One-Way%20Analysis%20of%20Variance

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One-Way Analysis of Variance

• Stat 700 Lecture 12
• 11/29/2001

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Illustrative Example

• Four chemical plants, producing the same product
  and owned by the same company, discharge
  effluents into streams in the vicinity of their
  locations. To check the extent of the pollution
  created by the effluents and to determine whether
  this varies from plant to plant, the company
  collected random samples of liquid waste, five
  specimens for each of the four plants.
• The data is presented in the table of the next
  slide.

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The Data
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Another Example

• College students were assigned to various study
  methods in an experiment to determine the effect
  of study technique on learning. The data
  presented in the next table was generated to be
  consistent with summary quantities found in the
  paper The Effect of Study Techniques, Study
  Preferences and Familiarity on Later Recall.
  The study methods compared were reading only,
  reading and underlining, and reading and taking
  notes. One week after studying the paper Love
  in Infant Monkeys students were given an exam
  on the article.

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The Data on Learning
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Testing Equality of Several Population Means
The setting is that there are p normal
populations each with variance . This
assumption of equal variances is called the
homoscedasticity assumption. The ith normal
population has mean mi, i1,2,p. The goal is to
test the null hypothesis that the p population
means are all identical, versus the alternative
hypothesis that there are at least two means
which are different.
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Sample Data
For the ith population, we observe a random
sample of size ni, i1,2,,p. The data for this
sample therefore consists of Yi1, Yi2, , Yin(i).
The sample data for the p samples could then be
summarized in tabular form as follows
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Data in Tabular Form
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Test Procedure
The test procedure for testing the equality of
the p population means is based on the
F-distribution, and is usually called the one-way
analysis of variance. The test statistic is given
by
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ANOVA Representation
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Formulas
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Example Using Effluents Data
We present and illustrate the analysis using
Minitab. In the next slide is the output from the
Minitab analysis. We will illustrate how this is
done in class.
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Output from Minitab
One-way Analysis of Variance Analysis of
Variance Source DF SS MS
F P Factor 3 0.4649 0.1550
5.20 0.011 Error 16 0.4768
0.0298 Total 19 0.9417
Individual 95 CIs For Mean
Based on Pooled
StDev Level N Mean StDev
--------------------------------- A
5 1.5680 0.1366 (---------------) B
5 1.7720 0.2160
(---------------) C 5 1.5460
0.1592 (--------------) D 5
1.9160 0.1689
(--------------)
--------------------------------- Pooled
StDev 0.1726 1.40 1.60
1.80 2.00
Conclusion The p-value of .011 is quite small,
so there is indication that at least two
population means are different.
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Calculations from Excel
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Example for RCB(Two-Analysis of Variance)

• A study was conducted to compare the effects of
  three levels of digitalis on the levels of
  calcium in the heart muscles of dogs. A
  description of the actual experimental procedure
  is omitted, but it is sufficient to note that the
  general level of calcium uptake varies from one
  animal to another so that comparison of digitalis
  levels (treatments) had to be blocked on heart
  muscles. That is, the tissue for a heart muscle
  was regarded as a block, and comparisons of the
  three treatments were made within a given muscle.
  The calcium uptakes for the three levels of
  digitalis, A, B, and C, were compared based on
  the heart muscles of the four dogs.

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The Raw Data from Study
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