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.