2. Review
Analisis Variansi dan Efek Utama
• Analisis variansi dengan 1 efek utama dikenal sebagai
analisis variansi satu jalan
• Analisis variansi dengan 2 efek utama dikenal sebagai
analisis variansi dua jalan
• Analisis variansi dengan 3 efek utama dikenal sebagai
analisis variansi tiga jalan
• Dan demikian seterusnya
3. Analisis variansi satu jalan hanya terdiri atas satu faktor
dengan dua atau lebih level
Analisis variansi dua jalan terdiri atas dua faktor, masing-
masing dengan dua atau lebih level
Faktor menghasilkan efek utama sehingga di sini
terdapat dua efek utama
4. Faktor Utama dan Interaksi
Dalam hal lebih dari satu faktor, faktor itu dapat saja saling
mempengaruhi atau tidak saling mempengaruhi
Apabila faktor itu tidak saling mempengaruhi maka kita memperoleh
dua faktor utama saja
Apabila faktor itu saling mempengaruhi, maka selain efek utama,
kita memperoleh lagi interaksi pada saling mempngaruhi itu
Dalam hal terdapat interaksi, kita memiliki efek utama dan interaksi
• Efek utama (dengan perbedaan rerata)
• Interaksi (dengan interaksi di antara faktror)
5. Variansi dan Efek Utama
Variansi sebelum ada efek
Variansi antara kelompok
Kelompok 1 (level 1)
Kelompok 2 (level 2)
Kelompok 3 (level 3)
Ada variansi dalam
kelompok pada kelompok
masing-masing
Ada variansi antara
kelompok
6. Variansi Sesudah Ada Efek Utama
Variansi antara kelompok
Variansi dalam kelompok tidak berubah
Variansi antara kelompok
menjadi besar:
Ada efek,
Paling sedikit ada satu
pasang rerata yang beda
8. So …Sources of variance
When we take samples from each population,
there will be two sources of variability
Within group variability - when we sample from a group
there will be variability from person to person in the
same group Sesatan
We will always have this form of variability because it is sampling
variability
Between group variability – the difference from group to
group Perlakuan
This form of variability will only exist if the groups are different
If the between group variability if large, the means of the two
groups is likely not the same
9. We can use the two types of variability to determine
if the means are likely different
How can we do this?
Look again at the picture
Blue arrow: within group, red arrow: between group
11. Eksperimen faktorial a x b melibatkan 2 faktor dimana
terdapat a tingkat faktor A dan b tingkat faktor B,
Eksperimen diulang r kali pada tiap-tiap tingkat faktor
kombinasi
Adanya replikasi inilah yang memungkinkan
terjadinya interaksi antara faktor A dan B
Rancangan Faktorial a x b
12. Interaction
Occurs When Effects of One Factor Vary According to Levels
of Other Factor
When Significant, Interpretation of Main Effects (A & B) Is
Complicated
Can Be Detected
In Data Table, Pattern of Cell Means in One Row Differs
From Another Row
In Graph of Cell Means, Lines Cross
The interaction between two factor A and B is the tendency
for one factor to behave differently, depending on the
particular level setting of the other variable.
Interaction describes the effect of one factor on the behavior
of the other. If there is no interaction, the two factors
behave independently.
13. A drug manufacturer has three
supervisors who work at each of three
different shift times. Do outputs of the
supervisors behave differently, depending
on the particular shift they are working?
Example
Supervisor 1 always does better
than 2, regardless of the shift.
(No Interaction)
Supervisor 1 does better earlier in the
day, while supervisor 2 does better at
night.
(Interaction)
14. Graphs of Interaction
Effects of Motivation (High or Low) & Training
Method (A, B, C) on Mean Learning Time
Interaction No Interaction
Average
Response
A B C
High
Low
Average
Response
A B C
High
Low
15. Interaksi X terhadap Y
• Tanpa interaksi (dua efek utama)
• Dengan interaksi (bentuk interaksi)
X1
X2
Y
Y
X1
X2
Y
17. Interaksi
• Interaksi terjadi apabila perbedaan rerata pada satu level (misalnya level 1)
tidak sama untuk dua level berbeda pada level 2 sehingga terjadi
perpotongan
Level 1
Level 2
Ada perpotongan karena tidak
sama
18. Two-Way ANOVA Assumptions
1. Normality
Populations are Normally Distributed
2. Homogeneity of Variance
Populations have Equal Variances
3. Independence of Errors
Independent Random Samples are Drawn
19. Two-Way ANOVA
Null Hypotheses
1. No Difference in Means Due to Factor A
H0: 1.. = 2.. =... = a..
2.No Difference in Means Due to Factor B
H0: .1. = .2. =... = .b.
3.No Interaction of Factors A & B
H0: ABij = 0
20. Let xijk be the k-th replication at the i-th level of A
and the j-th level of B.
i = 1, 2, …,a j = 1, 2, …, b, k = 1, 2, …,r
The total variation in the experiment is measured by
the total sum of squares:
The a x b Factorial
Experiment
2
)
(
SS
Total x
xijk
ijk
ij
j
i
ijk
x
21. Variansi Total
ANAVA 2 Jalan
Partisi Variansi Total
JKS
JKA
Variansi A
Variansi Sesatan
Variansi Interaksi
JK(AB)
JKT
Variansi B
JKB
22. JKT dibagi menjadi 4 bagian :
JKA (Jumlah Kuadrat faktor A) : variansi
antara faktor A
JKB (Jumlah Kuadrat faktor B): variansi
antara faktor B
JK(AB) (Jumlah Kuadrat Interaksi): variansi
antara kombinasi tingkat faktor ab
JKS (Jumlah Kuadrat Sesatan)
S
AB
B
A
T JK
JK
JK
JK
JK
23. Xijk
Level i
Factor A
Level j
Factor B
Observation k
Faktor Faktor B
A 1 2 ... b
1 X111 X121 ... X1b1
X112 X122 ... X1b2
2 X211 X221 ... X2b1
X212 X222 ... X2b2
: : : : :
a Xa11 Xa21 ... Xab1
Xa12 Xa22 ... Xab2
25. Contoh : Pabrik Obat
Supervisor Pagi Siang Sore Ai
1 571
610
625
480
474
540
470
430
450
4650
2 480
516
465
625
600
581
630
680
661
5238
Bj 3267 3300 3321 9888
Supervisor pabrik obat bekerja pada 3 shift yang berbeda dan
hasil produksi dihitung pada 3 hari yang dipilih secara
random
a=2 b=3 r=3
26. Tabel ANAVA
db Total = Rataan Kuadrat
db Faktor A =
db faktor B=
db Interaksi =
db Sesatan ?
n –1 = abr - 1
a –1
(a-1)(b-1)
RKA= JKA/(k-1)
RKS =JKS/ab(r-1)
Sumber
Variansi
db JK RK F
A a -1 JKA JKA/(a-1) RKA/RKS
B b -1 JKB JKB/(b-1) RKB/RKS
Interaksi (a-1)(b-1) JK(AB) JK(AB)/(a-1)(b-1) RK(AB)/RKS
Sesatan ab(r-1) JKE JKS/ab(r-1)
Total abr -1 JKT
b –1
RKB = JKB/(b-1)
Dengan pengurangan
RK(AB) = JK(AB)/(a-1)(b-1)
27. Two-way ANOVA: Output versus Supervisor, Shift
Analysis of Variance for Output
Source DF SS MS F P
Supervis 1 19208 19208 26.68 0.000
Shift 2 247 124 0.17 0.844
Interaction 2 81127 40564 56.34 0.000
Error 12 8640 720
Total 17 109222
28. Tests for a Factorial
Experiment
We can test for the significance of both
factors and the interaction using F-tests
from the ANOVA table.
Remember that s 2 is the common
variance for all ab factor-level
combinations. MSE is the best estimate of
s 2, whether or not H 0 is true.
Other factor means will be judged to be
significantly different if their mean square
is large in comparison to MSE.
29. Tests for a Factorial Experiment
The interaction is tested first using F =
MS(AB)/MSE.
If the interaction is not significant, the
main effects A and B can be individually
tested using F = MSA/MSE and F =
MSB/MSE, respectively.
If the interaction is significant, the main
effects are NOT tested, and we focus on
the differences in the ab factor-level
means.
30. Source of
Variation
Degrees of
Freedom
Sum of
Squares
Mean
Square
F
A
(Row)
a - 1 SS(A) MS(A) MS(A)
MSE
B
(Column)
b - 1 SS(B) MS(B) MS(B)
MSE
AB
(Interaction)
(a-1)(b-1) SS(AB) MS(AB) MS(AB)
MSE
Error n - ab SSE MSE
Total n - 1 SS(Total)
Same as Other
Designs
31. The Drug Manufacturer
Two-way ANOVA: Output versus Supervisor, Shift
Analysis of Variance for Output
Source DF SS MS F P
Supervis 1 19208 19208 26.68 0.000
Shift 2 247 124 0.17 0.844
Interaction 2 81127 40564 56.34 0.000
Error 12 8640 720
Total 17 109222
The test statistic for the interaction is F = 56.34 with p-value = .000.
The interaction is highly significant, and the main effects are not
tested. We look at the interaction plot to see where the differences
lie.
33. Revisiting the
ANOVA Assumptions
1. The observations within each population are
normally distributed with a common variance
s 2.
2. Assumptions regarding the sampling
procedures are specified for each design.
•Remember that ANOVA procedures are fairly
robust when sample sizes are equal and when
the data are fairly mound-shaped.
34. Diagnostic Tools
1. Normal probability plot of residuals
2. Plot of residuals versus fit or residuals
versus variables
•Many computer programs have graphics
options that allow you to check the
normality assumption and the
assumption of equal variances.
35. Residuals
•The analysis of variance procedure takes
the total variation in the experiment and
partitions out amounts for several important
factors.
•The “leftover” variation in each data point
is called the residual or experimental error.
•If all assumptions have been met, these
residuals should be normal, with mean 0 and
variance s2.
36. If the normality assumption is valid, the
plot should resemble a straight line,
sloping upward to the right.
If not, you will often see the pattern fail
in the tails of the graph.
Normal Probability Plot
37. If the equal variance assumption is valid,
the plot should appear as a random
scatter around the zero center line.
If not, you will see a pattern in the
residuals.
Residuals versus Fits
