Measurement bias detection through factor analysis
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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

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Measurement Bias Detection Through Factor Analysis

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Measurement bias detection through factor analysis

Measurement Bias Detection Through Factor Analysis

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


Defining measurement bias

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

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

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

This paper…

  • Compared methods:

  • MGFA

  • RFA with LMS

  • RFA with RSP

  • Measurement bias

  • Uniform

  • Nonuniform

  • Violator

  • Dichotomous

  • Continous


Data generation rfa

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

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 design1

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

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

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

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 & 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

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

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

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

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

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

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

Discussion

  • Nonconvergence occurs with RFA/LMS


Non convergence

Non-convergence

  • Summary:


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