Chap009 Anova.ppt

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9-1

COMPLETECOMPLETEBUSINESSBUSINESS

STATISTICSSTATISTICS by by

AMIR D. ACZELAMIR D. ACZEL

&&JAYAVEL SOUNDERPANDIANJAYAVEL SOUNDERPANDIAN

66thth edition (SIEedition (SIE





9-3

! U"in# St$ti"ti%"

! he 'yothe"i" e"t o) An$*y"i" o) V$+i$n%e

! he heo+y $nd Co,-t$tion" o) ANOVA

! he ANOVA $b*e $nd E$,*e"

! /-+the+ An$*y"i"

! Mode*"0 /$%to+"0 $nd De"i#n"

! 1o23$y An$*y"i" o) V$+i$n%e

! 4*o%5in# De"i#n"

Analysis of VarianceAnalysis of Variance99



9-4

! E*$in the -+o"e o) ANOVA

! De"%+ibe the ,ode* $nd %o,-t$tion" behind ANOVA

! E*$in the te"t "t$ti"ti% F

! Cond-%t $ 21$y ANOVA

! Reo+t ANOVA +e"-*t" in $n ANOVA t$b*e

!

A*y -5ey te"t )o+ $i+21i"e $n$*y"i"! Cond-%t $ 721$y ANOVA

! E*$in b*o%5in# de"i#n"

! A*y te,*$te" to %ond-%t 21$y $nd 721$y ANOVA

LEARNING OBJECTIVESLEARNING OBJECTIVES99

After studying this chapter you should be able to: After studying this chapter you should be able to:



9-5

! ANOVA (AN$*y"i" O) VA+i$n%e i" $ "t$ti"ti%$*,ethod )o+ dete+,inin# the ei"ten%e o) di))e+en%e"$,on# "e8e+$* o-*$tion ,e$n".

ANOVA i" de"i#ned to dete%t di))e+en%e" $,on# ,e$n")+o, o-*$tion" "-b9e%t to di))e+ent treatments

ANOVA i" $ joint te"t

he e:-$*ity o) "e8e+$* o-*$tion ,e$n" i" te"ted"i,-*t$neo-"*y o+ 9oint*y.

ANOVA te"t" )o+ the e:-$*ity o) "e8e+$* o-*$tion,e$n" by *oo5in# $t t1o e"ti,$to+" o) the o-*$tion8$+i$n%e (hen%e0 analysis of variance.

9-1 Using Statistics9-1 Using Statistics



9-6

! In $n $n$*y"i" o) 8$+i$n%e;3e h$8e r indeendent +$ndo, "$,*e"0 e$%h one

%o++e"ondin# to $ o-*$tion "-b9e%t to $ di))e+enttreatment.

3e h$8e; n < n= n7= n>= ...=n+ tot$* ob"e+8$tion".

+ "$,*e ,e$n"; 0 7 0 > 0 ... 0 + he"e + "$,*e ,e$n" %$n be -"ed to %$*%-*$te $n e"ti,$to+ o)

the o-*$tion 8$+i$n%e. If the population means are equal, weexpect the variance among the sample means to be small .

+ "$,*e 8$+i$n%e"; "70 "7

70 ">70 ...0"+

7

he"e "$,*e 8$+i$n%e" %$n be -"ed to )ind $ pooled estimatoro) the o-*$tion 8$+i$n%e.

9-2 The Hypothesis Test of9-2 The Hypothesis Test of

Analysis of Variance Analysis of Variance



9-7

! 3e $""-,e independent random sampling )+o, e$%h o) ther o-*$tion"

! 3e $""-,e th$t the r o-*$tion" -nde+ "t-dy; $+e normally distributed 0

1ith ,e$n" µi th$t ,$y o+ ,$y not be e:-$*0

b-t 1ith equal variances, σ i2.

! 3e $""-,e independent random sampling )+o, e$%h o) ther o-*$tion"

! 3e $""-,e th$t the r o-*$tion" -nde+ "t-dy;

$+e normally distributed 0 1ith ,e$n" µi th$t ,$y o+ ,$y not be e:-$*0

b-t 1ith equal variances, σ i2.

µ µ7 µ>

σ

Po-*$tion Po-*$tion 7 Po-*$tion >

9-2 The Hypothesis Test of Analysis of9-2 The Hypothesis Test of Analysis of

Variance (continued) Assu!ptionsVariance (continued) Assu!ptions



9-8

he te"t "t$ti"ti% o) $n$*y"i" o) 8$+i$n%e;

/(+20 n2+ < E"ti,$te o) 8$+i$n%e b$"ed on ,e$n" )+o, + "$,*e"

E"ti,$te o) 8$+i$n%e b$"ed on $** "$,*e ob"e+8$tion"

h$t i"0 the te"t "t$ti"ti% in $n $n$*y"i" o) 8$+i$n%e i" b$"ed on the +$tio o)

t1o e"ti,$to+" o) $ o-*$tion 8$+i$n%e0 $nd i" the+e)o+e b$"ed on the F

di"t+ib-tion0 1ith (r!" de#+ee" o) )+eedo, in the n-,e+$to+ $nd (nr"

de#+ee" o) )+eedo, in the deno,in$to+.

he te"t "t$ti"ti% o) $n$*y"i" o) 8$+i$n%e;

/(+20 n2+

< E"ti,$te o) 8$+i$n%e b$"ed on ,e$n" )+o, + "$,*e"

E"ti,$te o) 8$+i$n%e b$"ed on $** "$,*e ob"e+8$tion"

h$t i"0 the te"t "t$ti"ti% in $n $n$*y"i" o) 8$+i$n%e i" b$"ed on the +$tio o)

t1o e"ti,$to+" o) $ o-*$tion 8$+i$n%e0 $nd i" the+e)o+e b$"ed on the F

di"t+ib-tion0 1ith (r!" de#+ee" o) )+eedo, in the n-,e+$to+ $nd (nr"

de#+ee" o) )+eedo, in the deno,in$to+.

he hyothe"i" te"t o) $n$*y"i" o) 8$+i$n%e;

'?; µ < µ7 < µ> < µ@ < ... µ+

'; Not $** µi (i < 0 ...0 + $+e e:-$*

he hyothe"i" te"t o) $n$*y"i" o) 8$+i$n%e;

'?; µ < µ7 < µ> < µ@ < ... µ+

'; Not $** µi (i < 0 ...0 + $+e e:-$*

9-2 The Hypothesis Test of Analysis of9-2 The Hypothesis Test of Analysis of

Variance (continued)Variance (continued)





9-10

3hen the n-** hyothe"i" i" )$*"e;

i" e:-$* to b-t not to 0

i" e:-$* to b-t not to 0

i" e:-$* to b-t not to 0 o+

0 0 $nd $+e $** -ne:-$*.

µ

µ

µ

µµ

µ

µ

µ

µµ

µ

µ

In $ny o) the"e "it-$tion"0 1e 1o-*d not ee%t the "$,*e ,e$n" to $** be ne$+*y

e:-$*. 3e 1o-*d ee%t the 8$+i$tion $,on# the "$,*e ,e$n" (bet1een

"$,*e to be *$+#e0 +e*$ti8e to the 8$+i$tion $+o-nd the indi8id-$* "$,*e ,e$n"

(1ithin "$,*e.

I) the n-** hyothe"i" i" )$*"e0 the n-,e+$to+ in the te"t "t$ti"ti% i" ee%ted to be

lar0 +e*$ti8e to the deno,in$to+;

F(r-1, n-r)=Estimat !" #arian$ %as& !n mans "r!m r sam'ls

Estimat !" #arian$ %as& !n all sam'l !%sr#ati!ns

In $ny o) the"e "it-$tion"0 1e 1o-*d not ee%t the "$,*e ,e$n" to $** be ne$+*y

e:-$*. 3e 1o-*d ee%t the 8$+i$tion $,on# the "$,*e ,e$n" (bet1een

"$,*e to be *$+#e0 +e*$ti8e to the 8$+i$tion $+o-nd the indi8id-$* "$,*e ,e$n"

(1ithin "$,*e.

I) the n-** hyothe"i" i" )$*"e0 the n-,e+$to+ in the te"t "t$ti"ti% i" ee%ted to be

lar0 +e*$ti8e to the deno,in$to+;

F(r-1, n-r)=Estimat !" #arian$ %as& !n mans "r!m r sam'ls

Estimat !" #arian$ %as& !n all sam'l !%sr#ati!ns

"hen the #ull Hypothesis $s %alse"hen the #ull Hypothesis $s %alse



9-11

!S-o"e 1e h$8e @ o-*$tion"0 )+o, e$%h o) 1hi%h 1ed+$1 $n indeendent +$ndo, "$,*e0 1ith n = n7 = n> =n@ < @. hen o-+ te"t "t$ti"ti% i";

F(-1, *-)= F(+,*) =

Estimat !" #arian$ %as& !n mans "r!m sam'ls

Estimat !" #arian$ %as& !n all * sam'l !%sr#ati!ns

!S-o"e 1e h$8e @ o-*$tion"0 )+o, e$%h o) 1hi%h 1ed+$1 $n indeendent +$ndo, "$,*e0 1ith n = n7 = n> =n@ < @. hen o-+ te"t "t$ti"ti% i";

F(-1, *-)= F(+,*) = Estimat !" #arian$ %as& !n mans "r!m sam'ls

Estimat !" #arian$ %as& !n all * sam'l !%sr#ati!ns

&'21

*+

*,

*&

*'

*

*2

*1

*%(&)

f ( % )

% .istri/ution 0ith and & .egrees of %reedo!

2*+9

α*&

he non+e9e%tion +e#ion ()o+ α<?.?in thi"

in"t$n%e i" / ≤ 7.B0 $nd the +e9e%tion +e#ion

i" / 7.B. I) the te"t "t$ti"ti% i" *e"" th$n

7.B 1e 1o-*d not +e9e%t the n-** hyothe"i"0$nd 1e 1o-*d %on%*-de the @ o-*$tion

,e$n" $+e e:-$*. I) the te"t "t$ti"ti% i"

#+e$te+ th$n 7.B0 1e 1o-*d +e9e%t the n-**

hyothe"i" $nd %on%*-de th$t the )o-+

o-*$tion ,e$n" $+e not e:-$*.

The A#VA Test Statistic forThe A#VA Test Statistic for r 'r ' 3opulations and3opulations and nn

&' &' Total Sa!ple /ser4ationsTotal Sa!ple /ser4ations



9-12

R$ndo,*y %ho"en #+o-" o) %-"to,e+" 1e+e "e+8ed di))e+ent tye" o) %o))ee $nd $"5ed to +$te the

%o))ee on $ "%$*e o) ? to ??; 7 1e+e "e+8ed -+e 4+$i*i$n %o))ee0 7? 1e+e "e+8ed -+e Co*o,bi$n

%o))ee0 $nd 77 1e+e "e+8ed -+e A)+i%$n2#+o1n %o))ee.

he +e"-*tin# te"t "t$ti"ti% 1$" / < 7.?7

R$ndo,*y %ho"en #+o-" o) %-"to,e+" 1e+e "e+8ed di))e+ent tye" o) %o))ee $nd $"5ed to +$te the

%o))ee on $ "%$*e o) ? to ??; 7 1e+e "e+8ed -+e 4+$i*i$n %o))ee0 7? 1e+e "e+8ed -+e Co*o,bi$n

%o))ee0 $nd 77 1e+e "e+8ed -+e A)+i%$n2#+o1n %o))ee.

he +e"-*tin# te"t "t$ti"ti% 1$" / < 7.?7

othe+".the)+o,t*y"i#ni)i%$ndi))e+",e$n" o-*$tion

theo) $nyth$t%on%*-de%$nnot1e$nd+e9e%ted0 be%$nnot?

'

A.>6?07

?7.7

A.>6?07>6>0>202

;i"?.?A<)o+ oint%+iti%$*&he><+

6><77=7?=7<n 77<>

n 7?<7

n 7<

n

e:-$*,e$n"th+ee$** Not;

'>7

;?

'

=<=

==−−=

==

F F

F F r nr

F

α

µ µ µ

&'21

*+

*,

*&

*'

*

*2

*1

*%

f ( % )

F Distribution with 2 and 60 Degrees of Freedom

α<?.?

e"t St$ti"ti%<7.?7 %(2,)*1&

56a!ple 9-156a!ple 9-1



9-13

he grand mean grand mean0 0 i" the ,e$n o) $** n < n= n7= n>=...= n+ ob"e+8$tion"

in allall + "$,*e".he grand mean grand mean0 0 i" the ,e$n o) $** n < n= n7= n>=...= n+ ob"e+8$tion"

in allall + "$,*e".

. 9

nto)+o,+-n" 9th-"0iF o-*$tion)+o,"$,*ein the oint 1ithd$t$thedenote"

9"-b"%+ithe +.to)+o,+-n"$ndnt0o+ t+e$t,e0 o-*$tionthedenote"i"-b"%+ithe

i. o-*$tion)+o,"$,*ee1ithin th 9 o"itionin ointd$t$ $+ti%-*$+ thei" i9

1he+e

< i9

<

; oint"d$t$$**o) ,e$nthe,e$n0#+$ndhe

i9

<

;+070>0...0<(ii"$,*eo) ,e$nhe

i

ni

n

in

j

r

ii

x

in

in

ji

x

r

i xn∑=

∑=

∑=

∑=

9- The Theory and the Co!putations9- The Theory and the Co!putations

of A#VAof A#VA The Grand MeanThe Grand Mean





9-15

3e de)ine $n $" the di))e+en%e bet1een $ d$t$ oint$nd it" "$,*e ,e$n. E++o+" $+e denoted by 0 $nd 1e h$8e;

3e de)ine $ $" the de8i$tion o) $ "$,*e ,e$n)+o, the #+$nd ,e$n. +e$t,ent de8i$tion"0 t $+e #i8en by;

i

error devi error devi ationation

treatment treatment deviationdeviation

e

0

he ANOVA +in%i*e "$y";

3hen the o-*$tion ,e$n" $+e not e:-$*0 the H$8e+$#e e++o+

(1ithin "$,*e i" +e*$ti8e*y ",$** %o,$+ed 1ith the H$8e+$#e

t+e$t,ent (bet1een "$,*e de8i$tion.

he ANOVA +in%i*e "$y";

3hen the o-*$tion ,e$n" $+e not e:-$*0 the H$8e+$#e e++o+

(1ithin "$,*e i" +e*$ti8e*y ",$** %o,$+ed 1ith the H$8e+$#e

t+e$t,ent (bet1een "$,*e de8i$tion.

The Theory and Co!putations of A#VAThe Theory and Co!putations of A#VA

Error DeviationError Deviation andand Treatment DeviationTreatment Deviation

iijij x xe −=

iijij x xe −=

x xt ii

−= x xt ii

−=



9-16

Con"ide+ d$t$ oint 7@<> )+o, t$b*e 2. he

,e$n o) "$,*e 7 i" .0 $nd the #+$nd ,e$n i"

6.?0 "o;e x x

t x x

#ot t e

#ot x x

7@ 7@ 7> A A

7 7 A 6 C?C @ AC

7@ 7 7@ A @ AC 6 ?C

7@ 7@> 6 C?C 6 ?C

= − = − == − = − =

= + = + =

= − = − =

. .

. . .

. . .

. .

o+

1&

7<.

< 6.?

7@<>

ot$* de8i$tion;

ot7@<7@2<6.?

+e$t,ent de8i$tion;

t7<72<@.

E++o+ de8i$tion;

e7@<7@27<.

he t!tal &#iati!n (T!ti i" the di))e+en%e bet1een $ d$t$ oint (/i $nd the #+$nd ,e$n (/;

T!ti=/i - /

/o+ $ny d$t$ oint i9;

T!t = t 5

h$t i";

T!tal 6#iati!n = Tratmnt 6#iati!n 5 Err!r 6#iati!n

he t!tal &#iati!n (T!ti i" the di))e+en%e bet1een $ d$t$ oint (/i $nd the #+$nd ,e$n (/;

T!ti=/i - /

/o+ $ny d$t$ oint i9;

T!t = t 5

h$t i";


The Theory and Co!putations ofThe Theory and Co!putations of

A#VA The A#VA The Total DeviationTotal Deviation



9-17


S1.ar& 6#iati!ns

&he tot$* de8i$tion i" the "-, o) the t+e$t,ent de8i$tion $nd the e++o+ de8i$tion;

= < ( (

Noti%e th$t the "$,*e ,e$n te+, ( %$n%e*" o-t in the $bo8e $ddition0 1hi%h

"i,*i)ie" the e:-$tion.

7 =

7< (

7(

7

t

i

e

ij

x

i

x x

ij

x

i

x

ij

x #ot

ij x

i

t i

eij

xi

x xij xi

#ot ij xij x

− + − = − =

− + −

= −

(

( 7 7


A#VA A#VA Squared DeviationsSquared Deviations



9-18

S.ms !" S.ar& 6#iati!ns

7

=7

< ni(

7(

7

SS& < SS&R = SSE

#ot ij j

n j

i

r nit ii

r eij j

n j

i

r

xij

x

j

n j

i

r x

i x

i

r x

ij x

i j

n j

i

r

7

7

==∑

=∑

=∑

=∑

=∑

−=∑

=∑ −

=∑ + −

=∑

=∑(

The Sum of Squares PrincipleThe Sum of Squares Principle

he tot$* "-, o) ":-$+e" (SS i" the "-, o) t1o te+,"; the "-, o)

":-$+e" )o+ t+e$t,ent (SSR $nd the "-, o) ":-$+e" )o+ e++o+ (SSE.

SS < SSR = SSE


he tot$* "-, o) ":-$+e" (SS i" the "-, o) t1o te+,"; the "-, o)

":-$+e" )o+ t+e$t,ent (SSR $nd the "-, o) ":-$+e" )o+ e++o+ (SSE.

SS < SSR = SSE





9-19

SSSS

SSRSSR SSESSE

SST ,e$"-+e" the total variation in the d$t$ "et0 the 8$+i$tion o) $** indi8id-$* d$t$

oint" )+o, the #+$nd ,e$n.

SST7 ,e$"-+e" the explained variation0 the 8$+i$tion o) indi8id-$* "$,*e ,e$n"

)+o, the #+$nd ,e$n. It i" th$t $+t o) the 8$+i$tion th$t i" o""ib*y ee%ted0 o+e*$ined0 be%$-"e the d$t$ oint" $+e d+$1n )+o, di))e+ent o-*$tion". It" the

8$+i$tion between #+o-" o) d$t$ oint".

SSE ,e$"-+e" unexplained variation0 the 8$+i$tion within e$%h #+o- th$t %$nnot be

e*$ined by o""ib*e di))e+en%e" bet1een the #+o-".

SST ,e$"-+e" the total variation in the d$t$ "et0 the 8$+i$tion o) $** indi8id-$* d$t$

oint" )+o, the #+$nd ,e$n.

SST7 ,e$"-+e" the explained variation0 the 8$+i$tion o) indi8id-$* "$,*e ,e$n"

)+o, the #+$nd ,e$n. It i" th$t $+t o) the 8$+i$tion th$t i" o""ib*y ee%ted0 o+e*$ined0 be%$-"e the d$t$ oint" $+e d+$1n )+o, di))e+ent o-*$tion". It" the

8$+i$tion between #+o-" o) d$t$ oint".

SSE ,e$"-+e" unexplained variation0 the 8$+i$tion within e$%h #+o- th$t %$nnot be

e*$ined by o""ib*e di))e+en%e" bet1een the #+o-".


Picturing The Sum of Squares PrinciplePicturing The Sum of Squares Principle



9-20

T8 n.m%r !" &rs !" "r&!m ass!$iat& it8 SST is (n - 1)2

n tot$* ob"e+8$tion" in $** + #+o-"0 *e"" one de#+ee o) )+eedo,

*o"t 1ith the %$*%-*$tion o) the #+$nd ,e$n

T8 n.m%r !" &rs !" "r&!m ass!$iat& it8 SST7 is (r - 1)2

+ "$,*e ,e$n"0 *e"" one de#+ee o) )+eedo, *o"t 1ith the

%$*%-*$tion o) the #+$nd ,e$n

T8 n.m%r !" &rs !" "r&!m ass!$iat& it8 SSE is (n-r)2

n tot$* ob"e+8$tion" in $** #+o-"0 *e"" one de#+ee o) )+eedo,

*o"t 1ith the %$*%-*$tion o) the "$,*e ,e$n )+o, e$%h o) + #+o-"

he de#+ee" o) )+eedo, $+e $dditi8e in the "$,e 1$y $" $+e the "-," o) ":-$+e";

d)(tot$* < d)(t+e$t,ent = d)(e++o+

(n 2 < (+ 2 = (n 2 +

T8 n.m%r !" &rs !" "r&!m ass!$iat& it8 SST is (n - 1)2

n tot$* ob"e+8$tion" in $** + #+o-"0 *e"" one de#+ee o) )+eedo,

*o"t 1ith the %$*%-*$tion o) the #+$nd ,e$n

T8 n.m%r !" &rs !" "r&!m ass!$iat& it8 SST7 is (r - 1)2+ "$,*e ,e$n"0 *e"" one de#+ee o) )+eedo, *o"t 1ith the

%$*%-*$tion o) the #+$nd ,e$n

T8 n.m%r !" &rs !" "r&!m ass!$iat& it8 SSE is (n-r)2

n tot$* ob"e+8$tion" in $** #+o-"0 *e"" one de#+ee o) )+eedo,

*o"t 1ith the %$*%-*$tion o) the "$,*e ,e$n )+o, e$%h o) + #+o-"

he de#+ee" o) )+eedo, $+e $dditi8e in the "$,e 1$y $" $+e the "-," o) ":-$+e";

d)(tot$* < d)(t+e$t,ent = d)(e++o+

(n 2 < (+ 2 = (n 2 +


A#VA A#VA Degrees of FreedomDegrees of Freedom



9-21

Re%$** th$t the %$*%-*$tion o) the "$,*e 8$+i$n%e in8o*8e" the di8i"ion o) the "-, o)

":-$+ed de8i$tion" )+o, the "$,*e ,e$n by the n-,be+ o) de#+ee" o) )+eedo,. hi"

+in%i*e i" $*ied $" 1e** to )ind the ,e$n ":-$+ed de8i$tion" 1ithin the $n$*y"i" o)

8$+i$n%e.

Man s.ar tratmnt (MST7):

Man s.ar rr!r (MSE):

Man s.ar t!tal (MST):

(Note th$t the $dditi8e +oe+tie" o) "-," o) ":-$+e" do n!t etend to the ,e$n

":-$+e". MS ≠ MSR = MSE.

Re%$** th$t the %$*%-*$tion o) the "$,*e 8$+i$n%e in8o*8e" the di8i"ion o) the "-, o)

":-$+ed de8i$tion" )+o, the "$,*e ,e$n by the n-,be+ o) de#+ee" o) )+eedo,. hi"

+in%i*e i" $*ied $" 1e** to )ind the ,e$n ":-$+ed de8i$tion" 1ithin the $n$*y"i" o)

8$+i$n%e.

Man s.ar tratmnt (MST7):

Man s.ar rr!r (MSE):

Man s.ar t!tal (MST):

(Note th$t the $dditi8e +oe+tie" o) "-," o) ":-$+e" do n!t etend to the ,e$n

":-$+e". MS ≠ MSR = MSE.

$%#&

%%#&

r = −(

$%' %%'

n r =

−(

$%# %%#

n= −(


A#VA A#VA The Mean SquaresThe Mean Squares



9-22

' $%'

' $%#&

ni i

r

i

(

$nd

(

( 1hen the n-** hyothe"i" i" t+-e

1hen the n-** hyothe"i" i" )$*"e

1he+e i" the ,e$n o) o-*$tion i $nd i" the %o,bined ,e$n o) $** + o-*$tion".

=

= +

−∑

−

=

σ

σ

µ µ σ

σ

µ µ

7

77

7

7

h$t i"0 the ee%ted ,e$n ":-$+e e++o+ (MSE i" "i,*y the %o,,on o-*$tion 8$+i$n%e(+e,e,be+ the $""-,tion o) e:-$* o-*$tion 8$+i$n%e"0 b-t the ee%ted t+e$t,ent "-, o)

":-$+e" (MSR i" the %o,,on o-*$tion 8$+i$n%e *-" $ te+, +e*$ted to the 8$+i$tion o) the

indi8id-$* o-*$tion ,e$n" $+o-nd the #+$nd o-*$tion ,e$n.

I" t8 n.ll 8;'!t8sis is tr. "o th$t the o-*$tion ,e$n" $+e $** e:-$*0 the "e%ond te+, in

the E(MSR )o+,-*$tion i" e+o0 $nd E(MSR i" e:-$* to the %o,,on o-*$tion 8$+i$n%e.

h$t i"0 the ee%ted ,e$n ":-$+e e++o+ (MSE i" "i,*y the %o,,on o-*$tion 8$+i$n%e(+e,e,be+ the $""-,tion o) e:-$* o-*$tion 8$+i$n%e"0 b-t the ee%ted t+e$t,ent "-, o)

":-$+e" (MSR i" the %o,,on o-*$tion 8$+i$n%e *-" $ te+, +e*$ted to the 8$+i$tion o) the

indi8id-$* o-*$tion ,e$n" $+o-nd the #+$nd o-*$tion ,e$n.

I" t8 n.ll 8;'!t8sis is tr. "o th$t the o-*$tion ,e$n" $+e $** e:-$*0 the "e%ond te+, in

the E(MSR )o+,-*$tion i" e+o0 $nd E(MSR i" e:-$* to the %o,,on o-*$tion 8$+i$n%e.


A#VA A#VA The Expected Mean SquaresThe Expected Mean Squares



9-23

3hen the n-** hyothe"i" o) ANOVA i" t+-e $nd $** r o-*$tion ,e$n" $+e

e:-$*0 MSR $nd MSE $+e t1o indeendent0 -nbi$"ed e"ti,$to+" o) the

%o,,on o-*$tion 8$+i$n%e σ7.

3hen the n-** hyothe"i" o) ANOVA i" t+-e $nd $** r o-*$tion ,e$n" $+e

e:-$*0 MSR $nd MSE $+e t1o indeendent0 -nbi$"ed e"ti,$to+" o) the

%o,,on o-*$tion 8$+i$n%e σ7.

On the othe+ h$nd0 1hen the n-** hyothe"i" i" )$*"e0 then MSR 1i** tend to

be *$+#e+ th$n MSE2

S! t8 rati! !" MST7 an& MSE $an % .s& as an in&i$at!r !" t8S! t8 rati! !" MST7 an& MSE $an % .s& as an in&i$at!r !" t8

.alit; !r in.alit; !" t8.alit; !r in.alit; !" t8 rr '!'.lati!n mans2'!'.lati!n mans2

T8is rati! (MST7<MSE) ill tn& t! % nar t! 1 i" t8 n.ll 8;'!t8sis is

tr., an& ratr t8an 1 i" t8 n.ll 8;'!t8sis is "als2 T8 ANOVA tst,

"inall;, is a tst !" 8t8r (MST7<MSE) is .al t!, !r ratr t8an, 12

On the othe+ h$nd0 1hen the n-** hyothe"i" i" )$*"e0 then MSR 1i** tend to

be *$+#e+ th$n MSE2

S! t8 rati! !" MST7 an& MSE $an % .s& as an in&i$at!r !" t8S! t8 rati! !" MST7 an& MSE $an % .s& as an in&i$at!r !" t8

.alit; !r in.alit; !" t8.alit; !r in.alit; !" t8 rr '!'.lati!n mans2'!'.lati!n mans2

T8is rati! (MST7<MSE) ill tn& t! % nar t! 1 i" t8 n.ll 8;'!t8sis is

tr., an& ratr t8an 1 i" t8 n.ll 8;'!t8sis is "als2 T8 ANOVA tst,

"inall;, is a tst !" 8t8r (MST7<MSE) is .al t!, !r ratr t8an, 12

56pected 7ean S8uares and the56pected 7ean S8uares and the

A#VA 3rinciple A#VA 3rinciple



9-24

Unde+ the $""-,tion" o) ANOVA0 the +$tio (MSRKMSE

o""e"" $n / di"t+ib-tion 1ith (r!" de#+ee" o) )+eedo, )o+

the n-,e+$to+ $nd (nr" de#+ee" o) )+eedo, )o+ thedeno,in$to+ 1hen the n-** hyothe"i" i" t+-e.

he te"t "t$ti"ti% in $n$*y"i" o) 8$+i$n%e;

( 2 0 2 F $%#& $%' r n r

=


A#VA A#VA The F Statistic The F Statistic

9 25



9-25

( 7

n

i

( 7

C+iti%$* oint ( < ?.?; G.6

'?

,$y be +e9e%ted $t the ?.? *e8e*

o) "i#ni)i%$n%e.

%%' xij

xi

j

n j

i

r

%%#& x

i

x

i

r

$%#&%%#&

r

$%' %%#&

n r

F

$%#&

$%'

= −

=

∑ =

=

∑

= −

=

∑ =

=−

=−

=

=−

= =

= = =

B

> B

B

G7 7

7 G

B

7 7>B 67

.

.

( .

.

( 0

.

.. .

α

Tratmnt (i) i Val. (/ i ) (/i -/i ) (/i -/i )3

+i$n#*e @ 27 @

+i$n#*e 7 2

+i$n#*e > B

+i$n#*e @ G 7 @

S:-$+e 7 ? 2. 7.7

S:-$+e 7 7 2?. ?.7S:-$+e 7 > 7 ?. ?.7

S:-$+e 7 @ > . 7.7

Ci+%*e > 2

Ci+%*e > 7 7 ? ?

Ci+%*e > > >

+ 1

Tratmnt (/i

-/) (/i

-/)3 ni(/

i -/)3

+i$n#*e 2?.? ?.G767G >.>?7@

S:-$+e @. 7.?BB7G G@.>?7@

Ci+%*e 2@.? 7@.?G7G B7.7@G@>

1*929991

9-' The A#VA Ta/le and 56a!ples9-' The A#VA Ta/le and 56a!ples

9 26



9-26

S!.r$ !"

Variati!n

S.m !"

S.ars

6rs !"

Fr&!m Man S.ar F 7ati!

Tratmnt SSR<. (+2<7 MSR<B. >B.67

Err!r SSE<B.? (n2+<G MSE<7.7

T!tal SS<B6. (n2<? MS<B.6

1

*+

*,

*&

*'

*

*2

*1

*%(2)

f ( % )

% .istri/ution for 2 and .egrees of %reedo!

*,&

?.?

Co,-ted te"t "t$ti"ti%<>B.67

he ANOVA Ta%l "-,,$+ie" the

ANOVA %$*%-*$tion".

In thi" in"t$n%e0 "in%e the te"t "t$ti"ti% i"#+e$te+ th$n the %+iti%$* oint )o+ $n α <

?.? *e8e* o) "i#ni)i%$n%e0 the n-**

hyothe"i" ,$y be +e9e%ted0 $nd 1e ,$y

%on%*-de th$t the ,e$n" )o+ t+i$n#*e"0

":-$+e"0 $nd %i+%*e" $+e not $** e:-$*.

he ANOVA Ta%l "-,,$+ie" the

ANOVA %$*%-*$tion".

In thi" in"t$n%e0 "in%e the te"t "t$ti"ti% i"#+e$te+ th$n the %+iti%$* oint )o+ $n α <

?.? *e8e* o) "i#ni)i%$n%e0 the n-**

hyothe"i" ,$y be +e9e%ted0 $nd 1e ,$y

%on%*-de th$t the ,e$n" )o+ t+i$n#*e"0

":-$+e"0 $nd %i+%*e" $+e not $** e:-$*.

A#VA Ta/le A#VA Ta/le

9 27



9-27

Te!plate utputTe!plate utput

9 28



9-28

C*-b Med h$" %ond-%ted $ te"t to dete+,ine 1hethe+ it" C$+ibbe$n +e"o+t" $+e e:-$**y 1e** *i5ed by

8$%$tionin# %*-b ,e,be+". he $n$*y"i" 1$" b$"ed on $ "-+8ey :-e"tionn$i+e (#ene+$* "$ti")$%tion0

on $ "%$*e )+o, ? to ?? )i**ed o-t by $ +$ndo, "$,*e o) @? +e"ondent" )+o, e$%h o) +e"o+t".

C*-b Med h$" %ond-%ted $ te"t to dete+,ine 1hethe+ it" C$+ibbe$n +e"o+t" $+e e:-$**y 1e** *i5ed by

8$%$tionin# %*-b ,e,be+". he $n$*y"i" 1$" b$"ed on $ "-+8ey :-e"tionn$i+e (#ene+$* "$ti")$%tion0

on $ "%$*e )+o, ? to ?? )i**ed o-t by $ +$ndo, "$,*e o) @? +e"ondent" )+o, e$%h o) +e"o+t".

S!.r$ !"

Variati!n

S.m !"

S.ars

6rs !"


Tratmnt SSR< 13> (+2< @ MSR< +**3 2

Err!r SSE<G>6 (n2+< MSE< *2+9

T!tal SS<76@ (n2< MS< *0*20*

7s!rt Man 7s'!ns (/ )i

-$de*o-e G

M$+tini:-e B

E*e-th+$ B>

P$+$di"e I"*$nd

St. L-%i$ G

SS<76@ SSE<G>6

%('2)

% .istri/ution 0ith ' and 2 .egrees of %reedo!

*+

*,

*&

*'

*

*2

*1

*

f ( % )

*'1

?.?

Co,-ted te"t "t$ti"ti%<B.?@

he +e"-*t$nt /+$tio i" *$+#e+ th$n

the %+iti%$* oint )o+

α < ?.?0 "o the

n-** hyothe"i" ,$y

be +e9e%ted.

he +e"-*t$nt /+$tio i" *$+#e+ th$n

the %+iti%$* oint )o+

α < ?.?0 "o the

n-** hyothe"i" ,$y

be +e9e%ted.

56a!ple 9-2 Clu/ 7ed56a!ple 9-2 Clu/ 7ed

9 29



9-29

S!.r$ !"

Variati!n

S.m !"

S.ars

6rs !"


Tratmnt SSR< >92+ (+2<> MSR< 39+21 G.7

Err!r SSE< 1>*120 (n2+< *+9 MSE<>@.@

T!tal SS< 19329 (n2<@7 MS< +*2>+

i8en the tot$* n-,be+ o) ob"e+8$tion" (n < @>0 the n-,be+ o) #+o-"

(+ < @0 the MSE (>@. @0 $nd the / +$tio (G.70 the +e,$inde+ o) the ANOVAt$b*e %$n be %o,*eted. he %+iti%$* oint o) the / di"t+ib-tion )o+ α < ?.?

$nd (>0 @?? de#+ee" o) )+eedo, i" >.G>. he te"t "t$ti"ti% in thi" e$,*e i"

,-%h *$+#e+ th$n thi" %+iti%$* oint0 "o the 8$*-e $""o%i$ted 1ith thi" te"t

"t$ti"ti% i" *e"" th$n ?.?0 $nd the n-** hyothe"i" ,$y be +e9e%ted.

i8en the tot$* n-,be+ o) ob"e+8$tion" (n < @>0 the n-,be+ o) #+o-"

(+ < @0 the MSE (>@. @0 $nd the / +$tio (G.70 the +e,$inde+ o) the ANOVAt$b*e %$n be %o,*eted. he %+iti%$* oint o) the / di"t+ib-tion )o+ α < ?.?

$nd (>0 @?? de#+ee" o) )+eedo, i" >.G>. he te"t "t$ti"ti% in thi" e$,*e i"

,-%h *$+#e+ th$n thi" %+iti%$* oint0 "o the 8$*-e $""o%i$ted 1ith thi" te"t

"t$ti"ti% i" *e"" th$n ?.?0 $nd the n-** hyothe"i" ,$y be +e9e%ted.

56a!ple 9- :o/ $n4ol4e!ent56a!ple 9- :o/ $n4ol4e!ent

9 30



9-30

D$t$ ANOVA

Do Not Re9e%t '? Sto

Re9e%t '?

he "$,*e ,e$n" $+e -nbi$"ed e"ti,$to+" o) the o-*$tion ,e$n".

he ,e$n ":-$+e e++o+ (MSE i" $n -nbi$"ed e"ti,$to+ o) the %o,,on

o-*$tion 8$+i$n%e.

/-+the+

An$*y"i"

Con)iden%e Inte+8$*"

)o+ Po-*$tion Me$n"

-5ey P$i+1i"e

Co,$+i"on" e"t

he ANOVA Di$#+$,he ANOVA Di$#+$,

9-& %urther Analysis9-& %urther Analysis

9 31



9-31

A ( 2 ?? %on)iden%e inte+8$* )o+ 0 the ,e$n o) o-*$tion i;iα µ

α

α 1he+e t i" the 8$*-e o) the di"t+ib-tion 1ith de#+ee" o)

)+eedo, th$t %-t" o)) $ +i#ht 2 t$i*ed $+e$ o) 7

.7

α x t $%'

ni

i

±7

t (n r

x t $%'

n x xi

i

i i± = ± = ±

± =± =± =± =± =

α

7

6?@ >

@?6 6

G 6 6 G7 ?@ 6B 6 6 6G ?@ G 6B> 6 6 66 ?@ B 6 6 6 G@ ?@ B 6G 6 6 BG ?@ 6

..

.

. . 0 .

. . 0 .

. . 0 .

. . 0 .

. . 0 .

7s!rt Man 7s'!ns (/ i)

-$de*o-e G

M$+tini:-e B

E*e-th+$ B>

P$+$di"e I"*$nd

St. L-%i$ G

SS < 76@ SSE < G>6

ni < @? n < ((@? < 7??

MSE < ?@.>

Confidence $nter4als for 3opulationConfidence $nter4als for 3opulation

7eans7eans

9 32



9-32

he T.?; Pairis C!m'aris!n te"t0 o+ @!nstl; Sini"i$ant 6i""rn$s (MS6) te"t0 $**o1" -" to

%o,$+e e8e+y $i+ o) o-*$tion ,e$n" 1ith $ "in#*e *e8e* o) "i#ni)i%$n%e.

It i" b$"ed on the st.&nti& ran &istri%.ti!n0 :0 1ith r $nd (nr" de#+ee" o) )+eedo,.

he %+iti%$* oint in $ -5ey P$i+1i"e Co,$+i"on" te"t i" the T.?; Critri!n:

1he+e ni i" the smallest o) the + "$,*e "ie".

he tst statisti$ i" the absolute value of the difference bet1een the $+o+i$te "$,*e ,e$n"0 $nd

t8 n.ll 8;'!t8sis is r$t& i" t8 tst statisti$ is ratr t8an t8 $riti$al '!int !" t8 T.?;

Critri!n

# q $%' n

i

= α

( ) Note th$t the+e $+e+

7 $i+" o) o-*$tion ,e$n" to %o,$+e. /o+ e$,*e0 i) < ;

'? '? '?

'6 '6 '6

=

−

= = =

≠ ≠ ≠

r

r

r

P

P( P

; ; ;

; ; ;

7 7>

6 7 6 > 7 >

6 7 6 > 7 >

µ µ µ µ µ µ

µ µ µ µ µ µ

The Tu;ey 3air0ise-Co!parisons TestThe Tu;ey 3air0ise-Co!parisons Test

9-33



9-33

he tst statisti$ )o+ e$%h $i+1i"e te"t i" the $b"o*-te di))e+en%e bet1een the $+o+i$te

"$,*e ,e$n".

i Re"o+t Me$n I. '?; µ = µ7 VI. '?; µ7 = µ@

-$de*o-e G '; µ ≠ µ7 '; µ7 ≠ µ@

7 M$+tini:-e B QG2BQ<@>.B QB2Q<6>.B

> E*e-th+$ B> II. '?; µ

= µ

>VII.

'

?; µ

7 = µ

@ P$+$di"e I". '; µ ≠ µ> '; µ7 ≠ µ

St. L-%i$ G QG2B>Q<6>.B QB2GQ<?>.B

III. '?; µ = µ@ VIII. '?; µ> = µ@

he %+iti%$* oint ?.? )o+ '; µ ≠ µ@ '; µ> ≠ µ@

+< $nd (n2+< QG2Q<7>.B QB>2Q<G>.B

de#+ee" o) )+eedo, i"; IV. '?; µ = µ IT. '?; µ> = µ

'; µ ≠ µ '; µ> ≠ µ

QG2GQ<@>.B QB>2GQ<7>.B

V. '?; µ7 = µ> T. '?; µ@ = µ

'; µ7 ≠ µ> '; µ@ ≠ µ

QB2B>Q<7>.B Q2GQ< 6>.B

7$t t8 n.ll 8;'!t8sis i" t8 a%s!l.t #al. !" t8 &i""rn$ %tn t8 sam'l mansis ratr t8an t8 $riti$al #al. !" T2 (he hyothe"e" ,$+5ed 1ith $+e +e9e%ted.

he tst statisti$ )o+ e$%h $i+1i"e te"t i" the $b"o*-te di))e+en%e bet1een the $+o+i$te

"$,*e ,e$n".

i Re"o+t Me$n I. '?; µ = µ7 VI. '?; µ7 = µ@

-$de*o-e G '; µ ≠ µ7 '; µ7 ≠ µ@

7 M$+tini:-e B QG2BQ<@>.B QB2Q<6>.B

> E*e-th+$ B> II. '?; µ = µ> VII. '?; µ7 = µ

@ P$+$di"e I". '; µ ≠ µ> '; µ7 ≠ µ

St. L-%i$ G QG2B>Q<6>.B QB2GQ<?>.B

III. '?; µ = µ@ VIII. '?; µ> = µ@

he %+iti%$* oint ?.? )o+ '; µ ≠ µ@ '; µ> ≠ µ@

+< $nd (n2+< QG2Q<7>.B QB>2Q<G>.B

de#+ee" o) )+eedo, i"; IV. '?; µ = µ IT. '?; µ> = µ

'; µ ≠ µ '; µ> ≠ µ

QG2GQ<@>.B QB>2GQ<7>.B

V. '?; µ7 = µ> T. '?; µ@ = µ

'; µ7 ≠ µ> '; µ@ ≠ µ

QB2B>Q<7>.B Q2GQ< 6>.B

7$t t8 n.ll 8;'!t8sis i" t8 a%s!l.t #al. !" t8 &i""rn$ %tn t8 sam'l mansis ratr t8an t8 $riti$al #al. !" T2 (he hyothe"e" ,$+5ed 1ith $+e +e9e%ted.

# q $%' n

i

=

= =

α

>G6A?@ @

@?> B.

..

The Tu;ey 3air0ise Co!parison TestThe Tu;ey 3air0ise Co!parison Test

The Clu/ 7ed 56a!pleThe Clu/ 7ed 56a!ple

9-34



9-34

3e +e9e%ted the n-** hyothe"i" 1hi%h %o,$+ed the ,e$n" o) o-*$tion"

$nd 70 $nd >0 7 $nd @0 $nd > $nd @. On the othe+ h$nd0 1e $%%eted the

n-** hyothe"e" o) the e:-$*ity o) the ,e$n" o) o-*$tion" $nd @0 $nd 0

7 $nd >0 7 $nd 0 > $nd 0 $nd @ $nd .

he b$+" indi%$te the th+ee #+o-in#" o) o-*$tion" 1ith o""ib*y e:-$*

,e$n"; 7 $nd >F 70 >0 $nd F $nd 0 @0 $nd .

3e +e9e%ted the n-** hyothe"i" 1hi%h %o,$+ed the ,e$n" o) o-*$tion"

$nd 70 $nd >0 7 $nd @0 $nd > $nd @. On the othe+ h$nd0 1e $%%eted the

n-** hyothe"e" o) the e:-$*ity o) the ,e$n" o) o-*$tion" $nd @0 $nd 0

7 $nd >0 7 $nd 0 > $nd 0 $nd @ $nd .

he b$+" indi%$te the th+ee #+o-in#" o) o-*$tion" 1ith o""ib*y e:-$*

,e$n"; 7 $nd >F 70 >0 $nd F $nd 0 @0 $nd .

µµ7µ> µ@µ

3icturing the <esults of a Tu;ey 3air0ise3icturing the <esults of a Tu;ey 3air0ise

Co!parisons Test The Clu/ 7ed 56a!pleCo!parisons Test The Clu/ 7ed 56a!ple

9-35



9-35

3icturing the <esults of a Tu;ey 3air0ise3icturing the <esults of a Tu;ey 3air0ise

Co!parisons Test The Clu/ 7ed 56a!pleCo!parisons Test The Clu/ 7ed 56a!ple

9-36



9 36

! A statisti$al m!&l i" $ "et o) e:-$tion" $nd $""-,tion"

th$t %$t-+e the e""enti$* %h$+$%te+i"ti%" o) $ +e$*21o+*d

"it-$tion

he one2)$%to+ ANOVA ,ode*; xij µ i)ε ij µ )α i)ε ij

1he+e εi9 i" the e++o+ $""o%i$ted 1ith the jth ,e,be+ o)

the ith o-*$tion. he e++o+" $+e $""-,ed to be

no+,$**y di"t+ib-ted 1ith ,e$n ? $nd 8$+i$n%e σ7.

! A statisti$al m!&l i" $ "et o) e:-$tion" $nd $""-,tion"

th$t %$t-+e the e""enti$* %h$+$%te+i"ti%" o) $ +e$*21o+*d

"it-$tion

he one2)$%to+ ANOVA ,ode*; xij µ i)ε ij µ )α i)ε ij

1he+e εi9 i" the e++o+ $""o%i$ted 1ith the jth ,e,be+ o)

the ith o-*$tion. he e++o+" $+e $""-,ed to be

no+,$**y di"t+ib-ted 1ith ,e$n ? $nd 8$+i$n%e σ7.

9-, 7odels %actors and .esigns9-, 7odels %actors and .esigns

9-37



9 37

! A )$%to+ i" $ "et o) o-*$tion" o+ t+e$t,ent" o) $ "in#*e 5ind. /o+

e$,*e;

One )$%to+ ,ode*" b$"ed on "et" o) +e"o+t"0 tye" o) $i+*$ne"0 !r

5ind" o) "1e$te+"

1o )$%to+ ,ode*" b$"ed on )i+, an& *o%$tion

h+ee )$%to+ ,ode*" b$"ed on %o*o+ an& "h$e an& "ie o) $n $d.

! /ied2E))e%t" $nd R$ndo, E))e%t"

A "i/&-""$ts m!&l i" one in 1hi%h the *e8e*" o) the )$%to+ -nde+

"t-dy (the t+e$t,ent" $+e fixed in advance. In)e+en%e i" 8$*id on*y)o+ the *e8e*" -nde+ "t-dy.

A ran&!m-""$ts m!&l i" one in 1hi%h the *e8e*" o) the )$%to+

-nde+ "t-dy $+e +$ndo,*y %ho"en )+o, $n enti+e o-*$tion o) *e8e*"

(t+e$t,ent". In)e+en%e i" 8$*id )o+ the enti+e o-*$tion o) *e8e*".

! A )$%to+ i" $ "et o) o-*$tion" o+ t+e$t,ent" o) $ "in#*e 5ind. /o+

e$,*e;

One )$%to+ ,ode*" b$"ed on "et" o) +e"o+t"0 tye" o) $i+*$ne"0 !r

5ind" o) "1e$te+"

1o )$%to+ ,ode*" b$"ed on )i+, an& *o%$tion

h+ee )$%to+ ,ode*" b$"ed on %o*o+ an& "h$e an& "ie o) $n $d.

! /ied2E))e%t" $nd R$ndo, E))e%t"

A "i/&-""$ts m!&l i" one in 1hi%h the *e8e*" o) the )$%to+ -nde+

"t-dy (the t+e$t,ent" $+e fixed in advance. In)e+en%e i" 8$*id on*y)o+ the *e8e*" -nde+ "t-dy.

A ran&!m-""$ts m!&l i" one in 1hi%h the *e8e*" o) the )$%to+

-nde+ "t-dy $+e +$ndo,*y %ho"en )+o, $n enti+e o-*$tion o) *e8e*"

(t+e$t,ent". In)e+en%e i" 8$*id )o+ the enti+e o-*$tion o) *e8e*".

9-, 7odels %actors and .esigns9-, 7odels %actors and .esigns

(Continued)(Continued)

9-38



9 38

! A $!m'ltl;-ran&!mi& &sin i" one in 1hi%h the

e*e,ent" $+e assigned to treatments completely at random.

h$t i"0 $ny e*e,ent %ho"en )o+ the "t-dy h$" $n e:-$*

%h$n%e o) bein# $""i#ned to $ny t+e$t,ent.! In $ %l!$?in &sin0 e*e,ent" $+e $""i#ned to t+e$t,ent"

$)te+ )i+"t bein# %o**e%ted into ho,o#eneo-" #+o-". In $ $!m'ltl; ran&!mi& %l!$? &sin0 $** ,e,be+" o) e$%h

b*o%5 (ho,o#eneo-" #+o- $+e +$ndo,*y $""i#ned to the t+e$t,ent

*e8e*".

In $ r'at& mas.rs &sin0 e$%h ,e,be+ o) e$%h b*o%5 i"

$""i#ned to $** t+e$t,ent *e8e*".

! A $!m'ltl;-ran&!mi& &sin i" one in 1hi%h the

e*e,ent" $+e assigned to treatments completely at random.

h$t i"0 $ny e*e,ent %ho"en )o+ the "t-dy h$" $n e:-$*

%h$n%e o) bein# $""i#ned to $ny t+e$t,ent.! In $ %l!$?in &sin0 e*e,ent" $+e $""i#ned to t+e$t,ent"

$)te+ )i+"t bein# %o**e%ted into ho,o#eneo-" #+o-". In $ $!m'ltl; ran&!mi& %l!$? &sin0 $** ,e,be+" o) e$%h

b*o%5 (ho,o#eneo-" #+o- $+e +$ndo,*y $""i#ned to the t+e$t,ent*e8e*".



56peri!ental .esign56peri!ental .esign

9-39



9 39

! In $ t!-a; ANOVA0 the e))e%t" o) t1o )$%to+" o+ t+e$t,ent" %$n be in8e"ti#$ted

"i,-*t$neo-"*y. 1o21$y ANOVA $*"o e+,it" the in8e"ti#$tion o) the e))e%t" o)

eithe+ )$%to+ $*one $nd o) the two factors together .

he e))e%t on the o-*$tion ,e$n th$t %$n be $tt+ib-ted to the *e8e*" o) eithe+ )$%to+alone

i" %$**ed $ main ""$t.

An intra$ti!n ""$t bet1een t1o )$%to+" o%%-+" i) the tot$* e))e%t $t "o,e $i+ o) *e8e*" o)

the t1o )$%to+" o+ t+e$t,ent" di))e+" "i#ni)i%$nt*y )+o, the "i,*e $ddition o) the t1o ,$in

e))e%t". /$%to+" th$t do not inte+$%t $+e %$**ed additive.

! h+ee :-e"tion" $n"1e+$b*e by t1o21$y ANOVA; A+e the+e $ny factor * main effects A+e the+e $ny factor + main effects A+e the+e $ny interaction effects between factors * and +

! /o+ e$,*e0 1e ,i#ht in8e"ti#$te the e))e%t" on 8$%$tione+" +$tin#" o) +e"o+t" by

*oo5in# $t )i8e di))e+ent +e"o+t" ()$%to+ A $nd )o-+ di))e+ent +e"o+t $tt+ib-te" ()$%to+

4. In $ddition to the )i8e ,$in )$%to+ A t+e$t,ent *e8e*" $nd the )o-+ ,$in )$%to+ 4

t+e$t,ent *e8e*"0 the+e $+e (@<7? inte+$%tion t+e$t,ent *e8e*".>

! In $ t!-a; ANOVA0 the e))e%t" o) t1o )$%to+" o+ t+e$t,ent" %$n be in8e"ti#$ted

"i,-*t$neo-"*y. 1o21$y ANOVA $*"o e+,it" the in8e"ti#$tion o) the e))e%t" o)

eithe+ )$%to+ $*one $nd o) the two factors together .

he e))e%t on the o-*$tion ,e$n th$t %$n be $tt+ib-ted to the *e8e*" o) eithe+ )$%to+alone

i" %$**ed $ main ""$t.

An intra$ti!n ""$t bet1een t1o )$%to+" o%%-+" i) the tot$* e))e%t $t "o,e $i+ o) *e8e*" o)

the t1o )$%to+" o+ t+e$t,ent" di))e+" "i#ni)i%$nt*y )+o, the "i,*e $ddition o) the t1o ,$in

e))e%t". /$%to+" th$t do not inte+$%t $+e %$**ed additive.

! h+ee :-e"tion" $n"1e+$b*e by t1o21$y ANOVA; A+e the+e $ny factor * main effects

A+e the+e $ny factor + main effects A+e the+e $ny interaction effects between factors * and +

! /o+ e$,*e0 1e ,i#ht in8e"ti#$te the e))e%t" on 8$%$tione+" +$tin#" o) +e"o+t" by

*oo5in# $t )i8e di))e+ent +e"o+t" ()$%to+ A $nd )o-+ di))e+ent +e"o+t $tt+ib-te" ()$%to+

4. In $ddition to the )i8e ,$in )$%to+ A t+e$t,ent *e8e*" $nd the )o-+ ,$in )$%to+ 4

t+e$t,ent *e8e*"0 the+e $+e (@<7? inte+$%tion t+e$t,ent *e8e*".>

9-+ T0o-"ay Analysis of Variance9-+ T0o-"ay Analysis of Variance

9-40



! xij- µ )α i) β j ) ( αβ) ij ) ε ij-

1he+e µ i" the o8e+$** ,e$nF

αi i" the e))e%t o) *e8e* i(i!,...,a" o) )$%to+ AF

β 9 i" the e))e%t o) *e8e* j(j!,...,b" o) )$%to+ 4F

(αβ) 99 i" the inte+$%tion e))e%t o) *e8e*" i $nd jF

ε 995 i" the e++o+ $""o%i$ted 1ith the 5th d$t$ oint )+o,

*e8e* i o) )$%to+ A $nd *e8e* j o) )$%to+ 4. ε 995 i" $""-,ed to be di"t+ib-ted no+,$**y 1ith ,e$n e+o

$nd 8$+i$n%e σ7 )o+ $** i, j, $nd - .

! xij- µ )α i) β j ) ( αβ) ij ) ε ij-

1he+e µ i" the o8e+$** ,e$nF

αi i" the e))e%t o) *e8e* i(i!,...,a" o) )$%to+ AF β 9 i" the e))e%t o) *e8e* j(j!,...,b" o) )$%to+ 4F

(αβ) 99 i" the inte+$%tion e))e%t o) *e8e*" i $nd jF

ε 995 i" the e++o+ $""o%i$ted 1ith the 5th d$t$ oint )+o,

*e8e* i o) )$%to+ A $nd *e8e* j o) )$%to+ 4. ε 995 i" $""-,ed to be di"t+ib-ted no+,$**y 1ith ,e$n e+o

$nd 8$+i$n%e σ7 )o+ $** i, j, $nd - .

The T0o-"ay A#VA 7odelThe T0o-"ay A#VA 7odel

9-41



4.a&l!.' Martini. El.t8ra

Para&is

Islan& St2 L.$ia

Frin&s8i' n11 n31 n+1 n)1 n*1

S'!rts n13 n33 n+3 n)3 n*3

C.lt.r n1+ n3+ n++ n)+ n*+

E/$itmnt n1) n3) n+) n)) n*)

/$%to+ A; Re"o+t

/ $ % t o + 4 ;

A t t + i b - t e

<esort

< a t i n g

=raphical .isplay of 5ffects

5leuthra

7artini8ue

St* >ucia

=uadeloupe

3aradise island

%riendship

56cite!entSportsCulture

E*e-th+$K"o+t" inte+$%tion;Co,bined e))e%t #+e$te+ th$n$dditi8e ,$in e))e%t"

Sports

%riendship

Attri/ute

<esort

56cite!ent

Culture

<ating

5leuthra

7artini8ue

St* >ucia

=uadeloupe

3aradise $sland

T0o-"ay A#VA .ata >ayoutT0o-"ay A#VA .ata >ayout

Clu/ 7ed 56a!pleClu/ 7ed 56a!ple

9-42



! Factor A main effects test Factor A main effects test '?; αi< ? )o+ $** i!,2,...,a

'; Not $** αi $+e ?

! Factor B main effects test:Factor B main effects test:'?; β 9< ? )o+ $** j!,2,...,b

'; Not $** βi $+e ?

! Test for AB! interactions:Test for AB! interactions:'?; (αβ)i9< ? )o+ $** i!,2,...,a $nd j!,2,...,b

'; Not $** (αβ)i9 $+e ?

! Factor A main effects test Factor A main effects test '?; αi< ? )o+ $** i!,2,...,a

'; Not $** αi $+e ?

! Factor B main effects test:Factor B main effects test:'?; β 9< ? )o+ $** j!,2,...,b

'; Not $** βi $+e ?

! Test for AB! interactions:Test for AB! interactions:'?; (αβ)i9< ? )o+ $** i!,2,...,a $nd j!,2,...,b

'; Not $** (αβ)i9 $+e ?

Hypothesis Tests a T0o-"ay A#VAHypothesis Tests a T0o-"ay A#VA

9-43



In $ t1o21$y ANOVA;

i95 <µ=αi= β 9 = (αβ)i95 = εi95

SS < SSR =SSE SS < SSA = SS4 =SS(A4=SSE

In $ t1o21$y ANOVA;

i95 <µ=αi= β 9 = (αβ)i95 = εi95

SS < SSR =SSE SS < SSA = SS4 =SS(A4=SSE

%%# %%#& %%'

x x x x x x

%%#& %%* %%+ %% *+

xi

x x j

x xij

xi

x j

x

= +

−∑∑∑ = −∑∑∑ + −∑∑∑

= + +

= − + −∑∑∑∑∑∑ + + + −∑∑∑

( ( (

(

( ( (

7 7 7

7 7 7

Su!s of S8uaresSu!s of S8uares

9-44



S!.r$ !"

Variati!n

S.m !"

S.ars

6rs

!" Fr&!m Man S.ar F 7ati!

/$%to+ A SSA $2 $%*

%%*

a=

− F

$%*

$%' =

/$%to+ 4 SS4 b2 $%+ %%+b

=−

F $%+ $%'

=

Inte+$%tion SS(A4 ($2(b2 $% *+

%% *+

a b(

(

( ( =

− − F

$% *+

$%' =

(

E++o+ SSE $b(n2 $%'

%%'

ab n=

−(

ot$* SS $bn2

A M$in E))e%t e"t; /($20$b(n2 4 M$in E))e%t e"t; /(b20$b(n2

(A4 Inte+$%tion E))e%t e"t; /(($2(b20$b(n2

The T0o-"ay A#VA Ta/leThe T0o-"ay A#VA Ta/le

9-45



S!.r$ !"

Variati!n

S.m !"

S.ars

6rs


Lo%$tion G7@ 7 7 G.@

A+ti"t 77>? 7 ?.>

Inte+$%tion G?@ @ 7? .B

E++o+ G767 G ?7

ot$* >7? G

α <?.?0 /(70G<@.GG ⇒ 4oth ,$in e))e%t n-** hyothe"e" $+e +e9e%ted.

α<?.?0 /(70G<7.@G ⇒ Inte+$%tion e))e%t n-** hyothe"e" $+e not +e9e%ted.α <?.?0 /(70G<@.GG ⇒ 4oth ,$in e))e%t n-** hyothe"e" $+e +e9e%ted.

α<?.?0 /(70G<7.@G ⇒ Inte+$%tion e))e%t n-** hyothe"e" $+e not +e9e%ted.

56a!ple 9-' T0o-"ay A#VA56a!ple 9-' T0o-"ay A#VA

(>ocation and Artist)(>ocation and Artist)

9-46



,&'21

*+

*,

*&

*'

*

*2

*1

*%

f ( % )

% .istri/ution 0ith 2 and 1 .egrees of %reedo!

%*1'*

α*1

Lo%$tion te"t "t$ti"ti%<G.@

A+ti"t te"t "t$ti"ti%<?.>

,&'21

*+

*,

*&

*'

*

*2

*1

* %

f ( % )

% .istri/ution 0ith ' and 1 .egrees of %reedo!

Inte+$%tion te"t "t$ti"ti%<.B

α*&

%*&2*'

Hypothesis TestsHypothesis Tests

9-47



im%alls In.alit; #i8e" $n -e+ *i,it on the t+-e +ob$bi*ity o) $t *e$"t

one ye I e++o+ in the th+ee te"t" o) $ t1o21$y $n$*y"i";

α ≤ 2 (2α (2α7 (2α>

im%alls In.alit; #i8e" $n -e+ *i,it on the t+-e +ob$bi*ity o) $t *e$"t

one ye I e++o+ in the th+ee te"t" o) $ t1o21$y $n$*y"i";

α ≤ 2 (2α (2α7 (2α>

T.?; Critri!n )o+ )$%to+ A;

1he+e the de#+ee" o) )+eedo, o) the q di"t+ib-tion $+e no1 a $nd ab(n!". Note th$t MSE i" di8ided by bn.

T.?; Critri!n )o+ )$%to+ A;

1he+e the de#+ee" o) )+eedo, o) the q di"t+ib-tion $+e no1 a $nd ab(n!". Note th$t MSE i" di8ided by bn.

# q $%'

bn=

α

4erall Significance >e4el and Tu;ey4erall Significance >e4el and Tu;ey

7ethod for T0o-"ay A#VA7ethod for T0o-"ay A#VA

9-48



Te!plate for a T0o-"ay A#VATe!plate for a T0o-"ay A#VA

9-49



S!.r$ !"

Variati!n

S.m !"

S.ars

6rs


/$%to+ A SSA $2 $%*

%%*

a=

− F $%*

$%' =

/$%to+ 4 SS4 b2 $%+

%%+

b=

− F

$%+

$%' =

/$%to+ C SSC %2 $%/

%%/

c=

− F

$%/

$%' =

Inte+$%tion

(A4

SS(A4 ($2(b2 $% *+

%% *+

a b(

(

( ( =

− − F

$% *+

$%' =

(

Inte+$%tion

(AC

SS(AC ($2(%2 $% */

%% */

a c(

(

( ( =

− − F

$% */

$%' =

(

Inte+$%tion

(4C

SS(4C (b2(%2

$% +/

%% +/

b c(

(

( ( = − − F

$% +/

$%' =

(

Inte+$%tion

(A4C

SS(A4C ($2(b2(%2 $% *+/

%% *+/

a b c(

(

( ( ( =

− − − F

$% *+/

$%' =

(

E++o+ SSE $b%(n2 $%'

%%'

abc n=

−(

ot$* SS $b%n2

56tension of A#VA to Three %actors56tension of A#VA to Three %actors

9-50



! he %$"e o) one d$t$ oint in e8e+y %e** +e"ent" $

+ob*e, in t1o21$y ANOVA.

! he+e 1i** be no de#+ee" o) )+eedo, )o+ the e++o+ te+,.

! 3h$t %$n be done! I) 1e %$n $""-,e th$t the+e $+e no inte+$%tion" bet1een

the ,$in e))e%t"0 then 1e %$n -"e SS(A4 $nd it"

$""o%i$ted de#+ee" o) )+eedo, ($ (b in *$%e o)

SSE $nd it" de#+ee" o) )+eedo,.

! 3e %$n then %ond-%t ,$in e))e%t" te"t" -"in# MS(A4.

! See the net "*ide )o+ the ANOVA t$b*e.

T0o-"ay A#VA 0ith neT0o-"ay A#VA 0ith ne

/ser4ation per Cell/ser4ation per Cell

9-51

T " A#VA i h



T0o-"ay A#VA 0ith neT0o-"ay A#VA 0ith ne

/ser4ation per Cell/ser4ation per Cell

S!.r$ !"Variati!n

S.m !"S.ars

6rs !"Fr&!m

Man S.ar F 7ati!

/$%to+ A SSA $ 2

/$%to+ 4 SS4 b 2

HE++o+ SS(A4 ($ (b

ot$* SS $b 2

−= a%%* $%*

6−=

b%%+ $%+

(( (( −−= ba *+%% *+ $%

( *+ $% $%* F =

( *+ $% $%+ F =

9-52



! A %l!$? i" $ ho,o#eneo-" "et o) "-b9e%t"0 #+o-ed to

,ini,ie 1ithin2#+o- di))e+en%e".

! A $!m'tl;-ran&!mi& &sin i" one in 1hi%h the

e*e,ent" $+e assigned to treatments completely at

random. h$t i"0 $ny e*e,ent %ho"en )o+ the "t-dy h$" $n

e:-$* %h$n%e o) bein# $""i#ned to $ny t+e$t,ent.

! In $ %l!$?in &sin0 e*e,ent" $+e $""i#ned to t+e$t,ent"

$)te+ )i+"t bein# %o**e%ted into ho,o#eneo-" #+o-".

In $ $!m'ltl; ran&!mi& %l!$? &sin0 $** ,e,be+" o) e$%h

b*o%5 (ho,o#eno-" #+o- $+e +$ndo,*y $""i#ned to the

t+e$t,ent *e8e*".



9- ?loc;ing .esigns9- ?loc;ing .esigns

9-53

7 d l f < d i d C l7 d l f < d i d C l t



! xij µ )α i) β j ) ε ij1he+e µ i" the o8e+$** ,e$nF

αi i" the e))e%t o) *e8e* i(i!,...,a" o) )$%to+ AF β 9 i" the e))e%t o) b*o%5 j(j!,...,b"F

εi9 i" the e++o+ $""o%i$ted 1ith xij

εi9 i" $""-,ed to be di"t+ib-ted no+,$**y 1ith

,e$n e+o $nd 8$+i$n%e σ7 )o+ $** i $nd j.

! xij µ )α i) β j ) ε ij1he+e µ i" the o8e+$** ,e$nF

αi i" the e))e%t o) *e8e* i(i!,...,a" o) )$%to+ AF

β 9 i" the e))e%t o) b*o%5 j(j!,...,b"F

εi9 i" the e++o+ $""o%i$ted 1ith xij

εi9 i" $""-,ed to be di"t+ib-ted no+,$**y 1ith

,e$n e+o $nd 8$+i$n%e σ7 )o+ $** i $nd j.

7odel for <ando!i@ed Co!plete7odel for <ando!i@ed Co!plete

?loc; .esign?loc; .esign

9-54

A#VA T /l f ?l ;i . iA#VA T /l f ?l ;i . i



A#VA Ta/le for ?loc;ing .esigns A#VA Ta/le for ?loc;ing .esigns

56a!ple 9-&56a!ple 9-&

Soure of !ariation Sum of S"uares df #ean S"uare F $atio

%&o's 2750 39 70(51 0(69

)reatments 2640 2 1320 12(93

*rror 7960 78 102(05)ota& 13350 119

α *1 %(2 +) '*

Soure of !ariation Sum of S"uares Degress of Freedom #ean S"uare F $atio

%&o's SS?> n - 1 7S?> SS?>(n-1) % 7S?>7S5

)reatments SST< r - 1 7ST< SST<(r-1) % 7ST<7S5

*rror SS5 (n -1)(r - 1)

)ota& SST nr - 1

7S5 SS5(n-1)(r-1)

9-55

T l t f th < d i dT l t f th < d i d



Te!plate for the <ando!i@edTe!plate for the <ando!i@ed

Co!plete ?loc; .esignCo!plete ?loc; .esign

Documents

Chap009 Anova.ppt