A. "Measuring the Net Cumulative Coverage of a Print Campaign," Journal of Advertising Reseorch, Vol. 4, No. 4 (December, 1964), pp. 23-28.

Riorden, John, An Introduction t o Combinatorial Analysis. New York: John Wiley and Sons, Inc.

Rust, Roland T. and Jay Klompmaker, "Improving the Estimation Procedure for the Beta Binomial TV Exposure model," Journal of Marketing Research, vo1.28 (november. 1981) pp. 442- 448.

Takacs, Lajos, "onthe Method of Inclusion and Exclusion," Journal of American statistical Association, 62. (March,

Waring, E.,"on the Principles of Translating Algebraic Quantities into Probable Relations and Annuities". Cambridge University, 1792.

1977). pp. 1-9.

(August, 1966) pp. 269-278.

43-52.

1967). pp. 102-113.

RESUME

Pour la premidre fois. un moddle de publicite' te'le'vise'e est de'veloppe' et teste' sur des donntes canadiennes.Ce nouveau modde, base' sur le the'ordme de Waring, fournit des estimations pre'cises concernant la porte'e et la distribution des frkquences de prototypes d'horaires publi- citaires.Les re'sultats obtenus par le moddle de Waring shverent meilleurs que l'approche usuelle du BBD-LD, de'veloppe'e pour la te'le'vision ame'ricaine.Compte tenu de ses re'sultats pre'cis, le modde de Waring repre'sente un progrks conside'rable dans la planification publiciraire au Canada.

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