1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
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
|
<!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=1.0">
<title>Example evaluation of FOCUS Laboratory Data L1 to L3 • mkin</title>
<!-- jquery --><script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js" integrity="sha256-FgpCb/KJQlLNfOu91ta32o/NMZxltwRo8QtmkMRdAu8=" crossorigin="anonymous"></script><!-- Bootstrap --><link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/twitter-bootstrap/3.3.7/css/bootstrap.min.css" integrity="sha256-916EbMg70RQy9LHiGkXzG8hSg9EdNy97GazNG/aiY1w=" crossorigin="anonymous">
<script src="https://cdnjs.cloudflare.com/ajax/libs/twitter-bootstrap/3.3.7/js/bootstrap.min.js" integrity="sha256-U5ZEeKfGNOja007MMD3YBI0A3OSZOQbeG6z2f2Y0hu8=" crossorigin="anonymous"></script><!-- Font Awesome icons --><link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.7.1/css/all.min.css" integrity="sha256-nAmazAk6vS34Xqo0BSrTb+abbtFlgsFK7NKSi6o7Y78=" crossorigin="anonymous">
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.7.1/css/v4-shims.min.css" integrity="sha256-6qHlizsOWFskGlwVOKuns+D1nB6ssZrHQrNj1wGplHc=" crossorigin="anonymous">
<!-- clipboard.js --><script src="https://cdnjs.cloudflare.com/ajax/libs/clipboard.js/2.0.4/clipboard.min.js" integrity="sha256-FiZwavyI2V6+EXO1U+xzLG3IKldpiTFf3153ea9zikQ=" crossorigin="anonymous"></script><!-- headroom.js --><script src="https://cdnjs.cloudflare.com/ajax/libs/headroom/0.9.4/headroom.min.js" integrity="sha256-DJFC1kqIhelURkuza0AvYal5RxMtpzLjFhsnVIeuk+U=" crossorigin="anonymous"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/headroom/0.9.4/jQuery.headroom.min.js" integrity="sha256-ZX/yNShbjqsohH1k95liqY9Gd8uOiE1S4vZc+9KQ1K4=" crossorigin="anonymous"></script><!-- pkgdown --><link href="../pkgdown.css" rel="stylesheet">
<script src="../pkgdown.js"></script><meta property="og:title" content="Example evaluation of FOCUS Laboratory Data L1 to L3">
<meta property="og:description" content="">
<meta name="twitter:card" content="summary">
<!-- 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>
<div class="container template-article">
<header><div class="navbar navbar-default navbar-fixed-top" role="navigation">
<div class="container">
<div class="navbar-header">
<button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-target="#navbar" aria-expanded="false">
<span class="sr-only">Toggle navigation</span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
</button>
<span class="navbar-brand">
<a class="navbar-link" href="../index.html">mkin</a>
<span class="version label label-default" data-toggle="tooltip" data-placement="bottom" title="Released version">0.9.49.6</span>
</span>
</div>
<div id="navbar" class="navbar-collapse collapse">
<ul class="nav navbar-nav">
<li>
<a href="../reference/index.html">Functions and data</a>
</li>
<li class="dropdown">
<a href="#" class="dropdown-toggle" data-toggle="dropdown" role="button" aria-expanded="false">
Articles
<span class="caret"></span>
</a>
<ul class="dropdown-menu" role="menu">
<li>
<a href="../articles/mkin.html">Introduction to mkin</a>
</li>
<li>
<a href="../articles/FOCUS_D.html">Example evaluation of FOCUS Example Dataset D</a>
</li>
<li>
<a href="../articles/FOCUS_L.html">Example evaluation of FOCUS Laboratory Data L1 to L3</a>
</li>
<li>
<a href="../articles/web_only/FOCUS_Z.html">Example evaluation of FOCUS Example Dataset Z</a>
</li>
<li>
<a href="../articles/web_only/compiled_models.html">Performance benefit by using compiled model definitions in mkin</a>
</li>
<li>
<a href="../articles/twa.html">Calculation of time weighted average concentrations with mkin</a>
</li>
<li>
<a href="../articles/web_only/NAFTA_examples.html">Example evaluation of NAFTA SOP Attachment examples</a>
</li>
</ul>
</li>
<li>
<a href="../news/index.html">News</a>
</li>
</ul>
<ul class="nav navbar-nav navbar-right"></ul>
</div>
<!--/.nav-collapse -->
</div>
<!--/.container -->
</div>
<!--/.navbar -->
</header><div class="row">
<div class="col-md-9 contents">
<div class="page-header toc-ignore">
<h1>Example evaluation of FOCUS Laboratory Data L1 to L3</h1>
<h4 class="author">Johannes Ranke</h4>
<h4 class="date">2019-09-18</h4>
<div class="hidden name"><code>FOCUS_L.Rmd</code></div>
</div>
<div id="laboratory-data-l1" class="section level1">
<h1 class="hasAnchor">
<a href="#laboratory-data-l1" class="anchor"></a>Laboratory Data L1</h1>
<p>The following code defines example dataset L1 from the FOCUS kinetics report, p. 284:</p>
<div class="sourceCode" id="cb1"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb1-1" title="1"><span class="kw"><a href="https://rdrr.io/r/base/library.html">library</a></span>(<span class="st">"mkin"</span>, <span class="dt">quietly =</span> <span class="ot">TRUE</span>)</a>
<a class="sourceLine" id="cb1-2" title="2">FOCUS_<span class="dv">2006</span>_L1 =<span class="st"> </span><span class="kw"><a href="https://rdrr.io/r/base/data.frame.html">data.frame</a></span>(</a>
<a class="sourceLine" id="cb1-3" title="3"> <span class="dt">t =</span> <span class="kw"><a href="https://rdrr.io/r/base/rep.html">rep</a></span>(<span class="kw"><a href="https://rdrr.io/r/base/c.html">c</a></span>(<span class="dv">0</span>, <span class="dv">1</span>, <span class="dv">2</span>, <span class="dv">3</span>, <span class="dv">5</span>, <span class="dv">7</span>, <span class="dv">14</span>, <span class="dv">21</span>, <span class="dv">30</span>), <span class="dt">each =</span> <span class="dv">2</span>),</a>
<a class="sourceLine" id="cb1-4" title="4"> <span class="dt">parent =</span> <span class="kw"><a href="https://rdrr.io/r/base/c.html">c</a></span>(<span class="fl">88.3</span>, <span class="fl">91.4</span>, <span class="fl">85.6</span>, <span class="fl">84.5</span>, <span class="fl">78.9</span>, <span class="fl">77.6</span>,</a>
<a class="sourceLine" id="cb1-5" title="5"> <span class="fl">72.0</span>, <span class="fl">71.9</span>, <span class="fl">50.3</span>, <span class="fl">59.4</span>, <span class="fl">47.0</span>, <span class="fl">45.1</span>,</a>
<a class="sourceLine" id="cb1-6" title="6"> <span class="fl">27.7</span>, <span class="fl">27.3</span>, <span class="fl">10.0</span>, <span class="fl">10.4</span>, <span class="fl">2.9</span>, <span class="fl">4.0</span>))</a>
<a class="sourceLine" id="cb1-7" title="7">FOCUS_<span class="dv">2006</span>_L1_mkin <-<span class="st"> </span><span class="kw"><a href="../reference/mkin_wide_to_long.html">mkin_wide_to_long</a></span>(FOCUS_<span class="dv">2006</span>_L1)</a></code></pre></div>
<p>Here we use the assumptions of simple first order (SFO), the case of declining rate constant over time (FOMC) and the case of two different phases of the kinetics (DFOP). For a more detailed discussion of the models, please see the FOCUS kinetics report.</p>
<p>Since mkin version 0.9-32 (July 2014), we can use shorthand notation like <code>"SFO"</code> for parent only degradation models. The following two lines fit the model and produce the summary report of the model fit. This covers the numerical analysis given in the FOCUS report.</p>
<div class="sourceCode" id="cb2"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb2-1" title="1">m.L1.SFO <-<span class="st"> </span><span class="kw"><a href="../reference/mkinfit.html">mkinfit</a></span>(<span class="st">"SFO"</span>, FOCUS_<span class="dv">2006</span>_L1_mkin, <span class="dt">quiet =</span> <span class="ot">TRUE</span>)</a>
<a class="sourceLine" id="cb2-2" title="2"><span class="kw"><a href="https://rdrr.io/r/base/summary.html">summary</a></span>(m.L1.SFO)</a></code></pre></div>
<pre><code>## mkin version used for fitting: 0.9.49.6
## R version used for fitting: 3.6.1
## Date of fit: Wed Sep 18 17:55:08 2019
## Date of summary: Wed Sep 18 17:55:08 2019
##
## Equations:
## d_parent/dt = - k_parent_sink * parent
##
## Model predictions using solution type analytical
##
## Fitted using 133 model solutions performed in 0.293 s
##
## Error model: Constant variance
##
## Error model algorithm: OLS
##
## Starting values for parameters to be optimised:
## value type
## parent_0 89.850000 state
## k_parent_sink 0.100000 deparm
## sigma 2.779827 error
##
## Starting values for the transformed parameters actually optimised:
## value lower upper
## parent_0 89.850000 -Inf Inf
## log_k_parent_sink -2.302585 -Inf Inf
## sigma 2.779827 0 Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters with symmetric confidence intervals:
## Estimate Std. Error Lower Upper
## parent_0 92.470 1.28200 89.740 95.200
## log_k_parent_sink -2.347 0.03763 -2.428 -2.267
## sigma 2.780 0.46330 1.792 3.767
##
## Parameter correlation:
## parent_0 log_k_parent_sink sigma
## parent_0 1.000e+00 6.186e-01 -1.712e-09
## log_k_parent_sink 6.186e-01 1.000e+00 -3.237e-09
## sigma -1.712e-09 -3.237e-09 1.000e+00
##
## Backtransformed parameters:
## Confidence intervals for internally transformed parameters are asymmetric.
## t-test (unrealistically) based on the assumption of normal distribution
## for estimators of untransformed parameters.
## Estimate t value Pr(>t) Lower Upper
## parent_0 92.47000 72.13 8.824e-21 89.74000 95.2000
## k_parent_sink 0.09561 26.57 2.487e-14 0.08824 0.1036
## sigma 2.78000 6.00 1.216e-05 1.79200 3.7670
##
## FOCUS Chi2 error levels in percent:
## err.min n.optim df
## All data 3.424 2 7
## parent 3.424 2 7
##
## Resulting formation fractions:
## ff
## parent_sink 1
##
## Estimated disappearance times:
## DT50 DT90
## parent 7.249 24.08
##
## Data:
## time variable observed predicted residual
## 0 parent 88.3 92.471 -4.1710
## 0 parent 91.4 92.471 -1.0710
## 1 parent 85.6 84.039 1.5610
## 1 parent 84.5 84.039 0.4610
## 2 parent 78.9 76.376 2.5241
## 2 parent 77.6 76.376 1.2241
## 3 parent 72.0 69.412 2.5884
## 3 parent 71.9 69.412 2.4884
## 5 parent 50.3 57.330 -7.0301
## 5 parent 59.4 57.330 2.0699
## 7 parent 47.0 47.352 -0.3515
## 7 parent 45.1 47.352 -2.2515
## 14 parent 27.7 24.247 3.4528
## 14 parent 27.3 24.247 3.0528
## 21 parent 10.0 12.416 -2.4163
## 21 parent 10.4 12.416 -2.0163
## 30 parent 2.9 5.251 -2.3513
## 30 parent 4.0 5.251 -1.2513</code></pre>
<p>A plot of the fit is obtained with the plot function for mkinfit objects.</p>
<div class="sourceCode" id="cb4"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb4-1" title="1"><span class="kw"><a href="https://rdrr.io/r/graphics/plot.html">plot</a></span>(m.L1.SFO, <span class="dt">show_errmin =</span> <span class="ot">TRUE</span>, <span class="dt">main =</span> <span class="st">"FOCUS L1 - SFO"</span>)</a></code></pre></div>
<p><img src="FOCUS_L_files/figure-html/unnamed-chunk-4-1.png" width="576"></p>
<p>The residual plot can be easily obtained by</p>
<div class="sourceCode" id="cb5"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb5-1" title="1"><span class="kw"><a href="../reference/mkinresplot.html">mkinresplot</a></span>(m.L1.SFO, <span class="dt">ylab =</span> <span class="st">"Observed"</span>, <span class="dt">xlab =</span> <span class="st">"Time"</span>)</a></code></pre></div>
<p><img src="FOCUS_L_files/figure-html/unnamed-chunk-5-1.png" width="576"></p>
<p>For comparison, the FOMC model is fitted as well, and the <span class="math inline">\(\chi^2\)</span> error level is checked.</p>
<div class="sourceCode" id="cb6"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb6-1" title="1">m.L1.FOMC <-<span class="st"> </span><span class="kw"><a href="../reference/mkinfit.html">mkinfit</a></span>(<span class="st">"FOMC"</span>, FOCUS_<span class="dv">2006</span>_L1_mkin, <span class="dt">quiet=</span><span class="ot">TRUE</span>)</a></code></pre></div>
<pre><code>## Warning in mkinfit("FOMC", FOCUS_2006_L1_mkin, quiet = TRUE): Optimisation did not converge:
## false convergence (8)</code></pre>
<div class="sourceCode" id="cb8"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb8-1" title="1"><span class="kw"><a href="https://rdrr.io/r/graphics/plot.html">plot</a></span>(m.L1.FOMC, <span class="dt">show_errmin =</span> <span class="ot">TRUE</span>, <span class="dt">main =</span> <span class="st">"FOCUS L1 - FOMC"</span>)</a></code></pre></div>
<p><img src="FOCUS_L_files/figure-html/unnamed-chunk-6-1.png" width="576"></p>
<div class="sourceCode" id="cb9"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb9-1" title="1"><span class="kw"><a href="https://rdrr.io/r/base/summary.html">summary</a></span>(m.L1.FOMC, <span class="dt">data =</span> <span class="ot">FALSE</span>)</a></code></pre></div>
<pre><code>## Warning in sqrt(diag(covar)): NaNs wurden erzeugt</code></pre>
<pre><code>## Warning in sqrt(1/diag(V)): NaNs wurden erzeugt</code></pre>
<pre><code>## Warning in cov2cor(ans$cov.unscaled): diag(.) had 0 or NA entries; non-
## finite result is doubtful</code></pre>
<pre><code>## mkin version used for fitting: 0.9.49.6
## R version used for fitting: 3.6.1
## Date of fit: Wed Sep 18 17:55:10 2019
## Date of summary: Wed Sep 18 17:55:10 2019
##
##
## Warning: Optimisation did not converge:
## false convergence (8)
##
##
## Equations:
## d_parent/dt = - (alpha/beta) * 1/((time/beta) + 1) * parent
##
## Model predictions using solution type analytical
##
## Fitted using 899 model solutions performed in 1.914 s
##
## Error model: Constant variance
##
## Error model algorithm: OLS
##
## Starting values for parameters to be optimised:
## value type
## parent_0 89.850000 state
## alpha 1.000000 deparm
## beta 10.000000 deparm
## sigma 2.779871 error
##
## Starting values for the transformed parameters actually optimised:
## value lower upper
## parent_0 89.850000 -Inf Inf
## log_alpha 0.000000 -Inf Inf
## log_beta 2.302585 -Inf Inf
## sigma 2.779871 0 Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters with symmetric confidence intervals:
## Estimate Std. Error Lower Upper
## parent_0 92.47 1.2800 89.730 95.220
## log_alpha 10.58 NaN NaN NaN
## log_beta 12.93 NaN NaN NaN
## sigma 2.78 0.4507 1.813 3.747
##
## Parameter correlation:
## parent_0 log_alpha log_beta sigma
## parent_0 1.00000 NaN NaN 0.01452
## log_alpha NaN 1 NaN NaN
## log_beta NaN NaN 1 NaN
## sigma 0.01452 NaN NaN 1.00000
##
## Backtransformed parameters:
## Confidence intervals for internally transformed parameters are asymmetric.
## t-test (unrealistically) based on the assumption of normal distribution
## for estimators of untransformed parameters.
## Estimate t value Pr(>t) Lower Upper
## parent_0 92.47 72.13000 1.052e-19 89.730 95.220
## alpha 39440.00 0.02397 4.906e-01 NA NA
## beta 412500.00 0.02397 4.906e-01 NA NA
## sigma 2.78 6.00000 1.628e-05 1.813 3.747
##
## FOCUS Chi2 error levels in percent:
## err.min n.optim df
## All data 3.619 3 6
## parent 3.619 3 6
##
## Estimated disappearance times:
## DT50 DT90 DT50back
## parent 7.249 24.08 7.249</code></pre>
<p>We get a warning that the default optimisation algorithm <code>Port</code> did not converge, which is an indication that the model is overparameterised, <em>i.e.</em> contains too many parameters that are ill-defined as a consequence.</p>
<p>And in fact, due to the higher number of parameters, and the lower number of degrees of freedom of the fit, the <span class="math inline">\(\chi^2\)</span> error level is actually higher for the FOMC model (3.6%) than for the SFO model (3.4%). Additionally, the parameters <code>log_alpha</code> and <code>log_beta</code> internally fitted in the model have excessive confidence intervals, that span more than 25 orders of magnitude (!) when backtransformed to the scale of <code>alpha</code> and <code>beta</code>. Also, the t-test for significant difference from zero does not indicate such a significant difference, with p-values greater than 0.1, and finally, the parameter correlation of <code>log_alpha</code> and <code>log_beta</code> is 1.000, clearly indicating that the model is overparameterised.</p>
<p>The <span class="math inline">\(\chi^2\)</span> error levels reported in Appendix 3 and Appendix 7 to the FOCUS kinetics report are rounded to integer percentages and partly deviate by one percentage point from the results calculated by mkin. The reason for this is not known. However, mkin gives the same <span class="math inline">\(\chi^2\)</span> error levels as the kinfit package and the calculation routines of the kinfit package have been extensively compared to the results obtained by the KinGUI software, as documented in the kinfit package vignette. KinGUI was the first widely used standard package in this field. Also, the calculation of <span class="math inline">\(\chi^2\)</span> error levels was compared with KinGUII, CAKE and DegKin manager in a project sponsored by the German Umweltbundesamt <span class="citation">(Ranke 2014)</span>.</p>
</div>
<div id="laboratory-data-l2" class="section level1">
<h1 class="hasAnchor">
<a href="#laboratory-data-l2" class="anchor"></a>Laboratory Data L2</h1>
<p>The following code defines example dataset L2 from the FOCUS kinetics report, p. 287:</p>
<div class="sourceCode" id="cb14"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb14-1" title="1">FOCUS_<span class="dv">2006</span>_L2 =<span class="st"> </span><span class="kw"><a href="https://rdrr.io/r/base/data.frame.html">data.frame</a></span>(</a>
<a class="sourceLine" id="cb14-2" title="2"> <span class="dt">t =</span> <span class="kw"><a href="https://rdrr.io/r/base/rep.html">rep</a></span>(<span class="kw"><a href="https://rdrr.io/r/base/c.html">c</a></span>(<span class="dv">0</span>, <span class="dv">1</span>, <span class="dv">3</span>, <span class="dv">7</span>, <span class="dv">14</span>, <span class="dv">28</span>), <span class="dt">each =</span> <span class="dv">2</span>),</a>
<a class="sourceLine" id="cb14-3" title="3"> <span class="dt">parent =</span> <span class="kw"><a href="https://rdrr.io/r/base/c.html">c</a></span>(<span class="fl">96.1</span>, <span class="fl">91.8</span>, <span class="fl">41.4</span>, <span class="fl">38.7</span>,</a>
<a class="sourceLine" id="cb14-4" title="4"> <span class="fl">19.3</span>, <span class="fl">22.3</span>, <span class="fl">4.6</span>, <span class="fl">4.6</span>,</a>
<a class="sourceLine" id="cb14-5" title="5"> <span class="fl">2.6</span>, <span class="fl">1.2</span>, <span class="fl">0.3</span>, <span class="fl">0.6</span>))</a>
<a class="sourceLine" id="cb14-6" title="6">FOCUS_<span class="dv">2006</span>_L2_mkin <-<span class="st"> </span><span class="kw"><a href="../reference/mkin_wide_to_long.html">mkin_wide_to_long</a></span>(FOCUS_<span class="dv">2006</span>_L2)</a></code></pre></div>
<div id="sfo-fit-for-l2" class="section level2">
<h2 class="hasAnchor">
<a href="#sfo-fit-for-l2" class="anchor"></a>SFO fit for L2</h2>
<p>Again, the SFO model is fitted and the result is plotted. The residual plot can be obtained simply by adding the argument <code>show_residuals</code> to the plot command.</p>
<div class="sourceCode" id="cb15"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb15-1" title="1">m.L2.SFO <-<span class="st"> </span><span class="kw"><a href="../reference/mkinfit.html">mkinfit</a></span>(<span class="st">"SFO"</span>, FOCUS_<span class="dv">2006</span>_L2_mkin, <span class="dt">quiet=</span><span class="ot">TRUE</span>)</a>
<a class="sourceLine" id="cb15-2" title="2"><span class="kw"><a href="https://rdrr.io/r/graphics/plot.html">plot</a></span>(m.L2.SFO, <span class="dt">show_residuals =</span> <span class="ot">TRUE</span>, <span class="dt">show_errmin =</span> <span class="ot">TRUE</span>,</a>
<a class="sourceLine" id="cb15-3" title="3"> <span class="dt">main =</span> <span class="st">"FOCUS L2 - SFO"</span>)</a></code></pre></div>
<p><img src="FOCUS_L_files/figure-html/unnamed-chunk-8-1.png" width="672"></p>
<p>The <span class="math inline">\(\chi^2\)</span> error level of 14% suggests that the model does not fit very well. This is also obvious from the plots of the fit, in which we have included the residual plot.</p>
<p>In the FOCUS kinetics report, it is stated that there is no apparent systematic error observed from the residual plot up to the measured DT90 (approximately at day 5), and there is an underestimation beyond that point.</p>
<p>We may add that it is difficult to judge the random nature of the residuals just from the three samplings at days 0, 1 and 3. Also, it is not clear <em>a priori</em> why a consistent underestimation after the approximate DT90 should be irrelevant. However, this can be rationalised by the fact that the FOCUS fate models generally only implement SFO kinetics.</p>
</div>
<div id="fomc-fit-for-l2" class="section level2">
<h2 class="hasAnchor">
<a href="#fomc-fit-for-l2" class="anchor"></a>FOMC fit for L2</h2>
<p>For comparison, the FOMC model is fitted as well, and the <span class="math inline">\(\chi^2\)</span> error level is checked.</p>
<div class="sourceCode" id="cb16"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb16-1" title="1">m.L2.FOMC <-<span class="st"> </span><span class="kw"><a href="../reference/mkinfit.html">mkinfit</a></span>(<span class="st">"FOMC"</span>, FOCUS_<span class="dv">2006</span>_L2_mkin, <span class="dt">quiet =</span> <span class="ot">TRUE</span>)</a>
<a class="sourceLine" id="cb16-2" title="2"><span class="kw"><a href="https://rdrr.io/r/graphics/plot.html">plot</a></span>(m.L2.FOMC, <span class="dt">show_residuals =</span> <span class="ot">TRUE</span>,</a>
<a class="sourceLine" id="cb16-3" title="3"> <span class="dt">main =</span> <span class="st">"FOCUS L2 - FOMC"</span>)</a></code></pre></div>
<p><img src="FOCUS_L_files/figure-html/unnamed-chunk-9-1.png" width="672"></p>
<div class="sourceCode" id="cb17"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb17-1" title="1"><span class="kw"><a href="https://rdrr.io/r/base/summary.html">summary</a></span>(m.L2.FOMC, <span class="dt">data =</span> <span class="ot">FALSE</span>)</a></code></pre></div>
<pre><code>## mkin version used for fitting: 0.9.49.6
## R version used for fitting: 3.6.1
## Date of fit: Wed Sep 18 17:55:11 2019
## Date of summary: Wed Sep 18 17:55:11 2019
##
## Equations:
## d_parent/dt = - (alpha/beta) * 1/((time/beta) + 1) * parent
##
## Model predictions using solution type analytical
##
## Fitted using 239 model solutions performed in 0.511 s
##
## Error model: Constant variance
##
## Error model algorithm: OLS
##
## Starting values for parameters to be optimised:
## value type
## parent_0 93.950000 state
## alpha 1.000000 deparm
## beta 10.000000 deparm
## sigma 2.275722 error
##
## Starting values for the transformed parameters actually optimised:
## value lower upper
## parent_0 93.950000 -Inf Inf
## log_alpha 0.000000 -Inf Inf
## log_beta 2.302585 -Inf Inf
## sigma 2.275722 0 Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters with symmetric confidence intervals:
## Estimate Std. Error Lower Upper
## parent_0 93.7700 1.6130 90.05000 97.4900
## log_alpha 0.3180 0.1559 -0.04149 0.6776
## log_beta 0.2102 0.2493 -0.36460 0.7850
## sigma 2.2760 0.4645 1.20500 3.3470
##
## Parameter correlation:
## parent_0 log_alpha log_beta sigma
## parent_0 1.000e+00 -1.151e-01 -2.085e-01 -7.637e-09
## log_alpha -1.151e-01 1.000e+00 9.741e-01 -1.617e-07
## log_beta -2.085e-01 9.741e-01 1.000e+00 -1.387e-07
## sigma -7.637e-09 -1.617e-07 -1.387e-07 1.000e+00
##
## Backtransformed parameters:
## Confidence intervals for internally transformed parameters are asymmetric.
## t-test (unrealistically) based on the assumption of normal distribution
## for estimators of untransformed parameters.
## Estimate t value Pr(>t) Lower Upper
## parent_0 93.770 58.120 4.267e-12 90.0500 97.490
## alpha 1.374 6.414 1.030e-04 0.9594 1.969
## beta 1.234 4.012 1.942e-03 0.6945 2.192
## sigma 2.276 4.899 5.977e-04 1.2050 3.347
##
## FOCUS Chi2 error levels in percent:
## err.min n.optim df
## All data 6.205 3 3
## parent 6.205 3 3
##
## Estimated disappearance times:
## DT50 DT90 DT50back
## parent 0.8092 5.356 1.612</code></pre>
<p>The error level at which the <span class="math inline">\(\chi^2\)</span> test passes is much lower in this case. Therefore, the FOMC model provides a better description of the data, as less experimental error has to be assumed in order to explain the data.</p>
</div>
<div id="dfop-fit-for-l2" class="section level2">
<h2 class="hasAnchor">
<a href="#dfop-fit-for-l2" class="anchor"></a>DFOP fit for L2</h2>
<p>Fitting the four parameter DFOP model further reduces the <span class="math inline">\(\chi^2\)</span> error level.</p>
<div class="sourceCode" id="cb19"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb19-1" title="1">m.L2.DFOP <-<span class="st"> </span><span class="kw"><a href="../reference/mkinfit.html">mkinfit</a></span>(<span class="st">"DFOP"</span>, FOCUS_<span class="dv">2006</span>_L2_mkin, <span class="dt">quiet =</span> <span class="ot">TRUE</span>)</a>
<a class="sourceLine" id="cb19-2" title="2"><span class="kw"><a href="https://rdrr.io/r/graphics/plot.html">plot</a></span>(m.L2.DFOP, <span class="dt">show_residuals =</span> <span class="ot">TRUE</span>, <span class="dt">show_errmin =</span> <span class="ot">TRUE</span>,</a>
<a class="sourceLine" id="cb19-3" title="3"> <span class="dt">main =</span> <span class="st">"FOCUS L2 - DFOP"</span>)</a></code></pre></div>
<p><img src="FOCUS_L_files/figure-html/unnamed-chunk-10-1.png" width="672"></p>
<div class="sourceCode" id="cb20"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb20-1" title="1"><span class="kw"><a href="https://rdrr.io/r/base/summary.html">summary</a></span>(m.L2.DFOP, <span class="dt">data =</span> <span class="ot">FALSE</span>)</a></code></pre></div>
<pre><code>## mkin version used for fitting: 0.9.49.6
## R version used for fitting: 3.6.1
## Date of fit: Wed Sep 18 17:55:12 2019
## Date of summary: Wed Sep 18 17:55:12 2019
##
## Equations:
## d_parent/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) *
## exp(-k2 * time)) / (g * exp(-k1 * time) + (1 - g) *
## exp(-k2 * time))) * parent
##
## Model predictions using solution type analytical
##
## Fitted using 572 model solutions performed in 1.217 s
##
## Error model: Constant variance
##
## Error model algorithm: OLS
##
## Starting values for parameters to be optimised:
## value type
## parent_0 93.950000 state
## k1 0.100000 deparm
## k2 0.010000 deparm
## g 0.500000 deparm
## sigma 1.413899 error
##
## Starting values for the transformed parameters actually optimised:
## value lower upper
## parent_0 93.950000 -Inf Inf
## log_k1 -2.302585 -Inf Inf
## log_k2 -4.605170 -Inf Inf
## g_ilr 0.000000 -Inf Inf
## sigma 1.413899 0 Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters with symmetric confidence intervals:
## Estimate Std. Error Lower Upper
## parent_0 93.9500 9.998e-01 91.5900 96.3100
## log_k1 3.1370 2.376e+03 -5616.0000 5622.0000
## log_k2 -1.0880 6.285e-02 -1.2370 -0.9394
## g_ilr -0.2821 7.033e-02 -0.4484 -0.1158
## sigma 1.4140 2.886e-01 0.7314 2.0960
##
## Parameter correlation:
## parent_0 log_k1 log_k2 g_ilr sigma
## parent_0 1.000e+00 5.155e-07 2.371e-09 2.665e-01 -6.849e-09
## log_k1 5.155e-07 1.000e+00 8.434e-05 -1.659e-04 -7.791e-06
## log_k2 2.371e-09 8.434e-05 1.000e+00 -7.903e-01 -1.262e-08
## g_ilr 2.665e-01 -1.659e-04 -7.903e-01 1.000e+00 3.241e-08
## sigma -6.849e-09 -7.791e-06 -1.262e-08 3.241e-08 1.000e+00
##
## Backtransformed parameters:
## Confidence intervals for internally transformed parameters are asymmetric.
## t-test (unrealistically) based on the assumption of normal distribution
## for estimators of untransformed parameters.
## Estimate t value Pr(>t) Lower Upper
## parent_0 93.9500 9.397e+01 2.036e-12 91.5900 96.3100
## k1 23.0400 4.303e-04 4.998e-01 0.0000 Inf
## k2 0.3369 1.591e+01 4.697e-07 0.2904 0.3909
## g 0.4016 1.680e+01 3.238e-07 0.3466 0.4591
## sigma 1.4140 4.899e+00 8.776e-04 0.7314 2.0960
##
## FOCUS Chi2 error levels in percent:
## err.min n.optim df
## All data 2.53 4 2
## parent 2.53 4 2
##
## Estimated disappearance times:
## DT50 DT90 DT50_k1 DT50_k2
## parent 0.5335 5.311 0.03009 2.058</code></pre>
<p>Here, the DFOP model is clearly the best-fit model for dataset L2 based on the chi^2 error level criterion. However, the failure to calculate the covariance matrix indicates that the parameter estimates correlate excessively. Therefore, the FOMC model may be preferred for this dataset.</p>
</div>
</div>
<div id="laboratory-data-l3" class="section level1">
<h1 class="hasAnchor">
<a href="#laboratory-data-l3" class="anchor"></a>Laboratory Data L3</h1>
<p>The following code defines example dataset L3 from the FOCUS kinetics report, p. 290.</p>
<div class="sourceCode" id="cb22"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb22-1" title="1">FOCUS_<span class="dv">2006</span>_L3 =<span class="st"> </span><span class="kw"><a href="https://rdrr.io/r/base/data.frame.html">data.frame</a></span>(</a>
<a class="sourceLine" id="cb22-2" title="2"> <span class="dt">t =</span> <span class="kw"><a href="https://rdrr.io/r/base/c.html">c</a></span>(<span class="dv">0</span>, <span class="dv">3</span>, <span class="dv">7</span>, <span class="dv">14</span>, <span class="dv">30</span>, <span class="dv">60</span>, <span class="dv">91</span>, <span class="dv">120</span>),</a>
<a class="sourceLine" id="cb22-3" title="3"> <span class="dt">parent =</span> <span class="kw"><a href="https://rdrr.io/r/base/c.html">c</a></span>(<span class="fl">97.8</span>, <span class="dv">60</span>, <span class="dv">51</span>, <span class="dv">43</span>, <span class="dv">35</span>, <span class="dv">22</span>, <span class="dv">15</span>, <span class="dv">12</span>))</a>
<a class="sourceLine" id="cb22-4" title="4">FOCUS_<span class="dv">2006</span>_L3_mkin <-<span class="st"> </span><span class="kw"><a href="../reference/mkin_wide_to_long.html">mkin_wide_to_long</a></span>(FOCUS_<span class="dv">2006</span>_L3)</a></code></pre></div>
<div id="fit-multiple-models" class="section level2">
<h2 class="hasAnchor">
<a href="#fit-multiple-models" class="anchor"></a>Fit multiple models</h2>
<p>As of mkin version 0.9-39 (June 2015), we can fit several models to one or more datasets in one call to the function <code>mmkin</code>. The datasets have to be passed in a list, in this case a named list holding only the L3 dataset prepared above.</p>
<div class="sourceCode" id="cb23"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb23-1" title="1"><span class="co"># Only use one core here, not to offend the CRAN checks</span></a>
<a class="sourceLine" id="cb23-2" title="2">mm.L3 <-<span class="st"> </span><span class="kw"><a href="../reference/mmkin.html">mmkin</a></span>(<span class="kw"><a href="https://rdrr.io/r/base/c.html">c</a></span>(<span class="st">"SFO"</span>, <span class="st">"FOMC"</span>, <span class="st">"DFOP"</span>), <span class="dt">cores =</span> <span class="dv">1</span>,</a>
<a class="sourceLine" id="cb23-3" title="3"> <span class="kw"><a href="https://rdrr.io/r/base/list.html">list</a></span>(<span class="st">"FOCUS L3"</span> =<span class="st"> </span>FOCUS_<span class="dv">2006</span>_L3_mkin), <span class="dt">quiet =</span> <span class="ot">TRUE</span>)</a>
<a class="sourceLine" id="cb23-4" title="4"><span class="kw"><a href="https://rdrr.io/r/graphics/plot.html">plot</a></span>(mm.L3)</a></code></pre></div>
<p><img src="FOCUS_L_files/figure-html/unnamed-chunk-12-1.png" width="700"></p>
<p>The <span class="math inline">\(\chi^2\)</span> error level of 21% as well as the plot suggest that the SFO model does not fit very well. The FOMC model performs better, with an error level at which the <span class="math inline">\(\chi^2\)</span> test passes of 7%. Fitting the four parameter DFOP model further reduces the <span class="math inline">\(\chi^2\)</span> error level considerably.</p>
</div>
<div id="accessing-mmkin-objects" class="section level2">
<h2 class="hasAnchor">
<a href="#accessing-mmkin-objects" class="anchor"></a>Accessing mmkin objects</h2>
<p>The objects returned by mmkin are arranged like a matrix, with models as a row index and datasets as a column index.</p>
<p>We can extract the summary and plot for <em>e.g.</em> the DFOP fit, using square brackets for indexing which will result in the use of the summary and plot functions working on mkinfit objects.</p>
<div class="sourceCode" id="cb24"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb24-1" title="1"><span class="kw"><a href="https://rdrr.io/r/base/summary.html">summary</a></span>(mm.L3[[<span class="st">"DFOP"</span>, <span class="dv">1</span>]])</a></code></pre></div>
<pre><code>## mkin version used for fitting: 0.9.49.6
## R version used for fitting: 3.6.1
## Date of fit: Wed Sep 18 17:55:14 2019
## Date of summary: Wed Sep 18 17:55:14 2019
##
## Equations:
## d_parent/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) *
## exp(-k2 * time)) / (g * exp(-k1 * time) + (1 - g) *
## exp(-k2 * time))) * parent
##
## Model predictions using solution type analytical
##
## Fitted using 373 model solutions performed in 0.803 s
##
## Error model: Constant variance
##
## Error model algorithm: OLS
##
## Starting values for parameters to be optimised:
## value type
## parent_0 97.800000 state
## k1 0.100000 deparm
## k2 0.010000 deparm
## g 0.500000 deparm
## sigma 1.017292 error
##
## Starting values for the transformed parameters actually optimised:
## value lower upper
## parent_0 97.800000 -Inf Inf
## log_k1 -2.302585 -Inf Inf
## log_k2 -4.605170 -Inf Inf
## g_ilr 0.000000 -Inf Inf
## sigma 1.017292 0 Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters with symmetric confidence intervals:
## Estimate Std. Error Lower Upper
## parent_0 97.7500 1.01900 94.5000 101.000000
## log_k1 -0.6612 0.10050 -0.9812 -0.341300
## log_k2 -4.2860 0.04322 -4.4230 -4.148000
## g_ilr -0.1229 0.03727 -0.2415 -0.004343
## sigma 1.0170 0.25430 0.2079 1.827000
##
## Parameter correlation:
## parent_0 log_k1 log_k2 g_ilr sigma
## parent_0 1.000e+00 1.732e-01 2.282e-02 4.009e-01 -6.872e-07
## log_k1 1.732e-01 1.000e+00 4.945e-01 -5.809e-01 3.200e-07
## log_k2 2.282e-02 4.945e-01 1.000e+00 -6.812e-01 7.673e-07
## g_ilr 4.009e-01 -5.809e-01 -6.812e-01 1.000e+00 -8.731e-07
## sigma -6.872e-07 3.200e-07 7.673e-07 -8.731e-07 1.000e+00
##
## Backtransformed parameters:
## Confidence intervals for internally transformed parameters are asymmetric.
## t-test (unrealistically) based on the assumption of normal distribution
## for estimators of untransformed parameters.
## Estimate t value Pr(>t) Lower Upper
## parent_0 97.75000 95.960 1.248e-06 94.50000 101.00000
## k1 0.51620 9.947 1.081e-03 0.37490 0.71090
## k2 0.01376 23.140 8.840e-05 0.01199 0.01579
## g 0.45660 34.920 2.581e-05 0.41540 0.49850
## sigma 1.01700 4.000 1.400e-02 0.20790 1.82700
##
## FOCUS Chi2 error levels in percent:
## err.min n.optim df
## All data 2.225 4 4
## parent 2.225 4 4
##
## Estimated disappearance times:
## DT50 DT90 DT50_k1 DT50_k2
## parent 7.464 123 1.343 50.37
##
## Data:
## time variable observed predicted residual
## 0 parent 97.8 97.75 0.05396
## 3 parent 60.0 60.45 -0.44933
## 7 parent 51.0 49.44 1.56338
## 14 parent 43.0 43.84 -0.83632
## 30 parent 35.0 35.15 -0.14707
## 60 parent 22.0 23.26 -1.25919
## 91 parent 15.0 15.18 -0.18181
## 120 parent 12.0 10.19 1.81395</code></pre>
<div class="sourceCode" id="cb26"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb26-1" title="1"><span class="kw"><a href="https://rdrr.io/r/graphics/plot.html">plot</a></span>(mm.L3[[<span class="st">"DFOP"</span>, <span class="dv">1</span>]], <span class="dt">show_errmin =</span> <span class="ot">TRUE</span>)</a></code></pre></div>
<p><img src="FOCUS_L_files/figure-html/unnamed-chunk-13-1.png" width="700"></p>
<p>Here, a look to the model plot, the confidence intervals of the parameters and the correlation matrix suggest that the parameter estimates are reliable, and the DFOP model can be used as the best-fit model based on the <span class="math inline">\(\chi^2\)</span> error level criterion for laboratory data L3.</p>
<p>This is also an example where the standard t-test for the parameter <code>g_ilr</code> is misleading, as it tests for a significant difference from zero. In this case, zero appears to be the correct value for this parameter, and the confidence interval for the backtransformed parameter <code>g</code> is quite narrow.</p>
</div>
</div>
<div id="laboratory-data-l4" class="section level1">
<h1 class="hasAnchor">
<a href="#laboratory-data-l4" class="anchor"></a>Laboratory Data L4</h1>
<p>The following code defines example dataset L4 from the FOCUS kinetics report, p. 293:</p>
<div class="sourceCode" id="cb27"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb27-1" title="1">FOCUS_<span class="dv">2006</span>_L4 =<span class="st"> </span><span class="kw"><a href="https://rdrr.io/r/base/data.frame.html">data.frame</a></span>(</a>
<a class="sourceLine" id="cb27-2" title="2"> <span class="dt">t =</span> <span class="kw"><a href="https://rdrr.io/r/base/c.html">c</a></span>(<span class="dv">0</span>, <span class="dv">3</span>, <span class="dv">7</span>, <span class="dv">14</span>, <span class="dv">30</span>, <span class="dv">60</span>, <span class="dv">91</span>, <span class="dv">120</span>),</a>
<a class="sourceLine" id="cb27-3" title="3"> <span class="dt">parent =</span> <span class="kw"><a href="https://rdrr.io/r/base/c.html">c</a></span>(<span class="fl">96.6</span>, <span class="fl">96.3</span>, <span class="fl">94.3</span>, <span class="fl">88.8</span>, <span class="fl">74.9</span>, <span class="fl">59.9</span>, <span class="fl">53.5</span>, <span class="fl">49.0</span>))</a>
<a class="sourceLine" id="cb27-4" title="4">FOCUS_<span class="dv">2006</span>_L4_mkin <-<span class="st"> </span><span class="kw"><a href="../reference/mkin_wide_to_long.html">mkin_wide_to_long</a></span>(FOCUS_<span class="dv">2006</span>_L4)</a></code></pre></div>
<p>Fits of the SFO and FOMC models, plots and summaries are produced below:</p>
<div class="sourceCode" id="cb28"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb28-1" title="1"><span class="co"># Only use one core here, not to offend the CRAN checks</span></a>
<a class="sourceLine" id="cb28-2" title="2">mm.L4 <-<span class="st"> </span><span class="kw"><a href="../reference/mmkin.html">mmkin</a></span>(<span class="kw"><a href="https://rdrr.io/r/base/c.html">c</a></span>(<span class="st">"SFO"</span>, <span class="st">"FOMC"</span>), <span class="dt">cores =</span> <span class="dv">1</span>,</a>
<a class="sourceLine" id="cb28-3" title="3"> <span class="kw"><a href="https://rdrr.io/r/base/list.html">list</a></span>(<span class="st">"FOCUS L4"</span> =<span class="st"> </span>FOCUS_<span class="dv">2006</span>_L4_mkin),</a>
<a class="sourceLine" id="cb28-4" title="4"> <span class="dt">quiet =</span> <span class="ot">TRUE</span>)</a>
<a class="sourceLine" id="cb28-5" title="5"><span class="kw"><a href="https://rdrr.io/r/graphics/plot.html">plot</a></span>(mm.L4)</a></code></pre></div>
<p><img src="FOCUS_L_files/figure-html/unnamed-chunk-15-1.png" width="700"></p>
<p>The <span class="math inline">\(\chi^2\)</span> error level of 3.3% as well as the plot suggest that the SFO model fits very well. The error level at which the <span class="math inline">\(\chi^2\)</span> test passes is slightly lower for the FOMC model. However, the difference appears negligible.</p>
<div class="sourceCode" id="cb29"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb29-1" title="1"><span class="kw"><a href="https://rdrr.io/r/base/summary.html">summary</a></span>(mm.L4[[<span class="st">"SFO"</span>, <span class="dv">1</span>]], <span class="dt">data =</span> <span class="ot">FALSE</span>)</a></code></pre></div>
<pre><code>## mkin version used for fitting: 0.9.49.6
## R version used for fitting: 3.6.1
## Date of fit: Wed Sep 18 17:55:15 2019
## Date of summary: Wed Sep 18 17:55:15 2019
##
## Equations:
## d_parent/dt = - k_parent_sink * parent
##
## Model predictions using solution type analytical
##
## Fitted using 142 model solutions performed in 0.301 s
##
## Error model: Constant variance
##
## Error model algorithm: OLS
##
## Starting values for parameters to be optimised:
## value type
## parent_0 96.60000 state
## k_parent_sink 0.10000 deparm
## sigma 3.16181 error
##
## Starting values for the transformed parameters actually optimised:
## value lower upper
## parent_0 96.600000 -Inf Inf
## log_k_parent_sink -2.302585 -Inf Inf
## sigma 3.161810 0 Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters with symmetric confidence intervals:
## Estimate Std. Error Lower Upper
## parent_0 96.440 1.69900 92.070 100.800
## log_k_parent_sink -5.030 0.07059 -5.211 -4.848
## sigma 3.162 0.79050 1.130 5.194
##
## Parameter correlation:
## parent_0 log_k_parent_sink sigma
## parent_0 1.000e+00 5.938e-01 3.440e-07
## log_k_parent_sink 5.938e-01 1.000e+00 5.885e-07
## sigma 3.440e-07 5.885e-07 1.000e+00
##
## Backtransformed parameters:
## Confidence intervals for internally transformed parameters are asymmetric.
## t-test (unrealistically) based on the assumption of normal distribution
## for estimators of untransformed parameters.
## Estimate t value Pr(>t) Lower Upper
## parent_0 96.440000 56.77 1.604e-08 92.070000 1.008e+02
## k_parent_sink 0.006541 14.17 1.578e-05 0.005455 7.842e-03
## sigma 3.162000 4.00 5.162e-03 1.130000 5.194e+00
##
## FOCUS Chi2 error levels in percent:
## err.min n.optim df
## All data 3.287 2 6
## parent 3.287 2 6
##
## Resulting formation fractions:
## ff
## parent_sink 1
##
## Estimated disappearance times:
## DT50 DT90
## parent 106 352</code></pre>
<div class="sourceCode" id="cb31"><pre class="sourceCode r"><code class="sourceCode r"><a class="sourceLine" id="cb31-1" title="1"><span class="kw"><a href="https://rdrr.io/r/base/summary.html">summary</a></span>(mm.L4[[<span class="st">"FOMC"</span>, <span class="dv">1</span>]], <span class="dt">data =</span> <span class="ot">FALSE</span>)</a></code></pre></div>
<pre><code>## mkin version used for fitting: 0.9.49.6
## R version used for fitting: 3.6.1
## Date of fit: Wed Sep 18 17:55:15 2019
## Date of summary: Wed Sep 18 17:55:15 2019
##
## Equations:
## d_parent/dt = - (alpha/beta) * 1/((time/beta) + 1) * parent
##
## Model predictions using solution type analytical
##
## Fitted using 224 model solutions performed in 0.458 s
##
## Error model: Constant variance
##
## Error model algorithm: OLS
##
## Starting values for parameters to be optimised:
## value type
## parent_0 96.600000 state
## alpha 1.000000 deparm
## beta 10.000000 deparm
## sigma 1.830055 error
##
## Starting values for the transformed parameters actually optimised:
## value lower upper
## parent_0 96.600000 -Inf Inf
## log_alpha 0.000000 -Inf Inf
## log_beta 2.302585 -Inf Inf
## sigma 1.830055 0 Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters with symmetric confidence intervals:
## Estimate Std. Error Lower Upper
## parent_0 99.1400 1.2670 95.6300 102.7000
## log_alpha -0.3506 0.2616 -1.0770 0.3756
## log_beta 4.1740 0.3938 3.0810 5.2670
## sigma 1.8300 0.4575 0.5598 3.1000
##
## Parameter correlation:
## parent_0 log_alpha log_beta sigma
## parent_0 1.000e+00 -4.696e-01 -5.543e-01 -2.563e-07
## log_alpha -4.696e-01 1.000e+00 9.889e-01 4.066e-08
## log_beta -5.543e-01 9.889e-01 1.000e+00 6.818e-08
## sigma -2.563e-07 4.066e-08 6.818e-08 1.000e+00
##
## Backtransformed parameters:
## Confidence intervals for internally transformed parameters are asymmetric.
## t-test (unrealistically) based on the assumption of normal distribution
## for estimators of untransformed parameters.
## Estimate t value Pr(>t) Lower Upper
## parent_0 99.1400 78.250 7.993e-08 95.6300 102.700
## alpha 0.7042 3.823 9.365e-03 0.3407 1.456
## beta 64.9800 2.540 3.201e-02 21.7800 193.900
## sigma 1.8300 4.000 8.065e-03 0.5598 3.100
##
## FOCUS Chi2 error levels in percent:
## err.min n.optim df
## All data 2.029 3 5
## parent 2.029 3 5
##
## Estimated disappearance times:
## DT50 DT90 DT50back
## parent 108.9 1644 494.9</code></pre>
</div>
<div id="references" class="section level1 unnumbered">
<h1 class="hasAnchor">
<a href="#references" class="anchor"></a>References</h1>
<div id="refs" class="references">
<div id="ref-ranke2014">
<p>Ranke, Johannes. 2014. “Prüfung und Validierung von Modellierungssoftware als Alternative zu ModelMaker 4.0.” Umweltbundesamt Projektnummer 27452.</p>
</div>
</div>
</div>
</div>
<div class="col-md-3 hidden-xs hidden-sm" id="sidebar">
<div id="tocnav">
<h2 class="hasAnchor">
<a href="#tocnav" class="anchor"></a>Contents</h2>
<ul class="nav nav-pills nav-stacked">
<li><a href="#laboratory-data-l1">Laboratory Data L1</a></li>
<li>
<a href="#laboratory-data-l2">Laboratory Data L2</a><ul class="nav nav-pills nav-stacked">
<li><a href="#sfo-fit-for-l2">SFO fit for L2</a></li>
<li><a href="#fomc-fit-for-l2">FOMC fit for L2</a></li>
<li><a href="#dfop-fit-for-l2">DFOP fit for L2</a></li>
</ul>
</li>
<li>
<a href="#laboratory-data-l3">Laboratory Data L3</a><ul class="nav nav-pills nav-stacked">
<li><a href="#fit-multiple-models">Fit multiple models</a></li>
<li><a href="#accessing-mmkin-objects">Accessing mmkin objects</a></li>
</ul>
</li>
<li><a href="#laboratory-data-l4">Laboratory Data L4</a></li>
<li><a href="#references">References</a></li>
</ul>
</div>
</div>
</div>
<footer><div class="copyright">
<p>Developed by Johannes Ranke.</p>
</div>
<div class="pkgdown">
<p>Site built with <a href="https://pkgdown.r-lib.org/">pkgdown</a> 1.4.1.</p>
</div>
</footer>
</div>
</body>
</html>
|
