EFA, CFA, and Amos
(You should have read the EFA chapter in Field, 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 influences 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.
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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
106
78
115
80
...
93
83
101
90
128
72
...
93
80
103
84
121
76
...
97
81
Latent
Variable B
Latent
Variable A
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
Observed
Variable
CFA, Amos - 2
"Causal"
Arrow
"Causal"
Arrow
Latent
Variable
Observed
Variable
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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 
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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
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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.
1st Factor
1st Factor
2st Factor
2st Factor
Note: Contrast codes should be used to make sure
the group-coding variables uncorrelated (assuming
equal sample sizes.)
Dependent
Variable
e
Interaction
Interaction
Interaction
Interaction
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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.)
e1
Obs 1
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. These are sometimes called cross loadings.
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.
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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.
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Path Diagrem of EFA model of NEO-FFI Big Five 60 item questionnaire.
(From Biderman, M. (2014). Against all odds: Bifactors in EFAs of Big Five Data. Part of
symposium: S. McAbee & M. Biderman, Chairs. Theoretical and Practical Advances in Latent
Variable Models of Personality. Conducted at the 29th annual conference of The Society for
Industrial and Organizational Psychology; Honolulu, Hawaii, 2014.
Crossloadings are in red.
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Path Diagrams vs the Table of Loadings.
(Crossloadings are in red in both representations.)
Pattern Matrixa
Factor
1
2
3
4
ne1
-.025
-.001
-.056
-.067
ne2
.086
.002
.021
-.026
ne3
.126
-.023
.133
.229
ne4
.054
.014
.112
.184
ne5
-.115
-.045
-.069
-.269
ne6
.042
.206
-.200
.085
ne7
.194
-.168
.046
-.203
ne8
.342
-.127
.087
.299
ne9
.270
.046
.092
.419
ne10
-.017
-.138
-.105
.135
ne11
.209
-.270
.009
.102
ne12
.140
-.170
-.103
-.070
na1
-.100
-.193
.081
.518
na2
.347
-.153
-.076
.421
na3
.091
-.123
-.050
.560
na4
-.139
.034
-.011
.504
na5
.298
.073
.033
.335
na6
.335
.157
-.070
.353
na7
.019
-.177
.067
.231
na8
.053
.029
-.114
.543
na9
.163
.115
-.120
.319
na10
-.057
-.123
.167
.594
na11
.028
.012
.055
.471
na12
.027
-.210
-.075
.534
nc1
.092
-.429
-.090
.183
nc2
.019
-.580
-.049
.055
nc3
-.030
-.376
-.037
.011
nc4
-.093
-.406
.052
.156
nc5
.026
-.716
-.025
-.156
nc6
.146
-.476
.100
.241
nc7
-.154
-.694
-.070
-.121
nc8
.092
-.528
.019
-.017
nc9
.040
-.573
.044
-.050
nc10
.021
-.720
.016
-.103
nc11
.035
-.551
-.067
.148
nc12
.065
-.628
.035
-.011
ns1
.501
.072
.145
-.231
ns2
.544
-.119
.132
-.044
ns3
.653
.037
-.026
-.069
ns4
.664
.025
-.141
.006
ns5
.660
.082
-.031
.200
ns6
.677
-.053
-.104
.011
ns7
.658
.035
.027
-.069
ns8
.552
.068
-.078
.265
ns9
.724
-.093
.134
-.021
ns10
.662
-.041
-.068
-.009
ns11
.563
-.266
.035
.030
ns12
.611
-.082
-.177
.075
no1
-.146
.334
.267
-.003
no2
.030
.366
.136
.061
no3
-.014
.044
.661
-.040
no4
.116
.037
.142
-.119
no5
-.010
-.090
.731
.102
no6
.029
.168
.217
.124
no7
-.064
.025
.207
.082
no8
.024
.233
.240
-.101
no9
-.008
-.012
.822
.049
no10
.052
.135
.536
.026
no11
-.026
-.111
.560
-.041
no12
.021
.131
.615
-.174
Extraction Method: Maximum Likelihood.
Rotation Method: Oblimin with Kaiser Normalization.
a. Rotation converged in 14 iterations.
5
.678
.268
.260
.492
.624
.563
.166
.432
.223
.327
.278
.483
.249
-.139
.125
-.099
.188
.031
.346
.292
-.211
.271
-.227
-.022
.008
-.086
-.140
.028
.057
-.052
.109
.110
.094
.044
.005
.018
-.031
.027
-.118
.189
.047
.046
.011
.012
.027
.220
-.132
-.003
-.103
.087
.046
-.008
-.166
.207
-.082
-.197
-.123
-.033
.110
.090
Whew – there are tons of crossloadings, most of them near 0. Can’t they just be assumed to be zero?
This kind of thinking leads to Confirmatory Factor Analysis Models.
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Confirmatory vs Exploratory Factor Analysis
In Exploratory Factor Analysis, as discussed above, 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-orthogonal-factor model is represented by the following diagram.
Obs 1
F1
Exploratory
F2
e4
Obs 2
e5
Obs 3
e6
Obs 4
e7
Obs 5
e8
Obs 6
e9
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 programs that do just EFA (except Mplus) won’t allow some loadings to be
fixed at predetermined values. And programs designed to do CFAs (except Mplus) canNOT
estimate the above EFA model. Amos and all CFA programs other than MPlus require that some of
the loadings be fixed and are unable to do EFAs such as the above models.
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
Confirmatory
F2
E1
Obs 2
E2
Obs 3
E3
Obs 4
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.)
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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.
Correlation of Y with X
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” or “Error of
measurement” variable, labeled “E”.
Mean of X
Mean of E
Variance of X
Variance of E
Intercept of X->Y regression
Slope of X->Y regression
Correlation 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 5 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.
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Dealing with underidentification . . .
Solution 1
0) 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
X
Y
Y = a + b*X
In this case, there are 5 quantities in the model that must be estimated – mean of X, variance of X,
intercept of equation, slope of equation, and correlation of E with Y. There are also 5 quantities that
can be estimated from the observed data. The model is said to be “just identified” or “completely
identified”. This means that every estimable quantity in the model corresponds in some way to one
quantity obtained from the data.
Or,
Solution 2
0) The mean of E is always assumed to be 0.
1) Fix covariance of E with Y at 1.
0, Variance
E
1
Mean, Variance
X
Y
Y=a+b*X
Underidentified models: Cannot be estimated.
Just identified models: Every model quantity is a function of some data quantity. But no parsimony.
Overidentified models: There are more data quantities than model quantities. It is said that you then
have “degrees of freedom” in your model. This is good. Relationships are being explained by fewer
model quantities than there are data quantities. This is parsimonious – what science is all about.
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Identification in CFA models – We’ll have to do this for the reasons outlined above.
Here’s a typical CFA two-factor model.
Identifying the residuals part of the CFA . . .
Solution 11. Fix all residual variances to 1.
0,1
E1
Obs 1
0,1
F1
Obs 2
E2
Obs 3
E3
0,1
0,1
E4
Obs 4
0,1
F2
Obs 5
E5
Obs 6
E6
0,1
or
Solution 2. Fix all E->O covariances to 1 and estimate variances of “E”s.
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
CFA, Amos - 13
I
recommend
this.
Printed on 2/7/2016
Insuring that the Factors part of the CFA is identified
1. Fix one of the loadings for each factor at 1 and estimate all factor variances
1
F1
1
F2
Obs 1
E1
Obs 2
E2
Obs 3
E3
Obs 4
E4
Obs 5
E5
Obs 6
E6
Or
2. Fix the variance of each factor at 1 and estimate all factor loadings.
Obs 1
1
F1
E1
Obs 2
E2
Obs 3
E3
Obs 4
E4
1
F2
Obs 5
E5
Obs 6
E6
I
recommend
this.
This probably seems quite arcane right now, and it is really the province of the mathematical
statisticians and programmers who discovered the algorithms that allow us to apply the models.
But we’ll use these conventions when actually applying models such as these.
The main point for us.
The above models tell us that the variation in 6 observed variables – Obs1, Obs2, Obs3, Obs4,
Obs5, Obs5 - is due to variation in two internal characteristics – F1 and F2.
So we have explained why there is variation in the observed variables – because of variation in F1
and F2. We have also explained why the variation in Obs1 to Obs3 is unrelated to the variation in
Obs4 to Obs6 – because F1 and F2 are uncorrelated.
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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
My
favorite.
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
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Real Life Example of a CFA model
Now consider a Big Five questionnaire – the NEO FFI
questionnaire, for example.
(The path diagram does not have the “residual” latent variables.
To have included them would have made the figure even “busier”
than it is.)
The path diagram explains variation in 60 behaviors in terms of
only 5 internal characteristics. It says that variation in Extrav1
through Extrav12 is due primarily to individual differences in E,
for example.
It also explains why Extrav1 is typically uncorrelated with Agree1,
etc – because E is uncorrelated with A.
Extrav1
Extrav2
Extrav3
Extrav4
Extrav5
Extrav6
Extrav7
Extrav8
Extrav9
Extrav10
Extrav11
Extrav12
Agree1
Agree2
Agree3
Agree4
Agree5
Agree6
Agree7
Agree8
Agree9
Agree10
Agree11
Agree12
Consc1
Consc2
Consc3
Consc4
Consc5
Consc6
Consc7
Consc8
Consc9
Consc10
Consc11
Consc12
Stabil1
Stabil2
Stabil3
Stabil4
Stabil5
Stabil6
Stabil7
Stabil8
Stabil9
Stabil10
Stabil11
Stabil12
Open1
Open2
Open3
Open4
Open5
OIpen6
Open7
Open8
Open9
Open10
Open11
Open12
CFA, Amos - 16
E
A
C
S
O
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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.
Other programs
LISREL
EQS
Amos
Mplus – the best
LAVAAN
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Creating an Amos analysis
1. Open Amos Graphics.
1b. File -> New
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 . . .”
6. Save the model. File -> Save As...
7. To run Amos, click on the
button.
8. Click on
to see output.
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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.
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Doing old things in a new way: Analyses we’ve done before, now
performed using Amos
The data used for this example are the valdatnm data in the htlm2\5510\datafiles folder. 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
b. Amos Input Path Diagram - Input Parameter Values
All variables are
exogenous.
(Note, I told Amos to estimate means for this analysis.)
c. Amos Output Path Diagram - Unstandardized (Raw) coefficients
The covariance of
p511g and Formula.
The mean and
variance of p511g.
The mean and variance
of 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.
CFA, Amos - 20
Printed on 2/7/2016
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
CFA, Amos - 21
Printed on 2/7/2016
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; Means not estimated
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(1-r2)=sqrt(1-.482) = sqrt(1-.23)
Correlation of p511g
error
= sqrt(.77)=.88
with Formula.
.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.
CFA, Amos - 22
Printed on 2/7/2016
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
CFA, Amos - 23
Printed on 2/7/2016
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.
CFA, Amos - 24
Printed on 2/7/2016
d. Amos Output Path Diagram - Standardized coefficients.
Standardized
partial regression
coefficients ,
sometimes called
betas..
Correlation of ugpa
and greq.
ugpa
error
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
(Early version of Amos without p values)
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:
-------------------------------p511g <----- ugpa
p511g <---- error
p511g <----- greq
Covariances:
-----------ugpa <-----> greq
Correlations:
-------------
Estimate
--------
S.E.
-------
0.048
0.047
0.000
0.015
0.004
0.000
C.R.
-------
Label
-------
3.140
12.329
3.869
Estimate
-------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
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.
CFA, Amos - 25
Printed on 2/7/2016
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
F
5.9 30
CFA, Amos - 26
Sig .
.01 1
Printed on 2/7/2016
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
CFA, Amos - 27
Printed on 2/7/2016
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, but
estimate it anyway.
Multiple R2
Note that .292 + .562 + .782 = 1.01 ~~ 1.
r2s partition variance since the variables are all
independent.
CFA, Amos - 28
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f. Amos Text Output – Results
CFA, Amos - 29
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Results continued . . .
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.
CFA, Amos - 30
Printed on 2/7/2016