1crmda.KU.edu

Todd D. LittleUniversity of Kansas

Director, Quantitative Training ProgramDirector, Center for Research Methods and Data Analysis

Director, Undergraduate Social and Behavioral Sciences Methodology MinorMember, Developmental Psychology Training Program

crmda.KU.eduColloquium presented 5-24-2012 @

University of Turku, Finland

Special Thanks to: Mijke Rhemtulla & Wei Wu

Factorial Invariance: Why It's Important and How to Test for It

Comparing Across Groups or Across Time• In order to compare constructs across two

or more groups OR across two or more time points, the equivalence of measurement must be established.

• This need is at the heart of the concept of Factorial Invariance.

• Factorial Invariance is assumed in any cross-group or cross-time comparison

• SEM is an ideal procedure to test this assumption.

Comparing Across Groups or Across Time

•Meredith provides the definitive rationale for the conditions under which invariance will hold (OR not)…Selection Theorem

• Note, Pearson originated selection theorem at the turn of the century

Which posits: if the selection process effects

only the true score variances of a set of

indicators, invariance will hold

Classical Measurement Theorem

Xi = Ti + Si + ei

Where, Xi is a person’s observed score on an item,Ti is the 'true' score (i.e., what we hope to measure),Si is the item-specific, yet reliable, component, andei is random error, or noise.

Note that Si and ei are assumed to be normally distributed (with mean of zero) and uncorrelated with each other. And, across all items in a domain, the Sis are uncorrelated with each other, as are the eis.

Selection Theorem on Measurement Theorem

X1 = T1 + S1 + e1

X2 = T2 + S2 + e2

X3 = T3 + S3 + e3

Selection Process

Levels Of Invariance• There are four levels of invariance:

1) Configural invariance - the pattern of fixed & free parameters is the same.2) Weak factorial invariance - the relative factor loadings are proportionally equal across groups.3) Strong factorial invariance - the relative indicator means are proportionally equal across groups.4) Strict factorial invariance - the indicator residuals are exactly equal across groups

(this level is not recommended).

The Covariance Structures Model

where...

Σ = matrix of model-implied indicator variances and covariances

Λ = matrix of factor loadings

Ψ = matrix of latent variables / common factor variances and covariances

Θ = matrix of unique factor variances (i.e., S + e and all covariances are usually 0)

The Mean Structures Model

where...

μ = vector of model-implied indicator means

τ = vector of indicator intercepts

Λ = matrix of factor loadings

α = vector of factor means

Factorial Invariance• An ideal method for investigating the degree of

invariance characterizing an instrument is multiple-group (or multiple-occasion) confirmatory factor analysis; or mean and covariance structures (MACS) models

• MACS models involve specifying the same factor model in multiple groups (occasions) simultaneously and sequentially imposing a series of cross-group (or occasion) constraints.

Some Equations

Configural invariance: Same factor loading pattern across groups, no constraints.

Weak (metric) invariance: Factor loadings proportionally equal across groups.

Strong (scalar) invariance: Loadings & intercepts proportionally equal across groups.

Strict invariance: Add unique variances to be exactly equal across groups.

( ) ( ) ( ) ( ) ( )g g g g g ( ) ( ) ( ) ( )g g g g

( ) ( ) ( )g g g ( ) ( ) ( )g g g

( ) ( ) ( )g g g ( ) ( )g g

( ) ( )g g ( ) ( )g g

Models and Invariance• It is useful to remember that all models

are, strictly speaking, incorrect. Invariance models are no exception.

"...invariance is a convenient fiction created to help insecure humans make sense out of a universe in which there may be no sense."

(Horn, McArdle, & Mason, 1983, p. 186).

Measured vs. Latent Variables• Measured (Manifest) Variables• Observable• Directly Measurable• A proxy for intended construct

• Latent Variables• The construct of interest• Invisible• Must be inferred from measured variables• Usually ‘Causes’ the measured variables (cf.

reflective indicators vs. formative indicators)• What you wish you could measure directly

Manifest vs. Latent Variables

• “Indicators are our worldly window into the latent space”

• John R. Nesselroade

Manifest vs. Latent Variables

X1

ξ1

Ψ11

X3X2

θ22θ11 θ33

λ11 λ21 λ31

Selection Theorem

X1

Group(Time) 1

Ψ11

X3X2

θ22θ11 θ33

λ11 λ21 λ31

X1

Group (Time) 2

Ψ11

X3X2

θ22θ11 θ33

λ11 λ21 λ31

Selection Influence

Estimating Latent Variables

17

X1

ξ1

Ψ11

X3X2

θ22θ11 θ33

λ11 λ21 λ31

Implied variance/covariance matrixX1 X2 X3

X1 11y1111 + θ11

X2 11y1121 21y1121 + θ22

X3 11y1131 21y1131 31y1131 + θ33

To solve for the parameters of a latent construct, it is necessary to set a scale (and make sure the parameters are identified)

Scale Setting and Identification

18

Three methods of scale-setting (part of identification process)

Arbitrary metric methods:•Fix the latent variance at 1.0; latent mean at 0• (reference-group method)•Fix a loading at 1.0; an indicator’s intercept at 0• (marker-variable method)

Non-Arbitrary metric method•Constrain the average of loadings to be 1 and the

average of intercepts at 0 • (effects-coding method; Little, Slegers, & Card, 2006)

1. Fix the Latent Variance to 1.0and Latent mean to 0.0)

19

X1

ξ1

1.0*

X3X2

θ22θ11 θ33

λ11 λ21 λ31

Implied variance/covariance matrixX1 X2 X3

X1 1112 + q11

X2 11 1 21 1212 + q22

X3 11 1 31 21 1 31 1312 + q33

Three methods of setting scale 1) Fix latent variance (Ψ11)

2. Fix a Marker Variable to 1.0(and its intercept to 0.0)

20

X1

ξ1

Ψ11

X3X2

θ22θ11 θ33

1.0* λ21 λ31

Implied variance/covariance matrixX1 X2 X3

X1y11 q11

X21y1121 21y1121

q22

X31y1131 21y1131 31y1131

q33

3. Constrain Loadings to Average 1.0(and the intercepts to average 0.0)

21

X1

ξ1

Ψ11

X3X2

θ22 θ33

λ11= 3-λ21-λ31 λ21 λ31

Implied variance/covariance matrixX1 X2 X3

X1(3-21- 31)y11(3-21- 31) + q11

X2(3-21- 31)y1121 21y1121 + q22

X3(3-21- 31) y1131 21y1131 31y1131 + q33

θ11

Configural invariance

1 2xx

1 65432

1* 1*

.64 .66 .71 .59 .55 .57

.09 .11 .07 .11 .07 .06

Group 1:

Group 2:

1 2xx

1 65432

1* 1*

.57 .61 .63 .63 .59 .60

.12 .10 .10 .11 .10 .07

Configural invariance

1 2xx

1 65432

1* 1*

.64 .76 .71 .59 .55 .57

.09 .11 .07 .11 .07 .06

Group 1:

Group 2:

1 2xx

1 65432

1* 1*

.57 .51 .63 .63 .59 .60

.12 .10 .10 .11 .10 .07

Configural invariance

1 2-.32

1 65432

1* 1*

.64 .66 .71 .56 .55 .57

.09 .11 .07 .11 .07 .06

Group 1:

Group 2:

1 2-.07

1 65432

1* 1*

.57 .61 .63 .63 .59 .60

.12 .10 .10 .11 .10 .07

Weak factorial invariance (equate λs across groups)

1 2PS(2,1)

1 65432

1* 1*PS(1,1) PS(2,2)

LY(1,1) LY(2,1) LY(3,1) LY(4,2) LY(5,2) LY(6,2)

TE(1,1) TE(2,2) TE(3,3) TE(4,4) TE(5,5) TE(6,6)

1 2PS(2,1)

1 65432

e ePS(1,1) PS(2,2)

=LY(1,1) =LY(2,1) =LY(3,1) =LY(4,2) =LY(5,2) =LY(6,2)

TE(1,1) TE(2,2) TE(3,3) TE(4,4) TE(5,5) TE(6,6)

Group 1:

Group 2:

Note: Variances are now Freed in group 2

F: Test of Weak Factorial Invariance

Positive Negative

.58 .59 .64 .62 .59 .61

-.07

.11 .10 .11 .10 .07

Model Fit: χ2(20, n=759)=49.0; RMSEA=.062(.040-.084); CFI=.99;

NNFI=.99

1* 1*

Great+ Glad

Unhappy+ Bad

Down + Blue

Terrible + Sad

Happy+ Super

Cheerful+ Good

.10 .07 .11 .07 .06

1.22

.85

-.33

.12

.09

(9.2.1.TwoGroup.Loadings.FactorID)

M: Test of Weak Factorial Invariance

Positive Negative

1* 1.02 1.11 1* .95 .97

-.03

.11 .10 .11 .10 .07

Model Fit: χ2(20, n=759)=49.0; RMSEA=.062(.040-.084); CFI=.99; NNFI=.99

.33 .39

Great+ Glad

Unhappy+ Bad

Down + Blue

Terrible + Sad

Happy+ Super

Cheerful+ Good

.10 .07 .11 .07 .06

.41 .33-.12

.12

.09

(9.2.1.TwoGroup.Loadings.MarkerID)

EF: Test of Weak Factorial Invariance

Positive Negative

.96 .98 1.06 1.03 .97 1.00

-.03

.11 .10 .11 .10 .07

Model Fit: χ2(20, n=759)=49.0; RMSEA=.062(.040-.084); CFI=.99; NNFI=.99

.36 .37

Great+ Glad

Unhappy+ Bad

Down + Blue

Terrible + Sad

Happy+ Super

Cheerful+ Good

.10 .07 .11 .07 .06

.44 .31-.12

.12

.09

(9.2.1.TwoGroup.Loadings.EffectsID)

• The results of the two-group model with equality constraints on the corresponding loadings provides a test of proportional equivalence of the loadings:

Results Test of Weak Factorial Invariance

Nested significance test:(χ2

(20, n=759) = 49.0) - (χ2(16, n=759) = 46.0) = Δχ2

(4, n=759) = 3.0, p > .50

The difference in χ2 is non-significant and therefore the constraints are supported. The loadings are invariant across the two age groups.

“Reasonableness” tests:

RMSEA: weak invariance = .062(.040-.084) versus configural = .069(.046-.093)

The two RMSEAs fall within one another’s confidence intervals.CFI: weak invariance = .99 versus configural = .99 The CFIs are virtually identical (one rule of thumb is ΔCFI <= .01 is acceptable).

(9.2.TwoGroup. Loadings)

• When we regress indicators on to constructs we can also estimate the intercept of the indicator.

• This information can be used to estimate the Latent mean of a construct

• Equivalence of the loading intercepts across groups is, in fact, a critical criterion to pass in order to say that one has strong factorial invariance.

Adding information about means

Adding information about means

1 2

1 65432

1* 1*AL(1)

TY(1) TY(2) TY(3) TY(4) TY(6)TY(5)

AL(2)

X

Adding information about means(9.3.0.TwoGroups.FreeMeans)

1 2

1 65432

3.143.07

2.992.85

3.072.98

1.701.72

1.531.58

1.551.55

0*0*

0*0*

Model Fit: χ2(20, n=759) = 49.0 (note that model fit does not change)

X

Strong factorial invariance (aka. loading invariance) – Factor Identification Method

(9.3.1.TwoGroups.Intercepts.FactorID)

1 2

1 65432

X

3.15 2.97 3.08 1.70 1.55 1.54

0*-.16

0*.04

Model Fit: χ2(24, n=759) = 58.4, RMSEA = .061(.041;.081), NNFI = .986, CFI = .989

.58 .59 .64 .62 .59 .61

-.07-.33

1*1.22

1*0.85

(9.3.1.TwoGroups.Intercepts.MarkerID)

1 2

1 65432

X

0* -.28 -.43 0* -.06 -.12

3.153.06

1.701.72

Model Fit: χ2(24, n=759) = 58.4, RMSEA = .061(.041;.081), NNFI = .986, CFI = .989

1* 1.03 1.11 1* .95 .97

-.03-.12

.33

.40.39.33

Strong factorial invariance (aka. loading invariance) – Marker Var. Identification Method

(9.3.1.TwoGroups.Intercepts.EffectsID)

1 2

1 65432

X

.23 -.05 -.18 .06 -.00 -.06

3.072.97

1.591.62

Model Fit: χ2(24, n=759) = 58.4, RMSEA = .061(.041;.081), NNFI = .986, CFI = .989

.95 .98 1.06 1.03 .97 1.00

-.03-.12

.36

.44.37.31

Strong factorial invariance (aka. loading invariance) – Effects Identification Method

Indicator mean = intercept + loading(Latent Mean)i.e., Mean of Y = intercept + slope (X)

For Positive Affect then:Group 1 (7th grade): Group 2 (8th grade):

Y = τ + λ (α) Y = τ + λ (α)3.14 ≈ 3.15 + .58(0) 3.07 ≈ 3.15 + .58(-.16) = 3.062.99 ≈ 2.97 + .59(0) 2.85 ≈ 2.97 + .59(-.16) = 2.883.07 ≈ 3.08 + .64(0) 2.97 ≈ 3.08 + .64(-.16) = 2.98

Note: in the raw metric the observed difference would be -.103.14 vs. 3.07 = -.072.99 vs. 2.85 = -.14 gives an average of -.10 observed3.07 vs. 2.97 = -.10

============== i.e. averaging: 3.07 - 2.96 = -.10

How Are the Means Reproduced?

_ _

The complete model with means, std’s, and r’s

Positive3

Negative4

-.07

1* 1*-.32

Positive1

Negative2

0* 0*

1.0*(in group 1)

1.0*(in group 1)

1.11(in group 2)

.92(in group 2)

X

-.16 (z=2.02)

.04 (z=0.53)

3.15 2.97 3.08 1.70 1.541.54

.62 .59 .61.64.58 .59

Estimated only in group 2! Group 1 = 0

(9.7.1.Phantom variables.With Means.FactorID)

Model Fit: χ2(24, n=759) = 58.4, RMSEA = .061(.041;.081), NNFI = .986, CFI=.989

The complete model with means, std’s, and r’s

Positive3

Negative4

-.07

1* 1*-.32

Positive1

Negative2

0* 0*

.58(in group 1)

.62(in group 1)

.64(in group 2)

.57(in group 2)

X

3.15 3.06

1.701.72

0* -.28 -.43 0* -.12-.06

1* .95 .971.111* 1.03

(9.7.2.Phantom variables.With Means.MarkerID)

Model Fit: χ2(24, n=759) = 58.4, RMSEA = .061(.041;.081), NNFI = .986, CFI=.989

The complete model with means, std’s, and r’s

Positive3

Negative4

-.07

1* 1*-.32

Positive1

Negative2

0* 0*

.60(in group 1)

.61(in group 1)

.67(in group 2)

.56(in group 2)

X

3.07 2.97

1.591.62

.23 -.05 -.18 .06 -.06-.00

1.03 .97 1.001.06.96 .98

(9.7.3.Phantom variables.With Means.EffectsID)

Model Fit: χ2(24, n=759) = 58.4, RMSEA = .061(.041;.081), NNFI = .986, CFI=.989

Cohen’s d = (M2 – M1) / SDpooled

where SDpooled = √[(n1Var1 + n2Var2)/(n1+n2)]

Effect size of latent mean differences

Cohen’s d = (M2 – M1) / SDpooled

where SDpooled = √[(n1Var1 + n2Var2)/(n1+n2)]

Latent d = (α2j – α1j) / √ψpooled

where √ψpooled = √[(n1 ψ1jj + n2 ψ2jj)/(n1+n2)]

Effect size of latent mean differences

Cohen’s d = (M2 – M1) / SDpooled

where SDpooled = √[(n1Var1 + n2Var2)/(n1+n2)]

Latent d = (α2j – α1j) / √ψpooled

where √ψpooled = √[(n1 ψ1jj + n2 ψ2jj)/(n1+n2)]

dpositive = (-.16 – 0) / 1.05where √ψpooled = √[(380*1 + 379*1.22)/(380+379)]

= -.152

Effect size of latent mean differences

5. Invariance of Variances

χ2 difference test

6. Invariance of Correlations/Covariances

χ2 difference test

3b or 7. Invariance of Latent Means

χ2 difference test

3. Invariance of InterceptsRMSEA/CFI difference Test

Comparing parameters across groups1. Configural Invariance

Inter-occular/model fit Test

4. Invariance of Variance/

Covariance Matrix

χ2 difference test

2. Invariance of LoadingsRMSEA/CFI difference Test

The ‘Null’ Model

44

• The standard ‘null’ model assumes that all covariances are zero – only variances are estimated

• In longitudinal research, a more appropriate ‘null’ model is to assume that the variances of each corresponding indicator are equal at each time point and their means (intercepts) are also equal at each time point (see Widaman & Thompson).

• In multiple-group settings, a more appropriate ‘null’ model is to assume that the variances of each corresponding indicator are equal across groups and their means are also equal across groups.

Byrne, B. M., Shavelson, R. J., & Muthén, B. (1989). Testing for the equivalence of factor covariance and mean structures: The issue of partial measurement invariance. Psychological Bulletin, 105, 456-466.

Cheung, G. W., & Rensvold, R. B. (1999). Testing factorial invariance across groups: A reconceptualization and proposed new method. Journal of Management, 25, 1-27.

Gonzalez, R., & Griffin, D. (2001). Testing parameters in structural equation modeling: Every “one” matters. Psychological Methods, 6, 258-269.

Kaiser, H. F., & Dickman, K. (1962). Sample and population score matrices and sample correlation matrices from an arbitrary population correlation matrix. Psychometrika, 27, 179-182.

Kaplan, D. (1989). Power of the likelihood ratio test in multiple group confirmatory factor analysis under partial measurement invariance. Educational and Psychological Measurement, 49, 579-586.

Little, T. D., Slegers, D. W., & Card, N. A. (2006). A non-arbitrary method of identifying and scaling latent variables in SEM and MACS models. Structural Equation Modeling, 13, 59-72.

MacCallum, R. C., Roznowski, M., & Necowitz, L. B. (1992). Model modification in covariance structure analysis: The problem of capitalization on chance. Psychological Bulletin, 111, 490-504.

Meredith, W. (1993). Measurement invariance, factor analysis and factorial invariance. Psychometrika, 58, 525-543.

Steenkamp, J.-B. E. M., & Baumgartner, H. (1998). Assessing measurement invariance in cross-national consumer research. Journal of Consumer Research, 25, 78-90.

References

45

46crmda.KU.edu

Todd D. LittleUniversity of Kansas

Director, Quantitative Training ProgramDirector, Center for Research Methods and Data Analysis

Director, Undergraduate Social and Behavioral Sciences Methodology MinorMember, Developmental Psychology Training Program

crmda.KU.eduColloquium presented 5-24-2012 @

University of Turku, Finland

Special Thanks to: Mijke Rhemtulla & Wei Wu

Factorial Invariance: Why It's Important and How to Test for It

47www.Quant.KU.edu

Update

Dr. Todd Little is currently at Texas Tech University

Director, Institute for Measurement, Methodology, Analysis and Policy (IMMAP)Director, “Stats Camp”

Professor, Educational Psychology and LeadershipEmail: yhat@ttu.edu

IMMAP (immap.educ.ttu.edu)Stats Camp (Statscamp.org)