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

1. Identify and gather data appropriate for factor analysis

2. Decide upon extraction approach and selection criteria

1. PCA vs. PAF1. Eigenvalue => 12. Scree Plot

3. Rotate extracted factors after deciding upon rotational approach

1. Varimax2. Oblimin

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

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 analyzedScree plotSpecific 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

Rotating Extracted Factors

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

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

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.

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’!

Return to Our Example

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

Varimax Output: Naming

Factor I:

Factor II:

Factor III:

Scree Plot

Creating Factor Scores

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

Or use an averagecompute extrinsic=(momoney+betterjb)/2

Or use the factor loadingscompute 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.

Extending / Using FA Results

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

ReliabilityReliability 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 timeInternal consistency reliability -- multiple items thought to measure the same construct should be correlated

Coefficient Alpha

A standard measure of internal consistency, developed by CronbachExpands 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

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

Reliability Example

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

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.

CFA results from a study of college faculty

Lack of personal time

Time pressure

Teaching load

F1

E

E

E

Household responsibilities

Child care

Children's problems

Faculty meetings

Committee work

Colleagues

F2

F3

E

E

E

E

E

E

Marital friction E

F4Res./publishing demands

Rev./promotion process

Subtle discrimination

E

E

E

Initial confirmatory factor analysis model. (Circles represent latent constructs [factors], rectangles are measured variables, and 'E' indicates residual variances. Factor loadings are indicated by single headed arrows, covariances among factors are indicated by two-headed arrows.)

Some Last Details: Standardize First

For Next Week

Read Pedhazur Ch 6 p119-131

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