aboutsummaryrefslogblamecommitdiff
path: root/docs/articles/prebuilt/2022_dmta_parent.html
blob: 952d028eb6566bbd791c6401e3f88579a5ca3379 (plain) (tree)
1
2
3
4
5
6
7
8
9
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969





                                                                   

                                                                                      
                                                                                                                             


                                                                                                                                                                                                                                                                                                      
                                                                                                                                                                                                      
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      





                                                                                                                                                                                                                                                                                                                                                                                                                    

                                                                          

    


                                                                                                  
 










                                                                                                                                                                                
     


























                                                                                                                                                                                                   
     

                                                           

           







                                                                                                                                                                                                                 




           

    
        


                                              
 
 


                                                                                                                                                    



                                                                   
                                            

                                                                                                                                                                                                                         
                                                                      













                                                                                                 
                                                                       

















































































































































































































































































































































































































































































































































































































































































































































































































































































                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  
                        






                        
                        
































































































                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      


                             

















                                


                             















































































                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   

                 


                
                
















                                                

                            







                                                    




                             
                        































                                                                                                                                                                                                                                                                                      
                                          


                                          










                                                                      
               




                                                

                                    





                                                                             
                     











                                           
                    


                                       








                                       


                          


                          


                              


                              


                           

                           

                              

                                                      



































































































                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
                                          


                                          








                                                       
                 

























































                                                                             
                                          


                                          








                                                       
                 



























































                                                                             
                                          


                                          








                                                       
                






























                                                                             
                                  






























                                                           
                                          


                                          








                                                       
                 
























                                                                             
                    


                                     







                                     


                     

                     

               



                                

               


                           

                              

                         







                                                              
                                          


                                          










                                                                      
                 





























                                                                             
                                      

                                      
                                      

































                                                           
                                          


                                          










                                                                      
                 




                                                

                                    





                                                                             
                     











                                           
                    


                                       









                                       


                          


                          


                                 



                                 


                           

                           


                                         
                                         







                                                            
                                          


                                          

          






































































                                                                             




                                                         
                                          


                                          

          








































































                                                                             















































                                                                                                                                                                                   
                                       
                                      
                                             

                        

                                                            

       



                                                          
                                                          
                                                          
 
                        
                             





                                                                         
                                               

                                          













                                                                                         



                                                                                                    

                                                                      

      

                                                                    



      
                                             



                                   
                                  



                                                                                                    
             



      



         
<!DOCTYPE html>
<!-- Generated by pkgdown: do not edit by hand --><html lang="en">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<meta charset="utf-8">
<meta http-equiv="X-UA-Compatible" content="IE=edge">
<meta name="viewport" content="width=device-width, initial-scale=1, shrink-to-fit=no">
<meta name="description" content="mkin">
<title>Testing hierarchical parent degradation kinetics with residue data on dimethenamid and dimethenamid-P • mkin</title>
<script src="../../deps/jquery-3.6.0/jquery-3.6.0.min.js"></script><meta name="viewport" content="width=device-width, initial-scale=1, shrink-to-fit=no">
<link href="../../deps/bootstrap-5.2.2/bootstrap.min.css" rel="stylesheet">
<script src="../../deps/bootstrap-5.2.2/bootstrap.bundle.min.js"></script><!-- Font Awesome icons --><link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.12.1/css/all.min.css" integrity="sha256-mmgLkCYLUQbXn0B1SRqzHar6dCnv9oZFPEC1g1cwlkk=" crossorigin="anonymous">
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.12.1/css/v4-shims.min.css" integrity="sha256-wZjR52fzng1pJHwx4aV2AO3yyTOXrcDW7jBpJtTwVxw=" crossorigin="anonymous">
<!-- bootstrap-toc --><script src="https://cdn.jsdelivr.net/gh/afeld/bootstrap-toc@v1.0.1/dist/bootstrap-toc.min.js" integrity="sha256-4veVQbu7//Lk5TSmc7YV48MxtMy98e26cf5MrgZYnwo=" crossorigin="anonymous"></script><!-- headroom.js --><script src="https://cdnjs.cloudflare.com/ajax/libs/headroom/0.11.0/headroom.min.js" integrity="sha256-AsUX4SJE1+yuDu5+mAVzJbuYNPHj/WroHuZ8Ir/CkE0=" crossorigin="anonymous"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/headroom/0.11.0/jQuery.headroom.min.js" integrity="sha256-ZX/yNShbjqsohH1k95liqY9Gd8uOiE1S4vZc+9KQ1K4=" crossorigin="anonymous"></script><!-- clipboard.js --><script src="https://cdnjs.cloudflare.com/ajax/libs/clipboard.js/2.0.6/clipboard.min.js" integrity="sha256-inc5kl9MA1hkeYUt+EC3BhlIgyp/2jDIyBLS6k3UxPI=" crossorigin="anonymous"></script><!-- search --><script src="https://cdnjs.cloudflare.com/ajax/libs/fuse.js/6.4.6/fuse.js" integrity="sha512-zv6Ywkjyktsohkbp9bb45V6tEMoWhzFzXis+LrMehmJZZSys19Yxf1dopHx7WzIKxr5tK2dVcYmaCk2uqdjF4A==" crossorigin="anonymous"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/autocomplete.js/0.38.0/autocomplete.jquery.min.js" integrity="sha512-GU9ayf+66Xx2TmpxqJpliWbT5PiGYxpaG8rfnBEk1LL8l1KGkRShhngwdXK1UgqhAzWpZHSiYPc09/NwDQIGyg==" crossorigin="anonymous"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/mark.js/8.11.1/mark.min.js" integrity="sha512-5CYOlHXGh6QpOFA/TeTylKLWfB3ftPsde7AnmhuitiTX4K5SqCLBeKro6sPS8ilsz1Q4NRx3v8Ko2IBiszzdww==" crossorigin="anonymous"></script><!-- pkgdown --><script src="../../pkgdown.js"></script><meta property="og:title" content="Testing hierarchical parent degradation kinetics with residue data on dimethenamid and dimethenamid-P">
<meta property="og:description" content="mkin">
<!-- mathjax --><script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js" integrity="sha256-nvJJv9wWKEm88qvoQl9ekL2J+k/RWIsaSScxxlsrv8k=" crossorigin="anonymous"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/config/TeX-AMS-MML_HTMLorMML.js" integrity="sha256-84DKXVJXs0/F8OTMzX4UR909+jtl4G7SPypPavF+GfA=" crossorigin="anonymous"></script><!--[if lt IE 9]>
<script src="https://oss.maxcdn.com/html5shiv/3.7.3/html5shiv.min.js"></script>
<script src="https://oss.maxcdn.com/respond/1.4.2/respond.min.js"></script>
<![endif]-->
</head>
<body>
    <a href="#main" class="visually-hidden-focusable">Skip to contents</a>
    

    <nav class="navbar fixed-top navbar-default navbar-expand-lg bg-light"><div class="container">
    
    <a class="navbar-brand me-2" href="../../index.html">mkin</a>

    <small class="nav-text text-default me-auto" data-bs-toggle="tooltip" data-bs-placement="bottom" title="Released version">1.2.6</small>

    
    <button class="navbar-toggler" type="button" data-bs-toggle="collapse" data-bs-target="#navbar" aria-controls="navbar" aria-expanded="false" aria-label="Toggle navigation">
      <span class="navbar-toggler-icon"></span>
    </button>

    <div id="navbar" class="collapse navbar-collapse ms-3">
      <ul class="navbar-nav me-auto">
<li class="nav-item">
  <a class="nav-link" href="../../reference/index.html">Reference</a>
</li>
<li class="active nav-item dropdown">
  <a href="#" class="nav-link dropdown-toggle" data-bs-toggle="dropdown" role="button" aria-expanded="false" aria-haspopup="true" id="dropdown-articles">Articles</a>
  <div class="dropdown-menu" aria-labelledby="dropdown-articles">
    <a class="dropdown-item" href="../../articles/mkin.html">Introduction to mkin</a>
    <div class="dropdown-divider"></div>
    <h6 class="dropdown-header" data-toc-skip>Example evaluations with (generalised) nonlinear least squares</h6>
    <a class="dropdown-item" href="../../articles/FOCUS_D.html">Example evaluation of FOCUS Example Dataset D</a>
    <a class="dropdown-item" href="../../articles/FOCUS_L.html">Example evaluation of FOCUS Laboratory Data L1 to L3</a>
    <a class="dropdown-item" href="../../articles/web_only/FOCUS_Z.html">Example evaluation of FOCUS Example Dataset Z</a>
    <div class="dropdown-divider"></div>
    <h6 class="dropdown-header" data-toc-skip>Example evaluations with hierarchical models (nonlinear mixed-effects models)</h6>
    <a class="dropdown-item" href="../../articles/prebuilt/2022_dmta_parent.html">Testing hierarchical parent degradation kinetics with residue data on dimethenamid and dimethenamid-P</a>
    <a class="dropdown-item" href="../../articles/prebuilt/2022_dmta_pathway.html">Testing hierarchical pathway kinetics with residue data on dimethenamid and dimethenamid-P</a>
    <a class="dropdown-item" href="../../articles/prebuilt/2023_mesotrione_parent.html">Testing covariate modelling in hierarchical parent degradation kinetics with residue data on mesotrione</a>
    <a class="dropdown-item" href="../../articles/prebuilt/2022_cyan_pathway.html">Testing hierarchical pathway kinetics with residue data on cyantraniliprole</a>
    <a class="dropdown-item" href="../../articles/web_only/dimethenamid_2018.html">Comparison of saemix and nlme evaluations of dimethenamid data from 2018</a>
    <a class="dropdown-item" href="../../articles/web_only/multistart.html">Short demo of the multistart method</a>
    <div class="dropdown-divider"></div>
    <h6 class="dropdown-header" data-toc-skip>Performance</h6>
    <a class="dropdown-item" href="../../articles/web_only/compiled_models.html">Performance benefit by using compiled model definitions in mkin</a>
    <a class="dropdown-item" href="../../articles/web_only/benchmarks.html">Benchmark timings for mkin</a>
    <a class="dropdown-item" href="../../articles/web_only/saem_benchmarks.html">Benchmark timings for saem.mmkin</a>
    <div class="dropdown-divider"></div>
    <h6 class="dropdown-header" data-toc-skip>Miscellaneous</h6>
    <a class="dropdown-item" href="../../articles/twa.html">Calculation of time weighted average concentrations with mkin</a>
    <a class="dropdown-item" href="../../articles/web_only/NAFTA_examples.html">Example evaluation of NAFTA SOP Attachment examples</a>
  </div>
</li>
<li class="nav-item">
  <a class="nav-link" href="../../news/index.html">News</a>
</li>
      </ul>
<form class="form-inline my-2 my-lg-0" role="search">
        <input type="search" class="form-control me-sm-2" aria-label="Toggle navigation" name="search-input" data-search-index="../../search.json" id="search-input" placeholder="Search for" autocomplete="off">
</form>

      <ul class="navbar-nav">
<li class="nav-item">
  <a class="external-link nav-link" href="https://github.com/jranke/mkin/" aria-label="github">
    <span class="fab fa fab fa-github fa-lg"></span>
     
  </a>
</li>
      </ul>
</div>

    
  </div>
</nav><div class="container template-article">




<div class="row">
  <main id="main" class="col-md-9"><div class="page-header">
      <img src="" class="logo" alt=""><h1>Testing hierarchical parent degradation kinetics with residue data on dimethenamid and dimethenamid-P</h1>
                        <h4 data-toc-skip class="author">Johannes
Ranke</h4>
            
            <h4 data-toc-skip class="date">Last change on 5 January
2023, last compiled on 16 November 2023</h4>
      
      <small class="dont-index">Source: <a href="https://github.com/jranke/mkin/blob/HEAD/vignettes/prebuilt/2022_dmta_parent.rmd" class="external-link"><code>vignettes/prebuilt/2022_dmta_parent.rmd</code></a></small>
      <div class="d-none name"><code>2022_dmta_parent.rmd</code></div>
    </div>

    
    
<div class="section level2">
<h2 id="introduction">Introduction<a class="anchor" aria-label="anchor" href="#introduction"></a>
</h2>
<p>The purpose of this document is to demonstrate how nonlinear
hierarchical models (NLHM) based on the parent degradation models SFO,
FOMC, DFOP and HS can be fitted with the mkin package.</p>
<p>It was assembled in the course of work package 1.1 of Project Number
173340 (Application of nonlinear hierarchical models to the kinetic
evaluation of chemical degradation data) of the German Environment
Agency carried out in 2022 and 2023.</p>
<p>The mkin package is used in version 1.2.6. It contains the test data
and the functions used in the evaluations. The <code>saemix</code>
package is used as a backend for fitting the NLHM, but is also loaded to
make the convergence plot function available.</p>
<p>This document is processed with the <code>knitr</code> package, which
also provides the <code>kable</code> function that is used to improve
the display of tabular data in R markdown documents. For parallel
processing, the <code>parallel</code> package is used.</p>
<div class="sourceCode" id="cb1"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="va"><a href="https://pkgdown.jrwb.de/mkin/">mkin</a></span><span class="op">)</span></span>
<span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="va"><a href="https://yihui.org/knitr/" class="external-link">knitr</a></span><span class="op">)</span></span>
<span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="va">saemix</span><span class="op">)</span></span>
<span><span class="kw"><a href="https://rdrr.io/r/base/library.html" class="external-link">library</a></span><span class="op">(</span><span class="va">parallel</span><span class="op">)</span></span>
<span><span class="va">n_cores</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/parallel/detectCores.html" class="external-link">detectCores</a></span><span class="op">(</span><span class="op">)</span></span>
<span><span class="kw">if</span> <span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/Sys.info.html" class="external-link">Sys.info</a></span><span class="op">(</span><span class="op">)</span><span class="op">[</span><span class="st">"sysname"</span><span class="op">]</span> <span class="op">==</span> <span class="st">"Windows"</span><span class="op">)</span> <span class="op">{</span></span>
<span>  <span class="va">cl</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/parallel/makeCluster.html" class="external-link">makePSOCKcluster</a></span><span class="op">(</span><span class="va">n_cores</span><span class="op">)</span></span>
<span><span class="op">}</span> <span class="kw">else</span> <span class="op">{</span></span>
<span>  <span class="va">cl</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/parallel/makeCluster.html" class="external-link">makeForkCluster</a></span><span class="op">(</span><span class="va">n_cores</span><span class="op">)</span></span>
<span><span class="op">}</span></span></code></pre></div>
</div>
<div class="section level2">
<h2 id="data">Data<a class="anchor" aria-label="anchor" href="#data"></a>
</h2>
<p>The test data are available in the mkin package as an object of class
<code>mkindsg</code> (mkin dataset group) under the identifier
<code>dimethenamid_2018</code>. The following preprocessing steps are
still necessary:</p>
<ul>
<li>The data available for the enantiomer dimethenamid-P (DMTAP) are
renamed to have the same substance name as the data for the racemic
mixture dimethenamid (DMTA). The reason for this is that no difference
between their degradation behaviour was identified in the EU risk
assessment.</li>
<li>The data for transformation products and unnecessary columns are
discarded</li>
<li>The observation times of each dataset are multiplied with the
corresponding normalisation factor also available in the dataset, in
order to make it possible to describe all datasets with a single set of
parameters that are independent of temperature</li>
<li>Finally, datasets observed in the same soil (<code>Elliot 1</code>
and <code>Elliot 2</code>) are combined, resulting in dimethenamid
(DMTA) data from six soils.</li>
</ul>
<p>The following commented R code performs this preprocessing.</p>
<div class="sourceCode" id="cb2"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="co"># Apply a function to each of the seven datasets in the mkindsg object to create a list</span></span>
<span><span class="va">dmta_ds</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/lapply.html" class="external-link">lapply</a></span><span class="op">(</span><span class="fl">1</span><span class="op">:</span><span class="fl">7</span>, <span class="kw">function</span><span class="op">(</span><span class="va">i</span><span class="op">)</span> <span class="op">{</span></span>
<span>  <span class="va">ds_i</span> <span class="op">&lt;-</span> <span class="va">dimethenamid_2018</span><span class="op">$</span><span class="va">ds</span><span class="op">[[</span><span class="va">i</span><span class="op">]</span><span class="op">]</span><span class="op">$</span><span class="va">data</span>                     <span class="co"># Get a dataset</span></span>
<span>  <span class="va">ds_i</span><span class="op">[</span><span class="va">ds_i</span><span class="op">$</span><span class="va">name</span> <span class="op">==</span> <span class="st">"DMTAP"</span>, <span class="st">"name"</span><span class="op">]</span> <span class="op">&lt;-</span>  <span class="st">"DMTA"</span>              <span class="co"># Rename DMTAP to DMTA</span></span>
<span>  <span class="va">ds_i</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/subset.html" class="external-link">subset</a></span><span class="op">(</span><span class="va">ds_i</span>, <span class="va">name</span> <span class="op">==</span> <span class="st">"DMTA"</span>, <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"name"</span>, <span class="st">"time"</span>, <span class="st">"value"</span><span class="op">)</span><span class="op">)</span> <span class="co"># Select data</span></span>
<span>  <span class="va">ds_i</span><span class="op">$</span><span class="va">time</span> <span class="op">&lt;-</span> <span class="va">ds_i</span><span class="op">$</span><span class="va">time</span> <span class="op">*</span> <span class="va">dimethenamid_2018</span><span class="op">$</span><span class="va">f_time_norm</span><span class="op">[</span><span class="va">i</span><span class="op">]</span>  <span class="co"># Normalise time</span></span>
<span>  <span class="va">ds_i</span>                                                       <span class="co"># Return the dataset</span></span>
<span><span class="op">}</span><span class="op">)</span></span>
<span></span>
<span><span class="co"># Use dataset titles as names for the list elements</span></span>
<span><span class="fu"><a href="https://rdrr.io/r/base/names.html" class="external-link">names</a></span><span class="op">(</span><span class="va">dmta_ds</span><span class="op">)</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/lapply.html" class="external-link">sapply</a></span><span class="op">(</span><span class="va">dimethenamid_2018</span><span class="op">$</span><span class="va">ds</span>, <span class="kw">function</span><span class="op">(</span><span class="va">ds</span><span class="op">)</span> <span class="va">ds</span><span class="op">$</span><span class="va">title</span><span class="op">)</span></span>
<span></span>
<span><span class="co"># Combine data for Elliot soil to obtain a named list with six elements</span></span>
<span><span class="va">dmta_ds</span><span class="op">[[</span><span class="st">"Elliot"</span><span class="op">]</span><span class="op">]</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/cbind.html" class="external-link">rbind</a></span><span class="op">(</span><span class="va">dmta_ds</span><span class="op">[[</span><span class="st">"Elliot 1"</span><span class="op">]</span><span class="op">]</span>, <span class="va">dmta_ds</span><span class="op">[[</span><span class="st">"Elliot 2"</span><span class="op">]</span><span class="op">]</span><span class="op">)</span> <span class="co">#</span></span>
<span><span class="va">dmta_ds</span><span class="op">[[</span><span class="st">"Elliot 1"</span><span class="op">]</span><span class="op">]</span> <span class="op">&lt;-</span> <span class="cn">NULL</span></span>
<span><span class="va">dmta_ds</span><span class="op">[[</span><span class="st">"Elliot 2"</span><span class="op">]</span><span class="op">]</span> <span class="op">&lt;-</span> <span class="cn">NULL</span></span></code></pre></div>
<p>The following tables show the 6 datasets.</p>
<div class="sourceCode" id="cb3"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="kw">for</span> <span class="op">(</span><span class="va">ds_name</span> <span class="kw">in</span> <span class="fu"><a href="https://rdrr.io/r/base/names.html" class="external-link">names</a></span><span class="op">(</span><span class="va">dmta_ds</span><span class="op">)</span><span class="op">)</span> <span class="op">{</span></span>
<span>    <span class="fu"><a href="https://rdrr.io/r/base/print.html" class="external-link">print</a></span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="fu"><a href="../../reference/mkin_long_to_wide.html">mkin_long_to_wide</a></span><span class="op">(</span><span class="va">dmta_ds</span><span class="op">[[</span><span class="va">ds_name</span><span class="op">]</span><span class="op">]</span><span class="op">)</span>,</span>
<span>      caption <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/paste.html" class="external-link">paste</a></span><span class="op">(</span><span class="st">"Dataset"</span>, <span class="va">ds_name</span><span class="op">)</span>,</span>
<span>      label <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/paste.html" class="external-link">paste0</a></span><span class="op">(</span><span class="st">"tab:"</span>, <span class="va">ds_name</span><span class="op">)</span>, booktabs <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span><span class="op">)</span></span>
<span>    <span class="fu"><a href="https://rdrr.io/r/base/cat.html" class="external-link">cat</a></span><span class="op">(</span><span class="st">"\n\\clearpage\n"</span><span class="op">)</span></span>
<span><span class="op">}</span></span></code></pre></div>
<table class="table">
<caption>Dataset Calke</caption>
<thead><tr class="header">
<th align="right">time</th>
<th align="right">DMTA</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="right">0</td>
<td align="right">95.8</td>
</tr>
<tr class="even">
<td align="right">0</td>
<td align="right">98.7</td>
</tr>
<tr class="odd">
<td align="right">14</td>
<td align="right">60.5</td>
</tr>
<tr class="even">
<td align="right">30</td>
<td align="right">39.1</td>
</tr>
<tr class="odd">
<td align="right">59</td>
<td align="right">15.2</td>
</tr>
<tr class="even">
<td align="right">120</td>
<td align="right">4.8</td>
</tr>
<tr class="odd">
<td align="right">120</td>
<td align="right">4.6</td>
</tr>
</tbody>
</table>
<table class="table">
<caption>Dataset Borstel</caption>
<thead><tr class="header">
<th align="right">time</th>
<th align="right">DMTA</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="right">0.000000</td>
<td align="right">100.5</td>
</tr>
<tr class="even">
<td align="right">0.000000</td>
<td align="right">99.6</td>
</tr>
<tr class="odd">
<td align="right">1.941295</td>
<td align="right">91.9</td>
</tr>
<tr class="even">
<td align="right">1.941295</td>
<td align="right">91.3</td>
</tr>
<tr class="odd">
<td align="right">6.794534</td>
<td align="right">81.8</td>
</tr>
<tr class="even">
<td align="right">6.794534</td>
<td align="right">82.1</td>
</tr>
<tr class="odd">
<td align="right">13.589067</td>
<td align="right">69.1</td>
</tr>
<tr class="even">
<td align="right">13.589067</td>
<td align="right">68.0</td>
</tr>
<tr class="odd">
<td align="right">27.178135</td>
<td align="right">51.4</td>
</tr>
<tr class="even">
<td align="right">27.178135</td>
<td align="right">51.4</td>
</tr>
<tr class="odd">
<td align="right">56.297565</td>
<td align="right">27.6</td>
</tr>
<tr class="even">
<td align="right">56.297565</td>
<td align="right">26.8</td>
</tr>
<tr class="odd">
<td align="right">86.387643</td>
<td align="right">15.7</td>
</tr>
<tr class="even">
<td align="right">86.387643</td>
<td align="right">15.3</td>
</tr>
<tr class="odd">
<td align="right">115.507073</td>
<td align="right">7.9</td>
</tr>
<tr class="even">
<td align="right">115.507073</td>
<td align="right">8.1</td>
</tr>
</tbody>
</table>
<table class="table">
<caption>Dataset Flaach</caption>
<thead><tr class="header">
<th align="right">time</th>
<th align="right">DMTA</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="right">0.0000000</td>
<td align="right">96.5</td>
</tr>
<tr class="even">
<td align="right">0.0000000</td>
<td align="right">96.8</td>
</tr>
<tr class="odd">
<td align="right">0.0000000</td>
<td align="right">97.0</td>
</tr>
<tr class="even">
<td align="right">0.6233856</td>
<td align="right">82.9</td>
</tr>
<tr class="odd">
<td align="right">0.6233856</td>
<td align="right">86.7</td>
</tr>
<tr class="even">
<td align="right">0.6233856</td>
<td align="right">87.4</td>
</tr>
<tr class="odd">
<td align="right">1.8701567</td>
<td align="right">72.8</td>
</tr>
<tr class="even">
<td align="right">1.8701567</td>
<td align="right">69.9</td>
</tr>
<tr class="odd">
<td align="right">1.8701567</td>
<td align="right">71.9</td>
</tr>
<tr class="even">
<td align="right">4.3636989</td>
<td align="right">51.4</td>
</tr>
<tr class="odd">
<td align="right">4.3636989</td>
<td align="right">52.9</td>
</tr>
<tr class="even">
<td align="right">4.3636989</td>
<td align="right">48.6</td>
</tr>
<tr class="odd">
<td align="right">8.7273979</td>
<td align="right">28.5</td>
</tr>
<tr class="even">
<td align="right">8.7273979</td>
<td align="right">27.3</td>
</tr>
<tr class="odd">
<td align="right">8.7273979</td>
<td align="right">27.5</td>
</tr>
<tr class="even">
<td align="right">13.0910968</td>
<td align="right">14.8</td>
</tr>
<tr class="odd">
<td align="right">13.0910968</td>
<td align="right">13.4</td>
</tr>
<tr class="even">
<td align="right">13.0910968</td>
<td align="right">14.4</td>
</tr>
<tr class="odd">
<td align="right">17.4547957</td>
<td align="right">7.7</td>
</tr>
<tr class="even">
<td align="right">17.4547957</td>
<td align="right">7.3</td>
</tr>
<tr class="odd">
<td align="right">17.4547957</td>
<td align="right">8.1</td>
</tr>
<tr class="even">
<td align="right">26.1821936</td>
<td align="right">2.0</td>
</tr>
<tr class="odd">
<td align="right">26.1821936</td>
<td align="right">1.5</td>
</tr>
<tr class="even">
<td align="right">26.1821936</td>
<td align="right">1.9</td>
</tr>
<tr class="odd">
<td align="right">34.9095915</td>
<td align="right">1.3</td>
</tr>
<tr class="even">
<td align="right">34.9095915</td>
<td align="right">1.0</td>
</tr>
<tr class="odd">
<td align="right">34.9095915</td>
<td align="right">1.1</td>
</tr>
<tr class="even">
<td align="right">43.6369893</td>
<td align="right">0.9</td>
</tr>
<tr class="odd">
<td align="right">43.6369893</td>
<td align="right">0.7</td>
</tr>
<tr class="even">
<td align="right">43.6369893</td>
<td align="right">0.7</td>
</tr>
<tr class="odd">
<td align="right">52.3643872</td>
<td align="right">0.6</td>
</tr>
<tr class="even">
<td align="right">52.3643872</td>
<td align="right">0.4</td>
</tr>
<tr class="odd">
<td align="right">52.3643872</td>
<td align="right">0.5</td>
</tr>
<tr class="even">
<td align="right">74.8062674</td>
<td align="right">0.4</td>
</tr>
<tr class="odd">
<td align="right">74.8062674</td>
<td align="right">0.3</td>
</tr>
<tr class="even">
<td align="right">74.8062674</td>
<td align="right">0.3</td>
</tr>
</tbody>
</table>
<table class="table">
<caption>Dataset BBA 2.2</caption>
<thead><tr class="header">
<th align="right">time</th>
<th align="right">DMTA</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="right">0.0000000</td>
<td align="right">98.09</td>
</tr>
<tr class="even">
<td align="right">0.0000000</td>
<td align="right">98.77</td>
</tr>
<tr class="odd">
<td align="right">0.7678922</td>
<td align="right">93.52</td>
</tr>
<tr class="even">
<td align="right">0.7678922</td>
<td align="right">92.03</td>
</tr>
<tr class="odd">
<td align="right">2.3036765</td>
<td align="right">88.39</td>
</tr>
<tr class="even">
<td align="right">2.3036765</td>
<td align="right">87.18</td>
</tr>
<tr class="odd">
<td align="right">5.3752452</td>
<td align="right">69.38</td>
</tr>
<tr class="even">
<td align="right">5.3752452</td>
<td align="right">71.06</td>
</tr>
<tr class="odd">
<td align="right">10.7504904</td>
<td align="right">45.21</td>
</tr>
<tr class="even">
<td align="right">10.7504904</td>
<td align="right">46.81</td>
</tr>
<tr class="odd">
<td align="right">16.1257355</td>
<td align="right">30.54</td>
</tr>
<tr class="even">
<td align="right">16.1257355</td>
<td align="right">30.07</td>
</tr>
<tr class="odd">
<td align="right">21.5009807</td>
<td align="right">21.60</td>
</tr>
<tr class="even">
<td align="right">21.5009807</td>
<td align="right">20.41</td>
</tr>
<tr class="odd">
<td align="right">32.2514711</td>
<td align="right">9.10</td>
</tr>
<tr class="even">
<td align="right">32.2514711</td>
<td align="right">9.70</td>
</tr>
<tr class="odd">
<td align="right">43.0019614</td>
<td align="right">6.58</td>
</tr>
<tr class="even">
<td align="right">43.0019614</td>
<td align="right">6.31</td>
</tr>
<tr class="odd">
<td align="right">53.7524518</td>
<td align="right">3.47</td>
</tr>
<tr class="even">
<td align="right">53.7524518</td>
<td align="right">3.52</td>
</tr>
<tr class="odd">
<td align="right">64.5029421</td>
<td align="right">3.40</td>
</tr>
<tr class="even">
<td align="right">64.5029421</td>
<td align="right">3.67</td>
</tr>
<tr class="odd">
<td align="right">91.3791680</td>
<td align="right">1.62</td>
</tr>
<tr class="even">
<td align="right">91.3791680</td>
<td align="right">1.62</td>
</tr>
</tbody>
</table>
<table class="table">
<caption>Dataset BBA 2.3</caption>
<thead><tr class="header">
<th align="right">time</th>
<th align="right">DMTA</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="right">0.0000000</td>
<td align="right">99.33</td>
</tr>
<tr class="even">
<td align="right">0.0000000</td>
<td align="right">97.44</td>
</tr>
<tr class="odd">
<td align="right">0.6733938</td>
<td align="right">93.73</td>
</tr>
<tr class="even">
<td align="right">0.6733938</td>
<td align="right">93.77</td>
</tr>
<tr class="odd">
<td align="right">2.0201814</td>
<td align="right">87.84</td>
</tr>
<tr class="even">
<td align="right">2.0201814</td>
<td align="right">89.82</td>
</tr>
<tr class="odd">
<td align="right">4.7137565</td>
<td align="right">71.61</td>
</tr>
<tr class="even">
<td align="right">4.7137565</td>
<td align="right">71.42</td>
</tr>
<tr class="odd">
<td align="right">9.4275131</td>
<td align="right">45.60</td>
</tr>
<tr class="even">
<td align="right">9.4275131</td>
<td align="right">45.42</td>
</tr>
<tr class="odd">
<td align="right">14.1412696</td>
<td align="right">31.12</td>
</tr>
<tr class="even">
<td align="right">14.1412696</td>
<td align="right">31.68</td>
</tr>
<tr class="odd">
<td align="right">18.8550262</td>
<td align="right">23.20</td>
</tr>
<tr class="even">
<td align="right">18.8550262</td>
<td align="right">24.13</td>
</tr>
<tr class="odd">
<td align="right">28.2825393</td>
<td align="right">9.43</td>
</tr>
<tr class="even">
<td align="right">28.2825393</td>
<td align="right">9.82</td>
</tr>
<tr class="odd">
<td align="right">37.7100523</td>
<td align="right">7.08</td>
</tr>
<tr class="even">
<td align="right">37.7100523</td>
<td align="right">8.64</td>
</tr>
<tr class="odd">
<td align="right">47.1375654</td>
<td align="right">4.41</td>
</tr>
<tr class="even">
<td align="right">47.1375654</td>
<td align="right">4.78</td>
</tr>
<tr class="odd">
<td align="right">56.5650785</td>
<td align="right">4.92</td>
</tr>
<tr class="even">
<td align="right">56.5650785</td>
<td align="right">5.08</td>
</tr>
<tr class="odd">
<td align="right">80.1338612</td>
<td align="right">2.13</td>
</tr>
<tr class="even">
<td align="right">80.1338612</td>
<td align="right">2.23</td>
</tr>
</tbody>
</table>
<table class="table">
<caption>Dataset Elliot</caption>
<thead><tr class="header">
<th align="right">time</th>
<th align="right">DMTA</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="right">0.000000</td>
<td align="right">97.5</td>
</tr>
<tr class="even">
<td align="right">0.000000</td>
<td align="right">100.7</td>
</tr>
<tr class="odd">
<td align="right">1.228478</td>
<td align="right">86.4</td>
</tr>
<tr class="even">
<td align="right">1.228478</td>
<td align="right">88.5</td>
</tr>
<tr class="odd">
<td align="right">3.685435</td>
<td align="right">69.8</td>
</tr>
<tr class="even">
<td align="right">3.685435</td>
<td align="right">77.1</td>
</tr>
<tr class="odd">
<td align="right">8.599349</td>
<td align="right">59.0</td>
</tr>
<tr class="even">
<td align="right">8.599349</td>
<td align="right">54.2</td>
</tr>
<tr class="odd">
<td align="right">17.198697</td>
<td align="right">31.3</td>
</tr>
<tr class="even">
<td align="right">17.198697</td>
<td align="right">33.5</td>
</tr>
<tr class="odd">
<td align="right">25.798046</td>
<td align="right">19.6</td>
</tr>
<tr class="even">
<td align="right">25.798046</td>
<td align="right">20.9</td>
</tr>
<tr class="odd">
<td align="right">34.397395</td>
<td align="right">13.3</td>
</tr>
<tr class="even">
<td align="right">34.397395</td>
<td align="right">15.8</td>
</tr>
<tr class="odd">
<td align="right">51.596092</td>
<td align="right">6.7</td>
</tr>
<tr class="even">
<td align="right">51.596092</td>
<td align="right">8.7</td>
</tr>
<tr class="odd">
<td align="right">68.794789</td>
<td align="right">8.8</td>
</tr>
<tr class="even">
<td align="right">68.794789</td>
<td align="right">8.7</td>
</tr>
<tr class="odd">
<td align="right">103.192184</td>
<td align="right">6.0</td>
</tr>
<tr class="even">
<td align="right">103.192184</td>
<td align="right">4.4</td>
</tr>
<tr class="odd">
<td align="right">146.188928</td>
<td align="right">3.3</td>
</tr>
<tr class="even">
<td align="right">146.188928</td>
<td align="right">2.8</td>
</tr>
<tr class="odd">
<td align="right">223.583066</td>
<td align="right">1.4</td>
</tr>
<tr class="even">
<td align="right">223.583066</td>
<td align="right">1.8</td>
</tr>
<tr class="odd">
<td align="right">0.000000</td>
<td align="right">93.4</td>
</tr>
<tr class="even">
<td align="right">0.000000</td>
<td align="right">103.2</td>
</tr>
<tr class="odd">
<td align="right">1.228478</td>
<td align="right">89.2</td>
</tr>
<tr class="even">
<td align="right">1.228478</td>
<td align="right">86.6</td>
</tr>
<tr class="odd">
<td align="right">3.685435</td>
<td align="right">78.2</td>
</tr>
<tr class="even">
<td align="right">3.685435</td>
<td align="right">78.1</td>
</tr>
<tr class="odd">
<td align="right">8.599349</td>
<td align="right">55.6</td>
</tr>
<tr class="even">
<td align="right">8.599349</td>
<td align="right">53.0</td>
</tr>
<tr class="odd">
<td align="right">17.198697</td>
<td align="right">33.7</td>
</tr>
<tr class="even">
<td align="right">17.198697</td>
<td align="right">33.2</td>
</tr>
<tr class="odd">
<td align="right">25.798046</td>
<td align="right">20.9</td>
</tr>
<tr class="even">
<td align="right">25.798046</td>
<td align="right">19.9</td>
</tr>
<tr class="odd">
<td align="right">34.397395</td>
<td align="right">18.2</td>
</tr>
<tr class="even">
<td align="right">34.397395</td>
<td align="right">12.7</td>
</tr>
<tr class="odd">
<td align="right">51.596092</td>
<td align="right">7.8</td>
</tr>
<tr class="even">
<td align="right">51.596092</td>
<td align="right">9.0</td>
</tr>
<tr class="odd">
<td align="right">68.794789</td>
<td align="right">11.4</td>
</tr>
<tr class="even">
<td align="right">68.794789</td>
<td align="right">9.0</td>
</tr>
<tr class="odd">
<td align="right">103.192184</td>
<td align="right">3.9</td>
</tr>
<tr class="even">
<td align="right">103.192184</td>
<td align="right">4.4</td>
</tr>
<tr class="odd">
<td align="right">146.188928</td>
<td align="right">2.6</td>
</tr>
<tr class="even">
<td align="right">146.188928</td>
<td align="right">3.4</td>
</tr>
<tr class="odd">
<td align="right">223.583066</td>
<td align="right">2.0</td>
</tr>
<tr class="even">
<td align="right">223.583066</td>
<td align="right">1.7</td>
</tr>
</tbody>
</table>
</div>
<div class="section level2">
<h2 id="separate-evaluations">Separate evaluations<a class="anchor" aria-label="anchor" href="#separate-evaluations"></a>
</h2>
<p>In order to obtain suitable starting parameters for the NLHM fits,
separate fits of the four models to the data for each soil are generated
using the <code>mmkin</code> function from the <code>mkin</code>
package. In a first step, constant variance is assumed. Convergence is
checked with the <code>status</code> function.</p>
<div class="sourceCode" id="cb4"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="va">deg_mods</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="st">"SFO"</span>, <span class="st">"FOMC"</span>, <span class="st">"DFOP"</span>, <span class="st">"HS"</span><span class="op">)</span></span>
<span><span class="va">f_sep_const</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mmkin.html">mmkin</a></span><span class="op">(</span></span>
<span>  <span class="va">deg_mods</span>,</span>
<span>  <span class="va">dmta_ds</span>,</span>
<span>  error_model <span class="op">=</span> <span class="st">"const"</span>,</span>
<span>  quiet <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span></span>
<span></span>
<span><span class="fu"><a href="../../reference/status.html">status</a></span><span class="op">(</span><span class="va">f_sep_const</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
<table class="table">
<thead><tr class="header">
<th align="left"></th>
<th align="left">Calke</th>
<th align="left">Borstel</th>
<th align="left">Flaach</th>
<th align="left">BBA 2.2</th>
<th align="left">BBA 2.3</th>
<th align="left">Elliot</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="left">SFO</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
<tr class="even">
<td align="left">FOMC</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
<tr class="odd">
<td align="left">DFOP</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
<tr class="even">
<td align="left">HS</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">C</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
</tbody>
</table>
<p>In the table above, OK indicates convergence, and C indicates failure
to converge. All separate fits with constant variance converged, with
the sole exception of the HS fit to the BBA 2.2 data. To prepare for
fitting NLHM using the two-component error model, the separate fits are
updated assuming two-component error.</p>
<div class="sourceCode" id="cb5"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="va">f_sep_tc</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/stats/update.html" class="external-link">update</a></span><span class="op">(</span><span class="va">f_sep_const</span>, error_model <span class="op">=</span> <span class="st">"tc"</span><span class="op">)</span></span>
<span><span class="fu"><a href="../../reference/status.html">status</a></span><span class="op">(</span><span class="va">f_sep_tc</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
<table class="table">
<thead><tr class="header">
<th align="left"></th>
<th align="left">Calke</th>
<th align="left">Borstel</th>
<th align="left">Flaach</th>
<th align="left">BBA 2.2</th>
<th align="left">BBA 2.3</th>
<th align="left">Elliot</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="left">SFO</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
<tr class="even">
<td align="left">FOMC</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">C</td>
<td align="left">OK</td>
</tr>
<tr class="odd">
<td align="left">DFOP</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">C</td>
<td align="left">OK</td>
</tr>
<tr class="even">
<td align="left">HS</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
</tbody>
</table>
<p>Using the two-component error model, the one fit that did not
converge with constant variance did converge, but other non-SFO fits
failed to converge.</p>
</div>
<div class="section level2">
<h2 id="hierarchichal-model-fits">Hierarchichal model fits<a class="anchor" aria-label="anchor" href="#hierarchichal-model-fits"></a>
</h2>
<p>The following code fits eight versions of hierarchical models to the
data, using SFO, FOMC, DFOP and HS for the parent compound, and using
either constant variance or two-component error for the error model. The
default parameter distribution model in mkin allows for variation of all
degradation parameters across the assumed population of soils. In other
words, each degradation parameter is associated with a random effect as
a first step. The <code>mhmkin</code> function makes it possible to fit
all eight versions in parallel (given a sufficient number of computing
cores being available) to save execution time.</p>
<p>Convergence plots and summaries for these fits are shown in the
appendix.</p>
<div class="sourceCode" id="cb6"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="va">f_saem</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/mhmkin.html">mhmkin</a></span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span><span class="va">f_sep_const</span>, <span class="va">f_sep_tc</span><span class="op">)</span>, transformations <span class="op">=</span> <span class="st">"saemix"</span><span class="op">)</span></span></code></pre></div>
<p>The output of the <code>status</code> function shows that all fits
terminated successfully.</p>
<div class="sourceCode" id="cb7"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="../../reference/status.html">status</a></span><span class="op">(</span><span class="va">f_saem</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
<table class="table">
<thead><tr class="header">
<th align="left"></th>
<th align="left">const</th>
<th align="left">tc</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="left">SFO</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
<tr class="even">
<td align="left">FOMC</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
<tr class="odd">
<td align="left">DFOP</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
<tr class="even">
<td align="left">HS</td>
<td align="left">OK</td>
<td align="left">OK</td>
</tr>
</tbody>
</table>
<p>The AIC and BIC values show that the biphasic models DFOP and HS give
the best fits.</p>
<div class="sourceCode" id="cb8"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">f_saem</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">1</span><span class="op">)</span></span></code></pre></div>
<table class="table">
<thead><tr class="header">
<th align="left"></th>
<th align="right">npar</th>
<th align="right">AIC</th>
<th align="right">BIC</th>
<th align="right">Lik</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="left">SFO const</td>
<td align="right">5</td>
<td align="right">796.3</td>
<td align="right">795.3</td>
<td align="right">-393.2</td>
</tr>
<tr class="even">
<td align="left">SFO tc</td>
<td align="right">6</td>
<td align="right">798.3</td>
<td align="right">797.1</td>
<td align="right">-393.2</td>
</tr>
<tr class="odd">
<td align="left">FOMC const</td>
<td align="right">7</td>
<td align="right">734.2</td>
<td align="right">732.7</td>
<td align="right">-360.1</td>
</tr>
<tr class="even">
<td align="left">FOMC tc</td>
<td align="right">8</td>
<td align="right">720.7</td>
<td align="right">719.1</td>
<td align="right">-352.4</td>
</tr>
<tr class="odd">
<td align="left">DFOP const</td>
<td align="right">9</td>
<td align="right">711.8</td>
<td align="right">710.0</td>
<td align="right">-346.9</td>
</tr>
<tr class="even">
<td align="left">HS const</td>
<td align="right">9</td>
<td align="right">714.0</td>
<td align="right">712.1</td>
<td align="right">-348.0</td>
</tr>
<tr class="odd">
<td align="left">DFOP tc</td>
<td align="right">10</td>
<td align="right">665.7</td>
<td align="right">663.6</td>
<td align="right">-322.9</td>
</tr>
<tr class="even">
<td align="left">HS tc</td>
<td align="right">10</td>
<td align="right">667.1</td>
<td align="right">665.0</td>
<td align="right">-323.6</td>
</tr>
</tbody>
</table>
<p>The DFOP model is preferred here, as it has a better mechanistic
basis for batch experiments with constant incubation conditions. Also,
it shows the lowest AIC and BIC values in the first set of fits when
combined with the two-component error model. Therefore, the DFOP model
was selected for further refinements of the fits with the aim to make
the model fully identifiable.</p>
<div class="section level3">
<h3 id="parameter-identifiability-based-on-the-fisher-information-matrix">Parameter identifiability based on the Fisher Information
Matrix<a class="anchor" aria-label="anchor" href="#parameter-identifiability-based-on-the-fisher-information-matrix"></a>
</h3>
<p>Using the <code>illparms</code> function, ill-defined statistical
model parameters such as standard deviations of the degradation
parameters in the population and error model parameters can be
found.</p>
<div class="sourceCode" id="cb9"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">f_saem</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span><span class="op">)</span></span></code></pre></div>
<table class="table">
<thead><tr class="header">
<th align="left"></th>
<th align="left">const</th>
<th align="left">tc</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="left">SFO</td>
<td align="left"></td>
<td align="left">b.1</td>
</tr>
<tr class="even">
<td align="left">FOMC</td>
<td align="left"></td>
<td align="left">sd(DMTA_0)</td>
</tr>
<tr class="odd">
<td align="left">DFOP</td>
<td align="left">sd(k2)</td>
<td align="left">sd(k2)</td>
</tr>
<tr class="even">
<td align="left">HS</td>
<td align="left"></td>
<td align="left">sd(tb)</td>
</tr>
</tbody>
</table>
<p>According to the <code>illparms</code> function, the fitted standard
deviation of the second kinetic rate constant <code>k2</code> is
ill-defined in both DFOP fits. This suggests that different values would
be obtained for this standard deviation when using different starting
values.</p>
<p>The thus identified overparameterisation is addressed by removing the
random effect for <code>k2</code> from the parameter model.</p>
<div class="sourceCode" id="cb10"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="va">f_saem_dfop_tc_no_ranef_k2</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/stats/update.html" class="external-link">update</a></span><span class="op">(</span><span class="va">f_saem</span><span class="op">[[</span><span class="st">"DFOP"</span>, <span class="st">"tc"</span><span class="op">]</span><span class="op">]</span>,</span>
<span>  no_random_effect <span class="op">=</span> <span class="st">"k2"</span><span class="op">)</span></span></code></pre></div>
<p>For the resulting fit, it is checked whether there are still
ill-defined parameters,</p>
<div class="sourceCode" id="cb11"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="../../reference/illparms.html">illparms</a></span><span class="op">(</span><span class="va">f_saem_dfop_tc_no_ranef_k2</span><span class="op">)</span></span></code></pre></div>
<p>which is not the case. Below, the refined model is compared with the
previous best model. The model without random effect for <code>k2</code>
is a reduced version of the previous model. Therefore, the models are
nested and can be compared using the likelihood ratio test. This is
achieved with the argument <code>test = TRUE</code> to the
<code>anova</code> function.</p>
<div class="sourceCode" id="cb12"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/stats/anova.html" class="external-link">anova</a></span><span class="op">(</span><span class="va">f_saem</span><span class="op">[[</span><span class="st">"DFOP"</span>, <span class="st">"tc"</span><span class="op">]</span><span class="op">]</span>, <span class="va">f_saem_dfop_tc_no_ranef_k2</span>, test <span class="op">=</span> <span class="cn">TRUE</span><span class="op">)</span> <span class="op">|&gt;</span></span>
<span>  <span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html" class="external-link">kable</a></span><span class="op">(</span>format.args <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/list.html" class="external-link">list</a></span><span class="op">(</span>digits <span class="op">=</span> <span class="fl">4</span><span class="op">)</span><span class="op">)</span></span></code></pre></div>
<table class="table">
<colgroup>
<col width="38%">
<col width="7%">
<col width="8%">
<col width="8%">
<col width="9%">
<col width="8%">
<col width="4%">
<col width="15%">
</colgroup>
<thead><tr class="header">
<th align="left"></th>
<th align="right">npar</th>
<th align="right">AIC</th>
<th align="right">BIC</th>
<th align="right">Lik</th>
<th align="right">Chisq</th>
<th align="right">Df</th>
<th align="right">Pr(&gt;Chisq)</th>
</tr></thead>
<tbody>
<tr class="odd">
<td align="left">f_saem_dfop_tc_no_ranef_k2</td>
<td align="right">9</td>
<td align="right">663.7</td>
<td align="right">661.8</td>
<td align="right">-322.9</td>
<td align="right">NA</td>
<td align="right">NA</td>
<td align="right">NA</td>
</tr>
<tr class="even">
<td align="left">f_saem[[“DFOP”, “tc”]]</td>
<td align="right">10</td>
<td align="right">665.7</td>
<td align="right">663.6</td>
<td align="right">-322.9</td>
<td align="right">0</td>
<td align="right">1</td>
<td align="right">1</td>
</tr>
</tbody>
</table>
<p>The AIC and BIC criteria are lower after removal of the ill-defined
random effect for <code>k2</code>. The p value of the likelihood ratio
test is much greater than 0.05, indicating that the model with the
higher likelihood (here the model with random effects for all
degradation parameters <code>f_saem[["DFOP", "tc"]]</code>) does not fit
significantly better than the model with the lower likelihood (the
reduced model <code>f_saem_dfop_tc_no_ranef_k2</code>).</p>
<p>Therefore, AIC, BIC and likelihood ratio test suggest the use of the
reduced model.</p>
<p>The convergence of the fit is checked visually.</p>
<div class="figure" style="text-align: center">
<img src="2022_dmta_parent_files/figure-html/convergence-saem-dfop-tc-no-ranef-k2-1.png" alt="Convergence plot for the NLHM DFOP fit with two-component error and without a random effect on 'k2'" width="864"><p class="caption">
Convergence plot for the NLHM DFOP fit with two-component error and
without a random effect on ‘k2’
</p>
</div>
<p>All parameters appear to have converged to a satisfactory degree. The
final fit is plotted using the plot method from the mkin package.</p>
<div class="sourceCode" id="cb13"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">plot</a></span><span class="op">(</span><span class="va">f_saem_dfop_tc_no_ranef_k2</span><span class="op">)</span></span></code></pre></div>
<div class="figure" style="text-align: center">
<img src="2022_dmta_parent_files/figure-html/plot-saem-dfop-tc-no-ranef-k2-1.png" alt="Plot of the final NLHM DFOP fit" width="864"><p class="caption">
Plot of the final NLHM DFOP fit
</p>
</div>
<p>Finally, a summary report of the fit is produced.</p>
<div class="sourceCode" id="cb14"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/saemix/man/summary-methods.html" class="external-link">summary</a></span><span class="op">(</span><span class="va">f_saem_dfop_tc_no_ranef_k2</span><span class="op">)</span></span></code></pre></div>
<pre><code>saemix version used for fitting:      3.2 
mkin version used for pre-fitting:  1.2.6 
R version used for fitting:         4.3.2 
Date of fit:     Thu Nov 16 04:59:35 2023 
Date of summary: Thu Nov 16 04:59:35 2023 

Equations:
d_DMTA/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
           time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
           * DMTA

Data:
155 observations of 1 variable(s) grouped in 6 datasets

Model predictions using solution type analytical 

Fitted in 4.4 s
Using 300, 100 iterations and 9 chains

Variance model: Two-component variance function 

Starting values for degradation parameters:
  DMTA_0       k1       k2        g 
98.71186  0.08675  0.01374  0.93491 

Fixed degradation parameter values:
None

Starting values for random effects (square root of initial entries in omega):
       DMTA_0 k1 k2 g
DMTA_0  98.71  0  0 0
k1       0.00  1  0 0
k2       0.00  0  1 0
g        0.00  0  0 1

Starting values for error model parameters:
a.1 b.1 
  1   1 

Results:

Likelihood computed by importance sampling
    AIC   BIC logLik
  663.7 661.8 -322.9

Optimised parameters:
               est.     lower     upper
DMTA_0    98.256267 96.286112 100.22642
k1         0.064037  0.033281   0.09479
k2         0.008469  0.006002   0.01094
g          0.954167  0.914460   0.99387
a.1        1.061795  0.863943   1.25965
b.1        0.029550  0.022529   0.03657
SD.DMTA_0  2.068581  0.427706   3.70946
SD.k1      0.598285  0.258235   0.93833
SD.g       1.016689  0.360057   1.67332

Correlation: 
   DMTA_0  k1      k2     
k1  0.0213                
k2  0.0541  0.0344        
g  -0.0521 -0.0286 -0.2744

Random effects:
            est.  lower  upper
SD.DMTA_0 2.0686 0.4277 3.7095
SD.k1     0.5983 0.2582 0.9383
SD.g      1.0167 0.3601 1.6733

Variance model:
       est.   lower   upper
a.1 1.06180 0.86394 1.25965
b.1 0.02955 0.02253 0.03657

Estimated disappearance times:
      DT50  DT90 DT50back DT50_k1 DT50_k2
DMTA 11.45 41.32    12.44   10.82   81.85</code></pre>
</div>
<div class="section level3">
<h3 id="alternative-check-of-parameter-identifiability">Alternative check of parameter identifiability<a class="anchor" aria-label="anchor" href="#alternative-check-of-parameter-identifiability"></a>
</h3>
<p>The parameter check used in the <code>illparms</code> function is
based on a quadratic approximation of the likelihood surface near its
optimum, which is calculated using the Fisher Information Matrix (FIM).
An alternative way to check parameter identifiability <span class="citation">(Duchesne et al. 2021)</span> based on a multistart
approach has recently been implemented in mkin.</p>
<p>The graph below shows boxplots of the parameters obtained in 50 runs
of the saem algorithm with different parameter combinations, sampled
from the range of the parameters obtained for the individual datasets
fitted separately using nonlinear regression.</p>
<div class="sourceCode" id="cb16"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="va">f_saem_dfop_tc_multi</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/multistart.html">multistart</a></span><span class="op">(</span><span class="va">f_saem</span><span class="op">[[</span><span class="st">"DFOP"</span>, <span class="st">"tc"</span><span class="op">]</span><span class="op">]</span>, n <span class="op">=</span> <span class="fl">50</span>, cores <span class="op">=</span> <span class="fl">15</span><span class="op">)</span></span></code></pre></div>
<div class="sourceCode" id="cb17"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/graphics/par.html" class="external-link">par</a></span><span class="op">(</span>mar <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">6.1</span>, <span class="fl">4.1</span>, <span class="fl">2.1</span>, <span class="fl">2.1</span><span class="op">)</span><span class="op">)</span></span>
<span><span class="fu"><a href="../../reference/parplot.html">parplot</a></span><span class="op">(</span><span class="va">f_saem_dfop_tc_multi</span>, lpos <span class="op">=</span> <span class="st">"bottomright"</span>, ylim <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">0.3</span>, <span class="fl">10</span><span class="op">)</span>, las <span class="op">=</span> <span class="fl">2</span><span class="op">)</span></span></code></pre></div>
<div class="figure" style="text-align: center">
<img src="2022_dmta_parent_files/figure-html/multistart-full-par-1.png" alt="Scaled parameters from the multistart runs, full model" width="960"><p class="caption">
Scaled parameters from the multistart runs, full model
</p>
</div>
<p>The graph clearly confirms the lack of identifiability of the
variance of <code>k2</code> in the full model. The overparameterisation
of the model also indicates a lack of identifiability of the variance of
parameter <code>g</code>.</p>
<p>The parameter boxplots of the multistart runs with the reduced model
shown below indicate that all runs give similar results, regardless of
the starting parameters.</p>
<div class="sourceCode" id="cb18"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="va">f_saem_dfop_tc_no_ranef_k2_multi</span> <span class="op">&lt;-</span> <span class="fu"><a href="../../reference/multistart.html">multistart</a></span><span class="op">(</span><span class="va">f_saem_dfop_tc_no_ranef_k2</span>,</span>
<span>  n <span class="op">=</span> <span class="fl">50</span>, cores <span class="op">=</span> <span class="fl">15</span><span class="op">)</span></span></code></pre></div>
<div class="sourceCode" id="cb19"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/graphics/par.html" class="external-link">par</a></span><span class="op">(</span>mar <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">6.1</span>, <span class="fl">4.1</span>, <span class="fl">2.1</span>, <span class="fl">2.1</span><span class="op">)</span><span class="op">)</span></span>
<span><span class="fu"><a href="../../reference/parplot.html">parplot</a></span><span class="op">(</span><span class="va">f_saem_dfop_tc_no_ranef_k2_multi</span>, ylim <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">0.5</span>, <span class="fl">2</span><span class="op">)</span>, las <span class="op">=</span> <span class="fl">2</span>,</span>
<span>  lpos <span class="op">=</span> <span class="st">"bottomright"</span><span class="op">)</span></span></code></pre></div>
<div class="figure" style="text-align: center">
<img src="2022_dmta_parent_files/figure-html/multistart-reduced-par-1.png" alt="Scaled parameters from the multistart runs, reduced model" width="960"><p class="caption">
Scaled parameters from the multistart runs, reduced model
</p>
</div>
<p>When only the parameters of the top 25% of the fits are shown (based
on a feature introduced in mkin 1.2.2 currently under development), the
scatter is even less as shown below.</p>
<div class="sourceCode" id="cb20"><pre class="downlit sourceCode r">
<code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/r/graphics/par.html" class="external-link">par</a></span><span class="op">(</span>mar <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">6.1</span>, <span class="fl">4.1</span>, <span class="fl">2.1</span>, <span class="fl">2.1</span><span class="op">)</span><span class="op">)</span></span>
<span><span class="fu"><a href="../../reference/parplot.html">parplot</a></span><span class="op">(</span><span class="va">f_saem_dfop_tc_no_ranef_k2_multi</span>, ylim <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">0.5</span>, <span class="fl">2</span><span class="op">)</span>, las <span class="op">=</span> <span class="fl">2</span>, llquant <span class="op">=</span> <span class="fl">0.25</span>,</span>
<span>  lpos <span class="op">=</span> <span class="st">"bottomright"</span><span class="op">)</span></span></code></pre></div>
<div class="figure" style="text-align: center">
<img src="2022_dmta_parent_files/figure-html/multistart-reduced-par-llquant-1.png" alt="Scaled parameters from the multistart runs, reduced model, fits with the top 25\% likelihood values" width="960"><p class="caption">
Scaled parameters from the multistart runs, reduced model, fits with the
top 25% likelihood values
</p>
</div>
</div>
</div>
<div class="section level2">
<h2 id="conclusions">Conclusions<a class="anchor" aria-label="anchor" href="#conclusions"></a>
</h2>
<p>Fitting the four parent degradation models SFO, FOMC, DFOP and HS as
part of hierarchical model fits with two different error models and
normal distributions of the transformed degradation parameters works
without technical problems. The biphasic models DFOP and HS gave the
best fit to the data, but the default parameter distribution model was
not fully identifiable. Removing the random effect for the second
kinetic rate constant of the DFOP model resulted in a reduced model that
was fully identifiable and showed the lowest values for the model
selection criteria AIC and BIC. The reliability of the identification of
all model parameters was confirmed using multiple starting values.</p>
</div>
<div class="section level2">
<h2 id="acknowledgements">Acknowledgements<a class="anchor" aria-label="anchor" href="#acknowledgements"></a>
</h2>
<p>The helpful comments by Janina Wöltjen of the German Environment
Agency are gratefully acknowledged.</p>
</div>
<div class="section level2">
<h2 id="references">References<a class="anchor" aria-label="anchor" href="#references"></a>
</h2>
<div id="refs" class="references csl-bib-body hanging-indent">
<div id="ref-duchesne_2021" class="csl-entry">
Duchesne, Ronan, Anissa Guillemin, Olivier Gandrillon, and Fabien
Crauste. 2021. <span>“Practical Identifiability in the Frame of
Nonlinear Mixed Effects Models: The Example of the in Vitro
Erythropoiesis.”</span> <em>BMC Bioinformatics</em> 22 (478). <a href="https://doi.org/10.1186/s12859-021-04373-4" class="external-link">https://doi.org/10.1186/s12859-021-04373-4</a>.
</div>
</div>
</div>
<div class="section level2">
<h2 id="appendix">Appendix<a class="anchor" aria-label="anchor" href="#appendix"></a>
</h2>
<div class="section level3">
<h3 id="hierarchical-model-fit-listings">Hierarchical model fit listings<a class="anchor" aria-label="anchor" href="#hierarchical-model-fit-listings"></a>
</h3>
<caption>
Hierarchical mkin fit of the SFO model with error model const
</caption>
<pre><code>
saemix version used for fitting:      3.2 
mkin version used for pre-fitting:  1.2.6 
R version used for fitting:         4.3.2 
Date of fit:     Thu Nov 16 04:59:26 2023 
Date of summary: Thu Nov 16 05:00:43 2023 

Equations:
d_DMTA/dt = - k_DMTA * DMTA

Data:
155 observations of 1 variable(s) grouped in 6 datasets

Model predictions using solution type analytical 

Fitted in 0.788 s
Using 300, 100 iterations and 9 chains

Variance model: Constant variance 

Starting values for degradation parameters:
 DMTA_0  k_DMTA 
97.2953  0.0566 

Fixed degradation parameter values:
None

Starting values for random effects (square root of initial entries in omega):
       DMTA_0 k_DMTA
DMTA_0   97.3      0
k_DMTA    0.0      1

Starting values for error model parameters:
a.1 
  1 

Results:

Likelihood computed by importance sampling
    AIC   BIC logLik
  796.3 795.3 -393.2

Optimised parameters:
              est.    lower   upper
DMTA_0    97.28130 95.71113 98.8515
k_DMTA     0.05665  0.02909  0.0842
a.1        2.66442  2.35579  2.9731
SD.DMTA_0  1.54776  0.15447  2.9411
SD.k_DMTA  0.60690  0.26248  0.9513

Correlation: 
       DMTA_0
k_DMTA 0.0168

Random effects:
            est.  lower  upper
SD.DMTA_0 1.5478 0.1545 2.9411
SD.k_DMTA 0.6069 0.2625 0.9513

Variance model:
     est. lower upper
a.1 2.664 2.356 2.973

Estimated disappearance times:
      DT50  DT90
DMTA 12.24 40.65

</code></pre>
<p></p>
<caption>
Hierarchical mkin fit of the SFO model with error model tc
</caption>
<pre><code>
saemix version used for fitting:      3.2 
mkin version used for pre-fitting:  1.2.6 
R version used for fitting:         4.3.2 
Date of fit:     Thu Nov 16 04:59:28 2023 
Date of summary: Thu Nov 16 05:00:43 2023 

Equations:
d_DMTA/dt = - k_DMTA * DMTA

Data:
155 observations of 1 variable(s) grouped in 6 datasets

Model predictions using solution type analytical 

Fitted in 2.428 s
Using 300, 100 iterations and 9 chains

Variance model: Two-component variance function 

Starting values for degradation parameters:
  DMTA_0   k_DMTA 
96.99175  0.05603 

Fixed degradation parameter values:
None

Starting values for random effects (square root of initial entries in omega):
       DMTA_0 k_DMTA
DMTA_0  96.99      0
k_DMTA   0.00      1

Starting values for error model parameters:
a.1 b.1 
  1   1 

Results:

Likelihood computed by importance sampling
    AIC   BIC logLik
  798.3 797.1 -393.2

Optimised parameters:
               est.     lower    upper
DMTA_0    97.271822 95.703157 98.84049
k_DMTA     0.056638  0.029110  0.08417
a.1        2.660081  2.230398  3.08976
b.1        0.001665 -0.006911  0.01024
SD.DMTA_0  1.545520  0.145035  2.94601
SD.k_DMTA  0.606422  0.262274  0.95057

Correlation: 
       DMTA_0
k_DMTA 0.0169

Random effects:
            est.  lower  upper
SD.DMTA_0 1.5455 0.1450 2.9460
SD.k_DMTA 0.6064 0.2623 0.9506

Variance model:
        est.     lower   upper
a.1 2.660081  2.230398 3.08976
b.1 0.001665 -0.006911 0.01024

Estimated disappearance times:
      DT50  DT90
DMTA 12.24 40.65

</code></pre>
<p></p>
<caption>
Hierarchical mkin fit of the FOMC model with error model const
</caption>
<pre><code>
saemix version used for fitting:      3.2 
mkin version used for pre-fitting:  1.2.6 
R version used for fitting:         4.3.2 
Date of fit:     Thu Nov 16 04:59:27 2023 
Date of summary: Thu Nov 16 05:00:43 2023 

Equations:
d_DMTA/dt = - (alpha/beta) * 1/((time/beta) + 1) * DMTA

Data:
155 observations of 1 variable(s) grouped in 6 datasets

Model predictions using solution type analytical 

Fitted in 1.28 s
Using 300, 100 iterations and 9 chains

Variance model: Constant variance 

Starting values for degradation parameters:
 DMTA_0   alpha    beta 
 98.292   9.909 156.341 

Fixed degradation parameter values:
None

Starting values for random effects (square root of initial entries in omega):
       DMTA_0 alpha beta
DMTA_0  98.29     0    0
alpha    0.00     1    0
beta     0.00     0    1

Starting values for error model parameters:
a.1 
  1 

Results:

Likelihood computed by importance sampling
    AIC   BIC logLik
  734.2 732.7 -360.1

Optimised parameters:
              est.   lower   upper
DMTA_0     98.3435 96.9033  99.784
alpha       7.2007  2.5889  11.812
beta      112.8745 34.8816 190.867
a.1         2.0459  1.8054   2.286
SD.DMTA_0   1.4795  0.2717   2.687
SD.alpha    0.6396  0.1509   1.128
SD.beta     0.6874  0.1587   1.216

Correlation: 
      DMTA_0  alpha  
alpha -0.1125        
beta  -0.1227  0.3632

Random effects:
            est.  lower upper
SD.DMTA_0 1.4795 0.2717 2.687
SD.alpha  0.6396 0.1509 1.128
SD.beta   0.6874 0.1587 1.216

Variance model:
     est. lower upper
a.1 2.046 1.805 2.286

Estimated disappearance times:
      DT50  DT90 DT50back
DMTA 11.41 42.53     12.8

</code></pre>
<p></p>
<caption>
Hierarchical mkin fit of the FOMC model with error model tc
</caption>
<pre><code>
saemix version used for fitting:      3.2 
mkin version used for pre-fitting:  1.2.6 
R version used for fitting:         4.3.2 
Date of fit:     Thu Nov 16 04:59:28 2023 
Date of summary: Thu Nov 16 05:00:43 2023 

Equations:
d_DMTA/dt = - (alpha/beta) * 1/((time/beta) + 1) * DMTA

Data:
155 observations of 1 variable(s) grouped in 6 datasets

Model predictions using solution type analytical 

Fitted in 2.793 s
Using 300, 100 iterations and 9 chains

Variance model: Two-component variance function 

Starting values for degradation parameters:
DMTA_0  alpha   beta 
98.772  4.663 92.597 

Fixed degradation parameter values:
None

Starting values for random effects (square root of initial entries in omega):
       DMTA_0 alpha beta
DMTA_0  98.77     0    0
alpha    0.00     1    0
beta     0.00     0    1

Starting values for error model parameters:
a.1 b.1 
  1   1 

Results:

Likelihood computed by importance sampling
    AIC   BIC logLik
  720.7 719.1 -352.4

Optimised parameters:
              est.    lower     upper
DMTA_0    99.10577 97.33296 100.87859
alpha      5.46260  2.52199   8.40321
beta      81.66080 30.46664 132.85497
a.1        1.50219  1.23601   1.76836
b.1        0.02893  0.02099   0.03687
SD.DMTA_0  1.61887 -0.03636   3.27411
SD.alpha   0.58145  0.17364   0.98925
SD.beta    0.68205  0.21108   1.15303

Correlation: 
      DMTA_0  alpha  
alpha -0.1321        
beta  -0.1430  0.2467

Random effects:
            est.    lower  upper
SD.DMTA_0 1.6189 -0.03636 3.2741
SD.alpha  0.5814  0.17364 0.9892
SD.beta   0.6821  0.21108 1.1530

Variance model:
       est.   lower   upper
a.1 1.50219 1.23601 1.76836
b.1 0.02893 0.02099 0.03687

Estimated disappearance times:
      DT50  DT90 DT50back
DMTA 11.05 42.81    12.89

</code></pre>
<p></p>
<caption>
Hierarchical mkin fit of the DFOP model with error model const
</caption>
<pre><code>
saemix version used for fitting:      3.2 
mkin version used for pre-fitting:  1.2.6 
R version used for fitting:         4.3.2 
Date of fit:     Thu Nov 16 04:59:27 2023 
Date of summary: Thu Nov 16 05:00:43 2023 

Equations:
d_DMTA/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
           time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
           * DMTA

Data:
155 observations of 1 variable(s) grouped in 6 datasets

Model predictions using solution type analytical 

Fitted in 1.752 s
Using 300, 100 iterations and 9 chains

Variance model: Constant variance 

Starting values for degradation parameters:
  DMTA_0       k1       k2        g 
98.64383  0.09211  0.02999  0.76814 

Fixed degradation parameter values:
None

Starting values for random effects (square root of initial entries in omega):
       DMTA_0 k1 k2 g
DMTA_0  98.64  0  0 0
k1       0.00  1  0 0
k2       0.00  0  1 0
g        0.00  0  0 1

Starting values for error model parameters:
a.1 
  1 

Results:

Likelihood computed by importance sampling
    AIC BIC logLik
  711.8 710 -346.9

Optimised parameters:
               est.     lower    upper
DMTA_0    98.092481 96.573899 99.61106
k1         0.062499  0.030336  0.09466
k2         0.009065 -0.005133  0.02326
g          0.948967  0.862080  1.03586
a.1        1.821671  1.604774  2.03857
SD.DMTA_0  1.677785  0.472066  2.88350
SD.k1      0.634962  0.270788  0.99914
SD.k2      1.033498 -0.205994  2.27299
SD.g       1.710046  0.428642  2.99145

Correlation: 
   DMTA_0  k1      k2     
k1  0.0246                
k2  0.0491  0.0953        
g  -0.0552 -0.0889 -0.4795

Random effects:
           est.   lower  upper
SD.DMTA_0 1.678  0.4721 2.8835
SD.k1     0.635  0.2708 0.9991
SD.k2     1.033 -0.2060 2.2730
SD.g      1.710  0.4286 2.9914

Variance model:
     est. lower upper
a.1 1.822 1.605 2.039

Estimated disappearance times:
      DT50 DT90 DT50back DT50_k1 DT50_k2
DMTA 11.79 42.8    12.88   11.09   76.46

</code></pre>
<p></p>
<caption>
Hierarchical mkin fit of the DFOP model with error model tc
</caption>
<pre><code>
saemix version used for fitting:      3.2 
mkin version used for pre-fitting:  1.2.6 
R version used for fitting:         4.3.2 
Date of fit:     Thu Nov 16 04:59:29 2023 
Date of summary: Thu Nov 16 05:00:43 2023 

Equations:
d_DMTA/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
           time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
           * DMTA

Data:
155 observations of 1 variable(s) grouped in 6 datasets

Model predictions using solution type analytical 

Fitted in 3.186 s
Using 300, 100 iterations and 9 chains

Variance model: Two-component variance function 

Starting values for degradation parameters:
  DMTA_0       k1       k2        g 
98.71186  0.08675  0.01374  0.93491 

Fixed degradation parameter values:
None

Starting values for random effects (square root of initial entries in omega):
       DMTA_0 k1 k2 g
DMTA_0  98.71  0  0 0
k1       0.00  1  0 0
k2       0.00  0  1 0
g        0.00  0  0 1

Starting values for error model parameters:
a.1 b.1 
  1   1 

Results:

Likelihood computed by importance sampling
    AIC   BIC logLik
  665.7 663.6 -322.9

Optimised parameters:
               est.     lower     upper
DMTA_0    98.347470 96.380815 100.31413
k1         0.064524  0.034279   0.09477
k2         0.008304  0.005843   0.01076
g          0.952128  0.909578   0.99468
a.1        1.068907  0.868694   1.26912
b.1        0.029265  0.022262   0.03627
SD.DMTA_0  2.065796  0.428485   3.70311
SD.k1      0.583703  0.251796   0.91561
SD.k2      0.004167 -7.832168   7.84050
SD.g       1.064450  0.397476   1.73142

Correlation: 
   DMTA_0  k1      k2     
k1  0.0223                
k2  0.0568  0.0394        
g  -0.0464 -0.0269 -0.2713

Random effects:
              est.   lower  upper
SD.DMTA_0 2.065796  0.4285 3.7031
SD.k1     0.583703  0.2518 0.9156
SD.k2     0.004167 -7.8322 7.8405
SD.g      1.064450  0.3975 1.7314

Variance model:
       est.   lower   upper
a.1 1.06891 0.86869 1.26912
b.1 0.02927 0.02226 0.03627

Estimated disappearance times:
      DT50  DT90 DT50back DT50_k1 DT50_k2
DMTA 11.39 41.36    12.45   10.74   83.48

</code></pre>
<p></p>
<caption>
Hierarchical mkin fit of the HS model with error model const
</caption>
<pre><code>
saemix version used for fitting:      3.2 
mkin version used for pre-fitting:  1.2.6 
R version used for fitting:         4.3.2 
Date of fit:     Thu Nov 16 04:59:28 2023 
Date of summary: Thu Nov 16 05:00:43 2023 

Equations:
d_DMTA/dt = - ifelse(time &lt;= tb, k1, k2) * DMTA

Data:
155 observations of 1 variable(s) grouped in 6 datasets

Model predictions using solution type analytical 

Fitted in 2.28 s
Using 300, 100 iterations and 9 chains

Variance model: Constant variance 

Starting values for degradation parameters:
  DMTA_0       k1       k2       tb 
97.82176  0.06931  0.02997 11.13945 

Fixed degradation parameter values:
None

Starting values for random effects (square root of initial entries in omega):
       DMTA_0 k1 k2 tb
DMTA_0  97.82  0  0  0
k1       0.00  1  0  0
k2       0.00  0  1  0
tb       0.00  0  0  1

Starting values for error model parameters:
a.1 
  1 

Results:

Likelihood computed by importance sampling
  AIC   BIC logLik
  714 712.1   -348

Optimised parameters:
              est.    lower    upper
DMTA_0    98.16102 96.47747 99.84456
k1         0.07876  0.05261  0.10491
k2         0.02227  0.01706  0.02747
tb        13.99089 -7.40049 35.38228
a.1        1.82305  1.60700  2.03910
SD.DMTA_0  1.88413  0.56204  3.20622
SD.k1      0.34292  0.10482  0.58102
SD.k2      0.19851  0.01718  0.37985
SD.tb      1.68168  0.58064  2.78272

Correlation: 
   DMTA_0  k1      k2     
k1  0.0142                
k2  0.0001 -0.0025        
tb  0.0165 -0.1256 -0.0301

Random effects:
            est.   lower  upper
SD.DMTA_0 1.8841 0.56204 3.2062
SD.k1     0.3429 0.10482 0.5810
SD.k2     0.1985 0.01718 0.3798
SD.tb     1.6817 0.58064 2.7827

Variance model:
     est. lower upper
a.1 1.823 1.607 2.039

Estimated disappearance times:
      DT50  DT90 DT50back DT50_k1 DT50_k2
DMTA 8.801 67.91    20.44   8.801   31.13

</code></pre>
<p></p>
<caption>
Hierarchical mkin fit of the HS model with error model tc
</caption>
<pre><code>
saemix version used for fitting:      3.2 
mkin version used for pre-fitting:  1.2.6 
R version used for fitting:         4.3.2 
Date of fit:     Thu Nov 16 04:59:29 2023 
Date of summary: Thu Nov 16 05:00:43 2023 

Equations:
d_DMTA/dt = - ifelse(time &lt;= tb, k1, k2) * DMTA

Data:
155 observations of 1 variable(s) grouped in 6 datasets

Model predictions using solution type analytical 

Fitted in 3.658 s
Using 300, 100 iterations and 9 chains

Variance model: Two-component variance function 

Starting values for degradation parameters:
  DMTA_0       k1       k2       tb 
98.45190  0.07525  0.02576 19.19375 

Fixed degradation parameter values:
None

Starting values for random effects (square root of initial entries in omega):
       DMTA_0 k1 k2 tb
DMTA_0  98.45  0  0  0
k1       0.00  1  0  0
k2       0.00  0  1  0
tb       0.00  0  0  1

Starting values for error model parameters:
a.1 b.1 
  1   1 

Results:

Likelihood computed by importance sampling
    AIC BIC logLik
  667.1 665 -323.6

Optimised parameters:
              est.    lower    upper
DMTA_0    97.76571 95.81350 99.71791
k1         0.05855  0.03080  0.08630
k2         0.02337  0.01664  0.03010
tb        31.09638 29.38289 32.80987
a.1        1.08835  0.88590  1.29080
b.1        0.02964  0.02257  0.03671
SD.DMTA_0  2.04877  0.42607  3.67147
SD.k1      0.59166  0.25621  0.92711
SD.k2      0.30698  0.09561  0.51835
SD.tb      0.01274 -0.10914  0.13462

Correlation: 
   DMTA_0  k1      k2     
k1  0.0160                
k2 -0.0070 -0.0024        
tb -0.0668 -0.0103 -0.2013

Random effects:
             est.    lower  upper
SD.DMTA_0 2.04877  0.42607 3.6715
SD.k1     0.59166  0.25621 0.9271
SD.k2     0.30698  0.09561 0.5183
SD.tb     0.01274 -0.10914 0.1346

Variance model:
       est.   lower   upper
a.1 1.08835 0.88590 1.29080
b.1 0.02964 0.02257 0.03671

Estimated disappearance times:
      DT50  DT90 DT50back DT50_k1 DT50_k2
DMTA 11.84 51.71    15.57   11.84   29.66

</code></pre>
<p></p>
</div>
<div class="section level3">
<h3 id="hierarchical-model-convergence-plots">Hierarchical model convergence plots<a class="anchor" aria-label="anchor" href="#hierarchical-model-convergence-plots"></a>
</h3>
<div class="figure" style="text-align: center">
<img src="2022_dmta_parent_files/figure-html/convergence-saem-sfo-const-1.png" alt="Convergence plot for the NLHM SFO fit with constant variance" width="864"><p class="caption">
Convergence plot for the NLHM SFO fit with constant variance
</p>
</div>
<div class="figure" style="text-align: center">
<img src="2022_dmta_parent_files/figure-html/convergence-saem-sfo-tc-1.png" alt="Convergence plot for the NLHM SFO fit with two-component error" width="864"><p class="caption">
Convergence plot for the NLHM SFO fit with two-component error
</p>
</div>
<div class="figure" style="text-align: center">
<img src="2022_dmta_parent_files/figure-html/convergence-saem-fomc-const-1.png" alt="Convergence plot for the NLHM FOMC fit with constant variance" width="864"><p class="caption">
Convergence plot for the NLHM FOMC fit with constant variance
</p>
</div>
<div class="figure" style="text-align: center">
<img src="2022_dmta_parent_files/figure-html/convergence-saem-fomc-tc-1.png" alt="Convergence plot for the NLHM FOMC fit with two-component error" width="864"><p class="caption">
Convergence plot for the NLHM FOMC fit with two-component error
</p>
</div>
<div class="figure" style="text-align: center">
<img src="2022_dmta_parent_files/figure-html/convergence-saem-dfop-const-1.png" alt="Convergence plot for the NLHM DFOP fit with constant variance" width="864"><p class="caption">
Convergence plot for the NLHM DFOP fit with constant variance
</p>
</div>
<div class="figure" style="text-align: center">
<img src="2022_dmta_parent_files/figure-html/convergence-saem-dfop-tc-1.png" alt="Convergence plot for the NLHM DFOP fit with two-component error" width="864"><p class="caption">
Convergence plot for the NLHM DFOP fit with two-component error
</p>
</div>
<div class="figure" style="text-align: center">
<img src="2022_dmta_parent_files/figure-html/convergence-saem-hs-const-1.png" alt="Convergence plot for the NLHM HS fit with constant variance" width="864"><p class="caption">
Convergence plot for the NLHM HS fit with constant variance
</p>
</div>
<div class="figure" style="text-align: center">
<img src="2022_dmta_parent_files/figure-html/convergence-saem-hs-tc-1.png" alt="Convergence plot for the NLHM HS fit with two-component error" width="864"><p class="caption">
Convergence plot for the NLHM HS fit with two-component error
</p>
</div>
</div>
<div class="section level3">
<h3 id="session-info">Session info<a class="anchor" aria-label="anchor" href="#session-info"></a>
</h3>
<pre><code>R version 4.3.2 (2023-10-31)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Debian GNU/Linux 12 (bookworm)

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

locale:
 [1] LC_CTYPE=de_DE.UTF-8       LC_NUMERIC=C              
 [3] LC_TIME=de_DE.UTF-8        LC_COLLATE=de_DE.UTF-8    
 [5] LC_MONETARY=de_DE.UTF-8    LC_MESSAGES=de_DE.UTF-8   
 [7] LC_PAPER=de_DE.UTF-8       LC_NAME=C                 
 [9] LC_ADDRESS=C               LC_TELEPHONE=C            
[11] LC_MEASUREMENT=de_DE.UTF-8 LC_IDENTIFICATION=C       

time zone: Europe/Berlin
tzcode source: system (glibc)

attached base packages:
[1] parallel  stats     graphics  grDevices utils     datasets  methods  
[8] base     

other attached packages:
[1] saemix_3.2 npde_3.3   knitr_1.42 mkin_1.2.6

loaded via a namespace (and not attached):
 [1] sass_0.4.6        utf8_1.2.3        generics_0.1.3    stringi_1.7.12   
 [5] lattice_0.21-8    digest_0.6.31     magrittr_2.0.3    evaluate_0.21    
 [9] grid_4.3.2        fastmap_1.1.1     rprojroot_2.0.3   jsonlite_1.8.4   
[13] DBI_1.1.3         mclust_6.0.0      gridExtra_2.3     purrr_1.0.1      
[17] fansi_1.0.4       scales_1.2.1      codetools_0.2-19  textshaping_0.3.6
[21] jquerylib_0.1.4   cli_3.6.1         rlang_1.1.1       munsell_0.5.0    
[25] cachem_1.0.8      yaml_2.3.7        tools_4.3.2       memoise_2.0.1    
[29] dplyr_1.1.2       colorspace_2.1-0  ggplot2_3.4.2     vctrs_0.6.2      
[33] R6_2.5.1          zoo_1.8-12        lifecycle_1.0.3   stringr_1.5.0    
[37] fs_1.6.2          MASS_7.3-60       ragg_1.2.5        pkgconfig_2.0.3  
[41] desc_1.4.2        pkgdown_2.0.7     bslib_0.4.2       pillar_1.9.0     
[45] gtable_0.3.3      glue_1.6.2        systemfonts_1.0.4 highr_0.10       
[49] xfun_0.39         tibble_3.2.1      lmtest_0.9-40     tidyselect_1.2.0 
[53] htmltools_0.5.5   nlme_3.1-163      rmarkdown_2.21    compiler_4.3.2   </code></pre>
</div>
<div class="section level3">
<h3 id="hardware-info">Hardware info<a class="anchor" aria-label="anchor" href="#hardware-info"></a>
</h3>
<pre><code>CPU model: AMD Ryzen 9 7950X 16-Core Processor</code></pre>
<pre><code>MemTotal:       64927712 kB</code></pre>
</div>
</div>
  </main><aside class="col-md-3"><nav id="toc"><h2>On this page</h2>
    </nav></aside>
</div>



    <footer><div class="pkgdown-footer-left">
  <p></p>
<p>Developed by Johannes Ranke.</p>
</div>

<div class="pkgdown-footer-right">
  <p></p>
<p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.0.7.</p>
</div>

    </footer>
</div>

  

  

  </body>
</html>

Contact - Imprint