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 suitableIntercepts – uniform biasFactor loadings – non-uniform bias (vary with
t)
Restricted Factor Analysis (RFA)
• Advantages of RFA over MGFA:V can be continuous or discrete, observed or
latentInvestigate 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 LMSRFA with RSP• Measurement biasUniformNonuniform• ViolatorDichotomousContinous
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
0b 0c
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!
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The MGFA method
• When v is dichotomous, regular MGFA• When v is continuous, dichotomize x by V• Using chi-square difference test with df=2Uniform : interceptsNonuniform: loadings
The RFA/LMS method
• V is modeled as latent variable:Single indicatorFix 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
0b0c
RFA/RSP method
• Replacing with , where is a random slope.
• Robust ML estimation• Chi-square test with S-B correction: : uniform bias : nonuniform bias
0b
0c
Single & iterative procedures• Single run procedure: test once for each item• Iterative procedure: 1)Locate the item with the largest chi-square
difference2)Free constrains on intercepts and factor
loadings for this item and test others3)Locate the item with the largest chi-sqaure
difference 4)…5)Stops when no significant results exist or half
are detected as biased
Results of MGFA – single run
• Shown in Table 2.• Conclusion:1.better with dichotomous than with
continuous V; 2.non-uniform bias is more difficult to detect
than uniform bias; 3.Type I error inflated.
Results of MGFA – iterative run
• Shown in Table 3.• Conclusion:1.Iterative procedure produces close power as
single run does.2.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:1.LMS and RSP produce almost equivalent
results. 2. larger power than MGFA with continuous V.3.More severely inflated Type I error rates
Results of RFA/LMS & RFA/RSP - iterative run
• Shown in Table 6.• Conclusion:1.Power is close to the single run2.Type I error rates are improved
Results of estimation bias - MGFA
• Shown in Table 7.• Conclusion:1.Bias in estimates is small2.Bias in SD is non-ignorable3.Smaller bias in estimates for dichotomous V
(dependent T&V)
Results of estimation bias - RFA
• Shown in Table 8 & 9• Conclusion:1.Similar results for LMS and RSP2.Small bias in estimates3.Non-ignorable bias in SD4.Smaller SE than MGFA5.Smaller bias in estimates than MGFA with
dependent T&V, continuous V.