Chapter 7 One Way ANOVA Model

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Chapter 7One Way ANOVA Model
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General Experimental Setting

• Investigator Controls One or More Independent
  Variables
• Called treatment variables or factors
• Each treatment factor contains two or more groups
  (or levels)
• Observe Effects on Dependent Variable
• Response to groups (or levels) of independent
  variable
• Experimental Design The Plan Used to Test
  Hypothesis

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Completely Randomized Design

• Experimental Units (Subjects) are Assigned
  Randomly to Groups
• Subjects are assumed homogeneous
• Only One Factor or Independent Variable
• With 2 or more groups (or levels)
• Analyzed by One-way Analysis of Variance (ANOVA)

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Randomized Design ExampleTraining example
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Example 1

• Comparing magazine covers
• A magazine publisher wants to compare three
  different styles of covers for a magazine that
  will be offered for sale at supermarket checkout
  lines. She assigns 60 stores at random to the
  three styles of covers and records the number of
• magazines that are sold in a one-week period.

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Example 1

• Design 2 had the highest average sales
• Purpose of ANOVA is to see if this observed
  difference is statistically significant

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Hypotheses of One-Way ANOVA

• All population means are equal
• No treatment effect (no variation in means among
  groups)
• At least one population mean is different (others
  may be the same!)
• There is a treatment effect
• Does not mean that all population means are
  different

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One-way ANOVA (No Treatment Effect)
The Null Hypothesis is True
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One-way ANOVA (Treatment Effect Present)
The Null Hypothesis is NOT True
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Example 2
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Example 2 Box Plots
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Example 2 Means Plot

• P-value lt0.0001
• Have enough evidence to reject the null
  hypothesis and conclude that the three groups of
  students do not all have the same mean scores

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One-way ANOVA Model
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ANOVA Table
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One-way ANOVA(Partition of Total Variation)
Total Variation SST
Variation Due to Treatment SSA
Variation Due to Random Sampling SSW

• Commonly referred to as
• Within Group Variation
• Sum of Squares Within
• Sum of Squares Error
• Sum of Squares Unexplained

• Commonly referred to as
• Among Group Variation
• Sum of Squares Among
• Sum of Squares Between
• Sum of Squares Factor
• Sum of Squares Explained
• Sum of Squares Treatment

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Total Variation
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Total Variation
(continued)
Response, X
Group 1
Group 2
Group 3
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Among-Group Variation (Factor)
Variation Due to Differences Among Groups.
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Among-Group Variation
(continued)
Response, X
Group 1
Group 2
Group 3
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Within-Group Variation (Error)
Summing the variation within each group and then
adding over all groups.
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Within-Group Variation
(continued)
Response, X
Group 1
Group 2
Group 3
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Within-Group Variation
(continued)
For c 2, this is the pooled-variance in the
t-Test.

• If more than 2 groups, use F Test.
• For 2 groups, use t-Test. F Test more limited.

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One-way ANOVA Summary Table
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One-way Analysis of VarianceF Test

• Evaluate the Difference among the Mean Responses
  of 2 or More (c ) Populations
• E.g. Several types of tires, oven temperature
  settings
• Assumptions
• Samples are randomly and independently drawn
• This condition must be met
• Populations are normally distributed
• F Test is robust to moderate departure from
  normality
• Populations have equal variances
• Less sensitive to this requirement when samples
  are of equal size from each population

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One-way ANOVAF Test Statistic

• Test Statistic
• MSF/MSE
• MSA is mean squares among
• MSW is mean squares within
• Degrees of Freedom

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Features of One-way ANOVA F Statistic

• The F Statistic is the Ratio of the Among
  Estimate of Variance and the Within Estimate of
  Variance
• The ratio must always be positive

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Features of One-way ANOVA F Statistic
(continued)

• If the Null Hypothesis is True
• df1 c -1 will typically be small
• df2 n - c will typically be large
• The ratio should be close to 1.
• If the Null Hypothesis is False
• The numerator should be greater than the
  denominator
• The ratio should be larger than 1

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Training example
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One-way ANOVA F Test Example
Machine1 Machine2 Machine3 25.40 23.40
20.00 26.31 21.80 22.20 24.10
23.50 19.75 23.74 22.75
20.60 25.10 21.60 20.40

• As production manager, you want to see if 3
  machines have different mean running times. You
  assign 15 similarly trained and experienced
  workers, 5 per machine, to the machines. At the
  .05 significance level, is there a difference in
  mean running times?

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One-way ANOVA Example Scatter Diagram
Machine1 Machine2 Machine3 25.40 23.40
20.00 26.31 21.80 22.20 24.10
23.50 19.75 23.74 22.75
20.60 25.10 21.60 20.40
Time in Seconds
27 26 25 24 23 22 21 20 19















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One-way ANOVA Example Computations
Machine1 Machine2 Machine3 25.40 23.40
20.00 26.31 21.80 22.20 24.10
23.50 19.75 23.74 22.75
20.60 25.10 21.60 20.40
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Summary Table
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One-way ANOVA Example Solution
Test Statistic Decision Conclusion

• H0 ?1 ?2 ?3
• H1 Not All Equal
• ? .05
• df1 2 df2 12
• Critical Value(s)

MSF
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5820
.
Ftest
?
?
?
25
6
.
MSE
9211
.
Reject at ? 0.05
? 0.05
There is evidence that at least one ? i differs
from the rest.
F
0
3.89
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ANOVA in JMP

• Declare treatments as nominal
• Then proceed as in the regression case
• Analyze ? Fit Y by X
• Lets take a look at an example

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Example using JMP Cigarette Filters

• Suppose we have the following data
• An experiment was designed to compare the amount
  of tar passing through four types of cigarette
  filters.

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Cigarette Filter JMP Output
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Multiple Comparison Procedures

• If H0 is rejected, we conclude that the means are
  different
• We would like to know exactly where the means
  differ
• This is where Multiple Comparison Procedures come
  into play
• We will examine a few Tukeys and Dunnetts

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The Tukey-Kramer Procedure

• Tells which Population Means are Significantly
  Different
• e.g., ?1 ?2 ? ?3
• 2 groups whose means may be significantly
  different
• Post Hoc (a posteriori) Procedure
• Done after rejection of equal means in ANOVA
• Pairwise Comparisons
• Compare absolute mean differences with critical
  range

f(X)
X
?
?
?

1
2
3
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The Tukey-Kramer Procedure Example

• 1. Compute absolute mean differences

Machine1 Machine2 Machine3 25.40 23.40
20.00 26.31 21.80 22.20 24.10
23.50 19.75 23.74 22.75
20.60 25.10 21.60 20.40
2. Compute Critical Range 3. All of the
absolute mean differences are greater than the
critical range. There is a significance
difference between each pair of means at the 5
level of significance.
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Tukeys Test in JMP

• In the output with the ANOVA table, click on the
  red triangle
• Select Compare Means
• Select Tukeys HSD

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Tukeys Test Cigarette Filter Example
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Tukeys Test Cigarette Filters

• Notice that from the JMP output, we see that
  levels connected by the same letter are NOT
  significantly different.
• Therefore, we conclude that the following pairs
  of filters are significantly different
• (1,2) and (1,4)

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Dunnetts Procedure

• This procedure allows us to test whether means
  are statistically different from the mean of a
  control group.
• This is only used with experiments in which there
  is a control group.

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Dunnetts Procedure in JMP

• Again, select Compare Means from the red triangle
  menu
• Select With Control, Dunnetts
• Select which treatment factor is the control
  group
• Click OK

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Dunnetts Cigarette Filters

• Suppose we conducted this experiment where Filter
  1 is the control
• We obtain the following output

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Dunnetts Cigarette Filters

• Based on the above output, we see that positive
  numbers indicate significant differences from the
  control
• Therefore, filters 2,3, and 4 are all
  significantly different from the control filter 1.