In evaluating second-order structure, I have faced two common approaches in applied studies:
1) Some fit the second-order structure in an explicitly hierarchical model: the first-order structure models the indicators and the second-order structure models the [implied] covariance between first-order latent factors.
2) Some fit the second-order structure on the first-order factors that are represented with sum scores (i.e., each subscale as a parcel).
To me, it seems that the first approach is more accurate (as it correctly models the measurement error), but I have some doubts about how to assess the fit of the second-order structure. Fit indices seem to put much more weight on the first-order structure. This goes to the extent that in a large model with a good first-order and a very poor second-order structure, the fit indices tend to show a good fit. Many authors consider the good fit of this hierarchical model as evidence of the good fit for second-order structure as well. (Some authors compare fit indices such as CFI and RMSEA from models with and without second-order factors and if there is little difference they conclude that the second-order structure is a good fit.)
Is this practice OK? And am I missing something here?
And is there any way to use the first approach and still calculate fit indices that exclusively evaluate the second-order structure?
(Something like calculating a chi-square for the discrepancy between the implied covariance matrix from the first-order structure and the implied covariance matrix from the second-order structure and using this for calculating other fit indices!)
Thank you in advance.
Those are really interesting and difficult questions. I would send this inquiry to the SEMNET email listserv (https://listserv.ua.edu/cgi-bin/wa?A0=SEMNET). I'm pretty sure you would get some good answers there from people who have dealt with questions of SEM model fit assessment for decades. You may even find some already published discussions on this topic in the SEMNET archives.
Dear Christian Geiser ,
Thank you for introducing the SEMNET list. I subscribed and posted the question there.
Perfect! I hope you get some good answers there. I'm curious about what people have to say about these questions myself.
Dear Ali Zia-Tohidi ,
using sum scores is not such a good idea. Quite often, sum scores are used to cover problems in the measurement model. I think that using factor scores could solve this problem. If you first use a correlated factors model and save the factor scores for each factor, you could then perform a first-order factor analysis based on the factor scores.
I have a vague recollection that there is a measure to assess the structural model but unfortunately, I cannot remember which one it was exactly. If I find out I'll let you know.
HTH, Karin
Dear Ali Zia-Tohidi,
Marsh and Hocevar (1985) proposed a target coefficient T that relates the fit (Chi²) of the model with correlated first order factors to the fit of the second order factor model. Thus, the target coefficient can be used to determine whether a poor model fit is caused by the first-order models or the second-order model.
Marsh, H. W., & Hocevar, D. (1985). Application of confirmatory factor analysis to the study of self-concept: First-and higher order factor models and their invariance across groups. Psychological bulletin, 97(3), 562.
See also
Cheung, D. (2000). Evidence of a single second-order factor in student ratings of teaching effectiveness. Structural Equation Modeling, 7(3), 442-460.
best
Christoph
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