TWO-WAY ANOVA

To test the significant effect of two independent

variables to one dependent variable, and to test the

significant interaction of the two independent

variables to the dependent variable.

The two independent variables are called “row”

factors (A factor) and “column” factor (B factor).

Testing the mean of rows, testing the means of

column, and testing the mean of cells in the same row

or the same column.

The hypothesis test on 2 x 2 factorial design

eperiment

LEARNING

METHOD A

LEARNING

METHOD B

SMALL CLASS

BIG CLASS

The hypothesis test on 2 x 2 factorial design

eperiment

LEARNING

METHOD A

LEARNING

METHOD B

SMALL CLASS

BIG CLASS

The hypothesis test on 2 x 2 factorial design

eperiment

LEARNING

METHOD A

LEARNING

METHOD B

SMALL CLASS

BIG CLASS

The hypothesis test on 2 x 2 factorial design

eperiment

LEARNING

METHOD A

LEARNING

METHOD B

SMALL CLASS

BIG CLASS

The hypothesis test on 2 x 2 factorial design

eperiment

LEARNING

METHOD A

LEARNING

METHOD B

SMALL CLASS

BIG CLASS

The hypothesis test on 2 x 2 factorial design

eperiment

LEARNING

METHOD A

LEARNING

METHOD B

SMALL CLASS

BIG CLASS

The hypothesis test on 2 x 2 factorial design

eperiment

LEARNING

METHOD A

LEARNING

METHOD B

SMALL CLASS

BIG CLASS

NOTASI DAN TATA LETAK DATA

1. Komponen komputasi a. Jumlah Kuadrat (JK) atau sum of square (SS)

1) Anava dua jalan sel sama Tabel jumlah AB

Faktor A

Faktor B Total

b1 b2 ... bq

a1 AB11 AB12 ... AB1q A1

a2 AB21 AB22 ... AB2q A2

... ... ... ... ... ...

ap ABp1 ABp2 ... ABpq Ap

Total B1 B2 ... Bq G

Untuk memudahkan perhitungan pada anava dua jalan sel sama, didefinisikan besaran-besaran (1), (2), (3), (4), dan (5) sebagai berikut:

N

G 2

1 j

j

np

B2

4

kji

ijkX,,

22 ji

ij

n

AB

,

2

5

i

i

nq

A2

3

)1()3( JKA )1()4( JKB

)4()3()5()1( JKAB )5(2 JKG

JKGJKABJKBJKAJKTJKT atau )1()2(

1) Anava dua jalan sel tak sama Tabel rerata dan jumlah rerata

Faktor A

Faktor B Total

b1 b2 ... bq

a1 11AB 12AB ... qAB1 A1

a2 21AB 22AB ... qAB 2 A2

... ... ... ... ... ...

ap 1pAB 2pAB ...

pqAB Ap

Total B1 B2 ... Bq G

Pada analisis variansi dua jalan dengan sel tak sama, didefinisikan notasi-notasi sebagai berikut:

ji

ijnN,

banyaknya seluruh data amatan

ijn banyaknya data amatan pada sel ij

hn rerata harmonik frekuensi seluruh sel =

ji ijn

pq

,

1

ij

k

ijk

k

ijkijn

X

XSS

2

2

= jumlah kuadrat deviasi data amatan pada sel ij

ijAB rerata pada sel ij

i

iji ABA jumlah rerata pada baris ke-i

j

ijj ABB jumlah rerata pada kolom ke-j

ij

ijABG jumlah rerata semua sel

Untuk memudahkan perhitungan, didefinisikan besaran-besaran (1), (2), (3), (4), dan (5) sebagai berikut:

pq

G 2

1

ij

ijSS2

i

i

q

A2

3

j

j

p

B2

4

ij

ijAB2

5

13 hnJKA

14 hnJKB

4351 hnJKAB 2JKG

JKGJKABJKBJKAJKT

Derajat kebebasan atau degrees of freedom (df) dkA = 𝑝 − 1 dkB = 𝑞 − 1 dkAB = 𝑝 − 1 (𝑞 − 1) dkG = 𝑁 − 𝑝𝑞 dkT = 𝑁 − 1

Rerata kuadrat atau mean square (MS)

𝑅𝐾𝐴 =𝐽𝐾𝐴

𝑑𝑘𝐴

𝑅𝐾𝐵 =𝐽𝐾𝐵

𝑑𝑘𝐵

𝑅𝐾𝐴𝐵 =𝐽𝐾𝐴𝐵

𝑑𝑘𝐴𝐵

𝑅𝐾𝐺 =𝐽𝐾𝐺

𝑑𝑘𝐺

Statistik uji

1) Untuk H0A adalah 𝐹𝑎 =𝑅𝐾𝐴

𝑅𝐾𝐺

2) Untuk H0B adalah 𝐹𝑏 =𝑅𝐾𝐵

𝑅𝐾𝐺

3) Untuk H0AB adalah 𝐹𝑎𝑏 =𝑅𝐾𝐴𝐵

𝑅𝐾𝐺

Daerah kritik 1) Daerah kritik untuk Fa adalah DKa = {𝐹|𝐹 > 𝐹𝛼 ;𝑝−1,𝑁−𝑝𝑞 }

2) Daerah kritik untuk Fb adalah DKb = {𝐹|𝐹 > 𝐹𝛼 ;𝑞−1,𝑁−𝑝𝑞 }

3) Daerah kritik untuk Fab adalah DKab = {𝐹|𝐹 > 𝐹𝛼 ; 𝑝−1 (𝑞−1),𝑁−𝑝𝑞 }

Keputusan uji 1) H0A ditolak apabila Fa ∈ DKa 2) H0B ditolak apabila Fb ∈ 𝐷𝐾𝑏 3) H0AB ditolak apabila Fab ∈ DKab

Source of variance

Sum of squares

Degrees of Freedom

Mean Square Test Statistics

Rows SSR p-1 𝑀𝑆𝑅 = 𝑆𝑆𝑅 𝑝 − 1 𝐹𝑎 = 𝑀𝑆𝑅 𝑀𝑆𝐸

Columns SSC q-1 𝑀𝑆𝐶 = 𝑆𝑆𝐶 𝑞 − 1 𝐹𝑏 = 𝑀𝑆𝐶 𝑀𝑆𝐸

Treatments SS(Tr) (p-1)(q-1) 𝑀𝑆(𝑇𝑟) = 𝑆𝑆(𝑇𝑟) 𝑝 − 1 (𝑞 − 1) 𝐹𝑎𝑏 = 𝑀𝑆(𝑇𝑟) 𝑀𝑆𝐸

Error SSE N-pq 𝑀𝑆𝐸 = 𝑆𝑆𝐸 𝑁 − 𝑝𝑞 -

Total SST N-1 - -

Rangkuman Analisis

EXAMPLE

A study was conducted to examine the effect of eating high protein (level I, level II, and level III) breakfast on adolescents' performance during a physical education physical fitness test. Boys and Girls were given a fitness test with high scores representing better performance. The test scores are recorded below.

Are there any significant main effects or an interaction effect ?

(use 5% level of significance)

Level I Level II Level III

Boys 4 4 2 4 7 4 6 5 2 3 3 1

Girls 9 8 6 6 9 8 8 9 6 5 5 2

SOLUTION

1. Hypotheses:

H0A : there is no difference in the effect of gender on the fitness score.

H1A : there is difference in the effect of gender on the fitness score.

H0B : there is no difference in the effect of protein level on the fitness score.

H1B : there is difference in the effect of protein level on the fitness score.

H0AB : there is no interaction between gender and protein level of the fitness

score.

H1AB : there is interaction between gender and protein level of the fitness score.

2. Level of significance = 5%

SOLUTION

3. Computation

Source of

variance

Sum of

square

Degrees of

freedom

Mean

squaresFtest Ftable Decision Conclusion

Gender 54 1 54 35,345 4,41 H0A is rejectedthere is difference in the effect

of gender on the fitness score

Protein level 52,75 2 26,375 17,264 3,55 H0B is rejected

there is difference in the effect

of protein level on the fitness

score

Treatments

/Interacton2,25 2 1,125 0,736 3,55

H0AB is

accepted

there is interaction between

gender and protein level of

the fitness score

Error 27,5 18 1,528 - - - -

Total 136,5 23 - - - - -

Level I Level II Level III Marginalmeans

Boys 3,5 5,5 2,25 3,75

Girls 7,25 8,5 4,5 6,75

Marginalmeans

5,375 7 3,375

HOMEWORK

No 12 on page , Statistika Untuk Penelitian

No 15.34 on page 386, Modern Elementary Statistics

Number 12

Ada perbedaan efektivitas ukuran kelas terhadap prestasi, dilihat dari rerata

marginalnya, kelas kecil lebih efektif daripada kelas besar.

Ada perbedaan efektivitas metode pembelajarn terhadap prestasi, dilihat dari

rerata marginalnya, metode diskusi lebih efektif daripada metode ceramah.

Ada interaksi antara ukuran kelas dan metode pembelajaran terhadap prestasi

Pada kelas kecil, metode diskusi sama efektifnya dengan metode ceramah,

sedangkan pada kelas besar, metode diskusi lebih efektif daripada metode

ceramah.

Dengan metode diskusi, pembelajaran di kelas kecil sama efektifnya dengan

pembelajaran di kelas besar, sedangkan dengan metode ceramah,

pembelajaran di kelas kecil lebih efektif daripada pembelajaran di kelas besar.

Problem

A study was conducted to know the effect of antibiotic dose (20 grams, 30 grams,

and 40 grams) which are given to patients (children, adult, and elderly) on the

recovery time. This study was conducted with 36 people as the sample who spreads

into the same number of each cell (4 people for each). Assuming that the pre-

conditions of analysis are all satisfied and the SSR = 158, 389; SSC = 1071, 056; SS(Tr) =

847, 444; and SST = 3589,889. Perform a two-way analysis of variance using the 5% of

level of significance.

Children Adult Elderly

20 grams 31,75 34 36,5

30 grams 29,75 34,25 34,75

40 grams 21 42,5 50