ANOVA Nanti Ganti Nama

Statistika TKM 2105Teknik Mesin

From t to F Logic of ANOVA Source of Variance The F ratio

Analysis of Variance (ANOVA)10


From t to F…

• In the independent samples t test, you learned how to use the t distribution to test the hypothesis of no difference between two population means.

• Suppose, however, that we wish to know about the relative effect of three or more different “treatments”?


From t to F…

• We could use the t test to make comparisons among each possible combination of two means.

• However, this method is inadequate in several ways.– It is tedious to compare all possible combinations of groups.

– Any statistic that is based on only part of the evidence (as is the case when any two groups are compared) is less stable than one based on all of the evidence.

– There are so many comparisons that some will be significant by chance.


From t to F…

• What we need is some kind of survey test that will tell us whether there is any significant difference anywhere in an array of categories.

• If it tells us no, there will be no point in searching further.

• Such an overall test of significance is the F test, or the analysis of variance, or ANOVA.


The logic of ANOVA

• Hypothesis testing in ANOVA is about whether the means of the samples differ more than you would expect if the null hypothesis were true.

• This question about means is answered by analyzing variances.– Among other reasons, you focus on variances because

when you want to know how several means differ, you are asking about the variances among those means.


Two Sources of Variability

• In ANOVA, an estimate of variability between groups is compared with variability within groups.

• Within-group variation is the variation due to chance (random sampling error) among individuals given the same treatment.

• Between-group variation is the variation among the means of the different treatment conditions due to chance (random sampling error) and treatment effects, if any exist.


Variability Between Groups

• There is a lot of variability from one mean to the next.• Large differences between means probably are not due to

chance.• It is difficult to imagine that all six groups are random

samples taken from the same population.• The null hypothesis is rejected, indicating a treatment effect

in at least one of the groups.


Variability Within Groups

• Same amount of variability between group means.

• However, there is more variability within each group.

• The larger the variability within each group, the less confident we can be that we are dealing with samples drawn from different populations.


The F Ratio

A N O VA (F)

W ithin-Groups VariationV a ria tion du e to ch a nce .

Betw een-Groups VariationV a ria tion du e to ch an ce

a n d tre a tm e nt e ffe c t (if a ny e x is tis ).

Total Variation Am ong Scores

bilityGroupVariaWithin

bilityGroupVariaBetween F



roupsy Within GVariabilit

Groupsy Between VariabilitF

1F



roupsy Within GVariabilit

Groupsy Between VariabilitF

1F


The F Ratio

s 2 (X X )2n 1

Variance

Degrees of Freedom

Sum of Squares


The F Ratio

A N O VA (F)

M ean Squares Within

W ithin-Groups VariationV a ria tion du e to ch a nce .

M ean Squares Betw een

Betw een-Groups VariationV a ria tion du e to ch an ce

a n d tre a tm e nt e ffe c t (if a ny e x is tis ).

Total Variation Am ong Scores

F MSbetween

MSwithin“mean squares

within”

“mean squares between”


The F Ratio

F MSbetween

MSwithin

MSwithin SSwithin

dfwithin

MSbetween SSbetween

dfbetween

“sum of squares between” “sum of squares within”

“degrees of freedom between” “degrees of freedom within”


The F Ratio

F MSbetween

MSwithin

MSbetween SSbetween

dfbetween

MSwithin SSwithin

dfwithin

SStotal SSbetween SSwithin

df total dfbetween dfwithin

“sum of squares total”

“degrees of freedom total”


The F Ratio: SS Between

Grand Total (add all of the scores together, then square the total)

Total number of subjects.N

G

n

TSSbetween

22

Find each group total, square it, and divide by the number of subjects in the group.

2)( grandgroupbetween XXnSS


The F Ratio: SS Within

Squared group total.

Number of subjects in each group.

n

TXSSwithin

22

Square each individual score and then add up all of the squared scores.

2)( groupwithin XXSS


The F Ratio: SS Total

SStotal X2 G 2

N

Grand Total (add all of the scores together, then square the total)

Total number of subjects.Square each score, then add all of the squared scores together.

)()()( 2groupgrandgroupgrandtotal XXXXXXSS


Non-directional Test

• In testing the hypothesis of no difference between two means, a distinction was made between directional and nondirectional alternative hypotheses.

• Such a distinction no longer makes sense when the number of means exceeds two.

• A directional test is possible only in situations where there are only two ways (directions) that the null hypothesis could be false.

• H0 may be false in any number of ways.– Two or more group means may be alike and the remainder differ, all

may be different, and so on.


Example 1

• A study compared the felt intensity of unrequited love among three groups: individuals who were currently experiencing unrequited love,

individuals who had previously experienced unrequited love and described their experiences retrospectively, and

individuals who had never experienced unrequited love but described how they thought they would feel if they were to experience it. Determine the significance of the difference among groups, using the .05 level of significance.

Imagined Retrospective Current7 12 86 8 105 9 126 11 10


Example 1

• State the research hypothesis.– Do ratings of the intensity of unrequited love differ

if a person is feeling it now, remembering how it felt, or imagining how it may feel?

• State the statistical hypothesis.

false. isH:

:

0

3210

AH

H


Example 1

• Set decision rule.

9)14()14()14()1()1()1(

2131groups ofnumber

05.

321

nnndf

df

within

between



Example 1

• Set the decision rule.

26.4

9

2

05.

crit

within

between

F

df

df


Example 1

• Calculate the test statistic.2X

Grand Total ∑ T: 104

Imagined Retrospective Current

7 49 12 144 8 64

6 36 8 64 10 100

5 25 9 81 12 144

6 36 11 121 10 100

T:24 146 T:40 410 T:40 408

n

TXSSwithin

22

2X2X


Example 1

• Calculate the test statistic.

Grand Total: 104

Imagined Retrospective Current

7 49 12 144 8 64

6 36 8 64 10 100

5 25 9 81 12 144

6 36 11 121 10 100

T:24 146 T:40 410 T:40 408

N

G

n

TSSbetween

22

2X 2X 2X


Example 1

61.922.2

34.21

22.29

20

34.212

67.42

within

between

within

withinwithin

between

betweenbetween

within

between

MS

MSF

df

SSMS

df

SSMS

MS

MSF


Example 1

• Determine if your result is significant.– Reject H0, 9.61>4.26

• Interpret your results.– There is a significant difference in the ratings of the intensity of

unrequited love depending on when (or if) the emotion was felt.

• ANOVA Summary Table– In the literature, the ANOVA results are often summarized in a table.

Source df SS MS F

Between Groups 2 42.67 21.34 9.61

Within Groups 9 20 2.22

Total 11 62.67


After the F Test

• When an F turns out to be significant, we know, with some degree of confidence, that there is a real difference somewhere among our means.

• But if there are more than two groups, we don’t know where that difference is.

• Post hoc tests have been designed for doing pair-wise comparisons after a significant F is obtained.


Example 2

• A psychologist interested in artistic preference randomly assigns a group of 15 subjects to one of three conditions in which they view a series of unfamiliar abstract paintings.

– The 5 participants in the “famous” condition are led to believe that these are each famous paintings.

– The 5 participants in the “critically acclaimed” condition are led to believe that these are paintings that are not famous but are highly thought of by a group of professional art critics.

– The 5 in the control condition are given no special information about the paintings.

Does what people are told about paintings make a difference in how well they are liked? Use the .01 level of significance.


Example 2

Famous Critically Acclaimed

No Information

10 5 4

7 1 6

5 3 9

10 7 3

8 4 3


Example 2

• State the research hypothesis.– Does what people are told about paintings make a

difference in how well they are liked?

• State the statistical hypothesis.

false. is H:

:

0

3210

AH

H


Example 2

• Set decision rule.

93.6

12)15()15()15()1()1()1(

2131groups ofnumber

01.

321

crit

within

between

F

nnndf

df



Example 2


No Information

10 100 5 25 4 16

7 49 1 1 6 36

5 25 3 9 9 81

10 100 7 49 3 9

8 64 4 16 3 9

T:40 338 T:20 100 T:25 151

2X 2X 2X

Grand Total: 85

N

GXSS total

22

33.10767.48158915

)85(151100338

2

totalSS


Example 2


No Information

10 100 5 25 4 16

7 49 1 1 6 36

5 25 3 9 9 81

10 100 7 49 3 9

8 64 4 16 3 9

T:40 338 T:20 100 T:25 151

2X 2X 2X

Grand Total: 85

N

G

n

TSSbetween

22

33.4367.4811258032015

)85(

5

25

5

20

5

40 2222

betweenSS


Example 2


No Information

10 100 5 25 4 16

7 49 1 1 6 36

5 25 3 9 9 81

10 100 7 49 3 9

8 64 4 16 3 9

T:40 338 T:20 100 T:25 151

2X 2X 2X

Grand Total: 85withinbetweentotal SSSSSS

withinSS 33.4333.107

6433.4333.107 withinSS


Example 2

06.433.5

67.21

33.512

64

67.212

33.43

within

between

within

withinwithin

between

betweenbetween

within

between

MS

MSF

df

SSMS

df

SSMS

MS

MSF


Example 2

• Determine if your result is significant.– Retain H0, 4.06<6.93

• Interpret your results.– People who are exposed to different kinds of

information (or no information) about a painting do not differ in their ratings of how much they like the painting.

ANOVA Nanti Ganti Nama

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