Chapter 6 Single Factor Experiments and ANOVA

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Chapter 6Single Factor Experiments and ANOVA
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Agenda

• Designing engineering experiments
• Completely randomized design
• The random effects model
• Randomized complete block design

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Designing Engineering Experiments

• An is a test or series of tests.
• The of an experiment plays a major role in
  the eventual solution of the problem.
• In a , experimental trials (or runs) are
  performed at all combinations of the factor
  levels.
• The (ANOVA) will be used as one of the
  primary tools for statistical data analysis.

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Designing Engineering Experiments

• Every experiment involves a sequence of
  activities
• the original hypothesis that motivates the
  experiment.
• the test performed to investigate the
  conjecture.
• the statistical analysis of the data from
  the experiment.
• 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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The Completely Randomized Single-Factor Experiment
An Example
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The Completely Randomized Single-Factor Experiment
An Example
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The Completely Randomized Single-Factor Experiment
An Example

• The levels of the factor are sometimes called
  .
• Each treatment has six observations or .
• The runs are run in order.

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

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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 .
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The Completely Randomized Single-Factor Experiment
The Analysis of Variance
We may describe the observations in Table 13-2 by
the linear statistical model (1)
The model could be written as
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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,
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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.
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The Completely Randomized Single-Factor Experiment
The Analysis of Variance
Definition
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The Completely Randomized Single-Factor Experiment
The Analysis of Variance
The ratio MSTreatments SSTreatments/(a 1) is
called the mean square for treatments.
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The Completely Randomized Single-Factor Experiment
The Analysis of Variance
The appropriate test statistic is
We would reject H0 if f0 gt f?,a-1,a(n-1)
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The Completely Randomized Single-Factor Experiment
The Analysis of Variance
Definition
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The Completely Randomized Single-Factor Experiment
The Analysis of Variance
Analysis of Variance Table
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Example 1
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Example 1
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Example 1
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Minitab Practice for Example 1

• Data file Example6_1.xls ( )
• Menu ?Stat ? ANOVA ?
• Response
• Factor
• ? Comparisons Select

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The Completely Randomized Single-Factor Experiment
Definition
For 20 hardwood, the resulting confidence
interval on the mean is
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The Completely Randomized Single-Factor Experiment
Definition
For the hardwood concentration example,
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The Completely Randomized Single-Factor Experiment
An Unbalanced Experiment
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The Completely Randomized Single-Factor Experiment
Multiple Comparisons Following the ANOVA
The (LSD) is
If the sample sizes are different in each
treatment
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Example 2
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Example 2
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Example 2
Results of Fishers LSD method in Example 2

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The Completely Randomized Single-Factor Experiment
Residual Analysis and Model Checking
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The Completely Randomized Single-Factor Experiment
Residual Analysis and Model Checking
Normal probability plot of residuals from the
hardwood concentration experiment.
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The Completely Randomized Single-Factor Experiment
Residual Analysis and Model Checking
Plot of residuals versus factor levels (hardwood
concentration).
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The Completely Randomized Single-Factor Experiment
Residual Analysis and Model Checking
Plot of residuals versus
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The Random-Effects Model
Fixed versus Random Factors
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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 .
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The Random-Effects Model
ANOVA and Variance Components
For a , the appropriate hypotheses to test
are
The ANOVA decomposition of total variability is
still valid
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The Random-Effects Model
ANOVA and Variance Components
The expected values of the mean squares are
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The Random-Effects Model
ANOVA and Variance Components
The estimators of the variance components are
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Example 3
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Example 3
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Example 3
The distribution of fabric strength. (a) Current
process, (b) improved process.
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Randomized Complete Block Designs
Design and Statistical Analyses
The is an extension of the paired t-test to
situations where the factor of interest has more
than two levels.
A randomized complete block design.
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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.
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Randomized Complete Block Designs
Design and Statistical Analyses
General procedure for a randomized complete block
design
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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

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Randomized Complete Block Designs
Design and Statistical Analyses
We are interested in testing
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Randomized Complete Block Designs
Design and Statistical Analyses
The mean squares are
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Randomized Complete Block Designs
Design and Statistical Analyses
The expected values of these mean squares are
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Randomized Complete Block Designs
Design and Statistical Analyses
Definition
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Randomized Complete Block Designs
Design and Statistical Analyses
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Example 4
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Example 4
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Example 4
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Example 4
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Minitab Practice for Example 4

• Data File Example 6_4 ( )
• Menu ? Stat ? ANOVA ?
• Response
• Model
• ? Options check
• ? Results check .

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Randomized Complete Block Designs
Multiple Comparisons Fishers Least Significant
Difference for Example 4
Results of Fishers LSD method.
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Randomized Complete Block Designs
Residual Analysis and Model Checking
Normal probability plot of residuals from the
randomized complete block design.
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Randomized Complete Block Designs
Residuals by treatment.
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Randomized Complete Block Designs
Residuals by block.
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Randomized Complete Block Designs
Residuals versus yij.