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 as easily used, as in the other approaches. One way of getting 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. Then, the only difference between the
latent variable and its indicator, is that (assuming that it is an
exogenous variable) it has a zero-mean. This will work for both the LMS-
and QML approach in most cases, except for two exceptions.
The LMS approach
For the LMS approach you can use the above mentioned approach in
almost all cases, except in the case where you wish to use an observed
variable as a moderating variable. In the LMS approach, you will usually
select one variable in an interaction term as a moderator. The
interaction effect is then estimated via numerical integration, at
n quadrature nodes of the moderating variable. This process
however, requires that the moderating variable has an error-term, as the
distribution of the moderating variable is modelled 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. Thus if only one of the variables are latent,
you should put the latent variable first in the interaction term. If
both are observed, you have to 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(m1, oneInt, method = "lms")
summary(lms2)
#> Estimating null model
#> EM: Iteration = 1, LogLik = -10816.13, Change = -10816.126
#> EM: Iteration = 2, LogLik = -10816.13, Change = 0.000
#>
#> modsem (version 1.0.2):
#> Estimator LMS
#> Optimization method EM-NLMINB
#> Number of observations 2000
#> Number of iterations 58
#> Loglikelihood -8087.2
#> Akaike (AIC) 16202.4
#> Bayesian (BIC) 16280.81
#>
#> Numerical Integration:
#> Points of integration (per dim) 24
#> Dimensions 1
#> Total points of integration 24
#>
#> Fit Measures for H0:
#> Loglikelihood -10816
#> Akaike (AIC) 21658.25
#> Bayesian (BIC) 21731.06
#> Chi-square 0.01
#> Degrees of Freedom (Chi-square) 1
#> P-value (Chi-square) 0.917
#> RMSEA 0.000
#>
#> Comparative fit to H0 (no interaction effect)
#> Loglikelihood change 2728.93
#> Difference test (D) 5457.85
#> Degrees of freedom (D) 1
#> P-value (D) 0.000
#>
#> R-Squared:
#> Y 0.510
#> R-Squared Null-Model (H0):
#> Y 0.343
#> R-Squared Change:
#> Y 0.167
#>
#> Parameter Estimates:
#> Coefficients unstandardized
#> Information expected
#> Standard errors standard
#>
#> Latent Variables:
#> Estimate Std.Error z.value P(>|z|)
#> Z =~
#> z1 1.000
#> z2 0.811 0.018 45.02 0.000
#> X =~
#> x1 1.000
#> Y =~
#> y1 1.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> Y ~
#> Z 0.587 0.033 18.04 0.000
#> X 0.574 0.029 20.07 0.000
#> Z:X 0.627 0.027 23.51 0.000
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> z1 1.009 0.024 42.79 0.000
#> z2 1.203 0.020 60.79 0.000
#> x1 1.023 0.024 42.76 0.000
#> y1 1.046 0.033 31.42 0.000
#> Y 0.000
#> Z 0.000
#> X 0.000
#>
#> Covariances:
#> Estimate Std.Error z.value P(>|z|)
#> Z ~~
#> X 0.211 0.025 8.54 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> z1 0.170 0.018 9.21 0.000
#> z2 0.160 0.013 12.71 0.000
#> x1 0.000
#> y1 0.000
#> Z 1.010 0.020 51.00 0.000
#> X 1.141 0.017 67.45 0.000
#> Y 1.284 0.043 29.98 0.000The QML approach
If you are using the latest GitHub version
The estimation of the QML approach is different from the LMS approach, and you do not have the same issue as in the LMS approach. Thus you don’t have 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 1.0.2):
#> 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 0.000If you are using the CRAN version
If you are using the latest CRAN version, there is a slight caveat,
in that all endogenous variables have to have atleast two indicators.
This is due to a transformation, and the approximation of the
distribution of the indicators in the endogenous variables. This problem
will likely be fixed in a later update, but as of now, latent endogenous
variable need at least two indicators. If a latent variable in the QML
approach can be expressed without using an interaction term, you can in
some cases use the ‘cov.syntax’ argument as a workaround. If this is the
case, see the vignette on interaction effects between two endogenous
variable in the LMS- and QML approach
(vignette("interaction_two_etas"))
m4 <- '
# Outer Model
X =~ x1 # X is observed
Z =~ z1 # Z is observed
Y =~ y1 + y2 # Y needs to be latent, needing atleast two indicators
# Inner model
Y ~ X + Z
Y ~ X:Z
'
qml4 <- modsem(m3, oneInt, method = "qml")
summary(qml4)
#> Estimating null model
#> Starting M-step
#>
#> modsem (version 1.0.2):
#> 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