### Stages in Structural Equation Modeling

```The Stages in
Structural Equation Model
 Stage 1) Assessing individual constructs
 Stage 2) Developing and assessing the
measurement model validity
 Stage 3) Specifying the structural model
 Stage 4) Assessing structural model validity
Stage 1:Assessing Individual constructs
 One of the most important steps is operationalization of
constructs (Hair, 2006).
 In attempting to ensure theoretical accuracy, researchers
many times have a number of established scales to choose
from. Nonetheless, eve with the wide usage of scales, the
researcher often is faced with the lack of an established
scale and must develop a new scale or substantially
modify an existing scale to new context.
 Therefore, in all of these conditions, the foundation for the
SEM analysis is how selects the items to measure the
constructs (Hair et al., 2006).
Type of Individual construct
Type1: Simple Individual Construct
Type3: Construct with 3
dimension based on
Second –Order CFA
Type2: Individual Construct with 3 dimension
First –Order CFA
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e7
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Single factor
F2
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X1
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Y2
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X1
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Independent Model
F1
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F2
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Interrelated model
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Goodness-of-Fit Criteria
 Goodness-of-Fit measures can be classified into three types (Ho, 2006):
(1) Absolute fit measures: These measures determine the degree to which the
proposed model predicts (Fits) the observed covariance matrix. Some commonly
used measures of absolute fit such as:
a) Chi-square: In SEM, the researcher is looking for significant differences between
the actual and predicted matrices. As such, the researcher does not wish to reject
the null hypothesis and, accordingly, the smaller the chi-square value, the better fit
of the model.
b) Normed Chi-square/df: Values close to 1.0 indicate good fit. values between 2.0
and 3.0 indicate reasonable fit.
c) Goodness-of-Fit Index(GFI): value > 0.90
d) Root Mean Square Error of Approximation(RMSEA); Hair, et al., (2006; p748)
recommended RAMSEA between .03 and .08. Kline (2011, p 206 and 2005, p139)
RAMSEA < .05 indicates close approximate fit, between .05 and .08 reasonable, and
RAMSEA > .10 suggest poor fit (cited from Browne & Cudek, 1993)
-e) Standardised Root Mean-square Residual (SRMR): values less than .10 are
generally considered favorable (Kline, 2005, p141)
Goodness-of-Fit Criteria
(2) Incremental fit measures: These measures compare the proposed
model to some baseline model, most often referred to as the
null or independence model. In the independence model, the
observed variables are assumed to be uncorrelated with each
other. Incremental fit measures have been proposed, such as:
-Tucker-Lewis Index (TLI) -Normed Fit Index (NFI)
-Relative Fit Index (RFI) -Incremental Fit Index (IFI)
-Comparative Fit Index (CFI)
 By convention, researchers have used incremental fit indices > 0.90 as
traditional cutoff values to indicate acceptable levels of model fit. the model
represents more than 90% improvement over the null or independence
model. In other words, the only possible improvement to the model is less
than 10%.
Goodness-of-Fit Criteria
(3) Parsimonious fit measures: In scientific research, theories
should be as simple, or parsimonious, as possible (Ho, 2006).
parsimonious fit measures relate the goodness-of-fit of the proposed
model to the number of estimated coefficients required to achieve
this level of fit. Such as:
 Parsimonious Normed Fit Index (PNFI): When comparing
between models, differences of 0.06 to 0.09 are proposed
to be indicative of substantial model differences (Williams
& Holahan, 1994).
 Akaike Information Criterion (AIC): The AIC is a
comparative measure between models with differing
numbers of constructs. AIC values closer to zero indicate
better fit and greater parsimony. A small AIC generally
occurs when small chi-square values are achieved with
fewer estimated coefficients.
Detecting outliers
 The Mahalanobis D2 statistics is commonly used to detect case-wise
outliers. It computes the overall centroid and computes the distance
of each observation from the centroid.
 The Mahalanobis D2 is known to follow a Chi-square distribution.
The degree of freedom is the number of variables that are being
tested.
 Usually the inverse chi-square value at a=0.001 is taken as the trash
hold value.
 For example there are 2 variables, then df =2
 Inverse Chi-squared value for a=0.001 at a df of 2 is: 13.186
 If the highest value in the MAH_1 column is 6.784, which is less
than 13.186. Thus, there are no major outliers in the file.
different method to handle missing values
 Listwise deletion: means a subject with missing values is
deleted in all calculations.
 Pairwise: means it is deleted subjects with missing data just
for calculations comprising that variable.
In using the Listwise deletion and Pairwise deletion may be
losing a large number of cases, and then could reduce the sample
size, therefore these two methods are not always suggested
(Schumacher & Lomax, 2010).
 The SEM softwares the option for handling the missing values
are replacing the missing value with means when only a
small number of missing values is present in the data set, or
employed regression imputations for handle missing values
when a moderate amount of missing data present in the data
(Schumacher & Lomax, 2010; Byrne, 2010).
Examples and hands on
For
Individual construct CFA
Stage 2) Developing and assessing the
measurement model validity
 While specified the scale items, it is essential to specify the
measurement model. In this stage, each latent construct to
be included in the model is identified and the measured
indicator variables (items) are assigned to latent constructs
(Hair et al., 2006).
 The measurement models specify how the latent variables
are measured in terms of the observed variables. In the
other word the measurement models are concerned with the
relations between observed and latent variables (Ho, 2006).
The Criteria's for Assessment
Goodness-Of-Fit (GOF) Indices
 Assessing the individual constructs, measurement model validity and structural
model validity depends on a number of Goodness-Of-Fit (GOF) indices
 Goodness-of-fit measures the extent to which the actual or observed covariance
input matrix corresponds with (or departs from) that predicted from the proposed
model (Ho, 2006).
 GOF measures such as :
Chi-Square (non sig), Goodness of Fit Indicator (GFI), Adjusted Goodness of Fit Indicator (AGFI),
Comparative Fit Index (CFI), Normed Fit Index (NFI), and Tucker Lewis Index (TLI) indicates a
good fit to the model at about .9 or greater.
Root Mean Square Error of Approximation (RMSEA) which a measure greater than .1 indicates a poor
fit, values ranging between 0.08 to 0.1 indicate mediocre fit, and values ranging between 0.03 and
0.08 are indicate better fit model.
The Criteria's for Assessment
Construct validity
Construct validity is the extent to which a set of measured items actually reflected the
theoretical latent construct those item are designed to measure. Thus, it deals with
the accuracy of measurement.
Construct validity is made up of four important components which they are:
1) Convergent validity: the items that are indicators of a specific construct should
converge or share a high proportion of variance in common, known as
convergent validity. The ways to estimate the relative amount of convergent
validity among item measures:
A good rule of thumb is that standardized loading estimates should be .5 or
higher, and ideally .7 or higher.
Variance Extracted (VE): is the average squared factor loading. A VE of .5 or
higher is a good rule of thumb suggesting adequate convergence. A VE less than
.5 indicates that on average, more error remains in the items than variance
explained by the latent factor structure impose on the measure (Haire et al.,
2006, p 777).
Construct Reliability: construct reliability should be .7 or higher to indicate
The Criteria's for Assessment
Construct validity
2) Discriminant Validity: the extent to which a construct is truly
distinct frame other construct. To test the discriminant validity
the VE for two factors should be grater than the square of the
correlation between the two factors to provide evidence of
discriminant validity.
3) Nomological Validity: is tested by examining wheather the
correlation among the constructs in a measurement theory
make sense. Constructs of interest should be related to other
constructs according to hypothesised ways derived from the
theory in which the construct is embeded, forming the
nomological net for that set of constructs (The matrix of
correlations can be useful in this assessment.
4) Face Validity: must be established prior to any theoretical
testing when using CFA. Without an understanding of every
item’s content or meaning. It is impossible to express an
correctly specify a measurement theory.
Examples and hands on
For
Measurement Model
Stage 3)Specifying the structural model
 Once the measurement model is specified and validated
with CFA, then the structural model represented by
specifying the set of relationships between constructs.
 The structural equation model is a comprehensive model
that specifies the pattern of relationships among
independent and dependent variables, either observed or
latent (Hair, et al., 2010; Ho, 2006; Landis et al., 2001).
Stage 4) Assessing structural model validity
 The assessing structural model validity focused
on two issues comprising;
 a) the overall and relative model fit, and
 b) the size, direction, and significance of the
structural parameter estimates, depicted with
one-headed arrows on a path diagram (Hair et
al., 2006).
Accounting for Error
 Thus, SEM provides estimates for two types of error variance
– error terms and residual terms:
 Error term: represents measurement error associated with
observed variables, i.e. the degree to which the observed
variables do not perfectly describe the latent construct of
interest (Hair, Black et al. 2006). Such measurement error
terms represent causes of variance due to unmeasured variables
as well as random measurement error (Garson 2006).
 Residual term: represents the error in the prediction of
endogenous factors from exogenous factors, that is to say, how
much variance in the associated endogenous variable was not
accounted for by influences in the model (Texas 2002).
Examples and hands on
For
Structural Model
Approaches to aggregation
 Total disaggregation: In a totally disaggregated model, each
item serves as an indicator for a construct.
 Partial disaggregation: In a partially disaggregated model,
several items are summed or averaged resulting in parcels.
These parcels are then used as indicators for constructs.
 Total aggregation: In a totally aggregated model, all of the
items for a scale are summed or averaged. The result is that if
only one scale is used to measure each construct, then there is
only one indicator per construct and the model is a path
analysis rather than a latent variable model.
If more than one scale was used to measure a construct, then it is
still possible to specify a latent variable, and each indicator is a
total scale score (Coffman and Maccall.um, 2005).
Parceling
 Parcels are aggregations (sums or averages) of several individual
items.
 Advantages of parcels (Coffman and Maccall.um, 2005) :
1) parcels generally have higher reliability than single items(Kishton
& W idaman, 1994).
2) A second advantage of using parcels rather than items as indicators
of latent variables involves the reduction in the number of
measured variables in a model. From this perspective, models with
parcels as indicators are likely to fit better than models with items
as indicators because the order of the parcel correlation matrix is
much smaller than the order of the item correlation matrix.
3) They can be used as an alternative to data transformations or
alternative estimation techniques when working with nonnormally
distributed variables. The most often used estimation method in
structural equation modeling, maximum likelihood, assumes
multivariate normality of the measured variables in the population.
If the measured variables are not multivariate normal, then
estimates of fit measures and estimates of standard errors of
parameters may not be accurate (Hu & Bentler, 1998).
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