chap11 discriminant analysis
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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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"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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"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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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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'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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*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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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
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