2. Bivariate linear regression
Y = a + BX
Multiple regression
Y = a + B1X1 + B2X2 + B2X3
Path Model
Y = B1X1 + B2X2 + e1
Structural Equations Model
F2 = B1F1 + e1
3. Example 1
2.63* Organization and
Delta1 Planning 2.201*
4.10*
Delta2 Student Interaction
2.83*
2.11*
Delta3 Evaluation 1.99*
Teacher
5.79* 3.80* Performance in
Instructional NSTP
Delta4
Methods 3.03*
2.93*
Delta5 Course Outcome 2.95*
2.45*
Delta6 Learner- 2.00*
centeredness
Delta7
2.74*
Communication X2=3.42, GFI=.97, RMSEA=.01
What do you call this model?
What analysis is used in this model?
What are the three things that you interpret in this
model?
4. Example 2 DELTA2 DELTA2
25.12* 17.69*
KC RC
7.22* 7.86*
Metacognition
2.12*
X2=4.51, GFI=.99, RMSEA=.00
5.00*
Critical
ZETA
Thinking
1
1.00 0.67* 0.86* 0.74* 0.40*
Inference Recognition Deduction Interpretation Evaluation of
of Arguments
Assumption
7.37* 2.02* 6.16* 5.02* 3.57*
EPSILON1 EPSILON2 EPSILON3 EPSILON4 EPSILON5
1. How many measurement models are shown?
2. How are the two measurement models linked in the figure?
3. Where are the errors located in the figure?
4. What the things interpreted in the figure?
5. Structural Equations Modeling
Goes beyond regression models: Test several variables
and latent constructs with their underlying manifest
variables.
Provide a way to test the specified set of relationships
among observed and latent variables as a whole and
allow theory testing for causality even when
independent variables are not experimentally
manipulated
Theentire model is tested if the data fits the specified
model.
Takes into account errors of measurement.
6. Variables in a Structural Model:
3. Manifest variables: Directly observed or measured.
Boxes are used to denote manifest variables.
4. Latent variables: Not directly observed; we learn about
them through the manifest variables that are supposed
to “represent” them. Ovals are used to denote latent
variables.
7. Symbols used in Structural Models
Manifest Variable
Latent Variable
Direction of an
effect/Parameter Estimate
Relationship/Covariance
8. Some Levels of Analyzing Structural Models:
2. Confirmatory factor analysis
3. Causal modeling or path analysis
• Independent/exogenous variables
• Dependent/endogenous variables
9. Confirmatory Factor Analysis
Example 3
2.63* Organization and
Delta1 Planning 2.201*
2.11*
Delta2 Learner-
Centeredness 2.95*
2.43* Set 1
Delta3 Evaluation 1.99*
2.74* 2.00* 1.00
Delta4 Communication
4.10*
Delta5 Student Interaction
2.83*
2.93* Set 2
Delta6 Course Outcome 3.03*
5.79* 3.80*
Delta7 Teaching method
Common Factor Model/Multifactor Model
1. What critical estimate is determined in a common factor model?
2. When do we say that set 1 and set 2 are common factors?
10. Confirmatory Factor Analysis
Example 4
2.11* Evaluation 1.996*
Delta1
2.74* 2.00* Set 1
Delta2 Communication
2.33* 1.00
Delta3 Learner- 2.20*
Centeredness Set 2
2.43* 2.95* 1.00
Delta4 Organization & Plan
1.00
4.10*
Delta5 Course Outcome
2.83*
5.79*
Delta6 Student Interaction 3.80* Set 3
2.93* 3.03*
Delta7 Teaching method
Common Factor Model/
Multifactor Model
11. Path Model (Example 5)
Self-efficacy
.48*
E
1
1.0
.28*
Deep Approach Metacognition
.20*
Surface Approach
χ2=10.03, df=3, χ2/df=3.34, GFI (.98), adjusted GFI (.92), RMSEA= .08
1. What type of variables are studies in a path model?
2. What is the difference between a path and a structural
model?
3. What is the similarity between a path and structural model?
4. Where is the error located?
12. Structural Equations Model (Example 6)
DELTA DELTA DELTA DELTA DELTA DELTA
2 3 4 5 6 7
100.43 71.46 57.11 34.94 71.92 88.10
*
Conditional *
Procedural *
Planni *
Monitori *
Information *
Debugging
Knowledge Knowledge Manageme Strategy
ng ng nt
6.88*
7.07* 9.25* 7.91*
7.24*
Declarative 9.03* Evaluati
Knowledge
6.27* on
Metacogniti 25.12
82.57 on * 78.39
*
DELTA *
DELTA
1 8
2.10*
5.19*
Critical
ZETA
Thinking
1
0.67* 0.86* 0.74* 0.40*
Inferen Recognition Deductio Interpretati Evaluation
ce of n on of
Assumption Argument
7.27* 2.06* 6.15* 5.03* s 3.57*
EPSILON EPSILON EPSILON EPSILON EPSILON
1 2 3 4 5
1. What variable is exogenous?
2. What variable is endogenous?
13. Path Model (Example 7 & 8)
E
1
1.0
Self-efficacy
Example 7 .17* .51* E
2
1.0
.30*
School Ability Metacognition
E
Self-efficacy
4
.51* 1.0
Metacognition
Example 8 School Ability
.30*
Self-efficacy
X
School Ability
1. What is being demonstrated by self-efficacy in the example 7?
2. What is the difference between example 7 and example 8?
15. Procedures (Summary)
2. Specify the measurement models for the
exogenous latent variables (i.e., which
manifest variables represent which latent
variables?)
3. Likewise, specify the measurement models
for the endogenous latent variables.
4. Specify the paths from the exogenous to the
endogenous latent variables.
16. DELTA: residuals of exogenous manifest
variables
EPSILON: residuals of endogenous manifest
variables
17. Testing for Goodness of Fit
If the entire model approximates the
population.
The degree to which the solution fit the data
would provide evidence for or against the prior
hypothesis.
A solution which fit well would lend support for
the hypothesis and provide evidence for
construct validity of the attributes and the
hypothesized factorial structure of the domain
as represented by the battery of attributes.
18. Goodness of Fit
Noncentrality Interval Estimation
Single Sample Goodness of fit Index
19. Goodness of Estimate Goodness of fit index Estimate
fit index required required
RMSEA .08 and below Joreskog GFI .90 and above
Bentler-Bonett (1980)
Normed Fit Index
James-Mulaik-Brett
Parsimonious Fit Index
RMS .08 and below Akaike Information Compare nested
Criterion models
McDonald’s .90 and above Schwarz's Bayesian Lowest value is
Index of Criterion the best fitting
Noncentrality model
Bollen's Rho
Population .90 and above Independence Model Close to 0, not
Gamma Index Chi-square and df significant
Browne-Cudeck Cross Requires two
Validation Index samples
20. Noncentrality Interval Estimation
Represents a change of emphasis in
assessing model fit. Instead of testing the
hypothesis that the fit is perfect, we ask
the questions (a) "How bad is the fit of
our model to our statistical population?"
and (b) "How accurately have we
determined population badness-of-fit
from our sample data."
21. Noncentrality Indices
Steiger-Lind RMSEA -compensates for
model parsimony by dividing the estimate
of the population noncentrality
parameter by the degrees of freedom.
This ratio, in a sense, represents a "mean
square badness-of-fit."
Values of the RMSEA index below .05
indicate good fit, and values below .01
indicate outstanding fit
22. Noncentrality Indices
McDonald's Index of Noncentrality-The
index represents one approach to
transforming the population noncentrality
index F* into the range from 0 to 1.
Good fit is indicated by values above .95.
23. Noncentrality Indices
The Population Gamma Index- an
estimate of the "population GFI," the
value of the GFI that would be obtained if
we could analyze the population
covariance matrix Σ.
For this index, good fit is indicated by
values above .95.
24. Noncentrality Indices
Adjusted Population Gamma Index
(Joreskog AGFI) - estimate of the
population GFI corrected for model
parsimony. Good fit is indicated by values
above .95.
25. Single Sample Goodness of fit Index
Joreskog GFI. Values above .95 indicate
good fit. This index is a negatively biased
estimate of the population GFI, so it
tends to produce a slightly pessimistic
view of the quality of population fit.
Joreskog AGFI. Values above .95
indicate good fit. This index is, like the
GFI, a negatively biased estimate of its
population equivalent.
26. Single Sample Goodness of fit Index
Akaike Information Criterion. This
criterion is useful primarily for deciding which
of several nested models provides the best
approximation to the data. When trying to
decide between several nested models, choose
the one with the smallest Akaike criterion.
Schwarz's Bayesian Criterion. This
criterion, like the Akaike, is used for deciding
among several models in a nested sequence.
When deciding among several nested models,
choose the one with the smallest Schwarz
criterion value.
27. Single Sample Goodness of fit Index
Browne-Cudeck Cross Validation Index.
Browne and Cudeck (1989) proposed a single
sample cross-validation index as a follow-up to
their earlier (Cudeck & Browne,1983). It
requires two samples, i.e., the calibration
sample for fitting the models, and the cross-
validation sample.
Independence Model Chi-square and df.
These are the Chi-square goodness-of-fit
statistic, and associated degrees of freedom, for
the hypothesis that the population covariances
are all zero.
28. Single Sample Goodness of fit Index
Bentler-Bonett (1980) Normed Fit Index.
measures the relative decrease in the discrepancy
function caused by switching from a "Null Model" or
baseline model, to a more complex model. This index
approaches 1 in value as fit becomes perfect. However,
it does not compensate for model parsimony.
Bentler-Bonett Non-Normed Fit Index. This
comparative index takes into account model parsimony.
Bentler Comparative Fit Index. This comparative
index estimates the relative decrease in population
noncentrality obtained by changing from the "Null
Model" to the k'th model.
29. Single Sample Goodness of fit Index
James-Mulaik-Brett Parsimonious Fit Index.
Compensate for model parsimony. Basically, it
operates by rescaling the Bentler-Bonnet Normed fit
index to compensate for model parsimony.
Bollen's Rho. This comparative fit index computes
the relative reduction in the discrepancy function
per degree of freedom when moving from the "Null
Model" to the k'th model.
Bollen's Delta. This index is similar in form to the
Bentler-Bonnet index, but rewards simpler models
(those with higher degrees of freedom).
30. Noncentrality Estimates
(SIR-NSTP Models)
Model 1 Model 2 Model 3
Point Lower Point Point
Lower Estim Upper 90 Estim Upper Lower Estim Upper
Noncentrality 90% at 90% % at 90% 90% at 90%
Fit indices CI e CI CI e CI CI e CI
Population
Noncentrality
Parameter 0.156 0.222 0.303 0.158 0.224 0.305 0.162 0.228 0.31
Steiger-Lind
RMSEA Index 0.106 0.126 0.147 0.11 0.131 0.153 0.121 0.144 0.168
McDonald
Noncentrality
Index 0.859 0.895 0.925 0.859 0.894 0.924 0.857 0.892 0.922
Population
Gamma Index 0.92 0.94 0.957 0.92 0.94 0.957 0.919 0.939 0.956
Adjusted
Populatio
n
Gamma Index 0.841 0.881 0.914 0.827 0.87 0.907 0.793 0.844 0.887
31. Example
(metacognition and critical thinking)
Sing Sample Fit Indices Model 1 Model 2
Joreskog GFI 0.926 0.915
Joreskog AGFI 0.841 0.879
Akaike Information Criterion 0.394 0.817
Schwarz's Bayesian Criterion 0.612 1.210
Browne-Cudeck Cross 0.398 0.831
Validation Index
Independence Model Chi- 786.533 1382.03
Square 4
Independence Model df 21.000 78.000
32. Single Sample Fit Indicers Model 1 Model 2
Bentler-Bonett Normed Fit 0.919 0.898
Index
Bentler-Bonett Non-Normed Fit 0.892 0.928
Index
Bentler Comparative Fit Index 0.933 0.941
James-Mulaik-Brett 0.569 0.737
Parsimonious Fit Index
Bollen's Rho 0.868 0.875
Bollen's Delta 0.934 0.941
33. Reference for the Goodness of fit
for CFA
StatSoft, Inc. (2005). STATISTICA electronic
manual. Tulsa OK: Author.
Arbuckle, J. L. (2005). Amos: 6.0 User’s
guide. USA: Amos Development Corp.
