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)
#> modsem (version 1.0.4, approach = ca):
#> 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)
#> modsem (version 1.0.4, approach = dblcent):
#> lavaan 0.6-19 ended normally after 465 iterations
#>
#> Estimator ML
#> Optimization method NLMINB
#> Number of model parameters 186
#>
#> Number of observations 2000
#>
#> Model Test User Model:
#>
#> Test statistic 10052.300
#> Degrees of freedom 1089
#> 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 55.251 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 72.867 0.000
#> PC =~
#> pc1 1.000
#> pc2 0.891 0.010 89.774 0.000
#> pc3 0.792 0.009 86.846 0.000
#> BEH =~
#> b1 1.000
#> b2 0.850 0.039 21.738 0.000
#> INT =~
#> ATT 1.000
#> SN 0.670 0.061 11.032 0.000
#> PBC =~
#> PC 1.000
#> PB 0.668 0.072 9.292 0.000
#> INTxPBC =~
#> ATTPC 1.000
#> ATTPB 0.786 0.029 26.650 0.000
#> SNPC 0.715 0.026 27.124 0.000
#> SNPB 0.562 0.022 25.714 0.000
#> ATTPC =~
#> att1pc1 1.000
#> att2pc1 0.896 0.012 76.955 0.000
#> att3pc1 0.784 0.011 71.763 0.000
#> att1pc2 0.889 0.010 86.420 0.000
#> att2pc2 0.796 0.014 58.748 0.000
#> att3pc2 0.688 0.012 55.935 0.000
#> att1pc3 0.786 0.009 82.801 0.000
#> att2pc3 0.703 0.012 57.546 0.000
#> att3pc3 0.610 0.011 54.822 0.000
#> ATTPB =~
#> att1pb1 1.000
#> att2pb1 0.902 0.012 73.281 0.000
#> att3pb1 0.790 0.011 69.090 0.000
#> att1pb2 0.911 0.013 69.779 0.000
#> att2pb2 0.821 0.016 50.353 0.000
#> att3pb2 0.719 0.015 49.305 0.000
#> att1pb3 0.769 0.012 63.518 0.000
#> att2pb3 0.699 0.014 49.233 0.000
#> att3pb3 0.609 0.013 47.566 0.000
#> SNPC =~
#> sn1pc1 1.000
#> sn2pc1 0.916 0.017 52.984 0.000
#> sn3pc1 0.773 0.016 48.814 0.000
#> sn1pc2 0.890 0.011 80.299 0.000
#> sn2pc2 0.815 0.018 44.884 0.000
#> sn3pc2 0.684 0.016 42.622 0.000
#> sn1pc3 0.804 0.010 76.966 0.000
#> sn2pc3 0.734 0.016 44.770 0.000
#> sn3pc3 0.624 0.015 42.476 0.000
#> SNPB =~
#> sn1pb1 1.000
#> sn2pb1 0.932 0.018 52.861 0.000
#> sn3pb1 0.807 0.016 50.349 0.000
#> sn1pb2 0.921 0.014 67.113 0.000
#> sn2pb2 0.862 0.020 43.061 0.000
#> sn3pb2 0.746 0.018 41.358 0.000
#> sn1pb3 0.782 0.013 61.764 0.000
#> sn2pb3 0.726 0.018 41.471 0.000
#> sn3pb3 0.636 0.016 39.286 0.000
#>
#> Regressions:
#> Estimate Std.Err z-value P(>|z|)
#> BEH ~
#> PBC 0.211 0.026 8.120 0.000
#> INT 0.199 0.023 8.696 0.000
#> INTxPBC 0.139 0.017 8.230 0.000
#>
#> Covariances:
#> Estimate Std.Err z-value P(>|z|)
#> .att1pc1 ~~
#> .att2pc2 0.000
#> .att2pc3 0.000
#> .att3pc2 0.000
#> .att3pc3 0.000
#> .att2pc1 ~~
#> .att1pc2 0.000
#> .att1pc2 ~~
#> .att2pc3 0.000
#> .att3pc1 ~~
#> .att1pc2 0.000
#> .att1pc2 ~~
#> .att3pc3 0.000
#> .att2pc1 ~~
#> .att1pc3 0.000
#> .att2pc2 ~~
#> .att1pc3 0.000
#> .att3pc1 ~~
#> .att1pc3 0.000
#> .att3pc2 ~~
#> .att1pc3 0.000
#> .att2pc1 ~~
#> .att3pc2 0.000
#> .att3pc3 0.000
#> .att3pc1 ~~
#> .att2pc2 0.000
#> .att2pc2 ~~
#> .att3pc3 0.000
#> .att3pc1 ~~
#> .att2pc3 0.000
#> .att3pc2 ~~
#> .att2pc3 0.000
#> .att1pc1 ~~
#> .att1pc2 0.228 0.015 15.643 0.000
#> .att1pc3 0.202 0.013 15.648 0.000
#> .att2pc1 0.192 0.012 16.158 0.000
#> .att3pc1 0.171 0.011 16.161 0.000
#> .att1pc2 ~~
#> .att1pc3 0.182 0.012 15.689 0.000
#> .att2pc2 0.206 0.011 19.546 0.000
#> .att3pc2 0.180 0.009 19.366 0.000
#> .att1pc3 ~~
#> .att2pc3 0.194 0.009 21.544 0.000
#> .att3pc3 0.165 0.008 20.892 0.000
#> .att2pc1 ~~
#> .att2pc2 0.265 0.013 19.849 0.000
#> .att2pc3 0.231 0.012 19.601 0.000
#> .att3pc1 0.157 0.010 16.172 0.000
#> .att2pc2 ~~
#> .att2pc3 0.204 0.011 19.396 0.000
#> .att3pc2 0.165 0.009 19.366 0.000
#> .att2pc3 ~~
#> .att3pc3 0.150 0.007 20.763 0.000
#> .att3pc1 ~~
#> .att3pc2 0.266 0.012 22.627 0.000
#> .att3pc3 0.238 0.011 22.542 0.000
#> .att3pc2 ~~
#> .att3pc3 0.208 0.009 22.293 0.000
#> .att1pb1 ~~
#> .att2pb2 0.000
#> .att2pb3 0.000
#> .att3pb2 0.000
#> .att3pb3 0.000
#> .att2pb1 ~~
#> .att1pb2 0.000
#> .att1pb2 ~~
#> .att2pb3 0.000
#> .att3pb1 ~~
#> .att1pb2 0.000
#> .att1pb2 ~~
#> .att3pb3 0.000
#> .att2pb1 ~~
#> .att1pb3 0.000
#> .att2pb2 ~~
#> .att1pb3 0.000
#> .att3pb1 ~~
#> .att1pb3 0.000
#> .att3pb2 ~~
#> .att1pb3 0.000
#> .att2pb1 ~~
#> .att3pb2 0.000
#> .att3pb3 0.000
#> .att3pb1 ~~
#> .att2pb2 0.000
#> .att2pb2 ~~
#> .att3pb3 0.000
#> .att3pb1 ~~
#> .att2pb3 0.000
#> .att3pb2 ~~
#> .att2pb3 0.000
#> .att1pb1 ~~
#> .att1pb2 0.178 0.011 16.118 0.000
#> .att1pb3 0.150 0.010 15.776 0.000
#> .att2pb1 0.196 0.012 16.080 0.000
#> .att3pb1 0.164 0.011 15.563 0.000
#> .att1pb2 ~~
#> .att1pb3 0.142 0.009 15.903 0.000
#> .att2pb2 0.228 0.012 19.763 0.000
#> .att3pb2 0.195 0.010 19.467 0.000
#> .att1pb3 ~~
#> .att2pb3 0.212 0.010 22.266 0.000
#> .att3pb3 0.184 0.008 22.055 0.000
#> .att2pb1 ~~
#> .att2pb2 0.201 0.010 19.667 0.000
#> .att2pb3 0.170 0.009 19.380 0.000
#> .att3pb1 0.151 0.010 15.533 0.000
#> .att2pb2 ~~
#> .att2pb3 0.158 0.008 19.210 0.000
#> .att3pb2 0.191 0.010 19.989 0.000
#> .att2pb3 ~~
#> .att3pb3 0.165 0.008 21.609 0.000
#> .att3pb1 ~~
#> .att3pb2 0.189 0.009 21.673 0.000
#> .att3pb3 0.162 0.008 21.467 0.000
#> .att3pb2 ~~
#> .att3pb3 0.149 0.007 21.111 0.000
#> .sn1pc1 ~~
#> .sn2pc2 0.000
#> .sn2pc3 0.000
#> .sn3pc2 0.000
#> .sn3pc3 0.000
#> .sn2pc1 ~~
#> .sn1pc2 0.000
#> .sn1pc2 ~~
#> .sn2pc3 0.000
#> .sn3pc1 ~~
#> .sn1pc2 0.000
#> .sn1pc2 ~~
#> .sn3pc3 0.000
#> .sn2pc1 ~~
#> .sn1pc3 0.000
#> .sn2pc2 ~~
#> .sn1pc3 0.000
#> .sn3pc1 ~~
#> .sn1pc3 0.000
#> .sn3pc2 ~~
#> .sn1pc3 0.000
#> .sn2pc1 ~~
#> .sn3pc2 0.000
#> .sn3pc3 0.000
#> .sn3pc1 ~~
#> .sn2pc2 0.000
#> .sn2pc2 ~~
#> .sn3pc3 0.000
#> .sn3pc1 ~~
#> .sn2pc3 0.000
#> .sn3pc2 ~~
#> .sn2pc3 0.000
#> .sn1pc1 ~~
#> .sn1pc2 0.235 0.014 17.052 0.000
#> .sn1pc3 0.211 0.012 17.095 0.000
#> .sn2pc1 0.103 0.007 15.796 0.000
#> .sn3pc1 0.092 0.006 15.750 0.000
#> .sn1pc2 ~~
#> .sn1pc3 0.178 0.011 16.493 0.000
#> .sn2pc2 0.106 0.006 18.376 0.000
#> .sn3pc2 0.091 0.005 18.007 0.000
#> .sn1pc3 ~~
#> .sn2pc3 0.103 0.005 20.203 0.000
#> .sn3pc3 0.087 0.004 19.503 0.000
#> .sn2pc1 ~~
#> .sn2pc2 0.261 0.013 20.103 0.000
#> .sn2pc3 0.228 0.011 19.894 0.000
#> .sn3pc1 0.083 0.005 15.378 0.000
#> .sn2pc2 ~~
#> .sn2pc3 0.206 0.010 20.014 0.000
#> .sn3pc2 0.087 0.005 18.174 0.000
#> .sn2pc3 ~~
#> .sn3pc3 0.082 0.004 19.532 0.000
#> .sn3pc1 ~~
#> .sn3pc2 0.255 0.011 23.013 0.000
#> .sn3pc3 0.227 0.010 22.859 0.000
#> .sn3pc2 ~~
#> .sn3pc3 0.198 0.009 22.679 0.000
#> .sn1pb1 ~~
#> .sn2pb2 0.000
#> .sn2pb3 0.000
#> .sn3pb2 0.000
#> .sn3pb3 0.000
#> .sn2pb1 ~~
#> .sn1pb2 0.000
#> .sn1pb2 ~~
#> .sn2pb3 0.000
#> .sn3pb1 ~~
#> .sn1pb2 0.000
#> .sn1pb2 ~~
#> .sn3pb3 0.000
#> .sn2pb1 ~~
#> .sn1pb3 0.000
#> .sn2pb2 ~~
#> .sn1pb3 0.000
#> .sn3pb1 ~~
#> .sn1pb3 0.000
#> .sn3pb2 ~~
#> .sn1pb3 0.000
#> .sn2pb1 ~~
#> .sn3pb2 0.000
#> .sn3pb3 0.000
#> .sn3pb1 ~~
#> .sn2pb2 0.000
#> .sn2pb2 ~~
#> .sn3pb3 0.000
#> .sn3pb1 ~~
#> .sn2pb3 0.000
#> .sn3pb2 ~~
#> .sn2pb3 0.000
#> .sn1pb1 ~~
#> .sn1pb2 0.163 0.010 16.601 0.000
#> .sn1pb3 0.143 0.009 16.410 0.000
#> .sn2pb1 0.104 0.006 16.170 0.000
#> .sn3pb1 0.093 0.006 15.771 0.000
#> .sn1pb2 ~~
#> .sn1pb3 0.131 0.008 16.144 0.000
#> .sn2pb2 0.101 0.006 17.420 0.000
#> .sn3pb2 0.091 0.005 17.177 0.000
#> .sn1pb3 ~~
#> .sn2pb3 0.099 0.005 20.252 0.000
#> .sn3pb3 0.094 0.005 20.442 0.000
#> .sn2pb1 ~~
#> .sn2pb2 0.192 0.010 19.812 0.000
#> .sn2pb3 0.166 0.009 19.549 0.000
#> .sn3pb1 0.082 0.005 15.386 0.000
#> .sn2pb2 ~~
#> .sn2pb3 0.154 0.008 19.408 0.000
#> .sn3pb2 0.083 0.005 17.108 0.000
#> .sn2pb3 ~~
#> .sn3pb3 0.080 0.004 19.531 0.000
#> .sn3pb1 ~~
#> .sn3pb2 0.177 0.008 21.670 0.000
#> .sn3pb3 0.158 0.007 21.493 0.000
#> .sn3pb2 ~~
#> .sn3pb3 0.150 0.007 21.690 0.000
#> INT ~~
#> PBC 0.034 0.033 1.045 0.296
#> INTxPBC -0.003 0.035 -0.099 0.921
#> PBC ~~
#> INTxPBC -0.120 0.039 -3.063 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 16.806 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 17.845 0.000
#> .pc2 0.165 0.008 20.574 0.000
#> .pc3 0.150 0.007 22.166 0.000
#> .b1 0.133 0.029 4.600 0.000
#> .b2 0.189 0.022 8.768 0.000
#> .att1pc1 0.495 0.022 22.813 0.000
#> .att2pc1 0.501 0.019 25.991 0.000
#> .att3pc1 0.468 0.017 28.208 0.000
#> .att1pc2 0.459 0.018 25.111 0.000
#> .att2pc2 0.448 0.016 27.871 0.000
#> .att3pc2 0.406 0.014 29.875 0.000
#> .att1pc3 0.397 0.015 26.508 0.000
#> .att2pc3 0.380 0.013 28.927 0.000
#> .att3pc3 0.343 0.011 30.642 0.000
#> .att1pb1 0.426 0.018 23.037 0.000
#> .att2pb1 0.419 0.016 25.558 0.000
#> .att3pb1 0.356 0.013 26.825 0.000
#> .att1pb2 0.427 0.017 25.557 0.000
#> .att2pb2 0.435 0.015 28.402 0.000
#> .att3pb2 0.364 0.012 29.496 0.000
#> .att1pb3 0.387 0.014 28.034 0.000
#> .att2pb3 0.347 0.012 29.384 0.000
#> .att3pb3 0.299 0.010 30.764 0.000
#> .sn1pc1 0.414 0.018 23.103 0.000
#> .sn2pc1 0.404 0.016 24.941 0.000
#> .sn3pc1 0.392 0.014 27.841 0.000
#> .sn1pc2 0.340 0.014 23.817 0.000
#> .sn2pc2 0.363 0.014 26.739 0.000
#> .sn3pc2 0.321 0.011 28.764 0.000
#> .sn1pc3 0.296 0.012 25.044 0.000
#> .sn2pc3 0.300 0.011 27.459 0.000
#> .sn3pc3 0.271 0.009 29.358 0.000
#> .sn1pb1 0.320 0.013 23.685 0.000
#> .sn2pb1 0.325 0.013 25.720 0.000
#> .sn3pb1 0.292 0.011 27.314 0.000
#> .sn1pb2 0.287 0.012 24.298 0.000
#> .sn2pb2 0.298 0.011 26.547 0.000
#> .sn3pb2 0.267 0.009 28.196 0.000
#> .sn1pb3 0.256 0.010 26.436 0.000
#> .sn2pb3 0.248 0.009 27.882 0.000
#> .sn3pb3 0.235 0.008 29.850 0.000
#> .ATT 0.342 0.094 3.641 0.000
#> .SN 0.229 0.043 5.317 0.000
#> .PB 0.688 0.062 11.176 0.000
#> .PC 0.503 0.127 3.970 0.000
#> .BEH 0.545 0.034 16.237 0.000
#> INT 1.058 0.104 10.185 0.000
#> PBC 1.201 0.137 8.775 0.000
#> INTxPBC 1.520 0.089 17.158 0.000
#> .ATTPC 0.942 0.055 17.182 0.000
#> .ATTPB 0.729 0.040 18.024 0.000
#> .SNPC 0.453 0.030 15.307 0.000
#> .SNPB 0.385 0.023 16.924 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)
#> modsem (version 1.0.4, approach = ca):
#> 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)
#> modsem (version 1.0.4, approach = dblcent):
#> lavaan 0.6-19 ended normally after 339 iterations
#>
#> Estimator ML
#> Optimization method NLMINB
#> Number of model parameters 125
#>
#> Number of observations 2000
#>
#> Model Test User Model:
#>
#> Test statistic 6227.020
#> Degrees of freedom 505
#> 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 82.767 0.000
#> att3 0.802 0.010 80.426 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 66.751 0.000
#> pbc3 0.793 0.013 63.368 0.000
#> BEH =~
#> b1 1.000
#> b2 0.864 0.029 30.237 0.000
#> INT =~
#> ATT 1.000
#> PBC 0.819 0.024 33.977 0.000
#> SN 0.702 0.021 33.482 0.000
#> INTxPBC =~
#> ATTPBC 1.000
#> SNPBC 0.717 0.017 42.130 0.000
#> PBCPBC 1.003 0.024 42.678 0.000
#> ATTPBC =~
#> att1pbc1 1.000
#> att2pbc1 0.896 0.010 94.261 0.000
#> att3pbc1 0.808 0.009 92.565 0.000
#> att1pbc2 0.899 0.011 79.087 0.000
#> att2pbc2 0.804 0.013 61.655 0.000
#> att3pbc2 0.727 0.012 61.499 0.000
#> att1pbc3 0.752 0.011 70.229 0.000
#> att2pbc3 0.675 0.012 55.856 0.000
#> att3pbc3 0.614 0.011 56.951 0.000
#> SNPBC =~
#> sn1pbc1 1.000
#> sn2pbc1 0.884 0.013 65.689 0.000
#> sn3pbc1 0.783 0.012 63.016 0.000
#> sn1pbc2 0.893 0.012 76.169 0.000
#> sn2pbc2 0.791 0.015 51.434 0.000
#> sn3pbc2 0.693 0.014 49.947 0.000
#> sn1pbc3 0.771 0.011 70.139 0.000
#> sn2pbc3 0.673 0.014 48.082 0.000
#> sn3pbc3 0.601 0.013 47.729 0.000
#> PBCPBC =~
#> pbc1pbc1 1.000
#> pbc2pbc1 0.902 0.010 86.187 0.000
#> pbc3pbc1 0.764 0.010 79.520 0.000
#> pbc2pbc2 0.812 0.018 44.917 0.000
#> pbc3pbc2 0.689 0.013 51.106 0.000
#> pbc3pbc3 0.590 0.015 40.172 0.000
#>
#> Regressions:
#> Estimate Std.Err z-value P(>|z|)
#> BEH ~
#> PBC 0.189 0.043 4.359 0.000
#> INT 0.187 0.045 4.140 0.000
#> INTxPBC 0.203 0.015 13.754 0.000
#>
#> Covariances:
#> Estimate Std.Err z-value P(>|z|)
#> .att1pbc1 ~~
#> .att2pbc2 0.000
#> .att2pbc3 0.000
#> .att3pbc2 0.000
#> .att3pbc3 0.000
#> .att2pbc1 ~~
#> .att1pbc2 0.000
#> .att1pbc2 ~~
#> .att2pbc3 0.000
#> .att3pbc1 ~~
#> .att1pbc2 0.000
#> .att1pbc2 ~~
#> .att3pbc3 0.000
#> .att2pbc1 ~~
#> .att1pbc3 0.000
#> .att2pbc2 ~~
#> .att1pbc3 0.000
#> .att3pbc1 ~~
#> .att1pbc3 0.000
#> .att3pbc2 ~~
#> .att1pbc3 0.000
#> .att2pbc1 ~~
#> .att3pbc2 0.000
#> .att3pbc3 0.000
#> .att3pbc1 ~~
#> .att2pbc2 0.000
#> .att2pbc2 ~~
#> .att3pbc3 0.000
#> .att3pbc1 ~~
#> .att2pbc3 0.000
#> .att3pbc2 ~~
#> .att2pbc3 0.000
#> .att1pbc1 ~~
#> .att1pbc2 0.123 0.008 15.877 0.000
#> .att1pbc3 0.108 0.007 15.606 0.000
#> .att2pbc1 0.188 0.011 16.599 0.000
#> .att3pbc1 0.166 0.010 16.500 0.000
#> .att1pbc2 ~~
#> .att1pbc3 0.098 0.006 15.298 0.000
#> .att2pbc2 0.211 0.011 19.889 0.000
#> .att3pbc2 0.194 0.010 20.282 0.000
#> .att1pbc3 ~~
#> .att2pbc3 0.237 0.010 23.982 0.000
#> .att3pbc3 0.204 0.009 23.565 0.000
#> .att2pbc1 ~~
#> .att2pbc2 0.156 0.008 20.453 0.000
#> .att2pbc3 0.144 0.007 20.721 0.000
#> .att3pbc1 0.153 0.009 16.556 0.000
#> .att2pbc2 ~~
#> .att2pbc3 0.130 0.006 20.338 0.000
#> .att3pbc2 0.174 0.009 19.943 0.000
#> .att2pbc3 ~~
#> .att3pbc3 0.185 0.008 23.276 0.000
#> .att3pbc1 ~~
#> .att3pbc2 0.133 0.006 20.972 0.000
#> .att3pbc3 0.119 0.006 20.773 0.000
#> .att3pbc2 ~~
#> .att3pbc3 0.107 0.005 20.310 0.000
#> .sn1pbc1 ~~
#> .sn2pbc2 0.000
#> .sn2pbc3 0.000
#> .sn3pbc2 0.000
#> .sn3pbc3 0.000
#> .sn2pbc1 ~~
#> .sn1pbc2 0.000
#> .sn1pbc2 ~~
#> .sn2pbc3 0.000
#> .sn3pbc1 ~~
#> .sn1pbc2 0.000
#> .sn1pbc2 ~~
#> .sn3pbc3 0.000
#> .sn2pbc1 ~~
#> .sn1pbc3 0.000
#> .sn2pbc2 ~~
#> .sn1pbc3 0.000
#> .sn3pbc1 ~~
#> .sn1pbc3 0.000
#> .sn3pbc2 ~~
#> .sn1pbc3 0.000
#> .sn2pbc1 ~~
#> .sn3pbc2 0.000
#> .sn3pbc3 0.000
#> .sn3pbc1 ~~
#> .sn2pbc2 0.000
#> .sn2pbc2 ~~
#> .sn3pbc3 0.000
#> .sn3pbc1 ~~
#> .sn2pbc3 0.000
#> .sn3pbc2 ~~
#> .sn2pbc3 0.000
#> .sn1pbc1 ~~
#> .sn1pbc2 0.163 0.008 19.281 0.000
#> .sn1pbc3 0.140 0.007 18.855 0.000
#> .sn2pbc1 0.092 0.006 15.534 0.000
#> .sn3pbc1 0.082 0.005 15.167 0.000
#> .sn1pbc2 ~~
#> .sn1pbc3 0.128 0.007 18.607 0.000
#> .sn2pbc2 0.103 0.006 18.569 0.000
#> .sn3pbc2 0.097 0.005 19.080 0.000
#> .sn1pbc3 ~~
#> .sn2pbc3 0.110 0.005 21.785 0.000
#> .sn3pbc3 0.099 0.005 21.584 0.000
#> .sn2pbc1 ~~
#> .sn2pbc2 0.137 0.007 19.800 0.000
#> .sn2pbc3 0.122 0.006 19.734 0.000
#> .sn3pbc1 0.074 0.005 15.077 0.000
#> .sn2pbc2 ~~
#> .sn2pbc3 0.110 0.006 19.347 0.000
#> .sn3pbc2 0.084 0.005 18.558 0.000
#> .sn2pbc3 ~~
#> .sn3pbc3 0.087 0.004 21.102 0.000
#> .sn3pbc1 ~~
#> .sn3pbc2 0.121 0.006 20.829 0.000
#> .sn3pbc3 0.106 0.005 20.481 0.000
#> .sn3pbc2 ~~
#> .sn3pbc3 0.094 0.005 20.111 0.000
#> .pbc1pbc1 ~~
#> .pbc2pbc2 0.000
#> .pbc3pbc2 0.000
#> .pbc3pbc3 0.000
#> .pbc2pbc1 ~~
#> .pbc3pbc3 0.000
#> .pbc3pbc1 ~~
#> .pbc2pbc2 0.000
#> .pbc2pbc2 ~~
#> .pbc3pbc3 0.000
#> .pbc1pbc1 ~~
#> .pbc2pbc1 0.268 0.014 19.031 0.000
#> .pbc3pbc1 0.240 0.012 19.187 0.000
#> .pbc2pbc1 ~~
#> .pbc2pbc2 0.302 0.014 22.286 0.000
#> .pbc3pbc1 0.106 0.006 17.710 0.000
#> .pbc3pbc2 0.132 0.006 20.914 0.000
#> .pbc2pbc2 ~~
#> .pbc3pbc2 0.244 0.011 21.890 0.000
#> .pbc3pbc1 ~~
#> .pbc3pbc2 0.152 0.006 23.891 0.000
#> .pbc3pbc3 0.272 0.011 24.933 0.000
#> .pbc3pbc2 ~~
#> .pbc3pbc3 0.243 0.010 24.501 0.000
#> INT ~~
#> INTxPBC -0.038 0.033 -1.150 0.250
#>
#> Variances:
#> Estimate Std.Err z-value P(>|z|)
#> .att1 0.152 0.008 18.217 0.000
#> .att2 0.168 0.008 21.469 0.000
#> .att3 0.147 0.006 22.691 0.000
#> .sn1 0.171 0.009 19.842 0.000
#> .sn2 0.165 0.008 21.850 0.000
#> .sn3 0.151 0.006 23.313 0.000
#> .pbc1 0.162 0.009 18.804 0.000
#> .pbc2 0.167 0.008 21.285 0.000
#> .pbc3 0.159 0.007 23.501 0.000
#> .b1 0.144 0.021 6.755 0.000
#> .b2 0.181 0.017 10.910 0.000
#> .att1pbc1 0.363 0.016 23.326 0.000
#> .att2pbc1 0.370 0.014 26.472 0.000
#> .att3pbc1 0.305 0.011 26.846 0.000
#> .att1pbc2 0.378 0.014 26.203 0.000
#> .att2pbc2 0.366 0.013 28.665 0.000
#> .att3pbc2 0.303 0.010 29.105 0.000
#> .att1pbc3 0.369 0.013 29.040 0.000
#> .att2pbc3 0.361 0.011 31.610 0.000
#> .att3pbc3 0.281 0.009 31.219 0.000
#> .sn1pbc1 0.306 0.012 25.602 0.000
#> .sn2pbc1 0.256 0.010 26.184 0.000
#> .sn3pbc1 0.228 0.008 27.105 0.000
#> .sn1pbc2 0.300 0.011 27.855 0.000
#> .sn2pbc2 0.243 0.009 27.943 0.000
#> .sn3pbc2 0.209 0.007 29.007 0.000
#> .sn1pbc3 0.260 0.009 29.414 0.000
#> .sn2pbc3 0.225 0.007 30.173 0.000
#> .sn3pbc3 0.185 0.006 30.379 0.000
#> .pbc1pbc1 0.646 0.030 21.568 0.000
#> .pbc2pbc1 0.306 0.011 27.014 0.000
#> .pbc3pbc1 0.290 0.010 29.693 0.000
#> .pbc2pbc2 0.618 0.025 24.778 0.000
#> .pbc3pbc2 0.271 0.009 30.899 0.000
#> .pbc3pbc3 0.482 0.018 27.155 0.000
#> .ATT 0.389 0.025 15.717 0.000
#> .SN 0.199 0.014 14.732 0.000
#> .PBC 0.300 0.018 16.350 0.000
#> .BEH 0.521 0.027 19.503 0.000
#> INT 0.995 0.049 20.233 0.000
#> INTxPBC 1.701 0.074 22.836 0.000
#> .ATTPBC 0.309 0.025 12.116 0.000
#> .SNPBC 0.193 0.015 12.939 0.000
#> .PBCPBC 0.206 0.027 7.713 0.000