5 Single-Level CFA Models for Interchangeable Raters

#### Loading of data set

load("chapter_5.RData")                                       

MPlus data file

R data file

Models of Section 5.1

I-SAT model

R Code

isat <- '
Flon11 ~ .con1*1
Flon12 ~ .con1*1
Flon21 ~ .con2*1
Flon22 ~ .con2*1
Flon31 ~ .con3*1
Flon32 ~ .con3*1
Fflou11 ~ .con4*1
Fflou12 ~ .con4*1
Fflou21 ~ .con5*1
Fflou22 ~ .con5*1

# Set observed variable variances equal across Friend 1 and Friend 2
Flon11 ~~ .con6*Flon11
Flon12 ~~ .con6*Flon12
Flon21 ~~ .con7*Flon21
Flon22 ~~ .con7*Flon22
Flon31 ~~ .con8*Flon31
Flon32 ~~ .con8*Flon32
Fflou11 ~~ .con9*Fflou11
Fflou12 ~~ .con9*Fflou12
Fflou21 ~~ .con10*Fflou21
Fflou22 ~~ .con10*Fflou22

# Set observed variable covariances equal across Friend 1 and Friend 2
# Covariances within each rater
Flon11 ~~ .con11*Flon21
Flon12 ~~ .con11*Flon22
Flon11 ~~ .con12*Flon31
Flon12 ~~ .con12*Flon32
Flon11 ~~ .con13*Fflou11
Flon12 ~~ .con13*Fflou12
Flon11 ~~ .con14*Fflou21
Flon12 ~~ .con14*Fflou22
Flon21 ~~ .con15*Flon31
Flon22 ~~ .con15*Flon32
Flon21 ~~ .con16*Fflou11
Flon22 ~~ .con16*Fflou12
Flon21 ~~ .con17*Fflou21
Flon22 ~~ .con17*Fflou22
Flon31 ~~ .con18*Fflou11
Flon32 ~~ .con18*Fflou12
Flon31 ~~ .con19*Fflou21
Flon32 ~~ .con19*Fflou22
Fflou11 ~~ .con20*Fflou21
Fflou12 ~~ .con20*Fflou22

# Covariances across raters
Flon11 ~~ .con21*Flon22
Flon21 ~~ .con21*Flon12
Flon11 ~~ .con22*Flon32
Flon31 ~~ .con22*Flon12
Flon11 ~~ .con23*Fflou12
Fflou11 ~~ .con23*Flon12
Flon11 ~~ .con24*Fflou22
Fflou21 ~~ .con24*Flon12
Flon21 ~~ .con25*Flon32
Flon31 ~~ .con25*Flon22
Flon21 ~~ .con26*Fflou12
Fflou11 ~~ .con26*Flon22
Flon21 ~~ .con27*Fflou22
Fflou21 ~~ .con27*Flon22
Flon31 ~~ .con28*Fflou12
Fflou11 ~~ .con28*Flon32
Flon31 ~~ .con29*Fflou22
Fflou21 ~~ .con29*Flon32
Fflou11 ~~ .con30*Fflou22
Fflou21 ~~ .con30*Fflou12

# Freely estimated monotrait-heteromethod covariances (convergent validities)
Flon11 ~~ Flon12
Flon21 ~~ Flon22
Flon31 ~~ Flon32
Fflou11 ~~ Fflou12
Fflou21 ~~ Fflou22 
'

fit <- sem(model=isat, data=chapter_5, estimator="mlr", missing="fiml")

summary(fit, fit.measures=T, standardized=T)

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I-NULL model

inull <- '
# Set observed variable means equal across Friend 1 and Friend 2
Flon11 ~ .con1*1
Flon12 ~ .con1*1
Flon21 ~ .con2*1
Flon22 ~ .con2*1
Flon31 ~ .con3*1
Flon32 ~ .con3*1
Fflou11 ~ .con4*1
Fflou12 ~ .con4*1
Fflou21 ~ .con5*1
Fflou22 ~ .con5*1

# Set observed variable variances equal across Friend 1 and Friend 2
Flon11 ~~ .con6*Flon11
Flon12 ~~ .con6*Flon12
Flon21 ~~ .con7*Flon21
Flon22 ~~ .con7*Flon22
Flon31 ~~ .con8*Flon31
Flon32 ~~ .con8*Flon32
Fflou11 ~~ .con9*Fflou11
Fflou12 ~~ .con9*Fflou12
Fflou21 ~~ .con10*Fflou21
Fflou22 ~~ .con10*Fflou22

# Set all observed variable covariances to zero
# Covariances within each rater
Flon11 ~~ 0*Flon21
Flon12 ~~ 0*Flon22
Flon11 ~~ 0*Flon31
Flon12 ~~ 0*Flon32
Flon11 ~~ 0*Fflou11
Flon12 ~~ 0*Fflou12
Flon11 ~~ 0*Fflou21
Flon12 ~~ 0*Fflou22
Flon21 ~~ 0*Flon31
Flon22 ~~ 0*Flon32
Flon21 ~~ 0*Fflou11
Flon22 ~~ 0*Fflou12
Flon21 ~~ 0*Fflou21
Flon22 ~~ 0*Fflou22
Flon31 ~~ 0*Fflou11
Flon32 ~~ 0*Fflou12
Flon31 ~~ 0*Fflou21
Flon32 ~~ 0*Fflou22
Fflou11 ~~ 0*Fflou21
Fflou12 ~~ 0*Fflou22

# Covariances across raters
Flon11 ~~ 0*Flon22
Flon21 ~~ 0*Flon12
Flon11 ~~ 0*Flon32
Flon31 ~~ 0*Flon12
Flon11 ~~ 0*Fflou12
Fflou11 ~~ 0*Flon12
Flon11 ~~ 0*Fflou22
Fflou21 ~~ 0*Flon12
Flon21 ~~ 0*Flon32
Flon31 ~~ 0*Flon22
Flon21 ~~ 0*Fflou12
Fflou11 ~~ 0*Flon22
Flon21 ~~ 0*Fflou22
Fflou21 ~~ 0*Flon22
Flon31 ~~ 0*Fflou12
Fflou11 ~~ 0*Flon32
Flon31 ~~ 0*Fflou22
Fflou21 ~~ 0*Flon32
Fflou11 ~~ 0*Fflou22
Fflou21 ~~ 0*Fflou12

# Monotrait-heteromethod covariances (convergent validities)
Flon11 ~~ 0*Flon12
Flon21 ~~ 0*Flon22
Flon31 ~~ 0*Flon32
Fflou11 ~~ 0*Fflou12
Fflou21 ~~ 0*Fflou22
'

fit <- sem(model=inull, data=chapter_5, estimator="mlr", missing="fiml")

summary(fit, fit.measures=T, standardized=T)

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Models of Section 5.3

Application of the models see 5.3.2

Correlated first-order factor model with strict invariance

R Code

cfofm <- '
# Correlated trait-rater factors TR_jk
# Loneliness, Friend 1 (TR_11)
# Trait factor loadings set equal across raters
flone1 =~ 1*Flon11 + .con1*Flon21 + .con2*Flon31

# Loneliness, Friend 2 (TR_12)
flone2 =~ 1*Flon12 + .con1*Flon22 + .con2*Flon32

# Flourishing, Friend 1 (TR_21)
flou1 =~ 1*Fflou11 + .con3*Fflou21

# Flourishing, Friend 2 (TR_22)
flou2 =~ 1*Fflou12 + .con3*Fflou22

# Intercepts set equal for the same variable across raters
# Implies equal latent means across raters
# because latent means are all fixed to zero by default in Mplus
Flon11 ~ .con4*1
Flon12 ~ .con4*1
Flon21 ~ .con5*1
Flon22 ~ .con5*1
Flon31 ~ .con6*1
Flon32 ~ .con6*1
Fflou11 ~ .con7*1
Fflou12 ~ .con7*1
Fflou21 ~ .con8*1
Fflou22 ~ .con8*1

# Measurement error variances set equal for the same variable across raters
Flon11 ~~ .con9*Flon11
Flon12 ~~ .con9*Flon12
Flon21 ~~ .con10*Flon21
Flon22 ~~ .con10*Flon22
Flon31 ~~ .con11*Flon31
Flon32 ~~ .con11*Flon32
Fflou11 ~~ .con12*Fflou11
Fflou12 ~~ .con12*Fflou12
Fflou21 ~~ .con13*Fflou21
Fflou22 ~~ .con13*Fflou22

# Set rater factor variances equal across raters within the same trait
flone1 ~~ .con14*flone1
flone2 ~~ .con14*flone2
flou1 ~~ .con15*flou1
flou2 ~~ .con15*flou2

# Constrain rater covariances between diFferent traits to be equal
flone1 ~~ .con16*flou1
flone2 ~~ .con16*flou2
flone1 ~~ .con17*flou2
flone2 ~~ .con17*flou1
'

fit <- sem(model=cfofm, data=chapter_5, estimator="mlr", missing="fiml")

summary(fit, fit.measures=T, standardized=T)

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Correlated first-order factor model with indicator-specific residual factors

R Code

cfofmis <- '

# Correlated trait-rater factors TR_jk
# Loneliness, Friend 1 (TR_11)
# Trait factor loadings set equal across raters
flone1 =~ 1*Flon11 + .con1*Flon21 + .con2*Flon31

# Loneliness, Friend 2 (TR_12)
flone2 =~ 1*Flon12 + .con1*Flon22 + .con2*Flon32

# Flourishing, Friend 1 (TR_21)
flou1 =~ 1*Fflou11 + .con3*Fflou21

# Flourishing, Friend 2 (TR_22)
flou2 =~ 1*Fflou12 + .con3*Fflou22

# Indicator-specific factors for indicators Y_ijk, i > 1
# All loadings set equal across raters (fixed to 1)
# Loneliness
I21 =~ 1*Flon21 + 1*Flon22
I31 =~ 1*Flon31 + 1*Flon32

# Flourishing
I22 =~ 1*Fflou21 + 1*Fflou22

# Set correlations between trait-rater and indicator-specific factors to zero
flone1 ~~ 0*I21 + 0*I31 + 0*I22
flone2 ~~ 0*I21 + 0*I31 + 0*I22
flou1 ~~ 0*I21 + 0*I31 + 0*I22
flou2 ~~ 0*I21 + 0*I31 + 0*I22

# Intercepts set equal for the same variable across raters
# Implies equal latent means across raters
# because latent means are all fixed to zero by default in Mplus
Flon11 ~ .con4*1
Flon12 ~ .con4*1
Flon21 ~ .con5*1
Flon22 ~ .con5*1
Flon31 ~ .con6*1
Flon32 ~ .con6*1
Fflou11 ~ .con7*1
Fflou12 ~ .con7*1
Fflou21 ~ .con8*1
Fflou22 ~ .con8*1

# Measurement error variances set equal for the same variable across raters
Flon11 ~~ .con9*Flon11
Flon12 ~~ .con9*Flon12
Flon21 ~~ .con10*Flon21
Flon22 ~~ .con10*Flon22
Flon31 ~~ .con11*Flon31
Flon32 ~~ .con11*Flon32
Fflou11 ~~ .con12*Fflou11
Fflou12 ~~ .con12*Fflou12
Fflou21 ~~ .con13*Fflou21
Fflou22 ~~ .con13*Fflou22

# Set rater factor variances equal across raters within the same trait
flone1 ~~ .con14*flone1
flone2 ~~ .con14*flone2
flou1 ~~ .con15*flou1
flou2 ~~ .con15*flou2

# Constrain rater covariances between diFferent traits to be equal
flone1 ~~ .con16*flou1
flone2 ~~ .con16*flou2
flone1 ~~ .con17*flou2
flone2 ~~ .con17*flou1
'

fit <- sem(model=cfofmis, data=chapter_5, estimator="mlr", missing="fiml")

summary(fit, fit.measures=T, standardized=T)

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Models of Section 5.4.1

Basic CTIM model with general trait factors

R Code

bCTIMg <- '
# General trait factors T_j
# Loneliness (T_1)
# Trait factor loadings set equal across raters
T1_flone =~ 1*Flon11 + l_T21*Flon21 + l_T31*Flon31 + 1*Flon12 + l_T21*Flon22 +
     l_T31*Flon32
     
# Flourishing (T_2)
T2_flou =~ 1*Fflou11 + l_T22*Fflou21 + 1*Fflou12 + l_T22*Fflou22

# Trait-specific rater (method) factors R_jk
# Rater factor loadings set equal across raters
# Loneliness, Friend 1 (R_11)
R1_flone =~ 1*Flon11 + l_R21*Flon21 + l_R31*Flon31

# Loneliness, Friend 2 (R_12)
R2_flone =~ 1*Flon12 + l_R21*Flon22 + l_R31*Flon32

# Flourishing, Friend 1 (R_21)
R1_flou =~ 1*Fflou11 + l_R22*Fflou21

# Flourishing, Friend 2 (R_22)
R2_flou =~ 1*Fflou12 + l_R22*Fflou22

# Intercepts set equal for the same variable across raters
Flon11 ~ a_11*1
Flon12 ~ a_11*1
Flon21 ~ a_21*1
Flon22 ~ a_21*1
Flon31 ~ a_31*1
Flon32 ~ a_31*1
Fflou11 ~ a_12*1
Fflou12 ~ a_12*1
Fflou21 ~ a_22*1
Fflou22 ~ a_22*1

# Measurement error variables set equal for the same variable across raters
Flon11 ~~ E_11*Flon11
Flon12 ~~ E_11*Flon12
Flon21 ~~ E_21*Flon21
Flon22 ~~ E_21*Flon22
Flon31 ~~ E_31*Flon31
Flon32 ~~ E_31*Flon32
Fflou11 ~~ E_12*Fflou11
Fflou12 ~~ E_12*Fflou12
Fflou21 ~~ E_22*Fflou21
Fflou22 ~~ E_22*Fflou22

# Set rater factor variances equal across raters within the same trait
R1_flone ~~ VarR_1*R1_flone
R2_flone ~~ VarR_1*R2_flone
R1_flou ~~ VarR_2*R1_flou
R2_flou ~~ VarR_2*R2_flou

# Constrain rater covariances between different traits to be equal
R1_flone ~~ CovR12*R1_flou
R2_flone ~~ CovR12*R2_flou

# Fix correlations between trait and rater factors to zero
T1_flone ~~ 0*R1_flone + 0*R2_flone + 0*R1_flou + 0*R2_flou
T2_flou ~~ 0*R1_flone + 0*R2_flone + 0*R1_flou + 0*R2_flou

# Fix non-admissible correlations between rater factors to zero
R1_flone ~~ 0*R2_flone + 0*R2_flou
R2_flone ~~ 0*R1_flou
R1_flou ~~ 0*R2_flou
'

fit <- sem(model=bCTIMg, data=chapter_5, estimator="mlr", missing="fiml")

summary(fit, fit.measures=T, standardized=T)

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Models of Section 5.4.3

Application of the model see 5.4.4

Restricted CTIM model

R Code

rCTIM <- '
# General trait factors T_j
# Loneliness (T_1)
# Trait factor loadings set equal across raters
T1_flone =~ 1*Flon11 + .con1*Flon21 + .con2*Flon31 + 1*Flon12 + .con1*Flon22 +
  .con2*Flon32
  
# Flourishing (T_2)
T2_flou =~ 1*Fflou11 + .con3*Fflou21 + 1*Fflou12 + .con3*Fflou22

# Trait-specific rater (method) factors R_jk
# Rater factor loadings set equal across raters
# and to corresponding trait factor loadings
# Loneliness, Friend 1 (R_11)
R1_flone =~ 1*Flon11 + .con1*Flon21 + .con2*Flon31

# Loneliness, Friend 2 (R_12)
R2_flone =~ 1*Flon12 + .con1*Flon22 + .con2*Flon32

# Flourishing, Friend 1 (R_21)
R1_flou =~ 1*Fflou11 + .con3*Fflou21

# Flourishing, Friend 2 (R_22)
R2_flou =~ 1*Fflou12 + .con3*Fflou22

# Intercepts set equal for the same variable across raters
# Implies equal latent means across raters
# because latent means are all fixed to zero by default in Mplus
Flon11 ~ .con4*1
Flon12 ~ .con4*1
Flon21 ~ .con5*1
Flon22 ~ .con5*1
Flon31 ~ .con6*1
Flon32 ~ .con6*1
Fflou11 ~ .con7*1
Fflou12 ~ .con7*1
Fflou21 ~ .con8*1
Fflou22 ~ .con8*1

# Measurement error variances set equal for the same variable across raters
Flon11 ~~ .con9*Flon11
Flon12 ~~ .con9*Flon12
Flon21 ~~ .con10*Flon21
Flon22 ~~ .con10*Flon22
Flon31 ~~ .con11*Flon31
Flon32 ~~ .con11*Flon32
Fflou11 ~~ .con12*Fflou11
Fflou12 ~~ .con12*Fflou12
Fflou21 ~~ .con13*Fflou21
Fflou22 ~~ .con13*Fflou22

# Set rater factor variances equal across raters within the same trait
R1_flone ~~ .con14*R1_flone
R2_flone ~~ .con14*R2_flone
R1_flou ~~ .con15*R1_flou
R2_flou ~~ .con15*R2_flou

# Constrain rater covariances between diFferent traits to be equal
R1_flone ~~ .con16*R1_flou
R2_flone ~~ .con16*R2_flou

# Fix correlations between traits and rater factors to zero
T1_flone ~~ 0*R1_flone + 0*R2_flone + 0*R1_flou + 0*R2_flou
T2_flou ~~ 0*R1_flone + 0*R2_flone + 0*R1_flou + 0*R2_flou

# Fix non-admissible correlations between rater factors to zero
R1_flone ~~ 0*R2_flone + 0*R2_flou
R2_flone ~~ 0*R1_flou
R1_flou ~~ 0*R2_flou
'

fit <- sem(model=rCTIM, data=chapter_5, estimator="mlr", missing="fiml")

summary(fit, fit.measures=T, standardized=T)

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Models of Section 5.4.5

Basic CTIM model with general trait factors and indicator-specific residual factors

R Code

bCTIMIS <- '
# General trait factors T_j
# Loneliness (T_1)
# Trait factor loadings set equal across raters
T1_flone =~ 1*Flon11 + .con1*Flon21 + .con2*Flon31 + 1*Flon12 + .con1*Flon22 +
  .con2*Flon32
  
# Flourishing (T_2)
T2_flou =~ 1*Fflou11 + .con3*Fflou21 + 1*Fflou12 + .con3*Fflou22

# Indicator-specific residual factors I_ij
# The first indicator (i = 1) is chosen as reference
# All loadings are fixed to 1 to set them equal across raters
# Loneliness, Indicator 2 (I_21)
I2_flone =~ 1*Flon21 + 1*Flon22

# Loneliness, Indicator 3 (I_31)
I3_flone =~ 1*Flon31 + 1*Flon32

# Flourishing, Indicator 2 (I_22)
I2_flou =~ 1*Fflou21 + 1*Fflou22

# Trait-specific rater (method) factors R_jk
# Rater factor loadings set equal across raters
# Loneliness, Friend 1 (R_11)
R1_flone =~ 1*Flon11 + .con4*Flon21 + .con5*Flon31

# Loneliness, Friend 2 (R_12)
R2_flone =~ 1*Flon12 + .con4*Flon22 + .con5*Flon32

# Flourishing, Friend 1 (R_21)
R1_flou =~ 1*Fflou11 + .con6*Fflou21

# Flourishing, Friend 2 (R_22)
R2_flou =~ 1*Fflou12 + .con6*Fflou22

# Intercepts set equal for the same variable across raters
Flon11 ~ .con7*1
Flon12 ~ .con7*1
Flon21 ~ .con8*1
Flon22 ~ .con8*1
Flon31 ~ .con9*1
Flon32 ~ .con9*1
Fflou11 ~ .con10*1
Fflou12 ~ .con10*1
Fflou21 ~ .con11*1
Fflou22 ~ .con11*1

# Measurement error variables set equal for the same variable across raters
Flon11 ~~ .con12*Flon11
Flon12 ~~ .con12*Flon12
Flon21 ~~ .con13*Flon21
Flon22 ~~ .con13*Flon22
Flon31 ~~ .con14*Flon31
Flon32 ~~ .con14*Flon32
Fflou11 ~~ .con15*Fflou11
Fflou12 ~~ .con15*Fflou12
Fflou21 ~~ .con16*Fflou21
Fflou22 ~~ .con16*Fflou22

# Set rater factor variances equal across raters within the same trait
R1_flone ~~ .con17*R1_flone
R2_flone ~~ .con17*R2_flone
R1_flou ~~ .con18*R1_flou
R2_flou ~~ .con18*R2_flou

# Constrain rater covariances between diFferent traits to be equal
R1_flone ~~ .con19*R1_flou
R2_flone ~~ .con19*R2_flou

# Trait and indicator-specific factors are uncorrelated
T1_flone ~~ 0*I2_flone + 0*I3_flone + 0*I2_flou
T2_flou ~~ 0*I2_flone + 0*I3_flone + 0*I2_flou

# Fix correlations between traits and rater factors to zero
T1_flone ~~ 0*R1_flone + 0*R2_flone + 0*R1_flou + 0*R2_flou
T2_flou ~~ 0*R1_flone + 0*R2_flone + 0*R1_flou + 0*R2_flou

# Indicator-specific factors are uncorrelated with rater factors
I2_flone ~~ 0*R1_flone + 0*R2_flone + 0*R1_flou + 0*R2_flou
I3_flone ~~ 0*R1_flone + 0*R2_flone + 0*R1_flou + 0*R2_flou
I2_flou ~~ 0*R1_flone + 0*R2_flone + 0*R1_flou + 0*R2_flou

# Fix non-admissible correlations between rater factors to zero
R1_flone ~~ 0*R2_flone + 0*R2_flou
R2_flone ~~ 0*R1_flou
R1_flou ~~ 0*R2_flou
'

fit <- sem(model=bCTIMIS, data=chapter_5, estimator="mlr", missing="fiml")

summary(fit, fit.measures=T, standardized=T)

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Restricted CTIM model with indicator-specific residual factors

R Code

rCTIMIS <- '
# General trait factors T_j
# Loneliness (T_1)
# Trait factor loadings set equal across raters
T1_flone =~ 1*Flon11 + .con1*Flon21 + .con2*Flon31 + 1*Flon12 + .con1*Flon22 +
  .con2*Flon32
  
# Flourishing (T_2)
T2_flou =~ 1*Fflou11 + .con3*Fflou21 + 1*Fflou12 + .con3*Fflou22
# Indicator-specific residual factors I_ij
# The first indicator (i = 1) is chosen as reference
# All loadings are fixed to 1 to set them equal across raters
# Loneliness, Indicator 2 (I_21)
I2_flone =~ 1*Flon21 + 1*Flon22

# Loneliness, Indicator 3 (I_31)
I3_flone =~ 1*Flon31 + 1*Flon32

# Flourishing, Indicator 2 (I_22)
I2_flou =~ 1*Fflou21 + 1*Fflou22

# Trait-specific rater (method) factors R_jk
# Rater factor loadings set equal across raters
# and to corresponding trait factor loadings
# Loneliness, Friend 1 (R_11)
R1_flone =~ 1*Flon11 + .con1*Flon21 + .con2*Flon31

# Loneliness, Friend 2 (R_12)
R2_flone =~ 1*Flon12 + .con1*Flon22 + .con2*Flon32

# Flourishing, Friend 1 (R_21)
R1_flou =~ 1*Fflou11 + .con3*Fflou21

# Flourishing, Friend 2 (R_22)
R2_flou =~ 1*Fflou12 + .con3*Fflou22

# Intercepts set equal for the same variable across raters
# Implies equal latent means across raters
# because latent means are all fixed to zero by default in Mplus
Flon11 ~ .con4*1
Flon12 ~ .con4*1
Flon21 ~ .con5*1
Flon22 ~ .con5*1
Flon31 ~ .con6*1
Flon32 ~ .con6*1
Fflou11 ~ .con7*1
Fflou12 ~ .con7*1
Fflou21 ~ .con8*1
Fflou22 ~ .con8*1

# Measurement error variances set equal for the same variable across raters
Flon11 ~~ .con9*Flon11
Flon12 ~~ .con9*Flon12
Flon21 ~~ .con10*Flon21
Flon22 ~~ .con10*Flon22
Flon31 ~~ .con11*Flon31
Flon32 ~~ .con11*Flon32
Fflou11 ~~ .con12*Fflou11
Fflou12 ~~ .con12*Fflou12
Fflou21 ~~ .con13*Fflou21
Fflou22 ~~ .con13*Fflou22

# Set rater factor variances equal across raters within the same trait
R1_flone ~~ .con14*R1_flone
R2_flone ~~ .con14*R2_flone
R1_flou ~~ .con15*R1_flou
R2_flou ~~ .con15*R2_flou

# Constrain rater covariances between diFferent traits to be equal
R1_flone ~~ .con16*R1_flou
R2_flone ~~ .con16*R2_flou

# Trait and indicator-specific factors are uncorrelated
T1_flone ~~ 0*I2_flone + 0*I3_flone + 0*I2_flou
T2_flou ~~ 0*I2_flone + 0*I3_flone + 0*I2_flou

# Fix correlations between traits and rater factors
# as well as indicator-specific factors to zero
T1_flone ~~ 0*R1_flone + 0*R2_flone + 0*R1_flou + 0*R2_flou
T2_flou ~~ 0*R1_flone + 0*R2_flone + 0*R1_flou + 0*R2_flou

# Indicator-specific factors are uncorrelated with rater factors
I2_flone ~~ 0*R1_flone + 0*R2_flone + 0*R1_flou + 0*R2_flou
I3_flone ~~ 0*R1_flone + 0*R2_flone + 0*R1_flou + 0*R2_flou
I2_flou ~~ 0*R1_flone + 0*R2_flone + 0*R1_flou + 0*R2_flou

# Fix non-admissible correlations between rater factors to zero
R1_flone ~~ 0*R2_flone + 0*R2_flou
R2_flone ~~ 0*R1_flou
R1_flou ~~ 0*R2_flou
'

fit <- sem(model=rCTIMIS, data=chapter_5, estimator="mlr", missing="fiml")

summary(fit, fit.measures=T, standardized=T)

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CTIM model with indicator-specific trait factors

R Code

CTIMIT <- '
# Indicator-specific trait factors T_ij
# Trait factor loadings set equal across raters
# (by fixing them to 1)
# Loneliness, Parcel 1 (T_11)
T11_flone =~ 1*Flon11 + 1*Flon12

# Loneliness, Parcel 2 (T_21)
T21_flone =~ 1*Flon21 + 1*Flon22

# Loneliness, Parcel 3 (T_31)
T31_flone =~ 1*Flon31 + 1*Flon32

# Flourishing, Parcel 1 (T_12)
T12_flou =~ 1*Fflou11 + 1*Fflou12

# Flourishing, Parcel 2 (T_22)
T22_flou =~ 1*Fflou21 + 1*Fflou22

# Trait-specific rater (method) factors R_jk
# Rater factor loadings set equal across raters
# Loneliness, Friend 1 (R_11)
R1_flone =~ 1*Flon11 + .con1*Flon21 + .con2*Flon31

# Loneliness, Friend 2 (R_12)
R2_flone =~ 1*Flon12 + .con1*Flon22 + .con2*Flon32

# Flourishing, Friend 1 (R_21)
R1_flou =~ 1*Fflou11 + .con3*Fflou21

# Flourishing, Friend 2 (R_22)
R2_flou =~ 1*Fflou12 + .con3*Fflou22

# Intercepts set equal for the same variable across raters
Flon11 ~ .con4*1
Flon12 ~ .con4*1
Flon21 ~ .con5*1
Flon22 ~ .con5*1
Flon31 ~ .con6*1
Flon32 ~ .con6*1
Fflou11 ~ .con7*1
Fflou12 ~ .con7*1
Fflou21 ~ .con8*1
Fflou22 ~ .con8*1

# Measurement error variables set equal for the same variable across raters
Flon11 ~~ .con9*Flon11
Flon12 ~~ .con9*Flon12
Flon21 ~~ .con10*Flon21
Flon22 ~~ .con10*Flon22
Flon31 ~~ .con11*Flon31
Flon32 ~~ .con11*Flon32
Fflou11 ~~ .con12*Fflou11
Fflou12 ~~ .con12*Fflou12
Fflou21 ~~ .con13*Fflou21
Fflou22 ~~ .con13*Fflou22

# Set rater factor variances equal across raters within the same trait
R1_flone ~~ .con14*R1_flone
R2_flone ~~ .con14*R2_flone
R1_flou ~~ .con15*R1_flou
R2_flou ~~ .con15*R2_flou

# Constrain rater covariances between diFferent traits to be equal
R1_flone ~~ .con16*R1_flou
R2_flone ~~ .con16*R2_flou

# Trait and rater factors are uncorrelated for the same trait
T11_flone ~~ 0*R1_flone + 0*R2_flone
T21_flone ~~ 0*R1_flone + 0*R2_flone
T31_flone ~~ 0*R1_flone + 0*R2_flone
T12_flou ~~ 0*R1_flou + 0*R2_flou
T22_flou ~~ 0*R1_flou + 0*R2_flou

# Rater factors are uncorrelated with other rater factors
# pertaining to the same trait
R1_flone ~~ 0*R2_flone
R1_flou ~~ 0*R2_flou

# Constrain non-substantial factor correlations to zero
T11_flone ~~ 0*R1_flou + 0*R2_flou
T21_flone ~~ 0*R1_flou + 0*R2_flou
T31_flone ~~ 0*R1_flou + 0*R2_flou
T12_flou ~~ 0*R1_flone + 0*R2_flone
T22_flou ~~ 0*R1_flone + 0*R2_flone
R1_flone ~~ 0*R2_flou
R2_flone ~~ 0*R1_flou
'

fit <- sem(model=CTIMIT, data=chapter_5, estimator="mlr", missing="fiml")

summary(fit, fit.measures=T, standardized=T)

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