Şekercioğlu, G. (2018). Measurement invariance: 

Concept and implementation. International Online 

Journal of Education and Teaching (IOJET), 5(3). 

609-634. 
http://iojet.org/index.php/IOJET/article/view/439/257  

Received:   30.05.2018 
Received in revised version: 25.06.2018 

Accepted:   30.06.2018 

 

MEASUREMENT INVARIANCE: CONCEPT AND IMPLEMENTATION 

 

Güçlü Şekercioğlu   

Akdeniz University, Turkey 

guclus@akdeniz.edu.tr 

 

Güçlü Şekercioğlu has had his Ph.D. in Measurement and Evaluation from Ankara 

University, Turkey. He is a teaching staff at Akdeniz University, Education Faculty. He 

conducts research in multivariate statistics, measurement invariance, differential item 

functioning and psychological measurement in culture context. 

 

Copyright by Informascope. Material published and so copyrighted may not be published 

elsewhere without the written permission of IOJET.  

http://iojet.org/index.php/IOJET/article/view/439/257
mailto:guclus@akdeniz.edu.tr
https://orcid.org/0000-0003-1806-7003


International Online Journal of Education and Teaching (IOJET) 2018, 5(3), 609-634. 

 

 

    

609 

MEASUREMENT INVARIANCE: CONCEPT AND 

IMPLEMENTATION 

 

Güçlü Şekercioğlu 

guclus@akdeniz.edu.tr  

 

Abstract 

An empirical evidence for independent samples of a population regarding measurement 

invariance implies that factor structure of a measurement tool is equal across these samples; 

in other words, it measures the intended psychological trait within the same structure. In this 

case, the evidence of construct validity would be strengthened within the frame of the scores 

obtained from the tool. When measurement invariance is not supported, the researchers 

should consider the possibility of the different factor designs for each group. Ignoring such a 

situation brings forward the probability about differentiation of the trait(s) measured by 

measurement tool for that/those group(s), so it causes to suspect the validity of the scores 

obtained from the tool. The aim of this study is to examine measurement invariance in the 

context of the conceptual foundations of multi-group confirmatory factor analysis, and 

discuss the subject through the results from two hypothetical data set that one supports 

measurement invariance, but the other does not. As a result of analysis performed in this 

direction, it is determined that the five-factor design derived from the first data set is equal 

across the groups in the majors of science, health, and social science. It is also concluded that 

the three-factor design obtained from the secondary data set is not equal for female and male 

groups. Besides, the exploratory factor analysis performed for female and male groups 

separately shows that the three-factor design of the tool is valid for females, but the number 

of factors was four in males. When the factor design for male group is examined, it is 

determined that the three items in the second factor separate significantly. That leads to the 

conclusion that it is crucial to test measurement invariance in studies regarding the 

determination of the psychometric properties of the tool.  

Keywords: measurement invariance, equality of factor structures, multi-group 
confirmatory factor analysis, structural equation modeling 

 

1. Introduction 

The major problem in behavioural and educational science studies, which aim developing 

the psychological measurement tools, cultural adaptation of a tool developed in another 

culture, using the tool for a different purpose or for a different sample, is to demonstrate the 

validity of the empirical evidence on the psychometric properties of the tool. In this direction 

the researchers, within the framework of these fundamental problems, are obliged to question 

whether the tool measures the trait(s) what it intends to measure properly and precisely. 

Further examination related to psychometric properties of measurement tool and all other 

analyses based on the scores obtained from the measurement tool (ANOVA, regression, etc.) 

has been carried out after the validity of the evidence put forward and decision are taken in 

this direction. According to Nunnally and Bernstein (1994), the validity of each usage must 

be documented by empirical evidence even though a measurement result may be valid for 

more than one purpose. Therefore, the test authors and users should not assume that the 

validity of evidence cannot change (Crocker and Algina, 1986).  

mailto:asliakkoyunlu@gmail.com


Şekercioğlu 

 
 
 

610 

One of the most important dimensions of the validity of scores obtained from 

psychological measurement tools is the construct validity. In the report of testing standards 

published in 1954 it was discussed that the concept of validity, actually all types of validity 

should be assembled under the roof of construct validity (Cronbach and Meehl, 1955; Jonson 

and Plake, 1998; Urbina, 2004; Westen and Rosenthal, 2005). Similarly, Kline (2000) states 

that the construct validity includes other approaches as well, thus all types of validity are 

related to the assessment of construct validity. The factor analysis is one of the most 

commonly used techniques in the studies which aim to determine the psychometric properties 

of a measurement tool in behavioural and educational science, in order to obtain evidence of 

construct validity. According to Büyüköztürk (2002, 2014) the factor analysis is a 

multivariate statistics, which aims to find and explore conceptually meaningful fewer new 

variables (factor, component) by bringing a large number of inter-related variables together. 

The factor analysis can be considered under two headings, which are exploratory factor 

analysis (EFA) and confirmatory factor analysis (CFA) discussed under the concept of 

structural equation modelling (SEM).  

CFA, which is based on testing of theories about the latent variables, and used in advanced 

research, is a very sophisticated technique (Ullman, 2001). In this analysis, a predefined and 

constrained construct is tested whether it is confirmed as a model. It is also occasionally used 

to mean the confirmation of the theoretical structure (Maruyama, 1998). In this context, the 

determination of the construct validity for CFA is emphasized as a very powerful method 

(Floyd and Wideman, 1995; Kline, 2005; Stapleton, 1997).  

Multi-group confirmatory factor analysis (multi-group CFA) is also a spesific practice 

area in CFA. This analysis enables to test the equality of structural parameters for more than 

one group simultaneously. In this context, the assessment of equality between the groups in 

terms of factor structure is also termed as measurement invariance. Additionally, examining 

the fitness of structure brings about the concept of testing population heterogeneity. It is 

possible to encounter different terms for different tests of measurement invariance tests 

including equality test of factor structures, metric and factorial invariance in the literature 

(Brown, 2006). 

Nowadays, interest of the researchers in social sciences towards measurement invariance 

is gradually increasing. In a plain defination, measurement invariance is the description of 

whether the structures of measurement tool are equal across individuals from different 

groups. This concept has a critical importance in comparing groups. When measurement 

invariance is not supported between the groups, it is not possible to interpret the findings that 

reveal differences concerning these groups. If the researcher does not have the evidence for 

measurement invariance, then the existence of different psychometric responses for scale 

items more than one group cannot be known. Measurement invariance analyses are used in 

intercultural comparison for groups speaking different languages in a culture, scale 

adaptation studies, the comparison of groups with different academic achievement, the 

comparison of employee groups in different areas of industry, comparisons based on gender 

and are also used to compare a control group and an experimental group in empirical research 

(Cheung and Rensvold, 2002). The frequently asked question on the use of psychological 

measurement tools is whether the factor design ensued as a result of factor analysis of the 

measurement tools valid for groups, which differentiated at such a level that may impact the 

measurement process concerning the ethnic characteristics, age or the way they respond to 

the items. In fact, the fundamental issue here is whether the measurement tool measures the 

same structural properties for different groups or not. When the factor structure is not equal 

across groups, naturally it is not possible to make meaningful comparisons between groups 

based on the factor scores. On the contrary, when measurement equivalence is supported 



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611 

empirically, it is concluded that the group differences are completely reflected in terms of 

latent traits evaluated by factors. In this concept, the studies, which aim at determining 

equality of the measurement tools’ factor structure, are becoming more and more substantial 

because cultural, developmental and contextual impacts related to the psychological 

structural traits have become well-known by the researchers recently (Floyd and Widaman, 

1995). In addition to social science studies, using multi-group CFA becomes increasingly 

common in other majors such as psychology, education, management and organization, 

marketing, and communication, especially ones which carried out studies based on cross-

cultural comparisons.  

As Jöreskog, Sörbom and Toit (2000) claim, the factor structure of developed or adapted 

scale based on fundamental data set obtained from different groups or samples can be tested 

whether it is equal for more than one group or not concerning the national, territorial, 

regional, cultural or socio-economic status of the groups. It is highly functional to test the 

equality of factor structures for a scale or different numbers of items for more than a group. 

Thus, factors or structural relationships can be tested simultaneously whether they are equal 

across different samples (Baumgartner and Steenkamp, 1998). 

According to Marcoulides and Schumacker (1996) in multi-group CFA, the question of “is 

each group measured under the same structure?” is investigated and this examination is 

carried out within the framework of the measurement model defined in advance. Similarly, 

Kline (2005) stated that, the focus of multi-group CFA is to test whether measurement 

invariance is supported for different groups within the same latent variables. This concept is 

defined as invariance of the psychometric properties of a scale across groups in the context of 

modelling in the literature of psychometry. 

Determining whether the measurement invariance is supported for different groups or not 

has a critical role in the development of psychometric properties of psychological 

measurement tools. That implies whether the items of the same structure and all structures 

can be used for the sub-groups of a population. Likewise, the subject of testing measurement 

invariance plays a crucial role in terms of defining the generalizability of psychological 

structure across groups with differinf variables such as different cultures, age groups and 

genders. The equality tests of latent means, which are included in the analysis group, shows 

similarity with the comparison of observed group averages through t-test and ANOVA 

(Brown, 2006). 

According to Byrne (2006), the researchers often seek answers to any of the following five 

questions for evidence related to the multi-group equality: (i) do certain structures of the 

items on the measurement tool work equally across different groups? In other words, does the 

measurement model have a group equality? (ii) is the factor structure of the tool or theoretical 

structure measured by multiple scales equal for each level of the group? (iii) are the paths of 

the experimental structures equal across the groups? (iv) does the latent means in the model 

for a particular structure vary between groups? and (v) is the factor structure of a 

measurement tool equal for independent samples of the population? The author particularly 

emphasizes that there could be a cross-validation study in his last question. The analysis 

results reach the conclusion that if the factor structures are not equal between the groups, the 

validity of interpretations based on a comparison of scores for these groups decreases. 

According to Brown (2006), the process steps below should be followed in the evaluation 

of the multi-group CFA and measurement invariance: (i) performing CFA for each group 

included in the analysis separately, (ii) testing the equality of structures simultaneously 

(factor loadings, factor correlations and error variances constant), (iii) testing the equality of 

the factor structures (factor loadings free; factor correlations and error variances constant), 



Şekercioğlu 

 
 
 

612 

(iv) testing the equality of factor structures and the error variances indicators (factor loadings 

and error variances free; factor correlations constant), (v) testing the equality errors variances 

of indicator (error variances free, factor loadings and factor correlations constant), (vi) testing 

the equality of factor variances, (vii) testing the equality of factor covariances (if more than 

one factor), and (viii) testing the equality of latent means. Hereunder, the first step is one of 

the multi-group CFA’s assumption. The processes between the second and fifth steps are 

about testing measurement invariance, and the processes between the sixth and eighth steps 

are about testing the population heterogeneity. 

1.1. Measurement Invariance Test and Models 

Before computing the multi-group CFA, first of all, correlation or covariance matrix of the 

groups in the same sample is evaluated by comparing each other. In other words, before 

setting up the configural invariance model (Model 1), the establishment of the test equality of 

covariance matrices (Model 0) must be made. If the equality of covariance matrices is 

provided for each group (𝛴𝑔 = 𝛴𝑔
′
) the configural invariance model can be developed and 

tested. The equality of covariance structures of the groups should be discussed only after the 

null hypothesis (H0) has been rejected. Subsequently, the models for other hypothesizes 

should be tested separately. The configural model derived from different groups should be 

defined in the same sample. Thus, the defined model for each group of multi-group analysis 

would be simultaneously tested. In this case, it is expected to see high fitness between 

correlation or covariance matrices of different groups (Brown, 2006; Byrne, 2006; Dunn, 

Everitt and Pickles, 1993; Vandenberg and Lenca, 2000). In general, the measurement 

invariance is tested with four basic models. These models are summarized in Table 1 

(adapted  from Cheung and Rensvold, 2002). 

Table 1. Measurement invariance models 
Models Hypothesis Hypothesis Name Symbolic Statement Process 

1 𝐻𝑓𝑜𝑟𝑚 Configural 
invariance 

Λ𝑓𝑜𝑟𝑚
(1)

∴ Λ𝑓𝑜𝑟𝑚
(2)

 Invariance is supported for all groups regarding 

construct and items. Factor loadings, factor 

correlations, and error variances are equal for all 

groups.  

    

2 𝐻Λ Weak Factorial 
Invariance (Metric 

Invariance) 

Λ(1) ∴ Λ(2) Invariance is supported for all groups regarding 
factor correlations and error variances. The factor 

loadings have been freed for groups. 

3 𝐻𝜆 Strong Factorial 
Invariance (Scalar 

Invariance) 

𝜆𝑖𝑗
(1)

∴ Λ𝑖𝑗
(2)

 Invariance is supported for all groups regarding 

factor correlations. The factor loadings and error 

variances have been freed for groups. 

4 𝐻Λ,Θ(𝛿) Strict Factorial 
Invariance (Residual 

Variance Invariance)  

Θ
𝛿
(1)

∴ Θ
𝛿
(2)

 Invariance is supported for all groups regarding 

factor loadings and correlations. The error 

variances have been freed for groups. 

1.1.1. Configural invariance (baseline model) 

Developing a configural invariance (also known as baseline model) begins with 

identifying and testing the model, which was developed within the framework of a specific 

hypothesis for each group. In this context, the number of sub-scales in configural invariance 

model for each group (e.g. factors), the positions of the items (e.g. which factors the items are 

loaded) and correlations between sub-scales (e.g. setting such factors covariance) are 

determined. Secondly, the validity of the configural invariance model is tested separately for 

each group. Ideally, the model is expected to well fit and significant. However, the evidence, 

which shows a well fit, provides the information to the researcher that only the factor 

structure is similar but does not give any certain information about the equality of factors for 

each group. The evidence act as a design for invariance tests to be carried out subsequently. 

This model has two important functions: (i) the parameters are tested simultaneously for all 

groups, (ii) equal initial value is generated for the integration of configural invariance model 



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613 

for testing (Byrne, 2008). Hence, the criterion, which will be obtained from further models to 

be tested, is occurred. In this model, invariance, regarding structure and items are supported 

for all groups (factor loadings, factor correlations and error variances are equal for all 

groups). When weak, strong or strict factorial invariance hypothesizes are rejected, the 

"factor structure is equal across all groups" hypothesis, which is developed within the 

framework of configural invariance, is accepted. 

1.1.2. Weak factorial invariance (metric invariance) 

In this model, the equality of factor loadings (λ), (Λ
1
 = Λ

2
 = … = Λ

G
) is tested for all 

groups. (Spini, 2003; Vandenberg and Lence, 2000). If the fit, which is obtained from weak 

factorial invariance test, is better than the fit of configural invariance, configural invariance 

model is rejected. In other words, it indicates that the equivalence is not supported. 

According to Byrne and Stewart (2006) although measurement units are identical for groups 

in terms of underlying factors (e.g. factor loads), it constitutes one of the constraints of this 

model because scaling (e.g. intercepts) is not identical. Therefore, Meredith (1993) describes 

this invariance level as weak factorial invariance. This invariance is tested, 

𝑀𝑔 ≅ �̂�𝑔�̂�𝑔
′ + Λ̂(�̂�𝑔�̂�𝑔

′ + Ψ̂𝑔)Λ̂
′ + Θ̂ 𝑔 = �̂�𝑔′  

with this equation (Widaman and Reise, 1997). 

1.1.3. Strong factorial invariance (scalar invariance) 

It is tested whether the regression constant (τ) of observed variables on the latent variables 

is equal across groups (τ
1
 = τ

2
 = … = τ

3
) (Schmitt and Kuljanin, 2008). In this model, there are a 

series of additional constraints described in weak factorial invariance. These additional constraints 

include the intercepts of the variables that are observed in the matrices �̂�𝑔 . If estimations are 
problematic in terms of invariance on groups, subscript g on matrix τ is removed. In this case, 

invariance is tested, 

𝑀𝑔 ≅ �̂��̂�
′ + Λ̂(�̂�𝑔�̂�𝑔

′ + Ψ̂𝑔)Λ̂
′ + Θ̂ 𝑔 = �̂�𝑔′  

with this equation (Widaman and Reise, 1997). 

1.1.4. Strict factorial invariance (residual variance invariance) 

In this last model of the measurement invariance, about error terms across the groups 𝐻Λ𝜙 

model limits ( 𝐻Λ𝜙𝜃 ) model equally ( 𝜃
1 = 𝜃2 = ⋯ =  𝜃𝐺 ). With the addition of this 

constraint, testing the hypothesis of equality of measurement errors becomes possible for 

independent samples of the population. If the error variances are equal, it means the items 

have equal reliability in terms of groups (Spini, 2003). Strict factorial invariance is also 

created through strong factorial invariance as it occurs in strong factorial invariance created 

through the weak factorial model constraints. These additional constraints are defined as 

strict factorial invariance, which contains unique factorial invariance in �̂� 𝑔  matrix or 

measurement errors. This invariance is tested, 

𝑀𝑔 ≅ �̂��̂�
′ + Λ̂(�̂�𝑔�̂�𝑔

′ + Ψ̂𝑔)Λ̂
′ + Θ̂ = �̂�𝑔′  

with this equation (Widaman and Reise, 1997). 

It should be noted that there are varrious classification in the related literature. Therefore, 

it is worth to consider following aspects suggested by Meredith (1993) and Dimitrov (2010), 

in the testing process of the equality of factor structures across groups, metric invariance is 

the general name of weak factorial invariance, strong factorial (scale invariance) and strict 

factorial invariance (invariance of error variance) models. However, there are some research 



Şekercioğlu 

 
 
 

614 

in literature that discuss the weak factorial invariance with the term of metric invariance. 

(Gregorich, 2006; Meade, Michels and Lautenschlager, 2007; Schmitt and Kuljanin, 2008; 

Spini, 2003; Vandenberg and Lance, 2000; Wu, Li and Zumbo, 2007). Besides, Cheung and 

Rensvold (2002) used the terms metric invariance on construct-level for weak factorial 

invariance, item-level metric invariance for strong factorial invariance and error variance 

invariance for strict factorial invariance. 

Multi-group CFA for measurement invariance can be computed with such software 

statistical programs like LISREL, Amos, SAS/STAT, Mplus and EQS. The analysis starts 

with the creation of separate covariance matrices for the levels of the groups. It can be carried 

out by typing the syntax analysis in LISREL program or by following the instructions 

prescribed by the program (Toit and Toit, 2001). Measurement invariance is carried out in 

four models. The syntax samples of these models are named as EX10A.SPL, EX10B.SPL, 

EX10C.SPL, and EX10D.SPL in LISREL program. In the first model (Model 1), also known 

as configural invariance model, factor loads of structure(s), correlations and error variance 

are assumed to be equal and the analysis is run in this regard. The configural invariance 

model, which is a fundamental model for the equality of factor structure, is developed with 

the hypothesis that factor structures are equal (H0=There is no difference between factor 

structures). In order to make comparisons with model defined in the analysis, a second 

alternative model named as weak factorial invariance model (Model 2) is analysed. In the 

weak factor invariance model, freeing the factor loads for each group, keeping the factor 

correlations and error variances constant are discussed. In the third alternative model strong 

factorial invariance (Model 3) factor loads and error variances for each level of the group are 

released, factor correlations are kept constant. The last and fourth model of measurement 

invariance is strict factor invariance model (Model 4). In this model while error variances are 

released, factor loads and factor correlations are kept constant (Byrne, 2010; Jöreskog and 

Sörbom, 1993; Toit and Toit, 2001).  

1.2. Model Comparisons in the Decision of Measurement Invariance 

In multi-group CFA invariance test, constrained and unconstrained model are compared. 

In terms of availability of different values for each group in constrained model, model 

parameters (e.g. factor loads) are not constrained in this model. The parameters have the 

same value for all groups in constrained model. When the fit of unconstrained model is better 

than the constrained one, it implies that constrained model is incorrect. In other words, if the 

unconstrained model fits better when the constrained parameters are released, they are 

allowed to get different values for each group, and the constrained model developed within 

the invariance hypothesis framework is rejected (Cheung and Rensvold, 2000). 

For comparisons of models with multi-group CFA in the studies in which measurement 

invariance is tested, it can be said that there are two widely used approaches in literature. The 

first one is the comparison between configural invariance model developed with the 

hypothesis that there is no difference in factor structure for each group and alternative models 

(e.g. weak factorial invariance, strong factorial invariance and strict factorial invariance 

models). Hereunder, the first comparison is made between configural invariance model and 

weak factorial invariance model (model 1 and 2), the second is between configural invariance 

model and strong factorial invariance model (model 1 and 3), and the last one is between 

configural invariance model and strict factorial invariance model (model 1 and 4). According 

to this approach, in the case of equality of fit between any alternative model and configural 

invariance or in the event of deterioration, the configural invariance model developed with 

the hypothesis that there is no difference in factor structure for each group is accepted. On the 

other hand, if the alternative model indexes differ from the configural invariance indexes 



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615 

significantly (in favour of alternative models), H0 hypothesis is rejected. In this case, the 

equality of factor structure and thus, the measurement invariance cannot be supported (Byrne, 

2010; Jöreskog and Sörbom, 1993; Toit and Toit, 2001). In the second approach, the 

comparisons are performed by following stepwise process. According to this, the analysis 

starts with less limited models and then the models are assessed by using nested 
2
 method 

(Brown, 2006).
 
Accordingly, in comparison to nested models; 𝐻𝑓𝑜𝑟𝑚>𝐻Λ>𝐻𝜆>𝐻Λ,Θ(𝛿) is used 

as base. In other words, comparisons are made between configural invariance model and 

weak factorial model (model 1 and 2), weak factorial invariance model and strong factorial 

invariance model (model 2 and 3), strong factorial invariance and strict factorial invariance 

(model 3 and 4). According to Van de Vijyer and Leung (1997) if the fit of nested models is 

equal, more constrained model is frequently accepted. If this is not the case, the equality 

hypothesis is rejected (as cited in Spini, 2003). Cheung and Rensvold (2002) also suggest 

another comparison containing only one difference from the first approach. Although the first 

two comparison is the same, the authors suggest a comparison between weak factorial 

invariance and strict factorial invariance (model 2 and 4). 

1.3. Decision Making of Measurement Invariance 

While deciding whether the factor structures are equal for each group, the significance 

level of 
2
 matrix is required and the level should above .05 value, in other words, a non-

significance value p is expected. This situation means that the covariance matrix of each of 

the defined groups do not differ significantly, thereby measurement invariance is supported. 

According to Jöreskog and Sörbom (1993), examples of acceptability of fit indices provided 

in Table 2 might be used for decision. 

Table 2. Acceptance of equality of factor structure in multi-group confirmatory factor 

analysis 

Problem 
2 df p value Decision 

A 38.08 10 0.000 Reject 

B 1.52 2 0.468 Accept 

C 8.77 4 0.067 Accept 

D 21.55 8 0.006 Reject 

E 38.22 11 0.000 Reject 

As seen in Table 2, models A, D, and E in which significance value p is a problem, are 

rejected whereas problem B and C are accepted. The criteria determined in the developing 

first years of multi-group CFA have been questioned over time. 
2
 has a possibility to 

increase its significance value if the number of samples increases, therefore, alternative 

models are investigated whether to accept the fit of factor structures within the model 

framework or not to assess the fit between covariance matrices. Among these, firstly, the 

value of 
2 

and degree of freedom should be compared. In this regard the 
2
 value obtained 

from the more constrained model, 
2
 value

 
from less constrained model and the “delta” value 

(delta means the difference and its symbols is ) which is between the degree of freedom are 

calculated. 
2
 and df values are determined with this calculation. The significance level of 


2
 value obtained from this determination, is controlled in the level of p<.01 or p<.05 by 

comparing the critical values in the distribution table of 
2 

(Byrne, 2010; Jöreskog, 1971; 

Kline, 2005; Lee and Leung, 1982; Steiger, 2007; Van den Bergh and Van Ranst, 1998). In 

this case, H0 and H1 hypotheses can be developed in the following format: 

H0: There is no significant difference between the more constrained model and 

less constrained models in terms of fit. 



Şekercioğlu 

 
 
 

616 

H1: There is a significant difference between the more constrained model and less 

constrained models in terms of fit. 

In this respect, if 
2
,
 
which is calculated on the basis of 

2
 differences in a particular df 

level, is less than critical table values, H0 is accepted. In other words, there is no significant 

difference between two models in terms of fit, therefore, the researcher can make a decision 

about measurement invariance based on 
2
. On the other hand, if 

2
, which is calculated 

on the basis of 
2
 differences in a particular df level, is more than critical table values, H0 is 

rejected. Thus, there is a significant difference between two models in terms of fit, and if this 

difference is in favour of the alternative hypothesis, the researcher can assume that 

measurement invariance is not provided on the basis of 
2
. 

In many studies in which analysis of SEM concept is applied, the distribution(s) may be 

remote from normal within certain tolerances. In the absence of normality in large samples, 


2
 value (S-B

2
) obtained from Satorra-Bentler correction produces close values to the 

2
 

that is produced when the number of people in the sample and the distribution of the 

produces is normal. S-B
2 

is a rather reliable statistical test used to evaluate covariance 

structure models in various distributions and sample sizes (Byrne, 2006; Everitt and Howell, 

2005). As in the other SEM analyses, such as multi-group CFA, which is carried out to obtain 

evidence of measurement invariance, S-B
2
 can only be calculated if the distribution of each 

group is far from the normal distribution. In multi-group CFA, which is carried out with the 

maximum likelihood method, Ts value should be calculated for S-B
2
 scaled difference in 

terms of evidence of measurement invariance between nested models. Ts is calculated 

Ts = (T0 – T1) / cd 

with this equation. T0 is the normal maximum likelihood 
2
 value for nested model, T1 is the 

normal maximum likelihood 
2
 value for comparison (less constrained model) model, and cd 

is the degree of difference test correction. cd is calculated 

cd = (d0 * c0) – (d1 * c1) / (d0 – d1) 

with this equation. d0 is the degree of freedom of nested model, d1 is the degree of freedom of 

comparison model, c0 is the correction degree of nested model, and c1 is the correction degree 

of comparison model. c0 and c1 are calculated 

c0 = T0 / T0
*
 and c1 = T1 / T1

*
 

with this equation. T0
*
 is S-B

2 
value of nested model, on the other hand T1

*
 is S-B

2 
value of 

comparison model. By comparing Ts, which is calculated for S-B
2
 difference degrees, with 

the critical values in 
2
 distribution table, it can be determined whether measurement 

invariance is supported (Brown, 2006; Satorra and Bentler, 2011). 

Recently, it is widely used as an alternative to utilize from fit indices as well as to evaluate 

the 
2
 differences among the models in many research due to a large number of n. According 

to Cheung and Rensvold (2002) it is inadvisable to reject null hypothesis in case of obtaining 

an insignificant 
2
 value. 

2
 is statistically sensitive test for large samples, however, it is not a 

practical test for model fit. In such case, alternative fit indices should be offered for 
2
. The 

comparative fit indices (CFI, NNFI / TLI, RMSEA etc.) are among the most frequently 

recommended ones. Within this framework, it is observed that many goodness of fit indices 

are commonly used together to evaluate general fit of the model and to report it. GFI’s are 

used as an alternative for 
2
 in multi-group CFA which is performed to determine whether 

the factor structures are equal or not. As in 
2
, the configural model whose factor loads, 

factor correlation and error variance are released in covariance matrices of groups, in other 



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617 

words, the model which is developed with the hypothesis that factor structures are equal, is 

the basic model like in alternative fit indices. For the evidence of measurement invariance, 

the differences between models can be evaluated with the comparison of indices such as 

RMSEA, CFI, Gamma Hat, Mc, IFI, AIC, EVCI, NFI, TLI, and SRMR. 

The fit values are expected to become better for the equality of factor structures when the 

parameters like factor loads and error variance in covariance matrices of the group are 

released together or one by one. With this regard, the differences are evaluated by comparing 

the indices (e.g. SRMR, CFI and RMSEA) between configural model and other 

alternative models or nested models. The configural model set up with the hypothesis that 

there is no significant difference between factor structures of each group is accepted if the fit 

indices of alternative models are lower than the ones in configural model. On the other hand, 

if the fit indices of other alternative model are higher than the ones in configural model or 

nested model, the fit across models is evaluated whether it differs significantly or not. 

Cheung and Rensvold (2000; 2002) suggested cut-off points for CFI significance level 

between modes in terms of measurement invariance after carrying out a study by using 

Monte Carlo method. Hereunder, when CFI–.01 is provided, then configural invariance 

model is accepted. In contrast to this situation, if CFI is between –.01 and –.02, there will be 

increasing doubt about invariance. If it is more than -.02 it can be said that the difference 

between constrained and unconstrained model will increase. In this situation, configural 

model is rejected. In this context, it is decided that the factor structures are not equal and 

therefore an alternative model should be sought. In addition, the critical values of Gamma 

hat and McDonald NFI are –.001 and –.02. 

Chen (2007) suggested cut-off points for decision of measurement invariance by 

considering situations like sample size of CFI, RMSEA and SRMR indices and sample sizes 

in groups after carrying out a study, which aimed at testing sensitivity of goodness of fit 

indices through Monte Carlo method. Accordingly, it can be concluded that measurement 

invariance cannot be supported (case of noninvariance) if sample size is small (n<300), 

sample sizes of groups are not equal, pattern of variance is the same, there is a relationship 

like CFI–.005, RMSEA.010 or SRMR.025 between groups in terms of weak 

factorial invariance test, and there is a relationship like CFI-.005, RMSEA.010 

or SRMR.005 between groups in terms of strong factorial invariance or strict factorial 

invariance. On the other hand, measurement invariance can be supported when sample size is 

sufficient (n>300), numbers of groups compared are equal, there is a relationship 

like CFI–.010, RMSEA.015 or SRMR.030 between groups in terms of weak 

factorial invariance test, and there is a relationship like CFI–.010, RMSEA.015 between 

groups in terms of strong factorial invariance or strict factorial invariance. 

An important point to be considered in assessing multi-group CFA comparison of the four 

basic models is type I and type II error possibilities. If the sample is small for a null 

hypothesis, type I error is likely occurred. However, if the sample is getting larger for 

alternative hypothesis, the difference of fit will be extended. In that case, type II error is 

likely occurred. For this reason, to minimize the type I and type II error possibility, the cut-

off points should be determined efficiently (Hu and Bentler, 1998). In their maximum 

possibility 
2
 studies which were performed with the indicators acting as continuous 

variables, French and Finch (2008) controlled type I error in the level of .01 and .05 between 

different models and sample numbers. The researchers revealed that the power of 
2
 has a 

positive correlation with sample size, indicator number of each factors and factor number. 

Meade and Bauer (2007) also extrapolated the same results about 
2
 (as cited in Sass, 



Şekercioğlu 

 
 
 

618 

Schmitt and Marsh, 2014). There is no doubt that this case is valid for other delta fit indices 

as well. However, this study didn’t include detailed discussions on that subject because it was 

beyond the scope. 

1.4. Objectives 

Researchers of behavioural and educational sciences provide evidence through a sample 

on the validity of scores obtained from developed or adapted psychological measurement 

tools. After revealing the psychometric properties of the measurement tools, measurement 

process can be practiced on an independent group in the same sample and various decisions 

may be taken by means of obtained scores in the same or a different study. The fact that a 

measurement tool with confirmed factor structure for a sample may not be valid for the 

independent sub-groups in the relevant sample is a probability that researchers should pay 

attention. In such a case, the validity of decisions to be taken with scores obtained from 

groups will be suspicions. Within this scope, this research aims to discuss the conceptual 

basis of multi-group CFA in measurement invariance in terms of basic concepts and to 

introduce the subject through two hypothetic data set that one supports measurement 

invariance, but the other doesn’t, for the researchers aiming to determine the psychometric 

properties of a measurement tool. Thereby, a new perspective will be introduced to the 

researchers aiming to determine the psychometric properties of measurement tools, 

suggestions about decisions to be taken for the tool without equalized factor structure will be 

asserted. Accordingly, the present study searches answers to the following research 

questions: 

1. Is the five-factor structure of measurement tool 1 equal across the groups of science, 
health and social sciences? 

2. Is the three-factor structure of measurement tool 2 equal across groups of males and 
females? 

This research is limited to measurement invariance (measurement of configural 

invariance, weak factorial invariance, strong factorial invariance and strict factorial 

invariance). The heterogeneity of the population (factor variance invariance, factor 

covariance invariance and latent means invariance) is not included in the research.  

2. Method 

This study examines the method of multi-group CFA for the evidence concerning 

measurement invariance through two data set consisting of equal and unequal factor 

structure. Considering the findings of the study, the current study has the characteristics of 

correlational research concerning equality of factor structure for independent groups in two 

samples and due to the discussions on generation of construct validity evidences. The 

correlational studies analyse the relationship between two or more variables without 

intervening in these variables under any circumstances. These studies are the ones that are 

effective on revealing the relationships and determining the levels of relationships between 

variables and provides necessary cues for conducting high-level research on these 

relationships (Büyüköztürk, Kılıç Çakmak, Akgün, Karadeniz and Demirel, 2012). 

2.1. Research Data 

The ready-made data was used in this study. They consist of two data set (equal and 

unequal factor structured) that the researchers collected them from his previous researches. 

The first hypothetic data set that measurement invariance is supported consist of 666 

undergraduate students. When the distribution of the participants is examined based on 

scientific major, 32.28% (n=215) science, 31.83% (n=212) of health and 35.89% (n=239) of 

social science. The other hypothesis data set that measurement invariance is not supported 



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619 

consist of 353 high school students. The distribution in terms of participants’ gender is as 

follows, 62.32% (n=220) female, 37.68% (n=133) male. 

2.2. Data Collection Tools 

The study consists of two hypothetic data set, which are the subjects of measurement 

invariance analyses and the scores obtained from two different measurement tools. Some 

items were emitted from the tool in line with the results of EFA and CFA that were run on the 

data set collected from the participants. Moreover, the factor design differed for male 

participants in the second data collection tool whose factor design was not equal. The main 

purpose of this study is not to determine or discuss the psychometric properties of 

aforementioned tools. However the present study focuses on presenting the multi-group CFA 

in terms of measurement invariance through two hypothetic data set in which measurement 

invariance both was supported and was not, and creating a new view of validity for the 

researchers who aim to measure the psychometric properties of a measurement tool. 

Therefore, it is not appropriate to give the names of the tools and sub-scales in view of the 

probability that because they can form basis for further studies. For this reason, the data 

collection tools were mentioned as measurement tool 1 and measurement tool 2, and limited 

information about the psychometric properties of the tools was given because it was not 

wanted to reveal the tool. 

Measurement tool 1 is a tool that consists of five sub-scales to measure an effective trait 

through using four point rating. In the original study, EFA and CFA were performed to 

determine psychometric properties of the tool in terms of gathering evidence about construct 

validity, concurrent validity was examined by comparing with a criterion score, to obtain 

reliability evidence for stability a test-retest method was run, and lastly to obtain reliability 

evidence for internal consistency, Cronbach alfa coefficients were calculated. In conclusion, 

it can be said that the scores obtained from measurement invariance tool 1 have a high level 

of validity. 

This study starts with EFA to obtain construct validity evidence through hypothetic data 

set of measurement tool 1. Before the factor analysis, it is determined that the scales have a 

normal distribution and there is no multicollinearity problem across items. Also, there is no 

missing value in hypothetic data set. As a result of EFA, it is determined that items of 

measurement tool 1 are gathered under five factors, and they are also under their own factors 

in parallel with the results of original study. Since an item had high factor loading in more 

than a factor, it was emitted from the analysis. Factor loading values of the items are between 

.40-.80. The contributions of items to the total variance are as follows; for first factor 

10.63%, for second factor 10.02%, for third factor 8.87%, for fourth factor 8.03%, for fifth 

factor 6.94% and the total variance explained is 44.49%. In CFA results, which was 

performed to produce additional evidence for construct validity, the standardized coefficients 

of items which had a significant t value may change between .32-.70, and the error variance 

values may change between .50-.90. As a result of the analysis, it is determined that fit 

indices are S-B
2
(366)=699.22, p=.000, 

2
/df=1.91, RMSEA=.037, GFI=.92, NNFI=.96 and 

SRMR=.049. It is observed that the Cronbach Alfa coefficients which were calculated to 

determine internal consistency of factor are for the first factor .75, for the second .78, for the 

third .72, for the fourth .69, for fifth .57. The total Cronbach Alfa coefficient of the tool is .84 

Measurement tool 2 is a tool that consists of three sub-scale to measure an affective trait 

through using four-rating scoring. In original study, EFA was performed to determine 

psychometric properties of tool and to obtain construct validity evidence, the discriminant 

validity was investigated in the direction of the scores collected from two different groups. 

Item-test correlations were calculated to determine item discrimination, test-retest method 



Şekercioğlu 

 
 
 

620 

was applied to obtain reliability for stability and Cronbach Alfa coefficients were calculated 

to obtain reliability for internal consistency. 

In this study, the analysis of measurement tool 2 through the hypothetical data set starts 

with EFA. Before the factor analysis, it is determined that the scales have a normal 

distribution and there is no multicollinearity problem between items. Also, there is no 

missing value in hypothetic data set. As a result of EFA, it is determined that the items are 

gathered under three factors. Some items are emitted from the analysis because they give low 

factor loading value (
2
<.32) or they are overlapped items. The factor loading of items ranges 

between .45-.75. The contributions of items to the total variance are as follows, for first factor 

21.04%, for second 17.98%, for third 8.98%, and total 48%. In CFA results, which was 

performed to produce an additional evidence for the construct validity, the standardized 

coefficients of items which have significant t value may change between .45-.74 and their 

error variance may change between .45-.80. As a result of the analysis, it is determined that 

fit indices are S-B
2
(227)=423.46, p=.000, 

2
/df=1.87, RMSEA=.050, GFI=.89, NNFI=.97 

and SRMR=.052. It is seen that Cronbach Alfa coefficients, which were calculated to 

determine internal consistency of factor are for the first factor .89, for the second factor .84, 

and for the third factor .65. It is not necessary to calculate the total point within the frame of 

theoretical and logical view, so the whole scale was not calculated by Cronbach Alfa 

coefficient. 

2.3. Data Analysis 

To find answers to the research questions of study, EFA, CFA, Cronbach Alfa analysis, 

covariance matrices equality test and multi-group CFA were performed. The factor analysis 

aims to find a few but significant new (common) unrelated variable by combining the 

variables related with each other in p-variable situation (p-dimensional space). In other 

words, the factor analysis is a method in which common components are determined and 

construct dependence is dispelled (Diekhoff, 1992; Gorsuch, 1974; Tatlıdil, 1992; Thompson, 

2004; Tucker and MacCallum, 1997). Factor analysis is a technique, which is used to confirm 

whether the items of a certain scale or sub-scale are gathered under a certain construct or 

factor (Gable and Wolf, 2001). Beyond reducing variable and naming the emerging factors, 

the EFA reveals whether the analysis results are similar to the structure of the theory 

(unobserved latent variables) that enables to figure out the behaviour. After the analysis, a 

query is made for determining whether the indicators, which are gathered under a certain 

factor, are indicators of theoretical construct. In CFA, it is firstly aimed to test and confirm 

the structural hypotheses regarding the relationships between variables. Within this frame, it 

is focused on examining the relationships between factors and variables, and the relationships 

between factors in this research through the hypothesis developed. Therefore, the researcher 

should have the information about the construct of variables that s/he defined in model before 

the analysis.  By this way, the model can be based on a strong theoretical or empirical basis 

(Raykov and Marcoulides, 2008; Stevens, 1996). Multi-group CFA, which is a special 

application of CFA can test the measurement and equality of construct models for multi-

groups (Brown, 2006). The factor loads of measurement tool consist of measurement 

properties related to the variables that include constants and error variances. The multi-group 

CFA makes comparison between two or more groups simultaneously possible by using 

covariance matrices that are calculated for each compared groups. Thus, measurement 

invariance or equivalence can be tested by putting equality constraints the parameters of 

groups (Harrington, 2009). 

For the model comparisons in the studies in which the measurement invariance is tested 

through the multi-group CFA, the first approach of two common approaches is the 



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621 

comparison between the structural model developed by the hypothesis that there is no 

meaningful difference between the factor structures for each compared group and the 

alternative models. In the second approach, the fit between the more constrained nested 

model and the least constrained comparison model is evaluated by following a stepwise 

process. Although researcher suggests that evaluation of difference between models should 

be made between nested models, the comparisons were made for each methods to increase 

sample numbers and ’s were evaluated in this study. 

Additionally, cut of points for factor loading in EFA are accepted as 
2
.32; 

2
 level of 

acceptance in hypothesis test for significance as .05; since n>300 in each data set the cut of 

points for multi-group CFA in measurement invariance run for three-model comparison as 

CFI-.01; as SRMR.03 for weak factorial invariance and as  SRMR.01 for strict 

factorial invariance. 

LISREL sample syntax for covariance matrices is in appendix 1, LISREL sample syntax 

for four models, which are based on for measurement invariance is in appendix 2. 

 

3. Findings 

The five-factor structure of measurement tool 1 was tested to determine measurement 

invariance with multi-group DFA for the groups in the majors of science, health and social 

science. Before giving the findings of measurement invariance, test statistics, normality tests 

and reliability coefficients are given in Table 3 in terms of basis assumption of analysis. 

 

Table 3. Test statistics, normality tests and reliability coefficients of science, health and 

social science groups 
Major Factor n Mean Median Mode s Range Skewness Kurtosis

 


1 

Science 1 215 17.52 18 24 4.68 18 -.429 -.686 .74 

 2 215 16.89 17 15 3.94 17 -.184 -.521 .82 

 3 215 21.28 22 23 4.01 19 -.490 -.224 .75 

 4 215 16.19 16 20 3.06 13 -.672 -.060 .72 

 5 215 15.42 16 17 3.31 15 -.796 .388 .65 

 Scale 215 87.29 88 90 12.43 66 -.521 .374 .85 

Health 1 212 17.07 18 20 4.80 18 -.422 -.681 .76 

 2 212 17.65 18 19 4.03 18 -.433 -.224 .79 

 3 212 20.97 21 23 4.42 20 -.637 .135 .73 

 4 212 16.09 17 20 3.07 13 -.605 -.129 .68 

 5 212 15.14 16 17 3.41 15 -.544 -.202 .62 

 Scale 215 86.91 89 97 13.26 66 -.536 .014 .85 

Social 1 239 17.79 18 24 4.49 18 -.374 -.766 .75 

 2 239 17.37 18 16 4.03 17 -.188 -.676 .76 

 3 239 21.89 22 21 4.17 17 -.406 -.674 .74 

 4 239 16.27 17 20 3.14 15 -.805 .507 72 

 5 239 14.57 14 13 3.36 15 -.313 -.225 .56 

 Scale 239 87.90 88 86 13.05 63 -.303 .014 .85 
1
 Cronbach Alfa internal consistency coefficient 

As can be seen in Table 3, measures of central tendency are relatively close to each other 

for the groups in the majors of science, health and social science in the level of both sub-scale 

scores and total scale scores. The fact that skewness and kurtosis coefficients are in the range 

of ∓1 indicate that the distribution is close to normal (Rosenthal and Rosnow, 2008). 
Although the coefficients are between ∓1, it can be said that all of the sub-scales and total 
scale score points are partly negatively skewed distribution. Accordingly, multi-group CFA, 

which was performed to determine whether measurement invariance was provided or not for 

all groups, was computed through asymptotic covariance matrix and S-B
2
 statistics was 



Şekercioğlu 

 
 
 

622 

used as base for model fit. On the other hand, it is seen that internal consistency coefficients 

of science, health and social science groups, which were calculated based on sub-scale and 

scale scores, are generally in an acceptable level. According to Nunnaly and Bernstein 

(1994), the reliability coefficient may be accepted for the research if the value is between .70-

.80. In all groups, .70 condition is fulfilled with the factors 1-2-3, and 4 at a level of scale. 

However, this acceptance cannot be provided at the level of factor 5 in all groups. It can be 

thought that the internal consistency coefficient of sub-scale is low because the number of 

items is low. The equality of covariance matrices in science, health and social science groups 

was tested before multi-group CFA.  

As a result of the analysis, index values of fit between covariance matrices related to the 

groups are shown in Table 4. 

Table 4. The equality of covariance matrices of science, health and social science groups 
Groups S-B

2
(df) p 

2
/df RMSEA GFI CFI SRMR 

Science, Health and Social 1025.2(870) .000 1.178 .028 (.020-.035) .91 .98 .060 

As can be seen in Table 4, S-B
2
 and degree of freedom are below 2, RMSEA is below 

.05, GFI is above .090, CFI is above .95 and SRMR is below .08. In this situation it can be 

said that there is a fit between three covariance matrices.  

Multi-group CFA findings for five-factor structure equality of the measurement tool 1 are 

given in Table 4 for science, health and social science groups. 

 

Table 5. Findings of multi-group confirmatory factor analysis for science, health and social 

science groups (maximum possibility) 
 S-B

2
(df)

1 MC
2
 

2
(df) 

2
/df 

2
/df CFI CFI SRMR SRMR Decision 

Science 522.35(366) – – 1.427 – .95 – .073 – – 

Health 477.91(366) – – 1.306 – .97 – .064 – – 

Social 536.72(366) – – 1.466 – .95 – .068 – – 

Model 1
A 

1735.47(1234) – – 1.406 – .95 – .079 – – 

Model 2
B 

1671.52(1176) M1–M2 63.95(58) 1.421 -.015 .95 0 .075 .004 H0Accept 

Model 3
C 

1859.75(1205) M1–M3 -124.28(29) 1.543 -.137 .94 .01 .081 -.002 H0Accept 
 

 M2–M3 -188.23(-29) – -.122 – .01 – -.006 H0Accept 

Model 4
D 

1920.04(1265) M1–M4 -184.57(-31) 1.518 -.112 .94 .01 .085 -.006 H0Accept 
 

 M3–M4 -60.29(-60) – .025 – 0 – -.004 H0Accept 
1
 p<.05 

2
 Model comparison (M=Model) 

A
 Configural Invariance (Factor loads, factor correlation and error variance are constant) 

B
 Weak Factorial Invariance (Factor loads, factor correlation and error variance are constant) 

C
 Strong Factor Invariance (Factor loads and error variance are free, factor correlation is constant) 

D
 Strong Factorial Invariance (Error variance is free, factor loads and factor correlation are constant) 

Firstly, when the fit indices obtained as a result of CFA which was performed separately 

for science, health and social science groups are examined, it can be said that fit indices 

obtained from each of the three groups largely meet the acceptance levels. Accordingly, it 

can be seen that S-B
2
 and the degree of freedom are below 2, CFI is equal to .95 or above 

this value and SRMR is below .08. After analysing the fit indices in general, it can be said 

that for five-factor structure of the measurement tool 1 was confirmed separately for science, 

health and social science groups. 

The configural model, which was developed with the hypothesis about there is no 

significant difference between factor loads, factor correlation and error variance for science, 

health and social science groups was tested to evaluate measurement invariance. The analysis 

results show that S-B
2
 and the degree of freedom are below 2, CFI is equal to .95 or above 

this value and SRMR is below .08. Also, after analysing the fit indices in general, it can be 

acceptable that fit indices of configural model meet the acceptance levels. 



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623 

When configural invariance (Model 1) and weak factorial invariance (Model 2) models are 

compared, it is seen that fit gets worse in terms of the ratio of S-B
2
 and df. In addition, it 

can be said that there is no change in CFI value and the change is not significant (n<.025) in 

SRMR. 

When configural invariance (Model 1) and strong factorial invariance (Model 3) models 

are compared in terms of the ratios of S-B
2
 and df, it’s seen that the fit gets worse. 

Besides it can be said that the fit between models gets worse regarding the CFI and SRMR 

values. On the other hand, when weak factorial invariance (Model 2) and strong factorial 

invariance (Model 3) models are compared in terms of the second approach, between the 

ratios of both S-B
2
 and df, CFI and SRMR values, it can be stated that the fit gets 

worse.  

Finally, when configural invariance (Model 1) and strict factorial invariance (Model 4) 

models are compared, it can be seen that the fit gets worse in terms of S-B
2
 and df ratios 

for both models. Besides it can be stated that the fit between models also gets worse in terms 

of CFI and SRMR values. On the other hand, according to the second approach when 

strong factorial invariance (Model 3) and strict factorial invariance (Model 4) models are 

compared, the fit in terms of S-B
2
 and df ratios, gets better. In this direction Ts value 

calculated for difference ratio of S-B
2
 is 57.3 and it is confirmed that this value is smaller 

than the critical value in 
2
 distribution table, 

2
diff(60)=79.08, p>.05. Therefore, it can be 

said that there is no significant difference between strong factorial invariance and strict 

factorial invariance models. In other respects, it can be stated that there is no change in CFI 

value and SRMR value in the direction of the fit gets worse. 

In the light of findings outlined above, among the four models, the model that works best 

based upon covariance matrices in the majors of science, health and social science is 

configural invariance model developed in assumption of the equality of factor structures. In 

this context, it is accepted that the five-factor structure of the measurement tool 1 is equal for 

relevant groups, in other words, measurement invariance is supported.  

Measurement invariance for the three-factor structure of measurement tool was tested 

through multi-group CFA for both female and male groups. Before giving findings about 

measurement invariance, first in line with the basic assumption of the analysis, test statistics 

related to relevant groups, test of normality and reliability coefficients were given in Table 6. 

Table 6. Test statistics, tests of normality and reliability coefficients for female and 

male groups 
Gender Factor n Mean Median Mode S Range Skewness Kurtosis

 


1 

Female 1 220 9.57 8 0 7.45 28 0.51 -0.84 .89 

 2 220 14.58 15 12 6.31 26 -0.05 -0.78 .83 

 3 220 4.30 4 3 3.14 12 0.55 -0.53 .68 

Male 1 133 8.99 7 3 7.39 30 0.82 -0.23 .87 

 2 133 13.84 12 9 7.13 27 0.29 -1.04 .84 

 3 133 4.55 4 3 3.01 12 0.44 -0.44 .61 
1
 Cronbach Alfa internal consistency coefficient 

As seen in Table 6, it can be stated that measures of central tendency for female and male 

groups in the level of both sub-scale and scale total points is close. It is stated that except one 

distribution of coefficient of skewness and kurtosis, all other distributions even though it is 

between 1, points, to some extent, are negative-skewed. Also it is seen in data set for male 

students that sub-scale points are out of 1; when kurtosis coefficient is calculated to 

kurtosis’ standard error, the obtained value is still out of 1.96. In this direction, multi-group 

CFA, which intends to find out whether the CFA and measurement invariance are confirmed 



Şekercioğlu 

 
 
 

624 

for each group, is performed over asymptotic covariance matrix and model fit was based on 

S-B
2
. On the other hand, it is seen that internal consistency coefficient calculated on female 

and male groups’ sub-scale and scale points is generally in an acceptable level. In all groups, 

.70 condition meets with 1
st
 and 2

nd
 factors in the scale level but in both groups, this 

acceptance cannot be met in 3
rd

 factor level. It can be concluded that the internal consistency 

coefficient of the relevant scale is low because the number of items (4 items) in the scale is 

low. 

The equality of covariance matrices for female and male groups was tested before the 

multi-group CFA. In the result of analysis made, index values regarding the fit between 

covariance matrices for these groups are presented in Table 7. 

Table 7. Equality of covariance for female and male groups 
Groups S-B

2
(df) p 

2
/df RMSEA GFI CFI SRMR 

Female & Male 374.04(276) .000 1.355 .042(.029-.053) .89 .99 .083 

As can be seen in Table 7, the ratio of S-B
2
 to the degree of freedom is below 2, RMSEA 

is below .05, GFI is below .90, CFI is above .95 and SRMR is above .08. It can be stated that 

when fit indices are assessed in general and GFI and SRMR indices are taken into account, 

the fit between two variances is moderate. 

The multi-group CFA findings related to the equality of the three-factor structure of the 

measurement tool 2 for female and male groups are shown in Table 8. 

Table 8. Multiple-group confirmatory factor analysis findings of female and male groups 

(maximum possibility) 
 S-B

2
(df)

1 MC
2 


2
(df) 

2
/df 

2
/df CFI CFI SRMR SRMR Decision 

Female 349.31(227) – – 1.539 – .98 – .061 –  

Male 298.51(227) – – 1.315 – .98 – .068 –  

Model 1
 

732.94(503) – – 1.457 – .97 – .100 –  

Model 2
 

694.38(480) M1–M2 38.56(23) 1.447 .010 .97 0 .083 .017 H0Accept 

Model 3
 

651.04(457) M1–M3 81.90(46) 1.425 .032 .99 -.02 .076 .024 H0Reject 
 

 M2–M3 43.34(23) – .022 – -.02 – .007 H0Reject 
Model 4

 
691.77(480) M1–M4 41.17(23) 1.441 .016 .98 -.01 .098 .002 H0Accept 

 
 M3–M4 -40.73(-23) – -.016 – .01 – -.022 H0Accept 

1
 p<.05 

2
 Model Comparison (M=Model) 

Firstly, it can be said when fit indices obtained as a result of CFA’s performed separately 

for female and male groups are analysed, fit indices of both groups generally meet the level 

of acceptance. According to this, it is seen that S-B
2
 and degree of freedom ratios are below 

2, CFI is above .95 and SRMR is below .08. It can be stated that when fit indices are assessed 

in general, the three-factor structure of the measurement tool is confirmed separately for 

female and male groups. 

To evaluate the measurement invariance, factor loads, factor correlations, and error 

variances for female and male groups of the three-factor structure are initially tested with the 

configural invariance model based on the hypothesis asserting that there is no significant 

difference among the variables stated. In the result of the analysis made the ratio of S-B
2
 to 

the degree of freedom is below 2, CFI is above .95 but SRMR is above .08. When fit indices 

are assessed in general, fit indices for the configural invariance model meet the level of 

acceptance in general.  

When configural invariance (Model 1) and weak factorial invariance (Model 2) models are 

compared, it is seen that the fit gets better in terms of S-B
2
 and df ratio for both models. 

In this direction Ts value calculated for difference ratio of S-B
2
 is 41.02 and this value is 

bigger than the critical value in 
2 

distribution table, 
2

diff(23)=35.17, p<.05. In other words, 



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625 

there is a significant difference between configural invariance and weak factorial invariance 

models. On the other hand, it is seen that there is no difference in CFI value, and the change 

in SRMR is not significant (<.025). When the findings are evaluated in general, based on 

the results asserting that the difference is not significant in two out of three fit indices, it is 

decided that fit indices of the configural invariance and the weak factorial invariance do not 

differ from each other.  

When configural invariance (Model 1) and strong factorial invariance (Model 3) models 

are compared, it is seen that the fit gets better in terms of S-B
2
 and df ratio for these two 

models. In this direction, Ts value calculated for difference ratio of S-B
2
 is 78.14 and it has 

been determined that this value is bigger than the critical value on the 
2
 distribution table, 


2

diff(46)=62.83, p<.05. Hence, there is a significant difference between configural invariance 

and strong factorial invariance models. Also, when Model 1 and 3 are compared, it can be 

said that CFI value (<-.01) and SRMR value (>.01) considerably change. On the other 

hand, according to the second approach, when weak factorial invariance (Model 2) and strong 

factorial invariance models are compared, the similar results are obtained with the first 

approach comparison.  According to this, the fit between S-B
2
 and df percentages gets 

better. In this direction Ts value calculated for difference ratio of S-B
2
 is 38.37 and this 

value is bigger than the critical value on the 
2
 distribution table, 

2
diff(23)=35.17, p<.05. In 

other words, there is a significant difference between weak factorial invariance and strong 

factorial invariance models. Also in the context of Model 2 and 3, CFI value (<-.01) and 

SRMR value (>.01) significantly differ. 

Finally, when configural invariance (Model 1) and strict factorial invariance (Model 4) 

models are compared, it is observed that the fit gets better in terms of S-B
2
 and df ratio 

for these two models. In this direction, Ts value calculated for difference ratio of S-B
2
 is 

37.5 and this value is bigger than the critical value on the 
2
 distribution table, 


2

diff(23)=35.17, p<.05. According to that, there is a significant difference between weak 

factorial invariance and strict factorial invariance models. But on the other hand it is seen that 

the change is not significant in CFI (=-.01) and SRMR values (<.01). When the findings 

are evaluated, since there is no significant difference in two of three fit indices, it was 

decided that fit indices of the strict factorial invariance model and the configural invariance 

model don’t differ from one another. According to the second approach, when strong 

factorial invariance (Model 3) and strict factorial invariance (Model 4) models are compared, 

it can be said that the fit between two models gets worse in terms of S-B
2
 and df ratio 

and CFI and SRMR values.  

The findings above reveal that the best working model is the strong factorial invariance 

model among these four models. Accordingly, it has been accepted that the three-factor 

structure of the measurement tool 2 is not equal for female and male groups, and 

measurement invariance cannot be supported. 

In this step, different exploratory factor analyses have been computed for the data set 

obtained from female and male groups. As a result of analysis carried out for the female 

group, it has been observed that the three factors structure of measurement tool 2 is valid for 

this group, there is no considerable difference in factor loads (.38 and .75) and the 

contribution (%49.56) of these factors to the total variance explained. On the other hand, the 

analysis results, which were performed for male group have revealed that the items have been 

gathered under four factors. As a result of analysis repeated for these four factors, it is 

revealed that three items belonging to the second factor is showed up as another factor. In the 

data set for male group, it has been seen that factor loads (.47 and .83) and the contribution of 



Şekercioğlu 

 
 
 

626 

the factors to the total variance (57.55%) goes up, however the increase in the total variance 

explained has occurred because of the rise in the number of factors. When the results of 

analysis are examined in general, the factor numbers for measurement tool 2 are three for the 

female group and four for the male group. 

4. Discussion and Conclusion 

Validity is a concept that is referred to inferences from trait(s) that it measures, but not 

behalf of the measurement tool’s name. Not only the ones who develop and adapt 

measurement tool but also the researchers who use the tool with a different purpose or 

different sample from its initial purpose have certain responsibilities to reveal scientific 

evidences about the validity. However, it may not be sufficient for determining the 

psychometric properties of the measurement tool through conventional methods in all 

circumstances. For a particular measurement tool and group, it is a problematic issue for 

construct validity whether the factor structures that were confirmed empirically by the factor 

analysis, have the same meaning for independent sub-groups in the sample. Therefore, it 

might be required for the researchers to test validity for different groups in the sample. It is 

thereby a crucial psychometric problem to test the equality for specific groups obtained factor 

structure design by explanatory and/or confirmatory factor analyses in scale development or 

adaptation studies. 

The researchers usually make comparisons across groups to create theoretical information 

or contribute to existent theoretical knowledge one and naturally want their decisions as 

correct as possible about the population that they wish to generalize according to their 

findings.  The researchers should examine whether the factor structure of the tool is equal 

across groups because these comparisons are usually made by scores of the measurement 

tool. In contrast, in case of inequality of the factor structures defined in the measurement tool 

across groups, the group scores of these structures do not mean the same. When the factor 

structures are equal among groups, factor design will be the same for groups and thus it can 

be evaluated that the collected group scores from scale or sub-scale are valid. That leads to 

develop a new additional empirical evidence for construct validity of the measurement tool. 

On the other hand, if psychometric properties of the measurement tools are not equal for 

groups, the factor structures will vary from one sub-group to another, the comparisons made 

with scores of measurement tool and the decisions about group will be faulty. Also if there is 

no empirical evidence, the contribution of the research results to the theory will be doubtful.  

In this respect, it is beneficial to develop a different perspective on the construct validity, in 

the framework of this basic problem, in scale development or adaptation studies in the majors 

of behavioural and educational sciences. Hence, the researcher has the responsibility to test 

the equality of test scores from factor structure of the tool and sub-scale for the compared 

groups. Therefore, the evaluations related to empirical evidence about the validity of 

measurement tool are never considered as “last word”. Validating a tool requires everlasting 

effort.  

This study aimed at developing a sample for what sort of decisions could be made for two 

different factor designs that either of them supports measurement invariance. In this respect, 

it was determined that the five-factor structure of the measurement tool 1, regarded as a first 

sample, supported measurement invariance for the undergraduate students in the majors of 

science, health, and social sciences. Based on EFA and CFA results, it can be claimed that 

the five-factor structure of the tool has high validity for the entire of the sample. With the 

multi-group CFA, the relevant factor design was approved to be separately valid for students 

of science, medical and social sciences. This result recommends that this factor design is 

valid for the whole sample and all of the science majors (any of these groups) like science, 



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627 

health, and social sciences separately, so that the measurement tool 1 has high construct 

validity for the groups that maintain their academic lives in different majors.  

As a second sample, the three-factor structure of the measurement tool 2 supports 

measurement invariance for high school students within the context of gender. In such cases, 

the researchers should consider the relevant factor structure as unequal for groups, therefore, 

they need to take into account the different factor designs or possibility of bias for each 

comparison. Therefore, the researchers are suggested to run the EFA for each group. In fact, 

it has been revealed that three factor structure is valid for females but not in males, in which 

the items are gathered under four factor in this study. It is required to give cues to make the 

findings more meaningful (reason of hiding names of scales are explained in method section). 

In the EFA that is run for males, the three items of the second factor named as stress sub-

scale have been observed to be gathered under a new factor. When these items are examined, 

three of them are about “negative reaction showed in blocking situation”. These items, which 

are symptoms of stress for females, are loaded under a new factor called “intolerantness to 

blocking” for males. This finding reveals that the structure for female and male groups are 

different, in other words, doesn’t have the same psychological meaning for these groups. The 

researchers can produce different forms of measurement tool in this situation (e.g. female 

form-male form) and suggest these forms for the ones who study in the sub-groups of the 

sample. Although this situation causes a problem for practicality, it plays a crucial role in the 

construct validity.  

The researchers can test whether the measurement tool, based on the theoretical basis of 

the trait which it intends to measure is equal for more than a group or a sample like age, 

gender, socio-economic level, class, education level, academic major, subcultures in a 

society, international comparisons, experimental researchers, and different occupational 

groups. It is surely beyond doubt that the evidences about whether the measurement 

invariance is supported for different groups will strengthen psychometric properties of the 

tool, and will therefore increase the validity of the results presented in the direction of the 

research findings.  As a result, the contribution to the production of theoretical knowledge or 

existing theoretical knowledge accumulation will enhance as well. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



Şekercioğlu 

 
 
 

628 

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Appendix 1: LISREL Syntax Example for Test of Covariance Matrices Equality  
SCALE I EQUALITY TEST OF COVARIANCE MATRIX (ACADEMIC MAJOR)  

Group SCIENCE: 

Observed Variables: V1-V30 

Covariance Matrix from File SCIENCE.COV 

Method of Estimation: Maximum Likelihood 

Iterations: 20 

Sample Size: 215 

Latent Variables: f1-f30 

Relationships:  

V1 = 1*f1 

V2 = 1*f2 

V3 = 1*f3 

V4 = 1*f4 

V5 = 1*f5 

V6 = 1*f6 

V7 = 1*f7 

V8 = 1*f8 

V9 = 1*f9 

V10 = 1*f10 

V11 = 1*f11 

V12 = 1*f12 

V13 = 1*f13 

V14 = 1*f14 

V15 = 1*f15 

V16 = 1*f16 

V17 = 1*f17 

V18 = 1*f18 

V19 = 1*f19 

V20 = 1*f20 

V21 = 1*f21 

V22 = 1*f22 

V23 = 1*f23 

V24 = 1*f24 

V25 = 1*f25 

V26 = 1*f26 

V27 = 1*f27 

V28 = 1*f28 

V29 = 1*f29 

V30 = 1*f30 

Set the Error Variances of V1-V30 to zero 

Group HEALTHCARE: 

Observed Variables: V1-V30 

Covariance Matrix from File HEALTHCARE.COV 

Method of Estimation: Maximum Likelihood 

Iterations: 20 

Sample Size: 212 

Latent Variables: f1-f30 

Group HUMANITIES: 

Observed Variables: V1-V30 

Covariance Matrix from File HUMANITIES.COV 

Method of Estimation: Maximum Likelihood 

Iterations: 20 

Sample Size: 239 

Latent Variables: f1-f30 

End of Problem 

 

Appendix 2: LISREL Syntaxes Example for Test of Measurement Invariance  
Model A 

Group 1: Testing Equality Of Factor Structures 

Model A: Factor Loadings, Factor Correlation, Error Variances Invariant 

Observed Variables: V1-V30 

Covariance Matrix from File SCIENCE.COV 

Asymptotic Covariance Matrix from File SCIENCE.ACM 

Method of Estimation: Maximum Likelihood 

Iterations: 20 

Sample Size: 215 

Latent Variables: Factor1 Factor2 Factor3 Factor4 Factor5 



International Online Journal of Education and Teaching (IOJET) 2018, 5(3), p-p. 

 

633 

Relationships:  

V2 V8 V15 V19 V23 V27=Factor1 

V7 V14 V18 V24 V25 V30=Factor2 

V1 V3 V4 V9 V11 V16 V22=Factor3 

V5 V12 V17 V20 V29=Factor4 

V10 V13 V21 V26 V28=Factor5 

Group 2: Testing Equality Of Factor Correlations 

Covariance Matrix from File HEALTHCARE.COV 

Asymptotic Covariance Matrix from File HEALTHCARE.ACM 

Method of Estimation: Maximum Likelihood 

Iterations: 20 

Sample Size: 212 

Group 3: Testing Equality Of Factor Correlations 

Covariance Matrix from File HUMANITIES.COV 

Asymptotic Covariance Matrix from File HUMANITIES.ACM 

Method of Estimation: Maximum Likelihood 

Iterations: 20 

Sample Size: 239 

End of Problem 

Model B 

Group 1: Testing Equality Of Factor Structures 

Model B: Factor Correlation and Error Variances Invariant 

Observed Variables: V1-V30 

Covariance Matrix from File SCIENCE.COV 

Asymptotic Covariance Matrix from File SCIENCE.ACM 

Method of Estimation: Maximum Likelihood 

Iterations: 20 

Sample Size: 215 

Latent Variables: Factor1 Factor2 Factor3 Factor4 Factor5 

Relationships:  

V2 V8 V15 V19 V23 V27=Factor1 

V7 V14 V18 V24 V25 V30=Factor2 

V1 V3 V4 V9 V11 V16 V22=Factor3 

V5 V12 V17 V20 V29=Factor4 

V10 V13 V21 V26 V28=Factor5 

Group 2: Testing Equality Of Factor Correlations 

Covariance Matrix from File HEALTHCARE.COV 

Asymptotic Covariance Matrix from File HEALTHCARE.ACM 

Method of Estimation: Maximum Likelihood 

Iterations: 20 

Sample Size: 212 

Relationships:  

V2 V8 V15 V19 V23 V27=Factor1 

V7 V14 V18 V24 V25 V30=Factor2 

V1 V3 V4 V9 V11 V16 V22=Factor3 

V5 V12 V17 V20 V29=Factor4 

V10 V13 V21 V26 V28=Factor5 

Group 3: Testing Equality Of Factor Correlations 

Covariance Matrix from File HUMANITIES.COV 

Asymptotic Covariance Matrix from File HUMANITIES.ACM 

Method of Estimation: Maximum Likelihood 

Iterations: 20 

Sample Size: 239 

V2 V8 V15 V19 V23 V27=Factor1 

V7 V14 V18 V24 V25 V30=Factor2 

V1 V3 V4 V9 V11 V16 V22=Factor3 

V5 V12 V17 V20 V29=Factor4 

V10 V13 V21 V26 V28=Factor5 

End of Problem 

Model C 

Group 1: Testing Equality Of Factor Structures 

Model C: Factor Correlation Invariant 

Observed Variables: V1-V30 

Covariance Matrix from File SCIENCE.COV 

Asymptotic Covariance Matrix from File SCIENCE.ACM 

Method of Estimation: Maximum Likelihood 

Iterations: 20 

Sample Size: 215 



Şekercioğlu 

    

634 

Latent Variables: Factor1 Factor2 Factor3 Factor4 Factor5 

Relationships:  

V2 V8 V15 V19 V23 V27=Factor1 

V7 V14 V18 V24 V25 V30=Factor2 

V1 V3 V4 V9 V11 V16 V22=Factor3 

V5 V12 V17 V20 V29=Factor4 

V10 V13 V21 V26 V28=Factor5 

Group 2: Testing Equality Of Factor Correlations 

Covariance Matrix from File HEALTHCARE.COV 

Asymptotic Covariance Matrix from File HEALTHCARE.ACM 

Method of Estimation: Maximum Likelihood 

Iterations: 20 

Sample Size: 212 

V2 V8 V15 V19 V23 V27=Factor1 

V7 V14 V18 V24 V25 V30=Factor2 

V1 V3 V4 V9 V11 V16 V22=Factor3 

V5 V12 V17 V20 V29=Factor4 

V10 V13 V21 V26 V28=Factor5 

Set the Error Variances of V1-V30 free 

Group 3: Testing Equality Of Factor Correlations 

Covariance Matrix from File HUMANITIES.COV 

Asymptotic Covariance Matrix from File HUMANITIES.ACM 

Method of Estimation: Maximum Likelihood 

Iterations: 20 

Sample Size: 239 

V2 V8 V15 V19 V23 V27=Factor1 

V7 V14 V18 V24 V25 V30=Factor2 

V1 V3 V4 V9 V11 V16 V22=Factor3 

V5 V12 V17 V20 V29=Factor4 

V10 V13 V21 V26 V28=Factor5 

Set the Error Variances of V1-V30 free 

End of Problem 

Model D 

Group 1: Testing Equality Of Factor Structures 

Model D: Factor Loadings and Factor Correlation Invariant 

Observed Variables: V1-V30 

Covariance Matrix from File SCIENCE.COV 

Asymptotic Covariance Matrix from File SCIENCE.ACM 

Method of Estimation: Maximum Likelihood 

Iterations: 20 

Sample Size: 215 

Latent Variables: Factor1 Factor2 Factor3 Factor4 Factor5  

Relationships:  

V2 V8 V15 V19 V23 V27=Factor1 

V7 V14 V18 V24 V25 V30=Factor2 

V1 V3 V4 V9 V11 V16 V22=Factor3 

V5 V12 V17 V20 V29=Factor4 

V10 V13 V21 V26 V28=Factor5 

Group 2: Testing Equality Of Factor Correlations 

Covariance Matrix from File HEALTHCARE.COV 

Asymptotic Covariance Matrix from File HEALTHCARE.ACM 

Method of Estimation: Maximum Likelihood 

Iterations: 20 

Sample Size: 212 

Set the Error Variances of V1-V30 free 

Group 3: Testing Equality Of Factor Correlations 

Covariance Matrix from File HUMANITIES.COV 

Asymptotic Covariance Matrix from File HUMANITIES.ACM 

Method of Estimation: Maximum Likelihood 

Iterations: 20 

Sample Size: 239 

Set the Error Variances of V1-V30 free 

End of Problem