Articles • mkin

The mkin package is used in version 1.2.11, which is contains the functions that were used for the evaluations. The saemix package is used as a backend for fitting the NLHM, but is also loaded to make the convergence plot function available.

For obtaining fits with only well-defined random effects, we update -the set of fits, excluding random effects that were ill-defined -according to the illparms function.

DFOP @@ -1805,7 +1807,9 @@ identifiable parameters k2 and g.

 dfop_pH <- saem(f_sep_const["DFOP", ], no_random_effect = c("meso_0", "log_k1"),
   covariates = pH,
-  covariate_models = list(log_k2 ~ pH, g_qlogis ~ pH))

+ covariate_models = list(log_k2 ~ pH, g_qlogis ~ pH), + center_covariates = "median" +)

The corresponding parameters for the influence of soil pH are beta_pH(log_k2) for the influence of soil pH on k2, and beta_pH(g_qlogis) for its influence on @@ -1834,9 +1838,9 @@ identifiable parameters k2 and g.

As the random effect for g is again ill-defined, the fit @@ -1913,7 +1918,7 @@ on the likelihood ratio test.

SFORB @@ -1961,7 +1966,8 @@ meso 8.346428 28.34437 8.532507 4.191901 8.753618

 sforb_pH <- saem(f_sep_const["SFORB", ], no_random_effect = c("meso_free_0", "log_k_meso_free_bound"),
   covariates = pH,
-  covariate_models = list(log_k_meso_free ~ pH, log_k_meso_bound_free ~ pH))

+ covariate_models = list(log_k_meso_free ~ pH, log_k_meso_bound_free ~ pH), + center_covariates = "median")

The DFOP model with pH influence on k2 and g and a random effect only on k2 is finally @@ -2386,7 +2393,7 @@ refer to the Appendix for a detailed listing.

@@ -2418,11 +2425,11 @@ further refined to make them fully identifiable.

Hierarchical SFO fit with constant variance


-saemix version used for fitting:      3.3 
-mkin version used for pre-fitting:  1.2.10 
-R version used for fitting:         4.5.0 
-Date of fit:     Wed May 14 05:12:46 2025 
-Date of summary: Wed May 14 05:13:35 2025 
+saemix version used for fitting:      3.4 
+mkin version used for pre-fitting:  1.2.11 
+R version used for fitting:         4.5.1 
+Date of fit:     Fri Sep 12 22:15:01 2025 
+Date of summary: Fri Sep 12 22:15:50 2025 
 
 Equations:
 d_meso/dt = - k_meso * meso
@@ -2432,7 +2439,7 @@ Data:
 
 Model predictions using solution type analytical 
 
-Fitted in 0.71 s
+Fitted in 0.602 s
 Using 300, 100 iterations and 3 chains
 
 Variance model: Constant variance 
@@ -2495,11 +2502,11 @@ meso 16.41 54.5
 Hierarchical FOMC fit with constant variance
 
 
-saemix version used for fitting:      3.3 
-mkin version used for pre-fitting:  1.2.10 
-R version used for fitting:         4.5.0 
-Date of fit:     Wed May 14 05:12:46 2025 
-Date of summary: Wed May 14 05:13:35 2025 
+saemix version used for fitting:      3.4 
+mkin version used for pre-fitting:  1.2.11 
+R version used for fitting:         4.5.1 
+Date of fit:     Fri Sep 12 22:15:02 2025 
+Date of summary: Fri Sep 12 22:15:50 2025 
 
 Equations:
 d_meso/dt = - (alpha/beta) * 1/((time/beta) + 1) * meso
@@ -2509,7 +2516,7 @@ Data:
 
 Model predictions using solution type analytical 
 
-Fitted in 0.842 s
+Fitted in 0.983 s
 Using 300, 100 iterations and 3 chains
 
 Variance model: Constant variance 
@@ -2578,11 +2585,11 @@ meso 14.8 74.64    22.47
 Hierarchical DFOP fit with constant variance
 
 
-saemix version used for fitting:      3.3 
-mkin version used for pre-fitting:  1.2.10 
-R version used for fitting:         4.5.0 
-Date of fit:     Wed May 14 05:12:47 2025 
-Date of summary: Wed May 14 05:13:35 2025 
+saemix version used for fitting:      3.4 
+mkin version used for pre-fitting:  1.2.11 
+R version used for fitting:         4.5.1 
+Date of fit:     Fri Sep 12 22:15:02 2025 
+Date of summary: Fri Sep 12 22:15:50 2025 
 
 Equations:
 d_meso/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
@@ -2594,7 +2601,7 @@ Data:
 
 Model predictions using solution type analytical 
 
-Fitted in 1.168 s
+Fitted in 1.202 s
 Using 300, 100 iterations and 3 chains
 
 Variance model: Constant variance 
@@ -2669,11 +2676,11 @@ meso 16.04 63.75    19.19   3.931    20.8
 Hierarchical SFORB fit with constant variance
 
 
-saemix version used for fitting:      3.3 
-mkin version used for pre-fitting:  1.2.10 
-R version used for fitting:         4.5.0 
-Date of fit:     Wed May 14 05:12:47 2025 
-Date of summary: Wed May 14 05:13:35 2025 
+saemix version used for fitting:      3.4 
+mkin version used for pre-fitting:  1.2.11 
+R version used for fitting:         4.5.1 
+Date of fit:     Fri Sep 12 22:15:02 2025 
+Date of summary: Fri Sep 12 22:15:50 2025 
 
 Equations:
 d_meso_free/dt = - k_meso_free * meso_free - k_meso_free_bound *
@@ -2686,7 +2693,7 @@ Data:
 
 Model predictions using solution type analytical 
 
-Fitted in 1.256 s
+Fitted in 1.557 s
 Using 300, 100 iterations and 3 chains
 
 Variance model: Constant variance 
@@ -2776,11 +2783,11 @@ meso 14.79 60.81     18.3        4.521        20.37
 Hierarchical HS fit with constant variance
 
 
-saemix version used for fitting:      3.3 
-mkin version used for pre-fitting:  1.2.10 
-R version used for fitting:         4.5.0 
-Date of fit:     Wed May 14 05:12:48 2025 
-Date of summary: Wed May 14 05:13:35 2025 
+saemix version used for fitting:      3.4 
+mkin version used for pre-fitting:  1.2.11 
+R version used for fitting:         4.5.1 
+Date of fit:     Fri Sep 12 22:15:03 2025 
+Date of summary: Fri Sep 12 22:15:50 2025 
 
 Equations:
 d_meso/dt = - ifelse(time <= tb, k1, k2) * meso
@@ -2790,7 +2797,7 @@ Data:
 
 Model predictions using solution type analytical 
 
-Fitted in 1.653 s
+Fitted in 1.61 s
 Using 300, 100 iterations and 3 chains
 
 Variance model: Constant variance 
@@ -2869,11 +2876,11 @@ meso   16   76    22.88    11.1   25.84
 Hierarchichal SFO fit with pH influence
 
 
-saemix version used for fitting:      3.3 
-mkin version used for pre-fitting:  1.2.10 
-R version used for fitting:         4.5.0 
-Date of fit:     Wed May 14 05:13:00 2025 
-Date of summary: Wed May 14 05:13:35 2025 
+saemix version used for fitting:      3.4 
+mkin version used for pre-fitting:  1.2.11 
+R version used for fitting:         4.5.1 
+Date of fit:     Fri Sep 12 22:15:15 2025 
+Date of summary: Fri Sep 12 22:15:50 2025 
 
 Equations:
 d_meso/dt = - k_meso * meso
@@ -2883,7 +2890,7 @@ Data:
 
 Model predictions using solution type analytical 
 
-Fitted in 1.343 s
+Fitted in 0.924 s
 Using 300, 100 iterations and 3 chains
 
 Variance model: Constant variance 
@@ -2913,15 +2920,15 @@ Likelihood computed by importance sampling
 Optimised parameters:
                        est.   lower   upper
 meso_0              91.3481 89.2688 93.4275
-log_k_meso          -6.6614 -7.9715 -5.3514
+log_k_meso          -3.2854 -3.4590 -3.1118
 beta_pH(log_k_meso)  0.5871  0.3684  0.8059
 a.1                  5.4750  4.7085  6.2415
 SD.log_k_meso        0.3471  0.2258  0.4684
 
 Correlation: 
                     meso_0  lg_k_ms
-log_k_meso           0.0414        
-beta_pH(log_k_meso) -0.0183 -0.9917
+log_k_meso           0.1797        
+beta_pH(log_k_meso) -0.0183 -0.2379
 
 Random effects:
                 est.  lower  upper
@@ -2932,9 +2939,9 @@ Variance model:
 a.1 5.475 4.709 6.242
 
 Backtransformed parameters:
-            est.     lower     upper
-meso_0 91.348139 8.927e+01 93.427476
-k_meso  0.001279 3.452e-04  0.004741
+           est.    lower    upper
+meso_0 91.34814 89.26880 93.42748
+k_meso  0.03743  0.03146  0.04452
 
 Covariates used for endpoints below:
       pH
@@ -2950,11 +2957,11 @@ meso 18.52 61.52
 Hierarchichal FOMC fit with pH influence
 
 
-saemix version used for fitting:      3.3 
-mkin version used for pre-fitting:  1.2.10 
-R version used for fitting:         4.5.0 
-Date of fit:     Wed May 14 05:13:03 2025 
-Date of summary: Wed May 14 05:13:35 2025 
+saemix version used for fitting:      3.4 
+mkin version used for pre-fitting:  1.2.11 
+R version used for fitting:         4.5.1 
+Date of fit:     Fri Sep 12 22:15:18 2025 
+Date of summary: Fri Sep 12 22:15:50 2025 
 
 Equations:
 d_meso/dt = - (alpha/beta) * 1/((time/beta) + 1) * meso
@@ -2964,7 +2971,7 @@ Data:
 
 Model predictions using solution type analytical 
 
-Fitted in 1.897 s
+Fitted in 1.901 s
 Using 300, 100 iterations and 3 chains
 
 Variance model: Constant variance 
@@ -2995,7 +3002,7 @@ Likelihood computed by importance sampling
 Optimised parameters:
                         est.      lower   upper
 meso_0             92.840646  90.750461 94.9308
-log_alpha          -2.206602  -3.494546 -0.9187
+log_alpha           1.114051   0.475668  1.7524
 beta_pH(log_alpha)  0.577505   0.369805  0.7852
 log_beta            4.214099   3.438851  4.9893
 a.1                 5.027768   4.322028  5.7335
@@ -3004,9 +3011,9 @@ SD.log_beta         0.374640   0.009252  0.7400
 
 Correlation: 
                    meso_0  log_lph bt_H(_)
-log_alpha          -0.0865                
-beta_pH(log_alpha) -0.0789 -0.8704        
-log_beta           -0.3544  0.3302  0.1628
+log_alpha          -0.3220                
+beta_pH(log_alpha) -0.0789  0.1148        
+log_beta           -0.3544  0.9709  0.1628
 
 Random effects:
                  est.      lower upper
@@ -3018,10 +3025,10 @@ Variance model:
 a.1 5.028 4.322 5.734
 
 Backtransformed parameters:
-          est.    lower    upper
-meso_0 92.8406 90.75046  94.9308
-alpha   0.1101  0.03036   0.3991
-beta   67.6332 31.15113 146.8404
+         est.  lower   upper
+meso_0 92.841 90.750  94.931
+alpha   3.047  1.609   5.769
+beta   67.633 31.151 146.840
 
 Covariates used for endpoints below:
       pH
@@ -3037,11 +3044,11 @@ meso 17.28 76.37    22.99
 Refined hierarchichal FOMC fit with pH influence
 
 
-saemix version used for fitting:      3.3 
-mkin version used for pre-fitting:  1.2.10 
-R version used for fitting:         4.5.0 
-Date of fit:     Wed May 14 05:13:08 2025 
-Date of summary: Wed May 14 05:13:35 2025 
+saemix version used for fitting:      3.4 
+mkin version used for pre-fitting:  1.2.11 
+R version used for fitting:         4.5.1 
+Date of fit:     Fri Sep 12 22:15:21 2025 
+Date of summary: Fri Sep 12 22:15:50 2025 
 
 Equations:
 d_meso/dt = - (alpha/beta) * 1/((time/beta) + 1) * meso
@@ -3051,7 +3058,7 @@ Data:
 
 Model predictions using solution type analytical 
 
-Fitted in 4.184 s
+Fitted in 3.35 s
 Using 300, 100 iterations and 3 chains
 
 Variance model: Constant variance 
@@ -3077,44 +3084,44 @@ Results:
 
 Likelihood computed by importance sampling
     AIC   BIC logLik
-  767.5 772.8 -377.7
+  766.5 771.8 -377.3
 
 Optimised parameters:
                       est.   lower   upper
-meso_0             93.0536 90.9771 95.1300
-log_alpha          -2.9054 -4.1803 -1.6304
-beta_pH(log_alpha)  0.6590  0.4437  0.8744
-log_beta            3.9549  3.2860  4.6239
-a.1                 4.9784  4.2815  5.6754
-SD.log_beta         0.4019  0.2632  0.5406
+meso_0             93.1950 91.1047 95.2853
+log_alpha           0.8125  0.3292  1.2957
+beta_pH(log_alpha)  0.5497  0.3490  0.7504
+log_beta            3.8464  3.2179  4.4750
+a.1                 4.9972  4.2976  5.6968
+SD.log_beta         0.3829  0.2499  0.5159
 
 Correlation: 
                    meso_0  log_lph bt_H(_)
-log_alpha          -0.0397                
-beta_pH(log_alpha) -0.0899 -0.9146        
-log_beta           -0.3473  0.2038  0.1919
+log_alpha          -0.3094                
+beta_pH(log_alpha) -0.0779  0.0841        
+log_beta           -0.3493  0.9533  0.1434
 
 Random effects:
               est.  lower  upper
-SD.log_beta 0.4019 0.2632 0.5406
+SD.log_beta 0.3829 0.2499 0.5159
 
 Variance model:
      est. lower upper
-a.1 4.978 4.281 5.675
+a.1 4.997 4.298 5.697
 
 Backtransformed parameters:
-           est.    lower    upper
-meso_0 93.05359 90.97713  95.1300
-alpha   0.05473  0.01529   0.1958
-beta   52.19251 26.73597 101.8874
+         est. lower  upper
+meso_0 93.195 91.10 95.285
+alpha   2.253  1.39  3.654
+beta   46.826 24.98 87.793
 
 Covariates used for endpoints below:
       pH
 50% 5.75
 
 Estimated disappearance times:
-     DT50  DT90 DT50back
-meso 17.3 82.91    24.96
+      DT50  DT90 DT50back
+meso 16.86 83.27    25.07
 
 
 
@@ -3122,11 +3129,11 @@ meso 17.3 82.91    24.96
 Hierarchichal DFOP fit with pH influence
 
 
-saemix version used for fitting:      3.3 
-mkin version used for pre-fitting:  1.2.10 
-R version used for fitting:         4.5.0 
-Date of fit:     Wed May 14 05:13:11 2025 
-Date of summary: Wed May 14 05:13:35 2025 
+saemix version used for fitting:      3.4 
+mkin version used for pre-fitting:  1.2.11 
+R version used for fitting:         4.5.1 
+Date of fit:     Fri Sep 12 22:15:25 2025 
+Date of summary: Fri Sep 12 22:15:50 2025 
 
 Equations:
 d_meso/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
@@ -3138,7 +3145,7 @@ Data:
 
 Model predictions using solution type analytical 
 
-Fitted in 2.18 s
+Fitted in 2.592 s
 Using 300, 100 iterations and 3 chains
 
 Variance model: Constant variance 
@@ -3168,24 +3175,24 @@ Likelihood computed by importance sampling
   769.1 777.1 -375.5
 
 Optimised parameters:
-                        est.    lower    upper
-meso_0             92.843344  90.8464 94.84028
-log_k1             -2.815685  -3.0888 -2.54261
-log_k2            -11.479779 -15.3203 -7.63923
-beta_pH(log_k2)     1.308417   0.6948  1.92203
-g_qlogis            3.133036   0.4657  5.80035
-beta_pH(g_qlogis)  -0.565988  -1.0394 -0.09262
-a.1                 4.955518   4.2597  5.65135
-SD.log_k2           0.758963   0.4685  1.04943
-SD.g_qlogis         0.005215  -9.9561  9.96656
+                       est.   lower    upper
+meso_0            92.843344 90.8464 94.84028
+log_k1            -2.815685 -3.0888 -2.54261
+log_k2            -3.956384 -4.4741 -3.43868
+beta_pH(log_k2)    1.308417  0.6948  1.92203
+g_qlogis          -0.121394 -0.5691  0.32627
+beta_pH(g_qlogis) -0.565988 -1.0394 -0.09262
+a.1                4.955518  4.2597  5.65135
+SD.log_k2          0.758963  0.4685  1.04943
+SD.g_qlogis        0.005215 -9.9561  9.96656
 
 Correlation: 
                   meso_0  log_k1  log_k2  b_H(_2) g_qlogs
 log_k1             0.2706                                
-log_k2            -0.0571  0.1096                        
-beta_pH(log_k2)    0.0554 -0.1291 -0.9937                
-g_qlogis          -0.1125 -0.5062 -0.1305  0.1294        
-beta_pH(g_qlogis)  0.1267  0.4226  0.0419 -0.0438 -0.9864
+log_k2            -0.0457 -0.0667                        
+beta_pH(log_k2)    0.0554 -0.1291 -0.5566                
+g_qlogis           0.1004 -0.4462 -0.4397  0.5042        
+beta_pH(g_qlogis)  0.1267  0.4226  0.0123 -0.0438  0.2029
 
 Random effects:
                 est.   lower upper
@@ -3197,11 +3204,11 @@ Variance model:
 a.1 4.956  4.26 5.651
 
 Backtransformed parameters:
-            est.     lower     upper
-meso_0 9.284e+01 9.085e+01 9.484e+01
-k1     5.986e-02 4.556e-02 7.866e-02
-k2     1.034e-05 2.221e-07 4.812e-04
-g      9.582e-01 6.144e-01 9.970e-01
+           est.    lower    upper
+meso_0 92.84334 90.84641 94.84028
+k1      0.05986  0.04556  0.07866
+k2      0.01913  0.01140  0.03211
+g       0.46969  0.36145  0.58085
 
 Covariates used for endpoints below:
       pH
@@ -3217,11 +3224,11 @@ meso 20.23 88.45    26.62   11.58   36.23
 Refined hierarchical DFOP fit with pH influence
 
 
-saemix version used for fitting:      3.3 
-mkin version used for pre-fitting:  1.2.10 
-R version used for fitting:         4.5.0 
-Date of fit:     Wed May 14 05:13:14 2025 
-Date of summary: Wed May 14 05:13:35 2025 
+saemix version used for fitting:      3.4 
+mkin version used for pre-fitting:  1.2.11 
+R version used for fitting:         4.5.1 
+Date of fit:     Fri Sep 12 22:15:30 2025 
+Date of summary: Fri Sep 12 22:15:50 2025 
 
 Equations:
 d_meso/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
@@ -3233,7 +3240,7 @@ Data:
 
 Model predictions using solution type analytical 
 
-Fitted in 2.424 s
+Fitted in 4.483 s
 Using 300, 100 iterations and 3 chains
 
 Variance model: Constant variance 
@@ -3260,41 +3267,41 @@ Results:
 
 Likelihood computed by importance sampling
     AIC   BIC logLik
-  765.1 772.3 -374.6
+  765.1 772.2 -374.5
 
 Optimised parameters:
-                     est.    lower    upper
-meso_0            93.3333  91.2427 95.42394
-log_k1            -1.7997  -2.9124 -0.68698
-log_k2            -8.1810 -10.1819 -6.18008
-beta_pH(log_k2)    0.8064   0.4903  1.12257
-g_qlogis           3.3513  -1.1792  7.88182
-beta_pH(g_qlogis) -0.8672  -1.7661  0.03177
-a.1                4.9158   4.2277  5.60390
-SD.log_k2          0.3946   0.2565  0.53281
+                     est.   lower    upper
+meso_0            93.1612 91.0766 95.24580
+log_k1            -1.8857 -2.8975 -0.87395
+log_k2            -3.5711 -3.8859 -3.25622
+beta_pH(log_k2)    0.8252  0.4952  1.15513
+g_qlogis          -1.5326 -2.6994 -0.36574
+beta_pH(g_qlogis) -0.8365 -1.7163  0.04333
+a.1                4.9218  4.2328  5.61070
+SD.log_k2          0.4011  0.2604  0.54184
 
 Correlation: 
                   meso_0  log_k1  log_k2  b_H(_2) g_qlogs
-log_k1             0.1730                                
-log_k2             0.0442  0.5370                        
-beta_pH(log_k2)   -0.0392 -0.4880 -0.9923                
-g_qlogis          -0.1536  0.1431 -0.1129  0.1432        
-beta_pH(g_qlogis)  0.1504 -0.3151 -0.0196 -0.0212 -0.9798
+log_k1             0.1734                                
+log_k2             0.0553  0.6421                        
+beta_pH(log_k2)   -0.0382 -0.5121 -0.5860                
+g_qlogis           0.0616 -0.8501 -0.7251  0.4941        
+beta_pH(g_qlogis)  0.1408 -0.3240 -0.2716 -0.0041  0.6264
 
 Random effects:
             est.  lower  upper
-SD.log_k2 0.3946 0.2565 0.5328
+SD.log_k2 0.4011 0.2604 0.5418
 
 Variance model:
      est. lower upper
-a.1 4.916 4.228 5.604
+a.1 4.922 4.233 5.611
 
 Backtransformed parameters:
-            est.     lower    upper
-meso_0 9.333e+01 9.124e+01 95.42394
-k1     1.654e-01 5.435e-02  0.50309
-k2     2.799e-04 3.785e-05  0.00207
-g      9.661e-01 2.352e-01  0.99962
+           est.    lower    upper
+meso_0 93.16118 91.07655 95.24580
+k1      0.15172  0.05516  0.41730
+k2      0.02813  0.02053  0.03853
+g       0.17762  0.06301  0.40957
 
 Covariates used for endpoints below:
       pH
@@ -3302,7 +3309,7 @@ Covariates used for endpoints below:
 
 Estimated disappearance times:
       DT50  DT90 DT50back DT50_k1 DT50_k2
-meso 18.37 73.52    22.13   4.192   23.99
+meso 18.47 74.92    22.55   4.569   24.64
 
 
 
@@ -3310,11 +3317,11 @@ meso 18.37 73.52    22.13   4.192   23.99
 Further refined hierarchical DFOP fit with pH influence
 
 
-saemix version used for fitting:      3.3 
-mkin version used for pre-fitting:  1.2.10 
-R version used for fitting:         4.5.0 
-Date of fit:     Wed May 14 05:13:23 2025 
-Date of summary: Wed May 14 05:13:35 2025 
+saemix version used for fitting:      3.4 
+mkin version used for pre-fitting:  1.2.11 
+R version used for fitting:         4.5.1 
+Date of fit:     Fri Sep 12 22:15:37 2025 
+Date of summary: Fri Sep 12 22:15:50 2025 
 
 Equations:
 d_meso/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
@@ -3326,7 +3333,7 @@ Data:
 
 Model predictions using solution type analytical 
 
-Fitted in 3.211 s
+Fitted in 3.213 s
 Using 300, 100 iterations and 3 chains
 
 Variance model: Constant variance 
@@ -3356,21 +3363,21 @@ Likelihood computed by importance sampling
   767.4 773.6 -376.7
 
 Optimised parameters:
-                   est.    lower   upper
-meso_0          93.3011  91.1905 95.4118
-log_k1          -2.1487  -2.7607 -1.5367
-log_k2          -8.1039 -10.4225 -5.7853
-beta_pH(log_k2)  0.7821   0.4126  1.1517
-g_qlogis        -1.0373  -1.9337 -0.1409
-a.1              5.0095   4.3082  5.7108
-SD.log_k2        0.4622   0.3009  0.6235
+                   est.   lower   upper
+meso_0          93.3011 91.1905 95.4118
+log_k1          -2.1487 -2.7607 -1.5367
+log_k2          -3.6066 -3.9305 -3.2828
+beta_pH(log_k2)  0.7821  0.4126  1.1517
+g_qlogis        -1.0373 -1.9337 -0.1409
+a.1              5.0095  4.3082  5.7108
+SD.log_k2        0.4622  0.3009  0.6235
 
 Correlation: 
                 meso_0  log_k1  log_k2  b_H(_2)
 log_k1           0.2179                        
-log_k2           0.0337  0.5791                
-beta_pH(log_k2) -0.0326 -0.5546 -0.9932        
-g_qlogis         0.0237 -0.8479 -0.6571  0.6123
+log_k2           0.0271  0.5067                
+beta_pH(log_k2) -0.0326 -0.5546 -0.5485        
+g_qlogis         0.0237 -0.8479 -0.6866  0.6123
 
 Random effects:
             est.  lower  upper
@@ -3381,11 +3388,11 @@ Variance model:
 a.1 5.009 4.308 5.711
 
 Backtransformed parameters:
-            est.     lower     upper
-meso_0 9.330e+01 9.119e+01 95.411751
-k1     1.166e-01 6.325e-02  0.215084
-k2     3.024e-04 2.975e-05  0.003072
-g      2.617e-01 1.263e-01  0.464832
+           est.    lower    upper
+meso_0 93.30113 91.19050 95.41175
+k1      0.11664  0.06325  0.21508
+k2      0.02714  0.01963  0.03752
+g       0.26168  0.12635  0.46483
 
 Covariates used for endpoints below:
       pH
@@ -3401,11 +3408,11 @@ meso 17.09 73.67    22.18   5.943   25.54
 Hierarchichal SFORB fit with pH influence
 
 
-saemix version used for fitting:      3.3 
-mkin version used for pre-fitting:  1.2.10 
-R version used for fitting:         4.5.0 
-Date of fit:     Wed May 14 05:13:26 2025 
-Date of summary: Wed May 14 05:13:35 2025 
+saemix version used for fitting:      3.4 
+mkin version used for pre-fitting:  1.2.11 
+R version used for fitting:         4.5.1 
+Date of fit:     Fri Sep 12 22:15:41 2025 
+Date of summary: Fri Sep 12 22:15:50 2025 
 
 Equations:
 d_meso_free/dt = - k_meso_free * meso_free - k_meso_free_bound *
@@ -3418,7 +3425,7 @@ Data:
 
 Model predictions using solution type analytical 
 
-Fitted in 2.649 s
+Fitted in 2.758 s
 Using 300, 100 iterations and 3 chains
 
 Variance model: Constant variance 
@@ -3455,24 +3462,24 @@ Likelihood computed by importance sampling
   768.8 776.8 -375.4
 
 Optimised parameters:
-                                  est.    lower   upper
-meso_free_0                    93.4204  91.3213 95.5195
-log_k_meso_free                -5.3742  -6.9366 -3.8117
-beta_pH(log_k_meso_free)        0.4232   0.1769  0.6695
-log_k_meso_free_bound          -3.4889  -4.9243 -2.0535
-log_k_meso_bound_free          -9.9797 -19.2232 -0.7362
-beta_pH(log_k_meso_bound_free)  1.2290  -0.2107  2.6687
-a.1                             4.9031   4.1795  5.6268
-SD.log_k_meso_free              0.3454   0.2252  0.4656
-SD.log_k_meso_bound_free        0.1277  -1.9459  2.2012
+                                  est.   lower   upper
+meso_free_0                    93.4204 91.3213 95.5195
+log_k_meso_free                -2.9408 -3.2344 -2.6471
+beta_pH(log_k_meso_free)        0.4232  0.1769  0.6695
+log_k_meso_free_bound          -3.4889 -4.9243 -2.0535
+log_k_meso_bound_free          -2.9130 -4.3047 -1.5212
+beta_pH(log_k_meso_bound_free)  1.2290 -0.2107  2.6687
+a.1                             4.9031  4.1795  5.6268
+SD.log_k_meso_free              0.3454  0.2252  0.4656
+SD.log_k_meso_bound_free        0.1277 -1.9459  2.2012
 
 Correlation: 
                                ms_fr_0 lg_k_m_ b_H(___) lg_k_ms_f_ lg_k_ms_b_
-log_k_meso_free                 0.1493                                       
-beta_pH(log_k_meso_free)       -0.0930 -0.9854                               
-log_k_meso_free_bound           0.2439  0.4621 -0.3492                       
-log_k_meso_bound_free           0.2188  0.1292 -0.0339   0.7287              
-beta_pH(log_k_meso_bound_free) -0.2216 -0.0797 -0.0111  -0.6566    -0.9934   
+log_k_meso_free                 0.3460                                       
+beta_pH(log_k_meso_free)       -0.0930 -0.4206                               
+log_k_meso_free_bound           0.2439  0.7749 -0.3492                       
+log_k_meso_bound_free           0.1350  0.6400 -0.2912   0.9346              
+beta_pH(log_k_meso_bound_free) -0.2216 -0.4778 -0.0111  -0.6566    -0.6498   
 
 Random effects:
                            est.   lower  upper
@@ -3484,11 +3491,11 @@ Variance model:
 a.1 4.903  4.18 5.627
 
 Backtransformed parameters:
-                       est.     lower    upper
-meso_free_0       9.342e+01 9.132e+01 95.51946
-k_meso_free       4.635e-03 9.716e-04  0.02211
-k_meso_free_bound 3.054e-02 7.268e-03  0.12829
-k_meso_bound_free 4.633e-05 4.482e-09  0.47894
+                      est.     lower    upper
+meso_free_0       93.42040 91.321340 95.51946
+k_meso_free        0.05282  0.039384  0.07085
+k_meso_free_bound  0.03054  0.007268  0.12829
+k_meso_bound_free  0.05431  0.013505  0.21845
 
 Covariates used for endpoints below:
       pH
@@ -3512,11 +3519,11 @@ meso 16.42 75.2    22.64        6.185        27.08
 Refined hierarchichal SFORB fit with pH influence
 
 
-saemix version used for fitting:      3.3 
-mkin version used for pre-fitting:  1.2.10 
-R version used for fitting:         4.5.0 
-Date of fit:     Wed May 14 05:13:30 2025 
-Date of summary: Wed May 14 05:13:35 2025 
+saemix version used for fitting:      3.4 
+mkin version used for pre-fitting:  1.2.11 
+R version used for fitting:         4.5.1 
+Date of fit:     Fri Sep 12 22:15:44 2025 
+Date of summary: Fri Sep 12 22:15:50 2025 
 
 Equations:
 d_meso_free/dt = - k_meso_free * meso_free - k_meso_free_bound *
@@ -3529,7 +3536,7 @@ Data:
 
 Model predictions using solution type analytical 
 
-Fitted in 3.186 s
+Fitted in 2.971 s
 Using 300, 100 iterations and 3 chains
 
 Variance model: Constant variance 
@@ -3567,20 +3574,20 @@ Likelihood computed by importance sampling
 
 Optimised parameters:
                             est.   lower   upper
-meso_free_0              93.3196 91.1633 95.4760
-log_k_meso_free          -6.1460 -7.4306 -4.8614
+meso_free_0              93.3196 91.1650 95.4743
+log_k_meso_free          -3.0207 -3.2655 -2.7759
 beta_pH(log_k_meso_free)  0.5435  0.3329  0.7542
-log_k_meso_free_bound    -3.8001 -5.2027 -2.3975
-log_k_meso_bound_free    -2.9462 -4.2565 -1.6359
-a.1                       5.0825  4.3793  5.7856
+log_k_meso_free_bound    -3.8001 -5.1809 -2.4193
+log_k_meso_bound_free    -2.9462 -4.2411 -1.6513
+a.1                       5.0825  4.3709  5.7940
 SD.log_k_meso_free        0.3338  0.2175  0.4502
 
 Correlation: 
                          ms_fr_0 lg_k_m_ b_H(___ lg_k_ms_f_
-log_k_meso_free           0.1086                           
-beta_pH(log_k_meso_free) -0.0426 -0.9821                   
-log_k_meso_free_bound     0.2513  0.1717 -0.0409           
-log_k_meso_bound_free     0.1297  0.1171 -0.0139  0.9224   
+log_k_meso_free           0.3556                           
+beta_pH(log_k_meso_free) -0.0422 -0.2041                   
+log_k_meso_free_bound     0.2513  0.6866 -0.0400           
+log_k_meso_bound_free     0.1292  0.5341 -0.0129  0.9214   
 
 Random effects:
                      est.  lower  upper
@@ -3588,14 +3595,14 @@ SD.log_k_meso_free 0.3338 0.2175 0.4502
 
 Variance model:
      est. lower upper
-a.1 5.082 4.379 5.786
+a.1 5.082 4.371 5.794
 
 Backtransformed parameters:
-                       est.     lower    upper
-meso_free_0       93.319649 9.116e+01 95.47601
-k_meso_free        0.002142 5.928e-04  0.00774
-k_meso_free_bound  0.022369 5.502e-03  0.09095
-k_meso_bound_free  0.052539 1.417e-02  0.19478
+                      est.     lower    upper
+meso_free_0       93.31965 91.165044 95.47425
+k_meso_free        0.04877  0.038177  0.06230
+k_meso_free_bound  0.02237  0.005623  0.08898
+k_meso_bound_free  0.05254  0.014392  0.19179
 
 Covariates used for endpoints below:
       pH
@@ -3619,11 +3626,11 @@ meso 16.87 73.16    22.02         7.12        26.34
 Hierarchichal HS fit with pH influence
 
 
-saemix version used for fitting:      3.3 
-mkin version used for pre-fitting:  1.2.10 
-R version used for fitting:         4.5.0 
-Date of fit:     Wed May 14 05:13:32 2025 
-Date of summary: Wed May 14 05:13:35 2025 
+saemix version used for fitting:      3.4 
+mkin version used for pre-fitting:  1.2.11 
+R version used for fitting:         4.5.1 
+Date of fit:     Fri Sep 12 22:15:48 2025 
+Date of summary: Fri Sep 12 22:15:50 2025 
 
 Equations:
 d_meso/dt = - ifelse(time <= tb, k1, k2) * meso
@@ -3633,7 +3640,7 @@ Data:
 
 Model predictions using solution type analytical 
 
-Fitted in 1.833 s
+Fitted in 2.254 s
 Using 300, 100 iterations and 3 chains
 
 Variance model: Constant variance 
@@ -3665,11 +3672,11 @@ Likelihood computed by importance sampling
 Optimised parameters:
                     est.   lower   upper
 meso_0          93.32599 91.4658 95.1862
-log_k1          -5.81463 -7.2710 -4.3583
+log_k1          -3.08497 -3.2830 -2.8870
 beta_pH(log_k1)  0.47472  0.2334  0.7160
-log_k2          -6.79633 -8.7605 -4.8322
+log_k2          -3.68267 -3.9077 -3.4577
 beta_pH(log_k2)  0.54151  0.2124  0.8706
-log_tb           3.24674  1.2470  5.2465
+log_tb           2.67815  2.3846  2.9717
 beta_pH(log_tb) -0.09889 -0.4258  0.2280
 a.1              4.49487  3.7766  5.2132
 SD.log_k1        0.37191  0.2370  0.5068
@@ -3678,12 +3685,12 @@ SD.log_tb        0.25353 -0.0664  0.5735
 
 Correlation: 
                 meso_0  log_k1  b_H(_1) log_k2  b_H(_2) log_tb 
-log_k1           0.0744                                        
-beta_pH(log_k1) -0.0452 -0.9915                                
-log_k2           0.0066 -0.0363  0.0376                        
-beta_pH(log_k2) -0.0071  0.0372 -0.0391 -0.9939                
-log_tb          -0.0238 -0.1483  0.1362 -0.3836  0.3696        
-beta_pH(log_tb)  0.0097  0.1359 -0.1265  0.3736 -0.3653 -0.9905
+log_k1           0.2301                                        
+beta_pH(log_k1) -0.0452 -0.2842                                
+log_k2          -0.0029 -0.0322 -0.0008                        
+beta_pH(log_k2) -0.0071  0.0000 -0.0391 -0.2655                
+log_tb          -0.1003 -0.2052  0.1180 -0.4197  0.1794        
+beta_pH(log_tb)  0.0097  0.1125 -0.1265  0.1894 -0.3653 -0.3449
 
 Random effects:
             est.   lower  upper
@@ -3696,11 +3703,11 @@ Variance model:
 a.1 4.495 3.777 5.213
 
 Backtransformed parameters:
-            est.     lower     upper
-meso_0 93.325994 9.147e+01 9.519e+01
-k1      0.002984 6.954e-04 1.280e-02
-k2      0.001118 1.568e-04 7.969e-03
-tb     25.706437 3.480e+00 1.899e+02
+           est.    lower    upper
+meso_0 93.32599 91.46575 95.18624
+k1      0.04573  0.03752  0.05574
+k2      0.02516  0.02009  0.03150
+tb     14.55810 10.85502 19.52445
 
 Covariates used for endpoints below:
       pH
@@ -3716,11 +3723,11 @@ meso 15.65 79.63    23.97   15.16   27.55
 Refined hierarchichal HS fit with pH influence
 
 
-saemix version used for fitting:      3.3 
-mkin version used for pre-fitting:  1.2.10 
-R version used for fitting:         4.5.0 
-Date of fit:     Wed May 14 05:13:35 2025 
-Date of summary: Wed May 14 05:13:35 2025 
+saemix version used for fitting:      3.4 
+mkin version used for pre-fitting:  1.2.11 
+R version used for fitting:         4.5.1 
+Date of fit:     Fri Sep 12 22:15:50 2025 
+Date of summary: Fri Sep 12 22:15:50 2025 
 
 Equations:
 d_meso/dt = - ifelse(time <= tb, k1, k2) * meso
@@ -3730,7 +3737,7 @@ Data:
 
 Model predictions using solution type analytical 
 
-Fitted in 1.852 s
+Fitted in 1.425 s
 Using 300, 100 iterations and 3 chains
 
 Variance model: Constant variance 
@@ -3762,9 +3769,9 @@ Likelihood computed by importance sampling
 Optimised parameters:
                    est.    lower   upper
 meso_0          93.3251 91.49823 95.1520
-log_k1          -5.6796 -7.08789 -4.2714
+log_k1          -3.0535 -3.24879 -2.8582
 beta_pH(log_k1)  0.4567  0.22400  0.6894
-log_k2          -6.6083 -8.33839 -4.8781
+log_k2          -3.7439 -3.94307 -3.5447
 beta_pH(log_k2)  0.4982  0.20644  0.7899
 log_tb           2.7040  2.33033  3.0777
 a.1              4.4452  3.73537  5.1551
@@ -3774,11 +3781,11 @@ SD.log_tb        0.5488  0.24560  0.8521
 
 Correlation: 
                 meso_0  log_k1  b_H(_1) log_k2  b_H(_2)
-log_k1           0.0740                                
-beta_pH(log_k1) -0.0453 -0.9912                        
-log_k2           0.0115 -0.0650  0.0661                
-beta_pH(log_k2) -0.0116  0.0649 -0.0667 -0.9936        
-log_tb          -0.0658 -0.1135  0.0913 -0.1500  0.1210
+log_k1           0.2233                                
+beta_pH(log_k1) -0.0453 -0.2955                        
+log_k2           0.0028 -0.0420  0.0126                
+beta_pH(log_k2) -0.0116  0.0112 -0.0667 -0.2097        
+log_tb          -0.0658 -0.1928  0.0913 -0.2843  0.1210
 
 Random effects:
             est.   lower  upper
@@ -3791,11 +3798,11 @@ Variance model:
 a.1 4.445 3.735 5.155
 
 Backtransformed parameters:
-            est.     lower     upper
-meso_0 93.325134 9.150e+01 95.152036
-k1      0.003415 8.352e-04  0.013962
-k2      0.001349 2.392e-04  0.007611
-tb     14.939247 1.028e+01 21.707445
+           est.    lower    upper
+meso_0 93.32513 91.49823 95.15204
+k1      0.04719  0.03882  0.05737
+k2      0.02366  0.01939  0.02888
+tb     14.93925 10.28132 21.70744
 
 Covariates used for endpoints below:
       pH
@@ -3812,13 +3819,13 @@ meso 14.69 82.45    24.82   14.69   29.29
 
 Session info
 
-R version 4.5.0 (2025-04-11)
+R version 4.5.1 (2025-06-13)
 Platform: x86_64-pc-linux-gnu
-Running under: Debian GNU/Linux 12 (bookworm)
+Running under: Debian GNU/Linux 13 (trixie)
 
 Matrix products: default
-BLAS:   /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.11.0 
-LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.11.0  LAPACK version 3.11.0
+BLAS:   /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.12.1 
+LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.12.1;  LAPACK version 3.12.0
 
 locale:
  [1] LC_CTYPE=de_DE.UTF-8       LC_NUMERIC=C              
@@ -3836,29 +3843,29 @@ attached base packages:
 [8] base     
 
 other attached packages:
-[1] rmarkdown_2.29  nvimcom_0.9-167 saemix_3.3      npde_3.5       
-[5] knitr_1.49      mkin_1.2.10    
+[1] rmarkdown_2.29  nvimcom_0.9-167 saemix_3.4      npde_3.5       
+[5] knitr_1.50      mkin_1.2.11    
 
 loaded via a namespace (and not attached):
- [1] gtable_0.3.6      jsonlite_1.9.0    dplyr_1.1.4       compiler_4.5.0   
- [5] tidyselect_1.2.1  gridExtra_2.3     jquerylib_0.1.4   systemfonts_1.2.1
- [9] scales_1.3.0      textshaping_1.0.0 readxl_1.4.4      yaml_2.3.10      
-[13] fastmap_1.2.0     lattice_0.22-6    ggplot2_3.5.1     R6_2.6.1         
-[17] generics_0.1.3    lmtest_0.9-40     MASS_7.3-65       htmlwidgets_1.6.4
-[21] tibble_3.2.1      desc_1.4.3        munsell_0.5.1     bslib_0.9.0      
-[25] pillar_1.10.1     rlang_1.1.5       cachem_1.1.0      xfun_0.51        
-[29] fs_1.6.5          sass_0.4.9        cli_3.6.4         pkgdown_2.1.1    
-[33] magrittr_2.0.3    digest_0.6.37     grid_4.5.0        mclust_6.1.1     
-[37] lifecycle_1.0.4   nlme_3.1-168      vctrs_0.6.5       evaluate_1.0.3   
-[41] glue_1.8.0        cellranger_1.1.0  codetools_0.2-20  ragg_1.3.3       
-[45] zoo_1.8-13        colorspace_2.1-1  tools_4.5.0       pkgconfig_2.0.3  
-[49] htmltools_0.5.8.1
+ [1] gtable_0.3.6       jsonlite_2.0.0     dplyr_1.1.4        compiler_4.5.1    
+ [5] tidyselect_1.2.1   dichromat_2.0-0.1  gridExtra_2.3      jquerylib_0.1.4   
+ [9] systemfonts_1.2.3  scales_1.4.0       textshaping_1.0.1  readxl_1.4.5      
+[13] yaml_2.3.10        fastmap_1.2.0      lattice_0.22-7     ggplot2_3.5.2     
+[17] R6_2.6.1           generics_0.1.4     lmtest_0.9-40      MASS_7.3-65       
+[21] htmlwidgets_1.6.4  tibble_3.3.0       desc_1.4.3         bslib_0.9.0       
+[25] pillar_1.10.2      RColorBrewer_1.1-3 rlang_1.1.6        cachem_1.1.0      
+[29] xfun_0.52          fs_1.6.6           sass_0.4.10        cli_3.6.5         
+[33] pkgdown_2.1.3      magrittr_2.0.3     digest_0.6.37      grid_4.5.1        
+[37] mclust_6.1.1       lifecycle_1.0.4    nlme_3.1-168       vctrs_0.6.5       
+[41] evaluate_1.0.3     glue_1.8.0         cellranger_1.1.0   farver_2.1.2      
+[45] codetools_0.2-20   ragg_1.4.0         zoo_1.8-14         colorspace_2.1-1  
+[49] tools_4.5.1        pkgconfig_2.0.3    htmltools_0.5.8.1 
 
 
 Hardware info
 
 CPU model: AMD Ryzen 9 7950X 16-Core Processor
-MemTotal:       64927780 kB
+MemTotal:       64933716 kB

log_k_meso	-6.66	-7.97	-5.35	-3.29	-3.46	-3.11
beta_pH(log_k_meso)

log_alpha	-2.21	-3.49	-0.92	1.11	0.48	1.75
beta_pH(log_alpha)
meso_0	93.05	90.98	95.13	93.20	91.10	95.29
log_alpha	-2.91	-4.18	-1.63	0.81	0.33	1.30
beta_pH(log_alpha)	0.66	0.44	0.87	0.55	0.35	0.75
log_beta	3.95	3.29	4.62	3.85	3.22	4.47
a.1	4.98	4.28	5.68	5.00	4.30	5.70
SD.log_beta	0.40	0.26	0.54	0.38	0.25	0.52

log_k_meso_free	-5.37	-6.94	-3.81	-2.94	-3.23	-2.65
beta_pH(log_k_meso_free)
log_k_meso_bound_free	-9.98	-19.22	-0.74	-2.91	-4.30	-1.52
beta_pH(log_k_meso_bound_free)
meso_free_0	93.32	91.16	95.48	91.17	95.47
log_k_meso_free	-6.15	-7.43	-4.86	-3.02	-3.27	-2.78
beta_pH(log_k_meso_free)
log_k_meso_free_bound	-3.80	-5.20	-2.40	-5.18	-2.42
log_k_meso_bound_free	-2.95	-4.26	-1.64	-4.24	-1.65
a.1	5.08	4.38	4.37	5.79

log_k1	-5.81	-7.27	-4.36	-3.08	-3.28	-2.89
beta_pH(log_k1)
log_k2	-6.80	-8.76	-4.83	-3.68	-3.91	-3.46
beta_pH(log_k2)
log_tb	3.25	1.25	5.25	2.68	2.38	2.97
beta_pH(log_tb)
log_k1	-5.68	-7.09	-4.27	-3.05	-3.25	-2.86
beta_pH(log_k1)
log_k2	-6.61	-8.34	-4.88	-3.74	-3.94	-3.54
beta_pH(log_k2)

Example evaluations with hierarchical models (nonlinear mixed-effects models)

Johannes Ranke

Last change 13 May 2025 -(rebuilt 2025-05-14)

Last change 12 September 2025 +(rebuilt 2025-09-12)

Session info

Hardware info