Analysis of Variance_bm [Compatibility Mode]

8/14/2019 Analysis of Variance_bm [Compatibility Mode]

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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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When And Why?

When do we use ANOVA?

When we would like to compare means from 3 or

more groups.

Why not use lots of t-tests?

Cant look at several independent variables at once

Inflates the Type I error

Need many t tests when the number of groups

increase. [See Andy Field handout]

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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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ANOVA

ANOVA adalah kaedah ujian

hipotesis bagi mengenalpasti

perbezaan min yang wujud dalam

dua ataupun lebih sample ujian.

Tujuan utama ANOVA adalah

menentukan sama ada perbezaan

sample disebabkan kesilapan proses

sample ataupun kesan rawatan yang

sistematik .

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JENIS-JENIS ANOVA

a. One way ANOVA

- yang melibatkan satu

pembolehubah bebas.

b. Two way ANOVA

- melibatkan dua pembolehubah

bebas.


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ANOVA SATU HALA

Ujian statisti yan! di!unaan

untu mencari perbe"aan diantara ti!a atau lebih min

#ipotesis nol ba!i ANOVA

ialah $min-min populasi

daripada tempat sampel-

sampel itu diambil adalah

sama%

H0: 1 = 2 = 3 = . . . k8

ANOVA SATU HALA

#ipotesis alternatif ialah $&euran!-uran!nya terdapat

satu pasan!an min populasi yan!

berbe"a%

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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Theory of ANOVA

We calculate how much variability there is

between scores.

-Total sum of squares (SST )

We then calculate how much of this variability

can be explained by the model we fit to the

data

-how much variability is due to the

experimental manipulation, Model Sum of

Squares (SSM)

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Theory of ANOVA

How much cannot be explained

How much is due to individual differences in

performance, Residual Sum of Squares (SSR )

We compare the amount of variability

explained by the model (experiment), to the

error in the model ( individual differences)

- The ratio is called the F-ratio

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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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One-Way ANOVA Assumptions

1. The populations have normal distributions.

2. The populations have the same variance

2(or standard deviation ).3. The samples are simple random samples.

4. The samples are independent of each other.

5. The different samples are from populationsthat are categorized in only one way.

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ANOVA methods require theF-distribution

1. The F-distribution is not symmetric; it isskewed to the right.

2. The values of F can be 0 or positive, theycannot be negative.

3. There is a different F-distribution for each pairof degrees of freedom for the numerator anddenominator.

4. Critical values of F can be found from Tables

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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-1

denominator 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 yan!di!unaan ialah taburan ('

&ala statisti yan!

di!unaan ialah nisbah (

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Relationships Among Components of ANOVA

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CALCULATING ANOVA

BY HAND

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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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F = =variance within samples

variance between samples

ni(xi - x)2

k -1

(ni - 1)si

(ni - 1)

2

Calculations with Unequal Sample Sizes

See Formula Booklet pp. 20-21

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ni(xi - x)2

k -1

(ni - 1)si

(ni - 1)

where x = mean of all sample scorescombined, also known as grand mean

2



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F = =variance within samplesvariance between samples

ni(xi - x)2

k -1

(ni - 1)si

(ni - 1)

where x = mean of all sample scores combined

k = number of population means being compared

2


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F = =variance within samplesvariance between samples

ni(xi - x)2

k -1

(ni - 1)si

(ni - 1)



ni = number of values in the ith sample

2


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ni(xi - x)2

k -1

(ni - 1)si

(ni - 1)




xi = mean of the ith sample

2


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ni(xi - x)2

k -1

(ni - 1)si

(ni - 1)




xi = mean of the ith sample

si = variance of values in the ith sample

2

2


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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(treatment) is a measure of the variationbetween the samples [see Formula 6.119ii)].

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).


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

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

= ni(xi - x)2

SS(treatment) is a measure of the variationbetween the samples.

It is computed in the following way:


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

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

= (ni - 1)si

2 2 2

2

2


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SS(total) = SS(treatment) + SS(error)


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Mean Squares (MS)

Sum of Squares SS(treatment) and SS(error)divided by corresponding number of degreesof freedom give the mean squares.

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MS (Treatment / Model / Group)

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

MS (treatment) =SS (treatment)

k - 1


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MS (Error or Residual)

MS (error) is mean square for error,obtained as follows:

MS (error) =SS (error)

N - k

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Mean Squares (MS)

MS (error) is mean square for errorobtained as follows:

MS (total) =SS (total)

N - 1

MS (error) =SS (error)

N - k

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Test Statistic for ANOVA with

Unequal Sample Sizes

Numerator df= k -1 Denominator df= N - k

F =MS (treatment)

MS (error)

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Example 1 (Equal sample sizes)

A researcher wishes to try three different

techniques to lower the blood pressure of

individuals 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. Afterfour 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

Analysis of Variance_bm [Compatibility Mode]

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