“Nikmat manakah yang kamu dustakan? “
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
 Analisis variansi satu jalan hanya terdiri atas satu faktor
dengan dua atau lebih level
 Analisis variansi dua jalan terdiri atas dua faktor, masingmasing dengan dua atau lebih level
 Faktor menghasilkan efek utama sehingga di sini
terdapat dua efek utama
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)
Variansi dan Efek Utama
Variansi sebelum ada efek
Kelompok 1 (level 1)
Kelompok 2 (level 2)
Kelompok 3 (level 3)
Variansi antara kelompok
Ada variansi dalam
kelompok pada kelompok
masing-masing
Ada variansi antara
kelompok
Variansi Sesudah Ada Efek Utama
Variansi dalam kelompok tidak berubah

 
Variansi antara kelompok
menjadi besar:

 



Variansi antara kelompok
Ada efek,
Paling sedikit ada satu
pasang rerata yang beda
Variansi Total

 
Dengan membuka batas semua
kelompok, diperoleh variansi total

 


Variansi total

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
 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
Rancangan
Percobaan
Random Lengkap
RRL
One-Way
Anova
(ANAVA 1 Jalan
Blok Random
RBRL
Two-Way
Anova
(ANAVA 2 Jalan
Faktorial
Rancangan Faktorial a x b



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
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.

Example
A drug manufacturer has three
supervisors who work at each of three
Supervisor 1 always does better
Supervisor 1 does better earlier in the
different
shift
times.
Do
outputs of the
than 2, regardless of the shift.
day, while supervisor 2 does better at
night.
supervisors
behave differently,
depending
(No Interaction)
(Interaction)
on the particular shift they are
working?
Graphs of Interaction
Effects of Motivation (High or Low) & Training
Method (A, B, C) on Mean Learning Time
Interaction
Average
Response
No Interaction
High
Average
Response
High
Low
A
B
C
Low
A
B
C
Interaksi X terhadap Y
• Tanpa interaksi (dua efek utama)
X1
Y
X2
Y
• Dengan interaksi (bentuk interaksi)
X1
Y
X2
• Tanpa interaksi
Y
X1
X2
X
• Ada interaksi
Y
interaksi
X2
X1
X
Interaksi
• Interaksi terjadi apabila perbedaan rerata pada satu level (misalnya level 1)
tidak sama untuk dua level berbeda pada level 2 sehingga terjadi
perpotongan
Ada perpotongan karena tidak
sama
Level 1
Level 2
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
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
The a x b Factorial
Experiment

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
xijk    i   j   ij   ijk

The total variation in the experiment is measured by
the total sum of squares:
Total SS  ( xijk  x )
2
ANAVA 2 Jalan
Partisi Variansi Total
Variansi Total
JKT
Variansi A
Variansi B
JKB
JKA
Variansi Interaksi
JK(AB)
Variansi Sesatan
JKS
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)
JK T  JK A  JK B  JK AB  JKS
Faktor
A
1
X111
1
X112
X211
2
X212
:
:
Xa11
a
Xa12
Faktor B
2
...
X121
...
X122
...
X221
...
X222 ...
:
:
Xa21 ...
Xa22 ...
b
X1b1
X1b2
X2b1
X2b2
:
Xab1
Xab2
Observation k
Xijk
Level i
Factor A
Level j
Factor B
Rumus-rumus
G2
CM 
n
dengan G   xijk
2
JK T   xijk
 CM
 Ai 2
JK A 
 CM dengan Ai  jumlah total faktorA tingkatke - i
br
JK B 
 B j2
JK AB 
ar
 CM dengan B j  jumlah total faktorB tingkatke - j
 ABij 2
 CM - JK A - JK B
r
dengan ABij  jumlah total faktorA tingkatke - i dan faktorB tingkatke - j
JK S  JK T - JK A - JK B - JK AB
Contoh : Pabrik Obat
Supervisor pabrik obat bekerja pada 3 shift yang berbeda dan
hasil produksi dihitung pada 3 hari yang dipilih secara
random
a=2
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
b=3
r=3
Tabel ANAVA
db Total = n –1 = abr - 1
db Faktor A = a –1
db faktor B=b –1
db Interaksi = (a-1)(b-1)
Rataan Kuadrat
RKA= JKA/(k-1)
RKB = JKB/(b-1)
RK(AB) = JK(AB)/(a-1)(b-1)
db Sesatan ?
Dengan pengurangan
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
Two-way ANOVA: Output versus Supervisor, Shift
Analysis of Variance for Output
Source
DF
SS
MS
Supervis
1
19208
19208
Shift
2
247
124
Interaction
2
81127
40564
Error
12
8640
720
Total
17
109222
F
26.68
0.17
56.34
P
0.000
0.844
0.000
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.
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.
Source of Degrees of Sum of
Variation Freedom 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
(a-1)(b-1)
(Interaction)
SS(AB)
Error
n - ab
SSE
Total
n-1
SS(Total)
MS(AB) MS(AB)
MSE
MSE
Same as Other
Designs
The Drug Manufacturer
Two-way ANOVA: Output versus Supervisor, Shift
Analysis of Variance for Output
Source
DF
SS
MS
Supervis
1
19208
19208
Shift
2
247
124
Interaction
2
81127
40564
Error
12
8640
720
Total
17
109222
F
26.68
0.17
56.34
P
0.000
0.844
0.000
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.
The Drug Manufacturer
Supervisor 1 does better
earlier in the day, while
supervisor 2 does better at
night.
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.
Diagnostic Tools
•Many computer programs have graphics
options that allow you to check the
normality assumption and the
assumption of equal variances.
1. Normal probability plot of residuals
2. Plot of residuals versus fit or residuals
versus variables
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.
Normal Probability Plot
 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.
Residuals versus Fits
 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.