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: Often1 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.
Since the item variances are restricted to 1 the labda’s are scaled. This means the labdas squared reflect the variance explained by the common factor F1 in the item Xi

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.