Matakuliah
Tahun
Versi
: I0214 / Statistika Multivariat
: 2005
: V1 / R1
Pertemuan 15
Analisis Ragam Peubah Ganda
(MANOVA III)
1
Learning Outcomes
Pada akhir pertemuan ini, diharapkan
mahasiswa akan mampu :
• Mahasiswa dapat menerangkan konsep
dasar analisis ragam peubah ganda
(manova)  C2
• Mahasiswa dapat menghitung manova
satu klasifikasi  C3
• Mahasiswa dapat melakukan uji Fisher
dan uji Bartlette  C3
2
Outline Materi
• Konsep dasar analisis ragam peubah
ganda (manova)
• Analisis ragam peubah ganda satu
klasifikasi
• Uji Fisher
• Uji Bartlette
3
<<ISI>>
Null Hypothesis
Univariate t-test:
H0 : 1 = 2
(population means are equal)
Multivariate case (2-group MANOVA):
 11   12 

 



 21   22 

H0 :      

 

  p1    p 2 

 

Main
assumptions:
(population mean vectors are equal)
normally
matrices across groups
distributed
DVs,
equal
covariance
<<ISI>>
Test Statistic for 2-group MANOVA
n1n 2
1

(
y

y
)
S
( y1  y 2 )
1
2
Hotelling’s T : T =
n1  n 2
2
2
n1 : sample size in first group
n2 : sample size in second group
y1
: vector of means of DVs in first group
y2
: vector of means of DVs in second group
S : pooled within-group covariance matrix
<<ISI>>
Hotelling’s T2 measures the between-group difference (y1  y 2 ) , which
is weighted by the within-group covariance matrix S-1. The test works
as follows: From Hotellings T2, form
n1  n 2  p  1 2
F = (n  n  2)p T
1
2
F is the test statistic for testing whether there is a significant group
difference with respect to the whole vector y of dependent variables. Fdistributed with p and (n1 + n2 -p - 1) degress of freedom
6
<<ISI>>
Tests of Significance
Wilks' Lambda
where Se represents the error SSCP matrix and Sh represents the
hypothesis SSCP matrix.
For Example
In a fixed effects model, Sw is the Se for all effects.
While in the randoms effects model Sab is the Se for the main effects and
Sw for the interaction.
If A is fixed and B is random th Sab is the Se for A main effect and Sw is the
Se for the B main effect and the interaction
7
<<ISI>>
Rao's F Approximation
degrees of Freedom
Special Note Concerning s
If either the numerator or the deminator of s = 0 set s = 1
8
<<ISI>>
Hotelling's Trace Criterion
Roy's Largest Latent Root
Pillai's Trace Criterion
9
<<ISI>>
Which of these is "best?“
Schatzoff (1966)
Roy's largest-latent root was the most sensitive when population
centroids differed along a single dimension, but was otherwise least
sensative.
Under most conditions it was a toss-up between Wilks' and Hotelling's
criteria.
Olson (1976)
Pillai's criteria was the most robust to violations of assumptions
concerning homogeneity of the covariance matrix.
Under diffuse noncentrality the ordering was Pillai, Wilks, Hotelling and
Roy.
Under concentrated noncentrality the ordering is Roy, Hotelling, Wilks
and Pillai.
Final "Best"
When sample sizes are very large the Wilks, Hotelling and Pillai become
asymptotically equivalent
10
<<ISI>>
11
<<ISI>>
12
<<ISI>>
13
Tabel Manova
Sumber Variasi
Perlakuan
Matriks Jumlah Kuadrat
dan Hasil Kali Silang
g
A

Derajat Bebas
g 1
n1  xl  x  xl  x 
l 1
Residual
g
D
g
nl
 nl  g

x

x
x

x



 lj l lj l
l 1
l 1 j 1
Total
(terkoreksi)
g
A D 
nl
 
l 1 j 1


xlj  x xlj  x 
g
 nl 1
l 1
14
Uji hipotesa
H0 : 1  2 
  g  0 menyangkut generalized variance.
H 0 ditolak bila generalized variance
D
 
kecil
A D

(  ditemukan oleh Wilks).
Distribusi yang eksak untuk
 diberikan dalam tabel
15
Tabel Distribusi Wilks Lamda
Jumlah
Variabel
p 1

Jumlah
Grup
Distribusi sampling data multivariat
g2
 nl  g

 g 1
  1  *
 
*
 



Fg 1,nl  g (  )
p2
g2
 nl  g  1   1  *

 
g 1


*
p 1
g2
 nl  p  1   1  *

 
*
p

 
p 1
g 3
 nl  p  2   1  *

 
p


*







F2( g 1),2(  nl  g 1)   
Fp ,nl  p 1   




 nl  p  2    
F2 p ,2
16
Bila H 0 benar dan nl  n besar:

 p  g 
 p  g   D
* 
   n  1  
  ln     n  1  
  ln 
 2 
 2   A  D





berdistribusi mendekati Khi – kuadrat dengan derajat bebas p  g  1 .
Jadi, untuk nl besar, H 0 ditolak pada tingkat signifikansi bila:

 p  g   D
   n  1  
  ln 
 2   A  D


2
   p ( g 1) ()

17
Jumlah Jumlah
Variabel Grup
Daerah penolakan H0
p 1
g2
 n1  g   1  * 

  *  Fg 1,ne  g (  )
 g  1    
p2
g2
 n1  g  1   1  *

 
 g  1   *
p 1
g2
 n1  p  1   1  * 

  *  Fp, n  p 1    
p
l

  
p 1
g 3
 n1  p  2   1  *

 
p

  *

 F

 2( g 1),2 nl  g 1 


 F2 p ,2 n  p 2    
l


Untuk nl besar.
H 0 ditolak dengan tingkat signifikansi  bila
pg 

*
2
  n 1
ln



p ( g 1) ( )

2 

18
<< CLOSING>>
• Sampai dengan saat ini Anda telah
mempelajari kosep dasar analisis ragam
peubah ganda, dan manova satu
klasifikasi
• Untuk dapat lebih memahami konsep
dasar analisis ragam peubah ganda dan
manova satu klasifikasi tersebut, cobalah
Anda pelajari materi penunjang,
website/internet dan mengerjakan latihan
19