Advertising: does the S-curve apply?

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Faculty Working Papers

ADVERTISING: DOES THE S-CURVE APPLY?

Johny K. Johansson

#72

College of Commerce and Business Administration

University of Illinois at Urbana-Champaign

FACULTY WORKING PAPERS
College of Commerce and Business Administration
University of Illinois at Urbana-Champaign
November 14, 1972

ADVERTISING: DOES THE S-CURVE APPLY?

Johny K. Johansson

#72

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Introduction

Almost all literature on advertising agrees on one point: the
effect of advertising upon sales or market shares gradually reaches a
saturation point, aftier which additional advertising does not have any
effect. Reasons for this characteristic of advertising can be found ;
at both the individual and the aggregate level. As a person gets more
and more exposure to advertising, the possibility of increasing the
response (purchase, say) is limited. Larger quantities can be bought
than before, but generally the response becomes a matter of purchase
or no purchase (see Bogart, 1967). At the aggregate level, as the
advertising increases, the additional prospects exposed are usually of
lower potential than earlier prospects (see Stigler, 1961) or already
exposed and committed prospects are reached (see Longman, 1971).

In the same writings on advertising, however, much discussion is
often expended on the question of whether there is a "threshold" effect
of advertising, that is, whether advertising will have a small effect
initially and then a "takeoff" as more advertising is don«. One reason
for this takeoff at the individual level would be that th^ individual
would need some repetitive stimuli before he/she acts. Against this
hypothesis stands the argument that the first exposure will always be
the most effective one, later exposures being less new and interesting
and largely serving a supportive role. At the aggregate level the
takeoff would occur because the "right" media for reaching the best
prospects would be available (this argument refers most often to
Network TV) , and because without a certain level the advertising ef-
fort would be on the whole "drowned out" by competing messages. Against
this hypothesis it has been argued that the choice of media vehicle is

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such that the prospects with the highest potential are reached first
anyway; and that although very small expenditures are hardly worth-
while, a small advertiser can concentrate his advertising in a few
vehicles and thus make a relatively rtrong impact (see Bogart, 1967).

.The balance of available evidence, as reviewed by Simon (1965,
1969) and Preston (1968) , seems to lean towards the "no- takeoff" hypo-
thesis. Thus, both authors when referring to sales or market shares
data find very little evidence supporting the sigmoid or S-shaped
curve and generally see the advertising response as exhibiting dimin-
ishing returns. On the other hand, surprisingly little empirical
research has attempted a direct fit of the sigmoid to available data.
Thus, no research incorporating the logistic directly is covered in
Simon's two reviews or in Preston's review of scale returns. One
explanation lies in the difficulty of obtaining nonexperlmental data
where observations occur in the increasing part of the curve. If man-
agement made their advertising decisions correctly, they would normally
not stop their advertising there. This cannot provide the complete
explanation, however, since a good deal of the research discussed by
Simon in his two reviews covers experimental or quasi-experimental
data, which presumably could be analyzed directly for a sigmoid fit.
Furthermore, even if entrepreneurs would generally not like to operate
on the increasing part of the curve, they might still find themselves
there. This could be the case if the effectiveness of the advertising
effort Is partlj" determined by the competitive advertising input, so
that what matters Is the firm's "advertising share relative to com-
petitors' shares". Tlien the entrepreneur would no longer be in full
control of his advertising input--and increases In the spending might

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-3-

very well be offset by Increases in competitive spending. That this
can happen is pointed out quite frequently--8ee, for example, Bass
(1970), Bogart (1967), and Telser (1964).

Perhaps the major reason for the lack of a direct approach to the
estimation of the sigmoid cur/e is the mathematical and statistical
problems encountered. In resolving the msthematical difficulties one
would generally complicate the statistical problems. Thus, the common
generalized logistic function requires iterative maximum likelihood
estimation techniques (see Oliver, 1956) which become quite expensive
considering the type of data and the number of variables usually
involved in advertising research. Conversely, the slmplar estimating
models are generally incapable of exhibiting the required S-shape.
Accordingly, there is a need for a sigmoid function that is possible
to estimate using relatively simple statistical techniques. Because
such a sigmoid function has been developed (see Johansson, 1972a, and
1972b) it is now possible to estimate directly the goodness of fit of
the S-curve relative to alternative simpler models of advertising
effects. That is the subject of this paper.
The Data

Before dealing with the specification of the actual estimating
models incorporating the sigmoid, it will be useful to have an account
of the data available.

The data consist of monthly surveys of product users' prefer-
ences, and purchases (present and past) with respect to the four big-
gest national brands in a consumer product class (non-seasonal) . Each
month a different sample of 1000 respondents representative of the
U.S.' population was selected. Usable returns, received within the

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-4-

first two weeks of mailing (later returns were eliminated because of
the overlap problem), in general numbered about 600. The four national
brands used in the analysis were observed during 13 consecutive months.

The surveys also provide information as to degree of usage of
product, whether or not the last brand purchased was bought with a
coupon, on a deal, or sale, or regular price, and what brands the
respondent had bought within the last six months. In addition to the
survey data, data on prices during the months observed were collected
independently through general trade information and price lists.

The advertising data consist of monthly brand expenditures in
four different media — Network TV, Spot TV, Magazines, and Newspapers
(non- retail advertising). The observations were compiled from the
records maintained by Leading National Advertisers, Inc., and, in the
case of Newspapers, from the "Blue Book" published by Media Records.
These data, as is well known, suffer from certain deficiencies. Not
all stations in the country report the Spot TV expenditures; for the
62% that do, gross one-time rates for length of commercial and daypart,
without discounts, are used, under the assumption that this will allow
for the missing data, Tna Magazine expenditures are based upon adver-
tising space and revenue analyses, and cover about 807o of total expen-
ditures in consumer magazines. The Newspaper data are in linage, not
dollars, and only about 75% of total newspaper advertising is covered
in the Blue Books. The data for Network TV, finally, are probably the
most accurate. Here expenditures refer to net time (after discount
for daypart) plus talent or program estimated cost. Because of the

existence of only three network companies, these data are relatively

^ 1
easy to record.

As the frequency of purchase within the product class is on the

average one purchase per month, it was deemed feasible to trace monthly

effects from the media advertising inputs. Even so, it was clear that

some distributed lag approach might be needed to account for a possible

2
cumulative advertising impact over time. It should be emphasized,

however, that these data do not allow a test of whether or not the

S-curve is applicable over the long run.

Because of the short time periods involved, it was decided that
media inputs should be kept separated in the regression runs. This
would allow different coefficients and different lags to emerge--
although in the long run possible media differences tend to disappear,
in the short run they are often assumed to exist (see, for example,
Bogart, 1967) . The total advertising impact is then measured as the
sum of the different effects over time and over media.
Model Specification and Estimation Method

Because the new version of a sigmoid function and its estimation
has been covered in detail elsewhere (see Johansson, 1972b) , only a

1
The reader is referred to Johansson (1972a) , Chapter 4, for a

more extensive discussion of the data. One point should be noted,

however; although the focus here is upon national brands, the monthly

expenditures on media advertising vary widely over time, so that in

fact we have observations throughout the relevant range,

2
Because the advertising data did not come from the surveys, the

restriction to a 13 month time period did not apply and lags would

thus not decrease the number of observations available.

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■6-

a brief account will be given here. It is shown there that the model
/n\ y 1 2 n

with y the dependent variable, and x. , i'"lj...,n, the Independent
variables, exhibits a sigaoid or S-shape whose skewness is determined
by the estimated parameters b. ,1=1, . , , ,n, and with a saturation level
of 1.0. For a non-unit saturation level, say k, the model becomes

b,b- b

fo\ y . ^2 n

k-y 12 n

Finally, a non-zero Intercept, say I, can be introduced by writing

b b^ b ■

(3) — i-:--- «= ax^ X2 ... x^

The features of skewness of the S-shape, a non-unit saturation level,
and a non-zero intercept are all desirable to allow for in estimating
the advertising effect. As one model alternative, however, a com-
pletely symmetric sigmoid is also desirable. This is achieved by
using a "loglinear" form of the above versions. Thus, the completely
symmetric S-shape is depicted for model (3) by

(4) m -^-:-L „ a + b^^j^ + b^x^ + ... 4- b^x^ .

In standing for the natural logarithm.

Possible estimation methods for the saturation level k and the
intercept I could be derived from the approaches suggested by Croxton
& Cowden, 1939, or Nelder, 1961, or Oliver, 1969, for different
versions of the logistic function. The amount of computation involved,
especially when a priori notions about k and I are rudimentary, is
considerable, however. Another and much simpler approach which

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-7-

suggested itself was to use the construction of the questionnaire in
developing an estimate of the appropriate values. Thus, the upper
limit on the proportion of purchasers of a particular brand was
clearly the number of triers of the brand during the last six months.
If all these triers purchased last time around then for that month the
saturation level would in fact have been reached. Note that here the
saturation level is allowed to vary between months, a feature which at
first might seem undesirable but in fact has some advantages over a
static level. For one, it tends to eliminate variations in observed
proportions due simply to random variations between consecutive
monthly samples. Second, it will in fact also allow for an actual
change in the saturation level over time, a phenomenon that sometimes
can be seen as desirable (see Nicosia, 1966) .

For the intercept I the questionnaire provides another measure--
the proportion of users having bought the brand last and the next to
last time. Again, the measure is in fact the lower bound on the pur-
chase proportion; and again a rationale behind the choice can be seen.
As adverting goes towards zero, one would assume that some users would
continue to buy, namely those that are brand loyal. Then the propor-
tion repeating purchase can be seen as a proxy variable for such
loyalty.

If we assume the saturation level k and the intercept I thus
known, a natural procedure for estimating the parameters of these
models would seem to be the ordinary least squares. Because of the
limitation of the variation in y, however, this is not the correct
procedure. The error variance will be a function of the value taken
by y; thus, there is a problem of heteroscedasticity. An approximate

-8-

expresslon for this error variance is developed in the case where y is
a proportion (the case dealt with here) in Johansson, 1972b. Conse-
quently, the procedure followed here consist of first dividing through
each observation by the square root of this error variance after which
ordinary least squares are applied (a special case of generalized
least squares) .
Further Model Specification

One logical way to see how well the generalized sigmoid fits the
data is to test it against alternative functions. Tliis was done by
specifying a set of models directly explaining the sample proportions
observed:

(10) PUR^^ ^ f^ <T^^AL^.t-l' ^^^i,t-r ^^^^it' *^^it'--"«^«i.t-j,,,

MAG

NET , . . . ,NET_, ^ ^ SPOT^^,.,.,SPOT_, ^^ NEWS
NEWS

l-*- J*- » •- JMRf '*■'' *■ » "'spot i.1.,....

i "= 1,...,4, and tal, . . , , 12,

with j subscripted by media representing the longest lag to be consid-
ered, and where the subscript i stands for brand, and t for month. The
variable definitions and measurement are as follows:

PUR = purchase; the proportion of product users who indicated that
they bought the brand last time (used as a proxy measure of
market share)

TRIAL = trial; the proportion of users indicating that they had tried
the brand within the last six months

PREF == preference; the proportion of users who said they liked the
brand better than any other brand

DEAL = deal; the proportion of users who did not buy their last
brand at regular price

MAG « magazine advertising share; the share computed relative to
total magazine advertising by the four national brands

NET ■= network TV advertising share, computed similarly to MAG

SPOT = spot TV advertising share, computed similarly to MAG

NEWS = newspaper advertising share, computed similarly to MAG

This model was estimated using a linear form, a semi- logarithmic
form and a double- logarithmic form. Each one of these three functions
can be seen as an alternative to the S-curve,

The "odds" model was used to specify the competing S-curve. The
following model
PUR.

ill)

trial'- PUR.^ ° f2.(gREg,,,.i>DEAL^^.MAG^^.....MAG .

^"^MAG

NET ,...,NET .SPOT ,..., SPOT .NEWS , . . . ,NEWS^ ^ ^

^"^ ^"^ JnET ^^ ^'^'^SPOT ^^ ^'*^'^NEWS),

i - 1,...,4, and t=l,...,12.

variables defined as above, was run as a loglinear and as a doublelog
form, exhibiting the symmetric and skewed sigmoid, respectively.

Finally, the odds model with the intercept term included was run:
PUR - REP.

^^2> iHA^^^^~pfe^'^^3'<^^^'^.t-L' DEAL.^.MAG^e-"^^i t- 1

it it » ■'MAG

NET , NET ,SFOT ,SPOr ,NEWS^ ....,
It i,t jj^g^ xt i.t-jgpQ^ it

NEWS

' -"news),

i'=l,...,4, and t»l,.,.,12,

where REP = repeat; the proportion of product users who indicated that
they bought the brand last time and the time before that. Again a
loglinear and a doublelog form were introduced.

-10-

Some remarks on the specifications are in order. First, the ab-
sence of price (absolute as well as relative price) is motivated by
the almost constant price levels maintained for the brands throughout
the period. Some price competition took place in the form of specials,
coupons, etc, which are picked up by the DEAL variable. TRIAL and

PREF are introduced so as not to ascribe to advertising some effect

3

that is due to experience and affect towards the brand from the past.

Some potentially important variables are left out of the models.
This includes product quality, distribution variables such as retail
coverage and intensity, and the influence of closely related offerings
under the same brand name. In general, these variables were assumed
constant for the period in question, an assumption warranted by prelim-
inary investigation. Not observed variables that could conceivably
change during the period — in-store promotions other than deals, adver-
tising not included, such as direct mail, for example--had to be seen
as randomized, an undesirable approach but without alternatives.

There is also a question of causability: Does advertising cause
purchases or do purchases cause advertising? Because of the short time
intervals dealt with (months) we would argue that actual purchases can-
not cause advertising. There is very little chance that feedback from
the market can be received that fast, and then acted upon. On the
other hand, one might visualize that expected purchases would affect
advertising. As Zellner et.al. (1966) show, however, when the causal

3
When TRIAL was introduced as the saturation level, the TRIAL var-

»le on the right hand side was eliminated for obvious reasons. '

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-11-

mechanism is non-deterministic this might not create any trouble;
purchases regressed upon advertising is still the correct model.
Final Model Specification

The early regression runs were aimed at specifying the models
more precisely. With the low number of observations available for
each brand, it was deemed very desirable to pool the observations In
some fashion. With the brands all being national and established, it
seemed justifiable to assume the coefficients for the trial, deal, and
preference variables to be very similar. With reference to the media
advertising variables, however, the basic heterogeneity (due to crea-
tive, vehicle, and other wlthin-media-differences) of the variables
forestalled a parallell argument. Thus, the initial runs allowed for
separate coefficients for each brand's media variables, but constrained

the trial, deal, and preference coefficients to be the same for each

4
brand. Although the low degrees of freedom made tests of the signifi-
cance of the coefficients weak the results were remarkably consistent.
The advertising coefficients were low and very similar across brands.

Because of the low number of observations (with a lagged dependent
variable Introduced, there were 12 data points per brand) only current
media advertising (4 media variables) was included in these runs. In
order to assess differences between brands for lagged advertising, all
media advertising for the preceding month was summed and Included as
one additional variable. Again, no great brand differences emerged in
terms of the advertising coefficients. Runs were made in parallel for

4
The approach used assigns dummy variables for each brand's

separate slope coefficient; it is well discussed by Gujarati (1970).

-12-

advertising expenditures and for advertising shares (where the denom-
inator consisted of the four brands' total advertising in the parti-
cular medium). No differences between brands obtained, although
shares tended to do better (In terms of signs and significance levels
of the coefficient estimates) than expenditures. Furthermore, running
heavy and light users separately uncovered no significant differences
(except for the intercept) between the two groups .

For the final runs, then, the four brands were pooled allowing
only a separate intercept (but no separate slope coefficients) for
each brand, in the usual analysis of covarlance approach. With the
lagging of the dependent variable one period, there were 12 observa-
tions per brand; with four brands pooled, there were 48 observations.
In addition, the heavy and light users were pooled, again allowing
for a separate intercept. Tlius, the number of observations available
for the final runs was 96. Advertising was measured as each brand's
share of the four brand's total expenditures for the medium in ques-
tion.

In the initial "pooled" runs, advertising at t and t-1 (8 media
variables) was introduced. The results generally showed no significant
impact, and at times the effect seemed to be negative. The introduc-
tion of longer time lags was deemed necessary, and a Koyck distributed
lag approach was attempted. The result was unsatisfactory, with no
significant impact for the lagged dependent variable (which also ruled
out an autoregressive model), and a more general lag structure seemed

Thus making it very possible that a brand's media input places
it on the increasing part of the response. curve.

-13-

needed. The one used was the poljmomlal distributed lag approach of
Almon's (1965). A constrained quadratic form from t to t-4 showed by
far the best fit and was the one used in the final runs. The con-
straints consisted of zero restrictions at t+1 and at t-5. With the
constraints incorporated, each media variable introduced in fact repre-
sented a moving sum of the shares from time t to t-4. The weights of
the sum were (in order) 1.0, 1.6, 1.8, 1.6, and l.O. Accordingly,

the advertising impact for each medium culminates at time. t-2, with a

6
smooth decline before and after that point.

Results

As a first step in analyzing the results, it seems natural to

determine which model is the "best". One attractive criterion for

model choice is the predictive test, which necessitates observations

in addition to those used for estimation. In this case, however, the

initial model refinement did in fact draw upon the whole data base.

As a consequence, short of generating new data, this avenue was closed.

The best alternative is to ask how well the different models explain

the variation obsex-ved in the available data. This leads to the use

of the measure of the coefficient of determination, the R-square.

2
One problem in using the R measure when choosing between alterna-
tive models which differ with respect to the form of the dependent
variable, is that the relative explanation of a transformed variable
is of less interest than the explanation of the variations in the
original dependent variable. Thus, in our case, we are less interested

6
Although each medium was allowed to take on a different lag

structure independent of other media, this structure turned out to be

the best one for all the four media.

-14-

in how well the purchase odds, say, are explained, and more concerned
about the explanation of the original sample proportion of purchasers.
The transformation to odds is only "auxiliary" for estimation pur-
poses. Similarly taking the logarithms of the dependent variable

2
before regressing will give us an R not immediately interpretable as

explaining the relative amount of variation in the original dependent
variable.

The solution to this problem lies in re trans forming the "pre-
dicted" observations on the regression (from which the residuals are
computed) back to a prediction of the original variables. Thus, if
the dependent variable is "logged" before regression, antilogs are
taken of the predicted observations. Similarly, for the odds and the
odds with the intercept, the predicted values are transformed back to
predictions of the original proportions, by solving (for the odds)
p/(k-p) = y, for p with k given and y representing the predicted
values from the regression run. Then the retransformed predicted data

series is correlated with the original observations and a new "cor-

2 7

rected" R is derived.

For the seven regression runs specified above, Table 1 gives the

R-square in terms of explanation of the original sample proportions

2
(the corrected R ). All values are adjusted for degrees of freedom.

Note that when antilogs are taken, the predicted values are not
really the expected values of the original observations: E(log x) is
not equal to log E(x)) . Also, the assumption of a given k in retrans-
forming the odds means that the actual observed k will have to be
introduced, thus adding information to the odds explanation (and simi-
larly for I in the odds with intercept model). However undesirable
such features are in making the comparisons between the different
models more difficult, the alternatives are worse (see Goldberger,
1964," p. 217).

-15-

One result indicated by these values Is that the double log £orm does
better than any other functional form, regardless of the measurement
of the dependent variable. The differences are not great, but the
consistency of the pattern Indicates that they are fairly robust.
Accordingly, the three doublelog models were tentatively selected as
"the best". Although the R-square values differ widely between them,
the additional information Introduced in the two odds models makes a
final choice of one model based upon this criterion alone somewhat
arbitrary. It was decided to investigate them all further.

The regression coefficients and computed advertising impact,
together with the beta coefficients from these three doublelog models
are presented in Table 2. The individual monthly media coefficients
are not presented — the total effect per medium is a summation of the
impact over months t to t-4. Although some Interesting hypotheses may
be deduced from the different media coefficients, for our purposes
here we will concentrate upon the total advertising impact. (The
negative media coefficients for the odds and odds with Intercept
models are insignificant at the .05 level). Furthermore, the other-
than-advertlslng variables, although interesting in themselves will
not be discussed, except that it should be noted that, judging from
the beta coefficients, the advertising is playing not Just a marginal
role in determining the variations in the purchase sample proportion.

So what do these results tell us about the applicability of the
S-curve? First it should be noticed that the choice of the doublelog
form over the loglinear function implies that the S-curve, if appro-
priate, will not be symmetric. Second, the impact of advertising in
the proportions model is .33; as is well known, the doublelog function

..,, . ... ,,^

-16-

wlth a coefficient between zero and one will have a positive but
decreasing slope. According to this model then, response to adver-

o
tising exhibits everywhere diminishing returns.

But which one of the three "best" models should we accept as the

2
one? Judging from the corrected R values, can one say that the intro-
duction of the intercept might be worth the effort, raising as it does

2 2

the R six percentage point? And what about the differences in R

between the odds model (.84) and the simpler proportions model (,63).
Should we say that the difference is large enough that the added infor-
mation introduced through the actual value of k when re trans forming is

more balanced? Even though before retrans formating the proportions

2
had a larger R ? And is there any evidence in Table 2 that would favor

the choice of one model over the other two?

It so turns out that a decision does not have to be made. For

when the double log forms of the odds and the odds with intercept

models are checked for their second partial derivatives with respect

to advertising, these derivatives are negative in the relevant regions,

indicating diminishing returns. To find the inflexion point for the

9
odds double log model, we set"

b,(k-2y)

(13) —^-Y~ ^ "" °'

that ts,

(14) h

i k - 2y

Q

Notice that this type of response quite often is represented by
the semilog function. The results in Table 1 indicate that perhaps
the double log form, which is more flexible, is to be preferred.

9
See Johansson, 1972b.

■17-

In words, at the inflexion point, the coefficient b has to equal
k/(k-2y). With y always larger than zero, this means that b has to
be larger than one. From Table 2 we see that b is less than one
(b » .07) indicating that the inflexion point in fact occurs where y
is negative. From (8) we see that the second partial derivative is,
in fact, negative, implying diminishing returns in the relevant region.

For the odds with intercept model the same condition becomes

b.(k-2y+I)
(15) -4:^ 1-0,

or

k-I

(16) b

i k-2y+I

Again, b would have to be greater than one for the inflexion
point to fall in the relevant region (to see this, set y equal to its
lower bound and solve for b ). From Table 2 we see that the inflexion
point occurs at negative y- values, and again the second derivative is
negative in the appropriate region. The S-curve does not correctly
describe these advertising effects.
Concluding Comments

Through the research presented In this paper we found that adver-
tising exhibits no sigmoid effect. The generality of this finding out-
side the given product class is not easy to assess. Further research
is clearly needed on this score. Because the approach developed is
veiry straightforward, there should be little problem in applying
similar techniques to other sets of advertising data.

If the generality of the results cannot yet be assessed, do we
know with certainty that the product class analyzed in fact does exhi-
bit no sigmoid effect? Not quite, really. We eliminated the private

-18-

brands--they may encounter some threshold effect. Also, over longer
time periods a threshold might in fact obtain. Indeed, it has to be
pointed out that these two conditions together imply that we are deal-
ing with quite small movements in the market shares of the different
brands. In the final analysis this might be the reason for adver-
tising not exhibiting any threshold stage in these data.

TABLE 1.

THE COEFFICIENTS OF DETERMINATION

REGRESSAND

FUNCTIONAL FORM

ADJUSTED R^

Purchase

Linear

.58

(Sample

Proportion)

Semilog

.56

Double log

.63

Purchase

Log linear

.83

(Odds)

Double log

.84

Purchase

Log linear

.88

(Odds

With

Double log

.90

Intercept)

rv

TABLE 2

REGRESSION RESULTS FOR THE THREE DOUBLE LOG MODELS
(Significance at the .05 level is indicated by a star)

N. Regreasand
Regressors x^

PURCHASE (SAMPI£

PROPORTION) 1

1

PURCHASE
(ODDS)

PURCHASE (ODDS
WITH INTERCEPT)

Coefficient

(Standard

error

Beta

Coefficient

(Standard

error)

Beta

Coefficient

(Standard

error)

Beta

Magazine Adv.

.04
(.05)

.29

.01
(.03)

.07

.002
(.03)

.01

Network TV Adv.

.11*
(.05)

.32

-.04
(.04)

.12

-.03
(.05)

.08

Spot TV Adv.

.11*
(.06)

.57

.05
(.05)

.24

.07
(.05)

.36

Newspaper Adv.

.07*
(.03)

.47

.05*
(.02)

.34

.08*
(.02)

.50

ADVERTISING TOTAL

.33*
(.07)

.07*
(.03)

.12*
(.04)

DEAL

.37*
(.09)

.54

.18
(.10)

.23

.31*
(.10)

.41

TRIAL ^

-.26
(.17)

.27

PREFERENCE ^_ ^

.21
(.11)

.29

,07
(.09)

.11

.03
(.11)

.06

CONSTANTS :
Overall

1.53*
(.35)

1.38
(1.73)

-5.08*
(1.68)

Brand 1

-.99
(.71)

.95

-4.39
(3.23)

.51

-4.21
(3.07)

.52

Brand 2

-.04
(.24)

.04

-.99
(1.55)

.01

-1.73
(1.62)

.21

Brand 3

-.19
(.23)

.18

-1.08
(1.66)

.13

-1.79
(1.72)

.22

Heavy/ Light Users

-.37*
(.11)

.41

-4.79*
(.76)

.65

-2.07*
(.86)

.30

Adjusted R

Durbin-Watson

No. of Observations

.56 (reg
.63 (cor
1.94
96

res sand
rected)

.50 (regr.
.84 (corr(
1.94
96

ess and]
Bcted)

.30 (reg
.90 (cor
2.10
96

res sand)
rected)

■;J

«.. I

• V^

! :

REFERENCES

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tM

Oliver, F. R., "Aspects of Maximum Likelihood Estimation of the

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