Statistics 400 - Lecture 18

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Statistics 400 - Lecture 18
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• Last Day ANOVA Example, Paired Comparisons
• Today Re-visit boys shoes...Randomized Block
  Design

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Example (Boys Shoes)

• Company ran an experiment to determine if a new
  synthetic material is better than the existing
  one used for making the soles of boys' shoes
• Experiment was run to see if the new, cheaper
  sole wears at the same rate at which the soles
  wear out

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Example (Boys Shoes)

• 10 boys were selected at random
• Each boy was given a pair of shoes
• Each pair had 1 shoe with the old sole (Sole A)
  and 1 shoe with the new sole (sole B)
• For each pair of shoes, the sole type was
  randomly assigned to the right or left foot

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Data
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• Can we use a 2-sample t-test or ANOVA here?
• Would the 2-sample t-test or ANOVA detect a
  significant difference?

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Paired or Matched Pairs T-test

• Situation
• Two measurements made on same experimental unit
• Compute difference (say B-A) in observation on
  the same experimental unit
• Analyze differences using a 1-sample t-test
• Because we analyze the differences using a
  1-sample t-test, what must we assume about the
  difference?

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Data
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Analyzing the Data
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• Just looked at comparing means for two treatments
  applied to the same experimental unit (see boys
  shoes example)
• Used a matched pairs T-test and analyzed the
  differences to see if there was a significant
  difference in the treatment means
• When more than 2 treatments are applied to the
  same experimental unit, the experiment is called
  a randomized block experiment

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Example

• An experiment was performed to investigate the
  impact of soil salinity on the growth of salt
  marsh plants (C. Schwarz, 2001)
• Plots of land at 4 agriculture field stations
  were used to grow plants in this environment
• Six different amounts of salt (in ppm) are to be
  investigated
• The plots of land were divided into 6 smaller
  plots
• Each of the 6 smaller plots were treated with a
  different amount of salt and the bio-mass at the
  end of several months recorded

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Data
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• The application of the 6 treatments to the
  smaller plots are done randomly
• Like the Boys Shoes Example, each experimental
  unit has received more than 1 treatment
• Here each unit receives 6 treatments

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Plot of Bio-mass for Each Treatment
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Plot of Bio-mass for each plot
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Observations

• Notice that the 4th plot gives smaller results
  than the other plots
• Due to a block (plot effect)
• Similar to the way a boy wears his shoes
• Are the observations independent?

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Randomized Block Design

• Situation
• Have k treatments
• Have b blocks
• Each of the k treatments appears in each of the b
  blocks
• The treatments within block are assigned to the
  within block units in random order

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Structure of Data

• Have k treatments in b blocks
• Denote ith treatment from the jth block as yij

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Model

• Model for comparing k treatments from a
  randomized block design
• for i
  1, 2, , k and j 1, 2, , b
• where is the overall mean, and
• is the i th treatment effect
• is the jth block effect
• eij has a distribution
• Want to test

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ANOVA Table for the Bio-Mass Example
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What are the F-Tests?
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Why do this?
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• General Rule Block what you can, randomize what
  you cannot