Workshop on Structural equation modeling in AMOS (Analysis and Moment of Structure)

1. STRUCTURAL
EQUATION MODELING
IN AMOS
Balaji.P

2. Basic Concept
 Partial Correlation:
Correlation between Y and X1 where effects of
X2 have been removed from X1 but not from Y
is partial correlation

3. Interpretation of Partial
correlation
 Part correlation squared is the unique amount of total
variance explained.
 Sum of part correlations squared does NOT equal R
square because of overlapping variance.
 Multi collinearity: Existence of substantial
correlation among a set of independent
variables.

4. Latent variable
Latent variables:
representation of the
variance shared among
the variables
common variance
without error or specific
variance

5. Structural Equation Modeling
 Structural equation modeling (SEM), as a
concept, is a combination of statistical techniques
such as exploratory factor analysis and multiple
regression.
 The purpose of SEM is to examine a set of
relationships between one or more Independent
Variables (IV) and one or more Dependent

6. Goals of SEM
 To understand the patterns of
correlation/covariance among a set of variables.
 To explain as much of their variance as possible
with the model specified .

8. How SEM is different from traditional
approach?
 Multiple equations can be estimated
simultaneously
 Non-recursive models are possible
 Correlations among disturbances are possible
 Formal specification of a model is required
 Measurement and structural relations are
separated, with relations among latent variables
rather than measured variables
 Assessing of model fit is not as straightforward

9. Types of SEM models
 Path analysis
 Confirmatory factor analysis.
 Structural regression model
 Latent change model

10. Approach to SEM analysis
 Review the relevant theory and research literature to support model
specification
 Specify a model (e.g., diagram, equations)
 Determine model identification
 Collect data
 Conduct preliminary descriptive statistical analysis
 Estimate parameters in the model (Model Estimation)
 Assess model fit
 Model Respecification
 Interpret and present results.

11. Components of SEM
Latent variables, factors, constructs
Observed variables, measures, indicators,
manifest variables
Direction of influence, relationship from one variable
to another
Association not explained within the model

12. Important Definition
 A measured variable (MV) is a variables that is
directly measured.
 A latent variable could be defined as whatever its
multiple indicators have in common with each
other. It isn't measured directly.
 Relationships between variables are of three
types such as Association (Correlation,
covariance), direct effect and indirect effect.

13. Path Analysis
 Extension of multiple regression allowing us to
consider more than one dependent variable at a time
and more importantly, allowing variables to be both
Dependent and Independent variables.
 B is dependent as well as independent variable
(mediating variable).

14. Path Analysis
Once the data is available, conduction of path
analysis is straightforward:
 Draw a path diagram according to the theory.
 Conduct one or more regression analyses.
 Compare the regression estimates (B) to the
theoretical assumptions or (Beta) other studies.
 If needed, modify the model by removing or adding
connecting paths between the variables and redo
stages 2 and 3.

15. Examples of Path Analysis
illness = p5p3 fitness + p5p4 stress + p5p1
exercise + p5p2 hardy + e5;

16. Examples of Path Analysis
fitness = p3p1 exercise + e3,
stress = p4p2 hardy + e4,
illness = p5p3 fitness + p5p4 stress + e5;

17. Software’s (SEM)
 LISREL
 AMOS
 EQS
 MPLUS
 SAS

18. AMOS (Analysis of Moment
Structures)
 Starting AMOS Graphics

19. Reading Data in to AMOS

20. Filename Data.sav

21. AMOS can read
 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,

22. Data File

23. List variable in data set

24. Drawing in AMOS (Draw observed
variable)
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.
Click Draw
unobserved

25. Direct effect
Click path icon of direct
effect and click respective
independent variable drag
up to dependent variable

26. Touch up variable
It gives neat look to our
model. Click the touch up
variable and click respective
observed or unobserved
which you want to look
neat.

27. Add unique variable
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.)

28. Naming the variables in AMOS
Click list all the variable. Drag
the variable from list and put
directly in to observed variable.

29. Naming the variables in AMOS
 double click on the objects in the path
diagram. The Object Properties dialog box
appears.
And
Click on the Text tab and
enter the name of the
variable in the Variable
name field.

30. Regression Weight
Normally for error
value, regression
weight takes the
value of 1.

31. Performing the analysis in
AMOS
 For our example, check the Minimization
history, Standardized estimates, and Squared
multiple correlations boxes.
 To run AMOS, click on the Calculate estimates
icon on the toolbar.
 AMOS will want to save this problem to a file.

32. 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 icon on
the toolbar.

33. Text output

34. 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 standardized
estimates by click one or the other in the Parameter
Formats panel next to your drawing area

35. Standardized vs. Unstandardized
 Standardized coefficients can be compared across variables
within a model.
 Standardized coefficients reflect not only the strength of the
relationship but also variances and covariance's of variables
included in the model as well of variance of variables not
included in the model and subsumed under the error term.
 Standardized parameter estimates are transformations of
unstandardized estimates that remove scaling and can be
used for informal comparisons of parameters throughout the
model.

36. Standardized vs. Unstandardized
 Unstandardized parameter estimates retain
scaling information of variables and can only be
interpreted with reference to the scales of the
variables.
 A correlation matrix standardizes values and loses
the metric of the scales.
 Therefore for correlation matrix, both standardizes
and unstandardized are same.

37. Graphics output

39. 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.

40. 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.

41. SEM could impacted by
 the requirement of sufficient sample size. A desirable goal is to have
a 20:1 ratio for the number of subjects to the number of model
parameters . However, a 10:1 may be a realistic target. If the ratio is
less than 5:1, the estimates may be unstable.
 measurement instruments
 multivariate normality
 parameter identification
 outliers
 missing data
 interpretation of model fit indices

42. Model Identification
 A model is identified if:
 It is theoretically possible to derive a unique estimate
of each parameter
 The number of equations is equal to the number of
parameters to be estimated
 It is fully recursive (No feedback loop)

43. Model identification
 A model is over identified if:
 A model has fewer parameters than observations.
 There are more equations than are necessary for the
purpose of estimating parameters

44. Model identification
 A model is under identified or not identified if:
 It is not theoretically possible to derive a unique
estimate of each parameter
 There is insufficient information for the purpose of
obtaining a determinate solution of parameters.
 There are an infinite number of solutions may be
obtained

45. Model identification
 Determine the # of parameters you have.
 Formula: (v(v+1) / 2), where v= # of observed
variables
 Use of this formula, allows to see if trying to guess
more than the number of parameters the existing data
allows.
 Do not want to be JUST identified (cause lack of fit
indices) or UNDER identified, therefore looking to be
OVER-identified.
 Being OVER identified essentially means that there are more
available parameters than trying to estimate.

46. Example

47. Example

48. Example

49. Example

50. Model Estimation
 Maximum Likelihood
 Generalized and UnGeneralized least square
 2 stage and 3 stage least square

51. Model fit
 Model fit = sample data are consistent with the
implied model
 The smaller the discrepancy between the
implied model and the sample data, the better
the fit.
 Many fit indexes
 None are fallible (though some are better than others)

52. Fit indexes

53. Fit indexes

54. Model Respecification
What if the model does NOT fit?
 Model trimming and building
 LaGrange Multiplier test (add parameters)
 Wald test (drop parameters)
 Empirical vs. theoretical respecification
 What justification do you have to respecify?
 Consider equivalent models

55. Confirmatory factor analysis
 How it differs from the more commonly
encountered forms of factor analysis.
 What is factor analysis (FA)?
 have many variables and want to examine if they can be
explained by a smaller number of factors.
 No a priori hypothesis (impossible to even indicate a hunch
to the program) as to which variables will cluster together
on which factor.

56. Confirmatory factor analysis
 The major difference is that an a priori
hypothesis is essential:
 which variables grouped together as manifestations of
an underlying construct and fits the model.
 Like with path analysis, it can be helpful to draw
hypothesized relations in a diagram.

57. CFA is not model building
 With CFA, you stipulate where you think the variables
should load. Then, the program simply tells you
whether your model fits the data.
 If no fit, then there are few clues to guide you how to
shuffle the variables around to make the model better
fit the data.
 Note: Even if the model does fit, it does not guarantee that a
new arrangement of variables would be an even better fit.
 Therefore, one must really use theory, knowledge, or previous
research to guide your model, rather than rely on statistical
criteria.

58. Scaling
 Scaling factor: constrain one of the factor
loadings to 1 ( that variables called – reference
variable, the factor has a scale related to the
explained variance of the reference variable).
 fix factor variance to a constant ( ex. 1), so all
factor loadings are free parameters

59. CFA

60. Duplicate object
Select object which you
want to duplicate. Click
wherever you wanted
same variable.

61. Unobserved
Click unobserved icon
and draw like observed icon

62. Correlation or covariance

63. Unstandardized and Standardized
estimates

64.  Unstandardized solution
 Factor loadings =unstandardized regression coefficient
 Unanalyzed association between factors or errors=
covariances
 Standardized solution
 Unanalyzed association between factors or errors=
correlations
 Factor loadings =standardized regression coefficient (
structure coefficient)
 The square of the factor loadings = the proportion of
the explained ( common) indicator variance,
R2(squared multiple correlation)

65. Standardized regression model
 Inclusion of observed and latent variables
 Assessment both of relationship between
observed and latent variables.

66. Latent growth Analysis
 Can change in responses be tracked over
time?
 Latent Growth Curve Analysis