1. Advanced Statistical Methods: Continuous Variableshttp://statisticalmethods.wordpress.com Structural Equation Modeling_Part II Prof. Dr. Hab. K. M. Slomczynski

2. Equivalence of Measurement -1 - for comparative analysis: are measures are indeed comparable? Measurements can lack comparability because… – …different concepts are being compared (e.g. height vs. weight) – …a different measurement scale is used (weight in kilo vs. weightin pound) The concept of measurement equivalence deals with this issue: “whether or not, under different conditions of observing and studyingphenomena, measurement operations yield measures of the sameattribute” (Horn & McArdle 1992) Inequivalence if differences in the measurement scale donot reflect real differences

3. Equivalence of Measurement -2 Possible sources of measurement inequivalence (vande Vijver 1998): – Construct bias Some constructs are culture specific – Method bias Cultural traits can affect response style E.g. acquiescence E.g. choosing extreme response categories - Item bias Questionnaire translation Context specific meaning of certain items

4. Testing measurement equivalence - 1 Measurement equivalence can be tested my means ofMGCFA Several hierarchical levels of equivalence (cfr. Steenkamp & Baumgartner 1998) 1. Configural equivalence: – Identical factor structures over countries, but no equality constraintson the parameters – Same concept is measured in several countries – Yet: no score comparability! Group 1 Group 2 Group 3

5. Testing measurement equivalence – 2 2. Metric equivalence: – Equal slopes for all countries: - a 1unit increase in the latent variable has the same meaning across cultural groups; – Mean‐corrected scores can be compared over countries (e.g.regression coefficients, covariances)

6. Mean Structures – 1 For mean-comparisons to be valid, an higherlevel of equivalence is needed: scalar equivalence Introduction of a mean structure

7. Mean Structures – 2 New parameters in the model: – item intercepts (τ’s): predicted value for x when the latent mean = 0 – latent means (κ’s): mean value of the latent variable These new parameters imply a structure of the item means: Mean(X1) = τ1 +λκ => New pieces of information are involved in model estimation: the observed item means!

8. Mean Structures – 3 In single group CFA, including the mean structure isgenerally not very informative. Why? – Identification problem: 5 new parameters, 4 new pieces of information – Requires to add a restriction: κ = 0 – As a result: just‐identified; item intercepts = observed means MGCFA makes it possible to add cross‐group constraints, and to solve the identification problem (cfr. scalar invariance is needed for mean comparison)

9. Mean Structures – 4 Different possibilities to solve the identificationproblem. Most straightforward : – No computation of absolute means but relative toreference group – Fixing the latent mean of each construct in the firstgroup to zero; latent mean free in other groups – For at least one of the items, constrain the intercept tobe equal to the intercept in the first group (butpreferably more to guarantee scalar equivalence) Other methods of identification: Little, Slegers & Card (2006) in Structural equation modeling, pp. 59‐72

10. Mean Structures – 3 Exercise: 1 latent construct, 4 indicators, 2 groups. Is the mean structure identified if we use the method from previous slide? Pieces of information in the mean structure? 8 – Free intercept and mean parameters: • Latent mean of the second group: 1 • Intercept for x1 in group 1: 1 • Intercept for x1 in group 2: 0 • Intercepts for x2, x3 and x4 in group 1: 3 • Intercepts for x2, x3 and x4 in group 2: 3 • Total: 8 => JUST‐IDENTIFIED MEAN STRUCTURE!

11. Testing measurement equivalence – 3

12. Testing measurement equivalence – 4 Partial equivalence: meaningful comparisons arepossible if parameters for at least two items areequal across countries (cfr. Byrne, Shavelson &Muthen, 1989; but: De Beuckelaer & Swinnen. 2011) Unresolved issue: how to test the cross‐groupconstraints in practice? How to decide whether aconstraint is violated? Possible approaches: 1. Strictly confirmatory: Apply equivalence constraints;evaluate the overall model fit – Chi² test, RMSEA, CFI, TLI – But: overall fit indices are not always sensitiveto local misspecifications

13. Testing measurement equivalence – 5 2. Alternative models: fit several models (constrained,unconstrained, partially constrained); compareoverall model fit – Chi² difference test (but: sensitivity for large sample size and non‐normality) – Difference in alternative fit indices: ΔCFI; ΔRMSEA(Cheung & Rensvold 2002; Chen 2007); but: cut‐off pointsare arbitrary or based on limited simulation studies

14. Testing measurement equivalence – 6 3. Model generating approach: Start with aconstrained model, identifiy local misfit andadjust model accordingly – Modification indices – But again: sensitivity of the test to sample size

15. Functional equivalence • Functional equivalence of indicators vs. functional equivalence of constructs Functional equivalence of the indicator = the same meaning in compared populations Within country validity = emic Between country validity = etic