Interaction between latent variables using product indicators
modsem_pi.Rdmodsem_pi() is a function for estimating interaction effects between latent variables,
in structural equation models (SEMs), using product indicators.
Methods for estimating interaction effects in SEMs can basically be split into
two frameworks:
1. Product Indicator based approaches ("dblcent", "rca", "uca",
"ca", "pind"), and
2. Distributionally based approaches ("lms", "qml").
modsem_pi() is essentially a fancy wrapper for lavaan::sem() which generates the
necessary syntax and variables for the estimation of models with latent product indicators.
Use default_settings_pi() to get the default settings for the different methods.
Usage
modsem_pi(
model.syntax = NULL,
data = NULL,
method = "dblcent",
match = NULL,
match.recycle = NULL,
standardize.data = FALSE,
center.data = FALSE,
first.loading.fixed = FALSE,
center.before = NULL,
center.after = NULL,
residuals.prods = NULL,
residual.cov.syntax = NULL,
constrained.prod.mean = NULL,
constrained.loadings = NULL,
constrained.var = NULL,
res.cov.method = NULL,
res.cov.across = NULL,
auto.scale = "none",
auto.center = "none",
estimator = "ML",
group = NULL,
cluster = NULL,
run = TRUE,
na.rm = FALSE,
suppress.warnings.lavaan = FALSE,
suppress.warnings.match = FALSE,
rcs = FALSE,
rcs.choose = NULL,
rcs.res.cov.xz = rcs,
rcs.mc.reps = 1e+05,
rcs.scale.corrected = TRUE,
LAVFUN = lavaan::sem,
...
)Arguments
- model.syntax
lavaansyntax- data
dataframe
- method
method to use:
"dblcent"double centering approach (passed to
lavaan)."ca"constrained approach (passed to
lavaan)."rca"residual centering approach (passed to
lavaan)."uca"unconstrained approach (passed to
lavaan)."pind"prod ind approach, with no constraints or centering (passed to
lavaan).
- match
should the product indicators be created by using the match-strategy
- match.recycle
should the indicators be recycled when using the match-strategy? I.e., if one of the latent variables have fewer indicators than the other, some indicators are recycled to match the latent variable with the most indicators.
- standardize.data
should data be scaled before fitting model
- center.data
should data be centered before fitting model
- first.loading.fixed
Should the first factor loading in the latent product be fixed to one? Defaults to
FALSE, as this already happens inlavaanby default. IfTRUE, the first factor loading in the latent product is fixed to one. Manually in the generated syntax (e.g.,XZ =~ 1*x1z1).'- center.before
should indicators in products be centered before computing products.
- center.after
should indicator products be centered after they have been computed?
- residuals.prods
should indicator products be centered using residuals.
- residual.cov.syntax
should syntax for residual covariances be produced.
- constrained.prod.mean
should syntax for product mean be produced.
- constrained.loadings
should syntax for constrained loadings be produced.
- constrained.var
should syntax for constrained variances be produced.
- res.cov.method
method for constraining residual covariances. Options are
- "simple"
Residuals of product indicators with variables in common are allowed to covary freely. Defualt for most approches.
- "ca"
Residual covariances of product indicators are constrained according to the constrained approach.
- "equality"
Residuals of product indicators with variables in common are constrained to have equal covariances". Can be useful for models where the model is unidentifiable using
res.cov.method == "simple", (e.g., when there is an interaction between an observed and a latent variable).- "none"
Residual covariances between product indicators are not specificed (i.e., constrained to zero). Produces the same results as
constrained.cov.syntax = FALSE. Can be useful for models where the model is unidentifiable usingres.cov.method == "simple", (e.g., when there is an interaction between an observed and a latent variable).
- res.cov.across
Should residual covariances be specified/freed across different interaction terms. For example if you have two interaction terms
X:ZandX:Wthe residuals of the generated product indicatorsx1:z1andx1:w1may be correlated. IfTRUEresidual covariances are allowed across different latent interaction terms. IfFALSEresidual covariances are only allowed between product indicators which belong to the same latent interaction term.- auto.scale
methods which should be scaled automatically (usually not useful)
- auto.center
methods which should be centered automatically (usually not useful)
- estimator
estimator to use in
lavaan- group
group variable for multigroup analysis
- cluster
cluster variable for multilevel models
- run
should the model be run via
lavaan, ifFALSEonly modified syntax and data is returned- na.rm
should missing values be removed (case-wise)? Defaults to FALSE. If
TRUE, missing values are removed case-wise. IfFALSEthey are not removed.- suppress.warnings.lavaan
should warnings from
lavaanbe suppressed?- suppress.warnings.match
should warnings from
matchbe suppressed?- rcs
Should latent variable indicators be replaced with reliability-corrected single item indicators instead? See
relcorr_single_item.- rcs.choose
Which latent variables should get their indicators replaced with reliability-corrected single items? It is passed to
relcorr_single_itemas thechooseargument.- rcs.res.cov.xz
Should the residual (co-)variances of the product indicators created from the reliability-corrected single items (created if
rcs = TRUE) be specified and constrained before estimating the model? IfTRUEthe estimates for the constraints are approximated using a monte carlo simulation (see thercs.mc.repsargument). IfFALSEthe residual variances are not specified, which usually mean that all are constrained to zero.- rcs.mc.reps
Sample size used in monte-carlo simulation, when approximating the the estimates of the residual (co-)variances between the product indicators formed by reliabiliyt-corrected single items (see the
rcs.res.cov.xzargument).- rcs.scale.corrected
Should reliability corrected items be scale-corrected? If
TRUEreliability-corrected single items are corrected for differences in factor loadings between the items. Default isTRUE.- LAVFUN
Function used to estimate the model. Defaults to
lavaan::sem.- ...
arguments passed to
LAVFUN
Examples
library(modsem)
# For more examples, check README and/or GitHub.
# One interaction
m1 <- '
# Outer Model
X =~ x1 + x2 +x3
Y =~ y1 + y2 + y3
Z =~ z1 + z2 + z3
# Inner model
Y ~ X + Z + X:Z
'
# Double centering approach
est <- modsem_pi(m1, oneInt)
summary(est)
#> Estimating baseline model (H0)
#> modsem (version 1.0.12, approach = dblcent):
#>
#> Interaction Model Fit Measures (H1):
#> Loglikelihood -26807.61
#> Akaike (AIC) 53735.22
#> Bayesian (BIC) 54071.28
#> Chi-square 122.92
#> Degrees of Freedom 111
#> P-value (Chi-square) 0.207
#> RMSEA 0.007
#> CFI 1.000
#> SRMR 0.008
#>
#> Fit Measures for Baseline Model (H0):
#> Loglikelihood -27137.74
#> Akaike (AIC) 54393.48
#> Bayesian (BIC) 54723.93
#> Chi-square 783.18
#> Degrees of Freedom 112
#> P-value (Chi-square) 0.000
#> RMSEA 0.055
#> CFI 0.987
#> SRMR 0.141
#>
#> Comparative Fit to H0 (LRT test):
#> Chi-square diff 660.257
#> Degrees of freedom diff 1
#> P-value (LRT) 0.000
#>
#> R-Squared Interaction Model (H1):
#> Y 0.602
#> R-Squared Baseline Model (H0):
#> Y 0.397
#> R-Squared Change (H1 - H0):
#> Y 0.204
#>
#> lavaan 0.6-19 ended normally after 161 iterations
#>
#> Estimator ML
#> Optimization method NLMINB
#> Number of model parameters 60
#>
#> Number of observations 2000
#>
#> Model Test User Model:
#>
#> Test statistic 122.924
#> Degrees of freedom 111
#> P-value (Chi-square) 0.207
#>
#> Parameter Estimates:
#>
#> Standard errors Standard
#> Information Expected
#> Information saturated (h1) model Structured
#>
#> Latent Variables:
#> Estimate Std.Err z-value P(>|z|)
#> X =~
#> x1 1.000
#> x2 0.804 0.013 63.612 0.000
#> x3 0.916 0.014 67.144 0.000
#> Y =~
#> y1 1.000
#> y2 0.798 0.007 107.428 0.000
#> y3 0.899 0.008 112.453 0.000
#> Z =~
#> z1 1.000
#> z2 0.812 0.013 64.763 0.000
#> z3 0.882 0.013 67.014 0.000
#> XZ =~
#> x1z1 1.000
#> x2z1 0.805 0.013 60.636 0.000
#> x3z1 0.877 0.014 62.680 0.000
#> x1z2 0.793 0.013 59.343 0.000
#> x2z2 0.646 0.015 43.672 0.000
#> x3z2 0.706 0.016 44.292 0.000
#> x1z3 0.887 0.014 63.700 0.000
#> x2z3 0.716 0.016 45.645 0.000
#> x3z3 0.781 0.017 45.339 0.000
#>
#> Regressions:
#> Estimate Std.Err z-value P(>|z|)
#> Y ~
#> X 0.675 0.027 25.379 0.000
#> Z 0.561 0.026 21.606 0.000
#> XZ 0.702 0.027 26.360 0.000
#>
#> Covariances:
#> Estimate Std.Err z-value P(>|z|)
#> .x1z1 ~~
#> .x1z2 0.115 0.008 14.802 0.000
#> .x1z3 0.114 0.008 13.947 0.000
#> .x2z1 0.125 0.008 16.095 0.000
#> .x3z1 0.140 0.009 16.135 0.000
#> .x1z2 ~~
#> .x1z3 0.103 0.007 14.675 0.000
#> .x2z2 0.128 0.006 20.850 0.000
#> .x3z2 0.146 0.007 21.243 0.000
#> .x1z3 ~~
#> .x2z3 0.116 0.007 17.818 0.000
#> .x3z3 0.135 0.007 18.335 0.000
#> .x2z1 ~~
#> .x2z2 0.135 0.006 20.905 0.000
#> .x2z3 0.145 0.007 21.145 0.000
#> .x3z1 0.114 0.007 16.058 0.000
#> .x2z2 ~~
#> .x2z3 0.117 0.006 20.419 0.000
#> .x3z2 0.116 0.006 20.586 0.000
#> .x2z3 ~~
#> .x3z3 0.109 0.006 18.059 0.000
#> .x3z1 ~~
#> .x3z2 0.138 0.007 19.331 0.000
#> .x3z3 0.158 0.008 20.269 0.000
#> .x3z2 ~~
#> .x3z3 0.131 0.007 19.958 0.000
#> X ~~
#> Z 0.201 0.024 8.271 0.000
#> XZ 0.016 0.025 0.628 0.530
#> Z ~~
#> XZ 0.062 0.025 2.449 0.014
#>
#> Variances:
#> Estimate Std.Err z-value P(>|z|)
#> .x1 0.160 0.009 17.871 0.000
#> .x2 0.162 0.007 22.969 0.000
#> .x3 0.163 0.008 20.161 0.000
#> .y1 0.159 0.009 17.896 0.000
#> .y2 0.154 0.007 22.640 0.000
#> .y3 0.164 0.008 20.698 0.000
#> .z1 0.168 0.009 18.143 0.000
#> .z2 0.158 0.007 22.264 0.000
#> .z3 0.158 0.008 20.389 0.000
#> .x1z1 0.311 0.014 22.227 0.000
#> .x2z1 0.292 0.011 27.287 0.000
#> .x3z1 0.327 0.012 26.275 0.000
#> .x1z2 0.290 0.011 26.910 0.000
#> .x2z2 0.239 0.008 29.770 0.000
#> .x3z2 0.270 0.009 29.117 0.000
#> .x1z3 0.272 0.012 23.586 0.000
#> .x2z3 0.245 0.009 27.979 0.000
#> .x3z3 0.297 0.011 28.154 0.000
#> X 0.981 0.036 26.895 0.000
#> .Y 0.990 0.038 25.926 0.000
#> Z 1.016 0.038 26.856 0.000
#> XZ 1.045 0.044 24.004 0.000
#>
# \dontrun{
# The Constrained Approach
est_ca <- modsem_pi(m1, oneInt, method = "ca")
summary(est_ca)
#> Estimating baseline model (H0)
#> modsem (version 1.0.12, approach = ca):
#>
#> Interaction Model Fit Measures (H1):
#> Loglikelihood -24339.32
#> Akaike (AIC) 48746.65
#> Bayesian (BIC) 48937.08
#> Chi-square 60.40
#> Degrees of Freedom 56
#> P-value (Chi-square) 0.320
#> RMSEA 0.006
#> CFI 1.000
#> SRMR 0.020
#>
#> Fit Measures for Baseline Model (H0):
#> Loglikelihood -24669.20
#> Akaike (AIC) 49404.40
#> Bayesian (BIC) 49589.23
#> Chi-square 720.15
#> Degrees of Freedom 57
#> P-value (Chi-square) 0.000
#> RMSEA 0.076
#> CFI 0.972
#> SRMR 0.124
#>
#> Comparative Fit to H0 (LRT test):
#> Chi-square diff 659.749
#> Degrees of freedom diff 1
#> P-value (LRT) 0.000
#>
#> R-Squared Interaction Model (H1):
#> Y 0.594
#> R-Squared Baseline Model (H0):
#> Y 0.393
#> R-Squared Change (H1 - H0):
#> Y 0.201
#>
#> lavaan 0.6-19 ended normally after 286 iterations
#>
#> Estimator ML
#> Optimization method NLMINB
#> Number of model parameters 43
#> Row rank of the constraints matrix 9
#>
#> Number of observations 2000
#>
#> Model Test User Model:
#>
#> Test statistic 60.401
#> Degrees of freedom 56
#> P-value (Chi-square) 0.320
#>
#> Parameter Estimates:
#>
#> Standard errors Standard
#> Information Expected
#> Information saturated (h1) model Structured
#>
#> Latent Variables:
#> Estimate Std.Err z-value P(>|z|)
#> X =~
#> x1 (l_1_X) 1.000
#> x2 (l_2_X) 0.805 0.011 73.101 0.000
#> x3 (l_3_X) 0.912 0.012 76.636 0.000
#> Y =~
#> y1 (l_1_Y) 1.000
#> y2 (l_2_Y) 0.798 0.008 106.156 0.000
#> y3 (l_3_Y) 0.899 0.008 111.106 0.000
#> Z =~
#> z1 (l_1_Z) 1.000
#> z2 (l_2_Z) 0.813 0.011 74.076 0.000
#> z3 (l_3_Z) 0.878 0.011 76.574 0.000
#> XZ =~
#> x1z1 (l_11) 1.000
#> x2z2 (l_22) 0.654 0.010 62.981 0.000
#> x3z3 (l_33) 0.801 0.012 65.421 0.000
#>
#> Regressions:
#> Estimate Std.Err z-value P(>|z|)
#> Y ~
#> X (G_X_) 0.678 0.026 25.602 0.000
#> Z (G_Z_) 0.565 0.026 21.868 0.000
#> XZ (G_XZ) 0.715 0.026 27.094 0.000
#>
#> Covariances:
#> Estimate Std.Err z-value P(>|z|)
#> X ~~
#> Z (C_X_Z) 0.198 0.022 9.042 0.000
#> XZ (C_X_X) 0.000 NA
#> Z ~~
#> XZ (C_Z_) 0.000
#> .x1z1 ~~
#> .x2z2 0.000
#> .x3z3 0.000
#> .x2z2 ~~
#> .x3z3 0.000
#>
#> Intercepts:
#> Estimate Std.Err z-value P(>|z|)
#> XZ (M_XZ) 0.198 0.022 9.042 0.000
#> .x1 1.023 0.024 43.002 0.000
#> .x2 1.215 0.020 60.936 0.000
#> .x3 0.919 0.022 41.603 0.000
#> .y1 1.039 0.040 26.293 0.000
#> .y2 1.222 0.032 38.225 0.000
#> .y3 0.955 0.036 26.709 0.000
#> .z1 1.011 0.024 41.726 0.000
#> .z2 1.206 0.020 59.286 0.000
#> .z3 0.916 0.022 42.229 0.000
#> .x1z1 0.012 0.034 0.352 0.725
#> .x2z2 0.001 0.023 0.031 0.975
#> .x3z3 -0.007 0.028 -0.262 0.793
#>
#> Variances:
#> Estimate Std.Err z-value P(>|z|)
#> X (Vr_X) 0.976 0.029 33.284 0.000
#> .Y (Zt_Y) 0.987 0.038 25.711 0.000
#> Z (Vr_Z) 1.011 0.030 33.294 0.000
#> XZ (V_XZ) 1.026 0.032 32.214 0.000
#> .x1 (Vr_x1) 0.156 0.008 19.174 0.000
#> .x2 (Vr_x2) 0.163 0.006 25.240 0.000
#> .x3 (Vr_x3) 0.165 0.007 22.239 0.000
#> .y1 (Vr_y1) 0.159 0.009 17.881 0.000
#> .y2 (Vr_y2) 0.154 0.007 22.631 0.000
#> .y3 (Vr_y3) 0.164 0.008 20.686 0.000
#> .z1 (Vr_z1) 0.164 0.008 19.449 0.000
#> .z2 (Vr_z2) 0.159 0.007 24.247 0.000
#> .z3 (Vr_z3) 0.160 0.007 22.514 0.000
#> .x1z1 (V_11) 0.343 0.011 30.345 0.000
#> .x2z2 (V_22) 0.236 0.006 37.239 0.000
#> .x3z3 (V_33) 0.285 0.008 34.438 0.000
#>
#> Constraints:
#> |Slack|
#> Var_XZ - ((Var_X)*(Var_Z)+(Cov_X_Z)^2) 0.000
#> Cov_X_XZ - 0 0.000
#> Cov_Z_XZ - 0 0.000
#> V_11-(_1_X^2*(V_X)*V_1+_1_Z^2*(V_Z)*V_1+V 0.000
#> V_22-(_2_X^2*(V_X)*V_2+_2_Z^2*(V_Z)*V_2+V 0.000
#> V_33-(_3_X^2*(V_X)*V_3+_3_Z^2*(V_Z)*V_3+V 0.000
#> lambda_x1z1_XZ-(lambda_x1_X*lambda_z1_Z) 0.000
#> lambda_x2z2_XZ-(lambda_x2_X*lambda_z2_Z) 0.000
#> lambda_x3z3_XZ-(lambda_x3_X*lambda_z3_Z) 0.000
#> Mean_XZ - ((Cov_X_Z)) 0.000
#>
# }
# Theory Of Planned Behavior
tpb <- '
# Outer Model (Based on Hagger et al., 2007)
ATT =~ att1 + att2 + att3 + att4 + att5
SN =~ sn1 + sn2
PBC =~ pbc1 + pbc2 + pbc3
INT =~ int1 + int2 + int3
BEH =~ b1 + b2
# Inner Model (Based on Steinmetz et al., 2011)
# Covariances
ATT ~~ SN + PBC
PBC ~~ SN
# Causal Relationships
INT ~ ATT + SN + PBC
BEH ~ INT + PBC
BEH ~ INT:PBC
'
# Double centering approach
est_tpb <- modsem_pi(tpb, data = TPB)
summary(est_tpb)
#> Estimating baseline model (H0)
#> modsem (version 1.0.12, approach = dblcent):
#>
#> Interaction Model Fit Measures (H1):
#> Loglikelihood -34958.88
#> Akaike (AIC) 70073.77
#> Bayesian (BIC) 70510.64
#> Chi-square 207.61
#> Degrees of Freedom 222
#> P-value (Chi-square) 0.747
#> RMSEA 0.000
#> CFI 1.000
#> SRMR 0.010
#>
#> Fit Measures for Baseline Model (H0):
#> Loglikelihood -35025.66
#> Akaike (AIC) 70205.32
#> Bayesian (BIC) 70636.59
#> Chi-square 341.17
#> Degrees of Freedom 223
#> P-value (Chi-square) 0.000
#> RMSEA 0.016
#> CFI 0.998
#> SRMR 0.048
#>
#> Comparative Fit to H0 (LRT test):
#> Chi-square diff 133.551
#> Degrees of freedom diff 1
#> P-value (LRT) 0.000
#>
#> R-Squared Interaction Model (H1):
#> INT 0.367
#> BEH 0.278
#> R-Squared Baseline Model (H0):
#> INT 0.367
#> BEH 0.211
#> R-Squared Change (H1 - H0):
#> INT -0.000
#> BEH 0.067
#>
#> lavaan 0.6-19 ended normally after 171 iterations
#>
#> Estimator ML
#> Optimization method NLMINB
#> Number of model parameters 78
#>
#> Number of observations 2000
#>
#> Model Test User Model:
#>
#> Test statistic 207.615
#> Degrees of freedom 222
#> P-value (Chi-square) 0.747
#>
#> Parameter Estimates:
#>
#> Standard errors Standard
#> Information Expected
#> Information saturated (h1) model Structured
#>
#> Latent Variables:
#> Estimate Std.Err z-value P(>|z|)
#> ATT =~
#> att1 1.000
#> att2 0.878 0.012 71.509 0.000
#> att3 0.789 0.012 66.368 0.000
#> att4 0.695 0.011 61.017 0.000
#> att5 0.887 0.013 70.884 0.000
#> SN =~
#> sn1 1.000
#> sn2 0.889 0.017 52.553 0.000
#> PBC =~
#> pbc1 1.000
#> pbc2 0.912 0.013 69.500 0.000
#> pbc3 0.801 0.012 65.830 0.000
#> INT =~
#> int1 1.000
#> int2 0.914 0.016 58.982 0.000
#> int3 0.808 0.015 55.547 0.000
#> BEH =~
#> b1 1.000
#> b2 0.960 0.030 31.561 0.000
#> INTPBC =~
#> int1pbc1 1.000
#> int2pbc1 0.931 0.015 63.809 0.000
#> int3pbc1 0.774 0.013 60.107 0.000
#> int1pbc2 0.893 0.013 68.173 0.000
#> int2pbc2 0.826 0.017 48.845 0.000
#> int3pbc2 0.690 0.015 45.300 0.000
#> int1pbc3 0.799 0.012 67.008 0.000
#> int2pbc3 0.738 0.015 47.809 0.000
#> int3pbc3 0.622 0.014 45.465 0.000
#>
#> Regressions:
#> Estimate Std.Err z-value P(>|z|)
#> INT ~
#> ATT 0.213 0.026 8.170 0.000
#> SN 0.177 0.028 6.416 0.000
#> PBC 0.217 0.030 7.340 0.000
#> BEH ~
#> INT 0.191 0.024 7.817 0.000
#> PBC 0.230 0.022 10.507 0.000
#> INTPBC 0.204 0.018 11.425 0.000
#>
#> Covariances:
#> Estimate Std.Err z-value P(>|z|)
#> ATT ~~
#> SN 0.629 0.029 21.977 0.000
#> PBC 0.678 0.029 23.721 0.000
#> SN ~~
#> PBC 0.678 0.029 23.338 0.000
#> .int1pbc1 ~~
#> .int1pbc2 0.126 0.009 14.768 0.000
#> .int1pbc3 0.102 0.007 13.794 0.000
#> .int2pbc1 0.104 0.007 14.608 0.000
#> .int3pbc1 0.091 0.006 14.109 0.000
#> .int1pbc2 ~~
#> .int1pbc3 0.095 0.007 13.852 0.000
#> .int2pbc2 0.128 0.007 19.320 0.000
#> .int3pbc2 0.119 0.006 19.402 0.000
#> .int1pbc3 ~~
#> .int2pbc3 0.110 0.006 19.911 0.000
#> .int3pbc3 0.097 0.005 19.415 0.000
#> .int2pbc1 ~~
#> .int2pbc2 0.152 0.008 18.665 0.000
#> .int2pbc3 0.138 0.007 18.779 0.000
#> .int3pbc1 0.082 0.006 13.951 0.000
#> .int2pbc2 ~~
#> .int2pbc3 0.121 0.007 18.361 0.000
#> .int3pbc2 0.104 0.005 19.047 0.000
#> .int2pbc3 ~~
#> .int3pbc3 0.087 0.005 19.180 0.000
#> .int3pbc1 ~~
#> .int3pbc2 0.139 0.007 21.210 0.000
#> .int3pbc3 0.123 0.006 21.059 0.000
#> .int3pbc2 ~~
#> .int3pbc3 0.114 0.005 21.021 0.000
#> ATT ~~
#> INTPBC 0.086 0.024 3.519 0.000
#> SN ~~
#> INTPBC 0.055 0.025 2.230 0.026
#> PBC ~~
#> INTPBC 0.087 0.024 3.609 0.000
#>
#> Variances:
#> Estimate Std.Err z-value P(>|z|)
#> .att1 0.167 0.007 23.528 0.000
#> .att2 0.150 0.006 24.693 0.000
#> .att3 0.160 0.006 26.378 0.000
#> .att4 0.163 0.006 27.649 0.000
#> .att5 0.159 0.006 24.930 0.000
#> .sn1 0.178 0.015 12.110 0.000
#> .sn2 0.156 0.012 13.221 0.000
#> .pbc1 0.145 0.008 18.440 0.000
#> .pbc2 0.160 0.007 21.547 0.000
#> .pbc3 0.154 0.007 23.716 0.000
#> .int1 0.158 0.009 18.152 0.000
#> .int2 0.160 0.008 20.345 0.000
#> .int3 0.167 0.007 23.414 0.000
#> .b1 0.186 0.018 10.058 0.000
#> .b2 0.135 0.017 8.080 0.000
#> .int1pbc1 0.266 0.013 20.971 0.000
#> .int2pbc1 0.292 0.012 24.421 0.000
#> .int3pbc1 0.251 0.010 26.305 0.000
#> .int1pbc2 0.290 0.012 24.929 0.000
#> .int2pbc2 0.269 0.010 26.701 0.000
#> .int3pbc2 0.253 0.009 29.445 0.000
#> .int1pbc3 0.223 0.009 24.431 0.000
#> .int2pbc3 0.234 0.008 27.633 0.000
#> .int3pbc3 0.203 0.007 29.288 0.000
#> ATT 0.998 0.037 27.138 0.000
#> SN 0.987 0.039 25.394 0.000
#> PBC 0.962 0.035 27.260 0.000
#> .INT 0.490 0.020 24.638 0.000
#> .BEH 0.455 0.023 20.068 0.000
#> INTPBC 1.020 0.041 24.612 0.000
#>
# \dontrun{
# The Constrained Approach
est_tpb_ca <- modsem_pi(tpb, data = TPB, method = "ca")
summary(est_tpb_ca)
#> Estimating baseline model (H0)
#> modsem (version 1.0.12, approach = ca):
#>
#> Interaction Model Fit Measures (H1):
#> Loglikelihood -32849.60
#> Akaike (AIC) 65817.20
#> Bayesian (BIC) 66147.66
#> Chi-square 124.54
#> Degrees of Freedom 130
#> P-value (Chi-square) 0.619
#> RMSEA 0.000
#> CFI 1.000
#> SRMR 0.024
#>
#> Fit Measures for Baseline Model (H0):
#> Loglikelihood -32918.35
#> Akaike (AIC) 65952.70
#> Bayesian (BIC) 66277.56
#> Chi-square 262.04
#> Degrees of Freedom 131
#> P-value (Chi-square) 0.000
#> RMSEA 0.022
#> CFI 0.996
#> SRMR 0.046
#>
#> Comparative Fit to H0 (LRT test):
#> Chi-square diff 137.501
#> Degrees of freedom diff 1
#> P-value (LRT) 0.000
#>
#> R-Squared Interaction Model (H1):
#> INT 0.369
#> BEH 0.268
#> R-Squared Baseline Model (H0):
#> INT 0.369
#> BEH 0.211
#> R-Squared Change (H1 - H0):
#> INT -0.000
#> BEH 0.057
#>
#> lavaan 0.6-19 ended normally after 266 iterations
#>
#> Estimator ML
#> Optimization method NLMINB
#> Number of model parameters 67
#> Row rank of the constraints matrix 8
#>
#> Number of observations 2000
#>
#> Model Test User Model:
#>
#> Test statistic 124.540
#> Degrees of freedom 130
#> P-value (Chi-square) 0.619
#>
#> Parameter Estimates:
#>
#> Standard errors Standard
#> Information Expected
#> Information saturated (h1) model Structured
#>
#> Latent Variables:
#> Estimate Std.Err z-value P(>|z|)
#> ATT =~
#> att1 (l_1_A) 1.000
#> att2 (l_2_A) 0.878 0.012 71.494 0.000
#> att3 (l_3_A) 0.789 0.012 66.355 0.000
#> att4 (l_4_) 0.695 0.011 61.005 0.000
#> att5 (l_5_) 0.887 0.013 70.871 0.000
#> SN =~
#> sn1 (l_1_S) 1.000
#> sn2 (l_2_S) 0.889 0.017 52.693 0.000
#> PBC =~
#> pbc1 (l_1_P) 1.000
#> pbc2 (l_2_P) 0.913 0.011 80.296 0.000
#> pbc3 (l_3_P) 0.795 0.011 75.450 0.000
#> INT =~
#> int1 (l_1_I) 1.000
#> int2 (l_2_I) 0.913 0.013 71.295 0.000
#> int3 (l_3_I) 0.797 0.012 66.791 0.000
#> BEH =~
#> b1 (l_1_B) 1.000
#> b2 (l_2_B) 0.962 0.031 30.667 0.000
#> INTPBC =~
#> int1p1 (l_11) 1.000
#> int2p2 (l_22) 0.833 0.012 66.643 0.000
#> int3p3 (l_33) 0.634 0.010 62.661 0.000
#>
#> Regressions:
#> Estimate Std.Err z-value P(>|z|)
#> INT ~
#> ATT (G_AT) 0.214 0.026 8.193 0.000
#> SN (G_SN) 0.177 0.028 6.427 0.000
#> PBC (G_PBC_I) 0.220 0.029 7.615 0.000
#> BEH ~
#> INT (G_INT_) 0.189 0.024 7.810 0.000
#> PBC (G_PBC_B) 0.231 0.022 10.631 0.000
#> INTP (G_INTP) 0.209 0.018 11.587 0.000
#>
#> Covariances:
#> Estimate Std.Err z-value P(>|z|)
#> ATT ~~
#> SN (C_ATT_S) 0.630 0.026 24.699 0.000
#> PBC (C_ATT_P) 0.680 0.023 29.948 0.000
#> SN ~~
#> PBC (C_PBC_S) 0.683 0.023 29.055 0.000
#> ATT ~~
#> INTP (C_ATT_I) 0.028 0.018 1.551 0.121
#> SN ~~
#> INTP (C_SN) -0.004 0.019 -0.199 0.843
#> PBC ~~
#> INTP (C_PBC_I) 0.000
#> .int1pbc1 ~~
#> .in22 0.000
#> .in33 0.000
#> .int2pbc2 ~~
#> .in33 0.000
#>
#> Intercepts:
#> Estimate Std.Err z-value P(>|z|)
#> INTPBC (M_IN) 0.480 0.014 33.491 0.000
#> .att1 1.014 0.024 41.999 0.000
#> .att2 1.007 0.021 46.960 0.000
#> .att3 1.016 0.020 51.448 0.000
#> .att4 0.999 0.018 55.647 0.000
#> .att5 0.992 0.022 45.664 0.000
#> .sn1 1.005 0.024 41.594 0.000
#> .sn2 1.010 0.022 46.638 0.000
#> .pbc1 0.997 0.024 42.323 0.000
#> .pbc2 0.985 0.022 44.751 0.000
#> .pbc3 0.991 0.020 50.566 0.000
#> .int1 1.014 0.022 46.839 0.000
#> .int2 1.012 0.020 50.167 0.000
#> .int3 1.005 0.018 54.971 0.000
#> .b1 0.997 0.022 45.295 0.000
#> .b2 1.015 0.021 49.088 0.000
#> .int1pb1 -0.014 0.029 -0.478 0.633
#> .int2pb2 0.011 0.025 0.450 0.652
#> .int3pb3 -0.005 0.020 -0.269 0.788
#>
#> Variances:
#> Estimate Std.Err z-value P(>|z|)
#> ATT (V_AT) 0.998 0.034 29.072 0.000
#> SN (V_SN) 0.990 0.037 26.938 0.000
#> PBC (V_PB) 0.970 0.026 37.161 0.000
#> .INT (Z_IN) 0.495 0.019 26.260 0.000
#> .BEH (Z_BE) 0.453 0.023 19.820 0.000
#> INTPBC (V_IN) 0.992 0.032 31.229 0.000
#> .att1 (Vr_t1) 0.167 0.007 23.528 0.000
#> .att2 (Vr_t2) 0.150 0.006 24.693 0.000
#> .att3 (Vr_t3) 0.160 0.006 26.378 0.000
#> .att4 (Vr_4) 0.163 0.006 27.649 0.000
#> .att5 (Vr_5) 0.159 0.006 24.929 0.000
#> .sn1 (Vr_s1) 0.178 0.015 12.118 0.000
#> .sn2 (Vr_s2) 0.156 0.012 13.235 0.000
#> .pbc1 (Vr_p1) 0.141 0.007 19.079 0.000
#> .pbc2 (Vr_p2) 0.160 0.007 22.885 0.000
#> .pbc3 (Vr_p3) 0.156 0.006 25.493 0.000
#> .int1 (Vr_n1) 0.152 0.008 19.481 0.000
#> .int2 (Vr_n2) 0.161 0.007 22.582 0.000
#> .int3 (Vr_n3) 0.169 0.006 26.510 0.000
#> .b1 (Vr_b1) 0.187 0.019 9.943 0.000
#> .b2 (Vr_b2) 0.134 0.017 7.874 0.000
#> .int1p1 (V_11) 0.279 0.010 29.251 0.000
#> .int2p2 (V_22) 0.260 0.008 33.952 0.000
#> .int3p3 (V_33) 0.208 0.005 37.854 0.000
#>
#> Constraints:
#> |Slack|
#> V_INTPBC-((2*C_ATT_PBC*G_ATT_INT*G_PBC_IN 0.000
#> Cov_PBC_INTPBC - 0 0.000
#> V_11-(_1_INT^2*(2*C_ATT_PBC*G_ATT_INT*G_P 0.000
#> V_22-(_2_INT^2*(2*C_ATT_PBC*G_ATT_INT*G_P 0.000
#> V_33-(_3_INT^2*(2*C_ATT_PBC*G_ATT_INT*G_P 0.000
#> lmbd_nt1pbc1_INTPBC-(lmbd_nt1_INT*_1_PBC) 0.000
#> lmbd_nt2pbc2_INTPBC-(lmbd_nt2_INT*_2_PBC) 0.000
#> lmbd_nt3pbc3_INTPBC-(lmbd_nt3_INT*_3_PBC) 0.000
#> M_INTPBC-((C_ATT_PBC*G_ATT_INT+C_PBC_SN*G 0.000
#>
# }