design analysis of single factor experiments variance

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Learning Objectives for Chapter 13

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

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.

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© 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

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.

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13-2: The Completely Randomized Single-Factor Experiment

13-2.1 An Example

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• The levels of the factor are sometimes called

treatments

• Each treatment has six observations or replicates

• The runs are run in random order.

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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

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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

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

linear statistical model:

The model could be written as

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Fixed-effects Model

The treatment effects are usually defined as deviations

from the overall mean so that:

Also,

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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.

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Definition

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The ratio MS Treatments = SS Treatments /(a – 1) is called the

mean square for treatments.

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The appropriate test statistic is

We would reject H 0 if f 0 > f ,a-1,a(n-1)

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Definition

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Analysis of Variance Table

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Example 13-1

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

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Definition

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

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Definition

For the hardwood concentration example,

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An Unbalanced Experiment

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13-2.3 Multiple Comparisons Following the ANOVA

The least significant difference (LSD) is

If the sample sizes are different in each treatment:

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Figure 13-2 Results of Fisher’s LSD method in Example 13-2

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13-2: The Completely Randomized Single-Factor Experiment 13-2.5 Residual Analysis and Model Checking

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Figure 13-4 Normal probability plot of

residuals from the hardwood

concentration experiment

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Figure 13-5 Plot of residuals versus

factor levels (hardwood

concentration)

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Figure 13-6 Plot of residuals versus

i

y

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13-3: The Random-Effects Model 13-3.1 Fixed versus Random Factors

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13-3: The Random-Effects Model 13-3.2 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.

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For a random-effects model , the appropriate

hypotheses to test are:

The ANOVA decomposition of total variability is

still valid:

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13-3: The Random-Effects Model

13-3.2 ANOVA and Variance Components

The expected values of the mean squares are

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The estimators of the variance components are

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

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

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13-3: The Random-Effects Model

Figure 13-8 The distribution of fabric strength (a) Current process, (b) improved

process.

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13-4: Randomized Complete Block Designs

13-4.1 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.

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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.

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General procedure for a randomized complete block

design:

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The appropriate linear statistical model:

We assume

• treatments and blocks are initially fixed effects

• blocks do not interact

•

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13-4: Randomized Complete Block Designs 13-4.1 Design and Statistical Analyses

We are interested in testing:

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The mean squares are:

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The expected values of these mean squares are:

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Definition

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

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

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13-4: Randomized Complete Block Designs 13-4.2 Multiple Comparisons

Fisher’s Least Significant Difference for Example 13-5

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13-4: Randomized Complete Block Designs 13-4.3 Residual Analysis and Model Checking

Figure 13-11 Normal probability

plot of residuals from the

randomized complete block

design.

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Figure 13-13 Residuals by block.

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Important Terms & Concepts of Chapter 13

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