Skip to contents

modsem_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

lavaan syntax

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 in lavaan by default. If TRUE, 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 using res.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:Z and X:W the residuals of the generated product indicators x1:z1 and x1:w1 may be correlated. If TRUE residual covariances are allowed across different latent interaction terms. If FALSE residual 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, if FALSE only 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. If FALSE they are not removed.

suppress.warnings.lavaan

should warnings from lavaan be suppressed?

suppress.warnings.match

should warnings from match be 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_item as the choose argument.

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? If TRUE the estimates for the constraints are approximated using a monte carlo simulation (see the rcs.mc.reps argument). If FALSE the 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.xz argument).

rcs.scale.corrected

Should reliability corrected items be scale-corrected? If TRUE reliability-corrected single items are corrected for differences in factor loadings between the items. Default is TRUE.

LAVFUN

Function used to estimate the model. Defaults to lavaan::sem.

...

arguments passed to LAVFUN

Value

modsem object

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
#> 
# }