Factorial Invariance: An Overview

What is factorial invariance?

When conducting multiple-group confirmatory factor analysis, one should establish factorial invariance. If a

researcher establishes factorial invariance, then he or she is measuring the same construct across groups or

across time. Factorial invariance is assumed when making a comparison between groups or between time

points. When doing multiple-group confirmatory factor analysis, this assumption can be tested directly.

Factorial invariance is also known as measurement equivalence and metric invariance.

What are the levels of factorial invariance?

There are four levels of factorial invariance. These levels include configural invariance, weak factorial

invariance, strong factorial invariance, and strict factorial invariance (the last type is not recommended). The

basic definitions of the four levels of factorial invariance are:

(1) Configural invariance: across groups, the pattern of fixed and free parameters is equivalent

(2) Weak factorial invariance: across groups, corresponding factor loadings are equivalent

(3) Strong factorial invariance: across groups, corresponding indicator means are equivalent

(4) Strict factorial invariance: across groups, corresponding indicator residuals are equivalent (this

level is not recommended because the criterion is too strict and is hard to establish in practice)

Establishing Factorial Invariance

for Multiple-Group

Confirmatory Factor Analysis

Timmons, A.C. (2010)

An overview of tests for establishing factorial invariance

Note: See the KUant Guides on configural invariance, weak factorial invariance, and strong factorial

invariance for more detailed information regarding how to establish factorial invariance

(1) Configural invariance: use X2, RMSEA, CFI, NNFI, and other fit indexes to determine whether

the combined models have good model fit. If so, this indicates configural invariance.

(2) Weak factorial invariance: compare the RMSEA values and RMSEA confidence intervals of the

configural and weak factorial invariance models. If the RMSEA values fall within one another’s

confidence intervals, this indicates weak factorial invariance. Also, examine the change in CFI for the

configural and weak factorial invariance models. If the change is less than .01, this indicates weak

factorial invariance (Cheung & Rensvold, 2002).

(3) Strong factorial invariance: compare the RMSEA values and RMSEA confidence intervals of the

weak and strong factorial invariance models. If the RMSEA values fall within one another’s confidence

intervals, this indicates strong factorial invariance. Also, examine the change in CFI for the weak and

strong factorial invariance models. If the change is less than .01, this indicates strong factorial

invariance (Cheung & Rensvold, 2002).

22.1

Configural Invariance

What is configural invariance?

Configural invariance means that the pattern of fixed and free parameters is equivalent across groups. For

example, suppose that a researcher is measuring positive and negative affect in seventh and eighth graders.

This is what a path diagram depicting positive and negative affect might look like:

Which parameters are fixed and which are free?

The parameters that are estimated are free. The parameters that are not estimated are fixed. In this diagram, Ψ21,

λ11, λ21, λ31, λ42, λ52, λ62, 11, 22, 33, 44, 55, and 66 are free. In contrast, Ψ11 and Ψ22 are fixed to one, and all

other parameters are fixed to zero.

Is the pattern of fixed and free parameters the same across groups?

If so, then there is configural invariance. Here is a graphical representation of configural invariance. Note how

both groups have the same pattern of fixed and free parameters:

Group one: seventh graders

\

Group two: eighth graders

How can one test for configural invariance?

One can test for configural invariance using software packages. For this guide, the code is in LISREL and

uses the fixed factor method of identification. See the Works Consulted section (listed at the end of the

guide) for information on other forms of identification. The code typically consists of two parts, which are

listed in a single syntax file. In the first part, specify the information for group one. In the second part,

specify the information for group two. For basic information on LISREL, see the LISREL 8.80 Syntax

KUant Guide.

Code for group one

How to determine whether there is configural invariance

From the above code, the program will generate model fit information for the two groups

combined. If the combined model has good fit, then there is configural invariance. The results for

this example are:

Configural: χ2

(16, n = 759) = 41.0

RMSEA = .070 (.047 - .095)

CFI = .99

NNFI = .98

Overall, the fit indexes indicate good model fit. Therefore, there is configural invariance. Next,

one should test for weak factorial invariance. See the KUant Guide on weak factorial invariance

for more information.

Code for group two

Note: parts one and two are included in a single syntax file

Weak Factorial Invariance

What is weak factorial invariance?

Weak factorial invariance means that across groups, corresponding factor loadings are equivalent. For

example, suppose that a researcher is measuring positive and negative affect in seventh and eighth graders.

This is what a path diagram depicting positive and negative affect might look like:

What are the factor loadings?

The factor loadings are represented by lambda. Therefore, λ11, λ21, λ31, λ42, λ52, and λ62 are the factor loadings.

They represent how strongly the indicators load onto the construct. To establish weak factorial invariance, the

factor loadings for group one should be equivalent to the factor loadings for group two.

Are the factor loadings equivalent across groups?

If so, then there is weak factorial invariance. Here is a graphical representation of weak factorial invariance:

How can one test for weak factorial invariance?

One can test for weak invariance using software packages. For this guide, the code is in LISREL and uses

the fixed factor method of identification. See the Works Consulted section (listed at the end of the guide)

for information on other forms of identification. The code typically consists of two parts, which are listed

in a single syntax file. In the first part, specify the information for group one. In the second part, specify the

information for group two. For basic information on LISREL, see the LISREL 8.80 Syntax KUant Guide.

Code for group one

How to determine whether there is weak factorial invariance

From the above code, the program will generate model fit information. First, compare the CFI

value with the CFI value of the configural invariance model. If there is a change of less than .01,

then this indicates weak factorial invariance (Cheung & Rensvold, 2002).

CFI (weak) = .99

CFI (configural) = .99

The change of less than .01 indicates that there is weak factorial invariance (Cheung &

Rensvold, 2002). Next, examine the RMSEA values and confidence intervals for the weak and

configural models.

If the RMSEA values fall within one another’s confidence intervals, then this indicates weak

factorial invariance. For example:

RMSEA (weak) = .062 (.040 - .084)

RMSEA (configural) = .070 (.047 - .095)

The RMSEA values fall within one another’s confidence intervals, indicating that there is weak

factorial invariance. The constraints do not result in a decrease in model fit. Next, one should test

for strong factorial invariance. See the KUant Guide on strong factorial invariance for more

information.

Code for group two

Note: parts one and two are included in a single syntax file

Strong Factorial Invariance

What is strong factorial invariance?

Strong factorial invariance means that across groups, corresponding indicator means are equivalent. For

example, suppose that a researcher is measuring positive and negative affect in seventh and eighth graders. This

is what a path diagram depicting positive and negative affect (with intercepts and alphas) might look like:

What are the indicator means?

The indicator means are approximately equal to the lambda values times the alpha values plus the tau values (

τ + λ (α)). To establish strong factorial invariance, the indicator means of both groups (the seventh and

eighth graders) should be equivalent.

Are the corresponding indicator means equal across groups?

If so, then there is strong factorial invariance. Here is a graphical representation of strong factorial

invariance:

How can one test for strong factorial invariance?

One can test for strong invariance using software packages. For this guide, the code is in LISREL and uses

the fixed factor method of identification. See the Works Consulted section (listed at the end of the guide)

for information on other forms of identification. The code typically consists of two parts, which are listed

in a single syntax file. In the first part, specify the information for group one. In the second part, specify the

information for group two. For basic information on LISREL, see the LISREL 8.80 Syntax KUant Guide.

How to determine whether there is strong factorial invariance

From the above code, the program will generate model fit information. First, compare the CFI value

with the CFI value of the weak factorial invariance model. If there is a change of less than .01, then

this indicates strong factorial invariance (Cheung & Rensvold, 2002).

CFI (strong) = .99

CFI (weak) = .99

The change of less than .01 indicates that there is strong factorial invariance (Cheung & Rensvold,

2002). Next, examine the RMSEA values and confidence intervals for the strong and weak factorial

invariance models.

If the RMSEA values fall within one another’s confidence intervals, then this indicates strong factorial

invariance. For example:

RMSEA (strong) = .062 (.042 - .082)

RMSEA (weak) = .062 (.040 - .084)

The RMSEA values fall within one another’s confidence intervals, indicating that there is strong

factorial invariance. The constraints do not result in a decrease in model fit. The last type of factorial

invariance, strict factorial invariance, is not recommended. After establishing configural invariance,

weak factorial invariance, and strong factorial invariance, proceed by testing differences in latent

parameters.

Note: A thank you to Dr. Todd Little for supplying data for this example, as well as for instruction and

input on this topic

Note: A thank you to Dr. Todd Little for supplying data for this example, as well as for

instruction and input on this topic

Works Consulted

Brown, T. A. (2006). Confirmatory factor analysis for applied research. New York: Guilford Press.

Cheung, G. W., & Rensvold, R. B. (2002). Evaluating goodness-of-fit indexes for testing measurement

invariance. Structural Equation Modeling, 9, 233–255.

Kline, R. B. (2011). Principles and practice of structural equation modeling (3rd ed.). New York: Guilford

Press.