Model Specification

Bivariate linear growth curve model used by the Portland Collective as the fulcrum of coordination has the following specification

where

i - individual
t - timepoint
o - outcome (phys/cog)
p - physical measure used in estimation
c - cognitive measure used in estimation
y_t_i - score for the individual i at time t on an outcome
β_0i - bivariate initial status / random intercept
β_1i - bivariate rate of change / random slope
u_0i - is residual of the random intercept of an outcome
u_1i - is residual of the random slope of an outcome
γ₀₀ - o_GAMMA_00 - average initial status / common intercept of an outcome
γ₁₀ - o_GAMMA_10 - average rate of change / common slope of an outcome
γ_0k - o_GAMMA_0k - effect of the kth predictor on the random intercept of an outcome
γ_1k - o_GAMMA_1k - effect of the kth predictor on the random slope of an outcome
_ppτ₀₀ - pp_TAU_00 - variance of physical intercept - p_cov_int
_ppτ₁₁ - pp_TAU_11 - variance of physical slope - p_cov_slope
_ccτ₁₁ - cc_TAU_11 - variance of cognitive slope
_ccτ₀₀ - cc_TAU_00 - variance of cogntive intercept - var_int_cog
_ppτ₀₁ - pp_TAU_01 - covariance btw physical intercept and physical slope
_pcτ₀₁ - pc_TAU_01 - covariance btw physical intercept and cognitive slope
_pcτ₀₀ - pc_TAU_00 - covariance btw physical intercept and cognitive intercept - I var_int_physical / corr_int
_pcτ₁₁ - pc_TAU_11 - covariance btw physical slope and cognitive slope - S - var_slope_physical / corr_slope
_pcτ₁₀ - pc_TAU_10 - covariance btw physical slope and cognitive intercept
_ccτ₁₀ - cc_TAU_10 - covariance btw cognitive slope and cognitive intercept
_pσ - p_SIGMA - variance of the physical residual
_cσ - c_SIGMA - variance of the cogntive residual
_pcσ - pc_SIGMA - covariance btw physcial residual and cogntive residual - R - p_cov_res / corr_residual

THe model of this type could be fitted with the following Mplus syntax. To exemplify, we'll look at the model b1_female_aeh_tug_speed_tug_digitsymbolsubstitutiontest.out from ILSE study. This ILSE was specified in Mplus as follows

  MODEL:  ip sp | p1-p3 AT time1-time3;
          ic sc | c1-c3 AT time1-time3;
          ip; sp; ic; sc;
          ip sp ic sc WITH ip sp ic sc;

          ip sp ic sc ON Haupt second others Bage height;

          p1-p3 (res_p);
          c1-c3 (res_c);
          p1-p3 pwith c1-c3(res_cov);

The output of this model is processed with MplusAutomation package and the following data object produced, containing the numeric results of the estimated model solution. The right-hand column contains alphanumeric names that map onto the algebraic specification of our bivariate growth model.

> message("Getting ", study, ", model ", i, ", ",out_file)
Getting ilse, model 97, b1_female_aeh_tug_speed_tug_digitsymbolsubstitutiontest.out
> model
          paramHeader  param    est     se  est_se    pval
1               IP.ON  HAUPT -0.468  0.866  -0.541   0.589 - p_GAMMA_01
2               IP.ON SECOND  0.005  1.232   0.004   0.997 - p_GAMMA_02
3               IP.ON OTHERS -0.607  0.968  -0.627   0.531 - p_GAMMA_03
4               IP.ON   BAGE -0.172  0.209  -0.821   0.411 - p_GAMMA_04
5               IP.ON HEIGHT -0.006  0.030  -0.216   0.829 - p_GAMMA_05
6               SP.ON  HAUPT  0.111  0.138   0.806   0.420 - p_GAMMA_11
7               SP.ON SECOND  0.455  0.295   1.541   0.123 - p_GAMMA_12
8               SP.ON OTHERS  0.079  0.155   0.513   0.608 - p_GAMMA_13
9               SP.ON   BAGE  0.007  0.023   0.315   0.753 - p_GAMMA_14
10              SP.ON HEIGHT  0.002  0.003   0.595   0.552 - p_GAMMA_15
11              IC.ON  HAUPT  5.416  3.414   1.586   0.113 - c_GAMMA_01
12              IC.ON SECOND 14.395 10.836   1.328   0.184 - c_GAMMA_02
13              IC.ON OTHERS 15.247  4.121   3.700   0.000 - c_GAMMA_03
14              IC.ON   BAGE -0.252  0.934  -0.270   0.787 - c_GAMMA_04
15              IC.ON HEIGHT  0.153  0.144   1.064   0.288 - c_GAMMA_05
16              SC.ON  HAUPT -0.152  0.296  -0.513   0.608 - c_GAMMA_11
17              SC.ON SECOND -0.482  1.130  -0.426   0.670 - c_GAMMA_12
18              SC.ON OTHERS -0.166  0.318  -0.523   0.601 - c_GAMMA_13
19              SC.ON   BAGE  0.049  0.055   0.886   0.376 - c_GAMMA_14
20              SC.ON HEIGHT  0.003  0.006   0.414   0.679 - c_GAMMA_15
21            IP.WITH     SP -0.055  0.076  -0.731   0.465 - pp_TAU_01
22            IP.WITH     IC -2.513  1.901  -1.322   0.186 - pc_TAU_00
23            IP.WITH     SC -0.133  0.134  -0.994   0.320 - pc_TAU_01
24            SP.WITH     IC  0.178  0.220   0.810   0.418 - pc_TAU_10  
25            SP.WITH     SC  0.001  0.017   0.078   0.938 - pc_TAU_11  
26            IC.WITH     SC -0.514  0.621  -0.827   0.408 - cc_TAU_10
27            P1.WITH     C1  0.172  0.516   0.334   0.738 - pc_SIGMA
28            P2.WITH     C2  0.172  0.516   0.334   0.738 - pc_SIGMA
29            P3.WITH     C3  0.172  0.516   0.334   0.738 - pc_SIGMA
30         Intercepts     P1  0.000  0.000 999.000 999.000
31         Intercepts     P2  0.000  0.000 999.000 999.000
32         Intercepts     P3  0.000  0.000 999.000 999.000
33         Intercepts     C1  0.000  0.000 999.000 999.000
34         Intercepts     C2  0.000  0.000 999.000 999.000
35         Intercepts     C3  0.000  0.000 999.000 999.000
36         Intercepts     IP  5.047  1.757   2.872   0.004 - p_GAMMA_00  
37         Intercepts     SP  0.151  0.216   0.700   0.484 - p_GAMMA_10  
38         Intercepts     IC 36.655  6.672   5.494   0.000 - c_GAMMA_00  
39         Intercepts     SC  0.062  0.451   0.137   0.891 - c_GAMMA_10
40 Residual.Variances     P1  1.649  0.277   5.943   0.000 - p_SIGMA    
41 Residual.Variances     P2  1.649  0.277   5.943   0.000 - p_SIGMA  
42 Residual.Variances     P3  1.649  0.277   5.943   0.000 - p_SIGMA  
43 Residual.Variances     C1 14.190  1.094  12.973   0.000 - c_SIGMA  
44 Residual.Variances     C2 14.190  1.094  12.973   0.000 - c_SIGMA  
45 Residual.Variances     C3 14.190  1.094  12.973   0.000 - c_SIGMA  
46 Residual.Variances     IP  1.557  0.691   2.255   0.024 - pp_TAU_00
47 Residual.Variances     SP  0.003  0.010   0.267   0.789 - pp_TAU_11
48 Residual.Variances     IC 96.073 12.242   7.848   0.000 - cc_TAU_00
49 Residual.Variances     SC  0.058  0.045   1.306   0.192 - cc_TAU_11

NOTE

When variances and covariances are requested explicitly, the output will include additional elements of which we need to be aware during extraction. See b1_female_aeh_pulmonary_speed_fev_digitsymbol.out for example of the output with this option.

> message("Getting ", study, ", model ", i, ", ",out_file)
Getting radc, model 55, b1_female_aeh_pulmonary_speed_fev_digitsymbol.out
> model
          paramHeader  param    est    se  est_se    pval
1               IP.ON   BAGE -0.022 0.002 -13.925   0.000
2               IP.ON   EDUC  0.014 0.004   3.866   0.000
3               IP.ON HEIGHT  1.619 0.221   7.328   0.000
4               SP.ON   BAGE  0.001 0.000   1.680   0.093
5               SP.ON   EDUC  0.000 0.001   0.026   0.979
6               SP.ON HEIGHT -0.070 0.040  -1.748   0.080
7               IC.ON   BAGE -0.007 0.011  -0.582   0.560
8               IC.ON   EDUC  0.255 0.030   8.640   0.000
9               IC.ON HEIGHT  2.966 1.286   2.306   0.021
10              SC.ON   BAGE -0.001 0.003  -0.336   0.737
11              SC.ON   EDUC  0.001 0.008   0.104   0.917
12              SC.ON HEIGHT  0.310 0.362   0.857   0.392
13            SP.WITH     IP  0.000 0.001  -0.082   0.935
14            IC.WITH     IP  0.108 0.035   3.131   0.002
15            IC.WITH     SP -0.004 0.008  -0.532   0.595
16            SC.WITH     IP  0.003 0.011   0.319   0.750
17            SC.WITH     SP  0.000 0.003   0.019   0.985
18            SC.WITH     IC  0.066 0.083   0.800   0.424
19            P1.WITH     C1 -0.013 0.007  -1.781   0.075
20            P2.WITH     C2 -0.013 0.007  -1.781   0.075
21            P3.WITH     C3 -0.013 0.007  -1.781   0.075
22            P4.WITH     C4 -0.013 0.007  -1.781   0.075
23            P5.WITH     C5 -0.013 0.007  -1.781   0.075
24          BAGE.WITH   EDUC -1.096 0.783  -1.401   0.161
25        HEIGHT.WITH   EDUC  0.038 0.008   4.849   0.000
26        HEIGHT.WITH   BAGE -0.111 0.020  -5.580   0.000
27              Means   EDUC  7.337 0.098  75.069   0.000
28              Means   BAGE  8.665 0.250  34.638   0.000
29              Means HEIGHT -0.001 0.002  -0.418   0.676
30         Intercepts     P1  0.000 0.000 999.000 999.000
31         Intercepts     P2  0.000 0.000 999.000 999.000
32         Intercepts     P3  0.000 0.000 999.000 999.000
33         Intercepts     P4  0.000 0.000 999.000 999.000
34         Intercepts     P5  0.000 0.000 999.000 999.000
35         Intercepts     C1  0.000 0.000 999.000 999.000
36         Intercepts     C2  0.000 0.000 999.000 999.000
37         Intercepts     C3  0.000 0.000 999.000 999.000
38         Intercepts     C4  0.000 0.000 999.000 999.000
39         Intercepts     C5  0.000 0.000 999.000 999.000
40         Intercepts     IP  1.611 0.032  50.425   0.000
41         Intercepts     SP -0.038 0.008  -4.589   0.000
42         Intercepts     IC  7.795 0.274  28.419   0.000
43         Intercepts     SC  0.003 0.075   0.038   0.970
44          Variances   EDUC  9.633 0.552  17.438   0.000
45          Variances   BAGE 63.201 2.822  22.392   0.000
46          Variances HEIGHT  0.006 0.000  12.355   0.000
47 Residual.Variances     P1  0.027 0.005   5.711   0.000
48 Residual.Variances     P2  0.027 0.005   5.711   0.000
49 Residual.Variances     P3  0.027 0.005   5.711   0.000
50 Residual.Variances     P4  0.027 0.005   5.711   0.000
51 Residual.Variances     P5  0.027 0.005   5.711   0.000
52 Residual.Variances     C1  3.848 0.165  23.327   0.000
53 Residual.Variances     C2  3.848 0.165  23.327   0.000
54 Residual.Variances     C3  3.848 0.165  23.327   0.000
55 Residual.Variances     C4  3.848 0.165  23.327   0.000
56 Residual.Variances     C5  3.848 0.165  23.327   0.000
57 Residual.Variances     IP  0.112 0.008  14.254   0.000
58 Residual.Variances     SP  0.000 0.001   0.635   0.525
59 Residual.Variances     IC  4.521 0.343  13.173   0.000
60 Residual.Variances     SC  0.014 0.032   0.443   0.658

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Model Specification

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