Confirmatory Factor Analysis

Question 1

Confirmatory factor analysis purposes

Answer 1

Validate the model

• is a non-iterative procedure
• applies the model developed in previous steps, to a new independent sample
• assess how well the model fits the new sample
• conclude adequate or inadequate

Question 2

Constraints

Answer 2

to make the model estimable we need to fix one parameter among items on each latent variable to a constant 1.0. This only applies to the unstandardised paths.

Question 3

CMIN

Answer 3

A measure of residual Chi-Square, the equivalent of the residual SS in regression. Smaller indicates the model ‘fits’ better.

CMIN/DF: values <2 or <5 are often used as desireable rules of thumb. Over this can indicate poor fit of the common factor model.

Question 4

Independence model:

Answer 4

Equivalent of the null model in regression, assumes the variables are all unrelated.

Question 5

Saturated model

Answer 5

Equivalent of the full additive model in regression, assumes every variable is related to every other variable.

Question 6

Default model

Answer 6

The model we have specified, means that the associations exists as per the arrows we include.

Question 7

NPAR

Answer 7

Number of parameters estimated in the model. Includes variances, covariance and path coefficients.

Question 8

CFI:

Answer 8

Comparative Fit Index. Assesses improvement in fit in the default model compared with the independence model (IM). Values >0.95 are desirable. Since the independence model cannot be an improvement over itself it’s CFI must be zero. The saturated model must be a complete improvement over the IM so it’s value must be 1.0. The value 0.765 indicates a poor fit.

Question 9

TLI

Answer 9

Tucker-Lewis Index. A variation on CFI that penalizes models for complexity. Relatively independent of sample size. Values >0.95 are thought of as desireable. The value of CFI=0.742 indicates a poor fit.

Question 10

RMSEA

Answer 10

Root mean square error of approximation. Assesses error in fit and hence small values are desirable. A common rule-of-thumb is RMSEA<0.05 is desirable. RMSEA has a standard error and hence confidence intervals and hypothesis tests can be constructed.

Question 11

Modification indices

Answer 11

Is a measure of how much CMIN will reduce by the inclusion of a path or covariance not currently included.

The MI has an associated statistical significance so we can determine the relative value and significance of the improvement.

Each path/covariance uses 1df. A Chi-Square test on 1df needs to exceed 3.8 for p<0.05.

As a general rule:

1. Large MIs for covariances between observed variables suggest there is an association between them that is in addition to the specified latent variable structure.
2. Large MIs for covariances between latent variables suggest those constructs are related (cf non-orthogonal rotation)
3. Large MIs for paths suggest a structural deficiency, such as cross loadings.

Question 12

CFA - reliability formula

Answer 12

true score variable/total variance

(sum paths)sqrd x factor / ((sum paths)sqrd x factor) + error = reliability

Question 13

1. A model needs to be:
2. Identified means the total df:
3. Just identified means total df:
4. Under identified means total df:
5. The total df is calculated as:
6. Parameters estimated are:
7. Error terms in CFA measure:

Answer 13

1. identified for us to obtain SEs and measures of model ‘fit’
2. > number of parameters estimated
3. = number of parameters estimated
4. < number of parameters estimated
5. sample moments - parameters estimated
6. are all path coeffieicents, variances and covariances
7. The amount of variance in the observed variables that is not related to the latent variable.