Psyc613 Multivariate Statistics -- Syllabus

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Instructor: William P. Dunlap Book: Tabachnick & Fidell (1989). Using multivariate statistics. Grading: Homework and Take Home Final Homework: All homework must be turned in by the end of the course.

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1. Overview of Multivariate Methods
   
   
   1. Techniques that treat multiple DV's simultaneously
      1. Control of Type I errors across DV's
      2. Perhaps a better understanding of complex relationships between DV's
   2. Brief outline of multivariate methods
      1. Hypothesis testing techniques
         1. Canonical correlation
         2. MANOVA & Discriminant analysis
         3. Relations to multiple regression
         4. Relations to ANOVA
         5. Relations to Chi-square
      2. Discriptive techniques
         1. Factor analysis
         2. Cluster analysis

2. Matrix Algebra (Just enough to get by)
   
   
   1. Matrices, vectors, & scalars
   2. Matrix addition & subtraction
   3. Transpose of a matrix
   4. Matrix multiplication
      1. Row by column
      2. Requirements for multiplication
      3. A*B does not equal B*A
   5. Raw data matrix, D; Matrix of standard scores, Z
      1. R=Z'Z(1/N)
      2. VC the variance-covariance matrix
      3. Simultaneous equations, C*x=y
   6. Matrix inverses (how to divide in matrix algebra)
      1. Determinants (Multivariate variances)
         1. 2X2 matrices
         2. 3X3 matrices (and larger)
         3. Cofactors and the change of sign
      2. The inverse, X-1
         1. Transpose of the cofactor matrix divided by determinant
         2. The identity matrix; X*X-1 = X-1*X = I
         3. Solutions to simultaneous equations; x=C-1*y
   7. Eigenvalues and eigenvectors
      1. Xv=cv; the defining equation
      2. Hotellings method
      3. Normalizing a vector
      4. Best fitting ellipses & ellipsoids
         1. Centile contours (centours)
         2. Axes of ellipsoids are eigenvectors
         3. Eigenvalues represent explained variance

3. Brief Review of Multiple Regression
   
   
   1. Why Multiple Regression is a Univariate Technique
   2. Concept of the Best Linear Composite of Predictors
   3. Normal Equations - Solution by Matrix Inversion
   4. Inverse of the Correlation Matrix
   5. Types of Multiple Regression
      1. Standard (preferred)
      2. Stepwise
   6. Problems
      1. Multivariate Normality
      2. Multicollinearity
      3. Singularity
      4. Outliers

4. Factor Analysis
   
   
   1. Principal Component Analysis
      1. Eigenvectors of Correlation Matrix are Factors
      2. Eigenvalues are the Variance Explained
      3. Factor Weights -- Eigenvectors Normalized to 1/Lambda
      4. Factor Loadings -- Eigenvectors Normalized to Lambda
      5. Number of Factors
         1. Eigenvalues greater than 1.0
         2. Interpretability
         3. Scree plot
         4. Parallel analysis
      6. Rotation of Factors
         1. Graphical Rotation
         2. Varimax Rotation
         3. Other Orthogonal Rotations
         4. Oblique Rotation
   2. Principal Factor Analysis
      1. The Concept of Error
      2. The Common Factor Model
         1. Reliability
         2. Communality
         3. Common Factors
         4. Specific Factors
         5. Estimating Reliabilities
         6. Iterating to Stable Communalities
   3. Kaiser's Alpha Factor Analysis (the Little Jiffy)
   4. Guttman's Image Covariance Analysis
   5. Rao's Maximum Likelihood Factor Analysis

5. Multivariate Analysis of Variance (MANOVA) & Discriminant Analysis
   
   
   1. General Purposes
      1. Protection of Type I Error Rate for All DVs
      2. Uncovering Composites of DVs that Maximally Discriminate
   2. Mechanics
      1. Find a Linear Composite of DVs that Maximizes F
      2. The B Matrix - Sums of Square And Crossproducts Between
      3. The W Matrix - SS & SCP Within Groups
      4. Eigenvectors & Eigenvalues of BW-1
      5. Wilks Lambda - |W|/|T|
   3. Difference Between MANOVA & Discriminant Analysis
      1. MANOVA concentrates on test of significance
      2. Discriminant Analysis focuses on the Linear Composite
   4. Interpretation of Results
      1. At least One DV Differentiates the Groups
      2. Interpretation of the Discriminant Function - Loadings
   5. Tests Subsequent
      1. Follow up with Univariate ANOVA
      2. Hierarchical tests with ANCOVA (not recommended)
   6. Profile Analysis
      1. A MANOVA in the form of Ss/AxB
      2. Scaling variables for profile analysis
   7. Problems of Classification with Discriminant Analysis

6. Canonical Correlation
   
   
   1. The Grandfather of Parametric Statistics
      1. Multiple Regression - One Variable on Either Side
      2. MANOVA - One Set is Dummy Predictors of Groups
      3. ANOVA & r - Just Simpler Cases
      4. Chi Square - Can be done with Canonical Correlation
   2. Mechanics
      1. The Matrix - XX-1*XY*YY-1*YX
      2. Eigenvalues Equal RC2
      3. Eigenvectors are Maximizing Linear Composites
   3. Interpretation
      1. A Two Sided Factor Analysis
      2. Protection Against Type I Errors
      3. A First Step in Investigating Relationships Among Sets
   4. Tests Subsequent - Multiple R and Simple r

7. Cluster Analysis
   
   
   1. Purposes
      1. Discovering Subgroupings of Subjects
      2. Numerical Taxonomy
   2. Types of Cluster Analysis
      1. Hierarchical Techniques
      2. Optimization-Partitioning
      3. Density or Mode Seeking
      4. Clumping
   3. Mechanics
      1. Q-Type Correlation Matrix
      2. A Factor Analysis on Subjects rather than Variables
      3. Distance Matrices
      4. Agregating Algorithms
      5. Importance of Standardizing Measures First
   4. Interpretation
      1. Warning - You will always get Clusters - Are They Real?
      2. Must Examine Clusters In Terms of Original Variables
      3. Must Remember This is a Discriptive Technique, Not a Test
   5. Non Correlational Factor Analyses on Measures

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