Post Hoc Tests on One-Way ANOVA

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Lesson 13 - 2

• Post Hoc Tests on One-Way ANOVA

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Objectives

• Perform the Tukey Test

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Vocabulary

• Post Hoc Latin for after this
• Multiple comparison methods to determine which
  means differ significantly
• Studentized Range Distribution q-test statistic
  distribution (Table 8)
• Experiment-wise error rate also known as the
  family-wise error rate is also called the level
  of significance, a
• Comparison-wise error rate is the probability
  of making a Type I error when comparing two means

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After a One-way ANOVA

• We have performed a one-way ANOVA test
• If we do not reject the null hypothesis, then
• There is not enough evidence that the population
  means are different
• There is not much else to do
• If we do reject the null hypothesis, then
• There is at least two population means that are
  different
• Which ones are they?

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Tukey Test

• The Tukey Test analyzes all pairs of population
  means
• H0 µi µj for all i and j where i ? j
• That is to say, the Tukey Test analyzes each of
• H0 µ1 µ2
• H0 µ1 µ3
• H0 µ1 µ4
• H0 µ2 µ3
• Etc

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Tukeys Test

• Computing the Q-test Statistic
• Arrange the sample means in ascending order.
• Compute the pair-wise differences, xi xj, where
  xi gt xj
• Compute the test statistic, q, for each pair-wise
  difference (listed below)
• Determine the critical value, qa,n-k,k
• If q qa,n-k,k , reject H0 µi µj and conclude
  that the means are significantly different
• Compare all pair-wise differences to identify
  which means are considered equal
• 
  x2 x1 q
  --------------------------
  s2 1 1
  
• 
  --- --- ---
• 
  2 n1 n2
• where x2 gt x1, n1 is sample size from population
  1, n2 is the sample size from population 2 and
  s2 is the MSE estimate of s2 from ANOVA

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Critical Value for Tukey

• qa,v,k
• The critical values for the test statistic q
  depend on three parameters
• The significance level, a, as usual
• The error degrees of freedom (from the ANOVA
  output), ?, which is n - k
• The total number of means compared, k
• These critical values are found in Table VIII

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ANOVA Analysis of Variance Table
Source of Variation Sum of Squares Degrees of Freedom Mean Squares F-testStatistic F Critical Value
Treatment S ni(xi x)2 k - 1 MST MST/MSE F a, k-1, n-k
Error S (ni 1)si2 n - k MSE
Total SST SSE n - 1
Use this as the estimate for s2
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Excel ANOVA Output

• For Test Statistic need the means, the nis and
  MSE
• For the Critical Value a, df (error) and k

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See Example 2 on page 694
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Summary and Homework

• Summary
• When a One-Way ANOVA indicates that not all the
  means are equal, we would want to identify which
  ones are equal and which ones are not
• The Tukey Test provides a method for that
• After performing The Tukey Test, we can conclude
  whether each mean is significantly different
  from, or can be considered equal to, each other
  mean
• Homework
• pg 696 699 1, 2, 3, 5, 6, 15, 18