Anova

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Analysis of Variance

and Covariance

16-1Copyright 2010 Pearson Education, Inc.

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Copyright 2010 Pearson Education, Inc. 16-2

Relationship Among Techniques

Analysis of variance (ANOVA)is used as a testof means for two or more populations. The nullhypothesis, typically, is that all means are equal.

Analysis of variance must have a dependent

variable that is metric (measured using aninterval or ratio scale.

There must also be one or more independent

variables that are all categorical (nonmetric.Categorical independent variables are also calledfactors.

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Relationship Among Techniques

A particular combination of factor levels, or categories, is called

a treatment.

One-way analysis of varianceinvolves only one categoricalvariable, or a single factor. !n one"way analysis of variance, atreatment is the same as a factor level.

!f two or more factors are involved, the analysis is termedn-way analysis of variance.

!f the set of independent variables consists of both categoricaland metric variables, the technique is called analysis ofcovariance (ANCOVA). !n this case, the categoricalindependent variables are still referred to as factors, whereasthe metric"independent variables are referred to ascovariates.

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Relationship Amongst Test Analysis of VarianceAnalysis of Covariance ! Regression

Fig.

16.1

One Independent One or More

Metric Dependent Variable

t Test

Binary

Variable

One-ay !nalysiso" Variance

One Factor

#-ay !nalysiso" Variance

More t$anOne Factor

!nalysis o"Variance

%ategorical&

Factorial

!nalysis o"%o'ariance

%ategorical

and Inter'al

(egression

Inter'al

Independent Variables

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Copyright 2010 Pearson Education, Inc. 16-)

One-"ay Analysis of Variance

#ar$eting researchers are often interested ine%amining the differences in the mean values ofthe dependent variable for several categories of asingle independent variable or factor. &ore%ample'

o the various segments differ in terms of theirvolume of product consumption)

o the brand evaluations of groups e%posed todifferent commercials vary)

*hat is the effect of consumers+ familiarity withthe store (measured as high, medium, and lowon preference for the store)

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#tatistics Associate$ with One-"ayAnalysis of Variance

eta%( %). The strength of the effects ofX

(independent variable or factor on Y(dependentvariable is measured by eta%( %). The value of

,varies between - and .

Fstatistic. The null hypothesis that thecategory means are equal in the population istested by an Fstatisticbased on the ratio ofmean square related toXand mean square

related to error.

&ean square. This is the sum of squaresdivided by the appropriate degrees of freedom.

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#tatistics Associate$ with One-"ayAnalysis of Variance

SSbetween. Also denoted asSSx, this is thevariation in Yrelated to the variation in themeans of the categories ofX. This representsvariation between the categories ofX, or theportion of the sum of squares in Yrelated toX.

SSwithin. Also referred to as SSerror, this is the

variation in Ydue to the variation within eachof the categories ofX. This variation is notaccounted for byX.

SSy. This is the total variation in Y.

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Con$ucting One-"ay ANOVA

Interpret t$e (es,lts

Identi"y t$e Dependent and IndependentVariables

Decopose t$e Total Variation

Meas,re t$e /ects

Test t$e 0ignicance

Fig. 16.2

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Copyright 2010 Pearson Education, Inc. 16-

Con$ucting One-"ay Analysis of Variance'ecompose the Total Variation

The total variation in Y, denoted bySSy, can bedecomposed into two components'

SSy SSbetween SSwithin

where the subscripts betweenand withinrefer to thecategories ofX. SSbetweenis the variation in Yrelated

to the variation in the means of the categories ofX.&or this reason, SSbetweenis also denoted as SSx.

SSwithinis the variation in Yrelated to the variationwithin each category ofX. SSwithinis not accounted

for byX. Therefore it is referred to as SSerror.

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Con$ucting One-"ay Analysis of Variance'ecompose the Total Variation

The total variation in Ymay be decomposed as'

SSy SSx SSerror

*here

Yi / individual observation

j / mean for categoryj

/ mean over the whole sample, or grand mean

Yij/ ith observation in thejth category

Y

SSy= (Yi-Y )2

i=1

N

SSx= n (Yj-Y)2

j=1

c

SSerror= i

n

(Yij-Yj)2

j

c

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'ecomposition of the Total Variation*One-"ay ANOVA

Independent Variable

Total

CategoriesSample

1 2 3 5 c

1 1 1 1 1

2 2 2 2 2& &

& &n n n n #

1 2 3 c

it$in

%ategoryVariation

7008it$in

Bet8een %ategory Variation 7 00bet8een

TotalVariation 700y

%ategory Mean

Table 16.1

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Con$ucting One-"ay Analysis of Variance

!n analysis of variance, we estimate two measures of

variation' within groups (SSwithin and between groups(SSbetween. Thus, by comparing the Yvarianceestimates based on between"group and within"groupvariation, we can test the null hypothesis.

&easure the +ffects

The strength of the effects ofXon Yare measured asfollows'

%/ SSx0SSy/ (SSy" SSerror0SSy

The value of varies between - and .

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Con$ucting One-"ay Analysis of VarianceTest #ignificance

!n one"way analysis of variance, the interest lies in testing thenull hypothesis that the category means are equal in thepopulation.

,-* ./ .% .0 11111111111 .c

1nder the null hypothesis, SSxand SSerrorcome from the same

source of variation. !n other words, the estimate of thepopulation variance of Y,

/ SSx0(c "

/ #ean square due toX

/ MSx

or

/ SSerror0(N" c

/ #ean square due to error

/ MSerror

Sy2

Sy2

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Con$ucting One-"ay Analysis of VarianceTest #ignificance

The null hypothesis may be tested by the Fstatistic

based on the ratio between these two estimates'

This statistic follows the F$istri2ution, with (c " and (N" c degrees of freedom (df.

F =

SSx/(c- 1)

SSerror/(N- c) =

MSxMSerror

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Con$ucting One-"ay Analysis of Variance3nterpret the Results

!f the null hypothesis of equal category means isnot re2ected, then the independent variable doesnot have a significant effect on the dependentvariable.

3n the other hand, if the null hypothesis isre2ected, then the effect of the independentvariable is significant.

A comparison of the category mean values willindicate the nature of the effect of the independentvariable.

ll i li i f

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3llustrative Applications of One-"ayAnalysis of Variance

*e illustrate the concepts discussed in thischapter using the data presented in Table 4..

The department store is attempting to determinethe effect of in"store promotion (5 on sales (6.

&or the purpose of illustrating hand calculations,the data of Table 4. are transformed in Table4.7 to show the store sales (Yij for each level ofpromotion.

The null hypothesis is that the category meansare equal'

,-* ./ .% .0

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+ffect of 4romotion an$ Clientele on #ales

Table 16.2

3ll i A li i f O "

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+55+CT O5 3N-#TOR+ 4RO&OT3ON ON #A6+#

#tore 6evel of 3n-store 4romotionNo1 ,igh &e$ium 6ow

Normali7e$ #ales

- 8 9

: 8 ;

7 - ; 4

< 8 : ;:.8-

The strength of the effects ofXon Yare measured asfollows'

/ SSx0SSy

/ -4.-4;089.84;/ -.9;

!n other words, 9;.? of the variation in sales (6 isaccounted for by in"store promotion (5, indicating amodest effect. The null hypothesis may now be tested.

/ ;.:

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&rom Table 9 in the @tatistical Appendi% wesee that for and ; degrees of freedom, thecritical value of Fis 7.79 for . ecausethe calculated value of Fis greater than thecritical value, we re2ect the null hypothesis.

*e now illustrate the analysis of varianceprocedure using a computer program. The

results of conducting the same analysis bycomputer are presented in Table 4.

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One-"ay ANOVA* +ffect of 3n-#tore4romotion on #tore #ales

Table16.4

Cell means

Level of Count Mean

Promotion

High (1) 10 8.300ediu! (2) 10 ".200

#o$ (3) 10 3.%00

&' 30 ".0"%

Source of Sum of df Mean F ratio Fprob.

Variation squares square

et$een groups 10".0"% 2 *3.033 1%.+ 0.000

(Pro!otion)

-ithin groups %+.800 2% 2.+*"

(Error)

&' 18*.8"% 2+ ".0+

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Assumptions in Analysis of Variance

The salient assumptions in analysis of variance canbe summariBed as follows'

. 3rdinarily, the categories of the independentvariable are assumed to be fi%ed. !nferences aremade only to the specific categories considered.

This is referred to as the fi8e$-effects mo$el.

. The error term is normallydistributed, with a 7eromeanand a constant variance. The error is notrelated to any of the categories ofX.

7. The error terms are uncorrelate$. !f the errorterms are correlated (i.e., the observations are notindependent, the Fratio can be seriouslydistorted.

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N-"ay Analysis of Variance

!n mar$eting research, one is often concerned with theeffect of more than one factor simultaneously. &ore%ample'

ow do advertising levels (high, medium, and lowinteract with price levels (high, medium, and low to

influence a brand+s sale)

o educational levels (less than high school, high schoolgraduate, some college, and college graduate and age(less than 79, 79"99, more than 99 affect consumptionof a brand)

*hat is the effect of consumers+ familiarity with adepartment store (high, medium, and low and storeimage (positive, neutral, and negative on preference forthe store)

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Consider the simple case of two factorsXandXhavingcategories cand c. The total variation in this case is

partitioned as follows'

SStotal/ SSdue toX> SSdue toX> SSdue to interaction of

XandX> SSwithin

or

The strength of the 2oint effect of two factors, called the overalleffect, or multiple %, is measured as follows'

multiple /

SSy = SSx 1+ SSx 2+ SSx 1x 2+ SSerror

(SSx 1+ SSx 2+ SSx 1x 2)/ SSy

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Thesignificance of the overall effectmay be tested by anFtest, as follows'

where

dfn / degrees of freedom for the numerator

/ (c" > (c" > (c" (c" / cc"

dfd / degrees of freedom for the denominator

/ N" cc

MS / mean square

F=(SSx1+ SSx2+ SSx 1x 2)/dfn

SSerror/dfd

=SSx 1,x 2,x 1x 2/ dfn

SSerror

/dfd

=MSx1,x2,x1x2MSerror

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!f the overall effect is significant, the ne%t step is to

e%amine the significance of the interactioneffect. 1nder the null hypothesis of no interaction,the appropriate Ftest is'

*here

dfn / (c" (c"

dfd / N" cc

F=SSx 1x 2/dfn

SSerror/dfd

=MSx1x2

MSerror

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Thesignificance of the main effect of eachfactormay be tested as follows forX.'

where

dfn / c."

dfd / N" c.c,

F=SSx 1/dfn

SSerror/dfd

= MSx1MSerror

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Two-"ay Analysis of Variance

Source of Sum of Mean Sig. of

Variation squares df square F F

Main Effects Promotion 10.0! " #$.0$$ #%.&" 0.000 0.##!

Coupon #$.$$$ 1 #$.$$$ ##.1!" 0.000 0."&0

Combined 1#'.%00 $ #$.1$$ #%.' 0.000

()o*)a+ $."! " 1.$$ 1.'0 0.""

interaction

Model 1".! # $".#$$ $$.## 0.000,esidual -error "$."00 "% 0.'!

(/(L 1.&! "' .%0'

2

Table16.)

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Two-"ay Analysis of Variance

Table 16.)9 cont.

Cell Means

Promotion Coupon Count Mean

ig2 3es # '."00

ig2 4o # !.%00

Medium 3es # !.00

Medium 4o # %.&00

Lo) 3es # #.%00Lo) 4o # ".000

(/(L $0

Factor Level Means

Promotion Coupon Count Mean

ig2 10 &.$00

Medium 10 ."00

Lo) 10 $.!00

3es 1# !.%00

4o 1# %.!$$

5rand Mean $0 .0!

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Analysis of Covariance

*hen e%amining the differences in the mean values of the

dependent variable related to the effect of the controlledindependent variables, it is often necessary to ta$e intoaccount the influence of uncontrolled independent variables.&or e%ample'

!n determining how different groups e%posed to differentcommercials evaluate a brand, it may be necessary to controlfor prior $nowledge.

!n determining how different price levels will affect ahousehold+s cereal consumption, it may be essential to ta$ehousehold siBe into account. *e again use the data of Table4. to illustrate analysis of covariance.

@uppose that we wanted to determine the effect of in"storepromotion and couponing on sales while controlling for theeffect of clientele. The results are shown in Table 4.4.

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Analysis of Covariance

Sum of MeanSig.

Source of Variation Squares df Square F of F

Covariance

Clientele 0.838 1 0.838 0.862

0.363Main eects

!romotion 106.06" 2 #3.033 #$.#$6 0.000

Coupon #3.333 1 #3.333 #$.8## 0.000

Com%ined 1#&.$00 3 #3.133 #$.6$& 0.000

2'(a) *nteraction

!romotion+ Coupon 3.26" 2 1.633 1.680 0.208

Model 163.#0# 6 2".2#1 28.028 0.000

,esidual -rror/ 22.362 23 0.&"2

TT 18#.86" 2& 6.$0&

Covariate ,a Coe4cient

Clientele '0.0"8

Table 16.6

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3ssues in 3nterpretation

!mportant issues involved in the interpretation of AD3VAresults include interactions, relative importance of

factors, and multiple comparisons.

3nteractions

The different interactions that can arise whenconducting AD3VA on two or more factors are shown

in &igure 4.7.

Relative 3mportance of 5actors

E%perimental designs are usually balanced, in thateach cell contains the same number of respondents.

This results in an orthogonal design in which thefactors are uncorrelated. ence, it is possible todetermine unambiguously the relative importance ofeach factor in e%plaining the variation in the dependentvariable.

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A Classification of 3nteraction +ffects

#oncrosso'er:%ase 3;

%rosso'er:%ase 4;

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4atterns of 3nteraction

Fig. 16.4

11

12 13

%ase 1& #o Interaction2221

11

12 13

2221

%ase 2& OrdinalInteraction

11

12 13

2

221

%ase 3& DisordinalInteraction& #oncrosso'er

11

12 13

2

2

21

%ase 4& DisordinalInteraction& %rosso'er

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The most commonly used measure in AD3VA isomega

square$ . This measure indicates what proportion of thevariation in the dependent variable is related to a particularindependent variable or factor. The relative contribution of afactorXis calculated as follows'

Dormally, is interpreted only for statistically significanteffects. !n Table 4.9, associated with the level of in"store promotion is calculated as follows'

/ -.99;

x2=

SSx - (dfx xMSerror)

SStotal+MSerror

p2

=106.067- (2 x 0.967)

185.867 + 0.967

=104.133

186.834

2

2

2

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Dote, in Table 4.9, that

SStotal / -4.-4; > 97.777 > 7.4; > 7. / 89.84;

Fi$ewise, the associated with couponing is'

/ -.8-

As a guide to interpreting , a large e%perimental effect producesan inde% of -.9 or greater, a medium effect produces an inde% ofaround -.-4, and a small effect produces an inde% of -.-. !nTable 4.9, while the effect of promotion and couponing are bothlarge, the effect of promotion is much larger.

2

c

2=

53.333 - (1 x 0.967)

185.867 + 0.967

=52.366

186.834

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3ssues in 3nterpretation - &ultiple comparisons

!f the null hypothesis of equal means is re2ected, we canonly conclude that not all of the group means are equal.*e may wish to e%amine differences among specificmeans. This can be done by specifying appropriatecontrasts,or comparisons used to determine which of the

means are statistically different.

A priori contrastsare determined before conducting theanalysis, based on the researcher+s theoretical framewor$.=enerally, a priori contrasts are used in lieu of the AD3VAFtest. The contrasts selected are orthogonal (they are

independent in a statistical sense.

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3ssues in 3nterpretation - &ultiple Comparisons

A posteriori contrastsare made after the analysis.These are generallymultiple comparison tests. Theyenable the researcher to construct generaliBed confidenceintervals that can be used to ma$e pairwise comparisons ofall treatment means. These tests, listed in order of

decreasing power, include least significant difference,uncan+s multiple range test, @tudent"Dewman"Geuls,Tu$ey+s alternate procedure, honestly significantdifference, modified least significant difference, and@cheffe+s test. 3f these tests, least significant difference is

the most powerful, @cheffe+s the most conservative.

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Repeate$ &easures ANOVA

3ne way of controlling the differences betweensub2ects is by observing each sub2ect undereach e%perimental condition (see Table 4.;.@ince repeated measurements are obtainedfrom each respondent, this design is referred

to as within"sub2ects design orrepeate$measures analysis of variance. Hepeatedmeasures analysis of variance may be thoughtof as an e%tension of the paired"samples ttest

to the case of more than two related samples.

'ecomposition of the Total Variation*

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'ecomposition of the Total Variation*Repeate$ &easures ANOVA

Independent Variable XSubject CategoriesTotal

No. Sample

X1 X2 X3 Xc

1 Y11 Y12 Y13 Y1c Y1

2 Y21 Y22 Y23 Y2c Y2

n Yn1 Yn2 Yn3 Ync YN

Y1 Y2 Y3 Yc Y

!et"een

#eople

Variation

$SSbet"een

people

Total

Variation

$SS%

&it'in #eople Categor% Variation $ SS"it'in people

Categor%

(ean

Table 16.*

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!n the case of a single factor with repeated measures, the

total variation, with nc " degrees of freedom, may be splitinto between"people variation and within"people variation.

SStotal SSbetween people SSwithin people

The between"people variation, which is related to thedifferences between the means of people, has n " degreesof freedom. The within"people variation hasn (c " degrees of freedom. The within"people variationmay, in turn, be divided into two different sources of

variation. 3ne source is related to the differences betweentreatment means, and the second consists of residual orerror variation. The degrees of freedom corresponding tothe treatment variation are c " , and those correspondingto residual variation are (c " (n ".

$ O

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Thus,

SSwithin people/ SSx> SSerror

A test of the null hypothesis of equal means may

now be constructed in the usual way'

@o far we have assumed that the dependentvariable is measured on an interval or ratio scale.

!f the dependent variable is nonmetric, however, adifferent procedure should be used.

F=SSx/(c- 1)

SSerror/(n- 1) (c- 1)=

MSxMSerror

N i A l i f V i

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Nonmetric Analysis of Variance

Nonmetric analysis of variancee%aminesthe difference in the central tendencies ofmore than two groups when the dependentvariable is measured on an ordinal scale.

3ne such procedure is thek-sample me$iantest. As its name implies, this is an e%tensionof the median test for two groups, which was

considered in Chapter 9.

N i A l i f V i

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Nonmetric Analysis of Variance

A more powerful test is the9rus:al-"allis one-way

analysis of variance. This is an e%tension of the #ann"*hitney test (Chapter 9. This test also e%amines thedifference in medians. All cases from the kgroups areordered in a single ran$ing. !f the kpopulations are thesame, the groups should be similar in terms of ran$s withineach group. The ran$ sum is calculated for each group.

&rom these, the Grus$al"*allis Hstatistic, which has a chi"square distribution, is computed.

The Grus$al"*allis test is more powerful than the k"samplemedian test as it uses the ran$ value of each case, notmerely its location relative to the median. owever, ifthere are a large number of tied ran$ings in the data, thek"sample median test may be a better choice.

&ultivariate Analysis of Variance

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&ultivariate Analysis of Variance

&ultivariate analysis of variance(&ANOVA)is similar to analysis of variance(AD3VA, e%cept that instead of one metricdependent variable, we have two or more.

!n #AD3VA, the null hypothesis is that thevectors of means on multiple dependentvariables are equal across groups.

#ultivariate analysis of variance isappropriate when there are two or moredependent variables that are correlated.

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