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ANALYSIS OF VARIANCE

(ANOVA)

Wah Mong Weh

Jabatan Matematik

IPG KSAH

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Aims and Objectives

Understand the basic principles ofANOVA

-why it is done?

- what it tells us?

To conduct one-way independent

ANOVA by hand

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Kenapa ? Kenapa tidak gunakan t-test

Tidak dapat membandingkan lebih dari 2pembolehubah pada masa yang sama.

Apabila lebih daripada satu ujian-tdijalankan, dengan setiap ujian-t ditetapkanpada aras signifikan p < 0.05,kebarangkalian membuat ralat jenis 1 akan

bertambah

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AOVA

One-way analysis of variance (ANOVA) is a

hypothesis testing technique that is used to

compare means from three or more

populations.

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UJIAN ANOVA

Ujian ANOVA (Analysis of Variances Test)adalah ujian yang paling banyak digunakandalam kajian sains tingkah laku.

Ujian ANOVA digunakan untuk membezakanskor-skor min bagi sampel-sampel kajian.

Ujian-t membenarkan perbandingan dua skormin sahaja, tetapi ujian ANOVA digunakanuntuk membandingkan dua, tiga, atau n skormin.

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AOVA Penggunaan ANOVA tertakluk kepada

syarat-syarat dibawah

-pembolehubah bersandar dalam skala selangatau nisbah

-pembolehubah tak bersandar/bebasmempunyai 2 atau lebih aras

-skor-skor pembolehubah bersandar

bertaburan secara normal dalam semua

kumpulan

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AOVA

Unsur utama dalam ujian AOVA ialah

varians.

1.Varians antara kumpulan (between-groups variance)

2. Varians dalam kumpulan (within

groups variance)

3. Nisbah-F (F-ratio) ialah nisbah varians

antara kumpulan dan Varians dalam

kumpulan8

AOVA

Nisbah-F yang signifikan menunjukkan

bahawa min-min populasi adalah tidaksama.

Taburan persampelan yang digunakan

ialah taburan F.

Skala statistik yang digunakan ialah

nisbah F

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ANOVA SATU HALA

Ujian statistik yang digunakan

untuk mencari perbezaan di

antara tiga atau lebih min

Hipotesis nol bagi ANOVA

ialah min-min populasi

daripada tempat sampel-sampel itu diambil adalah

sama

H0: 1 = 2 = 3 = . . . k10

ANOVA SATU HALA

Hipotesis alternatif ialah

Sekurang-kurangnya terdapat

satu pasangan min populasi yang

berbeza

H1: 1 2

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Testing the Hypothesis

The null hypothesis that the three(or more) population means areequal.

That is written as

H0: 1 = 2 = 3 = . . . kH1: At least one mean is different

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Test Statistic for One-Way ANOVA

A excessively large F test statistic isevidence against equal population means.Thus, the null will be rejected.

F = variance between samplesvariance within samples

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The variance is calculated intwo different ways and the

ratio of the two values is

formed.

W

B

MS

MSF=

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MSB, Mean Square Between, the variancebetween samples, measures the differences

related to the treatment given to each sample.

2.MSW

Mean Square Within, the variance

within samples, measures the differences

related to entries within the same sample. The

variance within samples is due to sampling

error.

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Critical Value of F

Right-tailed test

Degree of freedom with ksamples of thesame size n is given by:

numerator df = k-1denominator df= k(n -1) or N-k

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F distribution: General Shape

nonnegative values only

Not symmetric (skewed to the right)

F [3,16]

Illustrative graph for 4 sampleswith 5 members each.

k-1 = 4-1 = 3 N-k = 20-4= 16

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Taburan persampelan yang

digunakan ialah taburan F.

Skala statistik yang

digunakan ialah nisbah F

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Ujian ANOVA

Unsur utama dalam ujian ANOVA ialahvarians. Prosedur pengiraan padaasasnya ialah memperolehi duaanggaran perbezaan varians populasidari data kajian, dan membandingkankedua-dua anggaran tersebut

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Relationships Among Components of ANOVA

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CALCULATING ANOVA

BY HAND

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Ujian ANOVA

Ujian ANOVA dijalankan berdasarkanprosedur dibawah:

1. Menyatakan hipotesis nul

2. Menentukan aras signifikan

3. Menentukan normaliti semua taburansampel

4. Menentukan sama ada menolak atautidak menolak hipotesis nul

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EQUAL SAMPLE SIZES

2

xs

2

xns

2

ps

Variance between samples =

where = variance of sample means

Variance within samples =

Where = pooled variance (or themean of the sample variances)

2

ps

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Key Components of ANOVA Method

SS(total), or total sum of squares, is a

measure of the total variation (around x) inall the sample data combined.

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Key Components of ANOVAMethod

SS(total), or total sum of squares, is a measureof the total variation (around x) in all the sampledata combined and can be obtained using either

formula below.

SS(total) = (x - x)2

( )N

xxTotalSS

2

2)(

=

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SS(Between/treatment) is a measure of thevariation between the samples [see Formula6.119ii)].

In one-way ANOVA, SS(treatment) is

sometimes referred to as SS(model).

Because it is a measure of variability between

the sample means, it is also referred to as SS

(between groups) or SS (between samples).

Key Components of ANOVA Method

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SS(treatment) = n1(x1 - x)2 + n2(x2 - x)

2 + . . . nk(xk - x)2

= ni(xi - x)2

SS(between) is a measure of the variation betweenthe samples.

It is computed in the following way:

Key Components of ANOVA Method

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SS(within) is a sum of squares representing the variabilitythat is assumed to be common to all the populations beingconsidered [Also called SS(residual)]

SS(within) = (n1 -1)s1 + (n2 -1)s2 + (n3 -1)s3 . . . nk(xk -1)si

= (ni - 1)si

2 2 2

2

2

Key Components of ANOVA Method

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SS(total) = SS(Between) + SS(within)

Key Components of ANOVA Method

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MS (Between/Treatment / Model /Group)

MSTR / MSM / MSG is the mean square fortreatment or model or group and is obtained asfollows:

MS (between) =SS (between)

k - 1

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MS (Within/Error or Residual)

MS (within) is mean square forerror, obtained as follows:

MS (within) =SS (within)

N - k

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Test Statistic for ANOVA with

Unequal Sample Sizes

Numerator df= k -1

Denominator df= N - k

F =MS (between)

MS (within)

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Example 1 (Equal sample sizes)

A researcher wishes to try three different

techniques to lower the blood pressure ofindividuals diagnosed with high blood

pressure. The subjects are randomly assigned

to three groups; the first group takes

medication, the second group exercises, and

the third group follows a special diet. After

four weeks, the reduction in each person's

blood pressure is recorded. At =0.05, test the

claim that there is no difference among the

means. The data are shown

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Medication

Exercise Diet

10 6 5

12 8 9

9 3 12

15 0 8

13 2 4

Sample means

sample sd

sample var

11.8 3.8 7.62.3874 3.193 3.209

5.7 10.2 10.3