GRA 6020Multivariate Statistics

Factor Analysis

Ulf H. Olsson

Professor of Statistics

Ulf H. Olsson

EFA• Eigenvalue of factor j

• The total contribution of factor j to the total variance of the entire set of variables

• Comunality of variable i• The common variance of a variable. The portion of a variable’s total

variance that is accounted for by the common factors

Ulf H. Olsson

EFA-How many factors to retain

• Based on theory• Eigenvalues 1• Checking the rows in the pattern matrix

Ulf H. Olsson

Factor Solutions

• Principal Factor Method• Extracts factors such that each factor accounts for the maximum

possible amount of the variance contained in the set of variables being factored

• No distributional assumptions

• Maximum Likelihood• Will be treated in detail later• Multivariate normality

Ulf H. Olsson

Rotation of Factors

• The objective is• To achieve a simpler factor structure• To achieve a meaningful structure

• Orthogonal rotation

• Oblique Rotation

Ulf H. Olsson

Rotation

• Varimax• Major objective is to have a factor structure in which each

variable loads highly on one and only one factor.

• Quartimax• All the variables have a fairly high loading on one factor• Each variable should have a high loading on one other factor

and near zero loadings on the remaining factors

Ulf H. Olsson

Rotation

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•The rationale for rotation is very much akin to sharpening the focus of a microscope in order to see the details more clearly

Ulf H. Olsson

The CFA model

• In a confirmatory factor analysis, the investigator has such a knowledge about the factorial nature of the variables that he/she is able to specify that each xi depends only on a few of the factors. If xi does not depend on faktor j, the factor loading lambdaij is zero

Ulf H. Olsson

CFA

• If does not depend on then • In many applications, the latent factor represents

a theoretical construct and the observed measures are designed to be indicators of this construct. In this case there is only (?) one non-zero loading in each equation

ix j 0ij

ix

Ulf H. Olsson

CFA

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Ulf H. Olsson

CFA

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Ulf H. Olsson

CFA

• The covariance matrices:

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Ulf H. Olsson

CFA and ML

k is the number of manifest variables.

If the observed variables comes from a multivariate normal distribution, then

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Ulf H. Olsson

Testing Fit

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Ulf H. Olsson

Problems with the chi-square test• The chi-square tends to be large in large samples if the

model does not hold• It is based on the assumption that the model holds in the

population• It is assumed that the observed variables comes from a

multivariate normal distribution• => The chi-square test might be to strict, since it is based on

unreasonable assumptions?!

Ulf H. Olsson

Alternative test- Testing Close fit

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FRMSEA

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Ulf H. Olsson

How to Use RMSEA

• Use the 90% Confidence interval for EA• Use The P-value for EA• RMSEA as a descriptive Measure

• RMSEA< 0.05 Good Fit• 0.05 < RMSEA < 0.08 Acceptable Fit• RMSEA > 0.10 Not Acceptable Fit

Ulf H. Olsson

Other Fit Indices

• CN• RMR• GFI• AGFI• Evaluation of Reliability• MI: Modification Indices

Ulf H. Olsson

Nine Psychological Tests

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93

83

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62

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42

31

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