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Factor model of Ordered -Categorical Measures: Measurement Invariance

Explore the measurement invariance using ordinal indicators in this detailed factor model. Learn about identification criteria, measurement invariance, and model restrictions for group comparisons.

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Factor model of Ordered -Categorical Measures: Measurement Invariance

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  1. Factor model of Ordered-Categorical Measures: Measurement Invariance Michel Nivard Sanja Franic Dorret Boomsma Special thanks: Conor Dolan OpenMx “team”

  2. Ordinal factor model • This model works for ordered categorical indicators. • It assumes these indicators are indicators of an underlying continuous (and normally distributed) trait.

  3. Ordinal indicators • Q: do you get angry? • 0: Never • 1: Sometimes • 2: Often 0 1 2

  4. Ordinal indicators • The latent distribution is restricted too mean=0 and variance = 1. Given these constraints we can estimate the the threshold parameters indicated in red and blue here. Either: μ and Σ Or: the Thresholds are identified.

  5. Ordinal indicators factor model.

  6. Ordinal indicators factor model. • Expected (polychoric) covariance: • Cov(Xi) = Σ = i it+ i • Expected means/threshold: • E(Xi)= μ = τi + iκ • Restrictions for identification (alternatives, Millsap 2004) •  = 1 • diag(22t+ 2)=diag() • μ = 0 = τi + iκ •  = factor loadings •  = factor (co)variance matrix •  = residual variances ε1 to εn • τ = intercepts • κ = factor mean

  7. Model restrictions for measurement invariance. • Configural invariance: • Group 1: • 1= 11 1t+ 1 • Group 2: • 2= 22 2t+ 2 • Restrictions: • Thresholds are equal over groups. • diag(11t+ 1)=diag(22t+ 2)=diag() • Indicator means: Group1: 0 Group2: FREE • 2 = 1 = 1

  8. Model restrictions for measurement invariance. • metric invariance: • Group 1: • 1= 1t+ 1 • Group 2: • 2= 2t+ 2 • Restrictions: • Thresholds are equal over groups. • diag(11t+ 1)=diag(22t+ 2)=diag() • τ : group1: 0 group2: FREE • Κ: set to 0 • 1 = 1 (however 2 = FREE)

  9. Model restrictions for measurement invariance. • Strong invariance: • This concerns the means model. • E(Xi)= μ1 = 0 • E(Xi)= μ2 = κ and τ2 = 0 • Thresholds are equal over groups. • diag(11t+ 1)=diag(22t+ 2)=diag() • τ : group1: 0 group2: 0 • Κ: set to 0, free in group 2. • 1 = 1 (however 2 = FREE)

  10. Model restrictions for measurement invariance. • Strict invariance: • Can we equate the matrix  over groups. • However at this point  is not a free parameter: • diag(11t+ 1)=diag(22t+ 2)=diag() • So we must chose arbitrary values for  that are equal for both groups. • Replace the residuals algebra (diag(22t+ 2)=diag()) with a residuals matrix.

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