observed variables in the LMS- and QML approach
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)
#>
#> modsem (1.0.12) ended normally after 22 iterations
#>
#> Estimator LMS
#> Optimization method EMA-NLMINB
#> Number of model parameters 10
#>
#> Number of observations 2000
#>
#> Loglikelihood and Information Criteria:
#> Loglikelihood -9114.81
#> Akaike (AIC) 18249.62
#> Bayesian (BIC) 18305.62
#>
#> Numerical Integration:
#> Points of integration (per dim) 24
#> Dimensions 1
#> Total points of integration 24
#>
#> Fit Measures for Baseline Model (H0):
#> Loglikelihood -9369.23
#> 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 Inf
#> SRMR 0.000
#>
#> Comparative Fit to H0 (LRT test):
#> Loglikelihood change 254.42
#> Difference test (D) 508.85
#> Degrees of freedom (D) 1
#> P-value (D) 0.000
#>
#> R-Squared Interaction Model (H1):
#> Y 0.497
#> R-Squared Baseline Model (H0):
#> Y 0.335
#> R-Squared Change (H1 - H0):
#> Y 0.163
#>
#> 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 0.664 0.031 21.320 0.000
#> Z 0.481 0.028 16.940 0.000
#> X:Z 0.588 0.025 23.479 0.000
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> .x1 1.023 0.024 42.843 0.000
#> .z1 1.012 0.024 41.576 0.000
#> .y1 1.057 0.034 31.316 0.000
#>
#> Covariances:
#> Estimate Std.Error z.value P(>|z|)
#> X ~~
#> Z 0.210 0.026 7.970 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> .x1 0.100
#> .z1 0.000
#> .y1 0.000
#> X 1.040 0.036 28.861 0.000
#> Z 1.184 0.037 31.627 0.000
#> .Y 1.322 0.044 29.910 0.000If you forget to specify a measurement error for the indicator of the first variable in the interaction term, you will receive an error message.
m2 <- '
# 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
'
lms3 <- modsem(m2, oneInt, method = "lms")
#> Error: The variance of a moderating variable of integration has an indicator with zero residual variance!
#> This will likely not work with the LMS approach, see:
#> `vignette('observed_lms_qml', 'modsem')` for more information.
#>
#> The following indicators have zero residual variance:
#> -> x1Note: You only get an error message for
X/x1, since Z is not modelled as
a moderating variable in this example.
The 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)
#>
#> modsem (1.0.12) ended normally after 11 iterations
#>
#> Estimator QML
#> Optimization method NLMINB
#> Number of model parameters 10
#>
#> Number of observations 2000
#>
#> Loglikelihood and Information Criteria:
#> Loglikelihood -9117.07
#> Akaike (AIC) 18254.13
#> Bayesian (BIC) 18310.14
#>
#> Fit Measures for Baseline Model (H0):
#> Loglikelihood -9369.23
#> 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 Inf
#> SRMR 0.000
#>
#> Comparative Fit to H0 (LRT test):
#> Loglikelihood change 252.17
#> Difference test (D) 504.33
#> Degrees of freedom (D) 1
#> P-value (D) 0.000
#>
#> R-Squared Interaction Model (H1):
#> Y 0.470
#> R-Squared Baseline Model (H0):
#> Y 0.320
#> R-Squared Change (H1 - H0):
#> 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 21.256 0.000
#> Z 0.490 0.028 17.553 0.000
#> X:Z 0.544 0.023 23.950 0.000
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> .x1 1.023 0.024 42.828 0.000
#> .z1 1.011 0.024 41.562 0.000
#> .y1 1.066 0.034 31.638 0.000
#>
#> Covariances:
#> Estimate Std.Error z.value P(>|z|)
#> X ~~
#> Z 0.210 0.026 7.952 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> .x1 0.000
#> .z1 0.000
#> .y1 0.000
#> X 1.141 0.036 31.625 0.000
#> Z 1.184 0.037 31.625 0.000
#> .Y 1.399 0.044 31.624 0.000