David A. Kenny

April 9, 2016

**Multiple Latent Variable Models:
Confirmatory Factor Analysis**

**Standard Exploratory Factor
Analysis Model or EFA**

**Every measure loads on each
factor**

**either
uncorrelated (orthogonal) or correlated (oblique)**

**generally
factors are uncorrelated**

**Because with more than one
factor, the solution is not unique (i.e., underidentified), it can be rotated. **

**To test if k factors are sufficient to
explain the covariation between measures estimate the following loading matrix
(assuming k = 5)
with orthogonal or uncorrelated factors with unit variance:**

__Measure
1
2
3
4 5__

**
1
x
0
0 0
0**

**
2
x
x
0 0
0**

**
3
x
x
x 0
0**

**
4
x
x
x x
0**

**
5
x
x x
x
x**

**
6
x
x
x x
x**

**
7
x
x
x x
x**

**
8
x
x
x x
x**

**If a model with this loading structure is good fitting (see Measures of Fit), then k factors are sufficient. **

**Confirmatory Factor Analysis Model or CFA (an
alternative to EFA)**

** Typically, each variable
loads on one and only one factor.**

** Factors are correlated
(conceptually useful to have correlated factors).**

** Generally errors (or
uniquenesses) across variables are uncorrelated.**

** Variables in CFA are usually
called indicators.**

**Degrees of Freedom (df)
for CFA Models**

**Unknowns**

** Free loadings (do not
count marker variable or loadings set equal**)

** Error variances**

** Correlated errors**

** Factor variances**

** Factor correlations**

**Knowns: k(k + 1)/2**

**Identification**** **

**Given ***k*** factors, there must be ***k*^{2}** constraints.**

**Usually ***k*** of these constraints are scaling ones
(i.e., marker variables).**

**The standard EFA model
with two or more factors and all the loadings free is not identified.
This is why the solution can be rotated.**

**Standard
CFA model: Simple Structure**

**
Each measure or indicator loads on one and only one factor which implies no
double loadings.**

**
No correlated errors**

**
Latent variables correlated**

**Simple Structure CFA model
is identified:**

** If
there are, at least, two indicators per latent variable and the errors of those
two or more indicators are uncorrelated with each other and with at least one
other indicator on the other latent variables.**

**Testing in
CFA and Structural Equation Modeling**

**Principle of
nesting: Model A is said to be nested within Model B, if Model B is a
more complicated version of Model A. For example, a one-factor model is nested
within a two-factor as a one-factor model can be viewed as a two-factor model
in which the correlation between factors is perfect). **

**Relative fit of a nested
model: the chi square difference test, the smaller chi square and its degrees
of freedom are subtracted from the larger chi square and degrees of freedom.**

**In principle, the more
complicated model should fit for the test to be valid.**

**Discriminant Validity**

**Definition of poor
discriminant validity: The correlation between two factors is or is very close
to one or minus one.**

**
Consequences**

**
multicollinearity: If the factors are treated as
causes of a third factor, the high collinearity leads to very large standard
errors.**

**
problems of convergence** and inadmissabile solutions

**
Criteria: A correlation of .85 or larger in absolute value indicates poor
discriminant validity**

**
Test: Estimate a model that fixes the correlation to one (Do not use a
marker variable strategy, but instead fix factor variances to one.) or collapse
the two factors and see if the model fit worsens.**

**Example 1: Unpublished
Master’s Thesis of Julie Fenster: “Multidimensional measurement of
Religiousness/Spirituality for use in health research assessment developed by
the Fetzer Institute”**

**Three Latent Variables**

**Daily Spiritual
Experiences (DSE)**

**
I feel God’s presence.**

**
I am touched by the beauty of creation.**

**Private Religious
Practices (PRP)**

**
Private prayer.**

**
Read the Bible.**

**Positive Religious and
Spiritual Coping (PRSC)**

**
Think about life as part of a larger spiritual force.**

**
I look to God for strength, support and guidance. **

**Correlations**

** DSE
with PRSC = .869**

** PRP
with PRSC = .918**

** DSE
with PRP = .910 **

**See also “Exploring the
Dimensionality of "Religiosity" and "Spirituality" in the
Fetzer Multidimensional Measure” by J. A. Neff, ****Journal for the Scientific Study of
Religion, 45****, 449‑459.
**

**
Example 2: Salovey, P., & Rodin, J. (1984). Some antecedents and
consequences of social-comparison jealousy. ****Journal of Personality and Social
Psychology, 47****,
780-792.**

**
One latent variable model χ²(5) = 24.305**

**
Two latent variable model χ²(4) = 8.669 **

**
****chi square difference test: ****χ²****(1) =
15.636, p < .001**

**
****conclusion:
two latent variables are needed**

**Salovey & Rodin Example with Standardized Estimates**

**
Example 3: Braze, D., Katz, L., Magnuson, J. S., Mencl, W. E., Tabor, W., Van
Dyke, J. A., Gong, T., Johns, C. L., & Shankweiler, D. P. (2016).
Vocabulary does not complicate the Simple View of Reading. ****Reading and Writing, 29****, 435-451: In this paper, they show that
language comprehension (LC) and reading comprehension (RC) have poor
discriminant validity.**

**Braze et al. Example (Standardized Estimates)**

**Respecification****
(see Respecification
page for more detail)
Criteria
Empirical
(again see Respecification page for more details)
Correlation matrix
Modification indices (also called Lagranian multipliers)
The
estimated change in chi square if the parameter were freely estimated.
Standardized residuals
If
model is correctly specified, large values (greater than 1.96 in absolute
value) indicate correlations poor fitted.
In
my experience, these values tend to be conservative (i.e., too small).
Theoretical:
All respecifications require some rationale and that rationale should be
extended to other cases.**

**Types of
Respecifications****
**

** ****Resulting in a MORE COMPLEX
MODEL ****(i.e.,
more parameters)****
Another factor
Correlated errors
definition:
Variance not explained by theoretical constructs may covary across two
measures. Such covariance is referred to as a correlated error.
Double loadings
**

**Specialized
Issues****
How many indicators per factor?
2 is the minimum
3 is safer, especially
if factor correlations are weak
4 provides safety
5 or more is more than
enough (If too many indicators then combine indicators into sets)
What to do about “too many” indicators? Parcels or “testlets”
Definition: Adding (or
averaging) sets of indicators up to create a smaller number of indicators
Strategies
random
conceptually similar sets
sets that may contain items with correlated errors
Disadvantages of parceling
loss of information
possibility of specification error
that is missed and becomes undetectable
Advantages of parceling
smaller models (better
participant to parameter ratio)
more “normal” distributions of variables
usually better fit
Compromise
strategy: Run individual CFA on each latent variable and then parcel.**

**Single Indicators **

**
measures with no measurement error**

**
**Treat as variable in most programs but LISREL requires

**
fix loading to one **

**
free variance if exogenous or disturbance if endogenous **

**
fix error variance to zero **

**
do not correlate its "error variance" with anything **

**
measures with measurement error**

**
fix loading to one**

**
free variance if exogenous or disturbance variance if endogenous **

**
error variance**

**
fix to a known value (see Williams and Hazer) or**

**
find an instrumental variable**

**Models with Means**

**Strategy 1**

** fix
factor mean (if exogenous) or intercept (if endogenous) to zero**

** free
all indicator intercepts **

**Strategy 2**

** free
factor mean (if exogenous) or intercept (if endogenous)**

** fix
the marker variable’s intercept to zero**

** free
all other indicators’ intercepts**

**The model fit and other
parameter estimates (e.g., loadings) are the same for both strategies.
Most people find the second strategy simpler and easier to work with.**