Test Statistic for ANOVA

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Lesson 26a
F-test in ANOVA

• Test Statistic for ANOVA
• F-tables
• ANOVA table

Main Ideas
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• 1. Test Statistic
• If there are no differences among population
  means, then MST and MSE will have values that are
  about the same.
• If the population means are not all the same, MST
  will tend to be larger than MSE.
• Thus a measure of whether or not the means are
  equal is the F-statistic defined by

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• 2. Reference Distribution
• If the means of the populations are not all
  equal, the F-statistic will be significantly
  greater than 1.
• The reference distribution is called the
  F-distribution.
• To use the distribution, we must specify the d.f.
  of MST which is k - 1 (numerator d.f.) and the
  d.f. of MSE which is n1 n2 nk - k
  (denominator d.f.)

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• 3. Picture of an F-Distribution

Area .05
0
If F is here, reject Ho
If F is here, accept Ho
Cut-off point depends on d.f. of numerator and
denominator
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• 4. Segment of F-Tables (5 level)
• d.f. numerator
• 1 2 3 4
• ...
• 11 4.84 3.98 3.59 3.36 ...
• 12 4.75 3.89 3.49 3.26 ...
• 13 4.67 3.81 3.41 3.18 ...
• ...

d.f. denominator
If d.f. for numerator 2 and d.f of denominator
11, then we reject H0 if F gt 3.98.
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• Example 1. We will compute the F-statistic for
  the data we used in Lecture 25b.
• We have MST 19.65 with d.f. 2, and MSE 3.47
  with d.f. 11, so F 19.65/3.47 5.66.
• The critical value is 3.98. Thus, F falls in the
  reject H0 region, so we conclude that the means
  are significantly different from one another.

Pop. Observations Mean S.D. n 1 2
3 4 5 6 4.0 1.58 5 2 1 2 3 4
2.5 1.29 4 3 4 5 6 8 10 6.6 2.41 5
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• 5. Analysis of Variance Table
• The results are usually summarized in an analysis
  of variance table.
• An ANOVA table will contain
• Sources of variation (between and within group,
  or treatment and error)
• Sum of squares (SST, SSE)
• Degrees of freedom
• Mean squares (MST, MSE)
• F statistic
• p-value and perhaps a critical F value

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• 6. ANOVA Table from Excel
• The following table was produced by the data
  analysis tool ANOVA-Single Factor.
• Other spreadsheets and statistical packages have
  similar ANOVA outputs.

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If ANOVA shows us that the means are not all
equal, how do we know which ones are different
from the others?
Data analysis does not stop with ANOVA. We have
to follow up with a so-called multiple comparison
procedure. This is the topic of the next section.
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