If you are not familiar with SEM it is strongly recommended you work your way
through the
Path
Analysis tutorial and example prior to working through this tutorial and
example.
Continuation of the previous tutorial notes regarding SEM.
**XIII. Structural Equation Modeling****
(SEM) using PROC CALIS**
First, let's take a moment to review our **fictional*** model.
Our model consists of 13 manifest variables which are assumed to measure four
latent factors.
The first latent factor, Personality, is assumed to be measured by Extroversion (extro),
Openness (open), and Agreeableness (agree). The second latent factor,
Engagement, is assumed to be measured by Social Engagement (social), Cognitive
Engagement (cognitive), Physical Engagement (physical), and Cultural Engagement
(cultural). The third latent factor, Crystallized Intelligence,
is assumed to be measured by the established tests of Vocabulary (vocab),
Abstruse Analogies (abstruse), and Block Design (block). The fourth latent
factor, Fluid Intelligence, is assumed to be measured by the established tests
of Common Analogies (common), Letter Sets (sets), and Letter Series (series).
The general research question for our fictional longitudinal study concerns
whether or not certain personality traits cause persons to lead an engaged
lifestyle, and do these personality traits and leading an engaged lifestyle
prevent loss of cognitive functioning in late life (e.g. beyond 65 years of
age).
***Again; this is a fictional example; it includes simulated data and is not meant to be taken seriously as
a research finding supported by empirical evidence.** It is merely used here
for instructional example purposes.
If you are unfamiliar with standard path and structural equation models; there
are a few things you should take note of in our diagram that tend to be
seen in published materials displaying path models and structural equation
models. First, the use of squares or rectangles to denote observed or measured
variables (often referred to as manifest variables or indicator variables). Second, the use of
circles or ellipses to denote unobserved or latent variables (often referred to
as latent factors). Third, the use of straight, single headed arrows to denote causal relationships.
There are two types of causal relationships shown in most diagrams. The
assumed causal relationship between a latent factor and its indicator or
manifest variables which are often referred to as loadings or
factor loadings or factor coefficients. The hypothesized causal relationship between two latent factors
which are often referred to as paths or path loadings or path coefficients. And fourth, the use of curved, double-headed
arrows to refer to bi-directional relationships (often referred to as correlations or covariances). Specific hypotheses should be used to clarify what
the researcher expects to find (e.g. a positive bi-directional relationship or correlation
between Crystallized & Fluid Intelligences).
From this point forward, we will use the term **loading **to refer to a relationship
between a manifest variable and a latent factor. We will use the term **path** to
refer to the relationship between two latent factors.
Something to consider when conducting SEM is the recommendation of using a two
stage process to conduct the SEM (Anderson & Gerbing, 1988). When using two
stages, the first stage is used to verify the measurement model. Verification of
the measurement model in SEM can be considered analogous to conducting a
confirmatory factor analysis. The purpose is to ensure you are measuring
(adequately) what you believe you are measuring. Essentially, you are verifying
the factor structure. When verifying the measurement model, you are more
concerned with the error variances and the loadings than with the relationships
between latent factors (paths). Recall, the loadings are those which
represent the relationships between a factor and its manifest indicator
variables. The second stage involves actually testing the hypothesized causal
relationships between factors, also referred to as testing the structural model,
which is where you are interested more in the paths than the loadings. Recall,
the paths represent the hypothesized causal relationships between latent
factors.
# Anderson, J. C., & Gerbing, D. W. (1988). Structural equation modeling in
# practice: A review and recommended two-step
approach. *Psychological *
# *Bulletin, 103,* 411 - 423.
Below you will see the verified Measurement Model from the previous tutorial
Stage
1.
**Stage II: Testing the Structural model. **
Below you'll find a diagram which represents our structural model. Notice we
are now concerned with the relationships between latent factors, those paths
will be estimated. Also notice, the variances of
the factors are no longer fixed at 1. This time, we must fix the largest one of loadings
from each of the factors to 1 in order to set the scale of the factors and
estimate their variances (and disturbance terms). Disturbance terms (D2, D3, D4)
are new here
and represent the error associated with each factor which can be thought of as
any causal influence for that factor which was not included in the model. Also
notice there is no disturbance term specified for our one exogenous factor.
Disturbance terms are only specified for endogenous factors.
One of the key requirements of Path Analysis and SEM is overidentification. A model is said to be overidentified if it contains more
unique inputs (sometimes called informations) than the number of parameters
being estimated. In our example, we have 13 manifest variables. We can apply
the following formula to calculate the number of unique inputs:
(1)
number of unique inputs = (p ( p + 1 ) ) / 2
where p = the number of manifest variables. Given this formula and
our 13 manifest variables; we calculate 91 unique inputs or informations which
is greater than the number of parameters we are estimating. Looking at the
diagram, we see 1 covariance (C?), 9 loadings (L?), 5 paths (P?), 3 disturbance
terms (VAR?), 1 factor variance (VAR?) and 13 error
variances (VAR?). Adding these up, we get 32 parameters to be estimated.
You'll notice, as discussed above, that for our structural model, we have fixed
the largest loading for each latent factor to be 1 (L = 1). This is done to allow estimation of
the disturbance terms and the first factor's variance. Also notice that we can
no longer estimate the correlation/covariance between F3 and F4 (as was done
when verifying the measurement model), instead we estimate the correlation (C?)
between the disturbance terms of those two endogenous factors. Remember too that SEM requires
large sample sizes. Several general rules have been put forth as lowest
reasonable sample size estimates; at least 200 cases at a minimum, at least 5
cases per manifest or measured variable, at least 400 cases, at least 25 cases
per measured variable, 5 observations or cases per parameter to be estimated, 10
observations or cases per parameter to be estimated...etc. The bottom line is
this; SEM is powerful when done with adequately large samples -- the
larger the better. Another issue related to sample size, is the recommendation
of having at least 3 manifest variables for each latent factor (Anderson & Gerbing,
1988). Another consideration is that of remaining realistic when
setting out to study a particular phenomena with SEM in mind as the analysis. It
is often easy to develop some very sophisticated models containing a great number of
manifest and latent variables. However, complex models containing more than 20 manifest
variables can lead to confusion in interpretation and a lack of fit, as well as
convergence difficulty. Bentler and Chou (1987) recommend a limit of 20 manifest
variables.
# Bentler, P. M., & Chou, C. (1987). Practical issues in structural modeling. *
Sociological *
# *Methods & Research, 16,* 78 - 117.
The procedure for conducting SEM in SAS is PROC CALIS;
however, PROC CALIS needs to have the data fed to it. Here we will use the
correlation matrix with number of observations and standard deviations as was
done previously.
Using the the same values from the previous tutorial (number of observations, standard deviations,
& correlation matrix) when verifying the Measurement Model; we can proceed to the
Structural Model.
The syntax for testing our Structural Model is displayed below.
Note that the top half of the syntax simply enters the data and is identical to
what was used in verifying the Measurement Model. The second half
(beginning with PROC CALIS) is substantively different from that used in
verifying the Measurement Model; and this new syntax is used to test the
Structural Model. Notice we have left out the option for requesting the
modification indices.
**DATA sem1(TYPE=CORR);**
INPUT _TYPE_ $ _NAME_ $ V1-V13;
LABEL
V1 = 'extro'
V2 = 'open'
V3 = 'agree'
V4 = 'social'
V5 = 'cognitive'
V6 = 'physical'
V7 = 'cultural'
V8 = 'vocab'
V9 = 'abstruse'
V10 = 'block'
V11 = 'common'
V12 = 'sets'
V13 = 'series'
;
CARDS;
N . 750 750 750 750 750 750 750 750 750 750 750 750 750
STD . 9.0000 6.0000 5.2500 15.0000 7.5000 3.7500 11.2500 8.2500 3.0000 3.0000
6.7500 12.0000 7.5000
CORR V1 1.0000 . . . . . . . . . . . .
CORR V2 .3385 1.0000 . . . . . . . . . . .
CORR V3 .3056 .3388 1.0000 . . . . . . . . . .
CORR V4 .1196 .1842 .2111 1.0000 . . . . . . . . .
CORR V5 .1889 .1970 .1691 .3685 1.0000 . . . . . . . .
CORR V6 .1475 .2099 .1926 .3234 .4054 1.0000 . . . . . . .
CORR V7 .1932 .2264 .1664 .3044 .3044 .3254 1.0000 . . . . . .
CORR V8 .1070 .1755 .1349 .1967 .1843 .1495 .1951 1.0000 . . . . .
CORR V9 .1563 .2082 .1724 .2056 .1290 .1905 .1971 .3659 1.0000 . . . .
CORR V10 .2076 .2207 .2062 .2196 .2012 .2116 .2223 .3383 .3725 1.0000 . . .
CORR V11 .1900 .1784 .1298 .1168 .1816 .1581 .1841 .1483 .1996 .1932 1.0000 . .
CORR V12 .1443 .1288 .0980 .1573 .2387 .2020 .1554 .1452 .2484 .1819 .3399
1.0000 .
CORR V13 .2136 .1707 .1571 .1589 .1821 .1364 .1940 .1752 .2522 .2316 .3765 .3437
1.0000
;
PROC CALIS COVARIANCE CORR RESIDUAL;
LINEQS
V1 = LV1F1 F1 + E1,
V2 = F1 + E2,
V3 = LV3F1 F1 + E3,
V4 = LV4F2 F2 + E4,
V5 = F2 + E5,
V6 = LV6F2 F2 + E6,
V7 = LV7F2 F2 + E7,
V8 = LV8F3 F3 + E8,
V9 = F3 + E9,
V10 = LV10F3 F3 + E10,
V11 = LV11F4 F4 + E11,
V12 = LV12F4 F4 + E12,
V13 = F4 + E13,
F2 = PF2F1 F1 + D2,
F3 = PF3F1 F1 + PF3F2 F2 + D3,
F4 = PF4F1 F1 + PF4F2 F2 + D4;
STD
F1 = VARF1,
D2 = VARD2,
D3 = VARD3,
D4 = VARD4,
E1-E13 = VARE1-VARE13;
COV
D3 D4 = CD3D4;
VAR V1-V13;
RUN;
The PROC CALIS statement is followed by options. First, COVARIANCE tells SAS we
want to use the covariance matrix to perform the analysis. Even though we are
using the correlation matrix as our data input, SAS calculates the covariance
matrix for the PROC CALIS. That is why the number of observations and standard
deviations must be included with the correlations. The CORR option specifies that we want the output to
include the correlation matrix or covariance matrix on which the analysis is
run. The RESIDUAL option allows us to see the absolute and standardized
residuals in the output. The next part of the
syntax, LINEQS, provides SAS with the specific linear equations which specify
the loadings and paths we want estimated. The first of which can be read as: variable 1
equals factor 1
and the error variance associated with variable 1. In the second lineqs line;
you'll see how we have fixed the loading of V1 to F1 at 1.00. When a path or
loading is left our, CALIS fixes it as 1.00. Also notice the last lineqs
statement which can be read as factor 4 equals the estimated path between factor
4 and factor 1 plus the estimated path between factor 4 and factor 2, and the
error (disturbance term) associated with factor 4. Next, we see the STD lines which
specify which variances we want estimated. The variance of our only exogenous
factor (factor 1) is listed as VARF1. The variable error variances are being
estimated as VARE1 through VARE13 and the disturbance terms are listed as VARD2
through VARD4. Last, the COV statements specify all the covariances which need to be
estimated; here there is only one, which reflects the correlation between the
disturbance terms for factor 3 and factor 4. Then, the VAR line simply lists the variables to be used in the
analysis; V1 through V13. When creating syntax on your own; be very careful and
attentive as to where semi colons are placed.
The first page of the PROC CALIS output (below) consists of general information,
including the number of endogenous variables (any variable *with* a straight
single-headed arrow pointing at it) and the number of exogenous variables (any
variable *without* any straight single-headed arrows pointing to it).
The second page of the PROC CALIS output consist of a listing of the loading parameters
to be estimated (those associated with our manifest variables); essentially a review of the
top part of specified model from the CALIS
syntax. Again; we can see that each factor has one loading fixed at 1.00.
The third page of the output contains a listing of the latent variable
parameters to be estimated, the variances, and the covariances to be estimated.
The fourth page and fifth pages show the general components of the model (e.g. number of
variables, number of informations, number of parameters, etc.); as well as the
descriptive statistics and covariance matrix for the variables entered in the
model. The covariance matrix starts on the fourth page and continues onto
the fifth page.
The 6th page provides the initial parameter estimates.
The 7th page includes the iteration history. Often it is important to focus on
the last line of the Optimization results (left side of the bottom of the page)
which states whether or not convergence criterion was satisfied.
The 8th and 9th pages contain the predicted covariance matrix, which is used for
comparison to the matrix of association (original covariance matrix) to produce
residual values.
The 10th page displays fit indices. As you can see, a fairly comprehensive list
is provided. Please note that although Chi-square is displayed it should not be
used as an interpretation of goodness-of-fit due to the large sample sizes
necessary for SEM (which inflates the chi-square statistic to
the point of meaninglessness). Some of the more commonly reported fit indices
are the RMSEA (root mean square error of approximation), which when below .05
indicates good fit; the Schwarz's Bayesian Criterion (also called BIC; Bayesian
Information Criteria), where the smaller the value (i.e. below zero) the better the fit; and the Bentler & Bonnett's Non-normed Index (NNFI) as well as the Bentler & Bonnett's
normed fit index (NFI)--both of which should be greater than .90 and above to
indicate good fit.
Page 11 and 12 provide the **Raw** residual matrix and the ranking of the 10 largest
**Raw** residuals.
The 13th and 14th pages show the **Standardized** residual matrix and the 10 largest **
Standardized** residuals; we expect values close to zero which indicates good
fit. Any values greater than |2.00| indicates lack of fit and should be
investigated.
This is really the heart of evaluating goodness of fit; if fit is truly good,
you would see virtually no difference between the original covariance matrix and
the predicted covariance matrix (i.e. each of the residuals would be zero). Here
in this example, we have a few values greater than |2.00| which is a bit
worrisome.
The 15th page displays a sideways histogram of the distribution of the **
Standardized** residuals. Generally we expect to see a normal distribution of
residuals with no values greater than |2.00|. Here we see a couple of values in
each tail of the distribution of residuals which are slightly larger than we
would like to see.
The 16th page displays our loadings (coefficients) in Raw form, as well as *t*-values
and standard errors for the *t-*values associated with each. Remember
that *t*-values for coefficients (here loadings) are statistically significant (*p*
< .05, two-tailed) if their absolute value is greater than 1.96; meaning they
are significantly different from zero. If the *t-*value is greater than
2.58, then *p *< .01 and if the *t-*value is greater than 3.29, then
*p *< .001. It is also recommended that a review of the standard errors be
performed, as extremely small standard errors (those very close to zero) may
indicate a problem with fit associated with one variable being linearly
dependent upon one or more other variables. Here, all our *t*-values for
our factor loadings are greater than 3.29 and none of our standard errors are
noticeably low (i.e. less than .0099 for instance).
We also notice the four loadings we fixed at 1.000 (V2, V5, V9, & V13).
On the 17th page, we see estimated paths in raw scale, estimated variance
parameters and the estimated covariance between our F3 and F4 disturbance terms;
each with *t*-values and standard errors for the *t-*values. All of
our *t*-values were greater than 3.29 and therefore, significantly
different from zero (*p* < .001). Notice at
the top the variance estimates we find F1 as asked for in the syntax. This was
the only factor we estimated variance for because, it was our only exogenous
factor.
The 18th page provides **Standardized factor loadings** (coefficients; analogous to
β). These are what get
reported both in text descriptions and in figures/diagrams. You will also notice
they are identical to what we found in our verification of the Measurement Model
in the previous tutorial.
The 19th page displays our **standardized path estimates** (also analogous to
β and reported in text
descriptions and diagrams) and the 'Squared Multiple Correlations'. The R-square
column gives us an idea of how well our manifest variables reflect the latent
factors because, these values are interpreted as the percentage of variance
accounted for in our manifest variables by their respective factors. As an
example; we could interpret V1 (Extroversion) as having 30.004% of its variance
accounted for by F1 (Personality). You'll notice, the R²
value is simply the square of the standardized loading (e.g. .5481²
= .3004). Furthermore, if we square our E1 term (.8364; previous page), we get .6996,
which when added to the variance accounted for by the factor (.3004) gives us
the complete standardized variance of our V1 variable: 1.00. This confirms our
classical test theory perspective of observed score = true score (the amount of
the latent factor) + error.
If we had asked for modification indices, the 20th page would begin displaying
them. However we did not specify them here because, as discussed previously they
are based on chi-square and chi-square does not provide an honest measure of
goodness-of-fit.
Below you will find our completed Structural Model diagram with standardized
path coefficients.
Notice, the disturbance terms do not get flagged, even though we did have *t*-values
for them.
This concludes the tutorial on Structural Equation Modeling.
**Please realize,**
this tutorial is not an exhaustive review, merely an introduction. It is not
meant to be a replacement for one or several good textbooks. |