Two-way Analysis of Variance



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

The hypothesis test on 2 x 2 factorial design

eperiment

LEARNING

METHOD A

LEARNING

METHOD B

SMALL CLASS

The hypothesis test on

2 x 2 factorial design

eperiment

LEARNING

METHOD A

LEARNING

METHOD B

SMALL CLASS

The hypothesis test on 2 x 2 factorial design

eperiment

LEARNING

METHOD A

LEARNING

METHOD B

SMALL CLASS

The hypothesis test on 2 x 2 factorial design

eperiment

LEARNING

METHOD A

LEARNING

METHOD B

SMALL CLASS

The hypothesis test on 2 x 2 factorial design

eperiment

LEARNING

METHOD A

LEARNING

METHOD B

SMALL CLASS

The hypothesis test on 2 x 2 factorial design

eperiment

LEARNING

METHOD A

LEARNING

METHOD B

SMALL CLASS

The hypothesis test on 2 x 2 factorial design

eperiment

LEARNING

METHOD A

LEARNING

METHOD B

SMALL CLASS

1. Komponen komputasi

a. Jumlah Kuadrat (JK) atau sum of square (SS) 1) Anava dua jalan sel sama

Tabel jumlah AB Faktor

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

1) Anava dua jalan sel tak sama

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

 

banyaknya seluruh data amatan



ij

n banyaknya data amatan pada sel ij



h

n rerata harmonik frekuensi seluruh sel =



= jumlah kuadrat deviasi data amatan pada sel ij



A jumlah rerata pada baris ke-i





j ij j AB

B jumlah rerata pada kolom ke-j





ij ij

AB

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

 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



Statistik uji

1)

Untuk

H

_0A

adalah

𝐹

_𝑎

=

𝑅𝐾𝐴

𝑅𝐾𝐺

2)

Untuk

H

_0B

adalah

𝐹

_𝑏

=

𝑅𝐾𝐵

𝑅𝐾𝐺

3)

Untuk

H

_0AB

adalah

𝐹

_𝑎𝑏

=

𝑅𝐾𝐴𝐵

𝑅𝐾𝐺



Daerah kritik

1)

Daerah kritik untuk

F

_a

adalah

DK

_a

= {

𝐹

|

𝐹

>

𝐹

_{𝛼;𝑝−1,𝑁−𝑝𝑞}

}

2)

Daerah kritik untuk

F

_b

adalah

DK

_b

= {

𝐹

|

𝐹

>

𝐹

_{𝛼;𝑞−1,𝑁−𝑝𝑞}

}

3)

Daerah kritik untuk

F

_ab

adalah

DK

_ab

= {

𝐹

|

𝐹

>

𝐹

_{𝛼; 𝑝−1 (𝑞−1),𝑁−𝑝𝑞}

}



Keputusan uji

1)

H

_0A

ditolak apabila

F

_a

∈

DK

_a

Source of

variance

Sum of

squares

Degrees of

Freedom

Mean Square

Test Statistics

Rows

SSR

p-1

𝑀𝑆𝑅

₌

𝑆𝑆𝑅 𝑝 −

₁

𝐹

_𝑎

₌

𝑀𝑆𝑅 𝑀𝑆𝐸

Columns

SSC

q-1

𝑀𝑆𝐶

=

𝑆𝑆𝐶 𝑞 −

1

𝐹

_𝑏

=

𝑀𝑆𝐶 𝑀𝑆𝐸

Treatments

SS(Tr)

(p-1)(q-1)

𝑀𝑆

(

𝑇𝑟

) =

𝑆𝑆

(

𝑇𝑟

)

𝑝 −

1

(

𝑞 −

1)

𝐹

_𝑎𝑏

=

𝑀𝑆

(

𝑇𝑟

)

𝑀𝑆𝐸

Error

SSE

N-pq

𝑀𝑆𝐸

=

𝑆𝑆𝐸 𝑁 − 𝑝𝑞

-

Total

SST

N-1

-

-

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

SOLUTION

1. Hypotheses:

 H_0A : there is no difference in the effect of gender on the fitness score.

H_1A : there is difference in the effect of gender on the fitness score.

 H_0B : there is no difference in the effect of protein level on the fitness score.

H_1B : there is difference in the effect of protein level on the fitness score.

 H_0AB : there is no interaction between gender and protein level of the fitness

score.

H_1AB : there is interaction between gender and protein level of the fitness score.

SOLUTION

3. Computation

Source of variance

Sum of square

Degrees of freedom

Mean

squares Ftest Ftable Decision Conclusion

Gender 54 1 54 35,345 4,41 H_0A is rejected there is difference in the effect of gender on the fitness score

Protein level 52,75 2 26,375 17,264 3,55 H_0B is rejected

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

Treatments

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

H_0AB is accepted

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

Error 27,5 18 1,528 - - -

-Level I Level II Level III Marginal means

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

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

Marginal means

HOMEWORK

 No 12 on page , Statistika Untuk Penelitian

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,

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