AMOS – Analysis of Moment

Structures

Rick Zimmerman, Olga Dekhtyar

HIV Prevention Center

University of Kentucky
Overview
Overview of Structural Equation Models
(SEM)
Introduction to AMOS
 User Interface
 AMOS Graphics
Examples of using AMOS
 Predictors of Condom Use using latent
variables
Structural Equation Models
Structural Equation Modeling (SEM)
An extension of Regression and general
Linear Models

Also can fit more complex models, like

confirmatory factor analysis and
longitudinal data.
Structural Equation Modeling
Ability to fit
non-standard models,
databases with autocorrelated error
structures
 time series analysis
 Latent Curve Models,
databases with non-normally distributed
variables
databases with incomplete data.
T-test Family Tree of SEM
ANOVA
Multi-way
ANOVA Repeated
Measure
Designs Growth
Curve
Analysis
Latent
Growth
Multiple Path Structural Curve
Bivariate
Regression
Equation Analysis
Analysis
Correlation Modeling

Confirmatory
Factor Next Workshop:
Factor
Analysis November 9
Analysis
See you there!
Exploratory
Factor
Analysis
Structural Equation Modeling
(SEM)
Exogenous variables=independent
Endogenous variables =dependent
Observed variables =measured
Latent variables=unobserved
Structural Equation Graphs
.10 : R2
Observed
Error Variable

.15 : Loading

Latent
Variable
 Example: Condom Use Model
Observed variables for
Impulsive decision making Respondent Sex

IDMA1R IDMC1R IDME1R IDMJ1R

SEX1
Impulsive

Impulsive
Decision Making FRBEHB1
ISSUEB1
Legend Peer norms about
condoms SXPYRC1 Condom attitude
Observed Latent
Variables Variables

.15 Loadings Condom Use

 Example: Condom Use Model
Dependent
Independent

IDMA1R IDMC1R IDME1R IDMJ1R

SEX1
Impulsive

Independent
FRBEHB1
ISSUEB1
Legend Dependent
SXPYRC1 Dependent
Observed Latent
Variables Variables

.15 Loadings Dependent

 Example: Condom Use Model
eidm1 eidm2 eidm2 eidm4

IDMA1R IDMC1R IDME1R IDMJ1R

SEX1
Impulsive

efr1 FRBEHB1
ISSUEB1 eiss

Legend
SXPYRC1
Observed Latent
Variables Variables

.15 eSXYRC1
Loadings
 Example: Condom Use Model
eidm1 eidm2 eidm2 eidm4

IDMA1R IDMC1R IDME1R IDMJ1R

SEX1
Impulsive

efr1 FRBEHB1
ISSUEB1 eiss

Legend
SXPYRC1
Observed Latent
Variables Variables

.15 eSXYRC1
Loadings
 Example: Condom Use Model
eidm1 eidm2 eidm2 eidm4

.28 .24 .48 .45

IDMA1R IDMC1R IDME1R IDMJ1R
.49 .69 .67 -.06 SEX1
.53
Impulsive -.19 -.10
-.15 .13
.03
efr1 FRBEHB1 .05
ISSUEB1 eiss
.11
.38
Legend
SXPYRC1 .15
Observed Latent
Variables Variables

.15 eSXYRC1
Loadings
SEM Assumptions
A Reasonable Sample Size
a good rule of thumb is 15 cases per predictor in a
standard ordinary least squares multiple regression
analysis.
[ “Applied Multivariate Statistics for the Social Sciences”,
by James Stevens]
researchers may go as low as five cases per
parameter estimate in SEM analyses, but only if the
data are perfectly well-behaved
[Bentler and Chou (1987)]
Usually 5 cases per parameter is equivalent to 15
measured variables.
SEM Assumptions (cont’d)
Continuously and Normally Distributed
Endogenous Variables

NOTE: At this time AMOS CANNOT handle

not continuously distributed outcome
variables
SEM Assumptions (cont’d)
Model Identification
P is # of measured variables
[P*(P+1)]/2

Df=[P*(P+1)]/2-(# of estimated parameters)

If DF>0 model is over identified
If DF=0 model is just identified
If DF<0 model is under identified
Missing data in SEM
Types of missing data

MCAR
 Missing Completely at Random
MAR
 Missing at Random
MNAR
 Missing Not at Random
Handling Missing data in SEM
Listwise
Pairwise
Mean substitution
Regression methods
Expectation Maximization (EM) approach
Full Information Maximum Likelihood (FIML)**
Multiple imputation(MI)**

The two best methods: FIML and MI

SEM Software
Several different packages exist
 EQS, LISREL, MPLUS, AMOS, SAS, ...

Provide simultaneously overall tests of

 model fit
 individual parameter estimate tests
May compare simultaneously
 Regression coefficients
 Means
 Variances
even across multiple between-subjects groups
Introduction to
AMOS
AMOS Advantages
Easy to use for visual SEM ( Structural
Equation Modeling).
Easy to modify, view the model
Publication –quality graphics
AMOS Components
AMOS Graphics
 draw SEM graphs
 runs SEM models using graphs

AMOS Basic
 runs SEM models using syntax
 Starting AMOS Graphics

Start Programs  Amos 5  Amos Graphics

Reading Data into AMOS
File  Data Files
The following dialog appears:
Reading Data into AMOS
Click on File Name to specify the
name of the data file
Currently AMOS reads the following
data file formats:
 Access
 dBase 3 – 5
 Microsft Excel 3, 4, 5, and 97
 FoxPro 2.0, 2.5 and 2.6
 Lotus wk1, wk3, and wk4
 SPSS *.sav files, versions 7.0.2 through 13.0
(both raw data and matrix formats)
Reading Data into AMOS
Example USED for this workshop:
 Condom use and what predictors affect it

DATASET:
AMOS_data_valid_condom.sav
Drawing in AMOS
In Amos Graphics, a model can be
specified by drawing a diagram on the
screen 1. To draw an observed variable, click
"Diagram" on the top menu, and
click "Draw Observed." Move the
cursor to the place where you want
to place an observed variable and
click your mouse. Drag the box in
order to adjust the size of the box.
You can also use in the tool
box to draw observed variables.
2. Unobserved variables can be drawn
similarly. Click "Diagram" and
"Draw Unobserved." Unobserved
variables are shown as circles.
You may also use in the toolbox
to draw unobserved variables.
Drawing in AMOS
To draw a path, Click “Diagram” on the top menu and
click “Draw Path”.
Instead of using the top menu, you may use the Tool
Box buttons to draw arrows ( and ).
Drawing in AMOS
To draw Error Term to the observed and unobserved
variables.
Use “Unique Variable” button in the Tool Box. Click
and then click a box or a circle to which you want to
add errors or a unique variables.(When you use "Unique
Variable" button, the path coefficient will be automatically
constrained to 1.)
Drawing in AMOS
Let us draw:

1 1

1
Naming the variables in AMOS
double click on the objects in the path
diagram. The Object Properties dialog box
appears.

• OR
Click on the Text tab and
enter the name of the
variable in the Variable name
field:
Naming the variables in AMOS
Example: Name the variables

IDM SEX1

FRBEHB1 ISSUEB1

1 1

efr1 eiss

SXPYRC1

eSXPYRC1
Constraining a parameter in AMOS
The scale of the latent variable or variance of the
latent variable has to be fixed to 1.
Double click on the
arrow between EXPYA2
and SXPYRA2.
The Object Properties
dialog appears.
Click on the Parameters
tab and enter the value
“1” in the Regression
weight field:
Improving the appearance
of the path diagram
You can change the appearance of your path diagram
by moving objects around
To move an object, click on the Move icon on the
toolbar. You will notice that the picture of a little moving
truck appears below your mouse pointer when you
move into the drawing area. This lets you know the
Move function is active.
Then click and hold down your left mouse button on the
object you wish to move. With the mouse button still
depressed, move the object to where you want it, and
let go of your mouse button. Amos Graphics will
automatically redraw all connecting arrows.
Improving the appearance of the
path diagram
To change the size and shape of an object, first
press the Change the shape of objects icon on the
toolbar.
You will notice that the word “shape” appears
under the mouse pointer to let you know the
Shape function is active.
Click and hold down your left mouse button on the
object you wish to re-shape. Change the shape of
the object to your liking and release the mouse
button.
Change the shape of objects also works on two-
headed arrows. Follow the same procedure to
change the direction or arc of any double-headed
arrow.
Improving the appearance of the
path diagram
If you make a mistake, there are always
three icons on the toolbar to quickly bail you
out: the Erase and Undo functions.

To erase an object, simply click on the Erase

icon and then click on the object you wish to
erase.

To undo your last drawing activity, click on

the Undo icon and your last activity
disappears.
Each time you click Undo, your previous
activity will be removed.

If you change your mind, click on Redo to

restore a change.
Performing the
analysis in AMOS
View/Set Analysis
Properties and click on
the Output tab.

There is also an Analysis

Properties icon you can
click on the toolbar. Either
way, the Output tab gives
you the following options:
Performing the analysis in AMOS
For our example, check the Minimization history,
Standardized estimates, and Squared multiple
correlations boxes. (We are doing this because these
are so commonly used in analysis).

To run AMOS, click on the Calculate estimates

icon on the toolbar.
 AMOS will want to save this problem to a file.

 if you have given it no filename, the Save As

dialog box will appear. Give the problem a file
name; let us say, tutorial1:
Results
When AMOS has completed the
calculations, you have two options for
viewing the output:
 text output,
 graphics output.
For text output, click the View Text ( or
F10) icon on the toolbar.

Here is a portion of the text output for

this problem:
Results for Condom Use Model(see handout)
The model is recursive. Sample size = 893
Chi-square=12.88 Degrees of Freedom =3

Maximum Likelihood Estimates

Estimate S.E. C.R. P
FRBEHB1 <--- SEX1 -.28 .09 -2.98 .00
ISSUEB1 <--- SEX1 .30 .08 3.79 ***
FRBEHB1 <--- IDM -.38 .11 -3.29 ***
ISSUEB1 <--- IDM -.57 .10 -5.94 ***
SXPYRC1 <--- ISSUEB1 .16 .05 3.42 ***
SXPYRC1 <--- FRBEHB1 .49 .04 12.21 ***

Standardized Regression Weights: (Group number 1 - Default model)

Estimate
FRBEHB1 <--- SEX1 -.10
ISSUEB1 <--- SEX1 .12
FRBEHB1 <--- IDM -.11
ISSUEB1 <--- IDM -.19
SXPYRC1 <--- ISSUEB1 .11
SXPYRC1 <--- FRBEHB1 .38
 Results for Condom Use Model
Covariances: (Group number 1 - Default model)

Estimate S.E. C.R. P Label

SEX1 <--> IDM -.02 .01 -2.48 .01

Correlations: (Group number 1 - Default model)

Estimate

SEX1 <--> IDM -.08

Viewing the graphics output in AMOS
• To view the graphics output, click
the View output icon next to the
drawing area.

• Chose to view either unstandardized

or (if you selected this option)
standardized estimates by click one or
the other in the Parameter Formats
panel next to your drawing area:
Viewing the graphics output in AMOS

Unstandardized -.02
Standardized -.08

.17
.25

IDM SEX1 IDM SEX1

-.57 -.28 -.19 -.10

-.38 .30 -.11 .12

.02 .06

FRBEHB1 ISSUEB1 FRBEHB1 ISSUEB1

1.94
1 1 1.36

efr1 eiss efr1 eiss

.49 .16 .38 .11

.15
0.15 is the squared multiple
correlation between
SXPYRC1 SXPYRC1

1
2.80 Condom use and
eSXPYRC1 ALL OTHER variables eSXPYRC1
How to read the
Output in AMOS

See the handout_1

Modification of the Model
Search for the better model

Suggestions from: 1) theory

2) modification indices
using AMOS
Modifying the Model using AMOS
View/Set Analysis
Properties and click on
the Output tab.
Then check the
Modification indices
option
Modifying the Model using AMOS
Modification Indices (Group number 1 - Default model)

Covariances: (Group number 1 - Default model)

M.I. Par Change Parameter

increase
eiss <--> efr1 9.909 .171

Chi-square
decrease
Modifying the Model using AMOS
2.38, .17 -.02 1.45, .25
IDM SEX1
-.57 -.28
-.38 .30
3.74
5.58
FRBEHB1 ISSUEB1

0, 1.94 1 1 0, 1.36
efr1 .49 .16 eiss
.17

3.08 SXPYRC1
1 0, 2.80
eSXPYRC1

SEE Handout # 2 for the whole output

Examples using AMOS
Condom Use Model with missing values

Confirmatory Factor Analysis for

Impulsive Decision Making construct

Multiple group analysis

How to deal with non-normal data

Missing data in AMOS
Full Information Maximum Likelihood
estimation
• View/Set -> Analysis Properties and
click on the Estimation tab.
• Click on the button Estimate Means and
Intercepts. This uses FIML estimation

Recalculate the previous example with

data “AMOS_data.sav” with some
missing values
Missing data in AMOS
The standardized graphical output.
-.10

IDM SEX1

-.18 -.09
-.10 .12

.02 .05

FRBEHB1 ISSUEB1

efr1 eiss
.37 .08

.14

SXPYRC1

eSXPYRC1
Missing data in AMOS

Example: see the handout #3

Confirmatory Factor Analysis with
Impulsive Decision Making scale

Need to fix either the variance of the IDM1 factor or

one of the loadings to 1.
0, 0, 0, 0,
e1 e2 e3 e4
1 1 1 1
IDMA1R IDMC1R IDME1R IDMJ1R

idm1 0,
Confirmatory Factor Analysis with
Impulsive Decision Making scale
Multiple
Correlation
e1 e2 e3 e4

.30 .26 .47 .47

IDMA1R IDMC1R IDME1R IDMJ1R

.55 .51 .69 .69

idm1 Factor
Loadings

Chi-square = 11.621 Degrees of freedom = 2, p=0.003

CFI=0.994, RMSEA=0.042
Confirmatory Factor Analysis with
Impulsive Decision Making scale

What if want to compare two NESTED

models for Impulsive Decision Making
Model?
1) error variances equal for all 4 measured
variables
2) error variances are different
Confirmatory Factor Analysis with
Impulsive Decision Making scale:
the error variances are the same
Need to give names to the error variances, by double
clicking on the error variance. The Object properties
will appear, click on the Parameter and type the
name for the error variance( e1, e2...) in the
Variance box.
Confirmatory Factor Analysis with
Impulsive Decision Making scale
0, e1 0, e2 0, e3 0, e4
e1 e2 e3 e4
1 1 1 1

IDMA1R IDMC1R IDME1R IDMJ1R

1
0,

idm1
Confirmatory Factor Analysis with
Impulsive Decision Making scale:
error variances are the same
Click MODEL FIT , then Manage Models
In the Manage Models window, click on New.
In the Parameter Constraints segment of the window
type “e1=e2=e3=e4”

Now there are

two nested models
 Confirmatory Factor Analysis with
Impulsive Decision Making scale
error variances are the same error variances are different
0, .45 0, .58 0, .47 0, .43
0, .48 0, .48 0, .48 0, .48
e1 e2 e3 e4
e1 e2 e3 e4

1 1 1 1 1 1 1 1

2.18 2.44 2.24 2.28 2.18 2.44 2.24 2.28

IDMA1R IDMC1R IDME1R IDMJ1R IDMA1R IDMC1R IDME1R IDMJ1R

1.15 1.50 1.03 1.48

1.00 1.36 1.00 1.40
0, .19
0, .19

idm1
idm1

Chi-square = 56.826, Chi-square = 11.621,

df=5, p=0.000 df=3, p=0.003
 Confirmatory Factor Analysis with
Impulsive Decision Making scale:
error variances are the same
Compare Nested Models using Chi-square
difference test:
Model2( errors the same) Model1 ( errors are different)
Chi-square = 56.826, Chi-square = 11.621,
df=5, p=0.000 df=3, p=0.003

Chi-squaredifference=56.826-11.621=45.205
df=5-3=2
Chi-squarecritical value=5.99  Significant
Model 2 with Equal error variances fits WORSE
than Model 1
Confirmatory Factor Analysis with
Impulsive Decision Making scale:
error variances are the same

Nested Model Comparisons

Assuming model Error are free to be correct:

NFI IFI RFI TLI

Model DF CMIN P Delta-1 Delta-2 rho-1 rho2
Errors are the same 3 45.205 .000 .026 .026 .032 .032
Multiple group analysis
WHY: test the equality/invariance of the factor
loadings for two separate groups
HOW :
1) test the model to both groups separately to check
the entire model
2) the same model by multiple group analysis

Example: Do Males and Females can be fitted to

the same Condom USE model?

Need to have 2 separate data files for each group.

 data_boys and data_girls.
Multiple group analysis
• Select Manage Groups... from the
Model Fit menu.
• Name the first group “Girls”.
• Next, click on the New button to
add a second group to the analysis.
• Name this group “Boys”.
• AMOS 4.0 will allow you to
consider up to 16 groups per
analysis.

• Each newly created group is

represented by its own path diagram
Multiple group analysis
• Select File->Data Files... to
launch the Data Files dialog
box.

• For each group, specify the

relevant data file name.

• For this example, choose

the data_girls SPSS
database for the girls' group;

• choose the data_boys

SPSS database for the boys'
group.
Multiple group analysis
The following models fit to both groups (see handout) :
Click Model Fit and Multiple Groups.
This gives a –name
Unconstrained to everyare
all parameters parameter
different inineach
the group
model in each
group.
Measurement weights – regression loadings are the same in both
groups
Measurement intercepts – the same intercepts for both groups
Structural weights – the same regression loadings between the
latent var.
Structural intercepts – the same intercepts for the latent variables
Structural covariates – the same variances/covariance for the
latent var.
Structural residuals – the same disturbances
Measurement residuals – the same errors-THE MOST RESTRICTIVE
MODEL
 Example: Multiple group analysis
for Condom use Model
0, .48 0, .65 0, .47 0, .47
0, .47 0, .62 0, .44 0, .39
eidm1 eidm2 eidm3 eidm4
1 1 1 1 eidm1 eidm2 eidm3 eidm4
2.33 2.60 2.39 2.43 1 1 1 1
2.21 2.41 2.36 2.40
IDMA1R
IDMC1R
IDME1R
IDMJ1R
1.00 1.04
1.58 1.41 IDMA1R
IDMC1R
IDME1R
IDMJ1R
0, .18 1.00 1.14
1.56 1.45
0, .16
Impulsive
Impulsive

-.62 -.64
-.28 -.38
4.35 2.72
4.12 3.06

FRBEHB1 ISSUEB1 FRBEHB1 ISSUEB1

0, 2.12
1 1 0, 1.50
0, 1.81
1 1 0, 1.13

efr1 eiss eiss

.40 .11 efr1 .62 .26

3.63 2.16

SXPYRC1 SXPYRC1
1 1
0, 2.95 0, 2.56

eSXPYRC1 eSXPYRC1

Boys UNCONSTRAINED MODEL Girls

 Example: Multiple group analysis for
Condom use Model
0, .47 0, .63 0, .43 0, .40
0, .48 0, .64 0, .48 0, .46 eidm1 eidm2 eidm3 eidm4
eidm1 eidm2 eidm3 eidm4 1 1 1 1
2.21 2.41 2.36 2.40
1 1 1 1
2.33 2.60 2.39 2.43
IDMA1R IDMC1R IDME1R IDMJ1R
IDMA1R IDMC1R IDME1R IDMJ1R
1.57
1.57
1.08 1.42
1.08 1.42
1.00
1.00
Impulsive Impulsive
0, .18 0, .16

-.50 -.50
-.45 -.45

4.35 2.72
4.12 3.06
FRBEHB1 ISSUEB1
FRBEHB1 ISSUEB1
1 1 0, 1.51

0, 2.14 .40 .11 1 1 0, 1.12

efr1 eiss
0, 1.81 .62 .26 eiss
3.62 efr1

SXPYRC1 2.16

1 SXPYRC1
0, 2.95

eSXPYRC1 1
0, 2.56

Boys Measurement weights Girls eSXPYRC1

Example: Multiple group analysis for
Condom use Model
see handout

Since Measurement Weights model is

nested within Unconstrained .

Chi-square difference test computed to

test the null hypothesis that the regression
weights for boys and girls are the same.
However, the variances and covariance are
different across groups.
Example: Multiple group analysis for
Condom use Model

Chi-squarediff =68.901-65.119=2.282
df=29-26=3  NOT SIGNIFICANT

FIT of the Measurement Weights model is

not significantly worse than Unconstrained
Handling non-normal data:
Verify that your variables are not distributed
joint multivariate normal

Assess overall model fit using the Bollen-

Stine corrected p-value

Use the bootstrap to generate parameter

estimates, standard errors of parameter
estimates, and significance tests for
individual parameters
Handling non-normal data: checking for
normality
To verify that the data is not
normal. Check the Univariate
SKEWNESS and KURTOSIS for
each variable .
• View/Set -> Analysis Properties
and click on the Output tab.

•Click on the button Tests for

normality and outliers
 Handling non-normal data: checking for
normality
Assessment of normality

Variable min max skew c.r. kurtosis c.r.

IDM 1.182 3.727 .381 4.649 .496 3.025
SEX1 1.000 2.000 .182 2.222 -1.967 -11.997
FRBEHB1 1.000 6.000 -.430 -5.245 -.778 -4.748
ISSUEB1 1.000 4.000 -.431 -5.259 -1.387 -8.462
SXPYRC1 2.000 7.000 -.937 -11.436 -.715 -4.360
Multivariate -3.443 -6.149

Critical ratio of +/- 2 for skewness and kurtosis

statistical significance of NON-NORMALLITY
Multivariate kurtosis >10  Severe Non-normality
Handling non-continuous data:
Bootstrapping
Use Bootstrapping

Bootstrapping generates an estimate of the sampling

distribution from the available data and computes the
p-values and construct confidence intervals.
Bootstrapping in AMOS generates random covariance
matrices from the sample covariance matrix assuming
multivariate normality
Handling non-continuous data:
Bootstrapping
Bootstrapping is useful for estimating standard errors
for statistics with complex distributions, for which
there is no practical approximate
However, Some limitations include:
 The “population” in nonparametric bootstrapping is merely
the researcher’s sample
 If the researcher’s sample is small, unrepresentative, or the
observations are not independent, resembling from it can
 magnify
 the effects of these features (see Rodgers, 1999)
 Bootstrap analyses are probably biased in small samples
(just as
 they are in other methods)—that is, bootstrapping is not a
“cure”
Handling non-normal data:Bollen-Stine
bootstrapping p-value•View/Set -> Analysis Properties
and click on the Bootstrap tab.
Check Perform bootstrap and Bollen-
Stine bootstrap

BOLLEN_STINE BOOTSTRAP performed

only for dataset without any missing
values

(see handout #6:

amos_data_valid_condom.sav)
 Handling non-normal data:Bollen-Stine
bootstrapping p-value

The model fits better than expected in 496 samples out of 500 samples

(500-496)/500=0.010
So, p-value=0.01 < 0.05 - Model does not fit to the data very well
Handling non-normal data:Bollen-Stine
bootstrapping p-value
Overall Model Fit:
Chi-square=12.88;
Degrees of freedom = 3

The expected(mean) value of

Chi-square is 2.929.
The mean value of Chi-
square (2.929) serves as
the critical chi-square
value against which the
obtained chi-square of
12.88 is compared
In our example, results from the Bollen-Stine are the same as results
for the overall model.
Handling non-normal data: Bootstrapping
Standard Errors
Bootstrapping can be used
to evaluate the estimates,
by computing the
Standard Errors of the
estimates
UnSELECT Bollen-Stine
Bootstrap
and Select Percentile
Confidence Intervals and
Bias-corrected confidence
intervals
Handling non-normal data: Bootstrapping
Standard Errors
Estimates using ML
Relationship between
Condom use and Peer
Norms about Condom
is 0.487, with
S.E.=0.04,
Almost the same
Bootstrap estimates
estimate produced by
Bootstrap, 0.488 with
S.e=0.042
 Handling non-normal data: Bootstrapping
Standard Errors
90% Percentile Method 90% Bias Corrected Percentile method

Hope to see similar results for the estimates

NOTE: BOOTSTRAP option works ONLY with COMPLETE data

Handling non-normal data: Bootstrapping
Standard Errors

NOTE: BOOTSTRAP option works ONLY with COMPLETE data

if missing is less than 5% , it is defensible to use LISTWISE
deletion
Sample size should be reasonably large with 200 for SEMs that
contain latent variables ( by Nevitt and Hancock, 1998)
Thank You!

See you in a week!

Upper critical values of chi-square
distribution
Degree of freedom Chi-square critical value

1 3.841
2 5.991
3 7.815
4 9.488
5 11.070
6 12.592
7 14.067
8 15.507
9 16.919
10 18.307
11 19.675