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
High protein, Beef
High protein, Cereal
High protein, Pork
Low protein, Beef
Low protein, Cereal
Low protein, Pork

3. Gains in weight (grams) for rats under six diets
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 Protein
Low protein
Source
 Beef
 Cereal
 Pork
Beef
Cereal
Pork
Diet 1 2 3 4 5 6 73 98 94 90
107 49
102 74 79 76 95 82
118 56 96 97
104
111 64 80 86 81 88 51
100
108 72
106 87 77 91 67 70
117
120 89 61 92
105 78 58
Median
103.0
87.0
100.0
82.0
84.5
81.5
Mean
85.9
99.5
79.2
83.9
78.7
IQR
24.0
18.0
11.0
23.0
16.0
PSD
17.78
13.33
8.15
17.04
11.05
Variance
229.11
225.66
119.17
192.84
246.77
273.79
Std. Dev.
15.14
15.02
10.92
13.89
15.71
16.55

4. High Protein
Low Protein
Beef
Cereal
Pork
Cereal
Pork
Beef

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 F test – for comparing k means
Situation
We have k normal populations
Let mi and s 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.
s1 = s2 = … = sk = s

7. We want to test
against

8. The data Assume we have collected data from each of th k populations
Let xi1, xi2 , xi3 , … denote the ni observations from population i.
i = 1, 2, 3, … k.
Let

9. The pooled estimate of standard deviation and variance:

10. Consider the statistic comparing the sample means
where

11. To test
against
use the test statistic

12. Computing Formulae

13. Also

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

15. Then 1) 2) 3)

16. We reject if
Fa is the critical point under the F distribution with n1 degrees of freedom in the numerator and n2 degrees of freedom in the denominator

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

19. Hence

20. Thus
Thus since F > we reject H0

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

22. Anova Table Mean Square F-ratio Between k - 1 SSBetween MSBetween
Source
d.f.
Sum of Squares
Mean Square
F-ratio
Between
k - 1
SSBetween
MSBetween
MSB /MSW
Within
N - k
SSWithin
MSWithin
Total
N - 1
SSTotal

23. Diet Example

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

25. the F-test

27. Hence

28. Using SPSS
Note: The use of another statistical package such as Minitab is similar to using SPSS

29. Assume the data is contained in an Excel file

30. Each variable is in a column
Weight gain (wtgn)
diet
Source of protein (Source)
Level of Protein (Level)

31. After starting the SSPS program the following dialogue box appears:

32. If you select Opening an existing file and press OK the following dialogue box appears

33. The following dialogue box appears:

34. If the variable names are in the file ask it to read the names
If the variable names are in the file ask it to read the names. If you do not specify the Range the program will identify the Range:
Once you “click OK”, two windows will appear

35. One that will contain the output:

36. The other containing the data:

37. To perform ANOVA select Analyze->General Linear Model-> Univariate

38. The following dialog box appears

39. Select the dependent variable and the fixed factors
Press OK to perform the Analysis

40. The Output