Fi e , C.
A icle
Ne e -buye s o consume non-du ables
Sou h A ican Jou nal o Business Managemen
P o ided in Coope a ion wi h:
Uni e si y o S ellenbosch Business School (USB), Bell ille, Sou h A ica
Sugges ed Ci a ion: Fi e , C. (1984) : Ne e -buye s o consume non-du ables, Sou h A ican Jou nal
o Business Managemen , ISSN 2078-5976, A ican Online Scien i ic In o ma ion Sys ems (AOSIS),
Cape Town, Vol. 15, Iss. 3, pp. 155-161,
h ps://doi.o g/10.4102/sajbm. 15i3.1121
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/217866
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Ne e -buye s
o
consume
non-du ables
C.
Fi e
G adua e School
o
Business Adminis a ion, Uni e si y o he Wi wa e s and, Johannesbu g
In
his a icle he concep
o
ne e -buye s o consume non-
du ables is discussed. The adi ional Nega i e Binomial Dis ibu-
ion app oach o Eh enbe g o he ques ion is p esen ed.
P e ious-
ly unpublished wo k ca ied ou a he G adua e School o
Business Adminis a ion, Uni e si y o he Wi wa e s and,
Is
e iewed and hypo heses
a e
pu o wa d ha he
obse ed
la ge
ze o
cell
in
he pu chase equency dis ibu ions may
be
caused
by
he exis ence o a g oup o ne e -buye s o he p oduc , o by he
supe imposi ion
o
a leas wo dis inc buying popula ions,
p e iously iden i ied
as
b and-loyal and mul ib and/b and-swi chlng
households. The esul s
o
he esea ch aimed a es ing he i s
hypo hesis
a e
p esen ed he e. Two ca e ully moni o ed da a se s
we e modelled using ze o-augmen ed Nega i e Binomial
and
Sichel
dis ibu ions. The da a we e p e iously shown o exhibi he
necessa y mean households pu chase/consump ion s a iona i y. In-
di idual b ands
in
one da a se (pu chases
o
oile soap) we e
shown o ollow he p edic ions
o
he adi ional heo y - he
p opo ion
o
non-buye s dec easing wi h ime. In he second da a
se (consump ion
o
packaged soup) he p opo ion
o
non-
consume s o he b ands ell owa ds ze o
as
he leng h o he
ime pe iod s udied was inc eased, bu a a a e as e han ha
p edic ed by he heo y. The hypo hesis o he exis ence o
ne e -
buye s/use s
o
indi idual b ands in hese wo p oduc classes was
he e o e ejec ed.
S.
A .
J.
Bus. Mgm .
1984,
15:
155-
161
In
die a ikel wo d die konsep
an
nooi -kope s
an
nle-duu same
e b uike sgoede e besp eek. Eh enbe gse adisionele Nega ie ·
binomiaaldis ibusie-benade ing in e band me die aag is
oo ges el. Vo ige ongepublisee de we k ui ge oe by die
Nag aadsebes uu skool, Uni e si ei
an
die Wi wa e s and is
besp eek,
en
hipo eses wo d oo ges el da die waa genome g oo
nul-sel
in
die e koops- ekwensiedis ibusies miskien 'n ge olg
mag
wees
an
die bes aan an 'n g oep nooi -kope s
an
die p o-
duk,
o
as
'n ge olg
an
die opeenplasing
an
en mins e wee a -
sonde like kope populasies oo heen geiden i isee
as
handels-
naam-lojaal
en
handelsnaam-wlsselende gesinne.
Die
esul a e
an
die na o sing wa gedoen is om die ee s e hipo ese e oe s, wo d
gegee.
Twee so g uldig gekon olee de s elsels da a is
gemodellee me geb uik
an
die nul e mee de de Nega ie -
binomiaal-
en
Sichel-dis ibusies. lndi iduele handelsname in
een
da as esel (aankope
an
oile seep) olg die oo spelling
an
die
a-
disionele eo ie -die p opo sie
an
nie- e b uike s
neem
me e yd
a .
In
die weede da as elsel ( e b uik
an
e pak e sop) he die
p opo sie
an
nie- e b uike s
an
die handelsname na nul gedaal
me e lenging
an
die ydpe k wa bes udee is, maa een
'n
snelheid innlge as wa oo spel was deu die eo le.
Die
hipo ese
da daa nool -kope s/geb uike s
an
indi iduele handelsname
in
die wee p odukklasse bes aan he , is dus e we p.
S.-A . Tydsk . Bed y sl.
1984,
15:
155-161
C.
Fi e
G adua e
School
o
Business
Adminis a ion, Uni e si y
o
he
Wi -
wa e s and,
P.O. Box
31170,
B aam on ein
2017,
Republic
o
Sou h
A ica
Accep ed
May
1984
In oduc ion
In he ield
o
consume non-du ables,
an
in e es ing segmen
o
he ma ke , known as 'ne e -buye s', has caused some dis-
cussion in he li e a u e.
Eh enbe g (1972) claimed ha , gi en a su icien ly long
pe iod
o
ime, e e y consume
will
e en ually pu chase a p o-
duc a leas once,
and
so he ne e -buye does no exis .
I
a s udy shows a numbe
o
non-buye s in a pe iod
o
ime,
Eh enbe g belie es ha , had he ime
o
he s udy been longe ,
hese non-buye s would no ha e been ound. In ac , his
Nega i e Binomial Dis ibu ion (NBD) heo y which p oposes
ha , p o ided he ma ke
is
s a iona y equen ly pu chased
consume non-du ables can be modelled using he NBD,
(Equa ion
Al)
p edic s his ou come. As ime inc eases,
he
p opo ion
o
buye s
o
he p oduc as p edic ed by he NBD
ends
o
uni y (Equa ion A2). (A b ie desc ip ion
o
he
s a is ical dis ibu ions men ioned in he ex
is
gi en in
he
Appendix.)
Pu ely om a logical poin
o
iew
one
may seek
o
aise
issue wi h Eh enbe g
on
he ques ion
o
ne e -buye s. Non-
smoke s p obably ne e buy ciga e es, non-d i e s
who
do
no
own ehicles a e unlikely
o
pu chase pe ol.
P oduc
classes can he e o e be iden i ied in which, by logical conside-
a ions alone, one would expec
o
ind some ne e -buye s.
Ne e heless, one should concede
o
Eh enbe g
ha
such
examples p obably ep esen a small p opo ion
o
he con-
sume non-du ables ma ke .
Eh enbe g also claimed ha his model
is
as applicable
o
an
indi idual b and as i
is
o
he p oduc class as a whole.
Howe e , wi h espec
o
indi idual b ands he si ua ion
may
be qui e di e en . Eh enbe g and o he o e seas w i e s ha e,
wi h
ew
published excep ions, only es ed he NBD model in
i s uni a ia e o m. Whe e hey ound depa u es om
he
model, hey asc ibed hem
o
e ec s which hey labelled
' he
shel ing phenomenon'
o
' he
a iance disc epency'. They
belie ed his e ec -an obse ed discon inui y in he equen-
cy
o
hea ie pu chases
a
o
close
o
he numbe
o
uni s,
equal
o
he numbe
o
weeks
o
pe iod in he ime pe iod
o
he
s udy-
o
be caused by a small sec o
o
he ma ke pu -
chasing he p oduc egula ly.
Sichel (1975) in es iga ed his phenomenon using obse a-
ions on packaged-soup consump ion. By hei e y na u e,
being
consu.-np ion-based, such da a, acco ding
o
Eh enbe g's
heo y, ough no
o
exhibi he e ec desc ibed abo e.
Howe e , 'shel ing' was obse ed in he da a.
She also ound ha o he s a is ical es s e ealed signi i-
can depa u es om Eh enbe g's NBD model. As was subse-
1S6
quen ly demons a ed
by
Zachos
(l
977),
al hough uni a ia e
dis ibu ions p edic ed
by
NBD
heo y
may
be
accep ed
by
he
chi-squa e es , p edic ions
in
he bi a ia e onn
o
he
NBD
o en p o ide
e y
unsa is ac o y models.
As a esul , bo h
o
he own wo k and an ea lie s udy
by
Tucke
(1973),
Sichel
(1975)
sugges ed
ha he ue phen~
menon
o
epea -consump ion/buying
may
be ep esen ed
by
wo dis inc ly di e en popula ions:
(a) A ha d-co e
o
s able, b and-loyal households
which
con-
sume egula ly
in
acco dance
wi h
Eh enbe g's o iginal
model.' These he e o e
would
be
expec ed o
ollow
he
co ela ed
NBD
model (Equa ion
A3).
(b)
'A
ai numbe
o
b and-swi ching, mul ib and consum-
ing households.' This g oup
is
s a is ically cha ac e ized
by
he unco ela ed bi a ia e
NBD
(Equa ion
AS)
in
suc-
cessi e
ime pe iods.
She
pu o wa d he hypo hesis ha he obse ed bi a ia e
dis ibu ion a ose om he supe imposi ion
o
hese wo
popula ions. This hypo hesis
was
suppo ed
by
he wo k on
he a ia ion
o
ma ke pene a ion, numbe o
new
and
epea
consume s and co ela ion coe icien s
wi h
inc easing leng hs
o
he obse a ional pe iod. She ound ha he obse a ions
usually
lay
be ween he p edic ions
o
he co ela ed and
un-
co ela ed models.
Fi e & Ba ne
(l
979)
ound
simila
esul s
in
he oile soap
ma ke .
Joannides
(l
976)
onnula ed a bina y model
by
combining
he co ela ed and unco ela ed bi a ia e
NBDs.
He showed
ha
his
model could p oduce many
o
he obse ed shapes,
pa e ns (including a
shelO,
and
alues
o
pa ame e s
which
he o iginal
NBD
heo y could no .
In p ac ice, pu chase/ consump ion equency dis ibu ions
can be obse ed
in
which
he
size
o
he
z.e o
cell
appea s
exces-
si ely
la ge
when
compa ed o he dis ibu ion
as
a
whole.
This
sugges s
ha he dis ibu ion
is
no d awn om a
single
homogeneous popula ion, bu ha a
leas
wo
dis inc
popula-
ions a e in ol ed.
Joannides
(1976)
iden i ied
hese
popula ions
as
b and-loyal
and mul ib and, b and-swi ching households. Howe e , he
same dis ibu ion shape could
a ise
om he supe imposi ion
o
a buying popula ion
(which
may, o a
gi en
ime pe iod
con ain non-buye s who
will
buy
in
a u u e pe iod) and a
g oup
o
ne e -buye s (whose dis ibu ion would consis
o
a ce ain equency
in
he ze o
cell
only).
The objec i e
o
he esea ch epo ed
in
his a icle
was
o
es
hese hypo heses
by
in es iga ing he p opo ion
o
ne e -
buye s
in
wo equen ly pu chased Sou h A ican consume
non-du ables p oduc
classes.
Da a
The wo da a se s analysed
we e:
Soap
The da a e e ed o
as
'soap' o igina e om he oile soap
p oduc
class.
In Sou h A ica his p oduc
class
comp ised
mo e han hi y- i e di e en b ands, bu
was
domina ed
by
he wo leading b ands,
Lux
and Palmoli e.
The da a base used consis ed
o
a sample
o
614
Whi e
households' bimon hly
pu chases
o
oile
soap,
o e
he pe iod
Ap il o No embe
1978,
de ails
o
which
we e
collec ed
by
Ma ke Resea ch A ica using he consume panel me hod.
Goodman
(1979)
es ed he da a and concluded ha pu -
chasing a es o e he eigh -mon h pe iod
we e
s a iona y. The
obse a ions had one peculia i y wo hy
o
men ion. When
S.-A . Tydsk . Bed y sl.
1984,
15(3)
hey
we e
plo ed
in
he onn
o
equency dis ibu ions,
e en-
numbe ed pu chase equencies appea ed high
ela i e
o
he
odd-numbe ed pu chase equencies, gi ing a
zig-z.ag
shaped
dis ibu ion (Figu e
1).
This
was
a ibu ed
by
Goodman o
a consume p e e ence o pu chasing he p oduc
in
e en-
numbe ed lo s.
200
100
oL----~~~~~~:::::::c.-...:A::..
0 2 4 6 8
10
12
14
No.
OF
OOTS
PURCHASED
Figu e
1 Obse ed equency dis ibu ion o Palmoli e soap -
eigh
mon hs (N =
614)
Fo he pu pose
o
es ing a dis ibu ion
i
o his da a
i
was
he e o e necessa y o g oup he
cells.
The
z.e o
cell
was
allowed o s and alone, he one-plus- wo pu chase
equency
cells,
he h ee-plus- ou
cells,
e c., we e g ouped oge he .
Soup
In he ea ly
1970s
he packaged-soup ma ke
in
Sou h A ica
was
domina ed by he leading b and (he e designa ed b and
A)
ollowed a some dis ance by he only o he b and
wi h
a easonable ma ke sha e (designa ed b and
B).
Nine o he
b ands made up he es
o
he ma ke .
All
b ands
we e
p esen ed
in
uni onn packagings
o
ou o
six
se ings
each.
The da a e e ed o
as
'soup'
in
his pape
we e
collec ed
by
Sichel
(1975)
as
pa
o
a Sou h A ican Na ional Dus bin
Panel
Su ey.
Housewi es
we e
p o ided
wi h
a bin in o
which
hey disca ded emp y packe s
o
soup, he con en s
o
which
had been consumed by hei amilies. E e y wen y-eigh
days
(i.e.
a s anda d mon hly in e als) he
housewi es
we e
isi ed
by
in e iewe s who eco ded he con en s
o
hei bins.
In onna ion ela ing o
512
households' consump ion
o
six een
wen y-eigh day pe iods
was
colla ed.
Soup consump ion displayed a seasonal pa e n. I d opped
du ing sp ing and ose
in
au umn. Th ee sepa a e s a iona y
pe iods could hus
be
iden i ied:
Pe iods one h ough ou ( he i s s a iona y win e
pe iod):
-pe iods
se en
h ough en ( he s a iona y summe pe iod);
-pe iods hi een h ough six een ( he second s a iona y
win e pe iod).
Simila
mean consump ion
le els
we e
ound o he wo
win e
pe iods.
This se
o
da a di e s undamen ally om he oile soap
obse a ions
in
ha i con ains consump ion and no pu chase
da a. Thus idosync asies
o
indi idual households such
as
he
S.
A .
J. Bus. Mgm . 1984, 15(3)
equency
o
shopping
and
he numbe
o
uni s pu chased
on
each shopping occasion a e emo ed. Consump ion should
1so
he e o e p oduce a mo e uni o mly dis ibu ed se
o
da a.
Ne e -Buye s
Ne e -buye s
o
he p oduc ield
In
he oile
soap
p oduc ield, all
o
he 614 households in
he
sample
had
pu chased some soap du ing he eigh -mon h
pe iod
o
he su ey.
Thus
no
ne e -buye s
o
oile soap ex-
is ed in he sample.
Howe e , in he packaged-soup ma ke , a e wo win e
pe iods, 20
o
he 512 households had no
ye
consumed. Sichel
(1975)
concluded ha , because he e we e so ew non-consu-
me s, hese
20
households did no p o ide su icien e idence
o
he exis ence
o
ne e -consume s. This conclusion
is
accep ed
by he
au ho
and
he analysis below
is
hus de o ed
o
indi i-
dual b ands.
Ne e -buye s o indi idual b ands:
A s udy
o
he soap
and
soup
da a
e eals ha e en a e eigh
and
six een mon hs espec i ely, subs an ial p opo ions
o
he
households in he samples
had
no
pu chased/ consumed he
leading
b ands
(Table 1).
Table
1 Non-buye s
o
leading b ands
P opo ions
o
non-buye s
B and o e pe iod s udied
Lux 0,450
Palmoli e 0,492
Sunligh Mild 0,590
B eeze 0,651
Shield 0,694
Soup B and A 0,203
Soup B and B 0,309
A isual inspec ion
o
he
b and
pu chase equency dis ibu-
ions e ealed ce ain in e es ing phenomena. Fi s , in almos
all
cases
a kink
o
a la ening ou ,
and
some imes e en a sligh
maximum, occu ed a
abou
he second
o
hi d cell. In he
case
o
he oile
soap
dis ibu ions, he p oblem
o
he con-
sume s' a ou ing
o
e en-numbe ed pu chases (discussed
abo e) may ha e
been
con ibu ing owa ds his e ec . Second-
ly, in isually sweeping a
smoo h
cu e upwa ds h ough he
da a poin s,
one
would ha e expec ed such a cu e
o
in e cep
he equency axis a a
much
lowe poin
han
ha ac ually
eco ded.
An
example
om
eacq
da a
se
is
shown in Figu es
1
and
2.
The hypo hesis pu o wa d
is
ha hese wo e ec s a e caus-
ed by
an
'excess'
o
ze os in he i s cell
o
he dis ibu ions,
and
ha his excess in ac comp ises ne e -buye s/consume s
o
he
b and.
I
was an icipa ed ha he obse a ions could
bes
be model-
led using a ze o-augmen ed dis ibu ion (See
(c)
in Appendix).
Resul s
Analysis
o
each
o
he
da a
se s
is
discussed sepa a ely.
Analysis
o
he Soap Da a
Goodman
(1979)
ound
ha
he i
o
he Sichel dis ibu ion
(Sichel,
1971)
wi h
-y
=
-2,5
((b) in Appendix) was gene ally
supe io
o
he
NBD
wi h espec
o
he oile soap p oduc
class
(all
b ands
included).
As
a esul
o
his inding, she ex-
100
so
6 8
10
12
14
No.
O
OOTS
Coo.HO
Figu e
2 Obse ed equency dis ibu ion o combined
b ands
A
and
B
o
soup - i s ou
mon hs
(N = 614)
amined only he Sichel dis ibu ion when analysing he indi i-
dual b ands wi hin he p oduc ield.
Acco ding o Sichel (1980), indi idual b ands need no neces-
sa ily ollow he same dis ibu ion as ha pos ula ed o a o al
p oduc ield,
and
he e o e
Goodman's
(1979)
omission in
no s udying he ze o-augmen ed
NBD
as
a po en ial model
o he indi idual b ands
is
open
o
some c i icism.
This s udy he e o e began by a emp ing
o
i a ze o-
augmen ed
NBD
model
o
he indi idual
b and
da a
o
Lux
and
Palmoli e soaps. Ea lie esea ch (Fi e & Ba ne ,
1979)
had
shown ha o all ime pe iods s udied, he
chi-squa e
es
ejec ed he hypo hesis
o
an
o dina y NBD model o he da a
g ouped as sugges ed abo e.
Pa ame e es ima es o he model we e made
on
ung ouped
da a.
As
can
be
seen om Table 2, he ung ouped
da a
' ailed'
he
chi-squa e
es . A de ailed s udy
o
he es s e ealed ha
he 'odd-e en' pu chase phenomenon was esponsible o a
subs an ial p opo ion
o
he o al chi-squa e.
The
es s we e
hus epea ed using g ouped da a.
The
ze o-augmen ed Sichel
dis ibu ion
(-y
= - 0,5) was hen i ed
o
he same g ouped
da a
using he same ail-end g ouping
o
cells
o
expec ed e-
quency less
han
5,0.
F om
Table 2 i can
be
seen
ha , con a y
o
Goodman's
expec a ions, he be e
i
was p o ided by he ze o-augmen ed
NBD
model.
An
example
o
he i s ob ained
is
shown in
Figu e 3.
Sys ema ic de ia ions
o
he wo models we e ound.
The
Sichel dis ibu ion was i s
oo
low, hen
oo
high, hen
oo
low again, ela i e
o
he NBD. Acco ding
o
Sichel
(1980)
he
na u e
o
he de ia ions led one
o
he conclusion ha dec eas-
ing he alue
o
'Y
om -0,5
o
-1,5
o
-2,5 would
no
signi ican ly imp o e he i .
I
was he e o e concluded ha , in he oile
soap
ield, in-
di idual b and pu chase equencies
a e
bes modelled by he
uni a ia e ze o-augmen ed NBD.
One
o
he implica ions
o
he ze o-augmen ed model
is
ha
es ima es ob ained o
q,
he p opo ion
o
ne e -buye s in
he sample, should
be
independen
o
ime. Table 3 shows he
esul s ob ained om he i ing
o
bo h
he NBD
and
Sichel
ze o-augmen ed models. These es ima es a e ce ainly
no
cons an .
1S8
S.-A . Tydsk . Bed y sl.
1984,
15(3)
Table 2 Soap: Resul s
o
chi-squa e
es s
o
ze o-augmen ed
dis ibu ions
NBD NBD Sichel
('y
= - 0,5)
Ung ouped G ouped G ouped
Lax
1s
2 mon hs P(x2~
1,531
I
2)<
=0,465 • a
1s
4 mon hs P(x2
!l;:
16,379
I 5) < =
0,050
P(x2
~
2,023
I
2)
= 0,364 P(x2~ 2,548 I 2)=0,275
6 mon hs P(x2
!l;:29,982
I
8)<
=0,050 P(x2~5.747 I 3)=0,124 P(x2~
7,791
I 3)=0,051
8 mon hs P(x2~
19,5(,6
I 11)=0.052 P(;x2~7,385 I 5)=0,194 P(x2~ 11,422 I 5)<0,050
Palmoli e
1s
2 mon hs P(x2~
1,482
I 2)=0,477 a a
1s
4 mon hs P(x2
!l;:25,474
I 5)<0,050 P(x2
!l;:2,536
I
1)=0,lll
P(x2~ 2,916 I 1)=0,088
6 mon hs P(x2
!l;:28,420
I 7)<0,050 P(x2
!l;:5,991
I 3)=0,112 P(x2
!l;:
8,398 I
3)
<
0,050
8 mon hs P(x2~21,031 I 9)<0.050 P(x2~9,181 I 4)=0,057
P(;x2!l;:
12,931
I 4)<0,050
'No
deg ees
o
l llom
le o chi-squa es s
l10
240
21)
11kl
;
150
z
a
~120
a
OOSERVED
o
ZEl{)-AUH:NTED
NBD
• ZERO-~NTED
smn
(y=-0.5)
'JO
ZERO-AUMNTED
NIil :
9=0.709
q=0.371
k=168
P(JCH
3855)
=0.194
60
ZER>-AUMNTED
S10£L:
8=0.814
q:O
406
p:1.87
(y=-0.5
a
PRICH)
P b11.422
l5l<O
05
J)
0 0 2 4 6 8
10
12 14
16
Nn
CF
L lTS
PLR:HASED
Fl&a e
3 Obse ed and i ed ze o-augmen ed dis ibu ions o Lux soap -eigh mon hs (N = 614)
Table 3 Soap: Es ima es
o
p opo ion
o
ne e -
buye s q de i ed om
ze o-augmen ed models
Lux
1s
2 mon hs
1s 4 mon hs
6 mon hs
8 mon hs
Palmoli e
1s 2 mon hs
1s 4 mon hs
6 mon hs
8 mon hs
P opo ion
o
ne e -buye s
NBD
Siebel
0,634 0,635
0,498 0,515
0,410 0,440
0,37) 0,407
0,686 0,686
0,474 0,503
0,450 0,485
0,376 0,430
I
ne e -buye s do
exis ,
one would expec he obse ed p o-
po ion
o
non-buye s
o
a p oduc o e en ually
le el
ou
a
he p opo ion
o
ne e -buye s in he sample, whe eas he p o-
po ion
o
non-buye s as p edic ed by Eh enbe g's heo y
would end o ze o wi h leng hening ime pe iods (Eh enbe g,
1972).
Figu e 4
is
an
example
o
he obse ed and expec ed p o-
po ions
o
non-buye s
o
Lux soap. The expec ed p opo -
ions we e de i ed using Equa ion A4. The pa ame e es ima es
we e made by he me hod
o
maximum likelihood using he
obse a ions
o
he i s wo-mon h pe iod.
In no case did he obse ed p opo ion
lie
abo e ha p e-
dic ed by Eh enbe g's heo y. As a esul
o
hese indings,
he hypo hesis
o
he exis ence
o
ne e -buye s o indi idual
b ands in he oile soap ma ke was ejec ed.
Analysis
o
he Soup Da a
In
his analysis he non-consume o e he en i e six een-mon h
pe iod
o
he su ey we e conside ed, despi e he ac ha he
da a did no exhibi s a iona i y o e he pe iod s udied.
Al hough he use
o
he six een-mon h pe iod
is
heo e ically
no co ec , i does a leas gi e some guidance as o he p o-
po ion
o
non-consume s o e a leng hy pe iod. As will
be
seen om Figu e
S,
he obse ed poin plo ed a six een

S. A .
J.
Bus. Mgm . 1984,
1S(3)
1,0
--o--
OBSERVED
-
EXPECTED
0.4
+-----,--"'T"""----,.----- ---
0 2 4 6 8
LE {iTH
OF
OBSERVATIONAL
PERIOD
I
11'.JNTHS
I
Figu e
4 P opo ion
o
non-buye s
o
Lux soap as a unc ion
o
leng h
o
obse a ional pe iod (N = 614)
mon hs
is
well
wi hin he gene al
sweep
o
he
cu e.
The consump ion equency dis ibu ions
o
he indi idual
b ands
we e
no modelled
excep
o he
case
o
he combined
b ands A and B o e he eigh s a iona y win e mon hs. The
esul s
o
he
chi-squa e
es s on he
wo
ze o-augmen ed
models
we e:
NBD
Sichel
P<x2
~
2S,
732 I
27)
=
0,531
P(x
2~33,304 I 26)=0,166
The
NBD
model,
which
p o ided he be e i ,
ga e
an
es ima e
o
ze o o he p opo ion
o
ne e -consume s.
Because
o
his esul , he i ing
o
models
o he indi idual
b ands
was
omi ed, and he g aphs
o
obse ed
and
expec ed
( ia
Eh enbe g's NBD heo y) p opo ions
o
non-consume s
o
b and
A,
b and B and b ands A and B combined
we e
im-
media ely s udied.
The pa ame e es ima es o b ands A and B
we e
based
on a e ages
o
each
o
he
i s
ou mon hs' es ima es,
calcula ed using he me hod
o
momen s. Fo he combined
b ands he maximum likelihood pa ame e es ima es o he
i s
mon h
we e
used. I
was
ound ha he obse ed p o-
po ions
o
non-consume s app oached
ze o
mo e
apidly
han
he
expec ed
p opo ions as p edic ed
by
he Eh enbe g heo y.
An example
o
he obse ed and
expec ed
p opo ions
o
non-
consume s
is
shown
in
Figu e
5.
On he basis
o
his
e idence,
one mus he e o e
also
e-
jec he hypo hesis
o
he exis ence
o
ne e -consume s
o
in-
di idual b ands
in
he wo da a se s s udied.
Concluslons
Fo
he
wo
da a
samples
a ailable,
no
e idence
could
be
ound
o
suppo he hypo hesis ha he appa en
excessi e
size
o
he
ze o
cell
in
he pu chase/ consump ion
equency
dis ibu-
ion
was
due o he supe imposi ion
o
a buying/consuming
popula ion and a g oup
o
ne e -buye s/consume s.
The
q pa ame e s
o
he
z.e o-augmen ed
NBD
(which
i
was
hoped would p o ide an es ima e
o
he p opo ion
o
ne e -
buye s
o
oile soap)
we e
no cons an o e he a ious ime
pe iods s udied. In addi ion,
wi h
leng hening
ime
pe iods,
he
obse ed
p opo ion
o
non-buye s
dec eased
a
almos
ex-
ac ly
he same a e as ha p edic ed by he Eh enbe g
NBD
1,0
---o---
OBSERVED
......,.__
~XPECTED
159
0,0
+--,---.,..--.....----,--,--,---,---,---
0 2 4 6 8
1l
12 14
16
LE «iTH
O
OBSER ATIONAL
PERIOO
!l'llNTHSl
Figu e
5 P opo ion
o
non-consume s
o
b and A soup as a unc ion
o
leng h
o
obse a ional pe iod (N = S
12)
heo y.
This
heo y
does
no p o ide o he
exis ence
o
ne e -
buye s. A simila calcula ion
was
made o he indi idual
b ands
in
he soup p oduc
ield.
The e o e, on he
basis
o
he a ailable
e idence
in
wo p o-
duc
ields,
he ne e -buye /consume
hypo hesis
was
ejec ed.
The
Bina y
Model
o
Joannides
(1976)
should hus
be
con-
side ed
as
he
explana ion o he
obse ed
dis ibu ion
shapes.
He
showed
ha he ma ginal dis ibu ions ob ained by sum-
ming a co ela ed
NBD,
ep esen ing
loyal
epea -buye s, and
an unco ela ed
NBD,
ep esen ing b and-swi ching/mul i-
b and buye s, displayed
high
ze o
cell
equencies as
well
as
poin s
o
in lec ion o seconda y modes
close
o he ze o
cell.
Thus he
esul s
ob ained, oge he
wi h
a conside a ion
o
he Joannides
model,
lead
one o he conclusion ha pe haps
b and-disloyal buye s/ consume s a e be e ep esen ed
by
a
dis ibu ion
o
equencies
a he han he subse he eo (ze o
cell
equencies
only)
implied
by
he ne e -buye hypo hesis.
These
esul s
ha e
impo an implica ions o he ma ke e s
o
consume non-du ables. Ins ead
o
de o ing ime and
e -
o
in
pe suading
ne e -buye s
o
use
hei p oduc s
(in
a
gene ic
sense),
hey
would
be s ongly ad ised o unde ake
ad e ising
campaigns
aimed
a
wooing
exis ing
use s
o
o he
b ands
o
he p oduc in o ying hei pa icula b and.
Finally,
his
s udy
highligh ed
he
need
o
model
he
ield
o
consume
non-du able pu chase pa e ns
om
a mul ib and
pe spec i e,
a he han
by
looking
a
indi idual
b ands
in
isola-
ion.
Resea ch
in
his
a ea
has
been
unde aken and will
be
epo ed
on
in
he u u e.
Re e ences
Eh enbe g, A.S.C.
1972.
Repea buying: heo y
and
applica ions.
Ams e dam: No h Holland Publishing Co.
Fi e , C.
1980.
The de elopmen
o
new
ma hema ical models o
mul ib and buying. Unpublished M.B.A. disse a ion, Uni e si y
o
he Wi wa e s and, Johannesbu g.
Fi e , C. & Ba ne , A.E.
1979.
Tes ing he NBD epea buying
model o exposu e pe iods
o
a ying leng hs. Unpublished
Repo .
Goodman, M.A.
1979.
Mul ib and pu chasing beha iou - heo y
and applica ions. Unpublished M.B.A. disse a ion, Uni e si y
o
he Wi wa e s and, Johannesbu g.
Joannides, J.
1976.
A
new
dis ibu ion o epea consump ion. Un-
published M.B.A. disse a ion, Uni e si y
o
he Wi wa e s and,
Johannesbu g.
Mo ison, D.G.
1969.
Condi ional end analysis: a model ha allows
160
o
non-use s.
J.
Ma ke .
Res.,
VI,
342-346.
Sichel,
H.S.
1971.
On
a amily
o
disc e e dis ibu ions pa icula ly
sui ed
o
ep esen long- ailed equency da a. In P oceedings
o
he
Thi d Symposium
on
Ma hema ical S a is ics (N.F. Laubsche ,
ed.). S.A.
C.S.l.R.
(P e o ia),
Sl-97.
Sichel, H.S. 1973. S a is ical alua ion
o
diamondi e ous deposi s .
Jou nal
o
he Sou h A ican Ins i u e
o
Mining
and
Me alle gy,
23S-239.
Sichel, H.S. 1974.
On
a dis ibu ion ep esen ing sen ence leng h in
w i en
p ose.
Jou nal
o
he
Royal S a is ical Socie y, 137(1),
2S-34.
Sichel,
H.S.
1980. P i a e Communica ion.
Sichel, N. 197S. P oblem a eas in epea consump ion models.
Un-
published M.B.A. disse a ion, Uni e si y
o
he
Wi wa e s and,
Johannesbu g.
Tucke ,
A.E.
1973.
The
epea pu chase pa e ns
o
non-du able con-
sume goods in selec ed households. Unpublished M.B.A. disse -
a ion, Uni e si y
o
he Wi wa e s and, Johannesbu g.
Zachos, G. 1977. Non-du able consump ion pa e ns
o
indi idual
households. Unpublished M.B.A. disse a ion, Uni e si y
o
he
Wi wa e s and, Johannesbu g.
Appendix
A b ie
desc ip ion
o
he
s a is ical dis ibu ions
men ioned
in
he
ex
is
gi en
he e.
(a)
The NBD Repea -buying Model
(i)
Uni a ia e Case
The
model assumes ha
an
indi idual household's pu chases a e
cha ac e ized by a Poisson dis ibu ion.
In o de
o
s udy ma ke beha iou
as
a whole, he indi idual
household's dis ibu ions a c compounded using a
'Y
dis ibu ion o
he
indi idual a e age pu chase equencies pe uni ime (Sichel,
1971).
The
esul
is
he Nega i e Binomial Dis ibu ion (NBD):
Ax I
J
= [kl(k + m)lk
ex
+
k)
l ml(k + m)]X
(Al)
(k)
(x+
I)
whe e
xis
he
numbe
o
pu chases in ime ,
0<
m <
co
,0<
k <
oo
,
and
x = 0,1,2,
...
The p opo ion
o
buye s
o
he p oduc in ime
is
gi en by
b1 = 1 - (I
+ mlkF"
(A2)
whe e a buye
is
de ined
as
one
who buys a
leas
one uni du ing
ime pe iod
.
(ii)
The Bi a ia e Co ela ed NBD
The
bi a ia e model is ob ained by mixing he p oduc
o
wo
uni a ia e Poisson dis ibu ions wi h he
'Y
dis ibu ion, o o m
he
bi a ia e co ela ed NBD model.
In i s symme ic o m (whe e successi e pu chase pe iods a e
o
equal leng h) i can be w i en
J x
I
J
= (l+Umlk)-1c
(x+y+k)
[ ml(k+Um) +" (A3)
,Y (k) (x+
1)
e,,+
I)
whe e k>O, m
>0,
k( J
= k,
m( )=
m,
x=0,1,2,
...
,
y=0,1,2,
...
The ma ke pene a ion (p opo ion
o
buye s
o
he p oduc
in
ime
J
is
gi en
by
c" = l
-(1
+
mlk)-k
(A4)
(iii)
The
Unco ela ed Bi a ia e
NBD
Sichel (1980) de i ed he unco ela ed bi a ia e NBD
in
an
a emp
o
model he beha iou
o
p oduc -disloyal consume s (b and-
swi che s
and
mul ib and buye s) whose consump ions a e unco -
ela ed be ween wo ime pe iods. The unco ela ed NBD is ob ain-
ed om he p oduc
o
wo uni a ia e NBDs. I can
be
w i en
in
he o m
I
l
______________________
_
(b)
(c)
S.-A .
Tydsk .
Bed y sl. 1984,
1S(3)
k>O, m>O, k( )= k,
m( )= m,
x=0,1,2,
...
, y=0,1,2,
...
In he
case
o
his dis ibu ion,
he
p opo ion
o
buye s
is
,/' = I -
(1
+
ml
k) -
'"
The Sichel dis ibu ion
(A6)
This dis ibu ion was also de i ed by Sichel in 1971.
I
can
be
w i -
en in he o m
_
( =e)l
(a0/2)'
K
(a)
1/J(. )
-K (a( ' I -
0)) !
•
+}
l
= 0,1,2,
...
(A7)
I
is
de i ed
by
mixing he Poisson dis ibu ion wi h he
lexible
mixing
dis ibu ion
>.)
= j2 '{ =e)/
a0P
•
>.
~
-1 •
exp!
_ (
_!_)
>.
_ a2 e
(AS)
2K (a 'l
-0)
0 4
} '
In his dis ibu ion -
oo
<
-y
<
oo
, O < 0 < I
and
a
>0
a e he
h ee
pa ame e s
and
K_
(.)
is
he modi ied Bessel unc ion
o
he
second
kind
o
o de
(.).
, ,
( ) ep esen s a amily
o
disc e e dis ibu ions. Many
o
he
be -
e known disc e e dis ibu ions such as he NBD, Poisson,
geome ic,
e c., a e special
o
limi ing o ms
o
1/J(. )
(Sichel, 1974).
Making
gamma
nega i e allows
one
o
gene a e
an
en i ely
new
se
o
disc e e dis ibu ions.
The
dis ibu ions a e ma hema ically
di -
icul o handle unless
'Y
is
a hal -in ege .
-y
could he e o e
be
se ,
a p io i,
o
-
O,S
and
alues
o
- l
,S
o
-2,S would
be
used
in
o de
o
y
and
imp o e
he
i
i
'Y
= -
O,S
is
no
sa is ac o y.
Two pa ame e s, a
and
0,
ha e he e o e
o
be es ima ed.
I
he
p opo ion
o
obse ed
equenci~
in
he
ze o cell is la ge (gene al-
ly abo e
S00 o),
he
me hod
o
equa ing mean
and
ze o cell p opo -
ions
is
ai ly e icien (Sichel, 1973).
I
his is
no
he
case, a
maxi-
mum
likelihood me hod
is
a ailable (Sichel, 1971). Howe e ,
his
me hod
is
e y complica ed, so when he
i.e o
cell
is
small,
he me hod
o
momen s is gene ally used (Sichel, 1974).
Ze o-augmen ed dis ibu ions
A la ge numbe
o
obse ed pu chase/consump ion equency
dis ibu ions ha e
an
(wha appea s
o
he eye), excessi ely la ge mo
cell. I espec i e
o
he
ue
cause
o
his e ec ,
one
equi es o
be
able
o
model such dis ibu ions.
In o de
o
do
so, i is assumed
ha
his cell
is
made up
o
wo
sepa a e g oups.
The
si ua ion
is
illus a ed g aphically
in
Figu e
Al·
An
implica ion
o
his
model
is
ha ,
i
a ha d co e
o
ne e -buyen
does exis ,
he
es ima e
o
IJ
should
be
independen
o
ime. The
alues
o
IJ
o di e en
b ands
wi hin a p oduc ield need no ,
o
cou se,
be
simila .
~
~q
1i
/
/
/
/
/
....
....
....
(
1-q
)do
(0)
--
0 1 2 3 i. S
-.,_
OF
I..HITS
PIR:HASEO
/
CON9.J £0
la11n
Al
A
g aphical
ep esen a ion
o
ne e -buye s
S.
A .
J. Bus.
Mgm .
1984, 15(3)
Mo ison (1969) showed ha he exis ence
o
ne e -buye s would
in oduce a sys ema ic bias in o he NBD model. He he e o e p o-
posed a model
o
he o m gi en abo e. In his model h ee
pa ame e s equi e es ima ion: he k and m pa ame e s
o
he NBD
and
lj, he p opo ion
o
ne e -buye s.
161
Sichel
(1980)
p oposed
a simila modd based
on
he
Sichel
dis nbu-
ion. He e, oo, h ee pa ame e s equi e es ima ion: a
and
0
o
he Sichel,
and
lj
he p opo ion
o
ne e -buye s.
The echniques
o
es ima ion
o
he pa ame e s o bo h
dis ibu-
ions a e discussed by Fi e (1980).