ANOVA Nanti Ganti Nama
-
Upload
iwan-irwan-arnol -
Category
Documents
-
view
238 -
download
4
description
Transcript of 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
Statistika TKM 2105Teknik Mesin
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”?
Statistika TKM 2105Teknik Mesin
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.
Statistika TKM 2105Teknik Mesin
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.
Statistika TKM 2105Teknik Mesin
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.
Statistika TKM 2105Teknik Mesin
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.
Statistika TKM 2105Teknik Mesin
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.
Statistika TKM 2105Teknik Mesin
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.
Statistika TKM 2105Teknik Mesin
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
Statistika TKM 2105Teknik Mesin
Two Sources of Variability
roupsy Within GVariabilit
Groupsy Between VariabilitF
1F
Statistika TKM 2105Teknik Mesin
Two Sources of Variability
roupsy Within GVariabilit
Groupsy Between VariabilitF
1F
Statistika TKM 2105Teknik Mesin
The F Ratio
s 2 (X X )2n 1
Variance
Degrees of Freedom
Sum of Squares
Statistika TKM 2105Teknik Mesin
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”
Statistika TKM 2105Teknik Mesin
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”
Statistika TKM 2105Teknik Mesin
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”
Statistika TKM 2105Teknik Mesin
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
Statistika TKM 2105Teknik Mesin
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
Statistika TKM 2105Teknik Mesin
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
Statistika TKM 2105Teknik Mesin
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.
Statistika TKM 2105Teknik Mesin
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
Statistika TKM 2105Teknik Mesin
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
Statistika TKM 2105Teknik Mesin
Example 1
• Set decision rule.
9)14()14()14()1()1()1(
2131groups ofnumber
05.
321
nnndf
df
within
between
Statistika TKM 2105Teknik Mesin
Statistika TKM 2105Teknik Mesin
Example 1
• Set the decision rule.
26.4
9
2
05.
crit
within
between
F
df
df
Statistika TKM 2105Teknik Mesin
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
Statistika TKM 2105Teknik Mesin
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
Statistika TKM 2105Teknik Mesin
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
Statistika TKM 2105Teknik Mesin
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
Statistika TKM 2105Teknik Mesin
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.
Statistika TKM 2105Teknik Mesin
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.
Statistika TKM 2105Teknik Mesin
Example 2
Famous Critically Acclaimed
No Information
10 5 4
7 1 6
5 3 9
10 7 3
8 4 3
Statistika TKM 2105Teknik Mesin
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
Statistika TKM 2105Teknik Mesin
Example 2
• Set decision rule.
93.6
12)15()15()15()1()1()1(
2131groups ofnumber
01.
321
crit
within
between
F
nnndf
df
Statistika TKM 2105Teknik Mesin
Statistika TKM 2105Teknik Mesin
Example 2
Famous Critically Acclaimed
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
Statistika TKM 2105Teknik Mesin
Example 2
Famous Critically Acclaimed
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
Statistika TKM 2105Teknik Mesin
Example 2
Famous Critically Acclaimed
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
Statistika TKM 2105Teknik Mesin
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
Statistika TKM 2105Teknik Mesin
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.