20231115 - CCA

Author: isaactpetersen

08-Structural-Equation-Modeling.Rmd

@@ -537,6 +537,99 @@ Thus, [formative constructs](#formativeConstruct) are challenging to use in a SE
 To estimate a [formative construct](#formativeConstruct) in a SEM framework, the [formative construct](#formativeConstruct) must be used in the context of a model that allows some constraints.\index{construct!formative}\index{structural equation modeling!model identification}\index{structural equation modeling!degrees of freedom}
 However, emerging techniques such as confirmatory composite analysis allow estimation of [formative](#formativeConstruct) latent factors [@Schuberth2023; @Yu2023].\index{construct!formative}\index{confirmatory composite analysis}
 
+Below is an example using confirmatory composite analysis (adapted from: https://confirmatorycompositeanalysis.com/tutorials-lavaan; archived at: https://perma.cc/7LSU-PTZR):
+
+```{r}
+formativeModel3_syntax <- '
+  # Specification of the reflective latent factor
+  reflective =~ y1 + y2 + y3 + y4
+  
+  # Specification of the associations between the observed variables v1 - v4
+  # and the emergent variable "formative" in terms of composite loadings.
+  
+  formative =~ NA*v1 + l11*v1+ l21*v2 + 1*v3 + l41*v4
+  
+  formative ~~ vformative*formative
+  
+  # Specification of the associations between the observed variables v1 - v4
+  # and their excrescent variables in terms of composite loadings.
+  
+  nu11 =~ 1*v1 + l22*v2 + l32*v3 + l42*v4
+  nu12 =~ 0*v1 + 1*v2 + l33*v3 + l43*v4
+  nu13 =~ 0*v1 + 0*v2 + l34*v3 + 1*v4
+  
+  nu11 ~~ vnu11*nu11
+  nu12 ~~ vnu12*nu12
+  nu13 ~~ vnu13*nu13
+  
+  # Specification of the covariances among the latent variables
+  formative ~~ reflective
+  
+  # The H-O specification assumes that the excrescent variables are uncorrelated.
+  # Therefore, the covariance between the excrescent variables is fixed to 0:
+  
+  nu11 ~~ 0*nu12 + 0*nu13
+  nu12 ~~ 0*nu13
+  
+  # Moreover, the H-O specification assumes that the excrescent variables are uncorrelated
+  # with the emergent and latent variables. Therefore, the covariances between
+  # the emergent and the excrescent varibales are fixed to 0:
+  
+  formative ~~ 0*nu11 + 0*nu12 + 0*nu13
+  
+  reflective =~ 0*nu11 + 0*nu12 + 0*nu13
+  
+  # In lavaan, the =~ command is originally used to specify a common factor model,
+  # which assumes that each observed variable is affected by a random measurement error.
+  # It is assumed that the observed variables forming composites are free from
+  # random measurement error. Therefore, the variances of the random measurement errors
+  # originally attached to the observed variables by the common factor model are fixed to 0:
+  
+  v1 ~~ 0*v1
+  v2 ~~ 0*v2
+  v3 ~~ 0*v3
+  v4 ~~ 0*v4
+  
+  # Specify weights:
+  w1:= (-l32 + l22*l33 + l34*l42 - l22*l34*l43)/(
+    1 - l11*l32 - l21*l33 + l11*l22*l33 - l34*l41 + l11*l34*l42 + 
+     l21* l34* l43 - l11* l22* l34* l43)
+  w2:= (-l33 + l34*l43)/(1 - l11*l32 - l21*l33 + 
+      l11*l22*l33 - l34*l41 + l11*l34*l42 + l21*l34*l43 - l11*l22*l34*l43)
+  w3:= 1/(1 - l11*l32 - l21*l33 + l11*l22*l33 -
+      l34*l41 + l11*l34*l42 + l21*l34*l43 - l11*l22*l34*l43)
+  w4:= -l34/(1 - l11*l32 - l21*l33 + l11*l22*l33 -
+      l34*l41 + l11*l34*l42 + l21*l34*l43 - l11*l22*l34*l43)
+  
+  vv1 := l11^2*vformative + vnu11
+  vv2 := l21^2*vformative + l22^2*vnu11 + vnu12
+  vv3 := vformative + l32^2*vnu11 + l33^2 * vnu12 + l34^2*vnu13
+  vv4 := l41^2*vformative + l42^2*vnu11 + l43^2*vnu12 + vnu13
+  
+  w1std := w1*(vv1/vformative)^(1/2)
+  w2std := w2*(vv2/vformative)^(1/2)
+  w3std := w3*(vv3/vformative)^(1/2)
+  w4std := w4*(vv4/vformative)^(1/2)
+'
+
+formativeModel3Fit <- sem(
+  formativeModel3_syntax,
+  data = PoliticalDemocracy,
+  missing = "ML",
+  estimator = "MLR")
+
+formativeModel3Parameters <- parameterEstimates(formativeModel3Fit)
+formativeModel3Parameters
+
+fitMeasures(formativeModel3Fit, "df")
+
+semPaths(
+  formativeModel3Fit,
+  what = "Std.all",
+  layout = "tree2",
+  edge.label.cex = 0.8)
+```
+
 ## Additional Types of SEM {#additionalSEMmodels}
 
 Up to this point, we have discussed SEM with dimensional constructs.\index{dimensional}