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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.
XIII. Structural Equation Modeling
(SEM) using PROC CALIS
First, let's take a moment to discuss and describe 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).
Please note: 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.
Stage I: Verifying the Measurement model.
Below you'll find a diagram which represents our measurement model. Notice we
are unconcerned with the relationships between latent factors, those paths are
free to vary (estimating the correlations or covariances) without specifying
hypothetical causal relationships between them. Also notice, the variances of
the factors are fixed at 1. Remember, the latent factors are unobserved and
therefore we do not have an idea of their variance, nor their scale. Another
thing to notice is that each observed score (i.e. manifest variable) is caused
by a latent factor which we believe we are measuring indirectly, and measurement
error. These error terms are shown in the diagram with arrows pointing toward
the manifest variables as classical test theory suggests (i.e. observed score =
true score + measurement error). This bears mentioning because it is one of the
reasons SEM is so popular; it allows us to model measurement error.
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 6 covariances (C?), 13 loadings (L?), and 13 error
variances (VAR?). Adding these up, we get 32 parameters to be estimated.
You'll notice that for our measurement model, we have specified the variance of
the latent factors to be 1 (VAR=1). This is done to allow estimation of all the
factor loadings. Remember too that path analysis and SEM require 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; path analysis and SEM are 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; with the suggestion of having 4 or more manifest variables for
each latent factor (Anderson & Gerbing, 1988). Having 4 allows you the
flexibility to delete one if you find it does not contribute meaningfully to a
latent factor or the model in general (e.g. it loads on more than one factor to
a meaningful extent). 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 complex models containing a great number of
manifest 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. There are three ways to
'feed' PROC CALIS the data, (1) a correlation matrix with the number of
observations and standard deviations for each variable, (2) a covariance matrix,
and (3) use of the raw data as input. Here we will use the correlation matrix
with number of observations and standard deviations. Use the Import Wizard to
import the
SEM Data file using the SPSS File (*.sav) source option and
member name semd.
Once imported, you can get the descriptive statistics and
correlations which you will need to run the SEM analysis.
PROC CORR DATA=semd;
RUN;
Note, there are 3 pages of output for the PROC CORR. The first page shows the
descriptive statistics, variable names, and variable labels.
The second and third pages show the correlation matrix.
Using the number of observations (n = 750), the standard deviations, and
the correlation matrix, you can proceed to the SEM.
The syntax for estimating or fitting our Measurement Model is displayed below.
Note that the top half of the syntax simply enters the data; the second half
(beginning with PROC CALIS) is used to conduct the SEM.
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 MODIFICATION;
LINEQS
V1 = LV1F1 F1 + E1,
V2 = LV2F1 F1 + E2,
V3 = LV3F1 F1 + E3,
V4 = LV4F2 F2 + E4,
V5 = LV5F2 F2 + E5,
V6 = LV6F2 F2 + E6,
V7 = LV7F2 F2 + E7,
V8 = LV8F3 F3 + E8,
V9 = LV9F3 F3 + E9,
V10 = LV10F3 F3 + E10,
V11 = LV11F4 F4 + E11,
V12 = LV12F4 F4 + E12,
V13 = LV13F4 F4 + E13;
STD
F1 = 1,
F2 = 1,
F3 = 1,
F4 = 1,
E1-E13 = VARE1-VARE13;
COV
F1 F2 = CF1F2,
F1 F3 = CF1F3,
F1 F4 = CF1F4,
F2 F3 = CF2F3,
F2 F4 = CF2F4,
F3 F4 = CF3F4;
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 MODIFICATION option tells SAS to print the
modification indices (e.g. Lagrange Multiplier Test). The next part of the
syntax, LINEQS, provides SAS with the specific linear equations which specify
the loadings we want estimated. The first of which can be read as: variable 1
equals factor 1
and the error variance associated with variable 1. Next, we see the STD lines which
specify which variances we want estimated. Notice here we have fixed the
variance of each factor at 1, as discussed above. The error variances are being
estimated as VARE1 through VARE13. Last, the COV statements specify all the covariances which need to be
estimated. Then, the VAR line simply lists the variables to be used in the
analysis; V1 through V13. Pay particular attention to the location of commas and
semi-colons.
*Please note; the first 3 pages of output was produced by the PROC CORR directly
after importing the data (above). Therefore, the references to page numbers of
output associated with the PROC CALIS will begin on the fourth page (p. 4) of
the total output file (e.g. page 1 of the PROC CALIS output actually has the
number 4 in the top right corner). The page number discrepancy is noted here
because all PROC CALIS procedures tend to produce several pages of output.
The first page of the PROC CALIS output 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). An important
point to notice here is that the factors are listed as exogenous variables, even
though they have arrows pointing to them, those arrows are double headed and
indicate relationship without causation. Meaning, the factors may be related,
but are not caused by anything specified in our model (more specifically by our
line equations). This may help clarify the use of the word exogenous --
which in SEM terminology refers to variables which have causes outside
the model.
The second and third pages of the PROC CALIS output consist of a listing of the parameters
to be estimated; essentially a review of the specified model from the CALIS
syntax.
The fourth 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 (p. 7) and continues onto
the fifth page (p. 8).
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).
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|.
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).
On the 17th page, we see estimated variance parameters and estimated "Covariances"
(which are really correlations);
each with t-values and standard errors for the t-values. Notice at
the top the variance estimates table, our factor variances are fixed (from our syntax) at 1.0000.
The 18th page provides Standardized factor loadings (coefficients; analogous to
β) and squared multiple
correlations. The 'Squared Multiple Correlations' 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 loading (e.g. .5481²
= .3004). Furthermore, if we square our E1 term (.8364), 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.
The 19th page again shows us the correlations between our factors (as was
displayed previously on the the bottom of the 17th page).
The 20th page begins the listing of the modification indices, which continues to
the end of the output. One should be careful when interpreting modification
indices and should do so only after carefully interpreting all the previous
output first. Modification indices generally take two forms; ones which
recommend the exclusion of a parameter from the specified model (the Wald Test) and ones which
recommend inclusion of a parameter to the model (the Lagrange Multiplier). Both types attempt to estimate
the change in chi-square associated with the recommendation being implemented
(i.e. increased goodness of fit). However, as mentioned above, chi-square is
generally not an acceptable measure of goodness-of-fit and therefore
modification indices should be treated with caution. For this reason, the
remaining pages of output (displaying modification indices) will not be shown
here.
The completed diagram of the verified Measurement Model is below. Note,
typically standardized factor loadings are reported (rather than
un-standardized) and error terms are generally not reported in diagrams -- as is
the case below.
Generally speaking, the output for any PROC CALIS will follow the same format
seen here for SEM analysis; for example, the order of the output's presentation
will be the same for the example in the next tutorial which shows the second
stage of SEM;
testing the structural model.
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.
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