Measurement Invariance for Latent Constructs in Multiple Populations: A Critical
View and Refocus
Tenko Raykov, George A. Marcoulides and Cheng-Hsien Li
The traditional Latent Variable Modeling (LVM) procedure in examining measurement invariance includes: (a) use nested multiple-group models to test factor loading invariance and, (b) if the factor loading invariance is possible, on multiple-group model assuming loading and intercept invariance across groups is fitted.
However, there are some limitations about the procedure.
Firstly, it lacks a complete test of loading invariance.
For model identification, one of factor loading, say 𝜆
1
is typically set equal to 1 for same indicator per factor in all groups. By doing so, an artificial metric of measuring the factor is introduced. In other words, it is in fact testing whether 𝜆
2
/𝜆
1
, 𝜆
3
/𝜆
1
, …, 𝜆 𝑠
/𝜆
1
are group invariant, rather than testing 𝜆
1
= 𝜆
2
=…= 𝜆 𝑠
which is our actual interest.
An alternative identification is to set each factor variance to be 1. However, again, it is not necessarily that the variance of the factor variance is 1, which hence introduces the incorrect metrix.
Secondly, it lacks a statistic test for intercept invariance.
When examining intercept invariance, the restriction 𝛼
1
= 𝛼
2
=…= 𝛼
𝐺
was applied
(Model 3). However, it frees the means of the factors in all G groups except one.
In another hand, when examining loading invariance, the restriction 𝛬
1
= 𝛬
2
=…= 𝛬
𝐺 was applied (Model 2). But, all factor means are fixed to be zeros.
Hence, Model 3 and Model 2 are not nested. Therefore, it is not appropriate to test the intercept invariance assumption by comparing the fit statistics between these two models.
A strategy was proposed:
In first step, the invariance of the set of factor indicators within each population is examined.
In second step, if the invariance was possible, a constrained model (Model 4) with the added constrain of intercept invariance 𝛼
1
= 𝛼
2
=…= 𝛼
𝐺
with zero factor means in one groups was fit. For model 4, just one of the groups variance is fixed at 1. At the same time, all loadings of all factors across G groups are constrained to be equal, and fits under normality this G-groups model to the covariance structure or to the mean structure with zero factor means and free mean intercepts in all groups (DON’T

UNDERSTAND!).
Comments:
(1) This article was not well written and was not easy to read and understand!
(2) The criticism of traditional LVM procedure was reasonable. However, the proposed strategy in this study seemed minor.
(3) Simulations investigating the performance of the proposed strategy under different conditions are desirable.