Factorial Invariance:

A Primer

Thomas J. Gross, Ph.D. University of Nebraska-Lincoln

Presented to

Tennessee State University

Psychology Department

Overview

• Confirmatory Factor Analysis (CFA)

• Longitudinal Invariance

• Issues with Short-forms

• Alabama Parenting Questionnaire-9

Confirmatory Factor Analysis

(CFA)

• How well measured variables (factors) represent a construct

• Confirm or reject

• Model Fit – how plausible a model is

• χ2

• Root Mean Square Error of Approximation (RMSEA)

• Standardized Root Mean Residual (SRMR)

• Tucker-Lewis Index (TLI): AKA Non-Normed Fit Index (NNFI)

• Comparative Fit Index (CFI)

Fit

Confirmatory Factor

Analysis

X1 X3 X2

θ22 θ11 θ33

λ11 λ21 λ31

Construct –

Latent Variable

Factors –

Manifest Variables

or Items

Ψ11

Factor Loadings –

Factors relationship to the Construct

Positive

Parenting

Praise Compliment Good job

Invariance

• The Same or Equal

• Measurement Invariance

• Longitudinal Invariance

• When would this be important?

• Has this measure shown to be invariant?

Invariance: 3 Levels

• Level 1:

• Configural Invariance

• Factors load on their presumed construct

• What does this mean longitudinally?

• What would you expect?

Invariance: 3 Levels

X1 X3 X2

θ22 θ11 θ33

λ11 λ21 λ31

Ψ11

Good model fit over time

Invariance: 3 Levels

• Level 2:

• Metric Invariance

• Factors load similarly on to the construct

• What does this mean longitudinally?

• What would you expect?

Invariance: 3 Levels

X1 X3 X2

θ22 θ11 θ33

λ11 λ21 λ31

Ψ11

Same Pattern across

factor loadings across

time

Invariance: 3 Levels

• Level 3:

• Scalar Invariance

• Factors means structure is similar

• What does this mean longitudinally?

• What would you expect?

Invariance: 3 Levels

X1 X3 X2

θ22 θ11 θ33

λ11 λ21 λ31

Ψ11

Same pattern of means

across time

Short-Forms of Assessments

• Correspondence to the full form

• Developmental periods

Alabama Parenting Questionnaire

(APQ)

• 42-item

• Parental Involvement (PI)

• Positive Parenting (PP)

• Inconsistent Discipline (ID)

• Poor Monitoring/Supervision (PS)

• Corporal Punishment (CP)

• Difficulty with progress monitoring

Alabama Parenting Questionnaire

(APQ)

• APQ-9

• Positive Parenting (PP)

• Inconsistent Discipline (ID)

• Poor Monitoring/Supervision (PS)

• 3 scales, 3 items each

• Is the youth version invariant over time?

APQ-9 Longitudinal Invariance

• Mid-8th, end 8th, end 9th grade

• Youth report of parenting

• Level 1: • What would we see if the APQ-9 has Configural Invariance?

• Fit stats?

• What does that mean?

• Level 2: • What would we see if the APQ-9 has Metric Invariance?

• Fit stats?

• What does that mean?

• Level 3: • What would we see if the APQ-9 has Scalar Invariance?

• Fit stats?

• What does that mean?

APQ-9 Longitudinal Invariance

MODEL χ2 df p RMSEA [90% CI] CFI TLI

NULL 3033.065 351 .000 NA NA NA

CONFIGURAL 392.482 261 .000 0.04 [0.031, 0.047] 0.951 0.934

METRIC 407.992 273 .000 0.039 [0.031, 0.047] 0.950 0.935

SCALAR 428.102 285 .000 0.040 [0.032, 0.047] 0.947 0.934

• Does the APQ-9 for Youth pass or fail?

PASS FAIL

APQ-9 Longitudinal Invariance

MODEL χ2 df p RMSEA [90% CI] CFI TLI

NULL 3033.065 351 .000 NA NA NA

CONFIGURAL 392.482 261 .000 0.04 [0.031, 0.047] 0.951 0.934

METRIC 407.992 273 .000 0.039 [0.031, 0.047] 0.950 0.935

SCALAR 428.102 285 .000 0.040 [0.032, 0.047] 0.947 0.934

• Does the APQ-9 for Youth pass or fail?

PASS FAIL

We don’t worry too much about Chi-

Squared because it is too sensitive to

sample size.

RMSEA < .10, and < .05. Low misfit

CFI > .90, or ≥ .95. Good fit.

TLI > .90. Adequate fit.

APQ-9 Longitudinal Invariance

T1 T1 T1 T2 T2 T2 T3 T3 T3

T1 T1 T1 T2 T2 T2 T3 T3 T3

T1 T1 T1 T2 T2 T2 T3 T3 T3

One more hurdle: Δχ2

• Δχ2 = χ12 – χ22

• Δdf = df1 – df2

• Look up value in χ2 table

• Non-significant result at .05-level

• METRIC vs. CONFIGURAL

• SCALAR vs METRIC

One more hurdle: Δχ2

MODEL VS. Δχ^2 Δdf p NULL -- -- -- -- CONFIGURAL -- -- -- --

METRIC CONFIGURAL 15.51 12 0.215

SCALAR METRIC 20.11 12 0.065

PASS FAIL

Next Foci

• Up next:

• Review matrix algebra

• The Greek

Σ = Λ Ψ Λ´ + Θ !!!!!!

• Syntax for the equations

• Level 4: Strict Invariance

• Multiple group models

• Partial invariance

Questions?

Review

• What is a CFA?

• Name important fit stats.

• What is invariance?

• Why do we care about invariance?

• How do we know if a measure is longitudinally invariant?

References

Dimitrov, D. M. (2010). Testing for factorial invariance in the context of construct validation. Measurement and Evaluation in Counseling and Development, 43, 121-149.

van de Schoot, R., Lugtig, P., & Hox, J. (2012). A checklist for testing measurement invariance. European Journal of Developmental Psychology, 9, 486-492.

• CFA Diagram from: Little, T.D. (June, 2014) Longitudinal Structural Equation Modeling. Texas Tech Institute for Measurement, Methodology, Analysis, & Policy, “Stats Camp.”

• Statistical Software: Muthén, B., & Muthén, L. (2012). Mplus user’s guide (7th ed.). Los Angeles, CA: Authors.

• Looking ahead: Milfont, T. L., & Fischer, R. (2010). Testing measurement invariance across groups: Applications in cross-cultural research. International Journal of Psychological Research, 3, 111-130.