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The Latent Moderated Structural Equations (LMS) and the Quasi Maximum Likelihood (QML) approach

Both the LMS- and QML approach works on most models, but interaction effects with endogenous can be a bit tricky to estimate (see the vignette. Both approaches (particularily the LMS approach) are quite computationally intensive, and are thus partly implemented in C++ (using Rcpp and RcppArmadillo). Additionally starting parameters are estimated using the double centering approach (and the means of the observed variables) are used to generate good starting parameters for faster convergence. If you want to see the progress of the estimation process you can use ´verbose = TRUE´.

A Simple Example

Here you can see an example of the LMS approach for a simple model. By default the summary function calculates fit measures compared to a null model (i.e., the same model without an interaction term).

library(modsem)
m1 <- '
# Outer Model
  X =~ x1
  X =~ x2 + x3
  Z =~ z1 + z2 + z3
  Y =~ y1 + y2 + y3

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

lms1 <- modsem(m1, oneInt, method = "lms")
summary(lms1, standardized = TRUE) # standardized estimates 
#> Estimating null model
#> EM: Iteration =     1, LogLik =   -17831.87, Change = -17831.875
#> EM: Iteration =     2, LogLik =   -17831.87, Change =      0.000
#> 
#> modsem (version 1.0.3):
#>   Estimator                                         LMS
#>   Optimization method                         EM-NLMINB
#>   Number of observations                           2000
#>   Number of iterations                               92
#>   Loglikelihood                               -14687.85
#>   Akaike (AIC)                                 29437.71
#>   Bayesian (BIC)                               29611.34
#>  
#> 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                          3144.02
#>   Difference test (D)                           6288.04
#>   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                             standardized
#>   Information                                  expected
#>   Standard errors                              standard
#>  
#> Latent Variables:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>   X =~ 
#>     x1               0.926                             
#>     x2               0.891      0.014    64.39    0.000
#>     x3               0.912      0.013    67.69    0.000
#>   Z =~ 
#>     z1               0.927                             
#>     z2               0.898      0.014    64.59    0.000
#>     z3               0.913      0.013    67.87    0.000
#>   Y =~ 
#>     y1               0.969                             
#>     y2               0.954      0.009   105.92    0.000
#>     y3               0.961      0.009   111.95    0.000
#> 
#> Regressions:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>   Y ~ 
#>     X                0.427      0.020    21.79    0.000
#>     Z                0.370      0.018    20.16    0.000
#>     X:Z              0.454      0.017    26.28    0.000
#> 
#> Covariances:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>   X ~~ 
#>     Z                0.199      0.024     8.43    0.000
#> 
#> Variances:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>     x1               0.142      0.007    19.27    0.000
#>     x2               0.206      0.009    23.86    0.000
#>     x3               0.169      0.008    21.31    0.000
#>     z1               0.141      0.008    18.34    0.000
#>     z2               0.193      0.009    22.39    0.000
#>     z3               0.167      0.008    20.52    0.000
#>     y1               0.061      0.003    17.93    0.000
#>     y2               0.090      0.004    22.72    0.000
#>     y3               0.077      0.004    20.69    0.000
#>     X                1.000      0.016    61.06    0.000
#>     Z                1.000      0.018    55.21    0.000
#>     Y                0.404      0.015    26.54    0.000

Here you can see the same example using the QML approach.

qml1 <- modsem(m1, oneInt, method = "qml")
summary(qml1)
#> Estimating null model
#> Starting M-step
#> 
#> modsem (version 1.0.3):
#>   Estimator                                         QML
#>   Optimization method                            NLMINB
#>   Number of observations                           2000
#>   Number of iterations                              111
#>   Loglikelihood                               -17496.22
#>   Akaike (AIC)                                 35054.43
#>   Bayesian (BIC)                               35228.06
#>  
#> 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                           335.66
#>   Difference test (D)                            671.32
#>   Degrees of freedom (D)                              1
#>   P-value (D)                                     0.000
#>  
#> R-Squared:
#>   Y                                               0.607
#> R-Squared Null-Model (H0):
#>   Y                                               0.395
#> R-Squared Change:
#>   Y                                               0.211
#> 
#> 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    63.96    0.000
#>     x3               0.914      0.013    67.80    0.000
#>   Z =~ 
#>     z1               1.000                             
#>     z2               0.810      0.012    65.12    0.000
#>     z3               0.881      0.013    67.62    0.000
#>   Y =~ 
#>     y1               1.000                             
#>     y2               0.798      0.007   107.57    0.000
#>     y3               0.899      0.008   112.55    0.000
#> 
#> Regressions:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>   Y ~ 
#>     X                0.674      0.032    20.94    0.000
#>     Z                0.566      0.030    18.96    0.000
#>     X:Z              0.712      0.028    25.45    0.000
#> 
#> Intercepts:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>     x1               1.023      0.024    42.89    0.000
#>     x2               1.215      0.020    60.99    0.000
#>     x3               0.919      0.022    41.48    0.000
#>     z1               1.012      0.024    41.57    0.000
#>     z2               1.206      0.020    59.27    0.000
#>     z3               0.916      0.022    42.06    0.000
#>     y1               1.038      0.033    31.45    0.000
#>     y2               1.221      0.027    45.49    0.000
#>     y3               0.955      0.030    31.86    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     8.24    0.000
#> 
#> Variances:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>     x1               0.158      0.009    18.14    0.000
#>     x2               0.162      0.007    23.19    0.000
#>     x3               0.165      0.008    20.82    0.000
#>     z1               0.166      0.009    18.34    0.000
#>     z2               0.159      0.007    22.62    0.000
#>     z3               0.158      0.008    20.71    0.000
#>     y1               0.159      0.009    17.98    0.000
#>     y2               0.154      0.007    22.67    0.000
#>     y3               0.164      0.008    20.71    0.000
#>     X                0.983      0.036    26.99    0.000
#>     Z                1.019      0.038    26.95    0.000
#>     Y                0.943      0.038    24.87    0.000

A more complicated example

Here you can see an example of a more complicated example using the model from the theory of planned behaviour (TPB), where there are two endogenous variables, where there is an interaction between an endogenous and exogenous variable. When estimating more complicated models with the LMS-approach, it is recommended that you increase the number of nodes used for numerical integration. By default the number of nodes is set to 16, and can be increased using the nodes argument. The argument has no effect on the QML approach. When there is an interaction effect between an endogenous and exogenous variable, it is recommended that you use at least 32 nodes for the LMS-approach. You can also get robust standard errors by setting robust.se = TRUE in the modsem() function.

Note: If you want the lms-approach to give as similar results as possible to mplus, you would have to increase the number of nodes (e.g., nodes = 100).

# ATT = Attitude, 
# PBC = Perceived Behavioural Control 
# INT = Intention 
# SN = Subjective Norms
# BEH = Behaviour
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 
  BEH ~ INT:PBC  
'

lms2 <- modsem(tpb, TPB, method = "lms", nodes = 32)
summary(lms2)
#> Estimating null model
#> EM: Iteration =     1, LogLik =   -26393.22, Change = -26393.223
#> EM: Iteration =     2, LogLik =   -26393.22, Change =      0.000
#> 
#> modsem (version 1.0.3):
#>   Estimator                                         LMS
#>   Optimization method                         EM-NLMINB
#>   Number of observations                           2000
#>   Number of iterations                               70
#>   Loglikelihood                               -23439.02
#>   Akaike (AIC)                                 46986.04
#>   Bayesian (BIC)                               47288.49
#>  
#> Numerical Integration:
#>   Points of integration (per dim)                    32
#>   Dimensions                                          1
#>   Total points of integration                        32
#>  
#> 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                          2954.20
#>   Difference test (D)                           5908.41
#>   Degrees of freedom (D)                              1
#>   P-value (D)                                     0.000
#>  
#> R-Squared:
#>   INT                                             0.364
#>   BEH                                             0.259
#> R-Squared Null-Model (H0):
#>   INT                                             0.367
#>   BEH                                             0.210
#> R-Squared Change:
#>   INT                                            -0.003
#>   BEH                                             0.049
#> 
#> Parameter Estimates:
#>   Coefficients                           unstandardized
#>   Information                                  expected
#>   Standard errors                              standard
#>  
#> Latent Variables:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>   PBC =~ 
#>     pbc1             1.000                             
#>     pbc2             0.914      0.013    68.52    0.000
#>     pbc3             0.802      0.012    65.02    0.000
#>   ATT =~ 
#>     att1             1.000                             
#>     att2             0.878      0.012    70.81    0.000
#>     att3             0.789      0.012    65.77    0.000
#>     att4             0.695      0.011    61.09    0.000
#>     att5             0.887      0.013    70.26    0.000
#>   SN =~ 
#>     sn1              1.000                             
#>     sn2              0.889      0.017    52.00    0.000
#>   INT =~ 
#>     int1             1.000                             
#>     int2             0.913      0.016    58.38    0.000
#>     int3             0.807      0.015    55.37    0.000
#>   BEH =~ 
#>     b1               1.000                             
#>     b2               0.959      0.033    29.28    0.000
#> 
#> Regressions:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>   INT ~ 
#>     PBC              0.218      0.030     7.36    0.000
#>     ATT              0.214      0.026     8.19    0.000
#>     SN               0.176      0.027     6.43    0.000
#>   BEH ~ 
#>     PBC              0.233      0.022    10.35    0.000
#>     INT              0.188      0.025     7.62    0.000
#>     PBC:INT          0.205      0.019    10.90    0.000
#> 
#> Intercepts:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>     pbc1             0.990      0.022    45.57    0.000
#>     pbc2             0.978      0.020    48.28    0.000
#>     pbc3             0.985      0.018    53.86    0.000
#>     att1             1.009      0.023    43.19    0.000
#>     att2             1.002      0.021    48.19    0.000
#>     att3             1.012      0.019    53.21    0.000
#>     att4             0.995      0.017    56.95    0.000
#>     att5             0.988      0.021    46.75    0.000
#>     sn1              1.001      0.023    42.73    0.000
#>     sn2              1.006      0.021    48.06    0.000
#>     int1             1.010      0.021    47.81    0.000
#>     int2             1.009      0.020    51.14    0.000
#>     int3             1.002      0.018    56.02    0.000
#>     b1               0.999      0.021    47.31    0.000
#>     b2               1.017      0.020    51.50    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              0.668      0.021    31.78    0.000
#>     SN               0.668      0.022    30.52    0.000
#>   ATT ~~ 
#>     SN               0.623      0.019    32.90    0.000
#> 
#> Variances:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>     pbc1             0.148      0.008    18.81    0.000
#>     pbc2             0.159      0.007    21.62    0.000
#>     pbc3             0.155      0.007    23.64    0.000
#>     att1             0.167      0.007    23.64    0.000
#>     att2             0.150      0.006    24.73    0.000
#>     att3             0.159      0.006    26.68    0.000
#>     att4             0.162      0.006    27.71    0.000
#>     att5             0.159      0.006    25.11    0.000
#>     sn1              0.178      0.015    11.97    0.000
#>     sn2              0.156      0.012    13.20    0.000
#>     int1             0.157      0.009    18.25    0.000
#>     int2             0.160      0.008    20.48    0.000
#>     int3             0.168      0.007    24.27    0.000
#>     b1               0.185      0.020     9.46    0.000
#>     b2               0.136      0.018     7.71    0.000
#>     PBC              0.947      0.017    55.23    0.000
#>     ATT              0.992      0.014    69.87    0.000
#>     SN               0.981      0.015    64.37    0.000
#>     INT              0.491      0.020    24.97    0.000
#>     BEH              0.456      0.023    19.46    0.000

qml2 <- modsem(tpb, TPB, method = "qml")
summary(qml2, standardized = TRUE) # standardized estimates
#> Estimating null model
#> Starting M-step
#> 
#> modsem (version 1.0.3):
#>   Estimator                                         QML
#>   Optimization method                            NLMINB
#>   Number of observations                           2000
#>   Number of iterations                               73
#>   Loglikelihood                               -26326.25
#>   Akaike (AIC)                                  52760.5
#>   Bayesian (BIC)                               53062.95
#>  
#> 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                            66.97
#>   Difference test (D)                            133.95
#>   Degrees of freedom (D)                              1
#>   P-value (D)                                     0.000
#>  
#> R-Squared:
#>   INT                                             0.366
#>   BEH                                             0.263
#> R-Squared Null-Model (H0):
#>   INT                                             0.367
#>   BEH                                             0.210
#> R-Squared Change:
#>   INT                                             0.000
#>   BEH                                             0.053
#> 
#> Parameter Estimates:
#>   Coefficients                             standardized
#>   Information                                  observed
#>   Standard errors                              standard
#>  
#> Latent Variables:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>   PBC =~ 
#>     pbc1             0.933                             
#>     pbc2             0.913      0.013    69.47    0.000
#>     pbc3             0.894      0.014    66.10    0.000
#>   ATT =~ 
#>     att1             0.925                             
#>     att2             0.915      0.013    71.56    0.000
#>     att3             0.892      0.013    66.38    0.000
#>     att4             0.865      0.014    61.00    0.000
#>     att5             0.912      0.013    70.85    0.000
#>   SN =~ 
#>     sn1              0.921                             
#>     sn2              0.913      0.017    52.61    0.000
#>   INT =~ 
#>     int1             0.912                             
#>     int2             0.895      0.015    59.05    0.000
#>     int3             0.866      0.016    55.73    0.000
#>   BEH =~ 
#>     b1               0.877                             
#>     b2               0.900      0.028    31.71    0.000
#> 
#> Regressions:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>   INT ~ 
#>     PBC              0.243      0.033     7.35    0.000
#>     ATT              0.242      0.030     8.16    0.000
#>     SN               0.199      0.031     6.37    0.000
#>   BEH ~ 
#>     PBC              0.289      0.028    10.37    0.000
#>     INT              0.212      0.028     7.69    0.000
#>     PBC:INT          0.227      0.020    11.32    0.000
#> 
#> Covariances:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>   PBC ~~ 
#>     ATT              0.692      0.030    23.45    0.000
#>     SN               0.695      0.030    23.08    0.000
#>   ATT ~~ 
#>     SN               0.634      0.029    21.70    0.000
#> 
#> Variances:
#>                   Estimate  Std.Error  z.value  P(>|z|)
#>     pbc1             0.130      0.007    18.39    0.000
#>     pbc2             0.166      0.008    21.43    0.000
#>     pbc3             0.201      0.008    23.89    0.000
#>     att1             0.144      0.006    23.53    0.000
#>     att2             0.164      0.007    24.71    0.000
#>     att3             0.204      0.008    26.38    0.000
#>     att4             0.252      0.009    27.65    0.000
#>     att5             0.168      0.007    24.93    0.000
#>     sn1              0.153      0.013    12.09    0.000
#>     sn2              0.167      0.013    13.26    0.000
#>     int1             0.168      0.009    18.11    0.000
#>     int2             0.199      0.010    20.41    0.000
#>     int3             0.249      0.011    23.55    0.000
#>     b1               0.231      0.023    10.12    0.000
#>     b2               0.191      0.024     8.10    0.000
#>     PBC              1.000      0.037    27.07    0.000
#>     ATT              1.000      0.037    26.93    0.000
#>     SN               1.000      0.040    25.22    0.000
#>     INT              0.634      0.026    24.64    0.000
#>     BEH              0.737      0.037    20.17    0.000