Techniques and Applications of Multivariate 2016. 9. 9.¢  Techniques and Applications of Multivariate

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  • 3. Factor Analysis

    Techniques and Applications of Multivariate Analysis

  • Lecture 3. Factor Analysis (FA)

    Lecture3-1 3.1 Introduction 3.2 Orthogonal Factor Model 3.3 Methods of Estimation: PCM vs MLM Lecture3-2 3.4 Factor Rotation 3.5 Factor Scores: WLSM vs RM 3.6 Example : Air-pollution Data 3.7 Strategy for Factor Analysis : Example 9.14

  • Gaelic English History Arithmetic Algebra Geometry

    Gaelic 1.00 . . . . .

    English 0.44 1.00 . . . .

    History 0.41 0.35 1.00 . . .

    Arithmetic 0.29 0.35 0.16 1.00 . .

    Algebra 0.33 0.32 0.19 0.59 1.00 .

    Geometry 0.25 0.33 0.18 0.47 0.46 1.00

    3.1 Introduction of FA

     Definition

    FA: technique for describing the covariance relationship among many variables in terms of a few factors which are underlying, but unobservable random quantities.

     Example (Ex. 9.8, p.502)

    Correlation matrix of 6 subjects

    Mathematical-ability factor

    Verbal-ability factor

  • 3.1 Introduction of FA

    History :  K. Pearson and Charles Spearman provided beginnings of FA in the early 20th century.

     Charles Spearman is known for being the one who coined the term factor analysis

    and actually used it to measure children’s cognitive performance.

     Spearman, C. (1904). “General intelligence” objectively determined and measured.

    "American Journal of Psychology", 15, 201–293.

  • 3.2 Orthogonal Factor Model

     Model with m common factors

     Properties

    Matrix of factor loadings

    Vector of specific factors

     Assumptions

    Common factors decomposition

    communality

    Specific variance

    Loading of the ith the variable on the jth factor

  • 3.3 Methods of Estimation: PCM vs MLM

    : Common factor decomposition

    with

  • 3.3 Methods of Estimation

    [step 3] Obtain the matrix of estimated factor loadings (m

  • 3.3 Methods of Estimation: PCM

     How do we select the number of factors m in PCM?

    for S, for R

  • 3.3 Methods of Estimation: PCM

     Example (Ex. 9.8, p.502)

    Correlation matrix of 6 subjects

     Program

    Gaelic English History Arithmetic Algebra Geometry Gaelic 1.00 . . . . . English 0.44 1.00 . . . . History 0.41 0.35 1.00 . . .

    Arithmetic 0.29 0.35 0.16 1.00 . . Algebra 0.33 0.32 0.19 0.59 1.00 .

    Geometry 0.25 0.33 0.18 0.47 0.46 1.00

  • 3.3 Methods of Estimation : Results of SAS

    They are the only eigenvalues greater than 1.

    2 factors account for a cumulative proportion of the total sample variance.

    General intelligence factor

    Bipolar factor: Half + and half -

    All communalities are nearly about 1

    All elements are small

  • 3.3 Methods of Estimation: MLM

    2) Algorithm for Maximum Likelihood Method

    [step 1] Given , consider the likelihood function

    [step 3] We have the ml estimators and mles of the communalities

    [step 2]With and

    obtain the maximization of the likelihood function subject to uniqueness condition

  • 3.3 Methods of Estimation: MLM

     How do we select the number of factors m in MLM?

     Test the hypotheses with an appropriate m.

    : Bartlett’s test statistic based on the chi-square approximation

     Residual matrix:

    The diagonal elements are zero and the other elements are small: m factor model is appropriate !

  • 3.3 Methods of Estimation: MLM

     Program

  • 3.3 Methods of Estimation: Comparison