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### Measurement Bias Detection Through Factor Analysis

Barendse, M. T., Oort, F. J. Werner, C. S., Ligtvoet, R., Schermelleh-Engel, K.

Defining measurement bias

- Violation of measurement invariance

Where V is violator

- If V is grouping variable, then MGFA is suitable
- Intercepts – uniform bias
- Factor loadings – non-uniform bias (vary with t)

Restricted Factor Analysis (RFA)

- Advantages of RFA over MGFA:
- V can be continuous or discrete, observed or latent
- Investigate measurement bias with multiple Vs.
- More precise parameter estimates and larger power
- Disadvantage of RFA:
- Not suited for nonuniform bias (interaction term)

Approaches for non-uniform bias

- RFA with latent moderated structural equations (LMS)

---- Simulation (categorical V) showed at least as good as MGFA

- RFA with random regression coefficients in structural equation modeling (RSP)

---- performance unknown

This paper…

- Compared methods:
- MGFA
- RFA with LMS
- RFA with RSP
- Measurement bias
- Uniform
- Nonuniform
- Violator
- Dichotomous
- Continous

Data generation (RFA)

- True model:
- Uniform bias: . Nonuniform bias:
- T and v are bivariate standard normal distributed with correlation r
- e is standard normal distributed
- u is null vector

Simulation Design

For continuous V:

- Type of bias (only on item 1):
- No bias (b=c=0),
- uniform bias(b=0.3,c=0),
- nonuniform bias (b=0,c=0.3),
- mixed bias (b=c=0.3)
- Relationship between T and V

Independent (r=0), dependent (r=0.5)

Simulation Design

For dichotomous V:

- V=-1 for group 1 and v=1 for group 2
- Model can be rewritten into
- Relationship between T and V:

Correlation varies!

The MGFA method

- When v is dichotomous, regular MGFA
- When v is continuous, dichotomize x by V
- Using chi-square difference test with df=2
- Uniform : intercepts
- Nonuniform: loadings

The RFA/LMS method

- V is modeled as latent variable:
- Single indicator
- Fix residual variance (0.01)
- Fix factor loading
- Three-factor model: T, V, T*V
- Robust ML estimation
- Chi-square test with S-B correction:

: uniform bias

: nonuniform bias

RFA/RSP method

- Replacing with , where is a random slope.
- Robust ML estimation
- Chi-square test with S-B correction:

: uniform bias

: nonuniform bias

Single & iterative procedures

- Single run procedure: test once for each item
- Iterative procedure:
- Locate the item with the largest chi-square difference
- Free constrains on intercepts and factor loadings for this item and test others
- Locate the item with the largest chi-sqaure difference
- …
- Stops when no significant results exist or half are detected as biased

Results of MGFA – single run

- Shown in Table 2.
- Conclusion:
- better with dichotomous than with continuous V;
- non-uniform bias is more difficult to detect than uniform bias;
- Type I error inflated.

Results of MGFA – iterative run

- Shown in Table 3.
- Conclusion:
- Iterative procedure produces close power as single run does.
- Iterative procedure produces better controlled Type I error rate.

Results of RFA/LMS & RFA/RSP - single run

- Shown in Table 4 and Table 5.
- Conclusion:
- LMS and RSP produce almost equivalent results.
- larger power than MGFA with continuous V.
- More severely inflated Type I error rates

Results of RFA/LMS & RFA/RSP - iterative run

- Shown in Table 6.
- Conclusion:
- Power is close to the single run
- Type I error rates are improved

Results of estimation bias - MGFA

- Shown in Table 7.
- Conclusion:
- Bias in estimates is small
- Bias in SD is non-ignorable
- Smaller bias in estimates for dichotomous V (dependent T&V)

Results of estimation bias - RFA

- Shown in Table 8 & 9
- Conclusion:
- Similar results for LMS and RSP
- Small bias in estimates
- Non-ignorable bias in SD
- Smaller SE than MGFA
- Smaller bias in estimates than MGFA with dependent T&V, continuous V.

Discussion

- Nonconvergence occurs with RFA/LMS

Non-convergence

- Summary:

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