EFA, CFA, and SEM
(You should have read the EFA chapter in T&F, the Structural Equation Modeling chapter, and be reading the Byrne
text.)
Observed Variable
A variable whose values are observable.
Examples: IQ Test scores (Scores are directly observable), GREV, GREQ, GREA, UGPA,
Minnesota Job Satisfaction Scale, Affective Commitment Scale, Gender, Questionnaire items.
Latent Variable
A variable, i.e., characteristic, presumed to exist, but whose values are NOT observable. A Factor
in Factor Analysis literature. A characteristic of people that is not directly observable.
Intelligence, Depression, Job Satisfaction, Affective Commitment, Tendency to display affective
state
No direct observation of values of latent variables is possible. Brain states? Brain chemistry?
Indicator
An observed variable whose values are assumed to be related to the values of a latent variable.
Reflective Indicator
An observed variable whose values are partially determined by, i.e., are influenced by or reflect,
the values of a latent variable. For example, responses to Conscientiousness items are assumed to
reflect a person’s Conscientiousness.
Formative Indicator
An observed variable whose values partially determine, i.e., cause or form, the values of a latent
variable.
Exogenous Variable (Ex = Out)
A variable whose values originate from / are caused by factors outside the model, i.e., are not
explained within the theory with which we’re working. That is, a variable whose variation we don’t
attempt to explain or predict by whatever theory we’re working with. Causes of exogenous variable
originate outside the model. Exogenous variables can be observed or latent.
Endogenous variable (En ~~ In)
A variable whose values are explained within the theory with which we’re working. We account for
all variation in the values of endogenous variables using the constructs of whatever theory we’re
working with. Causes of endogenous variables originate within the model.
Intro to SEM - 1
Printed on 2/6/2016
Basic SEM Path Analytic Notation
Observed variables are symbolized by squares or rectangles.
103
84
121
76
...
97
81
Observed
Variable
Latent Variables are symbolized by Circles or ellipses.
106
78
115
80
...
93
83
Latent
Variable
Values of individuals on
latent variables are not
observable, hence the
dimmed text.
Correlations or covariances between variables are represented by double-headed arrows.
"Cor / Cov"
Arrow
"Cor / Cov"
Arrow
Observed
Variable B
Observed
Variable A
101
90
128
72
...
93
80
103
84
121
76
...
97
81
Latent
Variable B
Latent
Variable A
106
78
115
80
...
93
83
104
79
114
79
...
92
81
"Causal" or "Predictive" or “Regression” relationships between variables are represented by single-headed
arrows
Latent
Variable
Latent
Variable
"Causal"
Arrow
Observed
Variable
Observed
Variable
"Causal"
Arrow
Latent
Variable
Intro to SEM - 2
Observed
Variable
"Causal"
Arrow
"Causal"
Arrow
Latent
Variable
Observed
Variable
Printed on 2/6/2016
Exogenous Observed Variables
"Correlation"
Arrow
"Causal"
Arrow
Observed
Variable
Observed
Variable
Exogenous variable connect to other variables in the model through either a “causal” arrow or a correlation
Exogenous Latent Variables
Latent
Variable
"Correlation"
Arrow
"Causal"
Arrow
Latent
Variable
Exogenous latent variables also connect to other variables in the model through either a “causal” arrow or a correlation
Endogenous Observed Variables "Causal"
Arrow
Observed
Variable
Endogenous Latent Variable
Random
error
Random
error
"Causal"
Arrow
Latent
Variable
Endogenous variables connect to other variables in the model by being on the “receiving” end of one or more “causal”
arrows. Specifically, endogenous variables are typically represented as being “caused” by 1) other variables in the
theory and 2) random error. Thus, 100% of the variation in every endogenous variable is accounted for by either
other variables in the model or random error. This means that random error is an exogenous latent variable in SEM
diagrams. Random error is a catch-all concept representing all “other” things that are affecting the endogenous variable.
Summary statistics associated with symbols
Mean, Variance
Mean, Variance
Observed
Variable
Latent
Variable
Our SEM program, Amos, prints means and variances above and to the right. Typically the mean and variance of latent
variables are fixed at 0 and 1 respectively, although there are exceptions to this in advanced applications.
"Correlation"
Arrow
"Causal"
Arrow
r or Covariance
B or 
Intro to SEM - 3
Printed on 2/6/2016
Path Diagrams of Analyses We’ve Done Previously
Following is how some of the analyses we’ve performed previously would be represented using path
diagrams.
1. Simple correlation between two observed variables.
rVQ
GRE-Q
GRE-V
2. Simple correlations between three observed variables.
rVA
rVQ
GRE-V
rQA
GRE-Q
GRE-A
3. Simple regression of an observed dependent variable onto one observed independent variable.
GRE-Q
B or 
P511G
Note that the
endogenous variable
is caused in part by
catch-all influences.
e
4. Multiple Regression of an observed dependent variable onto three observed independent variables.
GRE-V
GRE-Q
UPGA
BV or V
BQ or Q
P511G
e
Note that the
endogenous variable
is caused in part by
catch-all influences.
BU or U
Intro to SEM - 4
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ANOVA in SEM Models
Since ANOVA is simply regression analysis, the representation of ANOVA in SEM is merely as a
regression analysis. The key is to represent the differences between groups with group coding
variables, just as we did in 513 and in the beginning of 595 . . .
1) Independent Groups t-test
The two groups are represented by a single, dichotomous observed group-coding variable. It is the
independent variable in the regression analysis.
Dichotomous
variable representing
the two groups
Dependent
Variable
e
2) One Way ANOVA
The K groups are represented by K-1 group-coding variables created using one of the coding
schemes (although I recommend contrast coding). They are the independent variables in the
regression analysis. If contrast codes are used, the correlations between all the group coding
variables are 0, so no arrows between them need be shown.
1st Group-coding
contrast code
variable
2nd Group-coding
contrast code
variable.
Note: Contrast codes were used so Groupcoding variables are uncorrelated.
Dependent
Variable
e
. . . . .
(K-1)th Groupcoding contrast code
variable.
3) Factorial ANOVA.
Each factor is represented by G-1 group-coding variables created using one of the coding schemes.
The interaction(s) is/are represented by products of the group-coding variables representing the
factors. Again, no correlations between coding variables need be shown if contrast codes are used.
Note: Contrast codes should be used to make sure
the group-coding variables uncorrelated.
1st Factor
1st Factor
2st Factor
2st Factor
Dependent
Variable
e
Interaction
Interaction
Interaction
Interaction
Intro to SEM - 5
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Path Diagrams representing Exploratory Factor Analysis
1) Exploratory Factor Analysis solution with one factor.
The factor is represented by a latent variable with three or more observed indicators. (Three is the
generally recommended minimum no. of indicators for a factor.)
Obs 1
e1
Obs 2
e2
Obs 3
e3
F
Note that factors are exogenous. Indicators are endogenous. Since the indicators are endogenous,
all of their variance must be accounted for by the model. Thus, each indicator must have an error
latent variable to account for the variance in it not accounted for by the factor.
2) Exploratory Factor Analysis solution with two orthogonal factors.
Each factor is represented by a latent variable with three or more indicators. The orthogonality of
the factors is represented by the fact that there is no arrow connecting the factor symbols.
Let’s assume that Obs1, 2, and 3 are thought to be primary indicators of F1 and 4,5,6 of F2.
For exploratory factor analysis, each variable is allowed to load on all factors. Of course, the hope is
that the loadings will be substantial on only some of the factors and will be close to 0 on the others,
but the loadings on all factors are retained, even if they’re close to 0. The loadings that might be
close to 0 in the model are shown in red and orange.
Obs 1
e1
Obs 2
e2
Obs 3
e3
F1
Obs 4
e4
Obs 5
e5
Obs 6
e6
F2
Orthogonal factors represent uncorrelated aspects of behavior.
Note what is assumed here: There are two independent characteristics of behavior – F1 and F2.
Each one influences responses to all six items, although it is hoped that F1 influences primarily the
first 3 items and that F2 influences primarily the last 3 items.
If Obs 1 thru Obs 3 are one class of behavior and Obs 4 thru Obs 6 are a second class, then if the
loadings “fit” the expected pattern, this would be evidence for the existence of two independent
dispositions – that represented by F1 and that represented by F2.
Intro to SEM - 6
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3) Exploratory Factor Analysis solution with two oblique factors.
Each factor is represented by a latent variable with three or more indicators. The obliqueness of the
factors is represented by the fact that there IS an arrow connecting the factors.
Obs 1
e4
Obs 2
e5
Obs 3
e6
F1
Obs 4
e7
Obs 5
e8
Obs 6
e9
F2
Again, in exploratory factor analysis, all indicators load on all factors, even if the loadings are close
to zero.
Exploratory factor analysis (EFA) programs, such as that in SPSS, always report estimates of all
loadings.
This solution is potentially as important as the orthogonal solution, although in general, I think that
researchers are more interested in independent dispositions than they are in correlated dispositions.
But discovering why two dispositions are separate but still correlated is an important and potentially
rewarding task.
Intro to SEM - 7
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Confirmatory vs Exploratory Factor Analysis
In Exploratory Factor Analysis, the loading of every item on every factor is estimated. The analyst
hopes that some of those loadings will be large and some will be small. An EFA two-orthogonalfactor model is represented by the following diagram.
e4
Obs 1
Obs 2
e5
Obs 3
e6
F1
e7
Obs 4
Obs 5
e8
Obs 6
e9
F2
Note that there are arrows (loadings) connecting each variable to each factor. We have no
hypotheses about the loading values – we’re exploring – so we estimate all loadings and let them
lead us. No EFA programs (except that in Mplus) allow you to specify or fix loadings to predetermined values.
In contrast to the exploration implicit in EFA, a factor analysis in which some loadings are fixed at
specific values is called a Confirmatory Factor Analysis. The analysis is confirming one or more
hypotheses about loadings, hypotheses representing by our fixing them at specific (usually 0) values.
Unfortunately, EFA and CFA cannot be done using the same computer program except MPlus.
The problem is that all EFA programs except that in Mplus won’t allow some loadings to be fixed at
predetermined values. And CFA programs, except Mplus canNOT estimate the above model. Amos
and all CFA programs other than MPlus require that some of the loadings be fixed.
So, in many instances, you will have to employ both SPSS (for EFA) and AMOS (for CFA) in
exploring the interrelations between variables and factors. Often, analysts will use an EFA program
to estimate ALL loadings to all factors, then use an SEM program to perform a confirmatory factor
analysis, fixing those loadings that were close to 0 in the EFA to 0 in the CFA.
Obs 1
F1
Obs 2
E2
Obs 3
E3
Obs 4
F2
E1
E4
Obs 5
E5
Obs 6
E6
Note that in the above confirmatory model, loadings of indicators 4-6 on F1 are fixed at 0, as are
loadings of indicators 1-3 on F2. (The arrows are missing, therefore assumed to be zero.)
Intro to SEM - 8
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The Identification Problem
Mean, Variance
Consider the simple regression model . . .
E
Mean, Variance
Cov(E,Y)
X
Y
Y = a + b*X
Quantities which can be computed from the data . .
Mean of the X variable
Variance of the X variable
Mean of the Y variable.
Variance of the Y variable.
Intercept of X on Y
Slope of X->Y regression
Quantities in the diagram .
Remember that in SEM path diagrams, all the variance in every endogenous variable must be
accounted for. For that reason, the path diagram includes a latent “Other factors” variable, labeled
“E”.
Mean of X
Mean of E
Variance of X
Variance of E
Intercept of X-> regression
Slope of X->Y regression
Covariance of E with Y
Note: Mean and variance of Y are
not separately identified in the model
because they are assumed to be
completely determined by Y’s
relationship to X and to E.
Whoops! There are 6 quantities in the data but 7 in the model. There are too few quantities in the
data. The model is underidentified. – not identified enough - there aren't enough quantities from
the data to identify each model value.
Intro to SEM - 9
Printed on 2/6/2016
Dealing with underidentification . . .
The mean of E is always assumed to be 0.
1) Fix the variance of E to be 1.
So in this regression model, the path diagram will be
0, 1
E
Mean, Variance
Cov(E,Y)
X
Y
Y = a + b*X
In this case, the model is said to be “just identified” or “completely identified”. This means that
every estimable quantity in the model corresponds to one quantity obtained from the data.
Or,
2) Fix covariance of E with Y at 1.
Mean, Variance
E
1
Mean, Variance
X
Y
Y=a+B*X
Underidentified: Bad.
Just identified: OK
Overidentified: Great, you have degrees of freedom.
Intro to SEM - 10
Printed on 2/6/2016
Identification in CFA models
Here’s a typical CFA two-factor model.
Making the residuals part of the CFA identified . . .
1. Fix all residual variances to 1.
0,1
E1
Obs 1
0,1
F1
Obs 2
E2
Obs 3
E3
Obs 4
E4
Obs 5
E5
Obs 6
E6
0,1
0,1
0,1
F2
0,1
or
2. Fix all E-O covariances to 1.
1
Obs 1
F1
Obs 2
1
E2
Obs 3
1
E3
1
E4
1
E5
1
E6
Obs 4
F2
E1
Obs 5
Obs 6
Making the Factors part of the CFA identified
1. Fix one of the loadings for each factor at 1
Or
2. Fix the variance of each factor at 1.
Intro to SEM - 11
Printed on 2/6/2016
Examples
1. Fixing all variances.
1
E1
Obs 1
1
F1
1
E2
Obs 2
1
E3
Obs 3
E4
Obs 4
1
1
E5
Obs 5
F2
1
1
E6
Obs 6
2. Fixing residual loadings but Factor variances
1
Obs 1
1
1
Obs 2
F1
E1
E2
1
E3
Obs 3
1
E4
Obs 4
1
1
E5
Obs 5
F2
1
E6
Obs 6
3. Fixing residual loadings and factor loadings.
1
1
Obs 1
Obs 2
F1
1
E1
E2
1
E3
Obs 3
1
E4
Obs 4
1
1
F2
E5
Obs 5
1
Obs 6
E6
Intro to SEM - 12
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Programming with path diagrams: Introduction to Amos
Amos is an add-on program to SPSS that performs confirmatory factor analysis and structural
equation modeling.
It is designed to emphasize a visual interface and has been written so that virtually all analyses can
be performed by drawing path diagrams.
It also contains a text-based programming language for those who wish to write programs in the
command language.
The Amos drawing toolkit with functions of the most frequently used tools.
Observed variable tool
Tool to draw latent variables
Tool to draw regression arrows
Tool to draw correlation arrows
Tool to put text on the diagram
Tool to select all objects in diagram
Tool to select a single object
Tool to deselect all objects in diagram
Tool to copy an object
Tool to erase an object
Tool to move an object
Tool to tell Amos to run the diagram.
Intro to SEM - 13
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Creating an Amos analysis
1. Open Amos Graphics.
2. File -> Data Files . . .
(Because you have to connect the path diagram to a data file.)
3. Specify the name of the file that contains the raw or summary data.
a. Click on the [File Name] button.
b. Navigate to the file and double-click on it.
c. Click on the [OK] button.
In this example, I opened a file called IncentiveData080707.sav
4. Draw the desired path diagram using the appropriate drawing tools.
The example below is a simple
correlation analysis.
5. Name the variables by right-clicking on each object. And
choosing “Object Properties . . .”
Intro to SEM - 14
Printed on 2/6/2016
Amos Details
For most of the analyses you’ll perform using Amos, you should get in the habit of doing the
following . . .
View -> Analysis Properties -> Estimation
Check “Estimate means and intercepts”
View -> Analysis Properties -> Output
Check “Standardized estimates”
Check “Squared multiple correlations”
Remember that you must fix some parameter values to make the models identified.
Intro to SEM - 15
Printed on 2/6/2016
Doing old things in a new way: Analyses we’ve done before, now
performed using Amos
The data used for this example are the VALDAT data. We’ll simply look at the output here. Later,
we’ll focus on the menu sequences needed to get this output.
a. SPSS analysis of the correlation of FORMULA with P511G
Correlations
Correlations
Pearson
Correlation
Sig.
(2-tailed)
N
P511G
FORMULA
P511G
FORMULA
P511G
FORMULA
P511G
FORMULA
1.000
.502**
.502**
1.000
.
.000
.000
.
83
79
79
81
**. Correlation is significant at the 0.01 level
(2-tailed).
b. Amos Input Path Diagram - Input Parameter Values
(Note, I told Amos to estimate means for this analysis.)
All variables are
exogenous.
c. Amos Output Path Diagram - Unstandardized (Raw) coefficients
The mean and variance
of Formula.
The mean and
variance of p511g.
The covariance of
p511g and Formula.
c. Amos Path Diagram - Standardized coefficients
The correlation of p511g
and Formula.
Means and variances of
standardized variables are
not displayed, since they
are 0 and 1 respectively.
Intro to SEM - 16
Printed on 2/6/2016
Simple Regression Analysis: SPSS and Amos
The data used here are the VALDAT data.
a. SPSS Version 10 output
GET FILE='E:\MdbT\P595\Amos\valdatnm.sav'.
.
REGRESSION /MISSING LISTWISE /STATISTICS COEFF OUTS R ANOVA
/CRITERIA=PIN(.05) POUT(.10) /NOORIGIN /DEPENDENT p511g
/METHOD=ENTER formula .
Regression
Variables Entered/Removedb
Model
1
Variables
Entered
FORMULAa
Variables
Removed
Method
Enter
.
a. All requested variables entered.
b. Dependent Variable: P511G
Model Summary
Model
1
R
.480a
R Square
.230
Adjusted
R Square
.220
Std. Error of
the Estimate
4.725E-02
a. Predictors: (Constant), FORMULA
ANOVAb
Model
1
Regres sion
Residual
Total
Sum of
Squares
5.005E-02
.167
.217
df
1
75
76
Mean Square
5.005E-02
2.233E-03
F
22.420
Sig.
.000a
t
6.361
4.735
Sig.
.000
.000
a. Predic tors: (Constant), FORMULA
b. Dependent Variable: P511G
Coefficientsa
Model
1
(Constant)
FORMULA
Unstandardized
Coefficients
B
Std. Error
.496
.078
3.004E-04
.000
Standardi
zed
Coefficien
ts
Beta
.480
a. Dependent Variable: P511G
Intro to SEM - 17
Printed on 2/6/2016
b. Amos Input Path Diagram - Input parameter values
1
The model is underidentified
unless you fix the value of one
parameter. Fix either the
variance of the latent error
variable to 1 or the regression
weight to 1. Here, the
variance has been fixed.
error
p511g
formula
c. Amos Output Path Diagram - Unstandardized (Raw) coefficients
Variance
Mean of X
of
Formula.
The estimated
Variance of X
unstandardized (raw
score) relationship of
p511g .to Formula the slope, to 2 decimal
places.
1.00
error
.05
7203.60
formula
.00
p511g
Note that the fixed
parameter values
were not changed.
For what it's worth,
the estimated
unstandardized (raw
score) relationship of
p511g to the “other
factors” latent
variable.
d. Amos Output Path Diagram - Standardized coefficients
(View/Set -> Analysis Properties -> Output to get Amos to print Standardized estimates what a
pain!!)
Correlation of p511g with latent
“other factors”..= sqrt(1Correlation of p511g
error
r2)=sqrt(1-.482) = sqrt(1-.23) =
with Formula.
sqrt(.77)=.88
.88
.23
formula
.48
r2 for the model. You may
have to pull down [View/Set] > Analysis Properties ->
Output to ask for this to be
printed.
p511g
Note that .482 + .882 = 1. All of variance of p511g has been accounted for. We say that formula
and error partition the total variance of p511g.
Intro to SEM - 18
Printed on 2/6/2016
Two IV Regression Example - SPSS and Amos
The data here are the VALDATnm data. UGPA and GREQ are predictors of P511G.
a. SPSS output.
GET
FILE='G:\MdbT\P595\P595AL09-Amos\valdatnm.sav'.
DATASET NAME DataSet1 WINDOW=FRONT.
REGRESSION /DESCRIPTIVES MEAN STDDEV CORR SIG N /MISSING LISTWISE
/STATISTICS COEFF OUTS R ANOVA /CRITERIA=PIN(.05) POUT(.10) /NOORIGIN
/DEPENDENT p511g /METHOD=ENTER ugpa greq .
Regression
[DataSet1] G:\MdbT\P595\P595AL09-Amos\valdatnm.sav
Correla tions
Pe arson
Co rrelat ion
p5 11g
1.0 00
ug pa
.22 5
ug pa
.22 5
1.0 00
-.2 62
gre q
.32 2
-.2 62
1.0 00
p5 11g
Sig . (1-t ailed )
p5 11g
N
gre q
.32 2
.
.02 5
.00 2
ug pa
.02 5
.
.01 1
gre q
.00 2
.01 1
.
p5 11g
77
77
77
ug pa
77
77
77
gre q
77
77
77
Va riabl es Entere d/Rem ov e db
Mo del
1
Va riable s
Re move d
Va riable s En tered
gre q, ug pa a
.
Me thod
En ter
a. All requ ested vari ables ente red.
b. De pend ent V ariab le: p 511g
Model S umm ary
Mo del
1
R
.45 5 a
R S quare
.20 7
Std . Erro r of
the Estim ate
.04 828
Ad justed R S quare
.18 5
a. Pre dicto rs: (Consta nt), g req, ugpa
Coeffici ents a
Un stand ardized Co efficients
Mo del
1
B
Std . Erro r
(Co nstan
t)
.57 1
.06 9
ug pa
.04 8
.01 6
gre q
.00 0
.00 0
Sta ndardized
Co effici ents
Be ta
t
Sig .
8.2 28
.00 0
.33 2
3.0 98
.00 3
.41 0
3.8 17
.00 0
a. De pend ent V ariab le: p5 11g
Intro to SEM - 19
Printed on 2/6/2016
b. Amos Input Path Diagram - Input parameters.
1
The variance of the
(unobserved) error latent
variable must be specified at 1.
error
ugpa
Note that if the IVs are
correlated, you must specify
that they are correlated.
Otherwise, Amos will perform
the analysis assuming they're
uncorrelated.
p511g
greq
c. Amos Output Path Diagram - Unstandardized (Raw) coefficients
I forgot to check “Estimate means and intercepts, so no means are printed.)
Variance of ugpa
Raw partial
regression
coefficient
relating p511g
to ugpa
Covariance of ugpa and greq.
1.00
error
.13
ugpa
.05
Raw
Regression
coefficient
relating
p511g to
residual
effects.
.05
-8.54
p511g
7897.62
.00
greq
Raw partial
regression coefficient
relating p511g to
GREQ to 2 decimal
places.
Intro to SEM - 20
Printed on 2/6/2016
d. Amos Output Path Diagram - Standardized coefficients.
Standardized
partial regression
coefficients.
Correlation of ugpa
and greq.
error
ugpa
SQRT(1-R2)=sqrt(1-.21)
= sqrt(.79)=.89
.89
.33
-.26
.21
p511g
.41
greq
Multiple R2.
Note that .332 + .412 + .892 = 1.07 > 1.0. This is because r2s partition variance only when
variables are uncorrelated.
e. Amos Text Output - Details of input and minimization
Chi-square = 0.000
Degrees of freedom = 0
Probability level cannot be computed
Maximum Likelihood Estimates
---------------------------Regression Weights:
------------------p511g <----- ugpa
p511g <---- error
p511g <----- greq
Standardized Regression Weights:
--------------------------------
Estimate
-------0.048
0.047
0.000
S.E.
------0.015
0.004
0.000
C.R.
-------
Label
-------
3.140
12.329
3.869
Estimate
--------
p511g <----- ugpa
p511g <---- error
p511g <----- greq
Covariances:
-----------ugpa <-----> greq
Correlations:
-------------
0.332
0.891
0.410
Estimate
--------
S.E.
-------
C.R.
-------
-8.537
3.861
-2.211
S.E.
-------
C.R.
-------
Label
-------
Estimate
-------ugpa <-----> greq
Variances:
----------
-0.262
Estimate
--------
error
ugpa
greq
Squared Multiple Correlations:
------------------------------
1.000
0.134
7897.622
Estimate
--------
p511g
0.207
Intro to SEM - 21
0.022
1281.163
Label
-------
6.164
6.164
Note – No overall test of significance of R2.
This test is available in the ANOVA box in
SPSS.
Printed on 2/6/2016
Oneway Analysis of Variance Example - SPSS and Amos
The data for this example follow. They're used to introduce the 595 students to contrast coding. The
dependent variable is Job Satisfaction (JS). The research factor is Job, with three levels. It is
contrast coded by CC1 and CC2.
The data for this example are in ‘MdbT\P595\Amos\ OnewayegData.sav’
ID JS
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
JOB
6
7
8
11
9
7
7
5
7
8
9
10
8
9
4
3
6
5
7
8
2
1
1
1
1
1
1
1
2
2
2
2
2
2
2
3
3
3
3
3
3
3
CC1
CC2
.667
.667
.667
.667
.667
.667
.667
-.333
-.333
-.333
-.333
-.333
-.333
-.333
-.333
-.333
-.333
-.333
-.333
-.333
-.333
.000
.000
.000
.000
.000
.000
.000
.500
.500
.500
.500
.500
.500
.500
-.500
-.500
-.500
-.500
-.500
-.500
-.500
The rule for forming a contrast variable between two sets
of groups is
1st Value = No. of groups in 2nd set / Total no. of groups.
2nd Value= - No. of groups in 1st set / Total no. of groups.
3rd Value = 0 for all groups to be excluded.
1st Value of CC1 = 2 / 3 = .667.
So,
2nd Value of CC1 = - 1 / 3
1st Value of CC2 = 1 / 2 = .5
2nd Value of CC2 = -1 / 2 = -..5
3rd Value of CC2 = 0 to exclude Job 1.
a. SPSS Oneway output.
Oneway
ANOVA
JS
Su m of
Sq uares
Be tween
Gro ups
Wit hin G roup s
To tal
df
Me an S quare
40. 095
2
20. 048
60. 857
18
3.3 81
100 .952
20
Intro to SEM - 22
F
5.9 30
Sig .
.01 1
Printed on 2/6/2016
b. SPSS Regression Output.
regression variables = js cc1 cc2
/dependent = js /enter.
Regression
Va riable s Entered/Rem ov e db
Mo del
1
Va riable s
En tered
CC2, CC1 a
Va riable s
Re move d
.
Me thod
En ter
a. All requ ested varia bles ente red.
b. De pend ent V ariab le: JS
Model S umm ary
Mo del
1
R
.63 0 a
R S quare
.39 7
Ad justed R
Sq uare
.33 0
Std . Erro r of
the Estim ate
1.8 387
a. Pre dicto rs: (Consta nt), CC2, CC1
ANOVAb
Mo del
1
Su m of
Sq uares
Re gressi o
n
Re sidua l
To tal
df
Me an S quare
F
Sig .
a
40. 095
2
20. 048
60. 857
18
3.3 81
100 .952
20
5.9 30
.01 1
a. Pre dicto rs: (Consta nt), CC2, CC1
b. De pend ent V ariab le: JS
Coeffici ents a
Un stand ardized Co efficients
Mo del
1
B
Std . Erro r
(Co nstan
t)
6.9 52
.40 1
CC1
1.3 57
.85 1
CC2
3.0 00
.98 3
Sta ndardized
Co effici ents
Be ta
t
Sig .
17 .326
.00 0
.29 2
1.5 94
.12 8
.55 9
3.0 52
.00 7
a. De pend ent V ariab le: JS
Intro to SEM - 23
Printed on 2/6/2016
c. Amos Input Path Diagram.
This was prepared using Amos 3.6. I chose the "Estimate
means" option. This was not required, but it caused means to
be displayed.
d. Amos Output Path Diagram - Unstandardized (Raw) Coefficients
Intercept
Mean and variance.
e. Amos Output Path Diagram - Standardized Coefficients
Note that the correlation between
group coding variables must be
estimated. It's zero here because
they're contrast codes.
Multiple R2
Note that .292 + .562 + .782 = 1.01 ~~ 1.
r2s partition variance since the variables are all
independent.
Intro to SEM - 24
Printed on 2/6/2016
f. Amos Text Output - Results
Result (Default model)
Minimum was achieved
Chi-square = .00000
Degrees of freedom = 0
Probability level cannot be computed
Maximum Likelihood Estimates
Regression Weights: (Group number 1 - Default model)
Estimate
S.E.
P Label
C.R.
Coefficients Box – B coefficients only
JS <--- CC1 1.35714 .80749 1.68069 .09282
JS <--- CC2 3.00000 .93241 3.21747 .00129
Standardized Regression Weights: (Group number 1 - Default model)
Estimate
JS <--- CC1 .29179
JS <--- CC2 .55859
Coefficients Box – Standardized
coefficients
Means: (Group number 1 - Default model)
Estimate
S.E.
C.R.
P Label
CC1
.00033 .10541 .00316 .99748
CC2
.00000 .09129 .00000 1.00000
Descriptive statistics
Intercepts: (Group number 1 - Default model)
Estimate
JS
S.E.
C.R.
Coefficients box - Constant
P Label
6.95193 .38065 18.26308 ***
Covariances: (Group number 1 - Default model)
Estimate
CC1 <--> CC2
S.E.
C.R.
Not in SPSS
P Label
.00000 .04303 .00000 1.00000
Correlations: (Group number 1 - Default model)
Model summary
Estimate
CC1 <--> CC2 .00000
Variances: (Group number 1 - Default model)
Estimate
S.E.
C.R.
P Label
CC1
.22222 .07027 3.16228 .00157
CC2
.16667 .05270 3.16228 .00157
resid
2.89796 .91642 3.16228 .00157
Descriptive statistics
Squared Multiple Correlations: (Group number 1 - Default model)
Estimate
JS
.39717
Note that AMOS does not provide a test of
the null hypothesis that in the population, the
multiple R = 0. This test is provided in the
ANOVA box in SPSS.
Intro to SEM - 25
Printed on 2/6/2016