Chapter Thirteen: Principal Components and Factor Analysis

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Chapter Thirteen Principal Components and Factor
Analysis

• DBA 903
• Research Methods III
• J. Schreiber

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Goals of PCA or FA

• Summarize correlations
• Reduce multiple variables to fewer factors
• Provide definition for underlying process
• Test theory of underlying process

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Steps in PCA and FA

• Prepare correlation matrix
• Extract a set of factors
• Determine the number of factors
• Rotate factors
• Interpret Results
• Test construct validity

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Problems with PCA and FA

• No criterion variable to test the solution
• Infinite number of rotations/solutions available
• Associated with poorly conceived research

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Definitions

• Correlation matrices
• Observed
• Reproduced
• Residual
• FA versus PCA
• Variance analyzed
• Reason why variables are associated with
  factors/components

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Definitions

• Rotation
• Orthogonal
• Loading matrix
• Oblique
• Factor correlation matrix
• Loading matrix
• Structure matrix
• Pattern matrix

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Kinds of Research Questions

• Number of Factors
• Nature of Factors
• Importance of Solutions and Factors
• Testing Theory in FA
• Estimating scores on Factors

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Limitations

• Theoretical -- Decisions often based on
  ___________ reasons
• Practical
• Sample size and missing data
• Normality
• Linearity

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Limitations (continued)

• Practical
• Absence of outliers among cases
• Absence of multicollinearity and singularity
• Factorability of R
• Absence of outliers among variables

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Fundamental Elements

• Observed Correlation Matrix
• Eigenvalues
• Factor Loading Matrix
• Orthogonal Rotation (Varimax)

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Fundamental Elements

• Communalities
• Proportion of Variance
• Proportion of Covariance
• Factor Scores