Estabrook and Wu (2016)

[Farideh Alizadeh]

Dear Prof. Rosseel,

I’ll be thankful if you comment on this issue. I was recently introduced to Estabrook and Wu (2016). I think they argue that for testing metric MI, thresholds+loadings must be constrained, and for scalar MI, thresholds+loadings+intercepts must be constrained.

Is my understanding correct?  

Do we have both thresholds and intercepts in ordinal data? Aren’t they the same? 

Doesn’t metric and scalar become the same in this case?

To what extent is their argument accepted by the community?

 The paper was published in 2016.  How has it been received? Has it become a standard practice for ordinal data? Is deviation for this workflow considered wrong? 

Does it really change MI decisions?

Best

Farideh

Wu, H., & Estabrook, R. (2016). Identification of Confirmatory Factor Analysis Models of Different Levels of Invariance for Ordered Categorical Outcomes. Psychometrika, 81(4), 1014–1045. https://doi.org/10.1007/s11336-016-9506-0

[Terrence Jorgensen]

for testing metric MI, thresholds+loadings must be constrained

Yes, and compare that to a model that constrains only the thresholds to equality (and has no further constraints unnecessary for identification).

for scalar MI, thresholds+loadings+intercepts must be constrained.

 

Yes, and compare that to the model that constrains only the thresholds and loadings to equality.

Do we have both thresholds and intercepts in ordinal data?

Yes.

 

Aren’t they the same? 

No, a discrete item's threshold(s) link it to its latent response (assumed to be normally distributed).  The intercept is the latent response's expected value when the factor(s) predicting it = 0 (i.e., the same interpretation that a continuous item's intercept has, except the indicator is latent rather than observed).

Without equality constraints on thresholds (beyond minimal identification constraints), the intercepts and thresholds are not independently identifiable.

Doesn’t metric and scalar become the same in this case?

No, they are the same as with continuous items.  But they are only testable on the assumption of (at least part) threshold invariance.

 

To what extent is their argument accepted by the community?

Not enough attention is paid to that paper.  One tutorial was published that advertised how my software facilitated the correct approach, but more continue to be published based on incomplete or outright incorrect ideas that were disseminated in the 15 years prior to Wu & Estabrook's article.

Terrence D. Jorgensen    (he, him, his)
Assistant Professor, Methods and Statistics
Research Institute for Child Development and Education, the University of Amsterdam
http://www.uva.nl/profile/t.d.jorgensen