Stat 470-5

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Stat 470-5

• Today General Linear Model
• Assignment 1

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General Linear Model

• ANOVA model can be viewed as a special case of
  the general linear model or regression model
• Suppose have response, y, which is thought to be
  related to p predictors (sometimes called
  explanatory variables or regressors)
• Predictors x1, x2,,xp
• Model

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Example Rainfall (Exercise 2.16)

• In winter, a plastic rain gauge cannot be used to
  collect precipitation because it will freeze and
  crack. Instead, metal cans are used to collect
  snowfall and the snow is allowed to melt indoors.
  The water is then poured into a plastic rain
  gauge and a measurement recorded. An estimate of
  snowfall is obtained by multiplying this
  measurement by 0.44.
• One observer questions this and decides to
  collect data to test the validity of this
  approach
• For each rainfall in a summer, she measures (i)
  rainfall using a plastic rain gauge, (ii) using a
  metal can
• What is the current model being used?

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Example Rainfall (Exercise 2.16)
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Example Rainfall (Exercise 2.16)

• Seems to be a linear relationship
• Will use regression to establish linear
  relationship between x and y
• What should the slope be?

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Example Rainfall (Exercise 2.16)
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Example Rainfall (Exercise 2.16)
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Example Rainfall (Exercise 2.16)
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Example Rainfall (Exercise 2.16)
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Comments

• General linear model may have many predictors
• Is suitable for many situations
• Easily done in all stats packages

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Designs So Far

• Have considered 1-factor designs
• Paired comparisons (paired t-test)
• Completely randomized design (ANOVA)
• Frequently have more than one factor
• We will learn to design and analyze such
  experiments

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Example Penicillin Experiment

• Objective Compare four processes for making
  penicillin
• The raw material used in the process is thought
  to vary substantially from batch to batch
• Experiment Design
• Use five separately produced batches of raw
  material
• Divide each batch into four sub-batches
• Randomly assign each process to one sub-batch.
• Randomize the production order within each batch
  
• Measure the yield ()

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Blocking

• Paired comparisons (Section 2.1) is a special
  case of a Randomized Complete Block (RCB) design
• More generally
• Have k treatments
• have b blocks
• each of the k treatments is applied (in random
  order) to each block

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Blocking

• Units within a block are more homogeneous than
  units between blocks
• Can remove variability due to blocks (e.g., boy
  to boy variability) from the comparison of
  treatments

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Model

• i1, 2, , b
• j1, 2, ,k

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ANOVA Table
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Hypothesis Tests
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Multiple Comparisons
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Example Penicillin Experiment

• Objective Compare four processes for making
  penicillin
• The raw material used in the process is thought
  to vary substantially from batch to batch
• Experiment Design
• Use five separately produced batches of raw
  material
• Divide each batch into four sub-batches
• Randomly assign each process to one sub-batch.
• Randomize the production order within each batch
  
• Measure the yield ()
• This is a RCB design with
• b
• k

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Data Penicillin Example
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Yield versus Process (grouped by blocks)
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Observations

• Some consistent differences among batches
  generally, B1 high, B5 low
• No apparent consistent differences among
  processes

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ANOVA Randomized Block Design
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Conclusions

• F-value for Processes is not significant at
• F-value for Batches (P .04) is significant at
  indicates some differences among
  batches of raw material
• We suspected batch differences thats why the
  design was done this way. This result is no
  surprise or of particular interest, in this case.
• Which would you use?

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Diagnostic Checking

• Residual plots -- penicillin experiment
• To check Normality assumption
• plot all residuals dot chart, histogram, Normal
  prob. plot
• To check assumption of equal variances
• dot plot of residuals by Treatment
• dot plot of residuals by Block
• Other possible checks
• plot residuals vs. testing order
• plot residuals vs. other potential sources of
  variability
• e.g., vs. technician, or machine, etc.

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Randomized Block Design -- Summary

• Objective
• Compare several treatments for a factor
• eliminate source of variability from comparison
  of treatments
• broaden conclusions
• Experimental Method
• create b blocks each with a experimental units
• in each block, randomly assign each treatment to
  one experimental unit
• Analysis
• ANOVA Blocks, Treatments, Error are sources of
  variation

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Why Bother?

• Can remove variability due to blocks (e.g., boy
  to boy variability) from the comparison of
  treatments
• Removing source of variability often increases
  power to detect treatment differences
• Make comparisons on more homogeneous units

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Examples of Blocking Variables

• Blocks are units that can be sub-divided into
  sub-units
• Time
• Space
• People
• Batches