higher order interactions
higher_order_interactions.RmdAs of yet, the modsem package does not support using the
interaction operator : between two higher order constructs.
However, you can still attempt to estimate the interaction effect
between two higher order constructs by specifying the interaction term
in models using the product indicator approaches (higher order
constructs are not supported at all in the lms and
qml approaches yet). The modsem package will
then attempt to estimate the interaction effect between the two higher
order constructs.
Interaction between two higher order constructs
WARNING: Please note that the literature on higher order interactions in product indicator approaches is virtually non-existant, and you will likely need to experiment with different approaches to find one that works. As well as experiment with adding constraints to the model.
In modsem there are two datasets which are variants of
the Theory of Planned Behaviour (TPB) dataset. The
TPB_2SO contains two second order constructs,
INT (intention) which is a second order construct of
ATT (attitude) and SN (subjective norm), and
PBC (perceived behavioural control) which is a second order
construct of PC (perceived control) and PB
(perceived behaviour).
tpb <- '
# First order constructs
ATT =~ att1 + att2 + att3
SN =~ sn1 + sn2 + sn3
PB =~ pb1 + pb2 + pb3
PC =~ pc1 + pc2 + pc3
BEH =~ b1 + b2
# Higher order constructs
INT =~ ATT + SN
PBC =~ PC + PB
# Higher order interaction
INTxPBC =~ ATT:PC + ATT:PB + SN:PC + SN:PB
# Structural model
BEH ~ PBC + INT + INTxPBC
'
est_ca <- modsem(tpb, data = TPB_2SO, method = "ca")
summary(est_ca)
#> Estimating baseline model (H0)
#> Warning: Unable to estimate baseline model for models with higher-order
#> interaction terms!
#> modsem (version 1.0.11, approach = ca):
#>
#> Interaction Model Fit Measures (H1):
#> Loglikelihood -56477.03
#> Akaike (AIC) 113090.05
#> Bayesian (BIC) 113470.91
#> Chi-square 3479.48
#> Degrees of Freedom 309
#> P-value (Chi-square) 0.000
#> RMSEA 0.072
#> CFI 0.931
#> SRMR 0.076
#>
#> R-Squared Interaction Model (H1):
#> BEH 0.165
#>
#> lavaan 0.6-19 ended normally after 628 iterations
#>
#> Estimator ML
#> Optimization method NLMINB
#> Number of model parameters 96
#> Row rank of the constraints matrix 28
#>
#> Number of observations 2000
#>
#> Model Test User Model:
#>
#> Test statistic 3479.483
#> Degrees of freedom 309
#> 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|)
#> ATT =~
#> a (l_1_A) 1.000
#> a (l_2_A) 0.903 0.010 94.899 0.000
#> a (l_3_A) 0.787 0.009 89.747 0.000
#> SN =~
#> s (l_1_S) 1.000
#> s (l_2_S) 0.917 0.013 71.403 0.000
#> s (l_3_S) 0.804 0.012 67.969 0.000
#> PB =~
#> p (l_1_PB) 1.000
#> p (l_2_PB) 0.923 0.010 89.364 0.000
#> p (l_3_PB) 0.790 0.009 84.731 0.000
#> PC =~
#> p (l_1_PC) 1.000
#> p (l_2_PC) 0.889 0.009 101.651 0.000
#> p (l_3_PC) 0.787 0.008 97.811 0.000
#> BEH =~
#> b (l_1_B) 1.000
#> b (l_2_B) 0.848 0.043 19.772 0.000
#> INT =~
#> A (l_ATT_) 1.000
#> S (l_SN_) 0.646 0.076 8.547 0.000
#> PBC =~
#> P (l_PC) 1.000
#> P (l_PB) 0.650 0.081 7.985 0.000
#> INTxPBC =~
#> A (l_ATTPC) 1.000
#> A (l_ATTPB) 0.817 0.036 22.725 0.000
#> S (l_SNPC) 0.729 0.031 23.234 0.000
#> S (l_SNPB) 0.606 0.027 22.365 0.000
#> ATTPC =~
#> a (l_11_ATTPC) 1.000
#> a (l_22_ATTPC) 0.803 0.009 90.729 0.000
#> a (l_33_ATTPC) 0.620 0.007 86.179 0.000
#> ATTPB =~
#> a (l_11_ATTPB) 1.000
#> a (l_22_ATTPB) 0.834 0.010 83.614 0.000
#> a (l_33_ATTPB) 0.622 0.008 78.869 0.000
#> SNPC =~
#> s (l_11_SNPC) 1.000
#> s (l_22_SNPC) 0.815 0.011 71.600 0.000
#> s (l_33_SNPC) 0.633 0.009 68.160 0.000
#> SNPB =~
#> s (l_11_SNPB) 1.000
#> s (l_22_SNPB) 0.846 0.012 68.808 0.000
#> s (l_33_SNPB) 0.635 0.010 65.238 0.000
#>
#> Regressions:
#> Estimate Std.Err z-value P(>|z|)
#> BEH ~
#> PBC (G_PB) 0.221 0.031 7.155 0.000
#> INT (G_INT_) 0.209 0.029 7.224 0.000
#> INTPB (G_INTP) 0.158 0.019 8.137 0.000
#>
#> Covariances:
#> Estimate Std.Err z-value P(>|z|)
#> INT ~~
#> PBC (C_INT_P) 0.017 0.026 0.669 0.504
#> INTP (C_INT_I) -0.002 0.029 -0.083 0.934
#> PBC ~~
#> INTP (C_PB) -0.094 0.035 -2.712 0.007
#> .att1pc1 ~~
#> .at22 0.000
#> .at33 0.000
#> .att2pc2 ~~
#> .at33 0.000
#> .att1pb1 ~~
#> .at22 0.000
#> .at33 0.000
#> .att2pb2 ~~
#> .at33 0.000
#> .sn1pc1 ~~
#> .sn22 0.000
#> .sn33 0.000
#> .sn2pc2 ~~
#> .sn33 0.000
#> .sn1pb1 ~~
#> .sn22 0.000
#> .sn33 0.000
#> .sn2pb2 ~~
#> .sn33 0.000
#>
#> Intercepts:
#> Estimate Std.Err z-value P(>|z|)
#> .ATTP (M_ATTPC) 0.017 0.026 0.669 0.504
#> .ATTP (M_ATTPB) 0.011 0.017 0.668 0.504
#> .SNPC (M_SNPC) 0.011 0.017 0.668 0.504
#> .SNPB (M_SNPB) 0.007 0.011 0.667 0.505
#> .att1 1.008 0.025 40.614 0.000
#> .att2 1.002 0.023 43.736 0.000
#> .att3 1.012 0.021 49.282 0.000
#> .sn1 0.980 0.018 53.085 0.000
#> .sn2 0.986 0.018 56.087 0.000
#> .sn3 0.993 0.016 61.749 0.000
#> .pb1 1.010 0.024 41.515 0.000
#> .pb2 1.014 0.023 43.981 0.000
#> .pb3 1.015 0.020 50.248 0.000
#> .pc1 1.032 0.028 36.550 0.000
#> .pc2 1.023 0.026 39.909 0.000
#> .pc3 1.027 0.023 44.819 0.000
#> .b1 1.000 0.020 50.566 0.000
#> .b2 0.997 0.018 54.925 0.000
#> .at11 0.012 0.048 0.242 0.809
#> .at22 -0.016 0.039 -0.401 0.689
#> .at33 0.005 0.031 0.170 0.865
#> .at11 0.031 0.038 0.812 0.417
#> .at22 0.009 0.033 0.292 0.770
#> .at33 0.025 0.025 1.013 0.311
#> .sn11 0.021 0.034 0.605 0.545
#> .sn22 0.000 0.029 0.008 0.994
#> .sn33 0.006 0.023 0.282 0.778
#> .sn11 0.028 0.028 1.031 0.303
#> .sn22 0.008 0.024 0.344 0.731
#> .sn33 0.009 0.019 0.467 0.640
#>
#> Variances:
#> Estimate Std.Err z-value P(>|z|)
#> .AT (Vr_ATT) 0.306 0.088 3.482 0.000
#> .SN (Vr_SN) 0.190 0.037 5.088 0.000
#> .PB (Vr_PB) 0.619 0.054 11.411 0.000
#> .PC (V_PC) 0.469 0.120 3.907 0.000
#> .BE (Z_BE) 0.544 0.036 15.260 0.000
#> IN (Vr_INT) 0.752 0.091 8.252 0.000
#> PB (V_PBC) 0.958 0.123 7.760 0.000
#> IN (V_INTP) 1.297 0.089 14.646 0.000
#> .AT (V_ATTPC) 1.511 0.041 36.482 0.000
#> .AT (V_ATTPB) 1.084 0.031 35.464 0.000
#> .SN (V_SNPC) 0.719 0.022 32.194 0.000
#> .SN (V_SNPB) 0.516 0.016 31.825 0.000
#> .a1 (Vr_t1) 0.174 0.008 21.062 0.000
#> .a2 (Vr_t2) 0.186 0.007 24.851 0.000
#> .a3 (Vr_t3) 0.187 0.007 28.710 0.000
#> .s1 (Vr_s1) 0.177 0.007 24.784 0.000
#> .s2 (Vr_s2) 0.195 0.007 28.844 0.000
#> .s3 (Vr_s3) 0.192 0.006 32.240 0.000
#> .p1 (Vr_pb1) 0.161 0.009 18.864 0.000
#> .p2 (Vr_pb2) 0.191 0.008 23.432 0.000
#> .p3 (Vr_pb3) 0.178 0.007 26.465 0.000
#> .p1 (Vr_pc1) 0.167 0.009 18.483 0.000
#> .p2 (Vr_pc2) 0.185 0.008 22.968 0.000
#> .p3 (Vr_pc3) 0.165 0.007 24.405 0.000
#> .b1 (Vr_b1) 0.131 0.031 4.180 0.000
#> .b2 (Vr_b2) 0.191 0.023 8.211 0.000
#> .a1 (Vr_tt1pc1) 0.454 0.015 30.377 0.000
#> .a2 (Vr_tt2pc2) 0.404 0.011 36.058 0.000
#> .a3 (Vr_tt3pc3) 0.305 0.008 39.382 0.000
#> .a1 (Vr_tt1pb1) 0.377 0.012 30.603 0.000
#> .a2 (Vr_tt2pb2) 0.363 0.010 36.293 0.000
#> .a3 (Vr_tt3pb3) 0.270 0.007 40.454 0.000
#> .s1 (Vr_sn1pc1) 0.367 0.012 31.101 0.000
#> .s2 (Vr_sn2pc2) 0.334 0.009 36.194 0.000
#> .s3 (Vr_sn3pc3) 0.255 0.007 38.970 0.000
#> .s1 (Vr_sn1pb1) 0.291 0.009 32.171 0.000
#> .s2 (Vr_sn2pb2) 0.288 0.008 37.329 0.000
#> .s3 (Vr_sn3pb3) 0.214 0.005 40.765 0.000
#>
#> Constraints:
#> |Slack|
#> V_ATTPC-((_ATT_INT^2*V_INT+V_ATT)*(_PC_PB 0.000
#> V_11-(_1_ATT^2*(_ATT_INT^2*V_INT+V_ATT)*V 0.000
#> V_22-(_2_ATT^2*(_ATT_INT^2*V_INT+V_ATT)*V 0.000
#> V_33-(_3_ATT^2*(_ATT_INT^2*V_INT+V_ATT)*V 0.000
#> V_ATTPB-((_ATT_INT^2*V_INT+V_ATT)*(_PB_PB 0.000
#> V_11-(_1_ATT^2*(_ATT_INT^2*V_INT+V_ATT)*V 0.000
#> V_22-(_2_ATT^2*(_ATT_INT^2*V_INT+V_ATT)*V 0.000
#> V_33-(_3_ATT^2*(_ATT_INT^2*V_INT+V_ATT)*V 0.000
#> V_SNPC-((_SN_INT^2*V_INT+V_SN)*(_PC_PBC^2 0.000
#> V_11-(_1_SN^2*(_SN_INT^2*V_INT+V_SN)*V_1+ 0.000
#> V_22-(_2_SN^2*(_SN_INT^2*V_INT+V_SN)*V_2+ 0.000
#> V_33-(_3_SN^2*(_SN_INT^2*V_INT+V_SN)*V_3+ 0.000
#> V_SNPB-((_SN_INT^2*V_INT+V_SN)*(_PB_PBC^2 0.000
#> V_11-(_1_SN^2*(_SN_INT^2*V_INT+V_SN)*V_1+ 0.000
#> V_22-(_2_SN^2*(_SN_INT^2*V_INT+V_SN)*V_2+ 0.000
#> V_33-(_3_SN^2*(_SN_INT^2*V_INT+V_SN)*V_3+ 0.000
#> lmbd_tt1pc1_ATTPC-(lmbd_tt1_ATT*lmb_1_PC) 0.000
#> lmbd_tt2pc2_ATTPC-(lmbd_tt2_ATT*lmb_2_PC) 0.000
#> lmbd_tt3pc3_ATTPC-(lmbd_tt3_ATT*lmb_3_PC) 0.000
#> lmbd_tt1pb1_ATTPB-(lmbd_tt1_ATT*lmb_1_PB) 0.000
#> lmbd_tt2pb2_ATTPB-(lmbd_tt2_ATT*lmb_2_PB) 0.000
#> lmbd_tt3pb3_ATTPB-(lmbd_tt3_ATT*lmb_3_PB) 0.000
#> lmbd_sn1pc1_SNPC-(lmbd_sn1_SN*lmbd_p1_PC) 0.000
#> lmbd_sn2pc2_SNPC-(lmbd_sn2_SN*lmbd_p2_PC) 0.000
#> lmbd_sn3pc3_SNPC-(lmbd_sn3_SN*lmbd_p3_PC) 0.000
#> lmbd_sn1pb1_SNPB-(lmbd_sn1_SN*lmbd_p1_PB) 0.000
#> lmbd_sn2pb2_SNPB-(lmbd_sn2_SN*lmbd_p2_PB) 0.000
#> lmbd_sn3pb3_SNPB-(lmbd_sn3_SN*lmbd_p3_PB) 0.000
#> Mn_ATTPC-((Cv_INT_PBC*l_ATT_INT*_PC_PBC)) 0.000
#> Mn_ATTPB-((Cv_INT_PBC*l_ATT_INT*_PB_PBC)) 0.000
#> Mn_SNPC-((Cv_INT_PBC*lmb_PC_PBC*_SN_INT)) 0.000
#> Mn_SNPB-((Cv_INT_PBC*lmb_PB_PBC*_SN_INT)) 0.000
est_dblcent <- modsem(tpb, data = TPB_2SO, method = "dblcent")
summary(est_dblcent)
#> Estimating baseline model (H0)
#> Warning: Unable to estimate baseline model for models with higher-order
#> interaction terms!
#> modsem (version 1.0.11, approach = dblcent):
#>
#> Interaction Model Fit Measures (H1):
#> Loglikelihood -63633.46
#> Akaike (AIC) 127854.92
#> Bayesian (BIC) 129501.58
#> Chi-square 1995.66
#> Degrees of Freedom 981
#> P-value (Chi-square) 0.000
#> RMSEA 0.023
#> CFI 0.994
#> SRMR 0.057
#>
#> R-Squared Interaction Model (H1):
#> BEH 0.182
#>
#> lavaan 0.6-19 ended normally after 728 iterations
#>
#> Estimator ML
#> Optimization method NLMINB
#> Number of model parameters 294
#>
#> Number of observations 2000
#>
#> Model Test User Model:
#>
#> Test statistic 1995.657
#> Degrees of freedom 981
#> 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|)
#> ATT =~
#> att1 1.000
#> att2 0.908 0.011 83.766 0.000
#> att3 0.798 0.010 77.657 0.000
#> SN =~
#> sn1 1.000
#> sn2 0.909 0.016 55.250 0.000
#> sn3 0.813 0.015 53.511 0.000
#> PB =~
#> pb1 1.000
#> pb2 0.918 0.012 77.166 0.000
#> pb3 0.789 0.011 72.866 0.000
#> PC =~
#> pc1 1.000
#> pc2 0.891 0.010 89.773 0.000
#> pc3 0.792 0.009 86.846 0.000
#> BEH =~
#> b1 1.000
#> b2 0.851 0.039 21.727 0.000
#> INT =~
#> ATT 1.000
#> SN 0.670 0.061 11.022 0.000
#> PBC =~
#> PC 1.000
#> PB 0.667 0.072 9.279 0.000
#> INTxPBC =~
#> ATTPC 1.000
#> ATTPB 0.766 0.028 27.477 0.000
#> SNPC 0.712 0.025 28.039 0.000
#> SNPB 0.545 0.021 25.541 0.000
#> ATTPC =~
#> att1pc1 1.000
#> att2pc1 0.909 0.011 86.206 0.000
#> att3pc1 0.797 0.010 79.780 0.000
#> att1pc2 0.900 0.009 102.221 0.000
#> att2pc2 0.817 0.012 68.142 0.000
#> att3pc2 0.708 0.011 64.482 0.000
#> att1pc3 0.789 0.008 95.365 0.000
#> att2pc3 0.716 0.011 66.513 0.000
#> att3pc3 0.622 0.010 62.786 0.000
#> ATTPB =~
#> att1pb1 1.000
#> att2pb1 0.912 0.011 80.962 0.000
#> att3pb1 0.802 0.011 75.469 0.000
#> att1pb2 0.915 0.011 82.414 0.000
#> att2pb2 0.836 0.014 58.155 0.000
#> att3pb2 0.734 0.013 56.652 0.000
#> att1pb3 0.780 0.010 75.828 0.000
#> att2pb3 0.719 0.013 57.371 0.000
#> att3pb3 0.628 0.011 55.238 0.000
#> SNPC =~
#> sn1pc1 1.000
#> sn2pc1 0.908 0.016 57.477 0.000
#> sn3pc1 0.777 0.015 53.574 0.000
#> sn1pc2 0.902 0.010 93.729 0.000
#> sn2pc2 0.820 0.016 50.031 0.000
#> sn3pc2 0.697 0.015 47.864 0.000
#> sn1pc3 0.802 0.009 87.523 0.000
#> sn2pc3 0.727 0.015 49.543 0.000
#> sn3pc3 0.626 0.013 47.318 0.000
#> SNPB =~
#> sn1pb1 1.000
#> sn2pb1 0.923 0.016 56.507 0.000
#> sn3pb1 0.807 0.015 54.229 0.000
#> sn1pb2 0.926 0.012 78.526 0.000
#> sn2pb2 0.858 0.018 47.984 0.000
#> sn3pb2 0.748 0.016 46.473 0.000
#> sn1pb3 0.792 0.011 73.076 0.000
#> sn2pb3 0.728 0.016 46.470 0.000
#> sn3pb3 0.644 0.015 44.319 0.000
#>
#> Regressions:
#> Estimate Std.Err z-value P(>|z|)
#> BEH ~
#> PBC 0.211 0.026 8.108 0.000
#> INT 0.199 0.023 8.683 0.000
#> INTxPBC 0.142 0.017 8.209 0.000
#>
#> Covariances:
#> Estimate Std.Err z-value P(>|z|)
#> .att1pb1 ~~
#> .att1pb2 0.182 0.011 16.731 0.000
#> .att1pb3 0.156 0.009 16.541 0.000
#> .att1pc1 ~~
#> .att1pb1 0.137 0.011 12.232 0.000
#> .att1pc2 ~~
#> .att1pb1 0.116 0.010 11.632 0.000
#> .att1pc3 ~~
#> .att1pb1 0.102 0.009 11.380 0.000
#> .att1pb1 ~~
#> .att2pb1 0.206 0.012 17.027 0.000
#> .att3pb1 0.172 0.010 16.490 0.000
#> .sn1pb1 0.123 0.008 14.753 0.000
#> .sn2pb1 0.112 0.008 14.627 0.000
#> .sn3pb1 0.095 0.007 13.648 0.000
#> .att1pb2 ~~
#> .att1pb3 0.146 0.009 16.560 0.000
#> .att1pc1 ~~
#> .att1pb2 0.128 0.010 12.231 0.000
#> .att1pc2 ~~
#> .att1pb2 0.109 0.009 11.716 0.000
#> .att1pc3 ~~
#> .att1pb2 0.097 0.008 11.578 0.000
#> .att1pb2 ~~
#> .att2pb2 0.237 0.012 20.406 0.000
#> .att3pb2 0.202 0.010 20.075 0.000
#> .sn1pb2 0.117 0.008 15.317 0.000
#> .sn2pb2 0.111 0.007 15.625 0.000
#> .sn3pb2 0.100 0.006 15.476 0.000
#> .att1pc1 ~~
#> .att1pb3 0.102 0.009 11.259 0.000
#> .att1pc2 ~~
#> .att1pb3 0.088 0.008 10.862 0.000
#> .att1pc3 ~~
#> .att1pb3 0.077 0.007 10.598 0.000
#> .att1pb3 ~~
#> .att2pb3 0.213 0.010 22.209 0.000
#> .att3pb3 0.185 0.008 22.018 0.000
#> .sn1pb3 0.116 0.007 17.433 0.000
#> .sn2pb3 0.105 0.006 17.442 0.000
#> .sn3pb3 0.095 0.006 16.834 0.000
#> .att1pc1 ~~
#> .att1pc2 0.234 0.014 16.335 0.000
#> .att1pc3 0.209 0.013 16.392 0.000
#> .att2pc1 0.194 0.012 16.525 0.000
#> .att3pc1 0.172 0.010 16.401 0.000
#> .sn1pc1 0.109 0.008 13.102 0.000
#> .sn2pc1 0.100 0.008 12.856 0.000
#> .sn3pc1 0.091 0.007 13.046 0.000
#> .att1pc2 ~~
#> .att1pc3 0.188 0.011 16.450 0.000
#> .att2pc2 0.207 0.011 19.530 0.000
#> .att3pc2 0.182 0.009 19.397 0.000
#> .sn1pc2 0.126 0.008 16.654 0.000
#> .sn2pc2 0.122 0.007 16.977 0.000
#> .sn3pc2 0.099 0.006 15.802 0.000
#> .att1pc3 ~~
#> .att2pc3 0.199 0.009 21.852 0.000
#> .att3pc3 0.170 0.008 21.281 0.000
#> .sn1pc3 0.108 0.006 16.676 0.000
#> .sn2pc3 0.108 0.006 17.541 0.000
#> .sn3pc3 0.088 0.005 16.268 0.000
#> .att2pb1 ~~
#> .att2pb2 0.199 0.010 19.689 0.000
#> .att2pb3 0.168 0.009 19.330 0.000
#> .att2pc1 ~~
#> .att2pb1 0.159 0.011 15.145 0.000
#> .att2pc2 ~~
#> .att2pb1 0.139 0.009 14.889 0.000
#> .att2pc3 ~~
#> .att2pb1 0.130 0.008 15.491 0.000
#> .att2pb1 ~~
#> .att3pb1 0.161 0.010 16.545 0.000
#> .sn1pb1 0.120 0.008 15.431 0.000
#> .sn2pb1 0.107 0.007 15.020 0.000
#> .sn3pb1 0.094 0.007 14.406 0.000
#> .att2pb2 ~~
#> .att2pb3 0.157 0.008 19.192 0.000
#> .att2pc1 ~~
#> .att2pb2 0.145 0.010 14.691 0.000
#> .att2pc2 ~~
#> .att2pb2 0.126 0.009 14.314 0.000
#> .att2pc3 ~~
#> .att2pb2 0.116 0.008 14.779 0.000
#> .att2pb2 ~~
#> .att3pb2 0.196 0.010 20.486 0.000
#> .sn1pb2 0.113 0.007 15.494 0.000
#> .sn2pb2 0.104 0.007 15.500 0.000
#> .sn3pb2 0.091 0.006 14.944 0.000
#> .att2pc1 ~~
#> .att2pb3 0.122 0.008 14.355 0.000
#> .att2pc2 ~~
#> .att2pb3 0.104 0.008 13.775 0.000
#> .att2pc3 ~~
#> .att2pb3 0.098 0.007 14.458 0.000
#> .att2pb3 ~~
#> .att3pb3 0.165 0.008 21.586 0.000
#> .sn1pb3 0.101 0.006 16.782 0.000
#> .sn2pb3 0.090 0.005 16.458 0.000
#> .sn3pb3 0.083 0.005 16.282 0.000
#> .att2pc1 ~~
#> .att2pc2 0.262 0.013 19.764 0.000
#> .att2pc3 0.229 0.012 19.530 0.000
#> .att3pc1 0.158 0.010 16.408 0.000
#> .sn1pc1 0.102 0.008 13.253 0.000
#> .sn2pc1 0.088 0.007 12.537 0.000
#> .sn3pc1 0.081 0.006 12.743 0.000
#> .att2pc2 ~~
#> .att2pc3 0.201 0.010 19.307 0.000
#> .att3pc2 0.167 0.009 19.368 0.000
#> .sn1pc2 0.115 0.007 16.518 0.000
#> .sn2pc2 0.102 0.006 15.679 0.000
#> .sn3pc2 0.087 0.006 15.233 0.000
#> .att2pc3 ~~
#> .att3pc3 0.155 0.007 21.155 0.000
#> .sn1pc3 0.100 0.006 16.769 0.000
#> .sn2pc3 0.097 0.006 17.395 0.000
#> .sn3pc3 0.083 0.005 16.697 0.000
#> .att3pb1 ~~
#> .att3pb2 0.187 0.009 21.563 0.000
#> .att3pb3 0.160 0.008 21.286 0.000
#> .att3pc1 ~~
#> .att3pb1 0.145 0.009 15.712 0.000
#> .att3pc2 ~~
#> .att3pb1 0.122 0.008 15.099 0.000
#> .att3pc3 ~~
#> .att3pb1 0.114 0.007 15.560 0.000
#> .att3pb1 ~~
#> .sn1pb1 0.097 0.007 14.465 0.000
#> .sn2pb1 0.091 0.006 14.645 0.000
#> .sn3pb1 0.077 0.006 13.702 0.000
#> .att3pb2 ~~
#> .att3pb3 0.147 0.007 20.982 0.000
#> .att3pc1 ~~
#> .att3pb2 0.136 0.009 15.674 0.000
#> .att3pc2 ~~
#> .att3pb2 0.114 0.008 15.041 0.000
#> .att3pc3 ~~
#> .att3pb2 0.107 0.007 15.489 0.000
#> .att3pb2 ~~
#> .sn1pb2 0.095 0.006 15.008 0.000
#> .sn2pb2 0.089 0.006 15.239 0.000
#> .sn3pb2 0.080 0.005 15.087 0.000
#> .att3pc1 ~~
#> .att3pb3 0.115 0.008 15.300 0.000
#> .att3pc2 ~~
#> .att3pb3 0.098 0.007 14.827 0.000
#> .att3pc3 ~~
#> .att3pb3 0.093 0.006 15.443 0.000
#> .att3pb3 ~~
#> .sn1pb3 0.091 0.005 17.072 0.000
#> .sn2pb3 0.080 0.005 16.530 0.000
#> .sn3pb3 0.071 0.004 15.757 0.000
#> .att3pc1 ~~
#> .att3pc2 0.264 0.012 22.544 0.000
#> .att3pc3 0.236 0.010 22.462 0.000
#> .sn1pc1 0.087 0.007 12.683 0.000
#> .sn2pc1 0.076 0.006 12.061 0.000
#> .sn3pc1 0.069 0.006 12.168 0.000
#> .att3pc2 ~~
#> .att3pc3 0.205 0.009 22.185 0.000
#> .sn1pc2 0.101 0.006 16.349 0.000
#> .sn2pc2 0.094 0.006 16.178 0.000
#> .sn3pc2 0.079 0.005 15.459 0.000
#> .att3pc3 ~~
#> .sn1pc3 0.085 0.005 16.303 0.000
#> .sn2pc3 0.088 0.005 17.566 0.000
#> .sn3pc3 0.071 0.004 16.185 0.000
#> .sn1pb1 ~~
#> .sn1pb2 0.164 0.010 16.748 0.000
#> .sn1pb3 0.144 0.009 16.617 0.000
#> .sn1pc1 ~~
#> .sn1pb1 0.127 0.010 12.549 0.000
#> .sn1pc2 ~~
#> .sn1pb1 0.108 0.009 12.193 0.000
#> .sn1pc3 ~~
#> .sn1pb1 0.103 0.008 12.789 0.000
#> .sn1pb1 ~~
#> .sn2pb1 0.110 0.006 17.192 0.000
#> .sn3pb1 0.098 0.006 16.705 0.000
#> .sn1pb2 ~~
#> .sn1pb3 0.131 0.008 16.248 0.000
#> .sn1pc1 ~~
#> .sn1pb2 0.113 0.009 12.067 0.000
#> .sn1pc2 ~~
#> .sn1pb2 0.097 0.008 11.800 0.000
#> .sn1pc3 ~~
#> .sn1pb2 0.088 0.007 11.883 0.000
#> .sn1pb2 ~~
#> .sn2pb2 0.105 0.006 18.013 0.000
#> .sn3pb2 0.094 0.005 17.797 0.000
#> .sn1pc1 ~~
#> .sn1pb3 0.100 0.008 12.016 0.000
#> .sn1pc2 ~~
#> .sn1pb3 0.085 0.007 11.641 0.000
#> .sn1pc3 ~~
#> .sn1pb3 0.079 0.007 11.861 0.000
#> .sn1pb3 ~~
#> .sn2pb3 0.099 0.005 20.218 0.000
#> .sn3pb3 0.094 0.005 20.431 0.000
#> .sn1pc1 ~~
#> .sn1pc2 0.231 0.014 17.031 0.000
#> .sn1pc3 0.209 0.012 17.151 0.000
#> .sn2pc1 0.107 0.007 16.393 0.000
#> .sn3pc1 0.096 0.006 16.303 0.000
#> .sn1pc2 ~~
#> .sn1pc3 0.177 0.011 16.502 0.000
#> .sn2pc2 0.107 0.006 18.253 0.000
#> .sn3pc2 0.092 0.005 17.886 0.000
#> .sn1pc3 ~~
#> .sn2pc3 0.103 0.005 20.384 0.000
#> .sn3pc3 0.088 0.004 19.682 0.000
#> .sn2pb1 ~~
#> .sn2pb2 0.195 0.010 20.268 0.000
#> .sn2pb3 0.170 0.008 20.088 0.000
#> .sn2pc1 ~~
#> .sn2pb1 0.142 0.010 14.546 0.000
#> .sn2pc2 ~~
#> .sn2pb1 0.114 0.009 13.131 0.000
#> .sn2pc3 ~~
#> .sn2pb1 0.106 0.008 13.754 0.000
#> .sn2pb1 ~~
#> .sn3pb1 0.087 0.005 16.351 0.000
#> .sn2pb2 ~~
#> .sn2pb3 0.158 0.008 19.973 0.000
#> .sn2pc1 ~~
#> .sn2pb2 0.129 0.009 14.162 0.000
#> .sn2pc2 ~~
#> .sn2pb2 0.106 0.008 13.083 0.000
#> .sn2pc3 ~~
#> .sn2pb2 0.095 0.007 13.193 0.000
#> .sn2pb2 ~~
#> .sn3pb2 0.086 0.005 17.769 0.000
#> .sn2pc1 ~~
#> .sn2pb3 0.113 0.008 13.963 0.000
#> .sn2pc2 ~~
#> .sn2pb3 0.090 0.007 12.596 0.000
#> .sn2pc3 ~~
#> .sn2pb3 0.084 0.006 13.121 0.000
#> .sn2pb3 ~~
#> .sn3pb3 0.080 0.004 19.472 0.000
#> .sn2pc1 ~~
#> .sn2pc2 0.265 0.013 20.538 0.000
#> .sn2pc3 0.232 0.011 20.416 0.000
#> .sn3pc1 0.087 0.005 15.973 0.000
#> .sn2pc2 ~~
#> .sn2pc3 0.209 0.010 20.439 0.000
#> .sn3pc2 0.087 0.005 18.067 0.000
#> .sn2pc3 ~~
#> .sn3pc3 0.083 0.004 19.756 0.000
#> .sn3pb1 ~~
#> .sn3pb2 0.175 0.008 21.641 0.000
#> .sn3pb3 0.155 0.007 21.405 0.000
#> .sn3pc1 ~~
#> .sn3pb1 0.132 0.008 15.515 0.000
#> .sn3pc2 ~~
#> .sn3pb1 0.110 0.007 14.758 0.000
#> .sn3pc3 ~~
#> .sn3pb1 0.097 0.007 14.554 0.000
#> .sn3pb2 ~~
#> .sn3pb3 0.147 0.007 21.635 0.000
#> .sn3pc1 ~~
#> .sn3pb2 0.125 0.008 15.707 0.000
#> .sn3pc2 ~~
#> .sn3pb2 0.104 0.007 14.950 0.000
#> .sn3pc3 ~~
#> .sn3pb2 0.095 0.006 15.012 0.000
#> .sn3pc1 ~~
#> .sn3pb3 0.107 0.007 14.895 0.000
#> .sn3pc2 ~~
#> .sn3pb3 0.087 0.006 13.889 0.000
#> .sn3pc3 ~~
#> .sn3pb3 0.079 0.006 14.020 0.000
#> .sn3pc1 ~~
#> .sn3pc2 0.255 0.011 23.048 0.000
#> .sn3pc3 0.226 0.010 22.847 0.000
#> .sn3pc2 ~~
#> .sn3pc3 0.198 0.009 22.663 0.000
#> INT ~~
#> PBC 0.034 0.033 1.042 0.297
#> INTxPBC -0.002 0.035 -0.044 0.965
#> PBC ~~
#> INTxPBC -0.121 0.039 -3.071 0.002
#>
#> Variances:
#> Estimate Std.Err z-value P(>|z|)
#> .att1 0.164 0.009 18.496 0.000
#> .att2 0.150 0.008 19.739 0.000
#> .att3 0.161 0.007 23.292 0.000
#> .sn1 0.159 0.008 18.691 0.000
#> .sn2 0.172 0.008 21.687 0.000
#> .sn3 0.160 0.007 23.161 0.000
#> .pb1 0.146 0.009 16.805 0.000
#> .pb2 0.168 0.008 20.515 0.000
#> .pb3 0.159 0.007 23.189 0.000
#> .pc1 0.165 0.009 17.846 0.000
#> .pc2 0.165 0.008 20.574 0.000
#> .pc3 0.150 0.007 22.166 0.000
#> .b1 0.134 0.029 4.615 0.000
#> .b2 0.189 0.022 8.745 0.000
#> .att1pc1 0.505 0.021 23.530 0.000
#> .att2pc1 0.501 0.019 26.073 0.000
#> .att3pc1 0.466 0.017 28.157 0.000
#> .att1pc2 0.467 0.018 25.638 0.000
#> .att2pc2 0.446 0.016 27.728 0.000
#> .att3pc2 0.405 0.014 29.760 0.000
#> .att1pc3 0.407 0.015 27.249 0.000
#> .att2pc3 0.381 0.013 29.070 0.000
#> .att3pc3 0.345 0.011 30.813 0.000
#> .att1pb1 0.442 0.018 24.069 0.000
#> .att2pb1 0.428 0.016 26.217 0.000
#> .att3pb1 0.361 0.013 27.249 0.000
#> .att1pb2 0.440 0.017 26.371 0.000
#> .att2pb2 0.442 0.015 28.739 0.000
#> .att3pb2 0.367 0.012 29.701 0.000
#> .att1pb3 0.392 0.014 28.365 0.000
#> .att2pb3 0.345 0.012 29.208 0.000
#> .att3pb3 0.297 0.010 30.558 0.000
#> .sn1pc1 0.415 0.018 23.328 0.000
#> .sn2pc1 0.414 0.016 25.588 0.000
#> .sn3pc1 0.394 0.014 28.006 0.000
#> .sn1pc2 0.337 0.014 23.754 0.000
#> .sn2pc2 0.366 0.014 27.059 0.000
#> .sn3pc2 0.322 0.011 28.684 0.000
#> .sn1pc3 0.296 0.012 25.145 0.000
#> .sn2pc3 0.304 0.011 28.045 0.000
#> .sn3pc3 0.271 0.009 29.384 0.000
#> .sn1pb1 0.328 0.013 24.321 0.000
#> .sn2pb1 0.334 0.013 26.575 0.000
#> .sn3pb1 0.293 0.011 27.592 0.000
#> .sn1pb2 0.291 0.012 24.573 0.000
#> .sn2pb2 0.305 0.011 27.183 0.000
#> .sn3pb2 0.268 0.009 28.458 0.000
#> .sn1pb3 0.256 0.010 26.468 0.000
#> .sn2pb3 0.251 0.009 28.278 0.000
#> .sn3pb3 0.233 0.008 29.718 0.000
#> .ATT 0.342 0.094 3.634 0.000
#> .SN 0.229 0.043 5.316 0.000
#> .PB 0.689 0.061 11.210 0.000
#> .PC 0.500 0.127 3.928 0.000
#> .BEH 0.544 0.034 16.216 0.000
#> INT 1.058 0.104 10.177 0.000
#> PBC 1.204 0.137 8.768 0.000
#> INTxPBC 1.492 0.086 17.408 0.000
#> .ATTPC 0.869 0.050 17.215 0.000
#> .ATTPB 0.697 0.037 18.991 0.000
#> .SNPC 0.447 0.027 16.275 0.000
#> .SNPB 0.390 0.021 18.568 0.000Interaction between a first order and a higher order construct
In the TPB_1SO dataset, the INT construct
is a second order construct of ATT, SN and
PBC. In this example, we will estimate the interaction
between the INT (higher order construct) and
PBC (first order construct).
tpb <- '
# First order constructs
ATT =~ att1 + att2 + att3
SN =~ sn1 + sn2 + sn3
PBC =~ pbc1 + pbc2 + pbc3
BEH =~ b1 + b2
# Higher order constructs
INT =~ ATT + PBC + SN
# Higher order interaction
INTxPBC =~ ATT:PBC + SN:PBC + PBC:PBC
# Structural model
BEH ~ PBC + INT + INTxPBC
'
est_ca <- modsem(tpb, data = TPB_1SO, method = "ca")
summary(est_ca)
#> Estimating baseline model (H0)
#> Warning: Unable to estimate baseline model for models with higher-order
#> interaction terms!
#> modsem (version 1.0.11, approach = ca):
#>
#> Interaction Model Fit Measures (H1):
#> Loglikelihood -42117.62
#> Akaike (AIC) 84339.24
#> Bayesian (BIC) 84630.48
#> Chi-square 4246.90
#> Degrees of Freedom 178
#> P-value (Chi-square) 0.000
#> RMSEA 0.107
#> CFI 0.890
#> SRMR 0.135
#>
#> R-Squared Interaction Model (H1):
#> BEH 0.232
#>
#> lavaan 0.6-19 ended normally after 446 iterations
#>
#> Estimator ML
#> Optimization method NLMINB
#> Number of model parameters 73
#> Row rank of the constraints matrix 21
#>
#> Number of observations 2000
#>
#> Model Test User Model:
#>
#> Test statistic 4246.901
#> Degrees of freedom 178
#> 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|)
#> ATT =~
#> att1 (l_1_A) 1.000
#> att2 (l_2_A) 0.904 0.010 89.441 0.000
#> att3 (l_3_A) 0.801 0.009 85.442 0.000
#> SN =~
#> sn1 (l_1_S) 1.000
#> sn2 (l_2_S) 0.879 0.013 66.744 0.000
#> sn3 (l_3_S) 0.780 0.012 63.639 0.000
#> PBC =~
#> pbc1 (l_1_P) 1.000
#> pbc2 (l_2_P) 0.900 0.007 135.630 0.000
#> pbc3 (l_3_P) 0.776 0.006 125.111 0.000
#> BEH =~
#> b1 (l_1_B) 1.000
#> b2 (l_2_B) 0.863 0.033 26.043 0.000
#> INT =~
#> ATT (l_ATT_) 1.000
#> PBC (l_PBC_) 0.783 0.030 26.191 0.000
#> SN (l_SN_) 0.717 0.027 26.257 0.000
#> INTxPBC =~
#> ATTPB (l_ATTP) 1.000
#> SNPBC (l_SNP) 0.735 0.020 35.914 0.000
#> PBCPB (l_PBCP) 1.011 0.027 36.926 0.000
#> ATTPBC =~
#> att11 (l_11_A) 1.000
#> att22 (l_22_A) 0.813 0.009 87.006 0.000
#> att33 (l_33_A) 0.621 0.008 82.373 0.000
#> SNPBC =~
#> sn1p1 (l_11_S) 1.000
#> sn2p2 (l_22_S) 0.792 0.012 68.052 0.000
#> sn3p3 (l_33_S) 0.605 0.009 64.723 0.000
#> PBCPBC =~
#> pbc11 (l_11_P) 1.000
#> pbc22 (l_22_P) 0.810 0.012 67.815 0.000
#> pbc33 (l_33_P) 0.602 0.010 62.555 0.000
#>
#> Regressions:
#> Estimate Std.Err z-value P(>|z|)
#> BEH ~
#> PBC (G_PB) 0.249 0.052 4.775 0.000
#> INT (G_INT_) 0.160 0.056 2.838 0.005
#> INTPB (G_INTP) 0.221 0.016 13.492 0.000
#>
#> Covariances:
#> Estimate Std.Err z-value P(>|z|)
#> INT ~~
#> INTxPBC (C_IN) -0.019 0.025 -0.758 0.448
#> .att1pbc1 ~~
#> .att2pb2 0.000
#> .att3pb3 0.000
#> .att2pbc2 ~~
#> .att3pb3 0.000
#> .sn1pbc1 ~~
#> .sn2pbc2 0.000
#> .sn3pbc3 0.000
#> .sn2pbc2 ~~
#> .sn3pbc3 0.000
#> .pbc1pbc1 ~~
#> .pbc2pb2 0.000
#> .pbc3pb3 0.000
#> .pbc2pbc2 ~~
#> .pbc3pb3 0.000
#>
#> Intercepts:
#> Estimate Std.Err z-value P(>|z|)
#> .ATTPBC (M_AT) 0.422 0.013 32.832 0.000
#> .SNPBC (M_SN) 0.302 0.010 30.547 0.000
#> .PBCPBC (M_PB) 0.575 0.011 51.528 0.000
#> .att1 1.001 0.023 44.025 0.000
#> .att2 1.008 0.021 47.861 0.000
#> .att3 1.002 0.019 52.974 0.000
#> .sn1 0.974 0.018 55.116 0.000
#> .sn2 0.982 0.016 60.802 0.000
#> .sn3 0.991 0.015 67.883 0.000
#> .pbc1 0.983 0.021 47.193 0.000
#> .pbc2 0.988 0.020 49.207 0.000
#> .pbc3 0.998 0.018 54.375 0.000
#> .b1 1.150 0.020 57.082 0.000
#> .b2 1.132 0.018 61.428 0.000
#> .att1pb1 0.391 0.038 10.186 0.000
#> .att2pb2 0.330 0.032 10.249 0.000
#> .att3pb3 0.256 0.025 10.137 0.000
#> .sn1pbc1 0.262 0.029 9.112 0.000
#> .sn2pbc2 0.226 0.024 9.516 0.000
#> .sn3pbc3 0.177 0.019 9.454 0.000
#> .pbc1pb1 0.553 0.038 14.525 0.000
#> .pbc2pb2 0.501 0.032 15.560 0.000
#> .pbc3pb3 0.421 0.025 16.843 0.000
#>
#> Variances:
#> Estimate Std.Err z-value P(>|z|)
#> .ATT (Vr_ATT) 0.346 0.023 15.014 0.000
#> .SN (Vr_SN) 0.179 0.013 13.986 0.000
#> .PBC (Vr_PBC) 0.245 0.016 15.052 0.000
#> .BEH (Z_BE) 0.513 0.029 18.005 0.000
#> INT (Vr_INT) 0.539 0.027 19.889 0.000
#> INTPB (V_INTP) 1.535 0.076 20.148 0.000
#> .ATTPB (V_ATTP) 0.687 0.022 30.946 0.000
#> .SNPBC (V_SNP) 0.354 0.013 27.980 0.000
#> .PBCPB (V_PBCP) 0.662 0.026 25.764 0.000
#> .att1 (Vr_t1) 0.150 0.008 18.624 0.000
#> .att2 (Vr_t2) 0.165 0.007 22.170 0.000
#> .att3 (Vr_t3) 0.147 0.006 23.582 0.000
#> .sn1 (Vr_s1) 0.168 0.008 20.991 0.000
#> .sn2 (Vr_s2) 0.168 0.007 24.105 0.000
#> .sn3 (Vr_s3) 0.149 0.006 25.303 0.000
#> .pbc1 (Vr_p1) 0.293 0.009 30.965 0.000
#> .pbc2 (Vr_p2) 0.340 0.009 38.979 0.000
#> .pbc3 (Vr_p3) 0.327 0.007 44.262 0.000
#> .b1 (Vr_b1) 0.144 0.024 6.051 0.000
#> .b2 (Vr_b2) 0.181 0.018 9.880 0.000
#> .att11 (Vr_t11) 0.389 0.011 34.052 0.000
#> .att22 (Vr_t22) 0.378 0.010 39.469 0.000
#> .att33 (Vr_t33) 0.285 0.007 41.921 0.000
#> .sn1p1 (Vr_s11) 0.279 0.008 35.023 0.000
#> .sn2p2 (Vr_s22) 0.256 0.006 39.790 0.000
#> .sn3p3 (Vr_s33) 0.191 0.005 41.982 0.000
#> .pbc11 (Vr_p11) 0.423 0.015 28.080 0.000
#> .pbc22 (Vr_p22) 0.432 0.013 33.193 0.000
#> .pbc33 (Vr_p33) 0.334 0.009 35.936 0.000
#>
#> Constraints:
#> |Slack|
#> V_ATTPBC-((_ATT_INT^2*V_INT+V_ATT)*(_PBC_ 0.000
#> V_11-(_1_ATT^2*(_ATT_INT^2*V_INT+V_ATT)*V 0.000
#> V_22-(_2_ATT^2*(_ATT_INT^2*V_INT+V_ATT)*V 0.000
#> V_33-(_3_ATT^2*(_ATT_INT^2*V_INT+V_ATT)*V 0.000
#> V_SNPBC-((_SN_INT^2*V_INT+V_SN)*(_PBC_INT 0.000
#> V_11-(_1_SN^2*(_SN_INT^2*V_INT+V_SN)*V_1+ 0.000
#> V_22-(_2_SN^2*(_SN_INT^2*V_INT+V_SN)*V_2+ 0.000
#> V_33-(_3_SN^2*(_SN_INT^2*V_INT+V_SN)*V_3+ 0.000
#> V_PBCPBC-((_PBC_INT^2*V_INT+V_PBC)*(_PBC_ 0.000
#> V_11-(_1_PBC^2*(_PBC_INT^2*V_INT+V_PBC)*V 0.000
#> V_22-(_2_PBC^2*(_PBC_INT^2*V_INT+V_PBC)*V 0.000
#> V_33-(_3_PBC^2*(_PBC_INT^2*V_INT+V_PBC)*V 0.000
#> lmbd_tt1pbc1_ATTPBC-(lmbd_tt1_ATT*_1_PBC) 0.000
#> lmbd_tt2pbc2_ATTPBC-(lmbd_tt2_ATT*_2_PBC) 0.000
#> lmbd_tt3pbc3_ATTPBC-(lmbd_tt3_ATT*_3_PBC) 0.000
#> lmbd_sn1pbc1_SNPBC-(lmbd_sn1_SN*lm_1_PBC) 0.000
#> lmbd_sn2pbc2_SNPBC-(lmbd_sn2_SN*lm_2_PBC) 0.000
#> lmbd_sn3pbc3_SNPBC-(lmbd_sn3_SN*lm_3_PBC) 0.000
#> lmbd_pbc1pbc1_PBCPBC-(lmbd_p1_PBC*_1_PBC) 0.000
#> lmbd_pbc2pbc2_PBCPBC-(lmbd_p2_PBC*_2_PBC) 0.000
#> lmbd_pbc3pbc3_PBCPBC-(lmbd_p3_PBC*_3_PBC) 0.000
#> Mn_ATTPBC-((lmbd_ATT_INT*_PBC_INT*V_INT)) 0.000
#> Mn_SNPBC-((lmbd_PBC_INT*lm_SN_INT*V_INT)) 0.000
#> Mn_PBCPBC-((lmbd_PBC_INT^2*Vr_INT+V_PBC)) 0.000
est_dblcent <- modsem(tpb, data = TPB_1SO, method = "dblcent")
summary(est_dblcent)
#> Estimating baseline model (H0)
#> Warning: Unable to estimate baseline model for models with higher-order
#> interaction terms!
#> modsem (version 1.0.11, approach = dblcent):
#>
#> Interaction Model Fit Measures (H1):
#> Loglikelihood -42695.54
#> Akaike (AIC) 85803.08
#> Bayesian (BIC) 86956.87
#> Chi-square 565.72
#> Degrees of Freedom 424
#> P-value (Chi-square) 0.000
#> RMSEA 0.013
#> CFI 0.999
#> SRMR 0.018
#>
#> R-Squared Interaction Model (H1):
#> BEH 0.276
#>
#> lavaan 0.6-19 ended normally after 583 iterations
#>
#> Estimator ML
#> Optimization method NLMINB
#> Number of model parameters 206
#>
#> Number of observations 2000
#>
#> Model Test User Model:
#>
#> Test statistic 565.724
#> Degrees of freedom 424
#> 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|)
#> ATT =~
#> att1 1.000
#> att2 0.909 0.011 82.768 0.000
#> att3 0.802 0.010 80.425 0.000
#> SN =~
#> sn1 1.000
#> sn2 0.891 0.016 54.218 0.000
#> sn3 0.790 0.015 52.448 0.000
#> PBC =~
#> pbc1 1.000
#> pbc2 0.909 0.014 66.750 0.000
#> pbc3 0.793 0.013 63.370 0.000
#> BEH =~
#> b1 1.000
#> b2 0.865 0.029 30.297 0.000
#> INT =~
#> ATT 1.000
#> PBC 0.819 0.024 33.976 0.000
#> SN 0.702 0.021 33.484 0.000
#> INTxPBC =~
#> ATTPBC 1.000
#> SNPBC 0.717 0.017 41.167 0.000
#> PBCPBC 0.971 0.022 43.199 0.000
#> ATTPBC =~
#> att1pbc1 1.000
#> att2pbc1 0.897 0.010 92.034 0.000
#> att3pbc1 0.807 0.009 90.390 0.000
#> att1pbc2 0.908 0.010 87.991 0.000
#> att2pbc2 0.813 0.012 65.243 0.000
#> att3pbc2 0.733 0.011 65.091 0.000
#> att1pbc3 0.774 0.010 81.181 0.000
#> att2pbc3 0.694 0.011 60.977 0.000
#> att3pbc3 0.630 0.010 61.880 0.000
#> SNPBC =~
#> sn1pbc1 1.000
#> sn2pbc1 0.884 0.014 63.726 0.000
#> sn3pbc1 0.782 0.013 61.274 0.000
#> sn1pbc2 0.907 0.011 84.140 0.000
#> sn2pbc2 0.802 0.015 52.856 0.000
#> sn3pbc2 0.702 0.014 51.227 0.000
#> sn1pbc3 0.787 0.010 79.471 0.000
#> sn2pbc3 0.686 0.014 50.281 0.000
#> sn3pbc3 0.613 0.012 50.032 0.000
#> PBCPBC =~
#> pbc1pbc1 1.000
#> pbc2pbc1 0.909 0.010 89.205 0.000
#> pbc3pbc1 0.782 0.009 85.688 0.000
#> pbc2pbc2 0.827 0.017 47.996 0.000
#> pbc3pbc2 0.710 0.013 54.545 0.000
#> pbc3pbc3 0.616 0.014 45.151 0.000
#>
#> Regressions:
#> Estimate Std.Err z-value P(>|z|)
#> BEH ~
#> PBC 0.188 0.043 4.337 0.000
#> INT 0.188 0.045 4.173 0.000
#> INTxPBC 0.217 0.016 13.789 0.000
#>
#> Covariances:
#> Estimate Std.Err z-value P(>|z|)
#> .att1pbc1 ~~
#> .att1pbc2 0.122 0.008 15.676 0.000
#> .att1pbc3 0.108 0.007 15.545 0.000
#> .att2pbc1 0.201 0.013 16.047 0.000
#> .att3pbc1 0.178 0.011 16.033 0.000
#> .pbc1pbc1 0.256 0.018 13.950 0.000
#> .pbc2pbc1 0.118 0.009 12.940 0.000
#> .pbc3pbc1 0.098 0.008 12.237 0.000
#> .sn1pbc1 0.116 0.009 13.217 0.000
#> .sn2pbc1 0.100 0.008 12.831 0.000
#> .sn3pbc1 0.097 0.007 13.331 0.000
#> .att1pbc2 ~~
#> .att1pbc3 0.099 0.006 15.365 0.000
#> .att2pbc2 0.205 0.011 18.285 0.000
#> .att3pbc2 0.189 0.010 18.718 0.000
#> .pbc2pbc1 0.136 0.009 15.063 0.000
#> .pbc2pbc2 0.241 0.015 15.591 0.000
#> .pbc3pbc2 0.106 0.007 14.278 0.000
#> .sn1pbc2 0.123 0.008 15.415 0.000
#> .sn2pbc2 0.098 0.007 13.955 0.000
#> .sn3pbc2 0.098 0.007 15.102 0.000
#> .att1pbc3 ~~
#> .att2pbc3 0.232 0.010 22.926 0.000
#> .att3pbc3 0.199 0.009 22.457 0.000
#> .pbc3pbc1 0.144 0.008 18.050 0.000
#> .pbc3pbc2 0.130 0.007 17.741 0.000
#> .pbc3pbc3 0.232 0.012 18.949 0.000
#> .sn1pbc3 0.125 0.007 17.887 0.000
#> .sn2pbc3 0.107 0.006 17.035 0.000
#> .sn3pbc3 0.103 0.006 17.869 0.000
#> .att2pbc1 ~~
#> .att2pbc2 0.156 0.008 20.447 0.000
#> .att2pbc3 0.144 0.007 20.765 0.000
#> .att3pbc1 0.163 0.010 16.025 0.000
#> .pbc1pbc1 0.238 0.017 14.066 0.000
#> .pbc2pbc1 0.105 0.008 12.572 0.000
#> .pbc3pbc1 0.088 0.007 11.960 0.000
#> .sn1pbc1 0.111 0.008 13.687 0.000
#> .sn2pbc1 0.093 0.007 12.940 0.000
#> .sn3pbc1 0.092 0.007 13.718 0.000
#> .att2pbc2 ~~
#> .att2pbc3 0.130 0.006 20.314 0.000
#> .att3pbc2 0.169 0.009 18.418 0.000
#> .pbc2pbc1 0.121 0.008 14.699 0.000
#> .pbc2pbc2 0.223 0.014 15.709 0.000
#> .pbc3pbc2 0.097 0.007 14.256 0.000
#> .sn1pbc2 0.108 0.007 14.776 0.000
#> .sn2pbc2 0.090 0.006 14.023 0.000
#> .sn3pbc2 0.085 0.006 14.369 0.000
#> .att2pbc3 ~~
#> .att3pbc3 0.182 0.008 22.286 0.000
#> .pbc3pbc1 0.129 0.007 17.566 0.000
#> .pbc3pbc2 0.116 0.007 17.202 0.000
#> .pbc3pbc3 0.214 0.011 18.903 0.000
#> .sn1pbc3 0.116 0.006 17.956 0.000
#> .sn2pbc3 0.100 0.006 17.266 0.000
#> .sn3pbc3 0.093 0.005 17.549 0.000
#> .att3pbc1 ~~
#> .att3pbc2 0.132 0.006 20.977 0.000
#> .att3pbc3 0.118 0.006 20.694 0.000
#> .pbc1pbc1 0.205 0.015 13.672 0.000
#> .pbc2pbc1 0.093 0.007 12.446 0.000
#> .pbc3pbc1 0.078 0.007 11.904 0.000
#> .sn1pbc1 0.098 0.007 13.636 0.000
#> .sn2pbc1 0.084 0.006 13.195 0.000
#> .sn3pbc1 0.082 0.006 13.824 0.000
#> .att3pbc2 ~~
#> .att3pbc3 0.105 0.005 20.231 0.000
#> .pbc2pbc1 0.115 0.007 15.475 0.000
#> .pbc2pbc2 0.216 0.013 16.714 0.000
#> .pbc3pbc2 0.091 0.006 14.907 0.000
#> .sn1pbc2 0.096 0.007 14.798 0.000
#> .sn2pbc2 0.080 0.006 13.856 0.000
#> .sn3pbc2 0.077 0.005 14.399 0.000
#> .att3pbc3 ~~
#> .pbc3pbc1 0.116 0.007 17.724 0.000
#> .pbc3pbc2 0.106 0.006 17.648 0.000
#> .pbc3pbc3 0.186 0.010 18.584 0.000
#> .sn1pbc3 0.098 0.006 17.232 0.000
#> .sn2pbc3 0.087 0.005 16.906 0.000
#> .sn3pbc3 0.080 0.005 17.060 0.000
#> .pbc1pbc1 ~~
#> .pbc2pbc1 0.283 0.017 16.582 0.000
#> .pbc3pbc1 0.251 0.015 16.550 0.000
#> .sn1pbc1 ~~
#> .pbc1pbc1 0.201 0.014 14.829 0.000
#> .sn2pbc1 ~~
#> .pbc1pbc1 0.171 0.012 14.066 0.000
#> .sn3pbc1 ~~
#> .pbc1pbc1 0.163 0.011 14.496 0.000
#> .pbc2pbc1 ~~
#> .pbc2pbc2 0.299 0.015 19.338 0.000
#> .pbc3pbc1 0.111 0.007 15.091 0.000
#> .pbc3pbc2 0.130 0.007 17.842 0.000
#> .sn1pbc1 ~~
#> .pbc2pbc1 0.089 0.007 13.195 0.000
#> .sn1pbc2 ~~
#> .pbc2pbc1 0.088 0.007 13.471 0.000
#> .sn2pbc1 ~~
#> .pbc2pbc1 0.079 0.006 13.005 0.000
#> .sn2pbc2 ~~
#> .pbc2pbc1 0.080 0.006 13.607 0.000
#> .sn3pbc1 ~~
#> .pbc2pbc1 0.071 0.006 12.854 0.000
#> .sn3pbc2 ~~
#> .pbc2pbc1 0.077 0.005 14.169 0.000
#> .pbc2pbc2 ~~
#> .pbc3pbc2 0.241 0.013 18.860 0.000
#> .sn1pbc2 ~~
#> .pbc2pbc2 0.166 0.011 14.639 0.000
#> .sn2pbc2 ~~
#> .pbc2pbc2 0.152 0.010 14.913 0.000
#> .sn3pbc2 ~~
#> .pbc2pbc2 0.137 0.009 14.766 0.000
#> .pbc3pbc1 ~~
#> .pbc3pbc2 0.146 0.007 21.068 0.000
#> .pbc3pbc3 0.263 0.012 22.316 0.000
#> .sn1pbc1 ~~
#> .pbc3pbc1 0.078 0.006 12.972 0.000
#> .sn1pbc3 ~~
#> .pbc3pbc1 0.091 0.006 15.897 0.000
#> .sn2pbc1 ~~
#> .pbc3pbc1 0.067 0.005 12.407 0.000
#> .sn2pbc3 ~~
#> .pbc3pbc1 0.083 0.005 16.014 0.000
#> .sn3pbc1 ~~
#> .pbc3pbc1 0.062 0.005 12.611 0.000
#> .sn3pbc3 ~~
#> .pbc3pbc1 0.079 0.005 16.645 0.000
#> .pbc3pbc2 ~~
#> .pbc3pbc3 0.236 0.011 21.902 0.000
#> .sn1pbc2 ~~
#> .pbc3pbc2 0.075 0.005 13.720 0.000
#> .sn1pbc3 ~~
#> .pbc3pbc2 0.083 0.005 15.837 0.000
#> .sn2pbc2 ~~
#> .pbc3pbc2 0.067 0.005 13.755 0.000
#> .sn2pbc3 ~~
#> .pbc3pbc2 0.077 0.005 16.119 0.000
#> .sn3pbc2 ~~
#> .pbc3pbc2 0.062 0.004 13.867 0.000
#> .sn3pbc3 ~~
#> .pbc3pbc2 0.071 0.004 16.237 0.000
#> .sn1pbc3 ~~
#> .pbc3pbc3 0.153 0.009 17.396 0.000
#> .sn2pbc3 ~~
#> .pbc3pbc3 0.142 0.008 17.662 0.000
#> .sn3pbc3 ~~
#> .pbc3pbc3 0.129 0.007 17.642 0.000
#> .sn1pbc1 ~~
#> .sn1pbc2 0.163 0.009 19.134 0.000
#> .sn1pbc3 0.139 0.007 18.626 0.000
#> .sn2pbc1 0.099 0.006 15.359 0.000
#> .sn3pbc1 0.089 0.006 15.027 0.000
#> .sn1pbc2 ~~
#> .sn1pbc3 0.126 0.007 18.289 0.000
#> .sn2pbc2 0.101 0.006 17.396 0.000
#> .sn3pbc2 0.096 0.005 18.005 0.000
#> .sn1pbc3 ~~
#> .sn2pbc3 0.106 0.005 20.768 0.000
#> .sn3pbc3 0.095 0.005 20.530 0.000
#> .sn2pbc1 ~~
#> .sn2pbc2 0.136 0.007 19.542 0.000
#> .sn2pbc3 0.122 0.006 19.630 0.000
#> .sn3pbc1 0.079 0.005 14.861 0.000
#> .sn2pbc2 ~~
#> .sn2pbc3 0.109 0.006 19.185 0.000
#> .sn3pbc2 0.083 0.005 17.507 0.000
#> .sn2pbc3 ~~
#> .sn3pbc3 0.084 0.004 20.140 0.000
#> .sn3pbc1 ~~
#> .sn3pbc2 0.122 0.006 20.972 0.000
#> .sn3pbc3 0.107 0.005 20.599 0.000
#> .sn3pbc2 ~~
#> .sn3pbc3 0.096 0.005 20.403 0.000
#> INT ~~
#> INTxPBC -0.038 0.033 -1.147 0.252
#>
#> Variances:
#> Estimate Std.Err z-value P(>|z|)
#> .att1 0.152 0.008 18.217 0.000
#> .att2 0.168 0.008 21.468 0.000
#> .att3 0.147 0.006 22.692 0.000
#> .sn1 0.171 0.009 19.842 0.000
#> .sn2 0.165 0.008 21.851 0.000
#> .sn3 0.151 0.006 23.314 0.000
#> .pbc1 0.162 0.009 18.801 0.000
#> .pbc2 0.167 0.008 21.287 0.000
#> .pbc3 0.159 0.007 23.501 0.000
#> .b1 0.145 0.021 6.812 0.000
#> .b2 0.180 0.017 10.895 0.000
#> .att1pbc1 0.377 0.017 22.325 0.000
#> .att2pbc1 0.381 0.015 25.514 0.000
#> .att3pbc1 0.314 0.012 25.965 0.000
#> .att1pbc2 0.371 0.015 24.510 0.000
#> .att2pbc2 0.360 0.013 27.082 0.000
#> .att3pbc2 0.298 0.011 27.636 0.000
#> .att1pbc3 0.364 0.013 27.887 0.000
#> .att2pbc3 0.357 0.012 30.624 0.000
#> .att3pbc3 0.277 0.009 30.093 0.000
#> .sn1pbc1 0.315 0.013 25.133 0.000
#> .sn2pbc1 0.261 0.010 25.532 0.000
#> .sn3pbc1 0.232 0.009 26.660 0.000
#> .sn1pbc2 0.297 0.011 26.676 0.000
#> .sn2pbc2 0.240 0.009 26.744 0.000
#> .sn3pbc2 0.211 0.007 28.273 0.000
#> .sn1pbc3 0.253 0.009 28.195 0.000
#> .sn2pbc3 0.222 0.008 29.257 0.000
#> .sn3pbc3 0.183 0.006 29.644 0.000
#> .pbc1pbc1 0.680 0.036 19.050 0.000
#> .pbc2pbc1 0.310 0.013 23.088 0.000
#> .pbc3pbc1 0.287 0.011 25.366 0.000
#> .pbc2pbc2 0.611 0.028 21.890 0.000
#> .pbc3pbc2 0.264 0.010 26.146 0.000
#> .pbc3pbc3 0.470 0.019 24.836 0.000
#> .ATT 0.389 0.025 15.723 0.000
#> .SN 0.199 0.014 14.729 0.000
#> .PBC 0.300 0.018 16.350 0.000
#> .BEH 0.516 0.027 19.381 0.000
#> INT 0.994 0.049 20.231 0.000
#> INTxPBC 1.597 0.073 21.909 0.000
#> .ATTPBC 0.373 0.025 14.700 0.000
#> .SNPBC 0.224 0.015 15.118 0.000
#> .PBCPBC 0.349 0.025 13.718 0.000