13

Design and Analysis of

Single-Factor Experiments:

The Analysis of Variance

CHAPTER OUTLINE

1

Learning Objectives for Chapter 13

After careful study of this chapter, you should be able to do the

following:

1. Design and conduct engineering experiments involving a single factor with an

arbitrary number of levels.

2. Understand how the analysis of variance is used to analyze the data from

these experiments.

3. Assess model adequacy with residual plots.

4. Use multiple comparison procedures to identify specific differences between

means.

5. Make decisions about sample size in single-factor experiments.

6. Understand the difference between fixed and random factors.

7. Estimate variance components in an experiment involving random factors.

8. Understand the blocking principle and how it is used to isolate the effect of

nuisance factors.

9. Design and conduct experiments involving the randomized complete block

design.

2

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-1: Designing Engineering Experiments

Every experiment involves a sequence of activities:

1. Conjecture – the original hypothesis that motivates the

experiment.

2. Experiment – the test performed to investigate the

conjecture.

3. Analysis – the statistical analysis of the data from the

experiment.

4. 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.

3

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

13-2.1 An Example

4

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

13-2.1 An Example

5

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

13-2.1 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.

6

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

13-2.1 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

7

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

13-2.2 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.

8

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

13-2.2 The Analysis of Variance

We may describe the observations in Table 13-2 by the

linear statistical model:

The model could be written as

9

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

13-2.2 The Analysis of Variance

Fixed-effects Model

The treatment effects are usually defined as deviations

from the overall mean so that:

Also,

10

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

13-2.2 The Analysis of Variance

We wish to test the hypotheses:

The analysis of variance partitions the total variability

into two parts.

11

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

13-2.2 The Analysis of Variance

Definition

12

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

13-2.2 The Analysis of Variance

The ratio MSTreatments = SSTreatments/(a – 1) is called the

mean square for treatments.

13

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

13-2.2 The Analysis of Variance

The appropriate test statistic is

We would reject H0 if f0 > f,a-1,a(n-1)

14

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

13-2.2 The Analysis of Variance

Definition

15

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

13-2.2 The Analysis of Variance

Analysis of Variance Table

16

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

Example 13-1

17

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

Example 13-1

18

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

Example 13-1

19

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

20

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

Definition

For 20% hardwood, the resulting confidence interval on the mean is

21

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

Definition

For the hardwood concentration example,

22

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

An Unbalanced Experiment

23

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

13-2.3 Multiple Comparisons Following the ANOVA

The least significant difference (LSD) is

If the sample sizes are different in each treatment:

24

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

Example 13-2

25

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

Design and Analysis of

Single-Factor Experiments:

The Analysis of Variance

CHAPTER OUTLINE

1

Learning Objectives for Chapter 13

After careful study of this chapter, you should be able to do the

following:

1. Design and conduct engineering experiments involving a single factor with an

arbitrary number of levels.

2. Understand how the analysis of variance is used to analyze the data from

these experiments.

3. Assess model adequacy with residual plots.

4. Use multiple comparison procedures to identify specific differences between

means.

5. Make decisions about sample size in single-factor experiments.

6. Understand the difference between fixed and random factors.

7. Estimate variance components in an experiment involving random factors.

8. Understand the blocking principle and how it is used to isolate the effect of

nuisance factors.

9. Design and conduct experiments involving the randomized complete block

design.

2

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-1: Designing Engineering Experiments

Every experiment involves a sequence of activities:

1. Conjecture – the original hypothesis that motivates the

experiment.

2. Experiment – the test performed to investigate the

conjecture.

3. Analysis – the statistical analysis of the data from the

experiment.

4. 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.

3

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

13-2.1 An Example

4

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

13-2.1 An Example

5

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

13-2.1 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.

6

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

13-2.1 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

7

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

13-2.2 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.

8

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

13-2.2 The Analysis of Variance

We may describe the observations in Table 13-2 by the

linear statistical model:

The model could be written as

9

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

13-2.2 The Analysis of Variance

Fixed-effects Model

The treatment effects are usually defined as deviations

from the overall mean so that:

Also,

10

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

13-2.2 The Analysis of Variance

We wish to test the hypotheses:

The analysis of variance partitions the total variability

into two parts.

11

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

13-2.2 The Analysis of Variance

Definition

12

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

13-2.2 The Analysis of Variance

The ratio MSTreatments = SSTreatments/(a – 1) is called the

mean square for treatments.

13

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

13-2.2 The Analysis of Variance

The appropriate test statistic is

We would reject H0 if f0 > f,a-1,a(n-1)

14

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

13-2.2 The Analysis of Variance

Definition

15

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

13-2.2 The Analysis of Variance

Analysis of Variance Table

16

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

Example 13-1

17

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

Example 13-1

18

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

Example 13-1

19

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

20

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

Definition

For 20% hardwood, the resulting confidence interval on the mean is

21

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

Definition

For the hardwood concentration example,

22

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

An Unbalanced Experiment

23

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

13-2.3 Multiple Comparisons Following the ANOVA

The least significant difference (LSD) is

If the sample sizes are different in each treatment:

24

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13-2: The Completely Randomized Single-Factor Experiment

Example 13-2

25

© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

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