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Latent Variable Scores and Observational Residuals

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Karl G Joreskog
Karl G Joreskog
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Dag Sorbom
Dag Sorbom
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Fan Yang Wallentin at Uppsala University
Fan Yang Wallentin
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Latent Variable Scores and Observational Residuals
Karl G J¨oreskog
Dag S¨orbom
Fan Yang Wallentin
Uppsala University
February 24, 2006
There are several methods available for estimating latent variable scores
1
, e.g., Lawley &
Maxwell (1971, Chapter 8) or Bartholomew & Knott (1999, pp. 65–68). The two most com-
monly used methods for estimating latent variable scores are the regression method of Thomson
(1939) and the Bartlett method (Bartlett, 1937). In LISREL we use the procedure of Anderson
& Rubin (1956) as described in J¨oreskog (2000). This procedure has the advantage of producing
latent variable scores that have the same covariance matrix as the latent variables themselves.
In LISREL 8.8 we have added the possibility of estimating individual scores for all the er-
ror terms, measurement errors as well as structural errors, in any single-group LISREL model.
Following Bollen & Arminger (1991) we use the general term observational residuals for this.
The observational residuals depend on the method used for estimating the latent variable scores.
Bollen & Arminger (1991) gave formulas which are valid for any method of estimating latent
variable scores as linear combinations of observed variables. The results rep orted here are based
on latent variable scores estimated by the Anderson & Rubin method.
This paper describes how latent variable scores and observational residuals can be obtained
with LISREL 8.8 and illustrates their use with two examples.
1 Example 1: Interaction Model
The PRELIS system file KJUDD.PSF in the NSFEX subfolder contains data on the five variables
y, x
1
,x
2
,x
3
,x
4
generated according to the Kenny Judd model:
y
x
1
x
2
x
3
x
4
=
α
τ
1
τ
2
τ
3
τ
4
+
γ
1
γ
2
γ
3
100
λ
2
00
010
0 λ
4
0
ξ
1
ξ
2
ξ
1
ξ
2
+
ζ
δ
1
δ
2
δ
3
δ
4
(1)
See J¨oreskog & Yang (1996) for explanation of the terms and assumptions of this model. Note
that y is a nonlinear function of the latent variables ξ
1
and ξ
2
. This model was first considered
by Kenny & Judd (1984) and several metho ds have b een developed for estimating it, e.g., Marsh,
Wen, & Hau (2004) and references therein. Here we describe how the model can be estimated by
1
In classical exploratory factor analysis these are usually called factor scores
1
using latent variable scores. At the same time we illustrate how observational residuals can be
obtained.
We begin by estimating the linear part of the model, i.e., the model without the interaction
effect. This model is shown in the path diagram in Figure 1.
y
x
1
x
2
x
3
x
4
-
-
-
-
-
ξ
1
ξ
2
Q
Q
Q
Q
Q
Qk
Q
Q
Q
Q
Q
Qk
+
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
AK
Figure 1: The Linear Part of the Kenny-Judd Model
The data is in the PRELIS system file KJUDD.PSF. See J¨oreskog, et al. (2003, p. 168)
for information about PSF files and how to create them. The following SIMPLIS command file
KJ1.SPL estimates a standard linear confirmatory factor analysis model with two latent variables
ξ
1
and ξ
2
(called Ksi1 and Ksi2 in the SIMPLIS file) and estimates the latent variable scores and
all observational residuals at the same time.
Raw data from file KJUDD.PSF
Latent Variables: Ksi1 Ksi2
Relationships
Y=Ksi1 Ksi2
X1=1*Ksi1
X2=Ksi1
X3=1*Ksi2
X4=Ksi2
Path Diagram
PSFfile KJUDD.PSF
Estimate Residuals
End of Problem
In addition to the path diagram and the standard SIMPLIS output file KJ1.OUT, this run also
generates a new PSF file KJUDDnew.PSF containing all the following variables
Y X1 X2 X3 X4 Ksi1 Ksi2 R_Y R_X1 R_X2 R_X3 R_X4
The variables
2
Ksi1 Ksi2
contain the estimated latent variable scores of ξ
1
and ξ
2
. The variables
R_Y R_X1 R_X2 R_X3 R_X4
contain the estimated scores on ζ,δ
1
,δ
2
,δ
3
,δ
4
. LISREL automatically puts R_ in front of the
variable name to mean the residual or error term. Thus, R_varname means the residual of or
the error term on the variable varname. Since variable names in LISREL can contain at most
8 characters, it is advisable that varname contains at most 6 characters so that the extended
variable R_varname will contain at most 8 characters.
In the file KJ1.SPL it is the line
PSFfile KJUDD.PSF
that produces the latent variable scores and the lines
PSFfile KJUDD.PSF
Estimate Residuals
that pro duce the observational residuals. Obviously, the observational residuals cannot be pro-
duced without the latent variable scores. Thus, one can omit the line
Estimate Residuals
if one is only interested in latent variable scores.
With the file KJUDDnew.PSF displayed, one can plot Y against Ksi1 or Ksi2. These plots
clearly shows that there are nonlinear relationships.
The product variable ξ
1
ξ
2
can be computed and the nonlinear relationship between y and ξ
1
and ξ
2
can be estimated at the same time using the following PRELIS command file (KJ2.PR2)
Estimating the regression of Y on Ksi1, Ksi2 and Ksi1*Ksi2
SY=KJUDDnew.PSF
NE Ksi1Ksi2 = Ksi1*Ksi2
CA ALL
RG Y on Ksi1 Ksi2 Ksi1Ksi2
OU XU
The XU on the OU line is an option to skip the printing of univariate summary statistics.
The output KJ2.OUT gives the following estimated equation
Y = 1.085 + 0.198*Ksi1 + 0.482*Ksi2 + 0.458*Ksi1Ksi2
Standerr (0.0174) (0.0246) (0.0242) (0.0263)
Z-values 62.380 8.032 19.918 17.410
P-values 0.000 0.000 0.000 0.000
+ Error,
Error Variance = 0.271
3
which clearly shows that the estimate of γ
3
is highly significant.
Alternatively, one can use the estimated error term R_Y and regress this on Ksi1Ksi2 only,
using the following PRELIS syntax file (KJ3.PR2):
Estimating the regression of R_Y on Ksi1*Ksi2
SY=KJUDDnew.PSF
NE Ksi1Ksi2 = Ksi1*Ksi2
CO ALL
RG R_Y on Ksi1Ksi2
OU XU
This gives the following result
R_Y = - 0.0949 + 0.451*Ksi1Ksi2 + Error,
Standerr (0.0173) (0.0251)
Z-values -5.481 17.931
P-values 0.000 0.000
Error Variance = 0.271
Note that the two alternatives give the same estimates of the error variance and almost the same
estimates of γ
3
.
2 Example 2: Political Democracy
Bollen (1989, p. 17) presents a panel model of political democracy and industrialization in 75
countries. Bollen & Arminger (1991) used the same model in their discussion of observational
residuals. The mo del is shown in the path diagram in Figure 2.
The variables in the model are
• y
1
Freedom of press 1960
• y
2
Freedom of political opposition 1960
• y
3
Fairness of elections 1960
• y
4
Effectiveness of legislature 1960
• y
5
Freedom of press 1965
• y
6
Freedom of political opposition 1965
• y
7
Fairness of elections 1965
• y
8
Effectiveness of legislature 1965
• x
1
GNP per capita 1960
• x
2
Energy consumtion per capita 1960
• x
3
Percentage of labor force in industry 1960
4
Y
1
Y
2
Y
3
Y
4
Y
5
Y
6
Y
7
Y
8


η
1


η
2


ξ
1
X
3
X
2
X
1
3
-
H
H
H
H
H
H
H
Hj
Q
Q
Q
Q
Q
Q
Q
Qs
3
1
-
Q
Q
Q
Q
Q
Q
Q
Qs
7
S
S
S
S
S
S
S
Sw
?
Q
Q
Q
Q
Qk
+
-
-
-
Figure 2: Panel Model of Democracy and Industrialization
• η
1
Democracy in 1960 (Latent variable Dem60)
• η
2
Democracy in 1965 (Latent variable Dem65)
• ξ Level of industrialization in 1960 (Latent variable Indus)
y
1
– y
4
are taken as indicators of the latent variable Dem60 (Democracy 1960) and y
5
– y
8
are taken
as indicators of the latent variable Dem65 (Democracy 1965). x
1
– x
3
are taken as indicators of
the latent variable Indus (Industrialization 1960). Data on y
1
– x
3
are available for 75 developing
countries. These data are in the file POLIDEM.PSF.
2.1 Latent Variables Scores
The following SIMPLIS syntax file (BA1a.SPL) will estimate scores on Demo60, Demo65, and
Indus for each country in file POLIDEMnew.PSF. Slightly different versions of BA1a.SPL,
are used in sections 2.2, 2.3, and 2.4, namely BA1b.SPL, BA1c.SPL, and BA1d.SPL.
5
Industrialization-Democracy Example
Raw Data from file POLIDEM.PSF
Latent Variables: Dem60 Dem65 Indus
Relationships:
Y1= 1*Dem60
Y2-Y4 = Dem60
Y5 = 1*Dem65
Y6-Y8 = Dem65
X9 = 1*Indus
X10-X11 = Indus
Dem60 = Indus
Dem65 = Dem60 Indus
Set Dem60 -> Y2 = Dem65 -> Y6
Set Dem60 -> Y3 = Dem65 -> Y7
Set Dem60 -> Y4 = Dem65 -> Y8
Let the errors of Y5 and Y1 be correlated
let the errors of Y6 and Y2 be correlated
Let the errors of Y7 and Y3 be correlated
Let the errors of Y8 and Y4 be correlated
PSFfile POLIDEM.PSF
Path Diagram
End of Problem
The variables y
1
–y
4
are the same variables as y
5
–y
8
measured at two p oints in time. So Bollen &
Arminger (1991) assume that their loadings on η
1
and η
2
are the same. This is specified by the
lines
Set Dem60 -> Y2 = Dem65 -> Y6
Set Dem60 -> Y3 = Dem65 -> Y7
Set Dem60 -> Y4 = Dem65 -> Y8
The loadings of y
1
and y
5
are set to 1. Furthermore, they assume that the measurement errors of
corresponding y-variables are correlated. This is specified by the lines
Let the errors of Y5 and Y1 be correlated
let the errors of Y6 and Y2 be correlated
Let the errors of Y7 and Y3 be correlated
Let the errors of Y8 and Y4 be correlated
After this run is completed the file POLIDEMnew.PSF contains the following variables
Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 X1 X2 X3 Dem60 Dem65 Indus
It would be useful to know which country each row belongs to. We do not have this information.
However, one can construct a variable COUNTRY which runs from 1 to 75
2
This is done with the
following PRELIS syntax file (BA2.PR2) which at the same time constructs scores on another
latent variable Diff = Dem65 - Dem60.
2
The variable TIME is always available in PRELIS. It assigns values 1, 2,...,N to the cases. It is intended
mainly for time series. Hence its name TIME.
6
SY=POLIDEMnew.PSF
New COUNTRY=TIME
New Diff=Dem65-Dem60
CO ALL
Select COUNTRY Dem60 Dem65 Diff
OU RA=DEMDIFF.PSF
The file DEMDIFF.PSF contains the following variables
COUNTRY Dem60 Dem65 Diff
With this file one can do various things to find out which countries have most democracy or
which countries increased or decreased their democracy between 1960 and 1965. For example,
do a bivariate line plot of Diff against COUNTRY. This shows that country 2 increased democracy
most and country 30 had the largest decrease. This can also be seen by sorting Diff in descending
order. This shows that country 2 has a Diff value of 1.69 and country 30 has a Diff value of
-1.51. These are the best and worst countries. The second best and second worst countries are
the countries 22 and 34. These have Diff values of 1.39 and -1.38, respectively.
2.2 Observational Residuals
The observational residuals can be obtained by adding the line
Estimate Residuals
in the file BA1a.SPL, see file BA1b.SPL. Running this produces a new file POLIDEM-
new.PSF containing the following variables (Note that the previous filePOLIDEMnew.PSF
will be overwritten)
Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 X1 X2 X3 Dem60 Dem65 Indus
R_Y1 R_Y2 R_Y3 R_Y4 R_Y5 R_Y6 R_Y7 R_Y8 R_X1 R_X2 R_X3 R_Dem60 R_Dem65
For example, R_Y5 is the estimate of the measurement error
5
, R_X3 is the estimate of the
measurement error δ
3
, and R_Dem64 is the estimate of the structural error ζ
2
in the LISREL
model. One can use these variables in much the same way as described in the previous section.
For example, one can add the country code and plot any of the observational residuals against
COUNTRY or sort any of these residuals in ascending or descending values. This can be done for
the purpose of finding errors in the data or other outliers or to determine countries with largest
or smallest residuals.
2.3 Standardized Latent Variable Scores
One can also obtain estimates of the standardized latent variables. This is easy to do. Just delete
the 1* in three places in BA1a.SPL (see file BA1c.SPL) and rerun that file. This gives a
solution in which the three latent variables Dem60, Dem65, and Indus have unit variances. The
correlation matrix of the latent variables is given in the output file BA1c.OUT as
7
Covariance Matrix of Latent Variables
Dem60 Dem65 Indus
-------- -------- --------
Dem60 1.00
Dem65 0.94 1.00
Indus 0.45 0.56 1.00
Furthermore, the resulting file POLIDEMnew.PSF now contains estimates of the standardized
latent variables. Next run the following PRELIS syntax file (BA3.PR2):
SY=POLIDEMnew.PSF
Select Dem60 Dem65 Indus
OU MA=KM XU
This will compute the correlation matrix of the latent variable scores. The output file BA3.OUT
verifies that this correlation matrix is indeed equal to the correlation matrix of the latent variables
in the model as stated in the introduction:
Correlation Matrix
Dem60 Dem65 Indus
-------- -------- --------
Dem60 1.000
Dem65 0.945 1.000
Indus 0.449 0.560 1.000
PRELIS uses three decimals in the output whereas LISREL (SIMPLIS) uses two decimals by default.
One can change that by putting the line
Number of decimals: 3
in the SIMPLIS command file.
2.4 Standardized Observational Residuals
The latent variable scores and observational residuals considered in sections 2.1 and 2.2 depend
on the unit of measurement in the observed y- and x-variables. This is useful if these units
have some definite meaning. However, sometimes these units are the result of rather arbitrary
scaling of the observed variables, in which case it may be useful to standardize the observational
residuals in some way. This is discussed by Bollen & Arminger (1991) who consider different ways
of standardizing them.
One way to standardize the observational residuals is to rescale them such that they have zero
means and unit standard deviations in the sample. For this purpose we have added a command
to standardize variables in PRELIS:
SV varlist
8
will standardize the variables in varlist. One can then save the standadized variables in a sep-
arate data file. For example, to standardize the observational residuals in the file POLIDEM-
new.PSF obtained by running BA1b.SPL and save them in the file POLIDEMstdres.PSF,
use the following PRELIS command file
SY=POLIDEMnew.PSF
SE R_Y1-R_Dem65
SV ALL
OU RA=POLIDEMstdres.PSF XU
Another standardization is to standardize the residuals by answering the question: What would
the estimates of the residuals be if all variables, observed as well as latent, were standardized?
While it is easy to estimate a completely standardized solution in LISREL, using either SIMPLIS
or LISREL syntax, see p. 184 in J¨oreskog & S¨orbom (1999a) or p. 93 in J¨oreskog & S¨orbom
(1999b), it is not quite as easy to obtain estimates of the residuals in a standardized solution.
To obtain such standardized residuals, one must have the observed variables standardized in the
PSF file.
The following PRELIS syntax file
SY=POLIDEM.PSF
SV all
OU RA=POLIDEMstd.PSF XU
will standardize all the variables in POLIDEM.PSF and save the standardized variables in the
file POLIDEMstd.PSF.
To estimate the standardized observational residuals, one can now run BA1b.SPL with POLI-
DEM.PSF replaced by POLIDEMstd.PSF in two places, see file BA1d.SPL.
References
Anderson, T.W., & Rubin, H. (1956) Statistical inference in factor analysis. In Proceedings of
the Third Berkeley Symposium, Volume V. Berkeley: University of California Press.
Bartholomew, D., & Knott, M. (1999) Latent Variable Models and Factor Analysis. London:
Arnold.
Bartlett, M.S. (1937) The statistical conception of mental factors. British Journal of Psochology,
28,97–104.
Bollen, K.A. (1989) Structural Equations with Latent Variables. Wiley.
Bollen, K.A. & Arminger, G. (1991) Observational residuals in factor analysis and structural
equation models. Sociological Methodology, 21, 235–262.
J¨oreskog K. G. (2000) Latent variable scores.
Available at http://www.ssicentral.com/lisrel/advancedtopics.html.
J¨oreskog, K.G. & S¨orbom, D. (1999a) LISREL 8: Structural Equation Modeling with the SIMPLIS
Command Language. Lincolnwood, IL: Scientific Software International.
9
J¨oreskog, K.G. & S¨orbom, D. (1999b) LISREL 8 User’s Reference Guide. Lincolnwood, IL:
Scientific Software International.
J¨oreskog, K.G., S¨orbom, D., Du Toit, S., & Du Toit M. (2003) LISREL 8: New Statistical
Features. Third printing with revisions. Lincolnwoo d, IL: Scientific Software International.
J¨oreskog, K.G., & Yang, F. (1996). Nonlinear structural equation models: The Kenny-Judd
model with interaction effects. Pp. 57-88 in G.A. Marcoulides & R.E. Schumacker (Eds):
Advanced structural equation modeling: Issues and techniques. Lawrence Erlbaum Asso-
ciates, Publishers.
Kenny, D.A., & Judd, C.M. (1984) Estimating the nonlinear and interactive effects of latent
variables. Psychological Bulletin, 96, 201-210.
Lawley, D.N., & Maxwell, A.E. (1971) Factor Analysis as a Statistical Method, (2nd edition).
London: Butterworths.
Marsh, H.W., Wen, Z., & Hau, K.T. (2004) Structural equation models of latent interactions:
Evaluation of alternative strategies and indicator construction. Psychological Methods, 9,
275–300.
Thomson, G.H. (1939) The Factorial Nature of Human Ability. New York: Houghton-Mifflin.
10
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The aim of the study is to investigate the influence of training in multiple functions, communication of strategy, and feedback on performance on competitive advantage while assessing the mediating role of corporate image. A cross‑sectional survey design was used in this study. Three hundred and one permanent employees working in different organizations in the Ghanaian telecommunication service sector were sampled for this study. The respondents’ responses were analyzed using descriptive and inferential statistics (structural equation modeling partial least squares) with the aid of IBM SPSS Statistics (Version 26.0) and Smart PLS (Version 4.0). The results revealed that training in multiple functions, communication of strategies, and feedback on performance does not significantly relate to competitive advantage. Further, the results established that corporate image mediated the relationship between both communication of strategy and training in multiple functions and competitive advantage. However, corporate image did not mediate the link between feedback on performance and competitive advantage. These results imply that human resource managers should devise strategies to rapidly respond to situations in this current, competitive and ever-changing business environment. To the best of our knowledge, studies that have explored the mediating role of corporate image in the relationship between training in multiple functions, communication of strategy, and feedback on performance and competitive advantage in the telecommunication service sector are rare.
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... Our sample of 203 branches, and with 88 indicators, our sample was below the minimum requirement. Therefore, we simplified the dimensions by parceling variables (Rhemtulla 2016) and used latent variable scoring (Joreskog et al., 2006). In this case, the second-order confirmatory analysis model was transformed into a first-order confirmatory analysis model. ...
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... Our sample of 203 branches, and with 88 indicators, our sample was below the minimum requirement. Therefore, we simplified the dimensions by parceling variables (Rhemtulla 2016) and used latent variable scoring (Joreskog et al., 2006). In this case, the second-order confirmatory analysis model was transformed into a first-order confirmatory analysis model. ...
Understanding the Effects of Digital Transformation, Entrepreneurial Orientation, Readiness for Change, and Innovative Behavior on Performance
Article
Full-text available
  • Jan 2022
The fourth industrial revolution has led us to massive Internet use. Companies must go along with this revolution by adopting digital technologies for a quicker and more accurate response to the market. For this reason, entrepreneurship and digital transformation have come into play to make companies ready for change, to generate innovative behaviour, and eventually to increase performance. This study aims at investigating the effect of entrepreneurial orientation (EO) and digital transformation (DT) on organisational readiness for change (ORC), innovative behaviour (IB), and performance, particularly at the branch level (BP). We involve 203 branches of a state-owned pawnbroker company. We used a set of questionnaires to collect the data on EO, DT, ORC, IB, and BP. Such data were processed and analysed using structural equation modelling (SEM). This study found that DT positively and significantly affected ORC and IB, while EO only positively and significantly affected ORC. Both ORC and IB, in turn, positively and significantly affected BP. This study enriches our understanding of the importance EO, DT, ORC and IB in enhancing performance.
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Empirical Evaluation of Urban Sustainability from Underlying Causal Structures and Legacy Effects: The Prefecture Cities in China as a Case Study
Chapter
  • Oct 2023
One innovation separating the total socioenvironmental system (TSES) analytical framework from other coupled human–nature systems is its focus on examining causal relationships between ecological/environmental, socioeconomic, and global climate changes. This chapter advocates the concept of latent variables or hypothetical constructs to investigate causal relationships between ecological, socioeconomic, and climate changes. The adoption of hypothetical constructs enables researchers to examine interactions between a large number of ecological, socioeconomic, and climate change variables and their complex socioenvironmental causal structures. The capability to model multiple sets of variables in multiple dimensions (causal structures) is an innovative attempt to develop big models for examining big data in the context of socioenvironmental sustainability studies. The other TSES innovation is to explore spatial and temporal lagged (legacy) effects of the causal relationships between various socioenvironmental factors. This chapter addresses historical or legacy effects from individual variables to hypothetical system structures. Therefore, this chapter illustrates that the TSES analytical framework can handle complex causal structures through latent variables and their temporal lagged effects (i.e., legacy effects). This latent variable-based TSES framework is called the cross-lagged panel model (CLPM) for urban sustainability (US). This chapter introduces the literature and basic concepts concerning CLPM-US. It illustrates CLPM-US through a case study of urban sustainability of the 278 prefecture cities in China. The examinations include the formation of causal structures of urban sustainability, the exploration of temporal legacy effects of urban sustainability and their underlying socioenvironmental structures, the computation of urban sustainability scores and rankings, and the relationships between the sustainability rankings and the factor loadings of their underlying causal structures. Furthermore, this chapter provides an alternative approach, the multi-objective optimization problems (MOOPs) algorithm, to assess the relationships between the sustainability rankings and the loadings of their underlying structures. The total system approach based on the society and nature coevolution theory is vital for examining urban sustainability. The urban sustainability measurements involve all sectors of human society and the environment. Introducing causal constructs through latent variables in CLPM is a workable solution to describe the interactions between these causal structures that determine urban sustainability status. Moreover, urban sustainability is an evolving concept involving history, representing the current, and predicting the future. Lagged and cross-lagged effects of an urban system and its underlying causal structures are critical traits of urban system evolutionary dynamics. Furthermore, the measurement of urban sustainability is not a single composite score but involves a large set of trade-offs between sub-aims of urban sustainability.
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Psychosocial Factors Affecting Self-Regulated Learning among Indonesian Islamic College Students: The Mediating Role of Perception Feedback
Article
  • Jun 2023
In facing various problems during the online learning process, individuals need to have self-regulated learning to be able to undergo online lectures optimally and effectively. Hence, student’s knowledge increases, and learning objectives are achieved. This study aims to investigate the mediating effect of self-efficacy in the relationship between perception feedback on self-regulated learning in undergraduate students. Whether the mediation had varying effects on different age levels and sex differences was also examined. The sampling technique used in this study is non-probability sampling with 410 students. This study uses a quantitative approach with path analysis. In this study, researchers used confirmatory factor analysis (CFA) using Lisrel 8.8 to test the instrument's construct validity. The results showed a direct effect from the independent variables, namely self-efficacy, family support, and an indirect effect through mediator variables, namely self-efficacy, family support, friend support, and gender. The effect of self-efficacy and family support is partial mediation on self-regulated learning. In contrast, peer support and gender demographic variables have a perfect mediating effect on self-regulated learning with perception feedback as a mediator.
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Psychosocial risk factors identification in Mexican workers and RGIII validation
Article
Full-text available
  • Feb 2023
  • Work
Background: Since 2018, NOM-035-STPS-2018 has been applied in Mexico, focused on measuring psychosocial risk factors (PRFs) in workers and the Reference Guide III (RGIII) has been presented, however, research focused on its validation has been scarce, in very specific sectors and with small samples. Objective: Determine the levels of PRFs of five work centers and reliability and validity aspects of RGIII. Method: The RGIII was applied to 1458 workers (806 women and 652 men) from five workplace in the industrial sector of Ensenada (Mexico), and the level of risk of the PRFs was analyzed, as well as their reliability and validity through Exploratory Factor Analysis (EFA) and Confirmatory Factor Analysis (CFA). Results: The PRFs with medium, high and very high-risk levels are Workload, Lack of control over work and Workday. The RGIII presents adequate reliability with Cronbach's α, Alpha ordinal RHO and Omega of 0.93, 0.95 and 0.95, respectively. The EFA shows all five subscales maintain factor loadings greater than 0.43, although Leadership and relationships at work has better saturation values, and Work environment ended with only three items. The CFA indicates Leadership and work relationships with a Root Mean Square Error of Approximation (RMSEA) goodness of fit index of 0.072. Conclusion: The RGIII allows the identification and evaluation of the level of risk of PRFs. It complies with sufficient internal consistency. It does not have a clear factorial structure, because it does not meet the minimum values of goodness-of-fit indexes that would allow confirming the structure proposed in RGIII.
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Estimating the Nonlinear and Interactive Effects of Latent Variables
Article
Full-text available
  • Jul 1984
Describes a procedure that enables researchers to estimate nonlinear and interactive effects of latent variables in structural equation models. Given that the latent variables are normally distributed, the parameters of such models can be estimated. To do this, products of the measured variables are used as indicators of latent product variables. Estimation must be done using a procedure that allows nonlinear constraints on parameters. The procedure is demonstrated in 3 examples. The 1st 2 examples use artificial data with known parameter values. These parameters are successfully recovered by the procedure. The final complex example uses national election survey data. (14 ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)
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Observational Residuals in Factor Analysis and Structural Equation Models
Article
  • Jan 1991
In the last decade there has been a surge of interest in the role of outliers and residuals in statistical analyses. However, there is a surprising neglect of these topics in factor analysis and in other structural equation models with latent variables. In this paper we propose ways to calculate unstandardized residuals, derive standardized residuals, suggest tests of statistical significance for residuals, and illustrate the procedures with empirical and simulated data. Our significance tests for the residuals are based on asymptotic theory, but our examples indicate that the tests are helpful even with N's of 60 to 100.
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