GRA 6020Multivariate Statistics

Confirmatory Factor Analysis

Ulf H. Olsson

Professor of Statistics

Ulf H. Olsson

CFA

• The covariance matrices:

)(

)(

S

Ulf H. Olsson

CFA and ML

k is the number of manifest variables.

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

tkkdfwith

ondistributiaeapproximatwillSFN

)1(2

1

))(,()1( 2

kSStrSF log))(()(log))(,( 1

Ulf H. Olsson

Testing Fit

)(:

)(:0

AH

H

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

ondistributicentralNon 2

FunctionyDiscrepancPopulation

)0);/((0 ndfFMaxF

EAofestimateanisdf

FRMSEA

0

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

9

8

7

6

5

4

3

2

1

3

2

1

93

83

73

62

52

42

31

21

11

9

8

7

6

5

4

3

2

1

00

00

00

00

00

00

00

00

00

x

x

x

x

x

x

x

x

x

Ulf H. Olsson

Alternative Estimators

• Assuming multivariate normality• ML• GLS• ULS• If the model holds, ML and GLS are asymptotically

equivalente

Ulf H. Olsson

Alternative Estimators

))(())'(( 1 sUs

S: sample covariance

θ: parameter vector

σ(θ): model implied covariance

Ulf H. Olsson

Alternative Estimators

1, ghijOLSU

Ulf H. Olsson

Alternative Estimators

ghijghijGLS ssU ,

Ulf H. Olsson

Alternative Estimators

ghijghijMLU

,