Chapter 15 Statistical Pyramid

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Chapter 15 Statistical Pyramid
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One Factor Model
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One Factor Model
Dirty
Clean
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Relation to Regression and ANCOVA
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Example gas.sav

• One-Way ANOVA Example
• Gas Mileage -
• Does average gas mileage differ between the
  standard four-wheel drive pickup trucks made by
  Chevrolet, Dodge and Ford?
• Use the following miles per gallon data.
• Chevy Dodge Ford
• 15.2 14.8 15.1
• 15.4 14.4 14.3
• 14.8 14.3 14.6
• 14.4 14.1 13.9
• 14.7 14.4 14.6

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Two Factors

• Model(Clean)
• where
• are parameters
• are independent

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Important Features

• 1. The parameter is the mean response
  for the treatment in which factor A is at the I
  th level and factor B is at the j th level.
  This follows because
• 2. Since is a constant, the variance of
  is

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Example a2, b3
Cell Means b
a
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Definitions

• Main Effect A
• and the row average for the ith row is denoted by
  
• Main Effect B
• The overall mean learning time for all ages and
  both genders is denoted by , and is
  defined in the following equivalent fashion

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Another Model

• Dirty Model

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Definitions/ Old Fashion

• Main Effect A
• Main Effect B
• Interaction

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Definitions/ New

• Intercept u
• Main Effect A
• Main Effect B
• Interaction

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19.6 F Tests
Test for Interactions
The TS is
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Main Effects

• Test for Factor A Main Effects
• the TS is

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• Test for Factor B Main Effects
• the TS is

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Can we test Main Effects if Interaction is
Significant?

• Yes, but the interpretation may get complex.
• Sometimes it is necessary
• Example Three types of soil on my ranch. I have
  4 fertilizers to choose from. I am growing Kline
  grass. I must choose only 1 fertilizer.

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Estimated Means
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Estimated Means
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Estimated Means
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Estimated Means
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Paint Paving Example
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Y life of paint on roadway

• Paint
• YELLOW 1
• YELLOW 2
• WHITE 1
• WHITE 2
• Type
• Asphalt 1
• Asphalt 2
• Concrete

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Since there is no data can I formulate null
hypotheses at this point?

• Ho there is a difference in paints
• Ho there is a difference in type
• Ho there is an interaction between paint and type

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Since there is no data can I formulate null
hypotheses at this point?

• Ho there is NO difference in paints
• Ho there is NO difference in type
• Ho there is NO interaction between paint and type

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Population Parameters

• How many populations are generated by this
  experiment?
• What are the population parameters?
• What is the model?
• What are the usual assumptions?

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Design Setup
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What does it mean?

• Ho there is NO difference in paints
• Ho there is NO difference in type
• Ho there is NO interaction between paint and
  type

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What do we want to test?

• Paint Questions
• Yellow 1 versus Yellow 2
• White 1 versus White 2
• Yellow versus White
• Type Questions
• Asphalt 1 versus Asphalt 2
• Asphalt versus Concrete

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What do we want to test?

• Interaction Hypotheses
• Yellow By Asphalt
• White By Asphalt
• Color By Asphalt
• Yellow By Type
• White By Type
• Color By Type

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Nice questions , but .

• We can ONLY test questions about the populations
  parameters ( or the population distribution)
• Need a translator

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Translate the questions
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Data

• We have no data
• - Must we sample each population the same number
  of times
• How do we do the test?

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SPSS - Lmatrix

• Population Parameters
• Computer parameters

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