Stat 470-3

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Stat 470-3

• Today Will consider the one-way ANOVA model for
  comparing means of several treatments

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Example

• Issue shelf-life of pre-packaged meat
• Objective Compare four different packaging
  methods. Are there differences? Which packaging
  is best?
• T1. commercial plastic wrap
• T2. vacuum package
• T3. 1CO, 40O2, 59N
• T4. 100 CO2
• Factor Packaging
• Experimental units 12 steaks
• Experimental Design randomly assign 3 steaks to
  each packaging condition balanced completely
  randomized design with a 4, n 3
• Response count of bacteria after 9 days at 4oC
  (39oF)
• y log(bacteria count/cm2)

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DataAnalysis 1 Plot the Data
Notation k 4 Treatments ni 3 reps per
Treatment N 12 total observations
Eyeball Analysis Does it look like all of these
data could come from the same distribution? Or
from four different distributions?
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Experiments with a Single Factor Completely
Random Design

• Objective
• Determine if the mean response of a factor is the
  same at all levels
• If there is a difference, which levels differ?
• Method
• Have a single factors with k levels
• N experimental units available for the experiment
• N n1 n2nk
• Randomly assign treatments to different
  experimental units
• Conduct experiment
• Results yij, i1,,k j1,,ni

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Experiments with a Single Factor Completely
Random Design

• Model

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Sums of Squares
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Test Statistic
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ANOVA Table
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Summary

• We have found a statistic (F) which
• compares the variance among treatment means to
  the variance within treatments
• has a known distribution when all the treatment
  means are equal
• By comparing this F statistic to the F(k-1, N-k)
  distribution, we evaluate the strength of the
  evidence against the assumption of equal
  underlying treatment means

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Back to the Example

• Interpretation

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Comment

• When a 2 (two treatments), F for testing for no
  difference among treatments is equal to t2 in the
  two-sample (unpaired) t-test
• Out-of-Class Exercise.
• Demonstrate this equality by doing an ANOVA on
  the data in tomato plant problem.
• Compare percentiles in F and t tableswhat do you
  observe?
• For the mathematically inclined, demonstrate this
  equality algebraically

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NOTE It All Adds Up!

• It can be shown algebraically that
• Total SS Treatment SS Error SS
• Also, the degrees of freedom add up
• N-1 (k-1) (N-k)

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Exercise Out-of-Class

• By using the formulas in the ANOVA table, verify
  the above ANOVA table for the meat packaging data

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Estimation of Model Parameters Constraints

• The model is over-parameterized
• Have k types of observation
• Have (k1) parameters in the model
• k for the treatment effects
• 1 for the grand mean
• Need to impose constraints to get solution

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Constraints

• Sum to Zero Constraint
• Interpretation

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Constraints

• Baseline Constraint
• Interpretation

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Multiple Comparisons

• In previous example, we saw that there was a
  significant treatment effectso what?
• If an ANOVA is conducted and the analysis
  suggests that there is a significant treatment
  effect, then a reasonable question to ask is

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Multiple Comparisons

• Would like to see if there is a difference
  between treatments i and j
• Can use two-sample t-test statistic to do this
• For testing
  reject if
• Perform many of these tests

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Multiple Comparisons

• Perform many of these tests
• Error rate must be controlled

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Tukey Method

• Tests
• Confidence Interval

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