It’s a Dirichlet World

7/28/2019 Its a Dirichlet World

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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

[email protected]

JOHN DAWES

Ehrenberg-Ba Intitute

john.dawe@


CArL DriESENEr


carl.drieener@


LArS MEYEr-WAArDEN

EM Buine school

strabourg

[email protected]

LArA STOCCHi


Lara.stocchi@

Marketingscience.info

PHiLiP STErN

Loughborough Univerity

and Ehrenberg-Ba

Intitute

[email protected]

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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June 2012 JOURNAL OF ADVERTISING RESEARCH 205

Its a DIrIchlet WorlD

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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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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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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Jue 2012 JOURNAL OF ADVERTISING RESEARCH 209


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


8/12



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


9/12



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.


10/12

212 JOURNAL OF ADVERTISING RESEARCH Jue 2012


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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It’s a Dirichlet World

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