Degrees of freedom

[M-E-Rademaker]

There are still some open questions concerning the degrees of freedom calculation

• Check degrees of freedom calculation for results for common factor models with lavaan
• How should nonlinear terms be handled? Does including nonlinear terms affect the degrees of freedom calculation (other than that the number of dfs is diminished by one, since one more structural parameter has to be estimated).?

[M-E-Rademaker]

Comparison to lavaan is ok:

require(cSEM)
require(lavaan)
require(cSEM.DGP)

model_linear <- "
# Structural model
eta2 ~ eta1
eta3 ~ eta1 + eta2

# (Reflective) measurement model
eta1 =~ y11 + y12 + y13
eta2 =~ y21 + y22 + y23
eta3 =~ y31 + y32 + y33
"

dgp_linear_2 <- '
# Measurement model
A =~ 0.8*x11 + 0.8*x12 + 0.7*x13
B =~ 0.7*x21 + 0.8*x22
C =~ 0.7*x31 + 0.9*x32 + 0.7*x33
D =~ 0.9*x41 + 0.8*x42 + 0.8*x43

# Structural model
C ~ 0.1*A + 0.2*B
D ~ 0.3*A + 0.3*C

x11 ~~ 0.3*x12
'

model_linear_2 <- '
# Measurement model
A =~ x11 + x12 + x13
B =~ x21 + x22
C =~ x31 + x32 + x33
D =~ x41 + x42 + x43

# Structural model
C ~ A + B
D ~ A + C

x11 ~~ x12
'

myData <- cSEM.DGP::generateData(.model = dgp_linear_2, 
                                 .return_type = "data.frame",
                                 .empirical = FALSE,
                                 .N = 1000)

model_2nd <- "
# Path model / Regressions 
c4   ~ eta1
eta2 ~ eta1 + c4

# Reflective measurement model
c1   =~ y11 + y12 
c2   =~ y21 + y22 + y23 + y24
c3   =~ y31 + y32 + y33 + y34 + y35 + y36 + y37 + y38
eta1 =~ y41 + y42 + y43
eta2 =~ y51 + y52 + y53

# Composite model (second order)
c4   =~ c1 + c2 + c3
"

sem(model_linear, threecommonfactors) # 24 + 3 construct variances = 27
sem(model_linear_2, myData, auto.fix.first = FALSE, std.lv = TRUE) # 38 + 4 construct variances = 42
sem(model_2nd, dgp_2ndorder_cf_of_c) # 164 + 6 construct variances = 170

calculateDf(csem(threecommonfactors, model_linear))
calculateDf(csem(myData, model_linear_2))
calculateDf(csem(dgp_2ndorder_cf_of_c, model_2nd))