Constructing Approximate
Confidence Intervals for Parameters with Structural Equation Models
Structural equation modeling (SEM) may be used to
construct Wald (based on standard errors), likelihood-based and bootstrap
confidence intervals (CIs) for a variety of statistics and psychometric indices
(Cheung, 2009). By using the idea of phantom variables (Rindskopf, 1984), we
may create a path "p" to compute the required statistics. Then
CIs on "p" are equivalent to the CIs constructed on the
required statistics. Please note that the phantom variables (P and Q) are introduced to explain how to construct the CIs. It is
usually not necessary to introduce them in many SEM packages. There is also a
homepage on constructing
CIs on the indirect effects.
Example 1: Difference in Dependent Correlations
To construct a 95% CI on the
difference of the dependent correlations
, we may impose
. Then the 95% CI on p is the 95% CI on the difference
of the dependent correlations (also see Cheung & Chan, 2004).
Example 2: Squared Multiple R and Standardized Regression Coefficient
R2
is usually used to quantify the percentage of variance explained by all
predictors. We may construct a CI on the R2
by using
. Moreover, we may also construct a CI on the standardized
regression coefficient from JS to LS by using
.
Example 3: Coefficient
Alpha
To construct a CI on a coefficient alpha, we may
impose the following constraint: 
.
Example 4:
Reliability Estimate
To estimate the reliability coefficient, we impose the
following equality constraint, 
.
References
Cheung, M.W.L.
(2009). Constructing
approximate confidence intervals for parameters with structural equation models.
Structural Equation Modeling, 16, 267-294. (Mplus, LISREL and Mx
syntax, outputs and sample data are available here)
Cheung, M.W.L.,
& Chan, W. (2004). Testing dependent
correlation coefficients via structural equation modeling. Organizational Research Methods, 7, 206-223.
Rindskopf, D.
(1984). Using phantom and imaginary latent variables to parameterize
constraints in linear structural models. Psychometrika,
49, 37-47.