Two-way Analysis of Variance
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 LEARNING METHOD A METHOD B SMALL CLASS BIG CLASS
The hypothesis test on 2 x 2 factorial design
eperimentLEARNING LEARNING METHOD A METHOD B SMALL CLASS BIG CLASS
The hypothesis test on 2 x 2 factorial design
eperiment LEARNING LEARNING METHOD A METHOD B SMALL CLASS BIG CLASS
The hypothesis test on 2 x 2 factorial design
eperiment LEARNING LEARNING METHOD A METHOD B SMALL CLASS BIG CLASS
The hypothesis test on 2 x 2 factorial design
eperiment LEARNING LEARNING METHOD A METHOD B SMALL CLASS BIG CLASS
The hypothesis test on 2 x 2 factorial design
eperiment LEARNING LEARNING METHOD A METHOD B SMALL CLASS BIG CLASS
The hypothesis test on 2 x 2 factorial design
eperiment LEARNING LEARNING METHOD A METHOD B SMALL CLASS BIG CLASSNOTASI DAN TATA LETAK DATA
1.
5
4
k j i ijk
X , , 2
2
j i ij n AB , 2
i i nq A 2
3 ) ) 1 (
3 ( JKA ) ) 1 ( 4 ( JKB
) ) 4 ( ) 3 ( ) 5 (
1 ( JKAB
) 5 ( 2 JKG
j j np B 2
1
Komponen komputasi a.
1q A
Jumlah Kuadrat (JK) atau sum of square (SS) 1) Anava dua jalan sel sama
Tabel jumlah AB Faktor A Faktor B
Total b 1 b 2 ... b q a
1 AB
11 AB
12 ... AB
1 a
N G 2
2 AB
21 AB
22 ... AB
2q A
2 ... ... ... ... ... ... a p AB p1 AB p2 ... AB pq A p Total B
1 B 2 ... B q G Untuk memudahkan perhitungan pada anava dua jalan sel sama, didefinisikan besaran-besaran (1), (2), (3), (4), dan (5) sebagai berikut:
atau ) ) 1 ( 2 (
1) Anava dua jalan sel tak sama
ij ij
AB B jumlah rerata pada kolom ke-j
j ij j
i ij i AB A jumlah rerata pada baris ke-i
ij AB rerata pada sel ij
= jumlah kuadrat deviasi data amatan pada sel ij
X X SS 2 2
Tabel rerata dan jumlah rerata Faktor A
1 ij k ijk k ijk ij n
j i ij n pq ,
h n rerata harmonik frekuensi seluruh sel =
ij n banyaknya data amatan pada sel ij
N n , banyaknya seluruh data amatan
j i ij
Faktor B
Total b 1 b 2 ... b q a 1 11 AB 12 AB ... AB q 1 A 1 a 2 21 AB 22 AB ... AB q 2 A 2... ... ... ... ... ...
a p 1 AB p 2 AB p ... AB pq A p Total B 1 B 2 ... B q G Pada analisis variansi dua jalan dengan sel tak sama, didefinisikan notasi-notasi sebagai berikut:AB G jumlah rerata semua sel
Untuk memudahkan perhitungan, didefinisikan besaran-besaran (1), (2), (3), (4),
dan (5) sebagai berikut: pq G5
JKG 2
JKAB n
1 h
5
3
4
JKB n
4 h
1
JKA n
3 h
1
2
2
AB
ij ij
4
2
j j p B
3
2
i i q A
2
SS
ij ij
1
JKG JKAB JKB JKA JKT
Derajat kebebasan atau degrees of freedom (df) dkA = − 1 dkB = − 1 dkAB =
− 1 ( − 1) dkG = − dkT = − 1
Rerata kuadrat atau mean square (MS) = = = =
Statistik uji 1)
a
ab
= { | >
; −1 ( −1), −
} Keputusan uji
1) H
0A
ditolak apabila F
a
∈ DK
2) H
} 3)
0B
ditolak apabila F
b
∈ 3)
H
0AB
ditolak apabila F
ab
∈ DK
Daerah kritik untuk F ab adalah DK
; −1, −
Untuk H
Daerah kritik untuk F
0A
adalah =
2) Untuk H
0B
adalah =
3) Untuk H
0AB
adalah =
Daerah kritik 1)
a
= { | >
adalah DK
a
= { | >
; −1, −
} 2)
Daerah kritik untuk F
b
adalah DK
b
ab Rangkuman Analisis Source of Sum of Degrees of Mean Square Test Statistics variance squares Freedom Rows SSR p-1
= = − 1 Columns SSC q-1
= = − 1 Treatments SS(Tr) (p-1)(q-1)
= ( ) = ( ) − 1 ( − 1) ( ) Error SSE N-pq = −
- Total SST N-1
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.
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
Are there any significant main effects or an interaction effect ? (use 5% level of significance) SOLUTION 1.
Hypotheses: H : there is no difference in the effect of gender on the fitness score.
0A H : there is difference in the effect of gender on the fitness score.
1A H : there is no difference in the effect of protein level on the fitness score.
0B H : there is difference in the effect of protein level on the fitness score.
1B
H : there is no interaction between gender and protein level of the fitness
0AB score.
H : there is interaction between gender and protein level of the fitness score.
1AB
2. Level of significance = 5%
SOLUTION
3. Computation
Degrees of freedom Mean squares
Protein level 52,75 2 26,375 17,264 3,55 H
Error 27,5 18 1,528 - - - - Total 136,5 23 - - - - -
is accepted there is interaction between gender and protein level of the fitness score
0AB
2,25 2 1,125 0,736 3,55 H
Treatments /Interacton
is rejected there is difference in the effect of protein level on the fitness score
0B
is rejected there is difference in the effect of gender on the fitness score
F
Source of variance Sum of square
1 54 35,345 4,41 H
54
Decision Conclusion Gender
table
F
test
0A 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 5,375 7 3,375 means
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