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Education 795 Class Notes. Factor Analysis II Note set 7. Today’s Agenda. Announcements (ours and yours) Revisiting factor analysis Reliability Very Brief Intro to Confirmatory Factor Analysis. Revisiting Factor Analysis. The Necessary Steps.

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Education 795 Class Notes

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Education 795 class notes l.jpg

Education 795 Class Notes

Factor Analysis II

Note set 7

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Today’s Agenda

  • Announcements (ours and yours)

  • Revisiting factor analysis

  • Reliability

  • Very Brief Intro to Confirmatory Factor Analysis

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Revisiting Factor Analysis

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The Necessary Steps

  • Identify and gather data appropriate for factor analysis

  • Decide upon extraction approach and selection criteria

    • PCA vs. PAF

      • Eigenvalue => 1

      • Scree Plot

  • Rotate extracted factors after deciding upon rotational approach

    • Varimax

    • Oblimin

  • Before naming factors, cycle through steps 2 and 3 until you have achieved a reasonable statistical and conceptual solution

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Factor Selection Criteriaand Rotation

  • Tools to identify the appropriate number of factors:

    • In the interest of parsimony, n of factors should be less than the number of variables being analyzed

    • Scree plot

    • Specific theorized number


    • Amount of variance explained (Eigenvalue)


  • Varimax (Orthogonal) assumes factors will be uncorrelated.

  • Oblimin allows dependence between factors

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Rotating Extracted Factors

  • Unrotated factor matrix is only one of many possible ones; transformations can clarify meaning without changing the underlying relationships amongst the variables

  • Rotation is used to ease interpretation but it should be tied to theory!

  • Desire to approach “simple structure”

  • Orthogonal (Varimax) or oblique (Oblimin)?

  • Is it cheating to rotate?

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Interpreting and Naming Rotated Factors

  • Appropriate after cycling through various solutions and identifying the one that makes both statistical and conceptual sense

  • Naming should capture the essence of the variables that are most closely associated with each factor

  • Should take the relative strength of loading into account in naming factors

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Technical Details

  • Coefficients associated with unrotated factors can be interpreted like regression betas. Specifically, the square of the coefficient in the factor matrix indicates the proportion of variance of a given indicator that is accounted for by the factor.

  • The Factor Pattern Matrix contains the coefficients for the regression of each indicator on the factors.

  • The Factor Structure Matrix consists of the correlations between indicators and factors.

  • When the factors are uncorrelated, the two matrices are equal.

  • The eigenvalue is equal to the sum of the squared loadings of the indicators on the factor with which it is associated.

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Sampling and Sample Size

  • Probability sampling is necessary if one wants to generalize findings of EFA.

  • General Rule: at least 10 cases per variable in the factor analysis (Nunnally, 1978).

  • Many others disagree and just say, ‘Use large samples’!

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A factor analysis, employing a principal components extraction using the Eigenvalue > 1.0 criterion, identified three interpretable factors, explaining 46.5 percent of the common variance

After reviewing the results of the analysis, we named these three factors…

Return to Our Example

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Varimax Output: Naming

Factor I:

Factor II:

Factor III:

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Scree Plot

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Creating Factor Scores

  • A straightforward scale

    • compute extrinsic =momoney+betterjb

    • compute intrinsic=gainege+moculture+ improve+moculture+prepgrad.

  • Or use an average

    • compute extrinsic=(momoney+betterjb)/2

  • Or use the factor loadings

    • compute extrinsic=.84*momoney+.84*betterjb.

  • Be sure to represent ‘reversed’ items in creating scales:

    • If you have a negative sign in the factor group

      • recode Q4 (1=2) (2=1).

      • rerun the factor analysis.

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    Extending / Using FA Results

    • Validity

      • Whether a measurement instrument or technique measures what it is supposed to measure

    • Reliability

      • Reliability is a necessary but not sufficient condition for validity (a measure cannot be valid if it is not reliable but being reliable does not imply valid).

      • Reliability is the consistency or stability of a measure

        • Test-retest reliability -- consistency over time

        • Internal consistency reliability -- multiple items thought to measure the same construct should be correlated

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    Coefficient Alpha

    • A standard measure of internal consistency, developed by Cronbach

    • Expands the concepts of inter-item correlation averaging (add up all the correlations and divide by n), and split-half reliability (randomly divide the items measuring a single concept in half, compute total score for each half set of items, and then correlate them)

    • Mathematically equivalent to the average of all possible split-half estimates

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    Cronbach’s Alpha

    • Relatively low reliabilities OK and are tolerable in early phases of research.

    • Higher reliabilities are required when the measure is used to determine group differences (>.7) (Nunnally, 1978)

    • Very high reliabilities are needed for making important decisions about individuals (>.9) (Pedhazur, p. 109)

    • Ultimately it depends on how much error the researcher is willing to have

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    Reliability Example

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    Intro to Confirmatory Factor Analysis

    • Formulation of a model is a prerequesite for CFA—the aim is to “test” the model or assess the fit to the data

    • CFA is a submodel of Structural Equation Modeling

    • CFA is a measurement model of relations of indicators to factors as well as relations among factors

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    EFA vs. CFA

    • In EFA, all indicators have loadings; not necessarily so in CFA

    • Correlated factors are all or nothing in EFA. In CFA it is possible to specify that only some of the factors can be correlated.

    • In EFA, it is assumed that errors in indicators are not correlated. In CFA we can test this assumption.

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    CFA results from a study of college faculty

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    Some Last Details: Standardize First

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    For Next Week

    • Read Pedhazur Ch 6 p119-131

    • Readings to be handed out in class on Affirmative Action Case Study

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