Structural Equation Model[l]ing (SEM)
‘Structural equation modeling can best be described as a class of methodologies that
seeks to represent hypotheses about summary statistics derived from empirical
measurements in terms of a smaller number of “structural” parameters defined
by a hypothesized underlying model. ... Structural equation modeling represents
a melding of factor analysis and path analysis into one comprehensive statistical
methodology.’
(Kaplan, 2009;p. 1 & 3)
Why Structural Equation Modeling?
• Measurement moves from exploratory to confirmatory factor analysis
• Testing of complex “path” models
• Simultaneous consideration of both measurement and prediction
(Kelloway, 1998; p. 2-3)
Approach to SEM
(Kaplan, 2009; p. 9)
Theory
Model
Specification
Sample and
Measures
Estimation
Assessment
of fit
Modification
Discussion
A Typical SEM ‘Problem’
Appearance
Taste
Food Quality
Regression weights
Portion Size
Correlation
Friendly Employees
Satisfaction
Loadings
Competent Employees
Service Quality
Courteous Employees
= directly observable variable
= latent or unobservable or
construct or concept or
factor
Latent and Observable Variables
RLV
1
X1
2
FLV
3
X2
X3
4
1
X4
The indicators are considered to be influenced,
affected or caused by the underlying LV … a
change in the LV will be reflected in a change in
all indicators …there is a correspondence
between the LV and its indicators (i.e., the
indicators are seen as empirical surrogates for a
LV).
The underlying assumption is that the LV
theoretically exists, rather than being
constructed, and it is manifested through its
indicators. High correlation is expected
amongst the indicators.
The ’s are correlations
X1
2
3
X2
X3
4
X4
The indicators are viewed as causing rather
than being caused by the underlying LV … a
change in the LV is not necessarily
accompanied by a change in all its indicators,
rather if any one of the indicators changes,
then the latent variable would also change.
FLVs represent emergent constructs that are
formed from a set of indicators that may or
may not have theoretical rationale. Interdependence amongst the indicators is not
desired. Low correlation is expected amongst
the indicators.
The ’s are regression weights
RLVs
FLVs
Content validity
√
√
Face validity
√
√
Convergent validity
√
x
Discriminant validity
√
x
Criterion, concurrent and predictive validity
√
√
Test retest reliability, alternative form and scorer
√
√
Internal consistency (split half and Cronbach’s alpha)
√
x
Composite reliability
√
x
Confirmatory FA
√
x
Desirable
Not desirable
Validity (the attribute exists and variations in the
attribute produce variation in the measurement;
measures what it is supposed to measure)
Reliability (degree to which a measure is free from
random error; consistency)
Multicollinearity
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Exogenous (independent) and
Endogenous (dependent) Variables
Measurement or Outer model:
Relationship between the observed
variables and their LVs
1
x1
x2
x3
error1
error2
1
1
1
y1
2
y2
2
y3
y4
Structural or Inner model:
“Theoretically” grounded
relationships between LVs
x4
x5
x6
y5
y6
Mediation and Moderation
error 1
1
Education
Income
error 2
1
Satisfaction
Mediation
error 1
1
Partial Moderation
error 2
Income
1
Education
Full Moderation
error 1
Income
Education
1
Satisfaction
Satisfaction
Covariance (CB-SEM) and Partial
Least Squares (PLS-SEM)
“ML [CBSEM] is theory-oriented and emphasises the transition from
exploratory to confirmatory analysis. PLS is primarily intended for
causal-predictive analysis in situations of high complexity but low
theoretical information”
(Joreskog & Wold, 1982; 270)
“Such covariance-based SEM (CBSEM) focuses on estimating a set of
model parameters so that the theoretiacl covariance matrix implied by
the system of structural equations is as close as possible to the
empirical covariance matrix observed within the estimation sample. ...
Unline CBSEM PLS analysis does not work with latent variables, and
estimates model parameters to maximize the variance explained for all
endogenous constructs in the model through a series of ordinary least
squares (OLS) regressions.”
(Reinartz, Haenlein & Jenseler, 2009; 332)
CBSEM (LISREL, AMOS,
PLS (SmartPLS, PLS
EQS, Mplus)
Graph, XLSTAT)
Strong
‘Flexible’
Multivariate normality
Non-parametric
Large (at least 200)
Small (30-100)
Confirming theoretically
assumed relationships
Prediction and/or
identification of
relationships between
constructs
Number of indicators per construct
Depending on aggregation
(or parceling); ideally 4+
One or more (see
consistency at large)
Indicators to construct
Mainly reflective (can use
MIMIC for formative)
Both reflective and
formative
Depends on model
Always identified
Type of measurement
Interval or ration
(otherwise need PRELIS)
Categorical to ratio
Complexity of model
Large models (>100
indicators) problematic
Can deal with large models
Theory
Distribution assumptions
Sample size
Analytical focus
Improper solutions/factor indeterminacy (unique
solution)
CBSEM (Lisrel, AMOS,
PLS (SmartPLS, PLS
EQS, Mplus)
Graph, XLSTAT)
Consistent if no estimation
problems and confirmation
of assumptions
Consistency at large
(when indicators of each
construct and sample size
reach infinity)
Correlations between constructs
Can be modeled
Cannot be modeled
Correlations between errors
Can be modeled
Cannot be modeled
Available
Available
Structural model
independent of
measurement model
Structural and
measurement model
estimated simultaneously
Goodness-of-fit measures
Available for both for the
overall model and for
individual constructs
Limited for overall model
but available for individual
constructs
Statistical testing of estimates
Benchmarks from normal
distribution
Inference requires resampling (jackknife or
bootstrap) thus the term
‘soft’ modeling
Not directly estimated
Part of the analytical
approach
Possible to test
Possible to test
Available
Available
Parameter estimates
Assessment of measurement model
Estimation
LV scores
Higher order constructs
Response based segmentation
HBAT DATA
CUSTOMER SURVEY - HBAT is a manufacturer of paper products. Data from 100 randomly selected customers were collected on the
following variables.
Classification Variables/Emporographics
X1 - Customer Type: Length of time a particular customer has been buying from HBAT
(1 = Less than 1 year; 2 = Between 1 and 5 years; 3 = Longer than 5 years)
X2 - Industry Type: Type of industry that purchases HBAT’s paper products
(0 =Magazine industry; 1 = Newsprint industry)
X3 - Firm Size: Employee size
(0 = Small firm, fewer than 500 employees; 1 = Large firm, 500 or more employees)
X4 - Region: Customer location
(0 = USA/North America; 1 = Outside North America)
X5 - Distribution System: How paper products are sold to customers
(0 = Sold indirectly through a broker; 1 = Sold directly)
Perceptions of HBAT
Each respondent’s perceptions of HBAT on a set of business functions were measured on a graphic rating scale, where a 10 centimetre line
was drawn between the endpoints labelled “Poor” (for 0) and “Excellent” (for 10).
X6 - Product quality
X7 – E-Commerce activities/Web site
X8 – Technical support
X9 – Complaint resolution
X10 – Advertising
X11 – Product line
X12 – Salesforce image
X13 – Competitive pricing
X14 – Warranty and claims
X15 – New products
X16 – Ordering and billing
X17 – Price flexibility
X18 – Delivery speed
Outcome/Relationship Variables
For variables X19 to X21 a similar to the above scale was employed with appropriate anchors.
X19 – Satisfaction
X20 – Likelihood of recommendation
X21 – Likelihood of future purchase
X22 – Percentage of current purchase/usage level from HBAT
X23 – Future relationship with HBAT (0 = Would not consider; 1 = Would consider strategic alliance or partnership)

Conceptual Model
Delivery speed
Complain resolve
Customer
interface
Order & billing
Purchase level
Re-purchase
Satisfaction
Recommend
Product quality
Product line
Price flexibility
Value for
money
Competitive price
Sales image
E-commerce
Advertising
Market presence
CB-SEM
 Method ... Maximum Likelihood (ML) ... Generalised Least Squares
(GLS) ... Asymptotically Distribution Free (ADF) ...
 Sample size ...
 Multivariate normality ... Outliers ... Influential cases ...
 Missing cases ...
 Computer programs ... LISREL ... AMOS ... EQS ... MPlus ... Stata
... SAS ...
Confirmatory Factor Analysis
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Normality and Influential Cases
Measurement Model
The χ2 goodness of fit statistic between the observed
and estimated covariance matrices [HO : There is no
significant difference between the two matrices, HA :
There is significant difference between the two
matrices]… therefore, ideally we want to retain the
null hypothesis …
Unstandardised estimates (loadings) … can use
C.R. to test whether the estimate is significant
[HO = The estimate is not significantly different
from zero; HA = The estimate is significantly
[higher – or lower … depending on theory] than
zero]
Standardised estimates (loadings) … the size of
the estimate provides an indication of convergent
validity … should be at least > 0.50 and ideally >
0.70
Reliability
The most commonly reported test is Cronbach’s alpha ... but due to a
number of concerns recently report composite reliability ...
( ) 2
c 
( ) 2   var( )
λ is the standardised estimate (loading)
Var(ε) = 1- λ2
(Σ λ)2 =(.949+.919+.799)2
Σvar(ε) = (1-.949)2+ (1-.919)2+ (1-.799)2
which results in a composite reliability of 0.92
Value for money = 0.69
Market presence = 0.83
Customer interface :
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Validity
Convergent validity ... tested by Average Variance Extracted with a
benchmark of 0.50 ... Customer interface = 0.79; Value for money =
0.40; Market presence = 0.64
2
AVE  2
   var( )
Discriminant validity ... off diagonal bivariate correlations should be
notably lower that diagonal which represents the Square Root of AVE
Custo.Int
VFM
Cust.Int.
.889
VFM
.616
.632
Market Pr.
.270
-.061
Market pr.
.800
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Goodness of Fit
There are many indices ... some of the more commonly reported are
(should also consider sample size and number of variables in the
model):
 The χ2 goodness of fit statistic (ideally test should not be
significant … however, rarely this is the case and therefore often
overlooked)
 Absolute measures of fit: GFI > .90 and RMSEA < .08
 Incremental measures of fit (compared to a baseline model
which is usually the null model which assumes all variables are
uncorrelated): AGFI > .80, TLI > .90 and CFI > .90.
 Parsimonious fit (relates model fit to model complexity and is
conceptually similar to adjusted R2): normed χ2 values of χ2 :df
of 3:1, PGFI and PNFI higher values and Akalike information
criteria (AIC) smaller values
21
22
Summary solution after removing x17
23
Structural Model
Model Re-specification/Modification
 Look at residuals.
(benchmark 2.5)
 Look at modification indices
(relationship not in the model
that if added will improve
overall model χ2 value ... MI
> 4 indicate improvement)
Testing improvement in goodness:
Δχ2 = 15.84 – 5.07 = 10.77; Δdf = 7-6 = 1
Testing improvement in goodness:
Δχ2 = 15.84 – 5.07 = 10.77; Δdf = 7-6 =1
Higher Order
PLS-SEM




Method ... Ordinary Least Squares (OLS) ...
Sample size ...
Multivariate normality ... Bootstrap ... Jackknife ...
Missing cases ...
 Computer programs ... SmartPLS ... PLS-GUI ... VisualPLS ...
XLSTAT-Pls ... WarpPLS ... SPAD-PLS ...
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Measurement and Structural Models
An indicator should load
high with the
hypothesised latent
variable and low with the
other latent variables
For each block in the model with more than one manifest variable the quality of the
measurement model is assessed by means of the communality index. The communality of a
latent variable is interpreted as the average variance explained by its indicators (similar to R 2).
The redundancy index computed for each endogenous block, measures the portion of
variability of the manifest variables connected to an endogenous latent variable explained by
the latent variables directly connected to the block.
Testing significance
Since PLS makes no assumptions (e.g., normality) about the distribution of the parameters, either of the
following res-sampling approaches are employed.
 Bootstrapping • k samples are created of size n in order to obtain k estimates for each parameter.
• Each sample is created by sampling with replacement from the original data set.
• For each of the k samples calculate the pseudo-bootstrapping value.
• Calculate the mean of the pseudo-bootstrapping values as a proxy for the overall
 “population” mean.
• Treat the pseudo-bootstrapping values as independent and randomly distributed and calculate their
standard deviation and standard error.
• Use the bootstrapping t-statistic with n-1 degrees of freedom (n = number of samples) to test the
null hypothesis (significant of loadings, weights and paths).
• Jack-knifing • Calculate the parameter using the whole sample.
• Partition the sample into sub-samples according to the deletion number d.
• A process similar to bootstrapping is followed to test the null hypotheses
Recommendation – Use bootstrapping with ‘Individual sign changes’ option, k = number of valid observations
and n = 500+
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38
39
Model Evaluation
Predictive Power - R2: The interpretation is similar to that employed
under traditional multiple regression analysis, i.e. indicates the
amount of variance explained by the model. Examination of the
change in R2 can help to determine whether a LV has a substantial
effect (significant) on a particular dependent LV.
The following expression provides an estimate of the effect size of f2
and, using the guidelines provided by Cohen (1988), interpret an f2
of .02, .15 and .35 as respectively representing small, medium and
large effects.
2
2
R

R
f 2  included 2 excluded
1  Rincluded
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Removing the customer
interface → purchase
pathway results in an R2
of purchase of .585 ... f2
= (.664-.585)/(1-.664) =
0.23 which is a medium
to large effect.
In addition to the
significance we
should also examine
the relevance of
pathways ... > .20
coefficient
42
Predictive Relevance - Q2 [Stone, 1974; Geisser, 1975]: This relates to the
predictive sample reuse technique (PSRT) that represents a synthesis of
cross-validation and function fitting. In PLS this can be achieved through a
blindfolding procedure that “… omits a part of the data for a particular
block of indicators during initial parameter estimation and then attempts to
estimate the omitted part of the data by using the estimated parameters”.
Q2 = 1 – ΣE/ΣO
Where: ΣE = Sum square of prediction error [Σ(y – ye)2] for omitted data and
ΣO = Sum square of observed error [Σ(y – y)2] for remaining data
Q2 > 0 implies that the model has predictive relevance while Q2 < 0 indicates a
lack of predictive relevance.
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Higher Order
If the number of indicators for each of your two constructs are
approximately equal can use the method of repeated manifest variables.
The higher order factor that represents the two first order constructs is
created by using all the indicators used for the two first order constructs.
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Readings
Arbuckle, J. L. and Wothke, W. (1999), Amos 4.0 User’s Guide.
Chicago:Small Waters Corporation
Byrne, B.M. (2010), Structural Equation Modeling with AMOS, 2nd ed.,
New York:Toutledge
Hair, J. F., Anderson, R. E., Tatham, R.L. and Black, W. C. (1998),
Multivariate Data Analysis, 5th ed., New Jersey:Prentice Hall
Hair, J.F., Hult, G.T.M., Ringle, C.M. and Sarstedt, M. (2014), A
Primer on Partial Least Squares Structural Equation Modeling
(PLS-SEM), London:Sage Publ.
Kaplan, D. (2009), Structural Equation Modeling: Foundations and
Extensions, 2nd ed., London:Sage Publ.
Vinzi, V.E., Chin, W.W., Hensler, J. and Wang (eds) (2010), Handbook
of Partial Least Squares, London:Springer
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