David
A. Kenny

September
11, 2011

Being revised after 12 years.

Please send
comments and suggestions.

**Respecification of
Latent Variable Models**

****These
simplifications in the model do not usually improve the model's fit and are in purple.**** **

**Those in black without asterisks may improve the fit of the
model. **

**Step A: Is the measurement
model consistent with the data (i.e., good fitting)?**** **

**Yes, go to Step C.
No, go to Step B unless you have exhausted reasonable respecifications.
IF SO, THEN STOP!**

**Step B: Revise the measurement model. ****(Go to the CFA page
for other ideas.)**** **

** Strategies
for respecification**

**correlation**** matrix**

**modification**** index (Lagrangian
multiplier in EQS)**

**definition****: The approximate
increase in chi square if the parameter were free**

** zero implies parameter**

** already estimated**

** not identified**

** estimated as zero**

**sequence****: a large modification
index may be explained by some other specification error**

**Some modification indices (especially with Amos) may be implausible and
should be ignored**

**standardized**** residual: Difference
between predicted and observed covariance standardized**

** What
to respecify?**

**Factor loadings **

**Too small?****
Drop the measure?******
**

**Error variances **

**Too small or negative:** Heywood cases?**

**Set the error variances equal?** **(Measures should have the same metric and the covariance matrix
analyzed.) ********** **

Need fewer factors?

Poor discriminant validity?(Combine factors.)**

Make sure each factor has sufficient variance. (Drop factors)**

**Need more factors? **

**Run a maximum likelihood exploratory factor analysis to test the number
of factors (see how to do this). (Note that
this test presumes no correlated errors and so might be misleading if the true
model has correlated errors.)**

**Evaluate measures **

**Check to see if a
measure has large standardized residuals and modification indices (Lagrangian
multipliers in EQS). Consider dropping that measure.**

**Need correlated errors? **

**Considerations**

**Meaningfulness Rule: Only correlate errors for which there is a
theoretical rationale.**

**Transitivity Rule: If X _{1} is correlated with X_{2}
and X_{2} with X_{3}, then X_{1} should be correlated
with X_{3 }unless one pair is correlated for one reason and the other
for a different reason.**

**Generality Rule: If there is a reason for correlating the errors
between one pair of errors, then all pairs for which that reason applies should
also be correlated.**

**Sometimes correlated errors imply an additional factor. For
example, if X _{1}, X_{2 }and X_{3} all have correlated
errors, then consideration might be given to adding a factor on which the three
load.**

** **

**Single indicator
variables may not have correlated errors but indicators of other factors may
load on the single indicator "factor."**

**Which errors to
correlate?**

**large**** modification indices**

**large**** standardized residuals**

**Need measures to load
on more than one factor? (Read about
identification difficulties.)**

**When done return to
Step A. **

**Step C: If the structural
model is just-identified go directly to
Step D.**** **

**
Are the deleted paths in the structural model
actually zero? **

** No, go to Step D.**

**Yes, add in deleted paths (see how deleted paths are defined) and go to Step D. **

**Step D: Are the specified paths of the structural model needed?**** **

Yes, keep in model.

No, consider trimming from model.**

**Equivalent
Models**** **

**Even if one's model fits, there are
a myriad of other models that fit as well.
These equivalent models should
be considered. Realize that for any model, there always exist an
infinite number of models that fit exactly the same. Thus, although the
fit of the structural model confirms it, it in no way proves it to be uniquely
valid. **

**Respecification Strategies**** **

**There are two strategies to take in
the process of re-specifying a model. One can test a priori,
theoretically meaningful complications and simplifications of the model.
Alternatively, one can use empirical tests (e.g., modification indices and
standardized residuals) to respecify the model. ****All respecifications
should be theoretically meaningful**** and ideally a
priori. Too many empirically based respecifications likely lead to
capitalization on chance and over-fitting (unnecessary parameters added to the
model). Ideally, if many respecifications are made, a replication of the
model should be undertaken. Although a priori hypotheses deserve the
initial focus, an examination of empirical tests of
miss-specification are in order. But if model changes are made on
the basis of such tests, there still need to be some sort of theoretical
rationale for them. **

**Example**

** Consider the following study:
Reisenzein, R. (1986). A structural equation analysis of Weiner's attribution-affect model
of helping behavior. Journal of Personality and Social
Psychology, 50, 1123-1133.**

**The study asks UCLA
students to imagine that they are on a subway and they see a man lying on the
ground and he appears to blind (a white cane is next to him) or drunk (an empty
whiskey bottle is next do him). Do you
help this man? **

**Measurement Model**

**Control Anger**

** C _{1}:
controllable A_{1}: anger**

** C _{2}:
responsible A_{2}: irritated
**

** C _{3}:
fault A_{3}: aggravated**

**Sympathy Helping**

** S _{1}:
sympathy H_{1}: likelihood of helping**

** S _{2}:
pity H_{2}: certainty**

** S _{3}:
concern H_{3}: amount**

**Scenario: **

**single**** indicator dichotomy, no measurement
error, experimentally manipulated; 0 = blind and 1 = drunk **

**Structural Mod****el**

**Control = Scenario + U1**

** Anger
= Control + U3**

** ****Sympathy = Control + U2 **

**(disturbances**** of U2 and U3 correlated)
**

** Helping
= Anger + Sympathy + U4**

*Specified Model*

**The fit of the
specified model is **

**χ²(****60) = 87.507 p = .012 (fit: TLI =
.974 and RMSEA = .058)**

**Most investigators would stop
here. We adopt a different approach.**

**Note the df
of the measurement is 56 and the df of the structural model is 4, making the
total df 60.**

*Respecification of the Reisenzein
Model*

**Step A: Fit of the
Measurement Model or Just-identified Structural Model**

** χ²(56) =
81.63, p = .014 (TLI = .974; RMSEA = .058)**

**The fit is good, but it might be
improved. **

**Step B: Revised
Measurement Model**

**Let us examine at the largest
modification indices:**

**Covariances: M.I.
Par Change**

** E9 <-------------> Help 16.103 0.154**

**Regression
Weights: M.I. Par Change**

** S3 <------------- Help 10.581 0.198**

** S3 <---------------- H3 11.564 0.188**

** S3 <---------------- H1 11.500 0.187**

**The modification indices point to the S3 (concern) loading on the Help
factor.****
Because this makes sense, the measurement model is revised allowing for
this loading. (One might argue that S3
should be dropped as it is not a clean indicator.)**

**Test Revised
Measurement Model**

**χ²(****55) = 58.89, p = .335 (TLI = .996;
RMSEA = .023 and 0 is in the 90% confidence interval)**

**There are no Heywood
cases and all of factor correlations (except between Scenario and Control) are
less than .85. The fit of the
measurement model is deemed acceptable.
We can live with this large correlation because it is in essence a
manipulation check.**

**Step C: Test of the
Deleted Paths (paths assumed to be zero) **

** **

**The tests of the
deleted paths are as follows:**

__Cause Effect
Beta Z-Value__

**Scenario Anger .052 0.327**

**Scenario Sympathy .126 0.767**

**Scenario Helping -.354
-2.340**

**Control Helping
.186 0.836**

**So the only deleted
path needed is the path from Scenario to Helping. The path is negative indicating a willingness
to help the control person, the blind man, than the drunk
man. The combined test of the four
deleted paths is χ²(4) = 8.07, p < .10.**

**Step D: Test of the
Specified Paths and Correlation **

**All of the specified
paths are statistically significant.
However, the correlation between the disturbances of Anger and Sympathy
is not statistically significant (Z = 1.213).**

**Step E: Trimmed Model **

**The correlation
between the disturbances of Anger and Sympathy is set to zero.**

**
**

**
Factor Loadings**

** Factor **

**Variable Control
Sympathy Anger Helping**

**C1 .835 **

**C2 .858 **

**C3 .901 **

**S1 .955 **

**S2 .758 **

**S3 .574 .349 **

**A1 .885 **

**A2 .822 **

**A3 .841 **

**H1
.941 **

**H2
.923 **

**H3 .786**

Control .74 Sympathy .43 Anger .50 Help .50

*Go to the next SEM page.*

Go to the main SEM page.