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Multifactor CFA:
Measuring More Than One Factor
Higher Order Models
PRE 906: Structural Equation Modeling
Lecture #9 – April 1, 2015
PRE 906 SEM: Lecture 9--Multifactor CFA
Today’s Class
•
Confirmatory Factor Analysis with more than one factor:


Logic
Model Comparison
•
Including factors for “methods” effects
•
Structural models for multi-factor covariance matrices

•
Higher order structures
Including “Random Intercepts” for person effects
PRE 906 SEM: Lecture 9--Multifactor CFA
2
Key Questions for Today’s Lecture
PRE 906 SEM: Lecture 9--Multifactor CFA
3
Today’s Data
•
Data for today’s class come from a study of self-efficacy in
mathematics problem solving
•
Participants (n=225) answered a total of 24 survey items


Six math problems were shown…
…for each problem, participants were asked about their confidence in their
ability to correctly answer the problem
1.
2.
3.
4.
•
How sure are you that you can understand this mathematical problem?
How sure are you that you can determine a strategy to solve this problem?
How sure are you that you can determine the information required to solve this
problem?
How sure are you that you can solve this mathematical problem correctly?
Ratings were given on a scale from 0 to 100 in 10-unit
intervals (so we will assume these to be MVN)
PRE 906 SEM: Lecture 9--Multifactor CFA
4
Picking Up Where We Left Off
•
As a baseline analysis, let’s consider a model where we
measure one self-efficacy factor with all of the 24 items of
the survey:
PRE 906 SEM: Lecture 9--Multifactor CFA
5
Model 01: Single Factor Model Fit
•
Very poor model fit by any standard:
PRE 906 SEM: Lecture 9--Multifactor CFA
6
MULTIFACTOR CFA MODELS
PRE 906 SEM: Lecture 9--Multifactor CFA
7
Multifactor CFA Models
•
Multifactor CFA models are measurement models that
measure more than one latent trait simultaneously
•
The multiple factors represent theoretical constructs that
are best when they are defined a priori

For us that will be the self-esteem construct
Is it one thing? –results say no
 Is it four things? –that’s an empirical question…we’ll evaluate that next

•
Multiple factors can also come from modification of
models with single factors when model misfit is identified

Sometimes this shows up as design features of the study—such as our
multiple problems and multiple self-esteem questions
PRE 906 SEM: Lecture 9--Multifactor CFA
8
Sequence of for Measuring Multiple Factors
1.
Specify your measurement model for each factor alone


2.
For each: assess model fit, per factor, when possible
(remember…need 4+ indicators to test model fit)



3.
How many factors, which items load on which factors, and whether your need
any method factors or error covariances
For models with large numbers of items, you should start by modeling each
factor in its own analysis to make sure each factor fits its items
Global model fit: Does a one-factor model adequately fit each set of
indicators thought to measure the same latent construct?
Local model fit: Are any of the leftover covariances problematic? Any items
not loading well (or are too redundant) that you might drop?
(If the 1-factor model fits) Reliability/Info: Are your standardized loadings
practically meaningful?
Once your single-factor measurement models fit,
then consider the multiple factor measurement model
PRE 906 SEM: Lecture 9--Multifactor CFA
9
From One Factor to Four Factors
•
As our one-factor model of self-efficacy did not fit these
data, we must expand our thinking
•
Recall the self-efficacy items:
1.
2.
3.
4.
•
How sure are you that you can understand this mathematical problem?
How sure are you that you can determine a strategy to solve this problem?
How sure are you that you can determine the information required to solve
this problem?
How sure are you that you can solve this mathematical problem correctly?
Let’s start by constructing four separate one-factor models
PRE 906 SEM: Lecture 9--Multifactor CFA
10
Model 01a: SE1 Confidence in Understanding
•
For question #1: “How sure are you that you can understand this mathematical
problem?” we’ll define a single factor called “confidence in understanding”
•
Only responses to this item will be included in this analysis
•
Cannot compare model fit with previous analysis (Model 01)
PRE 906 SEM: Lecture 9--Multifactor CFA
11
Model 01a: Global Model Fit Evaluation
•
Model fit indices:
PRE 906 SEM: Lecture 9--Multifactor CFA
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Model 01a: Local Model Fit Evaluation
•
Normalized residual covariances:
PRE 906 SEM: Lecture 9--Multifactor CFA
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Model 01a: Item Parameter Evaluation
PRE 906 SEM: Lecture 9--Multifactor CFA
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Model 01b: SE2 Confidence in Strategy Development
•
For question #2: “How sure are you that you can determine a strategy to solve this
problem?” we’ll define a single factor called “confidence in strategy development”
•
Only responses to this item will be included in this analysis
•
Cannot compare model fit with previous analysis (Models 01 or 01a)
PRE 906 SEM: Lecture 9--Multifactor CFA
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Model 01b: Global Model Fit Evaluation
•
Model fit indices:
PRE 906 SEM: Lecture 9--Multifactor CFA
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Model 01b: Local Model Fit Evaluation
•
Normalized residual covariances:
PRE 906 SEM: Lecture 9--Multifactor CFA
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Model 01b: Item Parameter Evaluation
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Model 01c: SE3 Confidence in Information Finding
•
For question #3: “How sure are you that you can determine the information
required to solve this problem?” we’ll define a single factor called “confidence in
information finding”
•
Only responses to this item will be included in this analysis
•
Cannot compare model fit with previous analysis (Models 01, 01a, or 01b)
PRE 906 SEM: Lecture 9--Multifactor CFA
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Model 01c: Global Model Fit Evaluation
•
Model fit indices:
PRE 906 SEM: Lecture 9--Multifactor CFA
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Model 01c: Local Model Fit Evaluation
•
Normalized residual covariances:
PRE 906 SEM: Lecture 9--Multifactor CFA
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Model 01c: Item Parameter Evaluation
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Model 01d: SE4 Confidence in Problem Solving Ability
•
For question #4: “How sure are you that you can solve this mathematical problem
correctly?” we’ll define a single factor called “confidence in problem solving
ability”
•
Only responses to this item will be included in this analysis
•
Cannot compare model fit with previous analysis (Models 01, 01a, or 01b)
PRE 906 SEM: Lecture 9--Multifactor CFA
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Model 01d: Global Model Fit Evaluation
•
Model fit indices:
PRE 906 SEM: Lecture 9--Multifactor CFA
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Model 01d: Local Model Fit Evaluation
•
Normalized residual covariances:
PRE 906 SEM: Lecture 9--Multifactor CFA
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Model 01d: Item Parameter Evaluation
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Summarizing the Four Single Factor Models
•
For each of the four single-factor models we found:



All fit really well globally
All fit really well locally
All items seemed to load onto the factor
•
We can now proceed to building a four-factor model of self
efficacy using each of the four single-factor models
•
Starting with each factor by itself does not guarantee the
four-factor model will fit well
•
It does cut down on sources of where the four-factor
model may have misfit
PRE 906 SEM: Lecture 9--Multifactor CFA
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MULTI-FACTOR CONFIRMATORY FACTOR
ANALYSIS MODELS
PRE 906 SEM: Lecture 9--Multifactor CFA
28
CFA Model with Factor Means and Item Intercepts
But some parameters will have to be fixed to known
values for the model to be identified.
covF1,F
F1
λ11
λ21
λ31
κ1
κ2
2
F2
λ4
λ5
λ6
2
2
2
y1
y2
y3
y4
y5
y6
e1
e2
e3
e4
e5
e6
Measurement Model
for Items:
λ’s = factor loadings
e’s = error variances
μ’s = intercepts
PRE 906 SEM: Lecture 9--Multifactor CFA
μ
μ
2
1
μ
3
1
μ
μ
4 μ
5
6
Structural Model
for Factors:
F’s = factor variances
Cov = factor covariances
K’s = factor means
29
CFA Model Equations
•
Measurement model per item (numbered) for subject s:






y1s = μ1 + λ11F1s + 𝟎F2s + e1s
y2s = μ2 + λ21F1s + 𝟎F2s + e2s
y3s = μ3 + λ31F1s + 𝟎F2s + e3s
y4s = μ4 +
y5s = μ5 +
y6s = μ6 +
𝟎F1s + λ42F2s + e4s
𝟎F1s + λ52F2s + e5s
𝟎F1s + λ62F2s + e6s
Here is the general matrix equation
for these 6 item-specific equations:
𝐘 = 𝛍 + 𝛌𝐅 + 𝐞
where 𝐘, 𝛍, and 𝐞 = 6x1 matrices,
𝛌 = 6x2 matrix, and 𝐅 =2x1 matrix
PRE 906 SEM: Lecture 9--Multifactor CFA
You decide how many factors
and if each item has an estimated
loading on each factor or not.
Unstandardized loadings (𝛌) are
the linear slopes predicting the
item response (y) from the factor
(F). Thus, the model assumes a
linear relationship between the
factor and the item response.
Standardized loadings are the
slopes in a correlation metric
(Standardized Loading2 = R2).
Intercepts (𝛍) are expected item
responses (y) when all factors = 0.
30
CFA Model Implications
Items from same factor (room for misfit or mis-prediction):
•
•
•
Unstandardized solution: Covariance of y1 , y3 = 𝛌𝟏𝟏 ∗ 𝐕𝐚𝐫(𝐅𝟏) ∗ 𝛌𝟑𝟏
Standardized solution: Correlation of y1 , y3 = 𝛌𝟏𝟏 ∗ (𝟏) ∗ 𝛌𝟑𝟏  std load
ONLY reason for correlation is their common factor (local independence, LI)
Items from different factors (room for misfit or mis-prediction):
•
•
•
Unstandardized solution: Covariance of y1 , y6 = 𝛌𝟏𝟏 ∗ 𝐜𝐨𝐯𝐅𝟏, 𝐅𝟐 ∗ 𝛌𝟔𝟐
Standardized solution: Correlation of y1 , y6 = 𝛌𝟏𝟏 ∗ 𝐜𝐨𝐫𝐅𝟏, 𝐅𝟐 ∗ 𝛌𝟔𝟐  std load
ONLY reason for correlation is the correlation between factors (again, LI)
Variances are additive (and will be reproduced correctly):
•
Var(y1) = (λ112)*Var(F1) + Var(ei)  but note the imbalance of λ2 and e
PRE 906 SEM: Lecture 9--Multifactor CFA
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Model-Implied Item Covariance Matrix
•
Matrix equation: 𝚺 = 𝚲𝚽𝚲𝑇 + 𝚿
𝚺= model-predicted item
covariance matrix is
created from:
𝚲 = item factor loadings
𝚽= factor variances
and covariances
𝚲𝑻 = item factor loadings
transposed (~𝛌𝟐 )
𝚿 = item error variances
PRE 906 SEM: Lecture 9--Multifactor CFA
 2y1

  y2,y1

  y3,y1

 y4,y1
  y5,y1

  y6,y1

 11

 21
  31

 0
 0

 0
 y1,y2
 y1,y3
 y1,y4
 y1,y5
2y2
 y2,y3
 y2,y4
 y2,y5
 y3,y2
2y3
 y3,y4
 y3,y5
 y4,y2
 y4,y3
2y4
 y4,y5
 y5,y2
 y5,y3
 y5,y4
 2y5
 y6,y2
 y6,y3
 y6,y4
 y6,y5
 y1,y6 

 y2,y6 

 y3,y6 
 y4,y6 

 y5,y6 

2y6 
0 
0 
0   2F1  F1,F2   11  21  31 0


 42    F2,F1 2F2   0
0
0  42
 52 

 62 
 2e1 0

2
 0 e2

0
0

 0
0

 0
0

0
 0
0
0
0
0
0
0
2e3
0
0
0
2e4
0
0
0
2e5
0
0
0
0 

0 

0 
0 

0 

2e6 
32
0
 52
0 
 62 
Model-Implied Item Covariance Matrix
•
𝚺 = 𝚲𝚽𝚲𝑇 + 𝚿  Predicted Covariance Matrix
Items within Factor 1
2 2
2
 11
 F1  e1
11 2F1 21

2
2 2
2
  21 F111  21 F1   e2

2
2





31
F1
11
31
F1 21

 

 42 F2,F1 21
 42 F2,F1 11
  52 F2,F111  52 F2,F1 21

  62 F2,F111  62 F2,F1 21
The loadings control how
related items from the same
factor are predicted to be.
11 2F1 31
11 F2,F1 42
11 F2,F1 52
 21 2F1 31
 21 F2,F1 42
 21 F2,F1 52
2 2
2
 31
 F1  e2
 31 F2,F1 42
 31 F2,F1 52
 42 F2,F131
2
2
 42
 2F2  e4
 42  2F2 52
52 F2,F131
52 2F2  42
2
2
52
 2F2  e5
62 F2,F131
62 2F2  42
62 2F252
11 F2,F1 62 

 21 F2,F1 62 

 31 F2,F162 
 42 2F2 62 

2
 52 F2 62 
2
2 

 62
2F2  e6

Items within Factor 2
The only reason why items
from different factors should
be related is the covariance
between the two factors.
PRE 906 SEM: Lecture 9--Multifactor CFA
The loadings also control how
much of the item response is
due to factor versus error.
33
Factor Model Identification
•
Goal: Name that Tune  Reproduce observed item covariance matrix using as few
estimated parameters as possible

(Robust) Maximum likelihood used to estimate model parameters
Measurement Model: Item factor loadings, item intercepts, item error variances
 Structural Model: Factor variances and covariances, factor means


•
Global model fit is evaluated as difference between model-predicted matrix and
observed matrix (but only the covariances really contribute to misfit)
How many possible parameters can you estimate (total DF)?


Total DF depends on # ITEMS  𝐯 (NOT on # people)
Total number of unique elements in item covariance matrix
Unique item elements = each variance, each covariance, each mean
 Total unique elements = (v(v + 1) / 2) + v  if 4 items, then ((4*5)/2) + 4 = 14

•
Model degrees of freedom (df) = data input − model output

Model df = # possible parameters − # estimated parameters
PRE 906 SEM: Lecture 9--Multifactor CFA
34
Oops: Empirical Under-Identification
•
Did your model blow up (errors instead of output)?

•
Make sure each factor is identified (scale of factor mean and variance is set)
Sometimes you can set up your model correctly and it will still blow up because of
empirical under-identification

It’s not you; it’s your data – here are two examples of when these models should have
been identified, but weren’t because of an unexpected 0 relationship
F1 = ? Cov = F2 = ?
0
1 λ2
1 λ4
λ1 λ2
y1
1y
2
y3
2y
4
y11
1y
2
y3
e1
e2
e3
e4
e1
e2
e3
PRE 906 SEM: Lecture 9--Multifactor CFA
F1 = 1
0
35
Back to our example…
FOUR-FACTOR MODEL OF MATHEMATICS
PROBLEM SOLVING SELF-EFFICACY
PRE 906 SEM: Lecture 9--Multifactor CFA
36
Model 02a Lavaan Syntax: Each =~ Defines a Factor
PRE 906 SEM: Lecture 9--Multifactor CFA
37
Results: A Warning From the Analysis
•
Upon using the sem() function to estimate Model 02a, R
reported the following
•
Not positive definite = not invertible

•
Indication of some type of problem with the model

•
Not a valid solution if our factors are multivariate normally distributed
Empirical underidentification or model misspecification
This type of result would not be reportable

Look at the covariances…and the correlations they imply
PRE 906 SEM: Lecture 9--Multifactor CFA
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Solutions to Non-Positive Definite LV Covariance Matrices
•
The non-positive covariance matrix occurs in models with
multiple factor occasionally
•
Our factor model had a saturated covariance matrix
between factors

•
•
All factor covariances were estimated—making it harder to estimate
Fixes aren’t often obvious:

Can force a fix by making factors uncorrelated

Can look for sources of model misfit that may indicate omitted factors

Can specify a simpler factor covariance structure (such as a higher order factor
model)
In our example, we will try all three
PRE 906 SEM: Lecture 9--Multifactor CFA
39
Model 02b: Four-Factor Model with Uncorrelated Factors
•
Syntax—Note: Once factors are specified (using =~) they
act just like observable variables in the rest of the syntax

•
So ~~ represents factor variances and covariances
This model is very unrealistic—most if not all mental traits
are correlated


Would likely be a very big red flag to reviewers if you were to try to publish
this model in a journal article
I am presenting it in class to show how things work
PRE 906 SEM: Lecture 9--Multifactor CFA
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Model 02b: Global Model Fit Evaluation
PRE 906 SEM: Lecture 9--Multifactor CFA
41
Model 02b: Why Global Fit Was Terrible
•
When factors are fixed to have zero covariance the model
states that the items not measuring the same factor have zero
covariance
•
The same matrix for Model 02a (not shown) would not have
zeros throughout as factors were allowed to have non-zero
covariances (and correlations between items measuring
different factors)
PRE 906 SEM: Lecture 9--Multifactor CFA
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Model 02b: Inspecting Local Fit for Hints at Model Misspecification
•
Normalized Residuals:
•
Look at values different SE – problem is lack of factor covariance
PRE 906 SEM: Lecture 9--Multifactor CFA
43
Model 02a: Inspecting Local Fit for Hints at Model Misspecification
•
We will now use Model 02a to look at local fit to
determine where this model is misspecified
•
Look at items with same question index (Q) but different
factor index (SE)
PRE 906 SEM: Lecture 9--Multifactor CFA
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Model 02a: Inspecting Local Fit for Hints at Model Misspecification
•
The largest 40 modification indices from Model 02a
PRE 906 SEM: Lecture 9--Multifactor CFA
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What Model 02a and 02b Local Misfit Is Telling Us
•
From Model 02a Globally:


•
From Model 02b Globally and Locally:



•
Restricting factor covariances to zero leads to terrible global fit
Local fit suggest factor covariances are needed
Therefore, we looked at Model 02a’s local fit
From Model 02a Locally:


•
Overall four-factor model has a fit problem leading to factor correlations all
greater than one (non-positive definite factor covariance matrix)
Model has some type of misspecification
Items referring to the same mathematics question (e.g.,
Q1.SE1+Q1.SE2+Q1.SE3+Q1.SE4) had large positive normalized residual
covariances
Modification indices all showed these had the largest values (largest 36)
Conclusion: Model is misfitting on items that refer to the same
mathematics question
PRE 906 SEM: Lecture 9--Multifactor CFA
46
Ways to Improve the Fit of the Four-Factor Model
•
The four self-efficacy factors were composed of items that
all referenced a common stimulus: a mathematics problem

•
To address the additional dependencies we could consider
residual covariances between SE items referencing the
same mathematics question

•
The common stimulus added additional dependencies that the four-factor
model cannot accommodate
But…these indicate additional dimensionality
Instead, we can add a
“methods” factor for each
math question
PRE 906 SEM: Lecture 9--Multifactor CFA
47
Additional Multiple Factor Models
METHOD FACTORS
PRE 906 SEM: Lecture 9--Multifactor CFA
48
Method Factors
•
Method factors are latent constructs that summarize
variability due to differing testing methods


•
The method factors are typically uncorrelated with the
factors of interest

•
Most common in use in negatively worded items
In our example, these could represent the mathematics questions
The “method” items are then cross loaded onto each
In our example there are six different mathematics
questions, each with four SE items



Each mathematics question will be represented by a method factor
All four SE items that are asked about a question would then measure that
question’s method factor
As each math question is likely to be related (due to each measuring math),
we’ll let the methods factors correlate with themselves but not the self
efficacy factors (the factor of interest)
PRE 906 SEM: Lecture 9--Multifactor CFA
49
Model 03: Lavaan Syntax for Method Factors
•
The mathematics question method factors (Q1-Q6) are
coded as if they are regular factors
•
Covariances between method factors and all self esteem
factors are set to zero
PRE 906 SEM: Lecture 9--Multifactor CFA
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Model 03: Global Model Fit Indices
PRE 906 SEM: Lecture 9--Multifactor CFA
51
Model 03: Local Indices of Model Fit
•
No large normalized residual covariances
PRE 906 SEM: Lecture 9--Multifactor CFA
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Model 03: Comparison to Single-Factor Model 01
•
A typical comparison when estimating models with
multiple factors is that of the multiple factor model with a
single factor model

•
Testing the hypothesis that the model with a single factor fits as well as the
model with multiple factors
For us, that would be comparing our Model 03 (all items; 4
SE factors + 6 Math Question Factors) versus our Model 01
(all items; 1 SE factor)
PRE 906 SEM: Lecture 9--Multifactor CFA
53
Model 03: Path Diagram
PRE 906 SEM: Lecture 9--Multifactor CFA
54
Additional Considerations
•
The purpose of the instrument was to measure
mathematics self-efficacy as a unitary construct
•
The results from Model 03, however, have four factors of
mathematics self efficacy—not one
•
To try to get a unitary construct of mathematics selfefficacy, we can construct a higher-order factor of selfefficacy where the four SE factors are indicators
PRE 906 SEM: Lecture 9--Multifactor CFA
55
HIGHER ORDER FACTOR MODELS
PRE 906 SEM: Lecture 9--Multifactor CFA
56
Sequence of Steps in CFA or IFA
1.
Specify your measurement model(s)


2.
Assess model fit, per factor, when possible (if 4+
indicators)



3.
How many factors/thetas, which items load on which factors, and whether
your need any method factors or error covariances
For models with large numbers of items, you should start by modeling each
factor in its own analysis to make sure *each* factor fits its items
Global model fit: Does a one-factor model adequately fit each set of
indicators thought to measure the same latent construct?
Local model fit: Are any of the leftover covariances problematic? Any items
not loading well (or are too redundant) that you might drop?
Reliability/Info: Are your standardized loadings practically meaningful?
Once your single-factor measurement models are good,
it’s time to consider the (higher-order) structural model
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Higher-Order Factor Models
•
Purpose: What kind of higher-order factor structure best accounts for the
covariance among the measurement model factors (not items)?




•
In other words, what should the structural model among the factors look like?
Best-fitting baseline for the structural model has all possible covariances among the
lower-order measurement model factors  saturated structural model
Just as the purpose of the measurement model factors is to predict covariance among
the items, the purpose of the higher-order factors is to predict covariance among the
measurement model factors themselves
A single higher-order factor would be suggested by similar magnitude of correlations
across the measurement model factors
Note that distinctions between CFA, IFA, and other measurement models for
different item types are no longer relevant at this point


Factors and thetas are all multivariate normal latent variables, so a higher-order
model is like a CFA regardless of the measurement model for the items
Latent variables don’t have means apart from their items, so those are irrelevant
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Necessary Measurement Model Scaling to fit Higher-Order Factors
“Marker Item” for factor loadings
 Fix 1 item loading to 1
 Estimate factor variance
Var(F)=?
1
y1
e1
κ=0
PRE 906 SEM: Lecture 9--Multifactor CFA
λ21
λ31
y2
λ41
y3
e2
e3
μ2
μ3
μ1
1
μ4
y4
e4
Because it will become “factor variance
leftover” = “disturbance”, it can’t be a
fixed quantity (must be estimated)
“Z-Score” for item intercepts or
thresholds
 Fix factor mean to 0
 Estimate all intercepts/thresholds
All the factor means will be 0 and you
won’t need to deal with them in the
structural model anyway
59
Identifying a 3-Factor Structural Model
Option 1: 3 Correlated Factors
Measurement Model for Items:
item variances, covariances, and means
Structural Model for Factors:
factor variances and covariances, no means
Possible df = (12*13) / 2 + 12 = 90
Estimated df = 9𝛌 + 12𝛍 + 12𝛔𝟐𝐞 = 33
df = 90 – 33 = 57  over-identified
Possible df = (3*4) / 2 + 0 = 6
Estimated df = 3 variances + 3 covariances
df = 6 – 6 = 0  just-identified
covF1,F
covF1,F
Var(F1)=?
3
2
Var(F2)=?
Kκ
1 1==00
1
y1
μ1
e1
λ21
y2
μ2
e2
PRE 906 SEM: Lecture 9--Multifactor CFA
y3
e3
3
Var(F3)=?
κ2 = 0
1 λ62
λ31 λ41
μ3
covF2,F
y4
μ4
e4
y5
μ4
e5
y6
μ6
e6
κ3 = 0
λ72 λ82
y7
μ7
e7
1 λ103 λ113 λ123
y8
μ8
e8
y9
μ10 10
μ1111
μ12 12
e9
e10
e11
e12
μ9
y
y
60
y
Option 2a: 3 Factor “Indicators”
(Higher-Order Factor Variance = 1)
Same Measurement Model for Items:
Possible df = (12*13) / 2 + 12 = 90
Estimated df = 9𝛌 + 12𝛍 + 12𝛔𝟐𝐞 = 33
df = 90 – 33 = 57
 over-identified
New Structural Model for Factors:
Possible df = (3*4) / 2 + 0 = 6
Estimated df = 3𝛌 + 3𝛔𝟐𝐝
df = 6 – 6 = 0
 just-identified
Var(HF)=1
κHF = 0
λF
λF3
λF2
1
Leftover factor variances (part of factor not predicted
by higher-order factor) are called “disturbances”
Var(d1)=?
Var(d2)=?
κ1 = 0
κ2 = 0
F1
1
y1
μ1
e1
λ21
y2
μ2
e2
Var(d3)=?
1 λ62
λ31 λ41
y3
μ3
e3
y4
μ4
e4
κ3 = 0
F2
y5
μ5
e5
F3
λ72 λ82
y6
μ6
e6
y7
μ7
e7
1 λ103 λ113 λ123
y8
μ8
e8
y9
μ9
e9
y
μ1010
y
μ1111
e10
e11
If you only have 3 factors, both models will fit the same—the structural model
is just-identified, and thus the fit of a higher-order factor CANNOT be tested
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y
μ12 12
e12
Option 2b: 3 Factor “Indicators”
(using Marker Lower-Order Factor)
Same Measurement Model for Items:
Possible df = (12*13) / 2 + 12 = 90
Estimated df = 9𝛌 + 12𝛍 + 12𝛔𝟐𝐞 = 33
df = 90 – 33 = 57
 over-identified
New Structural Model for Factors:
Possible df = (3*4) / 2 + 0 = 6
Estimated df = 2𝛌 + 1𝛔𝟐𝐅 + 3𝛔𝟐𝐝
df = 6 – 6 = 0
 just-identified
Var(HF)=?
κHF = 0
1
λF3
λF2
Leftover factor variances (part of factor not predicted
by higher-order factor) are called “disturbances”
Var(d1)=?
Var(d2)=?
κ1 = 0
κ2 = 0
F1
1
y1
μ1
λ21
y2
μ2
e1
e2
Var(d3)=?
λ31
λ41
y3
μ3
e3
1
y4
μ4
e4
κ3 = 0
F2
y5
μ5
e5
λ62
λ72
y6
μ6
e6
F3
λ82
y7
μ7
e7
1
y8
μ8
e8
y9
μ9
e9
λ103
λ113
y10
μ10
e10
λ123
y11
μ11
e11
If you only have 3 factors, both models will fit the same—the structural model
is just-identified, and thus the fit of a higher-order factor CANNOT be tested
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y12
μ12
e12
Structural Model Identification:
2 Factor “Indicators”
Measurement Model for Items:
Possible df = (12*13) / 2 + 12 = 90
Estimated df = 8𝛌 + 12𝛍 + 12𝛔𝟐𝐞 = 32
df = 90 – 32 = 58  over-identified
Structural Model for Factors:
Possible df = (4*5) / 2 + 0 = 10
Estimated df = 4𝛌 + 0𝛔𝟐𝐅 + 1𝛔𝐅,𝐅 + 4𝛔𝟐𝐝
— OR —
Estimated df = 2𝛌 + 2𝛔𝟐𝐅 + 1𝛔𝐅,𝐅 + 4𝛔𝟐𝐝
df = 10 – 9 = 1  over-identified
However, this model will not be
identified structurally unless there is
covariance between the higher-order
factors
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Higher-Order Factor Identification
•
Possible structural df depends on # of measurement model factor
variances and covariances (NOT # items)

2 measurement model factors  Under-identified


They can be correlated, which would be just-identified… that’s it
3 measurement model factors  Just-identified
They can all be correlated OR a single higher-order factor can be fit
 Some # variance/disturbances per factor (so, 3 total) in either option
 Factor variances and covariances will be perfectly reproduced


4 measurement model factors  Can be over-identified
They can all be correlated (6 correlations required; just-identified)
 They can have a higher-order factor (4 loadings; over-identified)
 The fit of the higher-order factor can now be tested

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Examples of Structural Model Hypothesis Testing
•
Do I have a higher-order factor of my subscale factors?

If 4 or more subscale factors: Compare fit of alternative models


If 3 (or fewer) subscale factors: CANNOT BE DETERMINED

•
Saturated Baseline: All 6 factor covariances estimated freely
Alternative: 1 higher-order factor instead (so df=2)—is model fit WORSE?
Saturated baseline and alternative models will fit equivalently
Do I have need additional “method factors”?
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Model 04: Higher Order SE and Math Factors
•
We will add higher-order factors for Self-Efficacy and
Mathematics Items to our Model 03
•
When doing so, Model 03 (with the saturated factor
covariance matrix) now becomes our alternative model

•
No higher order factor model with the same Model 03 measurement model
can fit better than the saturated factor covariance matrix
Our higher order factors can only fit as well as the
saturated factor covariance matrix model
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Model 04: Lavaan Syntax
•
The higher order factors are defined with a =~ just like the
original factors

The terms on the right of the =~ are the original factors
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Model 04: Global Model Fit
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Model 04: Local Model Fit
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Model 04: Comparison with Model 03
•
The final model fit question: does Model 04 fit as well as
Model 03


•
If so: it is plausible that higher order factors exist
If not: no higher order factors
Using the likelihood ratio test:
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Model 04 Path Diagram
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At Long Last: Interpretation of Model Parameters
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At Long Last: Interpretation of Model Parameters
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At Long Last: Interpretation of Model Parameters
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At Long Last: Interpretation of Model Parameters
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At Long Last: Interpretation of Model Parameters
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WRAPPING UP
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Wrapping Up…
•
Fitting measurement and structural models are two separate issues:


Measurement model: Do my lower-order factors account for the
observed covariances among my ITEMS?
Structural model: Do higher-order factors account for the estimated covariances
among my measurement model FACTORS?

•
Figure out the measurement models FIRST, then structural models


•
A higher-order factor is NOT the same thing as a ‘total score’ though
Recommend fitting measurement models separately per factor, then bringing them
together once you have each factor/theta settled
This will help to pinpoint the source of misfit in complex models
Keep in mind that structural models may not be ‘unique’

Mathematically equivalent models can make very different theoretical statements, so
there’s no real way to choose between them if so…
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