TABLE 1.
Model Specification | CPI | GAPI | UPI | |
---|---|---|---|---|
1. Factor loadings of the product indicators on ξ1ξ2. For example, λX1X4 = λX1λX4 | Yes | Yes | No | |
2. E(ξ1ξ2) = Cov(ξ1, ξ2) | Yes | Yes | Yes | |
3. Var(ξ1ξ2) = Var(ξ1)Var(ξ2) + Cov2 (ξ1, ξ2) | Yes | No | No | |
4. Cov(ξ1,ξ1ξ2) = Cov(ξ2, ξ1ξ2) = 0 | Yes | No | No | |
5. Variances of the unique factors of the product indicators. For example, Var(δX1X4) = λX1 Var(ξ1)Var(δX4) + λX4 Var(ξ2)Var (δX1) + Var(δX1)Var(δX4) | Yes | Yes | No | |
6. Zero covariances between the unique factors of exogenous indicators and those of product indicators (assuming zero covariances among the unique factors of exogenous indicators) | Yes | Yes | Yes | |
7. Covariances between unique factors of product indicators that share the same exogenous indicators (assuming zero covariances among the unique factors of exogenous indicators). For example, Cov(δX1X4, δX1X5) = λX4λX5Var(ξ2)Var(δX1) | Yes | Yes | No | |
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8. Normality assumptions of exogenous indicators from model specification | Yes | More Liberal | Most Liberal |
Note. CPI means constrained product indicator approach. GAPI means generalized appended product indicator approach. UPI means unconstrained product indicator approach. As noted in 8, model constraints 3, 4, 5, 6, and 7 assume multivariate normality for the CPI approach. Constraint 7 is not a concern if the two exogenous latent variables have equal numbers of indicators. The distributional assumptions are relaxed for the GAPI and UPI approaches.