ANOVA Using Matrix Algebra

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ANOVAUsing Matrix Algebra
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Double Classification

• Analysis of Variance

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Improving Memory

• Population Patients Suffering from Korsakoff
  Psychosis
• Randomly Select Three Patients

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Experimental Treatments
Condition 1 Placebo Condition 2
Dexfermetrazine Condition 3 Physostigmine
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Repeated Measures Design
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Dependent Variable

• Number of Nonsense Syllabi Remembered

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Record Data
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Double Classification ANOVA
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Experimental Conditions
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Column Expanded Variance

• Due to Experimental Effects

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Column Expanded Variance

• Compute Columnwise DifferenceX - X
• Square and Sum the Positive Elements in the
  Resulting Matrix

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Matrix of Columnwise Differences
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Triangulate the Resulting Matrix
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Square and Sum the Matrix Elements
Column Expanded Variance 26
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The Subject Factor
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Row Expanded Variance

• Due to the Subject Factor

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Row Expanded Variance

• Compute Rowwise DifferencesX - X
• Square and Sum the Positive Elements in the
  Resulting Matrix

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Matrix of Rowwise Differences
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Triangulate the Resulting Matrix
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Square and Sum the Matrix Elements
Row Expanded Variance 122
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Total Expanded Variance

• Combine Separate Groups into One GroupT
• Compute the Total Pairwise DifferenceT - T
• Square and Sum the Positive Elements in the
  Resulting Matrix

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Total Group
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Matrix of Total Pairwise Differences
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Triangulate the Resulting Matrix
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Square and Sum the Matrix Elements
Total Expanded Variance 152
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Expanded Variance Components

• Total Expanded Variance 152
• Column Expanded Variance 26
• Row Expanded Variance 122
• Error Expanded Variance 152 -26 -122 4

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What percentage of variance in the dependent
variable was accounted forby the group
membership?
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Standard Variance Components
Divide each variance component by the total
variance.

• Total Standard Variance 152/152 1.00
• Column Standard Variance 26/152 .17
• Row Standard Variance 122 / 152 .80
• Error Standard Variance 4/152 .03

The group membership accounts for 17 of the
total variance.
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Standard Variance Components
Error
Column
Row
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Do the three groups differ on the dependent
variable?
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The F Ratio for Double Classification ANOVA
Column Standard Variance
Error Standard Variance
k is the number of columns n is the number of rows
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Compute the F ratio
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Double Classification ANOVA
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Comparison