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ReviewAnalisis 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, masing-masing 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

Variansi antara kelompokVariansi 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

Variansi Sesudah Ada Efek Utama

Variansi antara kelompokVariansi antara kelompok

Variansi dalam kelompok tidak berubah

Variansi antara kelompok menjadi besar:

Ada efek,Paling sedikit ada satu pasang rerata yang beda

Variansi Total

Variansi totalVariansi total

Dengan membuka batas semua kelompok, diperoleh variansi total

So …Sources of varianceWhen we take samples from each population,

there will be two sources of variabilityWithin 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 pictureBlue arrow: within group, red arrow:

between group

Rancangan Percobaan

One-Way Anova

(ANAVA 1 Jalan

Random Lengkap

RRL

Blok RandomRBRL

Two-Way Anova

(ANAVA 2 Jalan

Faktorial

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

InteractionOccurs When Effects of One Factor Vary According to

Levels of Other FactorWhen Significant, Interpretation of Main Effects (A &

B) Is ComplicatedCan Be Detected

In Data Table, Pattern of Cell Means in One Row Differs From Another Row

In Graph of Cell Means, Lines CrossThe interactioninteraction 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.

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)

Effects of Motivation (High or Low) & Training Method (A, B, C) on Mean Learning TimeInteractionInteraction No InteractionNo Interaction

AverageAverageResponseResponse

AA BB CC

HighHigh

LowLow

AverageAverageResponseResponse

AA BB CC

HighHigh

LowLow

Interaksi X terhadap Y

• Tanpa interaksi (dua efek utama)

• Dengan interaksi (bentuk interaksi)

X1

X2

Y

Y

X1

X2

Y

• Tanpa interaksi

• Ada interaksi

Y

X

X

Y

X1

X2

X1

X2

interaksi

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

Two-Way ANOVA Assumptions

1. NormalityPopulations are Normally Distributed

2. Homogeneity of VariancePopulations have Equal Variances

3. Independence of ErrorsIndependent Random Samples are Drawn

Two-Way ANOVA Null Hypotheses

1. No Difference in Means Due to Factor AH0: 1.. = 2.. =... = a..

2. No Difference in Means Due to Factor BH0: .1. = .2. =... = .b.

3. No Interaction of Factors A & BH0: ABij = 0

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 squarestotal sum of squares:

The a x b Factorial Experiment

2)( SS Total xxijk

ijkijjiijkx

Variansi Variansi Total Total

ANAVA 2 JalanPartisi Variansi Total

JKS

JKA

Variansi AVariansi A

Variansi SesatanVariansi SesatanVariansi InteraksiVariansi Interaksi

JK(AB)JK(AB)

JKTJKT

Variansi BVariansi B

JKBJKB

JKT dibagi menjadi 4 bagian : JKAJKA (Jumlah Kuadrat faktor A) :

variansi antara faktor A JKBJKB (Jumlah Kuadrat faktor B):

variansi antara faktor B JK(AB)JK(AB) (Jumlah Kuadrat Interaksi):

variansi antara kombinasi tingkat faktor ab

JKSJKS (Jumlah Kuadrat Sesatan)SAB BAT JK JKJK JK JK

XXiijj

kkLevel i Level i Factor Factor

AA

Level j Level j Factor Factor

BB

Observation kObservation k

FaktorFaktor Faktor BFaktor BAA 11 22 ...... bb11 XX111111 XX121121 ...... XX1b11b1

XX112112 XX122122 ...... XX1b21b2

22 XX211211 XX221221 ...... XX2b12b1

XX212212 XX222222 ...... XX2b22b2

:: :: :: :: ::aa XXa11a11 XXa21a21 ...... XXab1ab1

XXa12a12 XXa22a22 ...... XXab2ab2

Rumus-rumus

ABBATS

BA

2

AB

2

B

2A

2T

2

JK-JK-JK-JKJK

-ke tingkat Bfaktor dan -keA tingkat faktor aljumlah tot dengan

JK-JK- CMJK

-ke tingkat Bfaktor aljumlah totdengan CMJK

-keA tingkat faktor aljumlah tot dengan CMJK

CMJK

Gdengan G

CM

jiABr

AB

jBar

B

iAbrA

x

xn

ij

ij

jj

ii

ijk

ijk

Contoh : Pabrik Obat

Supervisor Pagi Siang Sore Ai

1 571610625

480474540

470430450

4650

2 480516465

625600581

630680661

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

Tabel ANAVAdb Total = Rataan Kuadratdb Faktor A = db faktor B=db Interaksi =

db Sesatan ?

n –1 = abr - 1a –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/RKSB b -1 JKB JKB/(b-1) RKB/RKS

Interaksi (a-1)(b-1) JK(AB) JK(AB)/(a-1)(b-1) RK(AB)/RKSSesatan ab(r-1) JKE JKS/ab(r-1)

Total abr -1 JKT

b –1RKB = JKB/(b-1)

Dengan pengurangan

RK(AB) = JK(AB)/(a-1)(b-1)

Two-way ANOVA: Output versus Supervisor, Shift

Analysis of Variance for Output Source DF SS MS F PSupervis 1 19208 19208 26.68 0.000Shift 2 247 124 0.17 0.844Interaction 2 81127 40564 56.34 0.000Error 12 8640 720Total 17 109222

Tests for a Factorial ExperimentWe can test for the significance of

both factors and the interaction using F-tests from the ANOVA table.

Remember that 2 is the common variance for all ab factor-level combinations. MSE is the best estimate of 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 ExperimentThe 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 ofSource ofVariationVariation

Degrees ofDegrees ofFreedomFreedom

Sum ofSum ofSquaresSquares

MeanMeanSquareSquare

FF

AA(Row)(Row)

a - 1a - 1 SS(A)SS(A) MS(A)MS(A) MS(A)MS(A)MSEMSE

BB(Column)(Column)

b - 1b - 1 SS(B)SS(B) MS(B)MS(B) MS(B)MS(B)MSEMSE

ABAB(Interaction)(Interaction)

(a-1)(b-1)(a-1)(b-1) SS(AB)SS(AB) MS(AB)MS(AB) MS(AB)MS(AB)MSEMSE

ErrorError n - abn - ab SSESSE MSEMSETotalTotal n - 1n - 1 SS(Total)SS(Total) Same as Other Same as Other

DesignsDesigns

The Drug ManufacturerTwo-way ANOVA: Output versus Supervisor, Shift

Analysis of Variance for Output Source DF SS MS F PSupervis 1 19208 19208 26.68 0.000Shift 2 247 124 0.17 0.844Interaction 2 81127 40564 56.34 0.000Error 12 8640 720Total 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.

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 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

1. Normal probability plot of residuals2. 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.

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 residualresidual or experimental errorexperimental error. •If all assumptions have been met, these residuals should be normalnormal, with mean 0 and variance 2.

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

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