it’s a dirichlet world
TRANSCRIPT
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DOI: 10.2501/JAR-52-2-203-213 June 2012 JOURNAL OF ADVERTISING RESEARCH 203
AN iNTrODuCTiON TO THE DiriCHLET WOrLD
Whether were predicting or analyzing market share,
the foundation for the analysis is the Dirichlet.
Greg Rogers, associate director o market
research, Procter & Gamble
Dirichlet-based models play a very important role in
our analytical services.
Phil Parker, vice-president product management
and development, Nielsen Company
Ehrenbergs models are a fundamental part of our
reporting to our consumer packaged goods clients
and retailers.
Michael Kruger, executive vice president
R&D at SymphonyIRI Group
Double Jeopardy (see Ehrenberg, Goodhardt,
and Barwise, 1990), which describes both consumer
attitudes and behaviors, is undoubtedly market-
ings most amous empirical law. It states that small
brands suer twiceewer people buy them, and
those who do buy them do so less oten. The Dou-
ble Jeopardy law aligns with many other patterns
in consumer brand choice, and these are so law-
like (holding across dierent product categories,
countries, and time) that they can be predicted by a
single unctional orm (Goodhardt, Ehrenberg, and
Chatfeld, 1984). This model, widely known as the
Dirichlet, describes the variation in individuals
loyalties across the category buying population.
The Dirichlet gives detailed insights into how
consumers behave and how brands compete (seeEhrenberg, Uncles, and Goodhardt, 2004), it is
widely used (see Bound, 2009, and Kennedy and
McColl, 2012 or practical guides to its applica
tion), and yet it is oten misunderstood. In particu
lar, critics claim it cannot adequately describe the
changes in loyalty typically observed in consumer
markets. In the current study, the authors
discuss how consumer loyalties revea
themselves;
examine the assumption o stable loyaltiesunderpinning the model; and
explain how the Dirichlet allows underlying
changes in loyalty to be detected and quantifed
A common misconception about the Dirichlet is
that it implies there is no such thing as loyalty and
that consumers are all alike. In act, the opposite
is true. The Dirichlet implies that consumers do
not randomly allocate their purchasing among al
its a Dchlet Wold
Modelng indvdals Loyaltes reveals How Bands
Compete, Gow, and Declne
BYrON SHArP
Ehrenberg Ba Intitute
byron.harp@
marketingcience.info
MALCOLM WriGHT
Maey Univerity and
Ehrenberg-Ba
Intitute
JOHN DAWES
Ehrenberg-Ba Intitute
john.dawe@
marketingcience.info
CArL DriESENEr
Ehrenberg-Ba Intitute
carl.drieener@
marketingcience.info
LArS MEYEr-WAArDEN
EM Buine school
strabourg
LArA STOCCHi
Ehrenberg-Ba Intitute
Lara.stocchi@
Marketingscience.info
PHiLiP STErN
Loughborough Univerity
and Ehrenberg-Ba
Intitute
The Dirichlet i one of the mot important theoretical achievement of marketing cience.
It provide inight into the ditribution of conumer loyaltie and i ued widely in indutry
for benchmarking and interpreting brand performance. The Dirichlet implication
run counter to ome well-entrenched marketing pedagogy and o, unurpriingly, it
has attracted criticism arguing that it cannot adequately reect the dynamic nature of
conumer choice. The author addre thee criticim by dicuing how conumer
loyaltie are manifeted and examining whether change in conumer loyaltie do, in
fact, dirupt Dirichlet buying pattern. To the bet of our dicipline knowledge, baed on
extenive empirical and theoretical work, brand compete in a Dirichlet world.
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204 JOURNAL OF ADVERTISING RESEARCH June 2012
ITs A DIRIcHLET WORLD
brands in a category but do so in a biased
ashion. In other words, all buyers avor
particular brands with their custom, all
buyers have their own particular loyal-
ties. Further, the Dirichlet assumes peopledier rom one another in both their rate
o category purchase and in the composi-
tion o their purchase repertoires. These
dierences result in great heterogeneity
between consumers, which the Dirichlet
describes very accurately.
This is neatly illustrated by the obser-
vation that other peoples supermarket
baskets look dierent rom ones own.
They have a collection o brands that looks
strange and unamiliar; each shopperhas a dierent repertoire. These loyalties
are quite enduring and, in a years time,
an individual households shopping bas-
ket still will look quite similar. It will not
look exactly the same on each shopping
trip, however, as consumers are polyga-
mously loyal to a number o brands in
most categories. Swapping between these
avored brands is normal and an everyday
occurrence or most shoppers. This does
not happen because consumers were per-suaded to change their mind about which
brand is best but rather as serendipitous
events (such as exposure to an advertise-
ment the night beore) play tiny, as-i ran-
dom, roles in nudging consumer choice.
As with any model, the Dirichlet is a
simplication o the world, but it describes
markets made up o real-world consumers
so well because the central assumptions
are close to reality. In eect these are as
ollows:
Loyalties are distributed across consum-
ers with little dierentiation between
brands, such that each brand in eect
sells to all category buyers rather than
a particular segment (see Uncles et al.,
2012). Most vegetarian pizzas are sold
to non-vegetarians, and Diet Coke still
competes very directly with regular
Coca-Cola.
Consumers do not alter their loyalties
oten, but they may revise their loyalties
over decade-long time spans, largely dueto changes in their livesas they grow
up, leave home, get married, have chil-
dren, move house, move jobs; economic
trends change income; technological
change makes some products obsolete.
Few consumers, however, will change
loyalties within typical brand-planning
periods. The assumption o stable loyal-
ties turns out to be largely true or most
o us, at least in the medium term and
oten longer. Certainly there are somebrands people buy all their lie and even
pass the loyalty on to their children.
A simple explanation weaves the
Dirichlets laws togetherthat brands
compete or custom primarily in terms o
mental and physical availability (Sharp,
2010). This Dirichlet world is one where
buyers are busy cognitive misers. They are
naturally loyal but polygamously so; their
mental and physical availability deter-mines the brands they loyally buy over
and over.
The Dirichlet world is also one where
marketing is very important, as brands
continually battle or attention. Branding
helps consumers exercise their natural
tendency to be loyal while clever creative
advertising reminds people to keep buy-
ing brands they rarely think about. Conse-
quently, a media strategy that maximizes
reach and coverage over time is needed.The search or persuasion break-
throughs that can change what consumers
think o a brand is a small part o mar-
ketingand oten a distraction rom the
important tasks. Rather, much o the eect
o marketing is to counter competitors,
to reresh and maintain existing loyalties,
and to maintain and increase the mental
and physical availability o a brand. The
timing o individuals exposures to par
ticular marketing activities orms part o
the vast random background o infuence
that nudge consumers, producing wobble
in an individuals brand purchases.Market share will change in a Dirichle
world i a brand secures additional men
tal and/or physical availability. This may
come about as a result o superior market
ing or through some innovation that leads
to real changes in loyalties and, hence
brand growth. Yet, the brand will still be
competing with the same rivals and sell
ing to the same sort o consumers. The
overall distribution o purchase propensi-
ties should still ollow the Dirichlet modelwhich provides benchmarks to help ana-
lyze these market changes (as shown by
McCabe, Stern, and Dacko, 2012).
Despite these well-established patterns
the Dirichlet is oten criticized as an unre-
alistic description o the brand switch
ing, variable loyalty, and non-stationarity
thought to characterize consumer mar
kets. Even when the Dirichlet conception
o loyalty is recognized as accurate, crit-
ics are prone to claim the model cannoexplain the ongoing changes observed in
most markets. Thereore, this study exam
ines and rebuts some o the most common
objections to the Dirichlet approach to
modeling markets. It shows that
individual brand switching lies within
ranges expected rom the Dirichlet model
aggregate loyalty to brands changes
very little over time; and
the parameters o the model are in actquite stable.
SEEiNG THrOuGH THE rANDOM
VAriATiON
Although the Dirichlet is less amous than
the laws that it predicts (such as the Dou-
ble Jeopardy or Duplication o Purchase
laws), it is used daily within marketing
corporations to benchmark brand metrics
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against the models stationary and non-
partitioned norms. For example, manag-
ers can determine whether their brands
purchase requency is high, low, or normal
given its market share. This is valuable inthe hands o knowledgeable managers.
Because the structure o Dirichlet mar-
kets is well understood, deviations rom
Dirichlet norms are easily diagnosed, and
their causes can usually be identifed.
The Dirichlet underpins many of our spe-
cialised marketing models because it tells
us what we expect to happen, we can then
quickly identify the exceptions which pro-
vide useful insights.Ian Hewitt, CMI director-marketing
science, Unilever
The Dirichlet allows the business to under-
stand which brands are unusual and differ
from the expected pattern. This provides
genuine learning about the category and
helps to dispel myths about which brands
are special or different.
Tom Lloyd, director, Metametrics UK
Without the Dirichlet benchmarks, con-
sumer buyingeven in a market where
no changes are occurringlooks chaotic,
almost completely random or subject
to enormous changes in preerence. For
example, brands in repertoire categories
such as grocery products seem to lose vast
numbers o their customers each year and
win most o these individuals back the next.
The Dirichlet is a powerul tool that allows
us to see through the as-i random noise todetermine what is really happening.
Due to the assumption o stable purchase
probabilities, the Dirichlet is a stationary
model in which brands do not grow or
decline and consumersalthough dierent
rom one anotherdo not change their loy-
alties. It does not assume, however, that we
each buy exactly the same this quarter (or
year) as we bought last quarter (or year).
Consumers are habitual but not robotic.
There are thousands o things that cause
variation in purchases rom period to
period, such as weather, a sports event, a
visit rom a riend, cooking a new meal,dropping a jar, price promotions, out-o-
stocks, competitions, sampling, seeing
an advertisement close to the shopping
encounter, bumping into a riend in a
supermarket, being late or dinner, a
trolley blocking the supermarket aisle or
some seconds, and so on. All can sway
purchase events but rarely produce any
ongoing change in loyalty.
In 1974, in the pages o theJournal of Mar-
keting Research, Frank Bass even speculatedthat the brain might have a stochastic ele-
ment. There is support or this in the way that
consumers reply to attitudinal questions (see
Castleberry, Barnard, Barwise, Ehrenberg,
and DallOlmo Riley, 1994; DallOlmo Riley,
Ehrenberg, and Castleberry, 1997; Rungie,
Laurent, DallOlmo Riley, Morrison, and
Roy, 2005). That is, individual respondents
oten do not say the same thing in sequen-
tial surveys, even i they are only separated
by a ew minutesperhaps simply becausememory does not deliver perect recall.
The Dirichlet assumes that this variation
is as-i random wobble around steady
ongoing loyalties. To illustrate how much
variation there can be, a hypothetical con
sumer in a Dirichlet market simulationshows a great deal o variation in her pur
chase (See Table 1). The authors do stress
that this is purely a simulated Dirichlet
purchase stream with fxed probabilities
it does not show an individual who is
changing her loyalties, rather just random
wobble around steady loyalties.
Gambling provides a useul analogy or
this variation around ongoing steady loy
alties; casino visitors have a steady ongo-
ing propensity to lose money; on somevisits, they actually make money, but the
long-run odds remain in the houses avor
Similarly, i we looked at a group o coin
tossers, we would see that very ew indi
viduals (i any) made a classic run o tosses
heads (H), tails (T): HTHTHT. Even in
quite long runs o coin tosses, we would
see many where heads were not 50 per
cent. This is all just random but, rather a
predictable wobble. In act, each coin has
a perectly divided (50/50) loyalty to bothheads and tails.
TABLE 1
Pure Dirihlet conumer: simulated choolate category
Brands
1 2 3 4 5 6 7 8 9 10
CategoryPurchases
1 1
2 1
3 1
4 1
5 1
6 1
7 1
8 1
9 1
10 1
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ITs A DIRIcHLET WORLD
The Dirichlet describes something
much more complex than coin tossing.
It assumes category purchase and brand
loyalties have steady probabilities but that
these vary between individuals. Also, anindividuals buying will exhibit stochas-
tic wobble around the steady-state prob-
abilities. In the long run, buyers with a 0.1
probability o buying the brand will buy it
on 10 percent o category purchase occa-
sions. This buying, however, will happen
in an as-if random ashion and indepen-
dently o the brand they bought on last
occasion. So, individual brand purchas-
ing or short periods may show stochastic
variation rather than change in underlyingpatterns o loyalty. Although individual
consumers category purchase rates and
brand loyalties might be quite stable, any-
one looking at their buying would fnd it
difcult to see these patterns.
Imagine consumers who have an ongo-
ing propensity to buy chocolate eight
times a year and to buy Snickers once o
every our chocolate bar purchases; we
would expect them (on average) to buy
a Snickers bar twice a year, but someyears they buy more chocolate and some
years less. It is quite reasonable that, even
though they did not buy any chocolate last
quarter, they still have an ongoing prob-
ability o buying it eight times a year on
averagealthough some years they buy
chocolate only once or twice, some years
12 times. On top o this, sometimes they
buy Snickers two o our purchases; some-
times none o our. So, this consumer can
conceivably buy six Snickers bars one yearand the ollowing year none. This sort o
variation is unremarkable and unlikely to
be noticed by the consumers themselves.
Tactical marketing certainly has an
eect on buying, with price promotions
massively aecting the purchase propen-
sities o those individuals who buy the cat-
egory during the promotion. I we see that
one o the brands in our repertoire is on
promotion, the probability leaps that we
will purchase that particular brand (ifwe
are in the market or that category); hence,
price promotions cause sales spikes. The
instant ater the purchase, however, theprobability o buying that brand next time
reverts to its normal steady propensity.
Price promotions are not unusual, buying
the brand is not unusual, so there is noth-
ing to change our opinion o the brand,
little to alter our memory structures. An
individuals purchase propensities, their
personal loyalties, remain intact with
plenty o random, predictably distributed
wobble.
The Dirichlet provides the ability tobenchmark and, thereby, see through, the
stochastic variation that can conceal the
underlying loyalties. Without knowledge
o the Dirichlet, it is easy to conuse ran-
dom wobble with real changes in the mar-
ket. For example:
A common mistake is to notice that a
customer who bought one year did not
buy again the ollowing year and to
iner that she or he has deected romthe brand (the bucket leaks). In real-
ity, these situations are typically not
deections but simply represent light
customers who do not buy every year.
Most churn, thereore, is simply light
customers sometimes buying and some-
times not. Many brandseven large
oneshave a majority o customers who
only buy the brand every 2 or so years
(Sharp, 2010). Due to stochastic wobble,
customers who buy inrequently willoccasionally skip a year. The Dirichlets
stationary benchmarks allow precise
estimates o how many o such lapsed
customers are really not lost at all (or
example, see Wright and Riebe, 2010;
and Riebe et al., orthcoming).
In 2009, the Chie Marketing Ofcer
Council issued a report announcing
a dramatic collapse in loyalty or U.S
packaged goods brands, due to the eco
nomic downturn in the previous year
(Pointer Media Network, 2009).Advertis
ing Age reported, For the average brandmore than hal o consumers52 per-
centwho were highly loyal to certain
package-goods brands in 2007 became
markedly less so last year.1 The resul
was purely due to the stochastic wob-
ble in individuals purchasing. I loyalty
could really collapse in such a manner in
the course o a single year, there would
be universal marketplace chaos.
In 2011, Catalina Marketing repeated
the same analysis and reported that lead-ing brands, once again, had lost almos
hal o their loyal customers in a single
year (even when they enjoyed revenue
growth). Catalina reported, Every year
brands experience a dramatic exodus o
previously loyal consumers, resulting in
signifcant reductions in potential vol
ume and share.2 Again, the Dirichle
tells us that this apparent change is per
ectly normal, due to the stochastic vari-
ation in purchase requency and brandchoice. Few consumers have changed
their loyalties, but stochastic wobble
can make it look that way. Without the
Dirichlets benchmarks or deection/
acquisition metrics, it is not possible to
separate the real churn rom the norma
stochastic wobble.
Several years earlier, in the pages o
the Journal of Advertising Research, A.L
Baldinger and J. Rubinson (1996) madea very similar observation, reporting
nearly identical results (and identi
cal surprise): We were surprised to
observe that only 53 percent o high loy
als to the brand remained high loyal to
1 http://adage.com/article/news/cpg-marketing-brand
loyalty-recession/137436/2 http://adage.com/article/news/catalina-major-packaged
goods-brands-lost-46-loyalists/229640/
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the brand a year later. Their nding led
the authors to mistakenly conclude that
attitudes determined who would switch
and who would not.
Stochastic wobble means that categoriz-
ing customers as heavies, loyals, or
deectors based on a period o their
purchasing will misclassiy many people.
Each o the preceding examples suered
rom this eect. I an analyst takes a group
o heavy buyers (e.g., the top 20 percent)
rom a customer database, they are likely
to note that, in a subsequent period, their
purchasing is lighter than beore, easily
leading to the incorrect conclusion thatreal changes in loyalty have occurred.
Stochastic wobble also means that
some o the people who were previously
classed as heavy simply had an upward
fuctuation in their purchase rate or the
base period, so regression-to-the-mean
occurs in the next period. The same eect
occurs with opposite valence or light
buyers, and it is normal that many zero
buyers rom one period may come into
the market the ollowing periodagain,regression to the mean. The Dirichlet can
quantiy the degree o regression to the
mean that will occur under a no-change
condition, enabling analysts to see how
much o the variation represents genuine
dynamic change.
Real changes in loyalties do occur, but,
without the Dirichlet, they are hard to
spot. A purchase run o AABB is likely
rom a person with a repertoire that ea-
tures 50 percent purchases o brand A and50 percent brand B, but it is tempting to
misinterpret such a purchase run as a con-
sumer who was loyal to brand A switching
to brand B. Without a benchmark such as
the Dirichlet, it is simply not possible to
sort the real changes in loyalties rom the
stochastic wobble.
An examination o repeat-purchase
levels rom month to month using the
Dirichlet, documented about 15-percent
annual erosion in repeat-rates or indi-
vidual brands (East and Hammond, 1996).
This change was largely due to consumers
changing the weight o a brand in theirrepertoire (adjusting their loyalties) rather
than wholesale switching rom one brand
to another.
This leads to the next question: i loyalty
to individual brands is more stable than
commonly thought, is loyalty eroding in
aggregate instead?
ErOSiON OF CONSuMEr LOYALTY WiTHiN
A CATEGOrY iS rArE AND SLOW
There is a long-held view that consumerloyalty in most established product cat-
egories is eroding, which is contrary to
the assumptions on which the Dirichlet is
based. For example, in 1992, J. S. Dubows
research suggested a decline in loyalty or
Coca-Cola dating back to the 1960s. More
recently another report stated To say that
brand loyalty is in decline today is, at the
very least, an understatement (Kaperer,
2005).
Industry publications oten report thatthe retail sector is being threatened by an
alarming decline in customer loyalty and
that shoppers are becoming less brand
loyal (Gerzema and Lebar, 2008; Lincoln,
2006) and, as noted earlier, Catalina Mar-
keting and the Chie Marketing Ocer
Council announced that loyalty erosion
and deection are increasing dramatically
(Pointer Media Network, 2009, page 2).
I brand loyalty, indeed is declining,
it would be a major issue or marketers.Lower loyalty would mean that, to stay
the same size, the average brand would
require a larger customer base but one
that buys the brand less regularly than in
the past. The important marketing task o
remindingand nudging buying pro-
pensitieswould be harder, and the scope
or customers to orget about any particu-
lar brand would be wider.
Past evidence, however, suggests brand
loyalty is quite stable over time. In 1984
T. Johnson, in the Journal of Advertising
Research examined 50 major brands in 20
U.S. product categories over a period oapproximately 8 years. He ound some
decline in loyalty or certain brands bu
noted the decline oten accompanied
category growth. That is, growth in the
category attracted new brands, which
broadened consumer repertoires. Johnson
concluded there was some evidence o loy
alty decline, but its magnitude was small
More recently, another study examined 21
categories in Holland or time periods o 1
to 2 years and concluded that there was little evidence o loyalty change (Dekimpe
Steenkamp, Mellens, Abeele, 1997).
This section now looks at loyalty change
over a long period, using more recent data
gathered across two countries.3 There were
some years or which data were unavaila-
ble or some o the U.K. categories. Where
actual data or beore and ater the missing
years were available, the missing values
were interpolated (See Table 2, italics).
Share o category requirements (SCRwas used as the loyalty measure. SCR is a
widely used and intuitively understand-
able empirical metric computed as the
average number o times buyers o brand
X buy that brand, divided by the average
number o times they buy the category
SCR is highly correlated with other loyalty
metrics, both behavioral and attitudinal. It
is also a loyalty metric that is predicted by
the Dirichlet.
The SCR was calculated or each brandover xed 52-week periods; the overal
average was then taken or the respec-
tive category. The process was repeated
or eight U.K. categories or periods
o between 11 and 13 years and or six
U.S. categories over 6 years (See Tables 2
and 3).
3 Kantar provided data for the United Kingdom; Nielsen
provided data for the United States.
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ITs A DIRIcHLET WORLD
The results show there is some gradual
decline in brand loyalty in some catego-
riesbut the magnitude is very small. The
biggest year-on-year loyalty change over
the period was the Yogurt category in theUnited Stateswith SCR declining by 0.8
percentage points per year. The next largest
change was or Bodysprays and Deodor-
ants in the United Kingdomon average,
only 0.5 percentage points per year. Both
these two categories have exhibited robust
growth over the time period.
Johnsons 1984 work also linked cat-
egory growth to a small decline in loyalty,
with the explanation that growth attracts
new brands, which, in turn, broadens
consumer choice. The other categories in
Tables 2 and 3 show zero, or almost zero,
loyalty change. Given the reported rise intemporary promotions, brand proliera-
tion, and the global nancial crisis, these
categories show remarkable stability in
average loyalty.
It should be noted that stable brand loy-
alty or a category, overall, does not mean
stability or each brand. For example,
some brandsin theory, at leastcould be
trending upward in loyalty, whereas others
trend downward. Loyalty metrics or indi
vidual brands, however, were ound to be
quite stable. Market share change, when it
occurred, was refected in changing brand
penetrations, with much smaller accom
panying changes in loyaltyquite consist-
ent with the Dirichlet models predictions
(Goodhardt et al., 1984).
MArkET STruCTurE iS LArGELY STABLE
FrOM PEriOD TO PEriOD
The Dirichlets assumption o stable con-
sumer loyalties means it represents a
stationary market, without substantia
category growth or change in the het
erogeneity o category purchase rates or
brand shares. The current study provides
evidence rom six product categories
over 2 years to show that the underlying
parameters o the modelM, Kand Phidescribed in Table 4are, in act, quite
stable.
This contrasts with other marketing sci
ence and econometric models that oten
show considerable instability or generate
parameters that are not stationary over
time. For example, unstable parameters
4 Data provided by Kantar.
TABLE 2
Loalt Metri for Eight Uk categorie 19982010
Aveage Band shae of eqements n
ths categoy:
Yea (52-wee peod JlyJne)Aveage anna
change n SCr98 99 00 01 02 03 04 05 06 07 08 09 10
Bodpra and Deodorant 27 29 28 27 27 27 30 25 25 22 22 0.5
Intant standard coffee 35 33 33 33 33 34 36 36 35 34 32 30 30 0.4
Tea Bag 34 34 34 34 34 34 34 34 35 33 33 33 31 0.3
Toothpate 27 26 27 27 26 26 27 26 25 24 25 24 23 0.3
Margarine 21 22 22 22 22 23 24 22 20 18 19 0.2
Breafat cereal 15 15 15 15 15 15 15 15 17 17 13 13 13 0.2
Laundr Detergent 31 31 31 30 30 29 33 34 34 33 30 31 30 0.1
Dog Food 19 20 19 19 20 21 20 19 21 20 19 0.0
Average ear-on-ear hange over all ategorie: 0.2% p.a.
TABLE 3
Loalt Metri for six U.s. categorie 20052010
Aveage Band shae of
eqements n ths categoy:
Yea (52-wee peod JlyJne)Aveage annal
change n SCr05 06 07 08 09 10
yogurt 24 22 23 21 20 20 0.8
shampoo 35 30 30 31 34 33 0.4
Deodorant 36 36 37 37 35 36 0.0
cat Food 19 21 20 18 21 20 0.2
Breafat cereal 11 11 12 11 11 12 0.2
Margarine 31 33 33 33 33 33 0.4
Average ear-on-ear hange over all ategorie: 0.06% p.a.
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Its a DIrIchlet WorlD
have been reported or the Bass model
(Tigert and Farivar, 1981), or exponential
trial-growth models (Fader, Hardie, and
Zeithammer, 2003), and or marketing
mix models estimated rom scanner data
(Blattberg and George, 1991; Montgomery
and Rossi, 1999; Bemmaor and Franses,
2005). There is also evidence o widespreadnon-stationarity in econometric time
series parameters (Stock and Watson,
1994).
The over-time stability oK, M, and Phi
is now examined (the standardized ver-
sion o the parameter S where Phi = 1/
(1 +S)). The parameter Awas not included
as the identity M = AK links these three
parameters.
The categories analyzed varied in pur-
chase rate and seasonality. These were:conectionary; breakast cereals; canned
beans; toothpaste; deodorant; and canned
soup. In each case, the top 30 brands were
includedaccounting or approximately
80 percent o the marketand 24 months
o data starting rom January were ana-
lyzed. (The one exception was canned
soup, or which only 23 months o data
were available.)
The authors ft the Dirichlet model to
each month o data or each category and
plotted time-series o the parameters and
then examined coefcients o variation
and conducted ormal tests o stationarity
(See Figures 1, 2, and 3).
TABLE 4
Deriptio of the Dirihlet Model Parameter
Parameter Meanng Manageral implcatons Theoretcal Stablty
K Expoet of the nBD ompoet of the model.
Model the ditributio of ategory purhae
rate.
ca be ued for utomer oetratio aalyi
ad aalyzig the ditributio of light eru heay
ategory buyer.
stable
A sale parameter of the nBD ompoet of
the model. Thi meaure the weight of
purhaig ... relatie to the heterogeeity i
ategory latet eletio rate (Drieeer
2005, p. 63).
Thi parameter how how eaoality ad demad
uctuations are reected in the weight (frequency) of
purhae. It a help atiipate expeted purhae
frequey ad peetratio.
varie proportioally
with time. varie with
eaoality
M Mea of the produt la purhaig rate, a
determied by Kad A. Thi i related to the
other parameter a M= AK.
The aerage buyig rate of all hopper (iludig o-
buyers) for a specied period.
varie proportioally
with time. varie with
eaoality
S sum of the idiidual brad alpha, meaureheterogeeity of latet brad eletio rate
(loyalty). Brad hare i alpha/S.
Deribe degree of diided loyalty geerally preeti the market. I it tadardized form (Phi), it proide
a deriptio of heterogeeity i loyalty.
stable
Confectionery
Cereals
Beans
Toothpaste
Deodorant
Soup
Note: Figure 1 plots the stability of the average buying rate Mparameter. This is the parameter in which
most variation is expected as it is responsive to seasonal and promotion ef fects. Figure 1 demonstrates
there was relatively little period-to-period variation in Malthough, unsurprisingly, in December (time 12),
people bought more confectionery and less cereal; beans and soup both show evidence of some broad
seasonal cycles.
0
0.5
1
1.5
2
2.5
3
1 2 3 4 5 6 7 8 9 10 1 1 12 13 1 4 15 16 1 7 18 19 2 0 21 22 2 3 24
Fgure 1 24-Moth Time serie of the MParameter
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210 JOURNAL OF ADVERTISING RESEARCH June 2012
ITs A DIRIcHLET WORLD
Linear plots are useul overviews but do
not provide a ormal analysis. Moreover
they will exhibit exactly the kind o sto-
chastic wobble described earlier. There
ore, ollowing traditional analysis o timeseries values (e.g., Bass and Leone, 1983)
the authors calculated the ratio o the
standard deviations to the means o the
parameters (the coefcient o variation)
This allows comparison o parameter val
ues across categories, as it is a measure o
dispersion that is independent o the units
o observation (See Table 5).
The grand average o all coefcients
o variation in Table 5 was 0.16, which is
low in the context o time series analysisNonetheless, as might be expected (See
Figure 2), the K parameter or toothpaste
is an outlier (the only one rom 18 coef-
cients o variation), with nearly triple the
coefcient o variation o the next most
volatile time series. Aside rom this outlier
dispersion around the average parameter
values is low.
To detect trends, the authors calculated
regression lines or the parameter time
series and, or this analysis, the 23 monthso data or the soup time series were
truncated to 12 months to avoid spuri-
ous trends rom seasonal variations. Only
three instances o non-stationarity were
ound. The intercepts and unstandardized
Note: The Kparameter shows the mix of light and heavy categories buyers. Here again, there is stability
except for toothpaste. The upward fluctuations in the Kparameter for toothpaste indicate the presence of a
greater proportion of lighter buyers (or conversely a lesser proportion of heavy buyers). Each one is
accompanied by a reduction in the rate of category buying and also less brand switching. Such a pattern of
change is consistent with a no-promotion period in a heavily promoted category; this would result in a
reduction in the overall buying rate, a reduction in the relative proportion of heavy buyers, and reduced
brand switching, all of which are seen in this category.
0
1
2
3
4
5
6
7
8
9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Confectioner y Cereals
Beans Toothpaste
Deodorant Soup
Figure 2 24-Month Time serie of the KParameter
0
0.2
0.4
0.6
0.8
1
1.2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Confectionery
Cereals
Beans
Toothpaste
Deodorant
Soup
Note: Figure 3 examines stability in the heterogeneity of brand choice using the Phi parameter. Again, there
is stability but with more period-to-period variation than typical for other parameters. The fluctuations in the
value of Phi for toothpaste match those already found for the Kparameter.
Figure 3 24-Month Time serie of the Phi Parameter
TABLE 5
Coefcients of Variation for
the Parameter M, K, and Phi
M KPhi
confetionery 0.09 0.07 0.23
cereal 0.07 0.05 0.09
Bean 0.08 0.11 0.04
Toothpate 0.07 0.70 0.26
Deodorant 0.15 0.10 0.16
soup 0.30 0.26 0.07
Average 0.12 0.22 0.14
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June 2012 JOURNAL OF ADVERTISING RESEARCH 211
Its a DIrIchlet WorlD
coecients or these three times series
were
Intercept Slope
Deodorant (M) 0.285 0.003
Beans (K) 0.308 0.003
Deodorant (Phi) 0.449 0.004
It is commonly agreed that non-stationarity
is present i a time series changes by more
than 5 percent over a 12-month period.
These slopes translate to annual changes
o 11 percent, 10 percent, and 9 per-
cent. Thus, though 3 o 18 time series o
Dirichlet model parameters showed non-
stationarity, the rate o change did notexceed 1 percent per month. It is notable
that, though the K parameter or tooth-
paste had a higher coecient o variation,
it showed no evidence o a trend in the
parameter value.
A more ormal approach to time series
stability is the Dickey-Fuller test. This
is usually applied to longer time series,
and test values are published only or
time series o length 25, 50, 100, and so
on. Nonetheless, the authors applied theDickey-Fuller test to their time series o
length 24 (23 or soup) and ound only two
cases o a deterministic trendor the Phi
parameter or deodorant (consistent with
the regression above) and the M param-
eter or cereal. The t values were 2.90 and
2.92, only just exceeding the published
critical value or time series o length 25,
so again these eects are not large.
Finally, using the augmented Dickey-
Fuller test allowed the authors to control
or twelth-order (annual) seasonal lag
and or drit, although initialization o thelag halved the data available or estima-
tion. In this case, no cases o deterministic
trend or drit were detected, and seasonal-
ity was detected only or the M parameter
or deodorant (t = 4.5) and the Kparameter
or toothpaste (t = 4.7), although this latter
nding likely was due to the combination
o extreme variability and a truncated data
set, rather than to true seasonality.
From this suite o tests, it is clear that
non-stationarity in parameters is both rareand minor, giving empirical support or
the assumption o stable loyalties. Either
consumer loyalties are very stable, or (less
plausibly) changes in one consumers loy-
alties are inversely matched by changes
in anothers. The model is robust to
minor seasonal variations and accurately
describes buying behavior over time.
CONCLuSiON
The Dirichlet is one o marketing sciencesmost important theoretical achievements
and also one o its most practical tools. It
provides detailed insight into the distri-
bution o consumer loyalties and is used
widely in industry or benchmarking and
interpreting brand perormance.
The insights derived rom the model
run counter to some well-entrenched mar-
keting mythology, and so, unsurprisingly,
the Dirichlet has disbelievers who argue
that the model cannot adequately refec
the dynamic nature o consumer choice.
This paper addresses some o the pri
mary misconceptions about the Dirichlethat have developed over the 30 (or so
years since the model was introduced to
the world. Although the Dirichlet is mos
commonly used to provide benchmarks
and predictions o brand perormance
metrics, it actually models the distribu-
tion o individual consumers loyalties
Brand-perormance metrics are calculated
rom these distributions o loyalties. As
shown, without using the Dirichlet, rea
changes in loyalty cannot be isolated romthe eects o stochastic variationand so
mistakes are easily made.
Although it is a model o stationary loy
altiesand, thereore, stationary brands
the Dirichlet can be used to analyze market
change through benchmarking. Nonethe
less, as the current paper demonstrates
such structural market change is rare. Both
average loyalty and the structural param-
eters o the Dirichlet show remarkable sta-
bility over time.This is not to say that the Dirichlet pro-
vides a perect explanation. Just as with
Newtonian mechanics and Boyles law
there are some well-documented excep
tions. It is a sign o a mature scientic theory
that such boundary conditions are known
Future research can be directed toward
explaining such conditions and possibly
improving or replacing the base theory
along the way. To compete, however, any
replacement model would need to showequivalent explanatory power over the wide
range o conditions or which the Dirichlet
already perorms exceptionally well.
Although the Dirichlet (like any model
does not represent the absolute truth
about markets, it is a suciently close
approximation that all marketers should
be very amiliar with Dirichlet behaviors
and loyalties.
The Dirichlets assumption of stable consumer
loyalties means it represents a stationary
market, without substantial category growth
or change in the heterogeneity of category
purchase rates or brand shares.
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212 JOURNAL OF ADVERTISING RESEARCH Jue 2012
ITs A DIRIcHLET WORLD
To the best of our disciplines knowl-
edgeand based on extensive empirical
and theoretical workbrands compete in
a Dirichlet world.
byRon shaRP i profeor of maretig iee at the
Uierit of south Autralia. He i diretor of the
Uierit Ehreberg-Ba Ititute whih i poored
b ma of the world leadig maretig orporatio,
uh a Turer Broadatig, coa-cola, Mar, cBs,
Uileer, ad niele. He i the author ofHow Brands
Grow(Oxford Uierit Pre). With Profeor Jerr Wid
at Wharto ad the Adertiig Reearh Foudatio
he i orgaizig a 2012 oferee aemblig law-
lie owledge about adertiig i the ew digital
eiromet.
maLCoLm WRight i profeor of maretig at Mae
Uierit ad adjut profeor at the Ehreberg-Ba
Ititute of the Uierit of south Autralia. He applie
empirial priiple to maretig problem ad ha
made iterrelated dioerie about brad loalt, the
ue of probabilit ale, ew produt foreatig, ad
optimizig the adertiig budget. He ha alo publihed
ma artile ritiall examiig the foudatio of
popular maretig owledge.
John daWes i a eior reearher at the Ehreberg-Ba
Ititute, Uierit of south Autralia. The Ititute
i poored b ma leadig orporatio iludig
Mar, Elder, EsPn, Geeral Motor, ad AnZ Ba. Hi
reearh iteret are priig ad promotio, brad
performae metri, ad ompetitie maret truture.
He regularl udertae reearh projet o thee
iue for idutr parter.
CaRL dRieseneR i a eior reearher at the Ehreberg-
Ba Ititute, Uierit of south Autralia. The
Ititute i poored b ma leadig orporatio
iludig Proter & Gamble, kraft, vodafoe, katar,
ad colgate-Palmolie. carl reearh iteret are
brad performae metri ad modelig. He regularl
udertae reearh projet o thee iue for
idutr parter.
LaRs meyeR-WaaRden i a profeor of maretig at the
EM Buie shool strabourg ad at the IAE-Graduate
shool of Maagemet Touloue. He ha publihed
reearh about cutomer Relatiohip Maagemet,
loalt, ad loalt program i aademi joural uh
a Journal of the Academy of Marketing Science
ad Journal of Retailing. Prior to joiig aademia, he
wored for ma ear a a maretig maager withi
the LORAL Group.
LaRa stoCChi i a reearh aoiate at the Ehreberg-
Ba Ititute at the Uierit of south Autralia ad
ha jut ubmitted a PhD i Maretig. Preioul,
he tudied at Uierit carlo cattaeo LIUc, i Ital
(Bahelor i Buie Admiitratio ad Potgraduate
tudie i Maretig). From Jul 2012, he will be
holdig a leturig poitio at the shool of Buie
ad Eoomi at the Uierit of Loughborough,
Uk. Her e area of reearh are: Dirihlet modelig,
aali of oumer behaior, traig brad
performae, traig brad equit ad brad aliee,
memor ad ogitie truture.
PhiLiP steRn i profeor of maretig at Loughborough
Uierit shool of Buie ad Eoomi ad
adjut Profeor at the Ehreberg-Ba Ititute. He i
oe of a log lit of Adrew Ehreberg PhD tudet
ad hi reearh i urretl foued o healthare.
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