This document discusses regression models, path models, and the output from AMOS software when conducting structural equation modeling (SEM). Regression models only include observed variables and assume independents are measured without error. Path models allow independents to be both causes and effects, and allow for error terms on endogenous variables. The AMOS output provides standardized and unstandardized regression weights, significance tests, and fit indexes to evaluate how well the specified model fits the sample data.

1. By Hui Bian
Office For Faculty Excellence
Fall 2011
1

2.  Regression models:
 only observed variables are modeled.
 only the dependent variable in regression has an error term. Independent
variables are assumed to be modeled without error.
 The partial coefficient for any independent variable controls for all other
independents, whether or not an actual causal effect is plausible.
 Path models:
 only observed variables without latent variables.
 Unlike regression models but like structural equation models, independents
can be both causes and effects of other variables.
 Only the endogenous variables in path models have error terms. Exogenous
variables in path models are assumed to be measured without error.
 Partial coefficients are calculated using only the independents in a direct path
to the endogenous variable.
2

3.  AMOS output
 Standardized regression weights:
 Structural or path coefficients in SEM. Standardized
estimates are used, for instance, when comparing direct
effects on a given endogenous variable in a single-group
study.
 Indicator variable regression weights. By convention, the
indicator variables should have standardized regression
weights of .7 or higher on the latent variable they
represent.
3

4.  AMOS output:
 Communalities.
 The Squared Multiple Correlation is the communality estimate for
an indicator variable.The communality measures the percent of
variance in a given indicator variable explained by its latent variable
(factor) and may be interpreted as the reliability of the indicator.
 If a variable has low theoretic importance and a low communality, it
may be targeted for removal in the model-modification.
 The communality is equal to the squared standardized regression
weight.This is why communalities are sometimes defined as the
squared factor loadings, where loadings are defined as the
standardized regression weights.
4

5.  AMOS output
 Unstandardized regression weights: are based on raw
data or covariance matrixes.
 When comparing across groups (across samples) and
groups have different variances, unstandardized
comparisons are preferred.
5

6.  AMOS output
 The critical ratio and significance of path coefficients.
When the critical ratio (CR) is > 1.96 for a regression
weight, that path is significant at the .05 level or better
(that is, its estimated path parameter is significant). In the
p-value column, three asterisks (***) indicate significance
smaller than .001.
 The critical ratio and the significance of factor covariances.
The significance of estimated covariances among the
latent variables are assessed in the same manner: if CR >
1.96, the factor covariance is significant.
6

7.  Purpose of this exercise is to show you how
AMOS estimates parameters in multiple
regression.
 Data used is from Schumacker and Lomax
(2004).
 We have three predictors and one dependent
variable.
7

8. 8
 Path diagram

9.  Draw path diagram using AMOS: File > New
 Three independent variables: IV1-IV3 (observed)
 One dependent variable: DV
 No latent variables in this model
 Three independent variables are correlated
9

10.  The single-headed arrows represent linear
dependencies. For example, the arrow leading from
IV1 to DV indicates that DV scores depend, in part,
on IV1.
 The variable error is enclosed in a circle because it is
not directly observed. Error represents much more
than random fluctuations in DV scores due to
measurement error.
10

11.  Model identification
 The variance of a variable, and any regression
weights associated with it, depends on the units in
which the variable is measured.
 Error is an unobserved variable, there is no natural
way to specify a measurement unit for it.
Assigning an arbitrary value to a regression weight
associated with error can be thought of as a way of
indirectly choosing a unit of measurement for
error. 11

12.  Model identification
 It is impossible to estimate the regression weight and
variance for the regression of DV on error,There is just
not enough information.
 We can solve this identification problem by fixing
either the regression weight applied to error in
predicting DV, or the variance of the error variable
itself, at an arbitrary, nonzero value. Let’s fix the
regression weight at 1.This will yield the same
estimates as conventional linear regression.
12

13.  Model identification
 Every unobserved variable presents this
identification problem, which must be resolved by
imposing some constraint that determines its unit
of measurement.
 Changing the scale unit of the unobserved error
variable does NOT change the overall model fit.
13

14.  Fix regression weight
 Right-click the arrow that points from error to DV and
choose Object Properties from the pop-up menu.
14
Type 1 inside Regression
weight box, which is under
Parameters tab.

15.  Before run data analysis
 Go to View > Analysis Properties > Click Output tab
15

16.  Text outputs
16

17.  4 sample variances and 6 sample covariances, for a
total of 10 sample moments, or use p (p+1)/2, we
have 4 observed variables, then the number of
sample distinct value is equal to 4(4+1)/2 = 10.
 3 regression paths, 4 model variances, and 3 model
covariances, for a total of 10 parameters that must
be estimated. Hence, the model has zero degrees of
freedom.
 Such a model is often called saturated or just-
identified.
17

18.  Text output
18
The standardized
regression weights and
the correlations are
independent of the units
in which all variables are
measured; therefore,
they are not affected by
the choice of
identification
constraints.

19.  Text output
19
Squared multiple
correlations are independent
of units of measurement.
Amos displays a squared
multiple correlation for each
endogenous variable.

20.  Graphics output (unstandardized estimates)
20

21.  Graphic output (standardized estimates)
21
Correlations

22.  Conclusion
 In this example, IV1, IV2, and IV3 account for 69% of
the variance of DV. IV1 and IV3 are significant
predictors.
22

23.  Path analysis model
 Only focus on relationships of multiple observed
variables
 Analysis of several regression equations
simultaneously.
 Use the same idea of model fitting and testing as any
SEM.
 Data used is still from Schumacker and Lomax’s book:
A beginner’s guide to structural equation modeling
(2004).
23

24.  The research question is whether the specified model is
supported by the sample data?
 Path diagram (please try to draw and identify this model)
24

25. 25
Chi-square
test is not
significant

26.  Unstandardized graphic output
26

27.  Standardized graphic output
27

28.  Choices of model fit indexes
 Reporting CMIN, RMSEA, and one of the baseline fit
measures.
 If there is model comparison, also report one of the
parsimony measures and one the information theory
measures.
28

29.  Model fit
29
The closer RMR is to 0, the better the model fit. Rule of thumb: RMR
should be < .10, or .08, or .06, or .05 or even .04.

30.  Model fit
30
Rule of thumb: a value of the RMSEA of about 0.05 or
less would indicate a close fit of the model in relation to the
degrees of freedom.

31.  Model fit
31
NFI values above .95 are good. RFI, IFI,TLI, and CFI values close
to 1 indicate a very good fit.

32.  Model fit
 Chi-square: χ2 = 1.25, df = 3, p = .74
 Root-mean-square error of approximation (RMSEA): it
is equal to 0.00 (<.05 is acceptable)
 Goodness-of-fit index (GFI): .997 (>.95 is acceptable)
32

33.  Model fit
33

34.  Path diagram
34

35.  This example: 4 tests: knowledge, value,
satisfaction, and performance. Each test was
randomly split into two halves, and each half was
scored separately.
 Measurement model
 The portion of the model that specifies how the
observed variables depend on the unobserved, or
latent, variables is sometimes called the
measurement model.
 The current model has four distinct measurement
submodels.
35

36.  The scores of the two split-half subtests,
1knowledge and 2knowledge, are hypothesized to
depend on the single underlying latent variable,
knowledge.
 According to the model, scores on the two
subtests may still disagree, owing to the
influence of measurement errors.
36

37.  Measurement model (e.g.)
 1knowledge and 2knowledge are called indicators of the
latent variable knowledge.
37
Measurement model

38.  Structural model
 The portion of the model that specifies how the latent
variables are related to each other is sometimes called the
structural model.
38
Structural model

39.  Model identification
 It is necessary to fix the unit of measurement of each
unobserved variable by suitable constraints on the
parameters.
 Find a single-headed arrow leading away from each
unobserved variable in the path diagram, and fix the
corresponding regression weight to an arbitrary value such
as 1.
 If there is more than one single-headed arrow leading
away from an unobserved variable, any one of them will
do.
39

40.  Text output
40
The hypothesis that
current Model is
correct is accepted.

41. 41
Standardized
regression
weights

42.  Reliability estimates
42

43.  The purpose of confirmatory factor analysis is to
test hypothesis about a factor structure.
 The theories come first.
 The model is derived from the theory.
 The model is tested for consistency with observed
data.
43

44.  Diagram of CFA model
44

45.  Two-factor model: spatial ability and verbal ability
 Three observed variables measure each construct.
 The relationship between the factor and its indicator
is represented by a factor loading.
 The measurement error represents other variation
for a particular observed variable.
 The variance of measurement error is estimated.
45

46.  Model summary
46

47.  Regression weights
47

48.  Path diagram with standardized estimates
displayed.
48

49.  Squared multiple correlations
49

50.  The squared multiple correlations can be interpreted
as follows:To take wordmean as an example:
 71% of its variance is accounted for by verbal ability.
 The remaining 29% of its variance is accounted for by the
unique factor e6.
 If e6 represented measurement error only, we could say
that the estimated reliability of wordmean is 0.71.
 0.71 is an estimate of a lower-bound on the reliability of
wordmean.
50

51.  Model fit
 Chi-square: χ2 = 7.85, df = 8, p = .45
 Root-mean-square error of approximation (RMSEA): it
is equal to 0.00 (<.05 is acceptable)
 Goodness-of-fit index (GFI): .966 (>.95 is acceptable)
51

52.  AMOS allows us to compare multiple samples
across the same measurement instrument or
multiple population groups (e.g., males vs.
females).
 We are going to use the data from IBM SPSS
company (the previous data for CFA).
 We want to test the equality of the factor
loadings for two separate groups of school
children, girls and boys.
52

53.  Before testing measurement invariance across
groups, we need test individual mode first.
 If consistency is found, then we will proceed to do
multiple groups testing.
 The goal of testing for measurement invariance is to
determine if the same SEM model is applicable
across groups.
53

54.  The general procedure is to test measurement
invariance between the unconstrained model for all
groups combined, then for a model with constrained
parameters (parameters are constrained to be equal
between the groups).
 If the chi-square difference statistic is not significant
between the original and constrained models, then we
conclude that the model has measurement invariance
across groups.
54

55.  Which parameters are constrained to be equal?
The selection of parameters to constrain
depends on our research questions.
 Invariant factor loadings
 Invariant structural relations among latent variables
 If lack of measurement invariance is found, the
meaning of the latent construct is shifting across
groups.
55

56.  First draw a diagram for a single group
56
By default, Amos Graphics assumes that both groups
have the same path diagram, so the path diagram does not have to
be drawn a second time for the second group.

57.  Select Manage Groups from the Analyze menu.
Name the first group Girls.
 Click on the New button to add a second group
to the analysis. Name this group Boys.
 Click the New button successively to add
additional groups as needed.
57

58. 58
Click
Type
Boys

59.  Select data sets:
 Use of the GroupingVariable and GroupValue buttons.
 Select the GroupingVariable > identify the grouping
variable within a database > Click the GroupValue
button > select which value of the grouping variable
represents the group of interest.
59

60.  File > Data files
60

61.  Open data files
61

62.  We will name the variances, covariances, and regression
weights in both the Girls and Boys models.
 We will name the parameters in Girls’ model first.
 Use the Object Properties dialog box. Uncheck the box
for “All groups”, so you can give the variances different
names in the two groups.
 To name these parameters for the Boys model, highlight
“boys” and go through the same procedure as before,
use different names for variances, covariance, and factor
loadings.
62

63.  There is a good way to name parameters. Go to Plugins >
Click Name Parameters.
63
Check Covariances,
Regression weights, and
Variances

64.  We give parameters different names for Boys
group. Here is a example:
64
Make sure All
groups is NOT
checked
New name for
Boys group

65.  For Girls group
65

66.  For Boys group
66

67.  Double-click on the Default Model label shown
on the left side of the path diagram window.
 The Manage Models window is open.
67

68.  You can rename the default model as something
meaningful (we name it Original model).
 Click New, a new model that imposes a set of
equality constraints on the default model such that
the unstandardized factor loadings are equal across
boys' and girls' groups (we name this model Equal
loading model).
 Identify the four pairs relevant factor loadings of
interest in the girls group and the boys group. By
double-clicking on c1 and then double-clicking on c2.
68

69. 69
 Manage models

70.  Model summary of two models
70

71.  Model parameters for Girls
71

72.  Model parameters for Boys
72

73.  Model fit: we can use CFI, NCP, and GFI because
they are independent of model complexity and
sample size.
73

74.  Model comparison
74

75.  The Chi-square difference of two models is 18.292-
16.480 = 1.812.
 The results from this model comparison (Chi-square =
1.812 with 4 DF, p =.77 ) suggest that imposing the
additional restrictions of four equal factor loadings
across the gender groups did not result in a statistically
significant worsening of overall model fit.
 AMOS assumes that the baseline model (our original
model) is true.The model (equal loading model) that
specifies a group-invariant factor pattern, is supported
by the sample data.
75

76.  Another way to do multiple group analysis
 The first step: set up groups (give names for each
group). Go to analyze > Manage Groups
 The second step: open data. Go to File > Data Files.
76

77.  Go to Analyze > Multiple-Group Analysis
77

78.  Four different models are obtained.
78
Four models test invariances of
measurement weights, structural
covariances, and measurement
residuals.

79.  Model for Girls (AMOS automatically assigns
names for each parameter).
79

80.  Model for Boys
80

81.  Model summary: original model (unconstrained)
81

82.  Model summary: Measurement weights model
82

83.  Model summary: Structural covariances model
83

84.  Model summary: Measurement residuals model
84

85.  Model fit
85

86.  Model comparisons
86

87.  If we find non-invariance across groups, the next
step is to know what is causing this within the
model.
 Usually, start with the factor loadings.
 Then, test structural weights.
87

88.  Nested model comparisons work by imposing a
constraint or set of multiple constraints on a
starting or less restricted model to obtain a more
restricted final model.
 Example: we want to compare the equality of
factor loadings with a CFA model.
88

89.  We want to test the equality of the Cubes factor loading and
the Sentence factor loading, as well as the Lozenges factor
loading and the Wordmean factor loading.
89
Test:
w1 = w3
w2 = w4

90.  Next, double-click on the section of the AMOS
diagram window labeled Default Model.
 Manage Models window is open.
90

91.  Model comparison
91

92.  Conclusion:
 The nested model comparison that assesses the worsening
of overall fit due to imposing the two restrictions on the
original model shows a statistically significant chi-square
value of 12.795 with 2 DF, resulting in a probability value of
.002.
 That the two models differ indicates that constraining the
parameters in the default model to obtain the equal
loadings model results in a substantial worsening of overall
model fit.
 Therefore, we reject the equal factor loadings model in
favor of the original model.
92

93.  The bootstrap technique
 It is a resampling procedure
 Multiple subsamples of the same size as the parent
sample are drawn randomly from the original data.
 Parameter estimates are computed for each
subsample.
93

94.  When we use bootstrapping
 Data fail to meet the assumption of multivariate
normality.
 Presence of excessive kurtosis.
 Data are from a moderately large sample.
94

95.  Example: path diagram
95

96.  Assess multivariate normality: Go toView >
Analysis Properties > CheckTest for normality
and outliers.
96

97.  Assess multivariate normality
97
The multivariate kurtosis
value of 13.167 is Mardia's
coefficient. Critical ratio
(c.r.)values of 1.96 or less
mean there is non-significant
kurtosis.Values of 7.979 >
1.96 mean there is significant
non-normality.

98.  Assess multivariate normality
98
1.Malanobis d-squared distance for a case, the
more it is improbably far from the solution
centroid under assumptions of normality.
2.The cases are listed in descending order of
d-square.
3.We may consider the cases with the highest
d-squared to be outliers and might delete
them from the analysis.
4.This should be done with theoretical
justification (ex., rationale why the outlier
cases need to be explained by a different
model).
5.After deletion, it may be the data will be
found normal by Mardia's coefficient when
model fit is re-run.

99.  Run bootstrapping: Go toView > Analysis
properties
99

100. 100
 Bootstrap ML estimates
1. The first column (SE) is Bootstrap
estimate of the standard error fro the
parameter.
2. The second column (SE-SE)is standard
error of bootstrap standard error itself.
3. The third column (Mean): is the mean
parameter estimate computed across
500 subsamples.
4. The fourth column (Bias): represents
the difference between the original
mean estimate and bootstrap mean
estimate.
5. The fifth column (SE-Bias): standard
error of the bias estimate.

101.  Bootstrap confidence intervals
101
1. It is bias-corrected confidence
interval.
2. If the range does not include
zero, that the hypothesis of the
parameter is equal to zero is
rejected

102.  It is used to study change.
 Latent growth analysis on individual and group
levels.
 The measurements are taken 3 or more times
(longitudinal data).
 Intercept
 The initial value, the average or mean of the outcome we
are interested in.
 Think about this: for each individual in the study,
everybody has an intercept of a certain value.
102

103.  Slope
 How much the curve grows over time, an average or
mean rate of growth.
 Each individual has a slope.
 Goal of LGM
 Understand the average change.
 Understand individual variation in change.
103

104.  Example: a longitudinal study with 4-time points
data set.We want to know the change of alcohol
drinking over time.
 a28, b28, c28, and d28 are variables we measured.
 The regression weights from intercept to measured
variables are fixed to 1. In this way, we establish the
initial level of alcohol drinking.
 The path values from slope to measured variables are
also fixed at a set of continuous values (time intervals).
104

105.  Diagram
105

106.  Fixing the values from the slope is how we
identify model growth.
 Parameters
 Mean and variance of intercept: Mean intercept is the
average start value.The variance of intercept reflects
the variation of individual start value.
106

107.  Parameters
 Mean and variance of slope: Mean slope is the average of
rate of change.The variance of the slope reflects the extent
to which individuals have different rates of change.
 Covariance: to test whether individuals who start higher
(higher intercepts) also change at a faster rate (higher
slope).If such a relationship exists, we expect the
covariance to be significant.
107

108.  From AMOS, choose Plugins > Growth Curve
Model.
 Enter the number of measures for the number of
time points.
 In this example, we would enter 4 for the number
of time points .
 ChooseView > Analysis Properties > Estimation
tab. Check the Estimate Means and Intercepts
check box.
108

109.  Model identifications
 Right-click on the latent variable circles (labeled ICEPT
and SLOPE by AMOS) and select Object Properties.
 Remove the 0 constraints on the means. Fix the
variance to zero for ICEPT and SLOPE.
 Right-click on each of the 4 error variance circles and
select Object Properties. Fix their mean values to 0
and set their variance values to 1.00.
109

110.  For each of the paths connecting the error circles to
the observed variables, replace the original value of
1.00 with the new parameter name.
 Add two new error terms to the ICEPT and SLOPE
latent variables.
 For the newly-created error terms, fix their mean
values to 0 and their variance values to 1.00.
 Next, replace the 1.00 values for the path arrows
connecting errors to ICEPT and Slope. Name the
newly freed parameters.
110

111.  Remove the covariance double-headed arrow
between ICEPT and SLOPE.
 Place the covariance double-headed arrow
between two new error terms and give a new
name to the covariance.
111

112.  Text output
112

113.  Text output
113
1.The mean intercept value of 1.35 indicates that the average starting amount of
alcohol drinking was 1.35 units.
2.The mean slope value was .11. It means the average rate of change is .11 units.
3.The correlation between the intercepts and the slopes was 2.09.
4.The means were statistically significant when tested with the null hypothesis that
their true values are zero in the population from which this sample was drawn.

114.  Diagram with estimated parameters
114

115.  The intercept indicates a statistical significant
mean alcohol use at the initial level (i.e., at
baseline) and the slope mean indicates a
significant average increase, via a liner functional
form.
 This alcohol use in adolescents is expected to
increase by .11 each studied time period,
beginning with an average score of 1.35.
115

116.  We also want to know the extent to which
adolescents in the sample vary around their group
average (mean) trajectories in alcohol use.
 This can be evaluated by looking at the variances.
 The corresponding variances ( .57 for intercept and
.30 for slope) are statistically significant, indicating
significant individual variability in the initial level and
rate of change (growth) in alcohol use across the
four waves of measurement.
116

117.  Hoyle, R. H. (1995). Structural equation modeling:
Concepts, issues, and applications.Thousand Oaks,
CA: Sage Publications, Inc.
 Raykov,T. & Marcoulides, G. A. (2000). A first course
in structural equation modeling. Mahwah, NJ:
Lawrence Erlbaum Associates, Inc.
 Schumacker, R. E. & Lomax, R. G. (2004). A
beginner’s guide to structural equation modeling.
Mahwah, NJ: Lawrence Erlbaum Associates, Inc.
 AMOS 19.o user’s guide. IBM SPSS, Chicago.
117

118. 118