chap11 discriminant analysis

7/17/2019 Chap11 Discriminant Analysis

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Statistics for Marketing & Consumer ResearchCopyright 2008 - Mario Mazzocchi 1

Discriminant analysis

Chapter 11


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2-groups discriminant analysis

iscriminant ana!ysis is a statistica! proce"ure#hich a!!o#s us to c!assify cases in separatecategories to #hich they $e!ong on the $asis of aset of characteristic in"epen"ent %aria$!es ca!!e"

predictorsordiscriminant variables

he target %aria$!e 'the one "etermining a!!ocationinto groups( is a qualitative 'nomina! or or"ina!(one) #hi!e the characteristics are measure" $y*uantitati%e %aria$!es+

DA!ooks at the "iscrimination $et#een t#o groups Multiple discriminant analysis 'M,( a!!o#s for

c!assification into three or more groups+


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Applications of DA

, is especia!!y usefu! to un"erstan" the

"ifferences an" factors !ea"ing consumers to make

"ifferent choices a!!o#ing themto "e%e!op

marketing strategies #hich take into properaccount the ro!e of the pre"ictors+

.amp!es

eterminants of customer !oya!ty Shopper profi!ing an" segmentation

eterminants of purchase an" non-purchase


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Example on the Trust data-set

/urchasers of chicken at the $utchers shop 'recor"e" in*uestion *8"(

Respon"ents may $e!ong to one of t#o groups those #ho purchase chicken at the $utchers shop those #ho "o not

iscrimination $et#een these groups through a set ofconsumer characteristics e.pen"iture on chicken in a stan"ar" #eek '*( age of the respon"ent '*1( #hether respon"ents agree 'on a se%en-point ranking sca!e( that $utchers

se!! safe chicken '*21"(

trust 'on a se%en-point ranking sca!e( to#ar"s supermarkets '*3$(

oes a !inear com$ination of these four characteristicsa!!o# one to "iscriminate $et#een those #ho $uy chickenat the $utchers an" those #ho "o not4


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Discriminant analysis(DA

#o groups on!y) thus a sing!e "iscriminating %a!ue'discriminating score(

5or each respon"ent a score is compute" using theestimate" !inear com$ination of the pre"ictors 'the

discriminant function( Respon"ents #ith a score a$o%e the "iscriminating %a!ue

are e.pecte" to $e!ong to one group) those $e!o# to theother group+

6hen the "iscriminant score is stan"ar"ize" to ha%e zeromean an" unity %ariance it is ca!!e" Z score

, a!so pro%i"es information a$out the "iscriminatingpo#er of each of the origina! pre"ictors


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Statistics for Marketing & Consumer ResearchCopyright 2008 - Mario Mazzocchi !

"ultiple discriminant analysis("DA(1

iscriminant ana!ysis may in%o!%e more than t#ogroups) in #hich case it is terme" multiplediscriminant analysis (MDA)+

.amp!e from the rust "ata-set epen"ent %aria$!e7 ype of chicken purchase" in a

typica! #eek) choosing among four categories7 value'goo" %a!ue for money() standard, organic an" luxury

/re"ictors7 age '*0() state" re!e%ance of taste '*2a()%a!ue for money '*2$( an" anima! #e!fare '*2k() p!usan in"icator of income '*90(


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Statistics for Marketing & Consumer ResearchCopyright 2008 - Mario Mazzocchi #

"ultiple discriminant analysis(2

:n this case there #i!! $e more than one

"iscriminant function+

he e.act num$er of "iscriminant functions is

e*ua! to either 'g-1() #heregis the num$er of

categories in c!assification or to k) the num$er of

in"epen"ent %aria$!es) hichever is the smaller

rust e.amp!e7 four groups an" fi%e e.p!anatory%aria$!es) the num$er of "iscriminant functions is

three 'that isg!" #hich is sma!!er than k#$(+


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Statistics for Marketing & Consumer ResearchCopyright 2008 - Mario Mazzocchi $

The output of "D%

Simi!arities #ith factor 'principa! component( ana!ysis the first "iscriminant function is the most re!e%ant for

"iscriminating across groups) the secon" is the secon" mostre!e%ant) etc+

the "iscriminant functions are a!so in"epen"ent) #hich means that

the resu!ting scores are non-corre!ate"+ ;nce the coefficients of the "iscriminant functions are estimate"

an" stan"ar"ize") they are interprete" in a simi!ar fashion to thefactor !oa"ings+

he !arger the stan"ar"ise" coefficients 'in a$so!ute terms() themore re!e%ant the respecti%e %aria$!es to "iscriminating $et#eengroups

here is no sing!e "iscriminant score in M, group means are compute" 'centroids( for each of the "iscriminant

functions to ha%e a c!earer %ie# of the c!assification ru!e


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Statistics for Marketing & Consumer ResearchCopyright 2008 - Mario Mazzocchi &

'unning discriminant analysis(to groups

0 1 1 2 2 3 3 4 4z x x x x = + + + +

Discriminant function

(Target variable: purchasers of chicken at

the butchers shop)

Discriminant score Predictors

weekly expenditure on chicken

age

safety of butchers chicken

trust in supermarkets

The discriminant coefficientsneed to be estimated


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Statistics for Marketing & Consumer ResearchCopyright 2008 - Mario Mazzocchi 1)

*isher+s linear discriminant analysis

he "iscrimant function is the starting point

#o key assumptions $ehin" !inear ,'a( the pre"ictors are norma!!y "istri$ute" what

type of fresh or froen

chicken do you &uy for

your households

home consumption$;alue chicken

'tandard chicken

r)anic chicken


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%tepise discriminant analysis

,s for !inear regression it is possi$!e to

"eci"e #hether a!! pre"ictors shou!" appear

in the e*uation regar"!ess of their ro!e in

"iscriminating 'the nter option( or a su$-

set of pre"ictors is chosen on the $asis of

their contri$ution to "iscriminating $et#een

groups 'the -tepise method(


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The step-ise method

1+ , one-#ay ,=;>, test is run on each of the pre"ictors) #here the targetgrouping %aria$!e "etermines the treatment !e%e!s+ he ,=;>, test pro%i"esa criterion %a!ue an" tests statistics 'usua!!y the ilks ambda)+ ,ccor"ing tothe criterion %a!ue) it is possi$!e to i"entify the pre"ictor #hich is mostre!e%ant in "iscriminating $et#een the groups

2+ he pre"ictor #ith the !o#est ilks ambda 'or #hich meets an a!ternati%eoptima!ity criterion( enters the "iscriminating function) pro%i"e" thep-%a!ue

is $e!o# the set thresho!" 'for e.amp!e @(+3+ ,n ,=C;>, test is run on the remaining pre"ictors) #here the co%ariates are

the target grouping %aria$!es an" the pre"ictors that ha%e a!rea"y entere"the mo"e!+ he ilks ambda is compute" for each of the ,=C;>, options+

+ ,gain) the criteria an" the p-%a!ue "etermine #hich %aria$!e 'if any( enterthe "iscriminating function 'an" possi$!y #hether some of the entere"%aria$!es shou!" !ea%e the mo"e!(+

+ he proce"ure goes $ack to step 3 an" continues unti! none of the e.c!u"e"%aria$!es ha%e ap-%a!ue $e!o# the thresho!" an" none of the entere"%aria$!es ha%e ap-%a!ue a$o%e the thresho!" 'the stopping rule is met(+


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Alternati/e criteria

Ane.p!aine" %ariance

Sma!!est 5 ratio

Maha!ano$is "istance Raos >


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n %,%%

The stepwise method

allows selection of

relevant predictors


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Statistics for Marketing & Consumer ResearchC i ht 2008 M i M hi 3!

0utput of the step-ise method&ariables in t'e (nalysis

1.000 -.272

1.000 -.241 .90

1.000 ,.-, .919

lease indicate your

)ross annual household

income ran)e

lease indicate your

)ross annual household

income ran)e

;alue for money

'tep1

2

olerance % to ?emo3e

ilks

chap11 discriminant analysis

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