Lecture 10 - Confirmatory factor analysis

Question 1

(!) Describe CFA

Answer 1

• Researcher has prior assumptions
• Theory-driven: EFA is data-driven
• Differ from EFA: Dont seek fewest poss. factors
• Verify factor structure of set of obs. variables
• Confirm specific structure of scale dim.
• Test if relation exist between variable & latent construct
• Show indicator reliability: Significance & corr. w. specific factor
• Confirm integrity of scale: Separated from other constructs
• Estimation require identification
• Estimate loadings so var-cov matrix can be reprod. backward
• Test different scales: Quality
• Extremely versatile method: Dansk: Alsidig

Question 2

Describe the parameters of the CFA model

Answer 2

Lambda: Factor loadings/parameters
Theta: Variances: Covariances
Xi: One factor variance
Phi: More factor variances
Kappa: Means for latent factors
Tau: Intercept for each indicator/item

Question 3

Describe the identification formula

Answer 3

General:
- Diff. between knowns & unknowns is degrees of freedom for model
- Used when testing distribution chi-square

Under-identified model:
- Known < unknowns

Just identified model:
- Knowns = unknowns

Over-identified model:
- Knowns > unknowns

Formula:
b = p(p+1)/2

Scale of factors:
- Marker = Ref. Factor scale
- Parameter not estimated if variance factor = 1

Question 4

Describe the CFA fitting function

Answer 4

Question 5

(!) Describe the reasons for CFA

Answer 5

Test if a relation exist between variable & construct:

Test errors & cross-loadings:
- Refine scales
- Eliminate items
- Notice phrasing problems

Allow model comparison:
- Statistic evaluate fit w. & without model restrictions
- Nested model: Scale quality & validity & common meth. var. bias

Restrict part of model between groups:
- Check if scale work for other groups

Question 6

Goodness of fit evaluation

Answer 6

Chi-squared test:
- Often not good approximation of distribution: Small dataset
- Can be used to compare nested models
- Inflated in big datasets: Hypothesis obviously untrue
- Basis for calculating other indices

Standardized root mean square residual/SRMR:
- Absolute fit
- Cut off point: Lower than or close to 0.08

Root mean square error of approximation / RMSEA:
- Cut off point: Lower than or close to 0.06

Comparative fit index / CFI:

Tucker-Lewis index / TLI:
- Cut off point: Higher than or close to 0.95

Akaike information criterion & Bayes information criterion / AIC / BIC:
- No cut-off points
- Smaller number better
- Can compare non-nested models

Question 7

(!) Describe reliability & validity of constructs

Answer 7

Reliability:
- Cronbach alpha
- Ability to uncover true construct: Accuracy
- Inter-connectedness between terms
- Should show in each sample: Internal consistency measure
- McDonald´s omega

___________

Validity:

General:
- Measure what we want to measure
- Constructs must differ
- Fit into nomological net: Predict relevant behavior/attitude

Validities with CFA:

Content validity:
- Item represent construct well?
- Items cover content domain?

Criterion validity:
- Construct predict criterion variable: DV
- Data collected same time or separate?

Construct-validity:
- Construct fit with existing ones?
- Discriminant validity: Factors separetable & uncorrelated
- Convergent validity: Variables correlate within factor
- Simple: Factors make sense

Question 8

(!) Describe Cronbach alpha & McDonald´s omega

Answer 8

Cronbach alpha:

General:
- Measure internal consistency: Correlation within group
- Measure scale reliability
- 0,6-0,7: Accepted
- Above 0,8: Good
- 0,757: Absorption
- Biased if assumptions not fulfilled
- Poss. Redundance if too high
- Popular measure for sum scales
- Increase if more items
- Proportion of scale variance = “True scale”
- Assume construct is parallel or at least tau-equivalent
- Need congeneric, tau-equivalent & parallel indicators proved
- Eg. Score high on Q1 = High on Q2

Congeneric:
- Independent indicators
- Indicator predict same factor

Tau-equivalent:
- Indicator predict same factor
- Indicator = factor loadings: Factor score the same amount

Parallel:
- Indicator measure same construct with same precision
- Require psychometrically interchangeable constructs: Same factor, factor loading & error variances

__________

McDonald’s omega:
- For Absorption: Omega = 0,765
- Dont assume parallel or tau-equivalent model
- Same interpretation as alpha: How much true variance scale explain

Question 9

(!) Describe invariance of CFA

Answer 9

General:
- Measure stableness of construct
- For comparing
- Relative or absolute diff. should be true diff.: Not measurement error

__________

Multi group CFA:

General:
- Separate samples
- Start with most lenient restriction & increase gradually

Configural invariance:
- Structure factor loadings
- Conceptualized the same way

Metric invariance with factor variance/covariance:
- Magnitude of factor loadings
- Poss. compare coefficient in regression analysis

Scalar invariance:
- Item intercepts
- Poss. compare absolute differences. ANOVA.

Factor covariance invariance:

Factor variance invariance:

Error variance:
- Structural equation modelling rather than regression if needed taken into account

Mean variance:
- Latent means
- Poss. merging groups to analyse within same sample