observed variables in the LMS- and QML approach
Source:vignettes/observed_lms_qml.Rmd
observed_lms_qml.RmdThe Latent Moderated Structural Equations (LMS) and the Quasi Maximum Likelihood (QML) Approach
In contrast to the other approaches, the LMS and
QML approaches are designed to handle latent variables
only. Thus, observed variables cannot be used as easily as in the other
approaches. One way to get around this is by specifying your observed
variable as a latent variable with a single indicator.
modsem() will, by default, constrain the factor loading to
1 and the residual variance of the indicator to
0. The only difference between the latent variable and its
indicator, assuming it is an exogenous variable, is that it has a
zero-mean. This approach works for both the LMS and
QML methods in most cases, with two exceptions.
The LMS Approach
For the LMS approach, you can use the above-mentioned
method in almost all cases, except when using an observed variable as a
moderating variable. In the LMS approach, you typically
select one variable in an interaction term as the moderator. The
interaction effect is then estimated via numerical integration at
n quadrature nodes of the moderating variable. However,
this process requires that the moderating variable has an error term, as
the distribution of the moderating variable is modeled as
,
where
is the expected value of
at quadrature point k, and
is the error term. If the error term is zero, the probability of
observing a given value of
will not be computable.
In most instances, the first variable in the interaction term is
chosen as the moderator. For example, if the interaction term is
"X:Z", "X" will usually be chosen as the
moderator. Therefore, if only one of the variables is latent, you should
place the latent variable first in the interaction term. If both
variables are observed, you must specify a measurement error (e.g.,
"x1 ~~ 0.1 * x1") for the indicator of the first variable
in the interaction term.
library(modsem)
# Interaction effect between a latent and an observed variable
m1 <- '
# Outer Model
X =~ x1 # X is observed
Z =~ z1 + z2 # Z is latent
Y =~ y1 # Y is observed
# Inner model
Y ~ X + Z
Y ~ Z:X
'
lms1 <- modsem(m1, oneInt, method = "lms")
# Interaction effect between two observed variables
m2 <- '
# Outer Model
X =~ x1 # X is observed
Z =~ z1 # Z is observed
Y =~ y1 # Y is observed
x1 ~~ 0.1 * x1 # Specify a variance for the measurement error
# Inner model
Y ~ X + Z
Y ~ X:Z
'
lms2 <- modsem(m2, oneInt, method = "lms")
summary(lms2)
#> Estimating null model
#> EM: Iteration = 1, LogLik = -9369.23, Change = -9369.232
#> EM: Iteration = 2, LogLik = -9369.23, Change = 0.000
#>
#> modsem (version 1.0.4):
#> Estimator LMS
#> Optimization method EM-NLMINB
#> Number of observations 2000
#> Number of iterations 38
#> Loglikelihood -6632.86
#> Akaike (AIC) 13285.71
#> Bayesian (BIC) 13341.72
#>
#> Numerical Integration:
#> Points of integration (per dim) 24
#> Dimensions 1
#> Total points of integration 24
#>
#> Fit Measures for H0:
#> Loglikelihood -9369
#> Akaike (AIC) 18756.46
#> Bayesian (BIC) 18806.87
#> Chi-square 0.00
#> Degrees of Freedom (Chi-square) 0
#> P-value (Chi-square) 0.000
#> RMSEA 0.000
#>
#> Comparative fit to H0 (no interaction effect)
#> Loglikelihood change 2736.38
#> Difference test (D) 5472.75
#> Degrees of freedom (D) 1
#> P-value (D) 0.000
#>
#> R-Squared:
#> Y 0.494
#> R-Squared Null-Model (H0):
#> Y 0.335
#> R-Squared Change:
#> Y 0.160
#>
#> Parameter Estimates:
#> Coefficients unstandardized
#> Information expected
#> Standard errors standard
#>
#> Latent Variables:
#> Estimate Std.Error z.value P(>|z|)
#> X =~
#> x1 1.000
#> Z =~
#> z1 1.000
#> Y =~
#> y1 1.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> Y ~
#> X 0.663 0.032 20.68 0.000
#> Z 0.482 0.029 16.55 0.000
#> X:Z 0.586 0.025 23.89 0.000
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> x1 1.023 0.025 40.20 0.000
#> z1 1.011 0.027 37.67 0.000
#> y1 1.057 0.036 29.02 0.000
#> Y 0.000
#> X 0.000
#> Z 0.000
#>
#> Covariances:
#> Estimate Std.Error z.value P(>|z|)
#> X ~~
#> Z 0.208 0.029 7.21 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> x1 0.100
#> z1 0.000
#> y1 0.000
#> X 1.028 0.035 29.37 0.000
#> Z 1.184 0.037 31.87 0.000
#> Y 1.323 0.044 30.17 0.000The QML Approach
The estimation process for the QML approach differs from
the LMS approach, and you do not encounter the same issue
as in the LMS approach. Therefore, you don’t need to
specify a measurement error for moderating variables.
m3 <- '
# Outer Model
X =~ x1 # X is observed
Z =~ z1 # Z is observed
Y =~ y1 # Y is observed
# Inner model
Y ~ X + Z
Y ~ X:Z
'
qml3 <- modsem(m3, oneInt, method = "qml")
summary(qml3)
#> Estimating null model
#> Starting M-step
#>
#> modsem (version 1.0.4):
#> Estimator QML
#> Optimization method NLMINB
#> Number of observations 2000
#> Number of iterations 11
#> Loglikelihood -9117.07
#> Akaike (AIC) 18254.13
#> Bayesian (BIC) 18310.14
#>
#> Fit Measures for H0:
#> Loglikelihood -9369
#> Akaike (AIC) 18756.46
#> Bayesian (BIC) 18806.87
#> Chi-square 0.00
#> Degrees of Freedom (Chi-square) 0
#> P-value (Chi-square) 0.000
#> RMSEA 0.000
#>
#> Comparative fit to H0 (no interaction effect)
#> Loglikelihood change 252.17
#> Difference test (D) 504.33
#> Degrees of freedom (D) 1
#> P-value (D) 0.000
#>
#> R-Squared:
#> Y 0.470
#> R-Squared Null-Model (H0):
#> Y 0.320
#> R-Squared Change:
#> Y 0.150
#>
#> Parameter Estimates:
#> Coefficients unstandardized
#> Information observed
#> Standard errors standard
#>
#> Latent Variables:
#> Estimate Std.Error z.value P(>|z|)
#> X =~
#> x1 1.000
#> Z =~
#> z1 1.000
#> Y =~
#> y1 1.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> Y ~
#> X 0.605 0.028 21.26 0.000
#> Z 0.490 0.028 17.55 0.000
#> X:Z 0.544 0.023 23.95 0.000
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> x1 1.023 0.024 42.83 0.000
#> z1 1.011 0.024 41.56 0.000
#> y1 1.066 0.034 31.64 0.000
#> Y 0.000
#> X 0.000
#> Z 0.000
#>
#> Covariances:
#> Estimate Std.Error z.value P(>|z|)
#> X ~~
#> Z 0.210 0.026 7.95 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> x1 0.000
#> z1 0.000
#> y1 0.000
#> X 1.141 0.036 31.62 0.000
#> Z 1.184 0.037 31.62 0.000
#> Y 1.399 0.044 31.62 0.000