Pertemuan 15

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Pertemuan 15

Matakuliah : I0214 / Statistika MultivariatTahun : 2005Versi : V1 / R1

Analisis Ragam Peubah Ganda(MANOVA III)

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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

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Outline Materi

• Konsep dasar analisis ragam peubah ganda (manova)

• Analisis ragam peubah ganda satu klasifikasi

• Uji Fisher

• Uji Bartlette

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Null Hypothesis

Univariate t-test:

H0 : 1 = 2 (population means are equal)

Multivariate case (2-group MANOVA):

H0 :

2p

22

12

1p

21

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(population mean vectors are equal)

Main assumptions: normally distributed DVs, equal covariance

matrices across groups

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Test Statistic for 2-group MANOVA

Hotelling’s T2 : T2 = )yy()yy(nn

nn21

121

21

21

S

n1 : sample size in first group

n2 : sample size in second group

1y : vector of means of DVs in first group

2y : vector of means of DVs in second group

S : pooled within-group covariance matrix

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Hotelling’s T2 measures the between-group difference )yy( 21 , which

is weighted by the within-group covariance matrix S-1. The test works

as follows: From Hotellings T2, form

F = 2

21

21 Tp)2nn(

1pnn

F is the test statistic for testing whether there is a significant group

difference with respect to the whole vector y of dependent variables. F-

distributed with p and (n1 + n2 -p - 1) degress of freedom

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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

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Rao's F Approximation

degrees of Freedom

                            Special Note Concerning s

If either the numerator or the deminator of s = 0 set s = 1

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Hotelling's Trace Criterion

                               Roy's Largest Latent Root

                        Pillai's Trace Criterion

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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

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Tabel Manova

Sumber Variasi Matriks Jumlah Kuadrat dan Hasil Kali Silang

Derajat Bebas

Perlakuan 1

1

g

l ll

A n x x x x

1g

Residual

1 1

lng

lj l lj ll j

D x x x x

1

g

ll

n g

Total (terkoreksi)

1 1

lng

lj ljl j

A D x x x x

1

1g

ll

n

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Uji hipotesa

0 1 2: 0gH menyangkut generalized variance.

0H ditolak bila generalized variance

D

A D

kecil

( ditemukan oleh Wilks).

Distribusi yang eksak untuk diberikan dalam tabel

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Tabel Distribusi Wilks Lamda

Jumlah Variabel

Jumlah Grup

Distribusi sampling data multivariat

1p

2g

*

1, ( )*

1

1 l g

lg n

n gF

g

2p 2g

*

2( 1),2( 1)*

1 1

1 l

lg n g

n gF

g

1p 2g

1

*

,*

1 1l p

lp n

n pF

p

1p 3g

2

*

2 ,2*

2 1l p

lp n

n pF

p

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Bila 0H benar dan ln n besar:

*1 ln 1 ln2 2

Dp g p gn n

A D

berdistribusi mendekati Khi – kuadrat dengan derajat bebas 1p g .

Jadi, untuk ln besar, 0H ditolak pada tingkat signifikansi bila:

2( 1)

1 ln ( )2 p g

Dp gn

A D

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Jumlah Variabel

Jumlah Grup

Daerah penolakan H0

1p

2g

*

11, ( )*

1

1 n geg

n gF

g

2p 2g

1

*1

2( 1),2*

1 1

1 n glg

n gF

g

1p 2g

1

*1

* ,

1 1n pl

p

n pF

p

1p 3g

*

12 ,2 2*

2 1lp n p

n pF

p

Untuk ln besar.

0H ditolak dengan tingkat signifikansi bila

* 2( 1)1 ln ( )

2 p gp g

n

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<< 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