1. Comparing k Populations Means – One way Analysis of Variance (ANOVA)

2. Example In this example we are looking at the weight gains (grams) for rats under six diets differing in level of protein (High or Low) and source of protein (Beef, Cereal, or Pork). –Ten test animals for each diet Diets 1.High protein, Beef 2.High protein, Cereal 3.High protein, Pork 4.Low protein, Beef 5.Low protein, Cereal 6.Low protein, Pork

3. Table Gains in weight (grams) for rats under six diets differing in level of protein (High or Low) and source of protein (Beef, Cereal, or Pork) Level High ProteinLow protein Source Beef Cereal PorkBeefCerealPork Diet 123456 7398949010749 1027479769582 1185696909773 10411198648086 8195102869881 10788102517497 100821087274106 877791906770 11786120958961 11192105785882 Median103.087.0100.082.084.581.5 Mean100.085.999.579.283.978.7 IQR24.018.011.018.023.016.0 PSD17.7813.338.1513.3317.0411.05 Variance229.11225.66119.17192.84246.77273.79 Std. Dev.15.1415.0210.9213.8915.7116.55

4. High ProteinLow Protein Beef Cereal Pork

5. Exploratory Conclusions Weight gain is higher for the high protein meat diets Increasing the level of protein - increases weight gain but only if source of protein is a meat source

6. The differences observed in the diets may due to chance (random variation) or they may be due to actual difference in the diets A confirmatory test of hypothesis will answer this question (with a 5% or 1% margin of error) We need confirmatory tests

7. One possible solution for comparing k populations Use the two sample t test to compare the means of each pair of populations. The number of tests in the example

8. The problem with this approach is the build up of the probability of type I error. (declaring a difference when it does not exist) Suppose that each test is performed using  = 0.05 This means that each test has a 5% chance of making a type I error. However in a group of tests (15) the chance that a type I error is made could be considerably higher than 5%.

9. A batter in baseball may have a 5% chance that he hits a home run each time If he comes to bat 15 times the chance that he will hit a home run at least one is actually 53.7% We need a single test that will detect a difference amongst the means. This test is called the F - test

10. The F test – for comparing k means Situation We have k normal populations Let  i and  denote the mean and standard deviation of population i. i = 1, 2, 3, … k. Note: we assume that the standard deviation for each population is the same.  1 =  2 = … =  k = 

11. We want to test against

12. The data Assume we have collected data from each of th k populations Let x i1, x i2, x i3, … denote the n i observations from population i. i = 1, 2, 3, … k. Let

13. The pooled estimate of standard deviation and variance:

14. Consider the statistic comparing the sample means where

15. To test against use the test statistic

16. Computing Formulae

18. Now Thus

19. To Compute F: Compute 1) 2) 3) 4) 5)

20. Then 1) 2) 3)

21. The sampling distribution of F The sampling distribution of the statistic F when H 0 is true is called the F distribution. The F distribution arises when you form the ratio of two  2 random variables divided by there degrees of freedom.

22. i.e. if U 1 and U 2 are two independent c 2 random variables with degrees of freedom n 1 and n 2 then the distribution of is called the F-distribution with 1 degrees of freedom in the numerator and 2 degrees of freedom in the denominator

24. Recall: To test against use the test statistic

25. We reject if F  is the critical point under the F distribution with 1 degrees of freedom in the numerator and 2 degrees of freedom in the denominator

26. Example In the following example we are comparing weight gains resulting from the following six diets 1.Diet 1 - High Protein, Beef 2.Diet 2 - High Protein, Cereal 3.Diet 3 - High Protein, Pork 4.Diet 4 - Low protein, Beef 5.Diet 5 - Low protein, Cereal 6.Diet 6 - Low protein, Pork

28. Hence

29. Thus Thus since F > 2.386 we reject H 0

30. The ANOVA Table A convenient method for displaying the calculations for the F-test

31. Sourced.f.Sum of Squares Mean Square F-ratio Betweenk - 1SS Between MS Between MS B /MS W WithinN - kSS Within MS Within TotalN - 1SS Total Anova Table

32. Diet Example

33. Equivalence of the F-test and the t-test when k = 2 the t-test

34. the F-test

36. Hence