8/14/2019 Analysis of Variance_bm [Compatibility Mode]
1/7
1
ANALYSIS OF VARIANCE
(ANOVA)
Wah Mong Weh
Jabatan Matematik
IPG KSAH
2
Aims and Objectives
Understand the basic principles ofANOVA
-why it is done?
- what it tells us?
To conduct one-way independent
ANOVA by hand
3
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]
4
AOVA
One-way analysis of variance (ANOVA) is a
hypothesis testing technique that is used to
compare means from three or more
populations.
5
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 .
6
JENIS-JENIS ANOVA
a. One way ANOVA
- yang melibatkan satu
pembolehubah bebas.
b. Two way ANOVA
- melibatkan dua pembolehubah
bebas.
8/14/2019 Analysis of Variance_bm [Compatibility Mode]
2/7
7
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
9
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
10
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)
11
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
12
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
8/14/2019 Analysis of Variance_bm [Compatibility Mode]
3/7
13
The variance is calculated intwo different ways and the
ratio of the two values is
formed'
W
B
MS
MSF =
14
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.
15
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.
16
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
17
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
18
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
8/14/2019 Analysis of Variance_bm [Compatibility Mode]
4/7
19
Taburan persampelan yan!di!unaan ialah taburan ('
&ala statisti yan!
di!unaan ialah nisbah (
20
Relationships Among Components of ANOVA
21
CALCULATING ANOVA
BY HAND
22
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
23
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
24
F = =variance within samples
variance between samples
ni(xi - x)2
k -1
(ni - 1)si
(ni - 1)
where x = mean of all sample scorescombined, also known as grand mean
2
Calculations with Unequal Sample Sizes
8/14/2019 Analysis of Variance_bm [Compatibility Mode]
5/7
25
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
Calculations with Unequal Sample Sizes
26
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
ni = number of values in the ith sample
2
Calculations with Unequal Sample Sizes
27
F = =variance within samples
variance 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
ni = number of values in the ith sample
xi = mean of the ith sample
2
Calculations with Unequal Sample Sizes
28
F = =variance within samples
variance 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
ni = number of values in the ith sample
xi = mean of the ith sample
si = variance of values in the ith sample
2
2
Calculations with Unequal Sample Sizes
29
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.
30
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)( =
8/14/2019 Analysis of Variance_bm [Compatibility Mode]
6/7
31
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).
Key Components of ANOVA Method
32
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:
Key Components of ANOVA Method
33
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
Key Components of ANOVA Method
34
SS(total) = SS(treatment) + SS(error)
Key Components of ANOVA Method
35
Mean Squares (MS)
Sum of Squares SS(treatment) and SS(error)divided by corresponding number of degreesof freedom give the mean squares.
36
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
8/14/2019 Analysis of Variance_bm [Compatibility Mode]
7/7
37
MS (Error or Residual)
MS (error) is mean square for error,obtained as follows:
MS (error) =SS (error)
N - k
38
Mean Squares (MS)
MS (error) is mean square for errorobtained as follows:
MS (total) =SS (total)
N - 1
MS (error) =SS (error)
N - k
39
Test Statistic for ANOVA with
Unequal Sample Sizes
Numerator df= k -1 Denominator df= N - k
F =MS (treatment)
MS (error)
40
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
41
Medication
Exercise Diet
10 6 5
12 8 9
9 3 12
15 0 8
13 2 4
Sample means
sample sd
sample var
Top Related