Analysis of Variance （ Part Ⅰ ）

2. Analysis of Variance （Part Ⅰ） March,7th,2005

3. Vocabulary(1/4) variance 方差 abbreviation 缩写 ANOVA 方差分析 one-way ANOVA 单因素方差分析

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

5. 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

6. Vocabulary(4/4) liver 肝脏 completely randomized design 完全随机设计 ratio比,比值 proportion(百分) 比

7. Analysis of Variance (ANOVA)

8. Variance —averagevariation of data

10. Contents • 1. Introduction • 2. An example in sport school • 3. A medical research on p.297 • 4. Steps of ANOVA • 5. Conditions for ANOVA application

11. 1. Introduction • The 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.

12. How about compare every pairs of and with t test or u test? and , and , and , and , and , and ,

13. The Probability with which we wrongly reject at least one true null hypothesis is, The risk that we actually take is higher than what we assumed!

14. Contents • 1. Introduction • 2. An example in sport school • 3. A medical research on p.297 • 4. Steps of ANOVA • 5. Conditions for ANOVA application

15. 2. 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.

16. 30 teenagers —— Future high-jump athletes in GD 1.1m

17. The situation, 2 years later A

18. The situation, 2 years later C B A

19. The situation in GX province

20. Contents • 1. Introduction • 2. An example in sport school • 3. A medical research on p.297 • 4. Steps of ANOVA • 5. Conditions for ANOVA application

21. 3. A medical research on p.297

22. Sum of Squares —summarizedvariation of data The following can be proved,

23. Calculation for sum of square

24. Decomposition of “degree of freedom” Since 16 individuals being drawn, Since 4 groups being studied, Just like decomposition of dispersion,

25. Mean Square MSw—average variation within all groups Actually, mean square is exactly the same concept as variance. What is the causation of MSW ?

26. Mean Square MSB—average variation between groups What is the causation of MSB ?

27. F-value— ratio of MSB over MSW What does F indicate if it is near to 1 ? What does F indicate if it is far away from 1 ?

28. F- distribution

29. Contents • 1. Introduction • 2. An example in sport school • 3. A medical research on p.297 • 4. Steps of ANOVA • 5. Conditions for ANOVA application

30. 4. 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 .

31. Analysis to Example 9-19 on p. 297 (1) Hypothesis & significance level. H0: 1=2= 3 = 4 H1: 1,2, 3 , 4 are not same in all =0.05

32. (2) Calculation of F-value.

33. (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?

34. 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.

35. When two sample means are compared,botht-testandANOVA can be used.(1) The conclusions will be equivalent.(2) The statistics being used have the following relationship:(3) Conditions for ANOVA application mentioned on p.295 are similar to those for t-test.

36. Contents • 1. Introduction • 2. An example in sport school • 3. A medical research on p.297 • 4. Steps of ANOVA • 5. Conditions for ANOVA application

37. 5. Conditions for ANOVA application • Individuals are drawn from the population randomly and independently. • The concerned populations all belong to normal distributions. • The variances in each populations are equal.