1. Chapter 9: The Analysis of Variance (ANOVA)

2. ANOVA: Examples
Do four different brands of gasoline have different effects on automobile efficiency?
Does the type of sugar solution (glucose, sucrose, fructose, mixture) have an effect on bacterial growth?
3) Does the resulting color density of a fabric depend on the amount of dye used?

3. ANOVA: Graphical

4. ANOVA test statistic

5. ANOVA test

6. F Distribution
Continuous_distributions/F_distribution.htm

7. P-value for an upper-tailed F test
shaded area=P-value = 0.05

8. Table IV F critical values

9. ANOVA Table: Formulas Source df SS MS (Mean Square) F Model (Between)
k – 1
Error (Within)
n – k
Total
n – 1

10. ANOVA Hypothesis test: Summary
H0: μ1 = μ2 =  = μk Ha: At least two μ’s are different Test statistic: P-value: area under the upper tail with an F distribution with (dfm, dfe) degrees of freedom

11. ANOVA: Example
A random sample of 15 healthy young men are split randomly into 3 groups of 5. They receive 0, 20, and 40 mg of the drug Paxil for one week. Then their serotonin levels are measured to determine whether Paxil affects serotonin levels. The data is on the next slide. Does Paxil affect serotonin levels in healthy young men?

12. ANOVA: Example (cont). Dose 0 mg 20 mg 40 mg 48.62 58.60 68.59 49.85
72.52
78.28
64.22
66.72
82.77
62.81
80.12
76.53
62.51
68.44
72.33

13. ANOVA: Example (cont)
1. Let 1 be the mean serotonin level for men receiving 0 mg of Paxil. Let 2 be the mean serotonin level for men receiving 20 mg of Paxil. Let 3 be the mean serotonin level for men receiving 40 mg of Paxil.

14. ANOVA: Example (cont)
2. H0: 1 = 2 = 3; mean serotonin levels are the same at all 3 dosage levels [or, mean serotonin levels are unaffected by Paxil dose] HA: The mean serotonin levels of the three groups are not all equal. [or, serotonin levels are affected by Paxil does]

15. ANOVA: Example (cont). Dose 0 mg 20 mg 40 mg 48.62 58.60 68.59 49.85
72.52
78.28
64.22
66.72
82.77
62.81
80.12
76.53
62.51
68.44
72.33
overall ni 5 15
x̅i
57.60
69.28
75.70
67.53 si
7.678
7.895
5.460
(ni-1)si2
235.78
249.32
119.24
604.34
ni(x̅i - x̅̅)2
492.56
15.36
333.96
841.88

16. ANOVA: Example (cont) Source df SS MS 4. F-Ratio 5. P-Value Model 2
841.88
420.94
Error 12
604.34
50.36
Total 14

17. Example: ANOVA (cont)
7. This does give strong support (0.001 < P < 0.01) to the claim that there is a difference in serotonin levels among the groups of men taking 0, 20, and 40 mg of Paxil.
This does give strong support (0.001 < P < 0.01) to the claim that Paxil intake affects serotonin levels in young men.

18. Example: ANOVA Effect plot

19. Problem with Multiple t tests

20. Overall Risk of Type I Error in Using Repeated t Tests at  = 0.05

21. Table IX: Tukey  = 0.05

22. Table IX: Tukey  = 0.01

23. Example: Tukey
A biologist wished to study the effects of ethanol on sleep time. A sample of 20 rants matched for age and other characteristics, was selected, and each rat was given an oral injection having a particular concentration of ethanol per body weight. The rapid eye movement (REM) sleep time for each rat was then recorded for a 24-hour period with the results in the table on the next slide. Does the data indicate that the true average REM sleep time depends of the concentration of ethanol?

24. Example: Tukey (cont) Treatment (conc. of ethanol) x̅i si 0 (control)
88.6
73.2
91.4
68.0
75.2
79.28
10.18
1 g/kg
63.0
53.9
69.2
50.1
71.5
61.54
9.34
2 g/kg
44.9
59.5
40.2
56.3
38.7
47.92
9.46
4 g/kg
31.0
39.6
45.3
25.2
22.7
32.76
9.56
Source df SS MS
F-Ratio
P-Value
Model 3
21.09
0.0001
Error 16
92.96
Total 19

25. Example: Tukey (cont) i - j x̅i - x̅j Difference? 0 – 1 17.74 yes
0 – 2
31.36
0 – 4
46.52
1 – 2
13.62
1 – 4
28.78
2 – 4
15.16
x̅4
x̅2
x̅1
x̅0
32.76
47.92
61.54
79.28

26. OUTPUT: Tukey cldiff
The GLM Procedure Tukey's Studentized Range (HSD) Test for sleep NOTE: This test controls the Type I experimentwise error rate. Alpha 0.05 Error Degrees of Freedom 16 Error Mean Square Critical Value of Studentized Range Minimum Significant Difference Comparisons significant at the 0.05 level are indicated by ***. Difference amount Between Simultaneous 95% Comparison Means Confidence Limits *** *** *** *** *** *** *** ***

27. OUTPUT: Tukey lines
The GLM Procedure Tukey's Studentized Range (HSD) Test for sleep NOTE: This test controls the Type I experimentwise error rate, but it generally has a higher Type II error rate than REGWQ. Alpha 0.05 Error Degrees of Freedom 16 Error Mean Square Critical Value of Studentized Range Minimum Significant Difference Means with the same letter are not significantly different. Tukey Grouping Mean N amount A B B C B C C

28. Critical values for Dunnett’s Method

29. Example: Dunnett
A biologist wished to study the effects of ethanol on sleep time. A sample of 20 rants matched for age and other characteristics, was selected, and each rat was given an oral injection having a particular concentration of ethanol per body weight. The rapid eye movement (REM) sleep time for each rat was then recorded for a 24-hour period with the results in the table on the next slide. Does the data indicate that the true average REM sleep time is different for the ingestion of any alcohol?

30. Example: Dunnett (cont)
Treatment (conc. of ethanol)
x̅i si
0 (control)
88.6
73.2
91.4
68.0
75.2
79.28
10.18
1 g/kg
63.0
53.9
69.2
50.1
71.5
61.54
9.34
2 g/kg
44.9
59.5
40.2
56.3
38.7
47.92
9.46
4 g/kg
31.0
39.6
45.3
25.2
22.7
32.76
9.56
Source df SS MS
F-Ratio
P-Value
Model 3
21.09
0.0001
Error 16
92.96
Total 19

31. Example: Dunnett (cont)
i - j
x̅i - x̅j
Difference?
0 – 1
17.74
yes
0 – 2
31.36
0 – 4
46.52

32. OUTPUT: Dunnett cldiff
The GLM Procedure
Dunnett's t Tests for sleep
NOTE: This test controls the Type I experimentwise error for comparisons of all treatments against a control.
Alpha
Error Degrees of Freedom
Error Mean Square
Critical Value of Dunnett's t
Minimum Significant Difference
Comparisons significant at the 0.05 level are indicated by ***.
Difference
amount Between Simultaneous 95%
Comparison Means Confidence Limits
***
***
***

33. Randomized Block Experiments
Researchers are interested in the effect that acid has on growth rate of alfalfa plants. To control sunlight, the randomized block procedure is used.

34. ANOVA Table: RCB Source df SS MS F Factor A (treatment) a – 1 Factor B
(block)
b - 1
Error (Within)
(a-1)(b-1)
SST – SSA - SSB
Total
n –1 = ab– 1

35. Example: RBD
An experiment was designed to study the performance of four different detergents in cleaning clothes. The clothes were measured on cleanness (higher= cleaner) after being stained with three different common stains and cleaned with the four detergents. We want to know which if any detergent is better.
Stain 1
Stain 2
Stain 3
Detergent 1 45 43 51
Detergent 2 47 46 52
Detergent 3 48 50 55
Detergent 4 42 37 49

36. Example: RBD (cont)
1. Let 1 be the mean cleaning value for detergent 1. Let 2 be the mean cleaning value for detergent 2. Let 3 be the mean cleaning value for detergent 3. Let 4 be the mean cleaning value for detergent .

37. Example: RBD (cont)
2. H0: 1 = 2 = 3 = 4; mean cleaning levels for the 4 detergents are the same. HA: At least two of the detergents have different cleaning levels.

38. Example: RBD (cont) Source df SS MS F P Value Factor A (soap) 3 110.92
36.97
Factor B (stain) 2
135.17
67.58
Error 6
18.83
3.14
Total 11
264.92
7. This data does give strong support (P = ) to the claim that there is difference in how well the four detergents clean stains.

39. Example: RBD 2
An experiment was designed to study the performance of four different detergents in cleaning clothes. The clothes were measured on cleanness (higher= cleaner) after being stained with three different common stains and cleaned with the four detergents. We want to know if the stains (blocks) have an effect.
Stain 1
Stain 2
Stain 3
Detergent 1 45 43 51
Detergent 2 47 46 52
Detergent 3 48 50 55
Detergent 4 42 37 49

40. Example: RBD 2 (cont)
1. Let 1 be the mean cleaning value for stain 1. Let 2 be the mean cleaning value for stain 2. Let 3 be the mean cleaning value for stain 3.

41. Example: RBD 2 (cont)
2. H0: 1 = 2 = 3 = 4; mean cleaning levels for the 4 detergents are the same. HA: At least two of the detergents have different cleaning levels.

42. Example: RBD 2 (cont) Source df SS MS F P Value Factor A (soap) 3
110.92
36.97
Factor B (stain) 2
135.17
67.58
Error 6
18.83
3.14
Total 11
264.92
7. This data does give strong support (P = ) to the claim that there is difference in how well the detergents clean the different stains.

43. Example: RBD (cont) i - j x̅i - x̅j Difference? 1 – 2 2.000 1 – 3
4.667
1 – 4
3.667
2 – 3
-2.667
2 – 4
5.667
yes
3 – 4
8.333
x̅4
x̅1
x̅2
x̅3
42.67
46.33
48.33
51.00