ANOVA Using Matrix Algebra

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ANOVAUsing Matrix Algebra
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Single Classification

• Analysis of Variance

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Experimental Study
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Improving Memory

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

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Random Assignment
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Experimental Treatments
Group 1 Placebo Group 2 Dexfermetrazine
Group 3 Physostigmine
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Dependent Variable

• Number of Nonsense Syllabi Remembered

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Record Data
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Single 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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Columnwise Differences
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Matrix of Columnwise Differences
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Triangulation of a Matrix

• Change the Negative Elements of a Skew Symmetric
  Matrix into Zero Elements

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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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Total Expanded Variance

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

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Total Group
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Total Pairwise Differences
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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
• Error Expanded Variance 152 -26 126

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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
• Error Standard Variance 126/152 .83

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Percentage Accounted For
The group membership accounts for 17 of the
total variance.
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Do the three groups differ on the dependent
variable?
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The Null Hypothesis
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The Alternative Hypothesis
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Statistical Inference

• Methods for generalizing from samples to the
  populations

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Degrees of Freedom

• The degrees of freedom associated with the column
  variance equal 3 - 1 2
• The degrees of freedom associated with the error
  variance equal 9 - 3 6

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The F ratio for Single Classification ANOVA
Column Standard Variance
Error Standard Variance
k is the number of groups n is the number of
subjects
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Compute the F ratio
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Single Classification ANOVA