Analysis of Variance ( Part Ⅰ )
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Transcript of Analysis of Variance ( Part Ⅰ )
Analysis of VarianceAnalysis of Variance
(( Part ⅠPart Ⅰ ))
March,7th,2005March,7th,2005
Vocabulary(1/4)Vocabulary(1/4)
variance 方差abbreviation 缩写ANOVA 方差分析one-way ANOVA 单因素方差分析
Vocabulary(2/4)Vocabulary(2/4)
invalid (依照律条而判为) 无效的permutation 排列combination 组合athlete 运动员
Vocabulary(3/4)Vocabulary(3/4)high jumper 跳高运动员square 平方sum of squares 平方(之)和
numerator 分子denominator 分母
Vocabulary(4/4)Vocabulary(4/4)
liver 肝脏completely randomized design
完全随机设计ratio 比 , 比值proportion ( 百分 ) 比
AnAnalysis alysis oof f VaVarianceriance
((ANOVAANOVA))
Variance — Variance — averageaverage variation of datavariation of data
1
12
2
2
2
nn
XX
n
XXs
ii
i
Table 9-14 Comparison of liver/body proportion (%) A B C D
2.62 2.82 2.91 3.92 2.23 2.76 3.02 3.00 2.36 2.43 3.28 3.32 2.40 2.73 3.18 3.04 2.4025 2.6850 3.0975 3.3200 2.8763 ( )
iX X
ContentsContents
1. Introduction2. An example in sport school3. A medical research on p.2974. Steps of ANOVA5. Conditions for ANOVA application
1. Introduction1. IntroductionThe t test and u test are only suitable
to compare two sample means. However, in medical practice, we
often try to compare more than two sample means.
In this situation, analysis of variance (ANOVA) is a powerful method.
How about compare every pairs ofHow about compare every pairs of and with and with tt test or test or uu test? test?
2X 3X
2X
iX
612
3424
C
and ,
and ,
and , and ,
and ,
and ,
1X1X
4X2X
3X
1X4X
3X 4X
jX
The Probability with which we The Probability with which we wrongly reject at least one true wrongly reject at least one true
null hypothesis is,null hypothesis is,
26.095.01
95.095.095.095.095.095.016
P
The risk that we actually take is The risk that we actually take is higher than what we assumed!higher than what we assumed!
ContentsContents
1. Introduction2. An example in sport school3. A medical research on p.2974. Steps of ANOVA5. Conditions for ANOVA application
2. An example in sport school2. An example in sport school
Thirty teenagers are chosen to be cultivated as high jumpers in 2000-09-01.
The three groups are guided by three different coaches.
1.1m
30 teenagers —— Future high-jump athletes in GD
The situation, 2 years laterThe situation, 2 years later
AA mX A 36.1
The situation, 2 years laterThe situation, 2 years later
AA
BB
CC
mX A 36.1
mX B 45.1
mX C 58.1
mX 47.1
The situation in GX provinceThe situation in GX province
mX A 42.1 mX B 46.1 mX C 50.1
mX 47.1
ContentsContents
1. Introduction2. An example in sport school3. A medical research on p.2974. Steps of ANOVA5. Conditions for ANOVA application
Table 9-14 Comparison of liver/body proportion (%) A B C D
2.62 2.82 2.91 3.92 2.23 2.76 3.02 3.00 2.36 2.43 3.28 3.32 2.40 2.73 3.18 3.04 2.4025 2.6850 3.0975 3.3200 2.8763 ( )
iX X
3. A medical research on 3. A medical research on pp.297.297
2)( xxSST
2)( xxnSS iiB
2)( iW xxSS
BWT SSSSSS
SSum of um of SSquaresquares — — summarizedsummarized variation of datavariation of data
The following can be proved,The following can be proved,
8194.2)( 22 CxxxSST
0276.2)(
)(2
2
Cn
xxxnSS
iiiB
7918.0)( 2 BTiW SSSSxxSS
365.132)( 2
N
xC
Calculation for Calculation for ssum of um of ssquare quare
1161 nT
Decomposition of “Decomposition of “degree of freedomdegree of freedom””
Since 16 individuals being drawn,Since 16 individuals being drawn,
Since 4 groups being studied,Since 4 groups being studied,141 kB
Just like decomposition of dispersion,Just like decomposition of dispersion,
416)1()1( knknBTW
0660.0416
7918.0)( 2
kn
XX
v
SSMS i
W
WW
Mean SquareMean Square MS MSww —— average variationaverage variation
within all groups within all groups
Actually, mean square is exactly theActually, mean square is exactly the same concept as variance.same concept as variance.
What is the causation of What is the causation of MSMSW W ? ?
6759.014
0276.2
1
)( 2
k
XXn
v
SSMS ii
B
BB
Mean SquareMean Square MS MSBB —— average variationaverage variation
between groups between groups
What is the causation of What is the causation of MSMSB B ? ?
24.100660.0
6759.0
W
B
MS
MSF
FF-value-value —— ratio of MS ratio of MSBB over MS over MSWW
What does What does FF indicate if it is near to 1 ? indicate if it is near to 1 ?
What does What does FF indicate if it is far away from 1 ? indicate if it is far away from 1 ?
FF - distribution- distribution
ContentsContents
1. Introduction2. An example in sport school3. A medical research on p.2974. Steps of ANOVA5. Conditions for ANOVA application
4. Steps of ANOVA4. Steps of ANOVA(1) Hypothesis & significance level.
H0: 1=2=…=k
H1: 1,2,…,k are not same in all
=0.05(2) Calculation of F value.
Construct a table of ANOVA(3) Make a judgment based on P value .
Analysis to Analysis to Example 9-19Example 9-19 on on pp. 297. 297
(1) Hypothesis & significance level.
H0: 1=2= 3 = 4
H1: 1,2, 3 , 4 are not same in all
=0.05
(2) Calculation of F-value.
The table for AONVA(example 9-19)
Source SS ν MS F P Among groups
(treatment) 2.0276 3 0.6759 10.24 <0.01
Within groups (error)
0.7918 12 0.0660
Total 2.8194 15
F0.05(3,12)=3.49, F0.01(3,12)=5.95
(3) Make a judgment based on P value .
Since P < 0.01, it can be concluded that the means of liver/body proportion are not same in all.
Question: Is the above mentioned means are sample means or population means?
Because the differences among the sample means are statistically significant, we can draw a conclusion that the population means are not same in all.
If we try to know in detail, which pair of them are different, which pair of them are equal, we need to do further analysis.
When two sample means are compared,When two sample means are compared, bothboth tt-test-test andand ANOVAANOVA can be used. can be used.
(1) The conclusions will be equivalent.(1) The conclusions will be equivalent.(2) The statistics being used have the foll(2) The statistics being used have the following relationship:owing relationship:
(3) Conditions for ANOVA application (3) Conditions for ANOVA application mmentioned on entioned on pp.295.295 are similar to those fo are similar to those for r tt-test.-test.
Ft
ContentsContents
1. Introduction2. An example in sport school3. A medical research on p.2974. Steps of ANOVA5. Conditions for ANOVA application
5. Conditions 5. Conditions for ANOVA application for ANOVA application(1) Individuals are drawn from the
population randomly and independently.
(2) The concerned populations all belong to normal distributions.
(3) The variances in each populations are equal.