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Education 795 Class Notes

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Education 795 Class Notes

Factor Analysis II

Note set 7

- Announcements (ours and yours)
- Revisiting factor analysis
- Reliability
- Very Brief Intro to Confirmatory Factor Analysis

- Identify and gather data appropriate for factor analysis
- Decide upon extraction approach and selection criteria
- PCA vs. PAF
- Eigenvalue => 1
- Scree Plot

- PCA vs. PAF
- 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

- 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
/CRITERIA = FACTORS(n)

- Amount of variance explained (Eigenvalue)
/CRITERIA = MINEIGEN(1.0)

- Varimax (Orthogonal) assumes factors will be uncorrelated.
- Oblimin allows dependence between 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?

- 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

- 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.

- 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’!

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…

Factor I:

Factor II:

Factor III:

- A straightforward scale
- compute extrinsic =momoney+betterjb
- compute intrinsic=gainege+moculture+ improve+moculture+prepgrad.

- compute extrinsic=(momoney+betterjb)/2

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

- If you have a negative sign in the factor group
- recode Q4 (1=2) (2=1).
- rerun the factor analysis.

- 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

- 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

- 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

- 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

- 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.

- Read Pedhazur Ch 6 p119-131
- Readings to be handed out in class on Affirmative Action Case Study