Introduction to Structural Equation Modeling

©drtamil@gmail.com 2020
Introduction to SEM
Dr Azmi M Tamil

Disclaimer
• I am not a trainer for SEM. I just introduce the
topic for my postgraduate students, as part of
a course module in Advanced Statistics.
• These notes are partially based on Prof Mohd.
Ayub Sadiq@Lin Naing’s 2011 lecture notes on
AMOS.
• Those using SEM in their thesis are advised to
attend other workshops specifically for SEM.

My Reference
A Beginner's Guide to
Structural Equation
Modeling. 4th edition.
Randall E.
Schumacker, Richard
G. Lomax

Glossary

Latent versus Measured Variables
• Latent variables – also known as construct or
factors. They are usually not directly observable
or measurable.
• Latent variables are indirectly observed or
measured or inferred from a set of items or
questions that we posed to respondents.
• The set of questions tend to be focused on the
latent trait that we want to measure such as
the level of fitness or the level of mental health.

Covariance versus Correlation
When comparing data samples from different populations,
• covariance is used to determine how much two random variables
vary together. Range from negative infinity to positive infinity.
• Whereas correlation is used to determine when a change in one
variable can result in a change in another. Range from negative 1 to
positive 1.
• Both covariance and correlation measure linear relationships
between variables. Scatter diagram is similar for both.

Co-variance
• Co-variance - the value of the product of the
deviations (variance) of two variates from
their respective means.
• Variance – deviations of a single variate from
the mean.

• Covariance is a measure of how much two random
variables vary together. It’s similar to variance, but
where variance tells you how a single variable
varies, co variance tells you how two variables vary
together.
– Xij & Xik are the random variables (of X & Y)
– xj = is the expected value (the mean μ) of the random
variable X.
– xk= is the expected value (the mean μ) of the random
variable Y.
– N = the number of items in the data set.
• Look at the formula, it is like variance, but not squared.
https://www.statisticshowto.datasciencecentral.com/covariance/

2015. Randall E. Schumacker, Richard G. Lomax. A Beginner's Guide to Structural Equation Modelling. 4th ed.

Pearson’s Correlation
• measures the degree of linear association
between two interval scaled variablese.g.,
between height and weight.
• r lies between -1 and 1. Values near 0 means no
(linear) correlation and values near ± 1 means
very strong (linear) correlation.

• Direction of correlation;
• r 2 =coefficient of determination.
• Coefficient of determination = the portion of
variability in one of the variables that can be
accounted for by variability in the second
variable.
Pearson’s Correlation
Positive and LinearNegative and Linear No correlation

History of SEM

Correlation & Factor Analysis
• Sir Francis Galton came out with the concept of co-
relation when studying height of sons & their fathers.
He died in 1911.
• 1896 – Karl Pearson developed the correlation formula.
• 1904 – Charles Spearman used correlation to develop
factor analysis technique. If a set of many items
correlated with one another, the responses could be
summed to yield a score. That set of items is called a
“construct”. Factor analysis is used to create a
measurement instrument or a construct.

CFA – to test a construct
• Confirmatory Factor Analysis (CFA) is done to test
a construct.
• The pioneers of CFA were Howe (1955), Anderson
& Rubin (1956) and Lawley (1958).
• Karl Gustav Joreskog published the first article on
CFA in 1969 and helped develop the first CFA
software programme (LISREL -
linear structural relations).
• Factor analysis create measurement instruments.
CFA is used to test the theoretical construct of
these instruments.

Path Models uses Correlation & Regression
• A biologist, Sewell Wright developed the path model
between 1918 till 1934, which uses correlation and
regression, to draw complex relationships between
observed variables. He used it for models of animal
behaviour.
• In 1950s, econometricians such as H. Wold used it for
simultaneous equation modelling.
• Sociologists such as D. Duncan & H.M. Blalock also used it
for the same purpose in the 1960s.
• Path analysis involves solving a set of simultaneous
regression equations that theoretically establish the
relationship among the observed variables in the path
models.

SEM combines Path & CFA
• SEM essentially combine path models with CFA
since SEM incorporates both latent and observed
variables.
• This model was initially known as the JKW model
due to the work of;
– Karl Gustav Joreskog (1973)
– Ward Keesling (1972) and
– David Wiley (1973)
• Since analysis was done using LISREL, it became
known as LInear Structural RELations model in
1973. Now it is better known as SEM.

Do we need to do SEM?
• Not all postgrad research need to do SEM.
• Usually only for those that need to validate their
model or questionnaire will use SEM.
• But the dependent variable (outcome) must be
interval or ratio.
– Interval scales are numeric scales in which we know
not only the arrangement order, but also able to
quantify the differences between the values.
– A ratio variable, has all the properties of an interval
variable, and also has a clear definition of what is “0”.

(Kindly read “On the Theory of Scales of Measurement”; S.S. Stevens, 1946 to better
understand the difference of nominal, ordinal, interval and ratio data.)

Ordinal or Interval?
• We have seen attempts by many researchers to claim that
their ordinal Likert scale is really interval data. Such as by
using a Likert-10 scale or Likert-100 scale. But since the
respondents are unable to differentiate the difference in
their answers, such efforts are futile.
• Computer applications are unable to differentiate between
ordinal and interval data unless defined by user. Some
software such as PRELIS, will determine the variable to be
ordinal if it has lesser than 15 distinct scale points (< 15
categories).
• A 15-point criterion allows Pearson correlation coefficient
to vary between +1.0. Otherwise the range will only be
between +0.5.

Sample Size for SEM
• Ding, Velicer & Harlow (1995) stated 100 to 150 is
the minimum satisfactory sample size.
• Boonsma (1983) recommended 400.
• Textbooks suggest either 10 or 20 per variable.
• Bentler & Chou (1987) suggested 5 per variable is
sufficient for normally distributed data. And 10
per variable for others.
• Most published research used 250 to 500
subjects.

Missing Data
• It is usually not possible to run SEM if there
are missing data issues. Therefore please
correct it before analysis using such methods.

Please explore data to ensure linearity

Why do SEM?
Hox, Joop & Bechger, Timo. (1999). An
Introduction to Structural Equation
Modeling. Family Science Review. 11.
http://joophox.net/publist/semfamre.pdf

Why do SEM?
• Combine regression with factor analysis
(latent)
• Confirmatory factor analysis (if using SPSS,
only EFA is available)
• Regression models
• Complex path models

Why do SEM?
• Confirmatory factor analysis

Why do SEM?
• Regression models

Why do SEM?
• Complex path models

SEM provides
• Convenient framework for statistical analysis
– Factor analysis
– Regression Analysis
– Discriminant Analysis
– Canonical Correlation
• Often visualized by a graphical path diagram.

Why more researchers do SEM?

1. Why SEM getting popular?

Why AMOS?
• AMOS has a student evaluation version of
AMOS 5.01, which does not expire.
• It used to be available; but not anymore.
• Since the aim is just to introduce postgrad students to
SEM, I’ll stick to the legal demo version.
• Google for “SPSS Amos Trial download version”
https://www-
01.ibm.com/marketing/iwm/iwmdocs/tnd/data/web/
en_US/trialprograms/G556357A25118V85.html
• Please download and install.

Basic Rules in AMOS
• Square box – measured
variable
• Oval box – unmeasured
variable
• Single headed arrow –
regression
• Double headed arrow -
correlation
knowledge
value
satisfaction
performance
e1
1

Spatial
VISPERCe11
1
CUBESe2
1
LOZENGESe3
1
Verbal
paragraphe4
SENTENCEe5
WORDMEANe6
1
1
1
1
Basic Rules in AMOS
• Square box – measured
variable
• Oval box – unmeasured
variable
• Single headed arrow –
regression
• Double headed arrow -
correlation

Regression models
Lin Naing 2011. Structural Equation
Modeling with SPSS AMOS Workshop
Lecture Notes. 21-23 October 2011.

How to do Regression Models
https://youtu.be/rNQw5RkGA6g

Un-
stan
dard
ised
4.11
knowledge
7.68
value
9.42
satisfaction
performance
3.98
6.89
3.54
1.03
.98
1.12
24.26
e1
1

Un-
stan
dard
ised
4.11
knowledge
7.68
value
9.42
satisfaction
performance
3.98
6.89
3.54
1.03
.98
1.12
24.26
e1
1
Mean Std. Deviation Variance
knowledge 19.73 2.035 4.14
value 19.95 2.781 7.73
satisfaction 20.02 3.080 9.49
performance 59.76 8.913 79.44
Descriptive Statistics

knowledge
value
satisfaction
.69
performance
.71
.81
.57
.23
.31
.39
e1
Stan
dard
ised

Complicated Path

How to do Complicated Path
https://youtu.be/JbElfduMufA

Confirmatory Factor Analysis

Data for this lesson.
• https://wp.me/p4mYLF-sV
• Download AMOS-Grnt_fem.sav
• Delete all the value labels using SPSS.

Modification Indices Parameters

How to display M.I. in AMOS
-----------------------------------------
To copy and paste to AMOS using "title"
-----------------------------------------
Chi-square (df) = cmin (df); P value (>=0.05) = p;
Relative Chi-Sq (<=2) = cmindf;
GFI(>=0.95) = gfi; AGFI(>=0.9) = agfi;
CFI(>=0.9) = cfi; Pratio = pratio;
RMSEA(<=0.08) = rmsea.
(format)
View the
video to see
how to
insert it.

How to do CFA
https://youtu.be/9PYBbg36iOs

RMSEA

Modification Index

Shortcut for CFA Using
Plugins in AMOS
Why draw the boxes when you can
just copy and paste from SPSS?

Download & Copy The Plugins
• http://statwiki.kolobkreations.com/index.php?title=Plugins
• MasterValidity - This plugin produces an HTML file with a correlation table
of constructs, including the square root of the AVE on the diagonal, the CR
and the AVE, as well as the less used MSV and MaxR. It also provides some
interpretation and indication of validity issues. When validity issues occur,
it also provides some recommendations. References for validity thresholds
are provided.
• ModelBias - This plugin automates the tedious job of testing the a model
for specific bias or common method bias by running multiple contrained
and unconstrained models through chi-square difference tests. The output
is an HTML file that includes a table of the results, as well as
interpretation, recommendations, and a reference.
• ModelFit - This plugin creates an HTML file with all the relevant model fit
measures, their thresholds, and an interpretation, as well as references for
the suggested thresholds.
• PatternMatrixBuilder - This plugin automates the tedious job of creating a
CFA from a pattern matrix. You can paste a pattern matrix from SPSS into
the plugin window and it will automatically generate your model for you.
All you have to do after that is to rename the latent factors appropriately.

Plugins for AMOS
https://drive.google.com/drive/folders/0B3T1TGdHG9aEbFg1eEpqOWtrR3c?usp=sharing

Install Into Your Plugins Folder

Please Unblock The DLL Files

How To Use The Plugins
• https://www.youtube.com/user/Gaskination/
search?query=plugin
• Let us re-try the previous CFA exercise using
this plugin.

• Download AMOS-Grnt_fem.sav
• Delete all the value labels using SPSS.
2010. Randall E. Schumacker, Richard G. Lomax. A Beginner's Guide to Structural Equation Modelling. 3rd ed.

https://youtu.be/Awf_wv4artU

Validation of Questionnaire
Using AMOS
Azmi Mohd Tamil

The Questionnaire
Validation of Questionnaire
Using AMOS

Sample of
an
instrument
to measure
QOL

Basic Concept
• In psychometry, an instrument (i.e.
questionnaires) with items are created to
measure a latent trait that is not usually
measurable in the normal physical way.
• For example, what if we want to measure
physical fitness? So we come up with items
that is related to measuring physical fitness.
• The following are sample measures of fitness
in ascending difficulty;

Sample Measures of Fitness
Physical Activity 1=Limit
ed a lot
2=A bit
limited
3=Not
limited
Bathing or dressing yourself
Bending, kneeling or stooping
Lifting or carrying groceries
Walking one block
Climb one flight of stairs
Walking several blocks
Climb several flight of stairs
Moderate activities such as moving a table
Walking a mile (1.6 km) or more
Vigorous activities such as running or strenuous sports.
i
n
c
r
e
a
s
i
n
g
d
i
f
f
• Minimum score 10 – very unfit, maximum score 30 – very fit.
• These items combined to measure a single factor/latent trait -> Fitness.

Sample of a Fitness Score
Physical Activity 1=Limit 2=A bit 3=Not Score
Bathing or dressing yourself  3
Bending, kneeling or stooping  2
Lifting or carrying groceries  2
Walking one block  3
Climb one flight of stairs  2
Walking several blocks  2
Climb several flight of stairs  1
Moderate activities such as moving a table  2
Walking a mile (1.6 km) or more  1
Vigorous activities such as strenuous sports.  1
Total Score 19
i
n
c
r
e
a
s
i
n
g
d
i
f
f
• Cut-off=(Maximum-Minimum)/2+Minimum=(30-10)/2+10=20.
• Score of 19 is below the cut-off point. Is the respondent unfit?
• The reliability of a `composite score’ can be checked by Cronbach α.

Aim of the current exercise
• To assess the psychometric properties of the
Mental & Physical Scale using AMOS.
• To ensure continuity, I shall use the same
dataset as the one we did with EFA and Rasch.
• 10 items on Physical Function.
• 5 items on Mental Function.

15 Variables, 108 Complete Dataset.

Physical Scores

Scoring for Physical
“Limited A Lot” “Limited A Little” “Not Limited At All”
1 2 3
Higher score indicate higher physical capability.
• Respondents scored 1 to 3 only.

Physical Function Questions
1
Vigorous activities, such as running, lifting heavy objects,
participating in strenuous sports
2
Moderate activities, such as moving a table, pushing a
vacuum cleaner, bowling, or playing golf
3 Lifting or carrying groceries
4 Climbing several flights of stairs
5 Climbing one flight of stairs
6 Bending, kneeling, or stooping
7 Walking more than a mile
8 Walking several blocks
9 Walking one block
10 Bathing or dressing yourself

Physical Scores
Score 1 2 3 4 5 6 7 8 9 10
1 37 22 9 13 10 12 26 20 21 11
2 61 49 24 52 15 38 52 34 44 14
3 24 47 86 55 92 68 39 63 52 93
Total 122 118 119 120 117 118 117 117 117 118
Higher Score, Better Physical Function
• Minimum score 10 – very unfit, maximum score 30 – very fit.
• These items combined to measure a single factor/latent trait -> Fitness.
Response
Limited a lot
Limited a bit
Not limited

Mental Scores

Scoring for Mental
• Negative 1, 4, 5, 8. Positive 2, 3, 6, 7, 9.
• The 5 questions used in this exercise are the 5
positive ones. The scores are not reversed.
• So higher score indicate better mental health.
All the time Most time A good bit Some time A little time None at all
1 2 3 4 5 6
1 Did you feel full of pep?
2 Have you been a very nervous person?
3 Have you felt so down in the dumps that nothing could cheer you up?
4 Have you felt calm and peaceful?
5 Did you have a lot of energy?
6 Have you felt downhearted and blue?
7 Did you feel worn out?
8 Have you been a happy person?
9 Did you feel tired?
+
+
+

Mental Function
Score 1 2 3 4 5
1 2 2 2 1 6
2 6 6 8 2 7
3 17 6 31 13 30
4 48 35 9 48 15
5 27 27 36 27 31
6 12 37 26 19 22
Total 112 113 112 110 111
Items Nervous In Dump Blue Worn Out Tired
Response
All the time
Most time
A good bit
Some time
A little time
None at all
• Minimum score 5 – poor mental health, maximum score 30 – good mental health.
• These items combined to measure a single factor/latent trait -> Mental Health.

VALIDATION OF INSTRUMENT
Exploratory Factor Analysis

• Download PF-MH-AMOS.sav and open in
SPSS. Delete all the labels, otherwise the
plugins will not work. Save the file and open
AMOS.
• Then follow these instructions -> Next slide.

https://youtu.be/rVrHhCz87YY

Analyze->Dimension Reduction>Factor

• KMO need to be 0.6 or
higher.
• Bartlett’s Test
significant means that
there is more than one
dimension.
Factor Analysis
Amount of variance from 15 items,
extracted by each factor is called
‘eigenvalue’ of each factor.

Factor Analysis - Options
• Easier to view the factor loadings of each
component once we removed the smaller values.
• Both PF & MH clearly split into two groups.

Copy Pattern Matrix Into AMOS

Automatically Drawn CFA

Manually Drawn CFA

https://youtu.be/JkZGWUUjdLg

Unstandardised

Standardised

Run Model Fit Measures

Run Master Validity Plugin

Conclusion
• CFA is used to test the theoretical construct of
these instruments.
• Based on Model Fit Measures & Master
Validity, the theoretical construct of these
instruments have been tested and found
excellent and acceptable.
• Further analysis could be done by following
examples in the attached video.

Introduction to Structural Equation Modeling

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