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The Basic Syntax

modsem introduces a new feature to the lavaan syntax—the semicolon operator (:). The semicolon operator works the same way as in the lm() function. To specify an interaction effect between two variables, you join them by Var1:Var2.

Models can be estimated using one of the product indicator approaches ("ca", "rca", "dblcent", "pind") or by using the latent moderated structural equations approach ("lms") or the quasi maximum likelihood approach ("qml"). The product indicator approaches are estimated via lavaan, while the lms and qml approaches are estimated via modsem itself.

A Simple Example

Here is a simple example of how to specify an interaction effect between two latent variables in lavaan.

m1 <- '
  # Outer Model
  X =~ x1 + x2 + x3
  Y =~ y1 + y2 + y3
  Z =~ z1 + z2 + z3
  
  # Inner Model
  Y ~ X + Z + X:Z 
'

est1 <- modsem(m1, oneInt)
summary(est1)
#> modsem (version 1.0.4, approach = dblcent):
#> 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 ~~                                             
#>    .x2z2              0.000                           
#>    .x2z3              0.000                           
#>    .x3z2              0.000                           
#>    .x3z3              0.000                           
#>  .x2z1 ~~                                             
#>    .x1z2              0.000                           
#>  .x1z2 ~~                                             
#>    .x2z3              0.000                           
#>  .x3z1 ~~                                             
#>    .x1z2              0.000                           
#>  .x1z2 ~~                                             
#>    .x3z3              0.000                           
#>  .x2z1 ~~                                             
#>    .x1z3              0.000                           
#>  .x2z2 ~~                                             
#>    .x1z3              0.000                           
#>  .x3z1 ~~                                             
#>    .x1z3              0.000                           
#>  .x3z2 ~~                                             
#>    .x1z3              0.000                           
#>  .x2z1 ~~                                             
#>    .x3z2              0.000                           
#>    .x3z3              0.000                           
#>  .x3z1 ~~                                             
#>    .x2z2              0.000                           
#>  .x2z2 ~~                                             
#>    .x3z3              0.000                           
#>  .x3z1 ~~                                             
#>    .x2z3              0.000                           
#>  .x3z2 ~~                                             
#>    .x2z3              0.000                           
#>  .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

By default, the model is estimated using the "dblcent" method. If you want to use another method, you can change it using the method argument.

est1 <- modsem(m1, oneInt, method = "lms")
summary(est1)
#> 
#> modsem (version 1.0.4):
#>   Estimator                                         LMS
#>   Optimization method                         EM-NLMINB
#>   Number of observations                           2000
#>   Number of iterations                               84
#>   Loglikelihood                               -14687.86
#>   Akaike (AIC)                                 29437.73
#>   Bayesian (BIC)                               29611.35
#>  
#> Numerical Integration:
#>   Points of integration (per dim)                    24
#>   Dimensions                                          1
#>   Total points of integration                        24
#>  
#> Fit Measures for H0:
#>   Loglikelihood                                  -17832
#>   Akaike (AIC)                                 35723.75
#>   Bayesian (BIC)                               35891.78
#>   Chi-square                                      17.52
#>   Degrees of Freedom (Chi-square)                    24
#>   P-value (Chi-square)                            0.826
#>   RMSEA                                           0.000
#>  
#> Comparative fit to H0 (no interaction effect)
#>   Loglikelihood change                          3144.01
#>   Difference test (D)                           6288.02
#>   Degrees of freedom (D)                              1
#>   P-value (D)                                     0.000
#>  
#> R-Squared:
#>   Y                                               0.596
#> R-Squared Null-Model (H0):
#>   Y                                               0.395
#> R-Squared Change:
#>   Y                                               0.201
#> 
#> Parameter Estimates:
#>   Coefficients                           unstandardized
#>   Information                                  expected
#>   Standard errors                              standard
#>  
#> Latent Variables:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>   X =~ 
#>     x1               1.000                             
#>     x2               0.804      0.018    45.23    0.000
#>     x3               0.915      0.015    62.37    0.000
#>   Z =~ 
#>     z1               1.000                             
#>     z2               0.810      0.014    55.97    0.000
#>     z3               0.881      0.014    64.52    0.000
#>   Y =~ 
#>     y1               1.000                             
#>     y2               0.799      0.009    85.43    0.000
#>     y3               0.899      0.010    88.71    0.000
#> 
#> Regressions:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>   Y ~ 
#>     X                0.677      0.039    17.21    0.000
#>     Z                0.572      0.038    14.98    0.000
#>     X:Z              0.712      0.032    22.46    0.000
#> 
#> Intercepts:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>     x1               1.026      0.025    41.65    0.000
#>     x2               1.218      0.020    61.14    0.000
#>     x3               0.922      0.024    38.69    0.000
#>     z1               1.016      0.031    32.90    0.000
#>     z2               1.209      0.029    42.26    0.000
#>     z3               0.920      0.027    34.22    0.000
#>     y1               1.046      0.024    43.95    0.000
#>     y2               1.228      0.022    54.98    0.000
#>     y3               0.962      0.025    39.06    0.000
#>     Y                0.000                             
#>     X                0.000                             
#>     Z                0.000                             
#> 
#> Covariances:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>   X ~~ 
#>     Z                0.198      0.032     6.27    0.000
#> 
#> Variances:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>     x1               0.160      0.010    15.38    0.000
#>     x2               0.163      0.008    19.89    0.000
#>     x3               0.165      0.009    19.18    0.000
#>     z1               0.166      0.010    16.74    0.000
#>     z2               0.160      0.009    17.18    0.000
#>     z3               0.158      0.009    18.54    0.000
#>     y1               0.160      0.010    16.78    0.000
#>     y2               0.154      0.008    20.01    0.000
#>     y3               0.163      0.009    18.39    0.000
#>     X                0.972      0.038    25.86    0.000
#>     Z                1.017      0.052    19.55    0.000
#>     Y                0.984      0.045    21.94    0.000

Interactions Between Two Observed Variables

modsem allows you to estimate interactions between not only latent variables but also observed variables. Below, we first run a regression with only observed variables, where there is an interaction between x1 and z2, and then run an equivalent model using modsem().

Using a Regression

reg1 <- lm(y1 ~ x1*z1, oneInt)
summary(reg1)
#> 
#> Call:
#> lm(formula = y1 ~ x1 * z1, data = oneInt)
#> 
#> Residuals:
#>     Min      1Q  Median      3Q     Max 
#> -3.7155 -0.8087 -0.0367  0.8078  4.6531 
#> 
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)  0.51422    0.04618  11.135   <2e-16 ***
#> x1           0.05477    0.03387   1.617   0.1060    
#> z1          -0.06575    0.03461  -1.900   0.0576 .  
#> x1:z1        0.54361    0.02272  23.926   <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 1.184 on 1996 degrees of freedom
#> Multiple R-squared:  0.4714, Adjusted R-squared:  0.4706 
#> F-statistic: 593.3 on 3 and 1996 DF,  p-value: < 2.2e-16

Using modsem

When you have interactions between observed variables, it is generally recommended to use method = "pind". Interaction effects with observed variables are not supported by the LMS and QML approaches. In some cases, you can define a latent variable with a single indicator to estimate the interaction effect between two observed variables in the LMS and QML approaches, but this is generally not recommended.

# Using "pind" as the method (see Chapter 3)
est2 <- modsem('y1 ~ x1 + z1 + x1:z1', data = oneInt, method = "pind")
summary(est2)
#> modsem (version 1.0.4, approach = pind):
#> lavaan 0.6-19 ended normally after 1 iteration
#> 
#>   Estimator                                         ML
#>   Optimization method                           NLMINB
#>   Number of model parameters                         4
#> 
#>   Number of observations                          2000
#> 
#> Model Test User Model:
#>                                                       
#>   Test statistic                                 0.000
#>   Degrees of freedom                                 0
#> 
#> Parameter Estimates:
#> 
#>   Standard errors                             Standard
#>   Information                                 Expected
#>   Information saturated (h1) model          Structured
#> 
#> Regressions:
#>                    Estimate  Std.Err  z-value  P(>|z|)
#>   y1 ~                                                
#>     x1                0.055    0.034    1.619    0.105
#>     z1               -0.066    0.035   -1.902    0.057
#>     x1z1              0.544    0.023   23.950    0.000
#> 
#> Variances:
#>                    Estimate  Std.Err  z-value  P(>|z|)
#>    .y1                1.399    0.044   31.623    0.000

Interactions Between Latent and Observed Variables

modsem also allows you to estimate interaction effects between latent and observed variables. To do so, simply join a latent and an observed variable with a colon (e.g., 'latent:observer'). As with interactions between observed variables, it is generally recommended to use method = "pind" for estimating the effect between latent and observed variables.

m3 <- '
  # Outer Model
  X =~ x1 + x2 + x3
  Y =~ y1 + y2 + y3
  
  # Inner Model
  Y ~ X + z1 + X:z1 
'

est3 <- modsem(m3, oneInt, method = "pind")
summary(est3)
#> modsem (version 1.0.4, approach = pind):
#> lavaan 0.6-19 ended normally after 45 iterations
#> 
#>   Estimator                                         ML
#>   Optimization method                           NLMINB
#>   Number of model parameters                        22
#> 
#>   Number of observations                          2000
#> 
#> Model Test User Model:
#>                                                       
#>   Test statistic                              4468.171
#>   Degrees of freedom                                32
#>   P-value (Chi-square)                           0.000
#> 
#> 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                0.803    0.013   63.697    0.000
#>     x3                0.915    0.014   67.548    0.000
#>   Y =~                                                
#>     y1                1.000                           
#>     y2                0.798    0.007  115.375    0.000
#>     y3                0.899    0.007  120.783    0.000
#>   Xz1 =~                                              
#>     x1z1              1.000                           
#>     x2z1              0.947    0.010   96.034    0.000
#>     x3z1              0.902    0.009   99.512    0.000
#> 
#> Regressions:
#>                    Estimate  Std.Err  z-value  P(>|z|)
#>   Y ~                                                 
#>     X                 0.021    0.034    0.614    0.540
#>     z1               -0.185    0.023   -8.096    0.000
#>     Xz1               0.646    0.017   37.126    0.000
#> 
#> Covariances:
#>                    Estimate  Std.Err  z-value  P(>|z|)
#>   X ~~                                                
#>     Xz1               1.243    0.055   22.523    0.000
#> 
#> Variances:
#>                    Estimate  Std.Err  z-value  P(>|z|)
#>    .x1                0.158    0.009   17.976    0.000
#>    .x2                0.164    0.007   23.216    0.000
#>    .x3                0.162    0.008   20.325    0.000
#>    .y1                0.158    0.009   17.819    0.000
#>    .y2                0.154    0.007   22.651    0.000
#>    .y3                0.164    0.008   20.744    0.000
#>    .x1z1              0.315    0.017   18.328    0.000
#>    .x2z1              0.428    0.019   22.853    0.000
#>    .x3z1              0.337    0.016   21.430    0.000
#>     X                 0.982    0.036   26.947    0.000
#>    .Y                 1.112    0.040   27.710    0.000
#>     Xz1               3.965    0.136   29.217    0.000

Quadratic Effects

Quadratic effects are essentially a special case of interaction effects. Thus, modsem can also be used to estimate quadratic effects.

m4 <- '
# Outer Model
X =~ x1 + x2 + x3
Y =~ y1 + y2 + y3
Z =~ z1 + z2 + z3

# Inner Model
Y ~ X + Z + Z:X + X:X
'

est4 <- modsem(m4, oneInt, method = "qml")
summary(est4)
#> 
#> modsem (version 1.0.4):
#>   Estimator                                         QML
#>   Optimization method                            NLMINB
#>   Number of observations                           2000
#>   Number of iterations                              104
#>   Loglikelihood                                -17496.2
#>   Akaike (AIC)                                  35056.4
#>   Bayesian (BIC)                               35235.62
#>  
#> Fit Measures for H0:
#>   Loglikelihood                                  -17832
#>   Akaike (AIC)                                 35723.75
#>   Bayesian (BIC)                               35891.78
#>   Chi-square                                      17.52
#>   Degrees of Freedom (Chi-square)                    24
#>   P-value (Chi-square)                            0.826
#>   RMSEA                                           0.000
#>  
#> Comparative fit to H0 (no interaction effect)
#>   Loglikelihood change                           335.68
#>   Difference test (D)                            671.35
#>   Degrees of freedom (D)                              2
#>   P-value (D)                                     0.000
#>  
#> R-Squared:
#>   Y                                               0.607
#> R-Squared Null-Model (H0):
#>   Y                                               0.395
#> R-Squared Change:
#>   Y                                               0.212
#> 
#> Parameter Estimates:
#>   Coefficients                           unstandardized
#>   Information                                  observed
#>   Standard errors                              standard
#>  
#> Latent Variables:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>   X =~ 
#>     x1               1.000                             
#>     x2               0.803      0.013   63.963    0.000
#>     x3               0.914      0.013   67.795    0.000
#>   Z =~ 
#>     z1               1.000                             
#>     z2               0.810      0.012   65.122    0.000
#>     z3               0.881      0.013   67.621    0.000
#>   Y =~ 
#>     y1               1.000                             
#>     y2               0.798      0.007  107.568    0.000
#>     y3               0.899      0.008  112.543    0.000
#> 
#> Regressions:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>   Y ~ 
#>     X                0.674      0.032   20.888    0.000
#>     Z                0.566      0.030   18.947    0.000
#>     X:X             -0.005      0.023   -0.207    0.836
#>     X:Z              0.713      0.029   24.554    0.000
#> 
#> Intercepts:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>     x1               1.023      0.024   42.894    0.000
#>     x2               1.216      0.020   60.994    0.000
#>     x3               0.919      0.022   41.484    0.000
#>     z1               1.012      0.024   41.575    0.000
#>     z2               1.206      0.020   59.269    0.000
#>     z3               0.916      0.022   42.062    0.000
#>     y1               1.042      0.038   27.683    0.000
#>     y2               1.224      0.030   40.158    0.000
#>     y3               0.958      0.034   28.101    0.000
#>     Y                0.000                             
#>     X                0.000                             
#>     Z                0.000                             
#> 
#> Covariances:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>   X ~~ 
#>     Z                0.200      0.024    8.238    0.000
#> 
#> Variances:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>     x1               0.158      0.009   18.145    0.000
#>     x2               0.162      0.007   23.188    0.000
#>     x3               0.165      0.008   20.821    0.000
#>     z1               0.166      0.009   18.340    0.000
#>     z2               0.159      0.007   22.621    0.000
#>     z3               0.158      0.008   20.713    0.000
#>     y1               0.159      0.009   17.975    0.000
#>     y2               0.154      0.007   22.670    0.000
#>     y3               0.164      0.008   20.711    0.000
#>     X                0.983      0.036   26.994    0.000
#>     Z                1.019      0.038   26.951    0.000
#>     Y                0.943      0.038   24.819    0.000

More Complicated Examples

Here is a more complex example using the theory of planned behavior (TPB) model.

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)
  INT ~ ATT + SN + PBC
  BEH ~ INT + PBC + INT:PBC  
'

# The double-centering approach
est_tpb <- modsem(tpb, TPB)

# Using the LMS approach
est_tpb_lms <- modsem(tpb, TPB, method = "lms")
#> Warning: It is recommended that you have at least 32 nodes for interaction
#> effects between exogenous and endogenous variables in the lms approach 'nodes =
#> 24'
summary(est_tpb_lms)
#> 
#> modsem (version 1.0.4):
#>   Estimator                                         LMS
#>   Optimization method                         EM-NLMINB
#>   Number of observations                           2000
#>   Number of iterations                               88
#>   Loglikelihood                               -23463.87
#>   Akaike (AIC)                                 47035.73
#>   Bayesian (BIC)                               47338.18
#>  
#> Numerical Integration:
#>   Points of integration (per dim)                    24
#>   Dimensions                                          1
#>   Total points of integration                        24
#>  
#> Fit Measures for H0:
#>   Loglikelihood                                  -26393
#>   Akaike (AIC)                                 52892.45
#>   Bayesian (BIC)                               53189.29
#>   Chi-square                                      66.27
#>   Degrees of Freedom (Chi-square)                    82
#>   P-value (Chi-square)                            0.897
#>   RMSEA                                           0.000
#>  
#> Comparative fit to H0 (no interaction effect)
#>   Loglikelihood change                          2929.36
#>   Difference test (D)                           5858.71
#>   Degrees of freedom (D)                              1
#>   P-value (D)                                     0.000
#>  
#> R-Squared:
#>   INT                                             0.361
#>   BEH                                             0.248
#> R-Squared Null-Model (H0):
#>   INT                                             0.367
#>   BEH                                             0.210
#> R-Squared Change:
#>   INT                                            -0.006
#>   BEH                                             0.038
#> 
#> Parameter Estimates:
#>   Coefficients                           unstandardized
#>   Information                                  expected
#>   Standard errors                              standard
#>  
#> Latent Variables:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>   PBC =~ 
#>     pbc1             1.000                             
#>     pbc2             0.911      0.023    39.06    0.000
#>     pbc3             0.802      0.016    51.13    0.000
#>   ATT =~ 
#>     att1             1.000                             
#>     att2             0.877      0.016    54.64    0.000
#>     att3             0.789      0.017    47.31    0.000
#>     att4             0.695      0.018    38.72    0.000
#>     att5             0.887      0.018    49.78    0.000
#>   SN =~ 
#>     sn1              1.000                             
#>     sn2              0.889      0.027    32.67    0.000
#>   INT =~ 
#>     int1             1.000                             
#>     int2             0.913      0.026    34.76    0.000
#>     int3             0.807      0.019    43.58    0.000
#>   BEH =~ 
#>     b1               1.000                             
#>     b2               0.961      0.044    21.59    0.000
#> 
#> Regressions:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>   INT ~ 
#>     PBC              0.217      0.045     4.78    0.000
#>     ATT              0.213      0.033     6.38    0.000
#>     SN               0.177      0.036     4.90    0.000
#>   BEH ~ 
#>     PBC              0.228      0.035     6.45    0.000
#>     INT              0.182      0.034     5.29    0.000
#>     PBC:INT          0.204      0.025     8.01    0.000
#> 
#> Intercepts:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>     pbc1             0.960      0.028    34.86    0.000
#>     pbc2             0.951      0.024    39.44    0.000
#>     pbc3             0.961      0.022    44.16    0.000
#>     att1             0.988      0.036    27.66    0.000
#>     att2             0.984      0.030    32.28    0.000
#>     att3             0.996      0.026    37.69    0.000
#>     att4             0.981      0.022    44.69    0.000
#>     att5             0.969      0.028    34.07    0.000
#>     sn1              0.980      0.031    31.89    0.000
#>     sn2              0.987      0.034    28.80    0.000
#>     int1             0.996      0.031    32.40    0.000
#>     int2             0.996      0.027    37.26    0.000
#>     int3             0.991      0.024    40.79    0.000
#>     b1               0.990      0.024    40.94    0.000
#>     b2               1.008      0.026    39.40    0.000
#>     INT              0.000                             
#>     BEH              0.000                             
#>     PBC              0.000                             
#>     ATT              0.000                             
#>     SN               0.000                             
#> 
#> Covariances:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>   PBC ~~ 
#>     ATT              0.658      0.037    17.84    0.000
#>     SN               0.657      0.043    15.32    0.000
#>   ATT ~~ 
#>     SN               0.616      0.047    13.06    0.000
#> 
#> Variances:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>     pbc1             0.147      0.012    12.31    0.000
#>     pbc2             0.164      0.010    16.73    0.000
#>     pbc3             0.154      0.010    15.21    0.000
#>     att1             0.167      0.011    15.71    0.000
#>     att2             0.150      0.010    15.45    0.000
#>     att3             0.159      0.009    17.26    0.000
#>     att4             0.163      0.008    19.28    0.000
#>     att5             0.159      0.009    16.97    0.000
#>     sn1              0.178      0.023     7.72    0.000
#>     sn2              0.156      0.017     9.32    0.000
#>     int1             0.157      0.013    12.37    0.000
#>     int2             0.160      0.011    14.88    0.000
#>     int3             0.168      0.009    18.87    0.000
#>     b1               0.186      0.026     7.28    0.000
#>     b2               0.135      0.025     5.45    0.000
#>     PBC              0.933      0.039    23.75    0.000
#>     ATT              0.985      0.056    17.72    0.000
#>     SN               0.974      0.058    16.66    0.000
#>     INT              0.491      0.032    15.36    0.000
#>     BEH              0.456      0.034    13.58    0.000

Here is an example that includes two quadratic effects and one interaction effect, using the jordan dataset. The dataset is a subset of the PISA 2006 dataset.

m2 <- '
ENJ =~ enjoy1 + enjoy2 + enjoy3 + enjoy4 + enjoy5
CAREER =~ career1 + career2 + career3 + career4
SC =~ academic1 + academic2 + academic3 + academic4 + academic5 + academic6
CAREER ~ ENJ + SC + ENJ:ENJ + SC:SC + ENJ:SC
'

est_jordan <- modsem(m2, data = jordan)
est_jordan_qml <- modsem(m2, data = jordan, method = "qml")
#> Warning: SE's for some coefficients could not be computed.
summary(est_jordan_qml)
#> 
#> modsem (version 1.0.4):
#>   Estimator                                         QML
#>   Optimization method                            NLMINB
#>   Number of observations                           6038
#>   Number of iterations                               86
#>   Loglikelihood                              -110520.22
#>   Akaike (AIC)                                221142.45
#>   Bayesian (BIC)                              221484.45
#>  
#> Fit Measures for H0:
#>   Loglikelihood                                 -110521
#>   Akaike (AIC)                                221138.58
#>   Bayesian (BIC)                              221460.46
#>   Chi-square                                    1016.34
#>   Degrees of Freedom (Chi-square)                    87
#>   P-value (Chi-square)                            0.000
#>   RMSEA                                           0.005
#>  
#> Comparative fit to H0 (no interaction effect)
#>   Loglikelihood change                             1.06
#>   Difference test (D)                              2.13
#>   Degrees of freedom (D)                              3
#>   P-value (D)                                     0.546
#>  
#> R-Squared:
#>   CAREER                                          0.513
#> R-Squared Null-Model (H0):
#>   CAREER                                          0.510
#> R-Squared Change:
#>   CAREER                                          0.003
#> 
#> Parameter Estimates:
#>   Coefficients                           unstandardized
#>   Information                                  observed
#>   Standard errors                              standard
#>  
#> Latent Variables:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>   ENJ =~ 
#>     enjoy1           1.000                             
#>     enjoy2           1.002      0.020   50.584    0.000
#>     enjoy3           0.894      0.020   43.669    0.000
#>     enjoy4           0.999      0.021   48.225    0.000
#>     enjoy5           1.047      0.021   50.399    0.000
#>   SC =~ 
#>     academic1        1.000                             
#>     academic2        1.104      0.028   38.946    0.000
#>     academic3        1.235      0.030   41.720    0.000
#>     academic4        1.254      0.030   41.829    0.000
#>     academic5        1.113      0.029   38.649    0.000
#>     academic6        1.198      0.030   40.357    0.000
#>   CAREER =~ 
#>     career1          1.000                             
#>     career2          1.040      0.016   65.181    0.000
#>     career3          0.952      0.016   57.839    0.000
#>     career4          0.818      0.017   48.358    0.000
#> 
#> Regressions:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>   CAREER ~ 
#>     ENJ              0.523      0.020   26.293    0.000
#>     SC               0.467      0.023   19.899    0.000
#>     ENJ:ENJ          0.026      0.022    1.221    0.222
#>     ENJ:SC          -0.040      0.046   -0.874    0.382
#>     SC:SC           -0.002      0.034   -0.049    0.961
#> 
#> Intercepts:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>     enjoy1           0.000                             
#>     enjoy2           0.000      0.005    0.032    0.975
#>     enjoy3           0.000      0.009   -0.035    0.972
#>     enjoy4           0.000                             
#>     enjoy5           0.000      0.006    0.059    0.953
#>     academic1        0.000      0.007   -0.016    0.988
#>     academic2        0.000      0.011   -0.014    0.989
#>     academic3        0.000      0.011   -0.039    0.969
#>     academic4        0.000      0.010   -0.022    0.983
#>     academic5       -0.001      0.011   -0.061    0.951
#>     academic6        0.001      0.011    0.064    0.949
#>     career1         -0.004      0.015   -0.246    0.806
#>     career2         -0.005      0.015   -0.299    0.765
#>     career3         -0.004      0.015   -0.255    0.798
#>     career4         -0.004      0.014   -0.269    0.788
#>     CAREER           0.000                             
#>     ENJ              0.000                             
#>     SC               0.000                             
#> 
#> Covariances:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>   ENJ ~~ 
#>     SC               0.218      0.009   25.477    0.000
#> 
#> Variances:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>     enjoy1           0.487      0.011   44.332    0.000
#>     enjoy2           0.488      0.011   44.407    0.000
#>     enjoy3           0.596      0.012   48.234    0.000
#>     enjoy4           0.488      0.011   44.561    0.000
#>     enjoy5           0.442      0.010   42.472    0.000
#>     academic1        0.644      0.013   49.816    0.000
#>     academic2        0.566      0.012   47.862    0.000
#>     academic3        0.474      0.011   44.316    0.000
#>     academic4        0.455      0.010   43.580    0.000
#>     academic5        0.565      0.012   47.696    0.000
#>     academic6        0.502      0.011   45.434    0.000
#>     career1          0.373      0.009   40.392    0.000
#>     career2          0.328      0.009   37.018    0.000
#>     career3          0.436      0.010   43.274    0.000
#>     career4          0.576      0.012   48.373    0.000
#>     ENJ              0.500      0.017   29.546    0.000
#>     SC               0.338      0.015   23.196    0.000
#>     CAREER           0.302      0.010   29.710    0.000

Note: Other approaches also work but may be quite slow depending on the number of interaction effects, particularly for the LMS and constrained approaches.