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
Bivariate
Correlation
ANOVA
Family Tree of SEM
Multi-way
ANOVA
Multiple
Regression
Factor
Analysis
Repeated
Measure
Designs
Path
Analysis
Growth
Curve
Analysis
Structural
Equation
Modeling
Confirmatory
Factor
Next Workshop:
Analysis
November 9
Exploratory
Factor
Analysis
See you there!
Latent
Growth
Curve
Analysis
Structural Equation Modeling
(SEM)
Exogenous variables=independent
Endogenous variables =dependent
Observed variables =measured
Latent variables=unobserved
Structural Equation Graphs
.10
Error
: R2
Observed
Variable
.15 : Loading
Latent
Variable
Example: Condom Use Model
Observed variables for
Impulsive decision making
IDMA1R
IDMC1R
IDME1R
Respondent Sex
IDMJ1R
SEX1
Impulsive
Impulsive
Decision Making
Legend
Observed
Variables
.15
FRBEHB1
Peer norms about
condoms
Latent
SXPYRC1
ISSUEB1
Condom attitude
Variables
Loadings
Condom Use
Example: Condom Use Model
Dependent
Independent
IDMA1R
IDMC1R
IDME1R
IDMJ1R
SEX1
Impulsive
Independent
Legend
Observed
Variables
.15
FRBEHB1
Dependent
Latent
Variables
Loadings
SXPYRC1
ISSUEB1
Dependent
Dependent
Example: Condom Use Model
eidm1
eidm2
eidm2
eidm4
IDMA1R
IDMC1R
IDME1R
IDMJ1R
SEX1
Impulsive
efr1
FRBEHB1
ISSUEB1
Legend
Observed
Variables
.15
Latent
Variables
Loadings
SXPYRC1
eSXYRC1
eiss
Example: Condom Use Model
eidm1
eidm2
eidm2
eidm4
IDMA1R
IDMC1R
IDME1R
IDMJ1R
SEX1
Impulsive
efr1
FRBEHB1
ISSUEB1
Legend
Observed
Variables
.15
Latent
Variables
Loadings
SXPYRC1
eSXYRC1
eiss
Example: Condom Use Model
eidm1
eidm2
eidm2
eidm4
.28
IDMA1R
.24
IDMC1R
.48
IDME1R
.45
IDMJ1R
.53
.49
.69
.67
Impulsive
-.15
efr1
.15
-.19
Latent
Variables
Loadings
SEX1
-.10
.13
.03
FRBEHB1
.38
Legend
Observed
Variables
-.06
.11
.05
ISSUEB1
SXPYRC1 .15
eSXYRC1
eiss
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
1. To draw an observed variable, click
screen
"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
eiss
efr1
SXPYRC1
1
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 twoheaded 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
FRBEHB1
ISSUEB1
FRBEHB1
ISSUEB1
SXPYRC1
SXPYRC1
<--<--<--<--<--<---
SEX1
SEX1
IDM
IDM
ISSUEB1
FRBEHB1
Estimate
-.28
.30
-.38
-.57
.16
.49
S.E.
.09
.08
.11
.10
.05
.04
C.R.
-2.98
3.79
-3.29
-5.94
3.42
12.21
P
.00
***
***
***
***
***
Standardized Regression Weights: (Group number 1 - Default model)
FRBEHB1
ISSUEB1
FRBEHB1
ISSUEB1
SXPYRC1
SXPYRC1
<--<--<--<--<--<---
SEX1
SEX1
IDM
IDM
ISSUEB1
FRBEHB1
Estimate
-.10
.12
-.11
-.19
.11
.38
Results for Condom Use Model
Covariances: (Group number 1 - Default model)
SEX1
<-->
Estimate
S.E.
C.R.
P
-.02
.01
-2.48
.01
IDM
Correlations: (Group number 1 - Default model)
Estimate
SEX1
<-->
IDM
-.08
Label
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
.17
Standardized
-.08
.25
IDM
IDM
SEX1
-.57
SEX1
-.19
-.28
-.38
-.10
-.11
.30
.12
.02
FRBEHB1
efr1
FRBEHB1
ISSUEB1
1.94
1
1
.49
efr1
0.15 is the squared multiple
SXPYRC1
1
2.80
eSXPYRC1
ISSUEB1
1.36
eiss
.16
.06
correlation between
Condom use and
ALL OTHER variables
.38
eiss
.11
.15
SXPYRC1
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.
eiss
<-->
efr1
Chi-square
decrease
9.909
Par Change
.171
Parameter
increase
Modifying the Model using AMOS
2.38, .17
-.02
1.45, .25
IDM
SEX1
-.57
-.38
-.28
.30
3.74
5.58
0, 1.94
1
efr1
FRBEHB1
.49
ISSUEB1
1
.16
.17
3.08
SXPYRC1
1
0, 2.80
eSXPYRC1
SEE Handout # 2 for the whole output
0, 1.36
eiss
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
efr1
ISSUEB1
.37
eiss
.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,
e2
e1
1
1
0,
0,
e4
e3
1
IDMA1R IDMC1R
IDME1R
1
IDMJ1R
1
idm1
0,
Confirmatory Factor Analysis with
Impulsive Decision Making scale
e1
e2
e3
.30
IDMA1R
.26
IDMC1R
.55
.51
idm1
e4
.47
IDME1R
.69
Multiple
Correlation
.47
IDMJ1R
.69
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
e1
1
0, e3
0, e4
e2
e3
e4
1
1
1
IDMA1R IDMC1R
1
IDME1R
0,
idm1
IDMJ1R
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, .48
0, .48
e1
0, .48
e2
1
e3
1
2.18
IDMA1R
2.44
1.00
IDME1R
1.15
1.50
2.28
IDMJ1R
1.36
0, .19
idm1
Chi-square = 56.826,
df=5, p=0.000
0, .47
e2
1
1
2.24
IDMC1R
e1
e4
1
0, .58
0, .43
0, .48
e3
1
2.18
IDMA1R
1
2.44
1
2.24
IDMC1R
1.00
e4
IDME1R
1.03
1.48
2.28
IDMJ1R
1.40
0, .19
idm1
Chi-square = 11.621,
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)
Chi-square = 56.826,
df=5, p=0.000
Model1 ( errors are different)
Chi-square = 11.621,
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:
Model
Errors are the same
P
NFI
Delta-1
3 45.205 .000
.026
DF
CMIN
IFI
RFI TLI
Delta-2 rho-1 rho2
.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.
Unconstrained
all parameters
different inineach
This
gives a –name
to everyare
parameter
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
eidm1 eidm2 eidm3 eidm4
1
2.33
1
2.60
1
2.39
0, .44
0, .39
eidm1 eidm2 eidm3 eidm4
1
2.43
1
2.21
IDMA1R
IDMC1R
IDME1R
IDMJ1R
1
2.41
1
2.36
1
2.40
IDMA1R
IDMC1R
IDME1R
IDMJ1R
1.58 1.41
1.00 1.04
0, .18
1.56 1.45
1.00 1.14
0, .16
Impulsive
Impulsive
-.64
-.62
-.38
-.28
4.35
2.72
FRBEHB1
4.12
ISSUEB1
0, 2.12
1
efr1
0, .62
FRBEHB1
0, 1.50
1
.40
eiss
.11
ISSUEB1
0, 1.81
1
efr1
.62
eiss
.26
2.16
SXPYRC1
SXPYRC1
1
1
0, 2.95
eSXPYRC1
UNCONSTRAINED MODEL
0, 1.13
1
3.63
Boys
3.06
0, 2.56
eSXPYRC1
Girls
Example: Multiple group analysis for
Condom use Model
0, .47
0, .48
eidm1
0, .64
eidm2
1
2.33
IDMA1R
0, .48
eidm3
1
2.60
IDMC1R
0, .63
eidm2
1
2.21
eidm4
1
2.39
IDME1R
eidm1
0, .46
1
2.43
0, .43
eidm3
1
2.41
0, .40
eidm4
1
2.36
1
2.40
IDMA1R IDMC1R IDME1R IDMJ1R
IDMJ1R
1.57
1.57
1.08
1.08
1.42
1.42
1.00
1.00
Impulsive
Impulsive
0, .16
0, .18
-.50
-.45
-.50
-.45
4.35
2.72
FRBEHB1
4.12
3.06
ISSUEB1
FRBEHB1
1
efr1
0, 2.14
1
.40
.11
1
eiss
efr1
3.62
ISSUEB1
0, 1.51
0, 1.81
.62
.26
0, 2.95
SXPYRC1
1
eSXPYRC1
Boys
eiss
2.16
SXPYRC1
1
Measurement weights
0, 1.12
1
Girls
0, 2.56
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 BollenStine 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
-3.443
-6.149
Multivariate
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 Chisquare (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
Bootstrap estimates
is 0.487, with
S.E.=0.04,
Almost the same
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