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
**

**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.
**

**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,**

** **** 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
**

**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)
**

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)

Note that making the model simpler, while often a very reasonable thing to do, does not improve the fit of the model.

**Specialized Issues**

** ****How many indicators per factor?
**

4

Wh

**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.**