a stochastic model of radio listening

A Stochastic Model of Radio ListeningAuthor(s): Roger A. LaytonSource: Journal of Marketing Research, Vol. 4, No. 3 (Aug., 1967), pp. 303-308Published by: American Marketing AssociationStable URL: http://www.jstor.org/stable/3149464 .

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ROGER A. LAYTON*

There has been considerable discussion of the validity of alternative measures of radio listening. Stochastic process models are used to explore the relationship between some common measures and, in the process, the sort of data that should

ideally be collected is specified.

A Stochastic Model of Radio Listening

INTRODUCTION

This article explores the differences between the diary and coincidental methods of assessing radio audiences and illustrates the usefulness of looking at radio and other media measurement problems with simple stochastic process models. An approach of first formulat- ing a probabilistic model of the basic phenomena and then collecting data that will provide estimates of the parameters of that model holds some promise of re- solving a number of much debated questions about the suitability of various measures. In addition, it focuses on those measurements that are essential in assessing, listening, viewing, or reading behavior.

THE BASIC MODEL

Radio listening is seen as a simple "on-off" process, where a radio is in one of the states at any point in time. Obviously, an extension to allow for the set's being tuned to any one of a number of stations would be useful; this can be done,1 but here this would unduly complicate the presentation of the basic ideas. No consideration is given to the possibility that a radio may be on with no one actually listening. Put another way, the model is concerned only with the sequence of on-off states for a specific radio.

The process under study is shown in Figure 1. In this figure, the xi' are the successive intervals

during which the radio is on, and the xi" are the corre- sponding off periods. It is obvious that these intervals will alternate; this is the reason for the name usually given to this kind of stochastic process-an alternating renewal process [1].

The xi are typically random variables, a reasonable assumption to make in dealing with radio, when pro- grams often vary in length and commonly run early or late on any station during the day. (With television, the situation is different as there is a much more success- ful attempt to keep to the hour or half hour breaks. This leads to a certain nonrandomness in the distribution of the xi and makes an analytical approach more difficult.)

In view of the above, it is assumed that the xi' are independently and identically distributed in an exponential manner. Similarly, the xi" are independently and identically distributed exponentially, but with a parameter that may be different from that of xi'. These are stringent assumptions and so some comment is necessary. The assumption of exponentially distributed variates materially simplifies the analysis. However, without too much difficulty, it would be possible to work with gamma distributions. A check of the available data drawn from diary records suggests that the exponential distribution is not unreasonable. It is impossible to be more precise as the diary is based on "at least eight minutes or more" listening in 15 minutes and thus misses short listening bursts such as the hourly news. If the actual distribution should appear untract- able, simulation techniques can always be used to analyze the process. This is an important possibility flowing out of this approach; it may be particularly useful in an analysis of television viewing where regu- larity complicates the distributions.

Besides the assumption of exponential distributions, a further assumption is that the parameters of the distributions are constant over time. In any practical application of the theory this is an important constraint, as the nature of the listening process can be expected to vary markedly during the course of a day, e.g., it depends on the shopping habits of housewives at specified times. Though this is a problem, the theory is still useful in predicting the effect of changes in the basic

* Roger A. Layton is professor of marketing, University of New South Wales, Sydney and is a consultant with Anderson Analysis Proprietary, Ltd.

1 The model would then take the form of a semi-Markov process.

Journal of Marketing Research, Vol. IV (August 1967), 303-8

303



304 JOURNAL OF MARKETING RESEARCH, AUGUST 1967

On-Off States for Radio Listening

Figure 1

SI I XI

SI I I

I I I I I I I I I

parameters; in any case, when there is sufficient doubt, the situation can be explored by simulation.

More precisely, it is assumed that the xi' are distributed exponentially with a density function pie-Px'X and the xi" are distributed exponentially with a density function p2e-p2x".

The xi' are independent and also independent of the xi".2

SOME PROPERTIES OF THE BASIC MODEL

Many of the results were derived by Cox [1] and will be simply mentioned. Perhaps the most important is the probability that the system will be in either the on (State 1) or the off (State 2) condition at time t after the start. This probability depends on whether the radio was on or off at time 0, but provided t is sufficiently large these initial conditions are unim-

portant. Assuming the radio was on at time t = 0, the prob-

abilities of its being on (pll(t)) or off (pl2(t)) at time t is given by:

(1) pn1(t) = P2 + Pl e(Pl+P2) t

Pl + P2 1 + P2

(2) pl2(t) = Pi

{1 - e-(Pl+P2)t}. P1 + P2

Similarly, assuming the radio was off at t = 0, the

probabilities of its being on (p21(t)) or off (p22(t)) at time t are:

(3) p21(t) = {2 1 - e-(P1+P2)t}, P1 + P2

P1 P2 -(Pl+P2)t (4) p22(t) = ?1 + e

P1 + P2 P1+ P2

Obviously, for large t, these will tend toward the limiting values,

P2 (5) pl = 21

(6) p2 P

Pl + P2

If t is large enough for (5) and (6) to apply, then these indicate the probabilities that the radio will be on or off at that time. This is obviously important when it comes to considering coincidental measures.

For some purposes it is also important to be able to

say something about the number of changes from on to off or vice versa that occur in a period of time of length t. For example, in Figure 2 the number of changes is 4.

It will also be noted in Figure 2, that the time to the first change is S,, to the second S2, and so on. These times are random variables with a distribution resulting from the sum of a number of exponentially distributed variates. The cumulative distribution associated with S, will be written K,(t), i.e.,

(7) Kr(t) = Pr (S, < t), = Pr {time up to the rth change is less

than or equal to t}.

Let Nt be the number of changes during t. Then,

(8) Pr {Nt < r} = 1 - Kr(t),

and

(9) Pr {N, = r} = K,(t) - Kri(t).

In particular, Pr {N, = 0} = Ko(t) -

Kl(t), = 1 -

K(t),

(10) Pr {N, = 1} = Ki(t) - K2(t),

Pr {N, = 2} = K2(t) - K3(t),

etc. Thus finding the distribution of Nt involves first

finding the distributions Ki(t), K2(t), etc. Remembering that the xi' and xi" are exponentially distributed, those distributions can be readily derived.

If the system starts with the radio on, S1 = xl', and

(11) Kl(t)

= 1 - e-t; if it is off, S1 = x2', and

(12) Kl(t)

= 1 - e-P2

Considering K2(t), note that S2 = Xl' ? Xl" (this does not depend on whether it starts on or off). Form- ing the distribution of the sum of these two exponentially distributed variates, and assuming pl 7 p2,

2 This is a further assumption that would need consideration in practice: to what extent will long-on periods be followed by long- off periods?



A STOCHASTIC MODEL OF RADIO LISTENING 305

K2(t) P1P2 11 - e"P P2 Pl pl1-

e} (13) 1

- {1 - e- . P2

Turning to K3(t), it is important to distinguish between the two initial states of the systems. Again assuming pi p2 ,if the system starts in the on state,

K3(t) = lP2 P1IeP1 (14)P2-

P1 (P2-

P,

1Pi (14) - -e e 2

1 e t? + 1+ 2

; P2 P1 P,1 P1 P2

if it starts in the off state,

SPP22 1

K3(t) = P12

2 Il Ie-P P2t

(15) \Pi- P2 P1- P2 P2

1 1 1P2 t 1 P- P2 - e-?t

- -- e-P t + + 2 * P1 P2 P2) P1P2

Though it is quite possible for this work to cover more than three changes, for the problems considered here, these results are adequate, i.e., for the common values of pi, P2 -' 0.2 for a 15-minute interval more than three changes are extremely unlikely.

Using (10), the distribution for Nt can now be written. Assuming on at t = 0:

(16) Pr {Nt = 01 = e-'t,

Pr {Nt = 1} = 1 - e-p t _

P1P2

(17) P2- P1

S{1- e-P {1 -eP2

Pr {Nt = 2} = K2(t)

P1P2 1 1

(18) - 1 1

P2 '-

Assuming off at t = 0:

(19) Pr {Nt = 0} = e-P2t

On-Off Patterns of Four Changes

Figure 2

On - , r ---

I I Off L

,

Time

to rth i

change L8. II

Pr { N = 1} = 1- e-P2t - P1P2 P2 - P1

(20) 1 {1 - e-P}- P1 P2

S- e-P2t}],

PrI{Nt = 21 = P1P2-pt

(21) P2 P1 P1

1 eP2t - {1 - e-P . P2

The values obtained from (16) to (21) depend on the sizes of pi , P2 , and t. If t is at all large, for example, more than three changes may be likely and the formulas would have to be extended. Under these circumstances, interest often centers on the average or expected number of changes during a period of length t. This can be found by considering the Laplace transform of the renewal function of the process.

Writing the transforms of the density functions for the on and the off periods as fi*(s) and f2*(s), Cox [1] shows that the transform of the mean number of off periods during time t, H2(t), assuming that the radio was on initially, is:

fi*(s)f2*(s) (22) H2* (s) =)f2 s{1 - fi*(s)f2*(s)

Similarly, the transform of the mean number of on periods, Hi(t), is:

fi*(s) (23) Hi*(s) = f*(s) s{1 - fi*(s)f2*(s) u

Substituting




fi*(s) P' and f2*(s) = P

pl+ s P2 + S

and inverting, leads to the results:

(24) Hi(t) = pl{p2(t - 1) + (Pl I p2)e-(Pl?P2)t},

(25) H2(t) = pi{p2(t - 1) + p-2e-(Pl+P2)t

Similar results hold if the radio were off initially. When t = 1 (a case of interest later), the following

equations can be found: Set on initially:

(26) Hi(1) = pie-(P1?P2) I + P2},

(27) H2(1) = plp2e-(P1P2)

Set off initially:

(28) HI(1) = pip2e-(P1P2),

(29) H2(1) = p2e(P1P2) {1 + pl}.

By addition, the mean number of changes is: Set on initially:

(30) Hi(1) + H2(1) = pielP{+P2) 1 + 2P2}. Set off initially:

(31) H1(1) + H2(1) = p2e-(P{1P2) 1 + 2p1}.

Considering now the probability that the radio is on or off at a point in time, i.e., (5) and (6), the average number of changes in unit interval is given by:

(32) 2 P1P2 e-(P1+P2) ? p1 -P2}. P1 + P2

With these results, it is now possible to explore the differences between the various methods of measuring radio audiences.

COMPARISON OF AUDIENCE MEASUREMENT PROCEDURES

The methods of interest here are the diary procedure and the coincidental check. The first means that chosen families have a diary for each radio, and they are asked

to note in the diary if the radio were on for eight minutes or more in 15 minutes. Since in practice, this criterion may be modified to have them record if the set were on at all in the quarter-hour period, both versions are considered. With the coincidental method, a personal or telephone interview is used to establish if a radio is on at a point in time. Though, much more information is usually obtained than the simple indication of whether a radio is on or off, this is disregarded here. Also, problems of sampling, e.g., probability sampling with the diary or an even rate of sampling, allowing for non- contacts, under the coincidental check, are crucial to the practical interpretation of the theory, but these are also disregarded.

To make use of the results of the previous section, a basic time unit has to be established. It is most convenient to suppose that one unit of time corresponds to a quarter hour (or, more generally, to whatever recording interval is used in the diary system). A check of diary records suggests that in many cases the average on or off period extends over five quarter hours, so that pi and p2 take on values near .2.

Some further assumptions are that pi and po are constant over the period of time being considered and, to avoid the aggregation problem, all listening groups have the same pi and p . Both assumptions could be relaxed if necessary, e.g., by assuming there is some functional relationship between the pi and time and that the pi follow some convenient bivariate distribution in the population.

If the process has settled down, the probability that a radio will be on by a coincidental check at time t is, by (5) and (6):

(33) P2

P1 + P2

and similarly, the probability of its being off, is:

(34) P1 p1 + p2

For the diary method consider the situation where a respondent records if the radio was on at some time during a period of length T.

Table 1 VALUES OF R

(T = 1)

P2

pi 4 2 1 / N/4 /8o 4 1.982 2.729 3.528 4.148 4.539 4.760 4.808 2 1.491 1.865 2.264 2.574 2.770 2.880 2.903 1 1.245 1.432 1.632 1.787 1.885 1.940 1.952

1.123 1.216 1.316 1.393 1.442 1.470 1.476

1/ 1.061 1.108 1.158 1.197 1.221 1.235 1.238 1.031 1.054 1.079 1.098 1.111 1.118 1.119

o10 1.025 1.043 1.063 1.079 1.088 1.094 1.095



A STOCHASTIC MODEL OF RADIO LISTENING 307

The probability of the radio's not being on at all during T is given by the product:

Pr (radio off at t) X Pr (radio remains off during inter-

val t, t + T) ( radio was off at t)

(35) P1 e-P2• pl + P2

The probability of recording that the radio was on at some time during the interval t, t + T is thus:

(36) 1 - P e- P1 + P2

This then is the probability that will be estimated by this diary recording. Comparing this with the proportion of radios found on by a coincidental check at time t leads to the ratio:

R = Diary (radio on at any time) estimate

Coincidental estimate

P1 -P2T 1- e

p2

P1 + P2

= 1 + (1 - e-P2T) P2

R will always be greater than or equal to one, i.e., the diary figures will normally be a little greater than the coincidental check results. The size of the ratio depends on the relative mean lengths of the on and off periods (a relatively long off period will increase R) and on the length of the interval, i.e., the greater T, the larger R.

This result has, of course, often been verified by empirical observation-diary audiences are usually larger than the coincidental check audience. This is not necessarily an error, simply a consequence of two different ways of measuring the same basic process. Which method is most useful depends on the user's purpose and will not always be the same for all potential users of audience data. If instead of trying to measure audiences more or less directly, more thought was given to measuring basic parameters such as pi and pj and to an empirical check of the distribution assumptions, it may be possible to derive either measure at will.

Some idea of the size of R for different values of pi and p2 can be seen in Table 1. Here T = 1, and it is assumed that the effect of the initial conditions has worn off. Using the values of pi and p2 noted earlier, R is about 1.18-the diary method would indicate an 18 percent bigger audience than the coincidental method. As the average length of an off period increases, i.e., as p2 decreases, the diary audience becomes rela-

Table 2 VALUES OF R*

P2

x= o

1 1.315 1.225 1.118 1.068 1.040 14 1.168 1.123 1.070 1.045 1.031 1j 1.088 1.066 1.039 1.027 1.020

1.060 1.045 1.027 1.019 1.014 Y8 1.045 1.034 1.021 1.015 1.011

x= Iz P1 1 i ' '8 1 1.234 1.145 1.046 1.000 0.975 /1 1.137 1.092 1.043 1.020 1.007 Y4 1.076 1.054 1.029 1.018 1.012

/N 1.053 1.038 1.022 1.014 1.010 Y8 1.041 1.030 1.017 1.011 1.008

P1 1 '2 '6 '8 1 1.000 0.925 0.855 0.825 0.808 ? 1.037 1.000 0.965 0.950 0.941 Y4 1.036 1.018 1.000 0.992 0.988 S1.029 1.017 1.005 1.000 0.997 /N 1.024 1.015 1.006 1.002 1.000

x= 34 P1 1 26 '8 1 0.806 0.759 0.720 0.704 0.695 ? 0.947 0.923 0.904 0.896 0.891 Y4 0.997 0.985 0.976 0.972 0.969

/N 1.005 0.997 0.991 0.988 0.987 Y8 1.007 1.001 0.996 0.994 0.993

tively larger, and the effect is most pronounced with very short on periods coupled with long off periods.

Cox [1] shows that the long run probabilities (33) and (34) hold when the distributions are not exponential. Writing F2(t) as the distribution function for the off periods, it can be shown that,

(38) R = 1 + Pl F2 (T), P2

and it will be noted that the same qualitative results seem to hold. If the off period distribution should be more highly skewed than the exponential, F(T) will be larger and R will go up, i.e., if there is a tendency for very short or very long off periods.

If the radio is recorded as being on, only if it was on for a fraction x of a recording interval, the analysis becomes somewhat more complicated. It will be assumed that there is at most one change during a recording interval; this can be relaxed if necessary.

If there are no changes during a recording interval, the probability that the radio is on for at least x of the interval is 1 if the radio were on at the beginning of the interval and zero otherwise.

Remembering that a recording interval is being taken as a time unit, if there is one change during the interval,




the probability that the radio is on for at least x of the interval is e-P" if it was on at the start and 1 - e-P2'(1-) if it was off at the start.

Using (16) to (21), the probability that the radio was on for at least x of the interval, assuming it was on initially, is,

e-Pi + (1 _ e- )e1;

and assuming it was off initially, is,

(1 - e-P2) (1 - e-P2(1-x))

The overall probability is thus

P2 e-pl e-plx e-pl(l+x) PI + P2

+ Pl 1 (1-x)

-- eP2 -P2(2-x)

P1 + P2 1.

Comparing this with P2/(Pl + P2) a new diary/coincidental check ratio R* can be established. This is tabu- lated in Table 2 for different values of x. As x increases, the ratio declines, this is to be expected as increasing x involves restricting the diary coverage and for large x the diary is not indicating as large an audience as the coincidental check. It appears that x = /1 gives a diary measurement that is close to the coincidental check. If the recording interval is a quarter hour, this suggests that a diary based on recorded listening for at least 8 of the 15 minutes should give results comparable with the other method.

CONCLUSION

This article developed from a debate over whether a diary or a coincidental check measurement gave a true indication of audience size. The audience obtained by the diary method was significantly larger than that obtained by the coincidental check-a difference that troubled advertisers. As a result of the analysis described here, it was seen that the difference was largely caused by alternative ways of assessing the same under- lying process.

This suggests that a variation in the more usual ap-

proaches to measuring radio audiences may be of value. Instead of a simple yes-no observation relating to a point in time or a short period, it may be more useful to take as a basic unit of observation a listening period- i.e., a period during which a person is listening con- tinuously to one or any station-and to collect data from each sample respondent on his listening periods for a week. Using these data, it would be possible to infer the value of an equivalent diary or coincidental check measurement, whichever is of more interest.

The model, as it stands, provides a theoretical basis for interpreting data on listening periods and could easily be extended to take into account switching patterns between stations. It would also be of value in looking for differences in listening behavior among the various population segments. These differences would show up in the estimates of the parameters of the listening period distribution. In short, a listening period has many of the attributes of a natural building block for the study of audience behavior, and the model described points to the kind of theoretical and practical analysis that can be based on the concept. An approach built on the concept of a listening period could have the additional ad- vantage of focusing the attention of media buyers on the basic qualities of a particular medium, rather than on the more or less technical problems of measurement.

The insights provided by the model in this simple case suggest that these may be of considerable value in a more general application to other media audience measurements. It may be particularly useful in studying magazine readership where there are many ideas about what constitutes a reader. Instead of debating the merits of differint definitions, it may be more productive if a stochastic model of the process of acquiring and then reading a magazine were constructed, with surveys directed toward obtaining the data needed to estimate the parameters of the stochastic process. This kind of model is being explored in television and magazine reading, and it is hoped that some results will be available soon.

REFERENCE

1. D. R. Cox, Renewal Theory, London: Methuen and Co., 1962.



a stochastic model of radio listening

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