1. 13 Design and Analysis of Single-Factor Experiments:
The Analysis of Variance
CHAPTER OUTLINE
Designing Engineering Experiments
The Random-Effects Model
Fixed versus random factors
ANOVA & variance components
Completely Randomized Single-Factor Experiments
Randomized Complete Block Design
Example: Tensile strength
Design & statistical analysis
Analysis of variance
Multiple comparisons
Multiple comparisons following the ANOVA
Residual analysis & model checking
Residual analysis & model checking
Dear Instructor: This file is an adaptation of the 4th edition slides for the 5th edition. It will be replaced as slides are developed following the style of the Chapters 1-7 slides.

2. Learning Objectives for Chapter 13
After careful study of this chapter, you should be able to do the following:
Design and conduct engineering experiments involving a single factor with an arbitrary number of levels.
Understand how the analysis of variance is used to analyze the data from these experiments.
Assess model adequacy with residual plots.
Use multiple comparison procedures to identify specific differences between means.
Make decisions about sample size in single-factor experiments.
Understand the difference between fixed and random factors.
Estimate variance components in an experiment involving random factors.
Understand the blocking principle and how it is used to isolate the effect of nuisance factors.
Design and conduct experiments involving the randomized complete block design.

3. 13-1: Designing Engineering Experiments
Every experiment involves a sequence of activities:
Conjecture – the original hypothesis that motivates the experiment.
Experiment – the test performed to investigate the conjecture.
Analysis – the statistical analysis of the data from the experiment.
Conclusion – what has been learned about the original conjecture from the experiment. Often the experiment will lead to a revised conjecture, and a new experiment, and so forth.

4. 13-2: The Completely Randomized Single-Factor Experiment
An Example

5. 13-2: The Completely Randomized Single-Factor Experiment
An Example

6. 13-2: The Completely Randomized Single-Factor Experiment
An Example
The levels of the factor are sometimes called treatments.
Each treatment has six observations or replicates.
The runs are run in random order.

7. 13-2: The Completely Randomized Single-Factor Experiment
An Example
Figure 13-1 (a) Box plots of hardwood concentration data. (b) Display of the model in Equation 13-1 for the completely randomized single-factor experiment

8. 13-2: The Completely Randomized Single-Factor Experiment
The Analysis of Variance
Suppose there are a different levels of a single factor that we wish to compare. The levels are sometimes called treatments.

9. 13-2: The Completely Randomized Single-Factor Experiment
The Analysis of Variance
We may describe the observations in Table 13-2 by the linear statistical model:
The model could be written as

10. 13-2: The Completely Randomized Single-Factor Experiment
The Analysis of Variance
Fixed-effects Model
The treatment effects are usually defined as deviations from the overall mean so that:
Also,

11. 13-2: The Completely Randomized Single-Factor Experiment
The Analysis of Variance
We wish to test the hypotheses:
The analysis of variance partitions the total variability into two parts.

12. 13-2: The Completely Randomized Single-Factor Experiment
The Analysis of Variance
Definition

13. 13-2: The Completely Randomized Single-Factor Experiment
The Analysis of Variance
The ratio MSTreatments = SSTreatments/(a – 1) is called the mean square for treatments.

14. 13-2: The Completely Randomized Single-Factor Experiment
The Analysis of Variance
The appropriate test statistic is
We would reject H0 if f0 > f,a-1,a(n-1)

15. 13-2: The Completely Randomized Single-Factor Experiment
The Analysis of Variance
Definition

16. 13-2: The Completely Randomized Single-Factor Experiment
The Analysis of Variance
Analysis of Variance Table

17. 13-2: The Completely Randomized Single-Factor Experiment
Example 13-1

18. 13-2: The Completely Randomized Single-Factor Experiment
Example 13-1

19. 13-2: The Completely Randomized Single-Factor Experiment
Example 13-1

21. 13-2: The Completely Randomized Single-Factor Experiment
Definition
For 20% hardwood, the resulting confidence interval on the mean is

22. 13-2: The Completely Randomized Single-Factor Experiment
Definition
For the hardwood concentration example,

23. 13-2: The Completely Randomized Single-Factor Experiment
An Unbalanced Experiment

24. 13-2: The Completely Randomized Single-Factor Experiment
Multiple Comparisons Following the ANOVA
The least significant difference (LSD) is
If the sample sizes are different in each treatment:

25. 13-2: The Completely Randomized Single-Factor Experiment
Example 13-2

26. 13-2: The Completely Randomized Single-Factor Experiment
Example 13-2

27. 13-2: The Completely Randomized Single-Factor Experiment
Example 13-2
Figure 13-2 Results of Fisher’s LSD method in Example 13-2

28. 13-2: The Completely Randomized Single-Factor Experiment
Residual Analysis and Model Checking

29. 13-2: The Completely Randomized Single-Factor Experiment
Residual Analysis and Model Checking
Figure 13-4 Normal probability plot of residuals from the hardwood concentration experiment.

30. 13-2: The Completely Randomized Single-Factor Experiment
Residual Analysis and Model Checking
Figure 13-5 Plot of residuals versus factor levels (hardwood concentration).

31. 13-2: The Completely Randomized Single-Factor Experiment
Residual Analysis and Model Checking
Figure 13-6 Plot of residuals versus

32. 13-3: The Random-Effects Model
Fixed versus Random Factors

33. 13-3: The Random-Effects Model
ANOVA and Variance Components
The linear statistical model is
The variance of the response is
Where each term on the right hand side is called a variance component.

34. 13-3: The Random-Effects Model
ANOVA and Variance Components
For a random-effects model, the appropriate hypotheses to test are:
The ANOVA decomposition of total variability is still valid:

35. 13-3: The Random-Effects Model
ANOVA and Variance Components
The expected values of the mean squares are

36. 13-3: The Random-Effects Model
ANOVA and Variance Components
The estimators of the variance components are

37. 13-3: The Random-Effects Model
Example 13-4

38. 13-3: The Random-Effects Model
Example 13-4

39. 13-3: The Random-Effects Model
Figure 13-8 The distribution of fabric strength. (a) Current process, (b) improved process.

40. 13-4: Randomized Complete Block Designs
Design and Statistical Analyses
The randomized block design is an extension of the paired t-test to situations where the factor of interest has more than two levels.
Figure 13-9 A randomized complete block design.

41. 13-4: Randomized Complete Block Designs
Design and Statistical Analyses
For example, consider the situation of Example 10-9, where two different methods were used to predict the shear strength of steel plate girders. Say we use four girders as the experimental units.

42. 13-4: Randomized Complete Block Designs
Design and Statistical Analyses
General procedure for a randomized complete block design:

43. 13-4: Randomized Complete Block Designs
Design and Statistical Analyses
The appropriate linear statistical model:
We assume
treatments and blocks are initially fixed effects
blocks do not interact

44. 13-4: Randomized Complete Block Designs
Design and Statistical Analyses
We are interested in testing:

45. 13-4: Randomized Complete Block Designs
Design and Statistical Analyses
The mean squares are:

46. 13-4: Randomized Complete Block Designs
Design and Statistical Analyses
The expected values of these mean squares are:

47. 13-4: Randomized Complete Block Designs
Design and Statistical Analyses
Definition

48. 13-4: Randomized Complete Block Designs
Design and Statistical Analyses

49. 13-4: Randomized Complete Block Designs
Example 13-5

50. 13-4: Randomized Complete Block Designs
Example 13-5

51. 13-4: Randomized Complete Block Designs
Example 13-5

52. 13-4: Randomized Complete Block Designs
Example 13-5

53. 13-4: Randomized Complete Block Designs
Minitab Output for Example 13-5

54. 13-4: Randomized Complete Block Designs
Multiple Comparisons
Fisher’s Least Significant Difference for Example 13-5
Figure Results of Fisher’s LSD method.

55. 13-4: Randomized Complete Block Designs
Residual Analysis and Model Checking
Figure Normal probability plot of residuals from the randomized complete block design.

56. 13-4: Randomized Complete Block Designs
Figure Residuals by treatment.

57. 13-4: Randomized Complete Block Designs
Figure Residuals by block.

58. 13-4: Randomized Complete Block Designs
Figure Residuals versus ŷij.

59. Important Terms & Concepts of Chapter 13
Analysis of variance (ANOVA) Blocking Completely randomized experiment Expected mean squares Fisher’s least significant difference (LSD) method Fixed factor Graphical comparison of means Levels of a factor Mean square Multiple comparisons Nuisance factors Random factor Randomization Randomized complete block design Residual analysis & model adequacy checking Sample size & replication in an experiment Treatment effect Variance component