From c322a8102a399cbb1fe38c4c4ca4485cea8bc4e8 Mon Sep 17 00:00:00 2001 From: Johannes Ranke Date: Tue, 7 May 2019 08:43:12 +0200 Subject: Fix hessian calculations Static documentation rebuilt by pkgdown --- R/mkinfit.R | 15 +- docs/articles/FOCUS_D.html | 50 +- .../FOCUS_D_files/figure-html/plot_2-1.png | Bin 13739 -> 14288 bytes docs/articles/FOCUS_L.html | 327 +++++----- docs/articles/web_only/FOCUS_Z.html | 219 ++++--- .../figure-html/FOCUS_2006_Z_fits_11b-1.png | Bin 20999 -> 22316 bytes docs/articles/web_only/NAFTA_examples.html | 691 +++++++++------------ docs/articles/web_only/benchmarks.html | 22 +- docs/articles/web_only/compiled_models.html | 12 +- docs/reference/NAFTA_SOP_2015.html | 32 +- docs/reference/NAFTA_SOP_Attachment.html | 32 +- docs/reference/logistic.solution.html | 18 +- docs/reference/mccall81_245T.html | 45 +- docs/reference/mkinfit.html | 176 +++--- docs/reference/mkinmod.html | 2 +- docs/reference/mkinparplot-1.png | Bin 17306 -> 16549 bytes docs/reference/mkinparplot.html | 2 +- docs/reference/mkinpredict.html | 4 +- docs/reference/mmkin.html | 4 +- docs/reference/nafta.html | 32 +- docs/reference/summary.mkinfit.html | 30 +- docs/reference/test_data_from_UBA_2014.html | 54 +- docs/reference/transform_odeparms.html | 90 +-- test.log | 23 +- tests/testthat/DFOP_FOCUS_C_messages.txt | 245 -------- tests/testthat/FOCUS_2006_D.csf | 2 +- vignettes/mkin_benchmarks.rda | Bin 795 -> 798 bytes 27 files changed, 963 insertions(+), 1164 deletions(-) diff --git a/R/mkinfit.R b/R/mkinfit.R index 664419be..5e7b5bd3 100644 --- a/R/mkinfit.R +++ b/R/mkinfit.R @@ -343,14 +343,15 @@ mkinfit <- function(mkinmod, observed, data_log_lik <- merge(observed[c("name", "time", "value", "std")], out_long, by = c("name", "time"), suffixes = c(".observed", ".predicted")) - # We only update likelihood and data during the optimisation, not during hessian calculations + if (OLS) { + nlogLik <- with(data_log_lik, sum((value.observed - value.predicted)^2)) + } else { + nlogLik <- - with(data_log_lik, + sum(dnorm(x = value.observed, mean = value.predicted, sd = std, log = TRUE))) + } + + # We update the current likelihood and data during the optimisation, not during hessian calculations if (update_data) { - if (OLS) { - nlogLik <- with(data_log_lik, sum((value.observed - value.predicted)^2)) - } else { - nlogLik <- - with(data_log_lik, - sum(dnorm(x = value.observed, mean = value.predicted, sd = std, log = TRUE))) - } assign("out_predicted", out_long, inherits = TRUE) assign("data_errmod", data_log_lik, inherits = TRUE) diff --git a/docs/articles/FOCUS_D.html b/docs/articles/FOCUS_D.html index 482e48f2..644a35b5 100644 --- a/docs/articles/FOCUS_D.html +++ b/docs/articles/FOCUS_D.html @@ -163,17 +163,13 @@

Confidence intervals for the parameter estimates are obtained using the mkinparplot function.

mkinparplot(fit)
-
## Warning in summary.mkinfit(object): Could not calculate correlation; no
-## covariance matrix

A comprehensive report of the results is obtained using the summary method for mkinfit objects.

-
summary(fit)
-
## Warning in summary.mkinfit(fit): Could not calculate correlation; no
-## covariance matrix
+
summary(fit)
## mkin version used for fitting:    0.9.49.4 
 ## R version used for fitting:       3.6.0 
-## Date of fit:     Tue May  7 08:09:11 2019 
-## Date of summary: Tue May  7 08:09:11 2019 
+## Date of fit:     Tue May  7 08:37:46 2019 
+## Date of summary: Tue May  7 08:37:46 2019 
 ## 
 ## Equations:
 ## d_parent/dt = - k_parent_sink * parent - k_parent_m1 * parent
@@ -181,7 +177,7 @@
 ## 
 ## Model predictions using solution type deSolve 
 ## 
-## Fitted using 153 model solutions performed in 0.398 s
+## Fitted using 389 model solutions performed in 1.018 s
 ## 
 ## Error model:
 ## Constant variance 
@@ -207,25 +203,37 @@
 ## m1_0     0 state
 ## 
 ## Optimised, transformed parameters with symmetric confidence intervals:
-##                   Estimate Std. Error Lower Upper
-## parent_0            99.600         NA    NA    NA
-## log_k_parent_sink   -3.038         NA    NA    NA
-## log_k_parent_m1     -2.980         NA    NA    NA
-## log_k_m1_sink       -5.248         NA    NA    NA
-## sigma                3.126         NA    NA    NA
+##                   Estimate Std. Error  Lower   Upper
+## parent_0            99.600    1.57000 96.400 102.800
+## log_k_parent_sink   -3.038    0.07626 -3.193  -2.883
+## log_k_parent_m1     -2.980    0.04033 -3.062  -2.898
+## log_k_m1_sink       -5.248    0.13320 -5.518  -4.977
+## sigma                3.126    0.35850  2.396   3.855
 ## 
 ## Parameter correlation:
-## No covariance matrix
+##                     parent_0 log_k_parent_sink log_k_parent_m1
+## parent_0           1.000e+00         6.067e-01      -6.372e-02
+## log_k_parent_sink  6.067e-01         1.000e+00      -8.550e-02
+## log_k_parent_m1   -6.372e-02        -8.550e-02       1.000e+00
+## log_k_m1_sink     -1.688e-01        -6.252e-01       4.731e-01
+## sigma              1.164e-09        -8.908e-10       1.652e-08
+##                   log_k_m1_sink      sigma
+## parent_0             -1.688e-01  1.164e-09
+## log_k_parent_sink    -6.252e-01 -8.908e-10
+## log_k_parent_m1       4.731e-01  1.652e-08
+## log_k_m1_sink         1.000e+00 -1.340e-10
+## sigma                -1.340e-10  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      99.600000      NA     NA    NA    NA
-## k_parent_sink  0.047920      NA     NA    NA    NA
-## k_parent_m1    0.050780      NA     NA    NA    NA
-## k_m1_sink      0.005261      NA     NA    NA    NA
-## sigma          3.126000      NA     NA    NA    NA
+##                Estimate t value    Pr(>t)     Lower     Upper
+## parent_0      99.600000  63.430 2.298e-36 96.400000 1.028e+02
+## k_parent_sink  0.047920  13.110 6.126e-15  0.041030 5.596e-02
+## k_parent_m1    0.050780  24.800 3.269e-23  0.046780 5.512e-02
+## k_m1_sink      0.005261   7.510 6.165e-09  0.004012 6.898e-03
+## sigma          3.126000   8.718 2.235e-10  2.396000 3.855e+00
 ## 
 ## FOCUS Chi2 error levels in percent:
 ##          err.min n.optim df
diff --git a/docs/articles/FOCUS_D_files/figure-html/plot_2-1.png b/docs/articles/FOCUS_D_files/figure-html/plot_2-1.png
index 577869e8..64da1d2e 100644
Binary files a/docs/articles/FOCUS_D_files/figure-html/plot_2-1.png and b/docs/articles/FOCUS_D_files/figure-html/plot_2-1.png differ
diff --git a/docs/articles/FOCUS_L.html b/docs/articles/FOCUS_L.html
index 23fa68c6..2cffb323 100644
--- a/docs/articles/FOCUS_L.html
+++ b/docs/articles/FOCUS_L.html
@@ -112,19 +112,17 @@
 
Since mkin version 0.9-32 (July 2014), we can use shorthand notation like "SFO" 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.

 
m.L1.SFO <- mkinfit("SFO", FOCUS_2006_L1_mkin, quiet = TRUE)
 summary(m.L1.SFO)

-
## Warning in summary.mkinfit(m.L1.SFO): Could not calculate correlation; no
-## covariance matrix

 
## mkin version used for fitting:    0.9.49.4 
 ## R version used for fitting:       3.6.0 
-## Date of fit:     Tue May  7 08:09:13 2019 
-## Date of summary: Tue May  7 08:09:13 2019 
+## Date of fit:     Tue May  7 08:37:48 2019 
+## Date of summary: Tue May  7 08:37:48 2019 
 ## 
 ## Equations:
 ## d_parent/dt = - k_parent_sink * parent
 ## 
 ## Model predictions using solution type analytical 
 ## 
-## Fitted using 41 model solutions performed in 0.088 s
+## Fitted using 133 model solutions performed in 0.284 s
 ## 
 ## Error model:
 ## Constant variance 
@@ -145,21 +143,25 @@
 ## None
 ## 
 ## Optimised, transformed parameters with symmetric confidence intervals:
-##                   Estimate Std. Error Lower Upper
-## parent_0            92.470         NA    NA    NA
-## log_k_parent_sink   -2.347         NA    NA    NA
-## sigma                2.780         NA    NA    NA
+##                   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:
-## No covariance matrix
+##                     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      NA     NA    NA    NA
-## k_parent_sink  0.09561      NA     NA    NA    NA
-## sigma          2.78000      NA     NA    NA    NA
+##               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
@@ -195,24 +197,26 @@
 ##    30   parent      2.9     5.251  -2.3513
 ##    30   parent      4.0     5.251  -1.2513

 
A plot of the fit is obtained with the plot function for mkinfit objects.

-
plot(m.L1.SFO, show_errmin = TRUE, main = "FOCUS L1 - SFO")

+
plot(m.L1.SFO, show_errmin = TRUE, main = "FOCUS L1 - SFO")

 

 
The residual plot can be easily obtained by

-
mkinresplot(m.L1.SFO, ylab = "Observed", xlab = "Time")

+
mkinresplot(m.L1.SFO, ylab = "Observed", xlab = "Time")

 

 
For comparison, the FOMC model is fitted as well, and the \(\chi^2\) error level is checked.

-
m.L1.FOMC <- mkinfit("FOMC", FOCUS_2006_L1_mkin, quiet=TRUE)

+
m.L1.FOMC <- mkinfit("FOMC", FOCUS_2006_L1_mkin, quiet=TRUE)

 
## Warning in mkinfit("FOMC", FOCUS_2006_L1_mkin, quiet = TRUE): Optimisation did not converge:
 ## false convergence (8)

-
plot(m.L1.FOMC, show_errmin = TRUE, main = "FOCUS L1 - FOMC")

+
plot(m.L1.FOMC, show_errmin = TRUE, main = "FOCUS L1 - FOMC")

 

-
summary(m.L1.FOMC, data = FALSE)

-
## Warning in summary.mkinfit(m.L1.FOMC, data = FALSE): Could not calculate
-## correlation; no covariance matrix

+
summary(m.L1.FOMC, data = FALSE)

+
## Warning in sqrt(diag(covar)): NaNs wurden erzeugt

+
## Warning in sqrt(1/diag(V)): NaNs wurden erzeugt

+
## Warning in cov2cor(ans$cov.unscaled): diag(.) had 0 or NA entries; non-
+## finite result is doubtful

 
## mkin version used for fitting:    0.9.49.4 
 ## R version used for fitting:       3.6.0 
-## Date of fit:     Tue May  7 08:09:14 2019 
-## Date of summary: Tue May  7 08:09:14 2019 
+## Date of fit:     Tue May  7 08:37:50 2019 
+## Date of summary: Tue May  7 08:37:50 2019 
 ## 
 ## 
 ## Warning: Optimisation did not converge:
@@ -224,7 +228,7 @@
 ## 
 ## Model predictions using solution type analytical 
 ## 
-## Fitted using 743 model solutions performed in 1.558 s
+## Fitted using 899 model solutions performed in 1.913 s
 ## 
 ## Error model:
 ## Constant variance 
@@ -247,23 +251,28 @@
 ## None
 ## 
 ## Optimised, transformed parameters with symmetric confidence intervals:
-##           Estimate Std. Error Lower Upper
-## parent_0     92.47         NA    NA    NA
-## log_alpha    10.58         NA    NA    NA
-## log_beta     12.93         NA    NA    NA
-## sigma         2.78         NA    NA    NA
+##           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:
-## No covariance matrix
+##           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      NA     NA    NA    NA
-## alpha     39440.00      NA     NA    NA    NA
-## beta     412500.00      NA     NA    NA    NA
-## sigma         2.78      NA     NA    NA    NA
+##           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
@@ -281,19 +290,19 @@
 

 Laboratory Data L2

 
The following code defines example dataset L2 from the FOCUS kinetics report, p. 287:

-
+
 

 

 SFO fit for L2

 
Again, the SFO model is fitted and the result is plotted. The residual plot can be obtained simply by adding the argument show_residuals to the plot command.

-
m.L2.SFO <- mkinfit("SFO", FOCUS_2006_L2_mkin, quiet=TRUE)
-plot(m.L2.SFO, show_residuals = TRUE, show_errmin = TRUE,
-     main = "FOCUS L2 - SFO")

+
m.L2.SFO <- mkinfit("SFO", FOCUS_2006_L2_mkin, quiet=TRUE)
+plot(m.L2.SFO, show_residuals = TRUE, show_errmin = TRUE,
+     main = "FOCUS L2 - SFO")

 

 
The \(\chi^2\) 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.

 
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.

@@ -303,24 +312,22 @@
 

 FOMC fit for L2

 
For comparison, the FOMC model is fitted as well, and the \(\chi^2\) error level is checked.

-
m.L2.FOMC <- mkinfit("FOMC", FOCUS_2006_L2_mkin, quiet = TRUE)
-plot(m.L2.FOMC, show_residuals = TRUE,
-     main = "FOCUS L2 - FOMC")

+
m.L2.FOMC <- mkinfit("FOMC", FOCUS_2006_L2_mkin, quiet = TRUE)
+plot(m.L2.FOMC, show_residuals = TRUE,
+     main = "FOCUS L2 - FOMC")

 

-
summary(m.L2.FOMC, data = FALSE)

-
## Warning in summary.mkinfit(m.L2.FOMC, data = FALSE): Could not calculate
-## correlation; no covariance matrix

+
summary(m.L2.FOMC, data = FALSE)

 
## mkin version used for fitting:    0.9.49.4 
 ## R version used for fitting:       3.6.0 
-## Date of fit:     Tue May  7 08:09:15 2019 
-## Date of summary: Tue May  7 08:09:15 2019 
+## Date of fit:     Tue May  7 08:37:51 2019 
+## Date of summary: Tue May  7 08:37:51 2019 
 ## 
 ## Equations:
 ## d_parent/dt = - (alpha/beta) * 1/((time/beta) + 1) * parent
 ## 
 ## Model predictions using solution type analytical 
 ## 
-## Fitted using 83 model solutions performed in 0.169 s
+## Fitted using 239 model solutions performed in 0.494 s
 ## 
 ## Error model:
 ## Constant variance 
@@ -343,23 +350,28 @@
 ## None
 ## 
 ## Optimised, transformed parameters with symmetric confidence intervals:
-##           Estimate Std. Error Lower Upper
-## parent_0   93.7700         NA    NA    NA
-## log_alpha   0.3180         NA    NA    NA
-## log_beta    0.2102         NA    NA    NA
-## sigma       2.2760         NA    NA    NA
+##           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:
-## No covariance matrix
+##             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      NA     NA    NA    NA
-## alpha       1.374      NA     NA    NA    NA
-## beta        1.234      NA     NA    NA    NA
-## sigma       2.276      NA     NA    NA    NA
+##          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
@@ -380,12 +392,10 @@
      main = "FOCUS L2 - DFOP")

 

 
summary(m.L2.DFOP, data = FALSE)

-
## Warning in summary.mkinfit(m.L2.DFOP, data = FALSE): Could not calculate
-## correlation; no covariance matrix

 
## mkin version used for fitting:    0.9.49.4 
 ## R version used for fitting:       3.6.0 
-## Date of fit:     Tue May  7 08:09:16 2019 
-## Date of summary: Tue May  7 08:09:16 2019 
+## Date of fit:     Tue May  7 08:37:53 2019 
+## Date of summary: Tue May  7 08:37:53 2019 
 ## 
 ## Equations:
 ## d_parent/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) *
@@ -394,7 +404,7 @@
 ## 
 ## Model predictions using solution type analytical 
 ## 
-## Fitted using 336 model solutions performed in 0.708 s
+## Fitted using 572 model solutions performed in 1.244 s
 ## 
 ## Error model:
 ## Constant variance 
@@ -419,25 +429,31 @@
 ## None
 ## 
 ## Optimised, transformed parameters with symmetric confidence intervals:
-##          Estimate Std. Error Lower Upper
-## parent_0  93.9500         NA    NA    NA
-## log_k1     3.1370         NA    NA    NA
-## log_k2    -1.0880         NA    NA    NA
-## g_ilr     -0.2821         NA    NA    NA
-## sigma      1.4140         NA    NA    NA
+##          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:
-## No covariance matrix
+##            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      NA     NA    NA    NA
-## k1        23.0400      NA     NA    NA    NA
-## k2         0.3369      NA     NA    NA    NA
-## g          0.4016      NA     NA    NA    NA
-## sigma      1.4140      NA     NA    NA    NA
+##          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
@@ -454,18 +470,18 @@
 

 Laboratory Data L3

 
The following code defines example dataset L3 from the FOCUS kinetics report, p. 290.

-
FOCUS_2006_L3 = data.frame(
-  t = c(0, 3, 7, 14, 30, 60, 91, 120),
-  parent = c(97.8, 60, 51, 43, 35, 22, 15, 12))
-FOCUS_2006_L3_mkin <- mkin_wide_to_long(FOCUS_2006_L3)

+
FOCUS_2006_L3 = data.frame(
+  t = c(0, 3, 7, 14, 30, 60, 91, 120),
+  parent = c(97.8, 60, 51, 43, 35, 22, 15, 12))
+FOCUS_2006_L3_mkin <- mkin_wide_to_long(FOCUS_2006_L3)

 

 

 Fit multiple models

 
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 mmkin. The datasets have to be passed in a list, in this case a named list holding only the L3 dataset prepared above.

-
# Only use one core here, not to offend the CRAN checks
-mm.L3 <- mmkin(c("SFO", "FOMC", "DFOP"), cores = 1,
-               list("FOCUS L3" = FOCUS_2006_L3_mkin), quiet = TRUE)
-plot(mm.L3)

+
# Only use one core here, not to offend the CRAN checks
+mm.L3 <- mmkin(c("SFO", "FOMC", "DFOP"), cores = 1,
+               list("FOCUS L3" = FOCUS_2006_L3_mkin), quiet = TRUE)
+plot(mm.L3)

 

 
The \(\chi^2\) 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 \(\chi^2\) test passes of 7%. Fitting the four parameter DFOP model further reduces the \(\chi^2\) error level considerably.

 

@@ -474,13 +490,11 @@
 Accessing mmkin objects
 
The objects returned by mmkin are arranged like a matrix, with models as a row index and datasets as a column index.

 
We can extract the summary and plot for e.g. the DFOP fit, using square brackets for indexing which will result in the use of the summary and plot functions working on mkinfit objects.

-
summary(mm.L3[["DFOP", 1]])

-
## Warning in summary.mkinfit(mm.L3[["DFOP", 1]]): Could not calculate
-## correlation; no covariance matrix

+
summary(mm.L3[["DFOP", 1]])

 
## mkin version used for fitting:    0.9.49.4 
 ## R version used for fitting:       3.6.0 
-## Date of fit:     Tue May  7 08:09:17 2019 
-## Date of summary: Tue May  7 08:09:17 2019 
+## Date of fit:     Tue May  7 08:37:55 2019 
+## Date of summary: Tue May  7 08:37:55 2019 
 ## 
 ## Equations:
 ## d_parent/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) *
@@ -489,7 +503,7 @@
 ## 
 ## Model predictions using solution type analytical 
 ## 
-## Fitted using 137 model solutions performed in 0.287 s
+## Fitted using 373 model solutions performed in 0.791 s
 ## 
 ## Error model:
 ## Constant variance 
@@ -514,25 +528,31 @@
 ## None
 ## 
 ## Optimised, transformed parameters with symmetric confidence intervals:
-##          Estimate Std. Error Lower Upper
-## parent_0  97.7500         NA    NA    NA
-## log_k1    -0.6612         NA    NA    NA
-## log_k2    -4.2860         NA    NA    NA
-## g_ilr     -0.1229         NA    NA    NA
-## sigma      1.0170         NA    NA    NA
+##          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:
-## No covariance matrix
+##            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      NA     NA    NA    NA
-## k1        0.51620      NA     NA    NA    NA
-## k2        0.01376      NA     NA    NA    NA
-## g         0.45660      NA     NA    NA    NA
-## sigma     1.01700      NA     NA    NA    NA
+##          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
@@ -553,7 +573,7 @@
 ##    60   parent     22.0     23.26 -1.25919
 ##    91   parent     15.0     15.18 -0.18181
 ##   120   parent     12.0     10.19  1.81395

-
plot(mm.L3[["DFOP", 1]], show_errmin = TRUE)

+
plot(mm.L3[["DFOP", 1]], show_errmin = TRUE)

 

 
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 \(\chi^2\) error level criterion for laboratory data L3.

 
This is also an example where the standard t-test for the parameter g_ilr 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 g is quite narrow.

@@ -563,32 +583,30 @@
 

 Laboratory Data L4

 
The following code defines example dataset L4 from the FOCUS kinetics report, p. 293:

-
FOCUS_2006_L4 = data.frame(
-  t = c(0, 3, 7, 14, 30, 60, 91, 120),
-  parent = c(96.6, 96.3, 94.3, 88.8, 74.9, 59.9, 53.5, 49.0))
-FOCUS_2006_L4_mkin <- mkin_wide_to_long(FOCUS_2006_L4)

+
FOCUS_2006_L4 = data.frame(
+  t = c(0, 3, 7, 14, 30, 60, 91, 120),
+  parent = c(96.6, 96.3, 94.3, 88.8, 74.9, 59.9, 53.5, 49.0))
+FOCUS_2006_L4_mkin <- mkin_wide_to_long(FOCUS_2006_L4)

 
Fits of the SFO and FOMC models, plots and summaries are produced below:

-
# Only use one core here, not to offend the CRAN checks
-mm.L4 <- mmkin(c("SFO", "FOMC"), cores = 1,
-               list("FOCUS L4" = FOCUS_2006_L4_mkin),
-               quiet = TRUE)
-plot(mm.L4)

+
# Only use one core here, not to offend the CRAN checks
+mm.L4 <- mmkin(c("SFO", "FOMC"), cores = 1,
+               list("FOCUS L4" = FOCUS_2006_L4_mkin),
+               quiet = TRUE)
+plot(mm.L4)

 

 
The \(\chi^2\) error level of 3.3% as well as the plot suggest that the SFO model fits very well. The error level at which the \(\chi^2\) test passes is slightly lower for the FOMC model. However, the difference appears negligible.

-
summary(mm.L4[["SFO", 1]], data = FALSE)

-
## Warning in summary.mkinfit(mm.L4[["SFO", 1]], data = FALSE): Could not
-## calculate correlation; no covariance matrix

+
summary(mm.L4[["SFO", 1]], data = FALSE)

 
## mkin version used for fitting:    0.9.49.4 
 ## R version used for fitting:       3.6.0 
-## Date of fit:     Tue May  7 08:09:17 2019 
-## Date of summary: Tue May  7 08:09:18 2019 
+## Date of fit:     Tue May  7 08:37:56 2019 
+## Date of summary: Tue May  7 08:37:56 2019 
 ## 
 ## Equations:
 ## d_parent/dt = - k_parent_sink * parent
 ## 
 ## Model predictions using solution type analytical 
 ## 
-## Fitted using 50 model solutions performed in 0.105 s
+## Fitted using 142 model solutions performed in 0.29 s
 ## 
 ## Error model:
 ## Constant variance 
@@ -609,21 +627,25 @@
 ## None
 ## 
 ## Optimised, transformed parameters with symmetric confidence intervals:
-##                   Estimate Std. Error Lower Upper
-## parent_0            96.440         NA    NA    NA
-## log_k_parent_sink   -5.030         NA    NA    NA
-## sigma                3.162         NA    NA    NA
+##                   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:
-## No covariance matrix
+##                    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      NA     NA    NA    NA
-## k_parent_sink  0.006541      NA     NA    NA    NA
-## sigma          3.162000      NA     NA    NA    NA
+##                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
@@ -637,20 +659,18 @@
 ## Estimated disappearance times:
 ##        DT50 DT90
 ## parent  106  352

-
summary(mm.L4[["FOMC", 1]], data = FALSE)

-
## Warning in summary.mkinfit(mm.L4[["FOMC", 1]], data = FALSE): Could not
-## calculate correlation; no covariance matrix

+
summary(mm.L4[["FOMC", 1]], data = FALSE)

 
## mkin version used for fitting:    0.9.49.4 
 ## R version used for fitting:       3.6.0 
-## Date of fit:     Tue May  7 08:09:18 2019 
-## Date of summary: Tue May  7 08:09:18 2019 
+## Date of fit:     Tue May  7 08:37:56 2019 
+## Date of summary: Tue May  7 08:37:56 2019 
 ## 
 ## Equations:
 ## d_parent/dt = - (alpha/beta) * 1/((time/beta) + 1) * parent
 ## 
 ## Model predictions using solution type analytical 
 ## 
-## Fitted using 68 model solutions performed in 0.139 s
+## Fitted using 224 model solutions performed in 0.453 s
 ## 
 ## Error model:
 ## Constant variance 
@@ -673,23 +693,28 @@
 ## None
 ## 
 ## Optimised, transformed parameters with symmetric confidence intervals:
-##           Estimate Std. Error Lower Upper
-## parent_0   99.1400         NA    NA    NA
-## log_alpha  -0.3506         NA    NA    NA
-## log_beta    4.1740         NA    NA    NA
-## sigma       1.8300         NA    NA    NA
+##           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:
-## No covariance matrix
+##             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      NA     NA    NA    NA
-## alpha      0.7042      NA     NA    NA    NA
-## beta      64.9800      NA     NA    NA    NA
-## sigma      1.8300      NA     NA    NA    NA
+##          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
diff --git a/docs/articles/web_only/FOCUS_Z.html b/docs/articles/web_only/FOCUS_Z.html
index c4c69b8f..3cda0261 100644
--- a/docs/articles/web_only/FOCUS_Z.html
+++ b/docs/articles/web_only/FOCUS_Z.html
@@ -131,94 +131,86 @@
 
plot_sep(m.Z.2a)

 

 
summary(m.Z.2a, data = FALSE)$bpar

-
## Warning in summary.mkinfit(m.Z.2a, data = FALSE): Could not calculate
-## correlation; no covariance matrix

-
##             Estimate se_notrans t value Pr(>t) Lower Upper
-## Z0_0      9.7015e+01         NA      NA     NA    NA    NA
-## k_Z0_sink 7.2231e-10         NA      NA     NA    NA    NA
-## k_Z0_Z1   2.2360e+00         NA      NA     NA    NA    NA
-## k_Z1_sink 4.8212e-01         NA      NA     NA    NA    NA
-## sigma     4.8041e+00         NA      NA     NA    NA    NA

+
##             Estimate se_notrans    t value     Pr(>t)    Lower    Upper
+## Z0_0      9.7015e+01   3.393176 2.8591e+01 6.4352e-21 91.66556 102.3642
+## k_Z0_sink 7.2231e-10   0.225254 3.2067e-09 5.0000e-01  0.00000      Inf
+## k_Z0_Z1   2.2360e+00   0.159134 1.4051e+01 1.1369e-13  1.95303   2.5600
+## k_Z1_sink 4.8212e-01   0.065454 7.3658e+00 5.1186e-08  0.40341   0.5762
+## sigma     4.8041e+00   0.637618 7.5345e+00 3.4431e-08  3.52677   6.0815

 
As obvious from the parameter summary (the  component of the summary), the kinetic rate constant from parent compound Z to sink is very small and the t-test for this parameter suggests that it is not significantly different from zero. This suggests, in agreement with the analysis in the FOCUS kinetics report, to simplify the model by removing the pathway to sink.

 
A similar result can be obtained when formation fractions are used in the model formulation:

-
+
 
## Successfully compiled differential equation model from auto-generated C code.

-
m.Z.2a.ff <- mkinfit(Z.2a.ff, FOCUS_2006_Z_mkin, quiet = TRUE)

+
m.Z.2a.ff <- mkinfit(Z.2a.ff, FOCUS_2006_Z_mkin, quiet = TRUE)

 
## Warning in mkinfit(Z.2a.ff, FOCUS_2006_Z_mkin, quiet = TRUE): Observations
 ## with value of zero were removed from the data

-
plot_sep(m.Z.2a.ff)

+
plot_sep(m.Z.2a.ff)

 

-
summary(m.Z.2a.ff, data = FALSE)$bpar

-
## Warning in summary.mkinfit(m.Z.2a.ff, data = FALSE): Could not calculate
-## correlation; no covariance matrix

-
##            Estimate se_notrans t value Pr(>t) Lower Upper
-## Z0_0       97.01488         NA      NA     NA    NA    NA
-## k_Z0        2.23601         NA      NA     NA    NA    NA
-## k_Z1        0.48212         NA      NA     NA    NA    NA
-## f_Z0_to_Z1  1.00000         NA      NA     NA    NA    NA
-## sigma       4.80411         NA      NA     NA    NA    NA

+
summary(m.Z.2a.ff, data = FALSE)$bpar

+
##            Estimate se_notrans t value     Pr(>t)    Lower    Upper
+## Z0_0       97.01488   3.301084 29.3888 3.2971e-21 91.66556 102.3642
+## k_Z0        2.23601   0.207078 10.7979 3.3309e-11  1.95303   2.5600
+## k_Z1        0.48212   0.063265  7.6207 2.8155e-08  0.40341   0.5762
+## f_Z0_to_Z1  1.00000   0.094764 10.5525 5.3560e-11  0.00000   1.0000
+## sigma       4.80411   0.635638  7.5579 3.2592e-08  3.52677   6.0815

 
Here, the ilr transformed formation fraction fitted in the model takes a very large value, and the backtransformed formation fraction from parent Z to Z1 is practically unity. Here, the covariance matrix used for the calculation of confidence intervals is not returned as the model is overparameterised.

 
A simplified model is obtained by removing the pathway to the sink. 

 
In the following, we use the parameterisation with formation fractions in order to be able to compare with the results in the FOCUS guidance, and as it makes it easier to use parameters obtained in a previous fit when adding a further metabolite.

-
Z.3 <- mkinmod(Z0 = mkinsub("SFO", "Z1", sink = FALSE),
-               Z1 = mkinsub("SFO"), use_of_ff = "max")

+
Z.3 <- mkinmod(Z0 = mkinsub("SFO", "Z1", sink = FALSE),
+               Z1 = mkinsub("SFO"), use_of_ff = "max")

 
## Successfully compiled differential equation model from auto-generated C code.

-
m.Z.3 <- mkinfit(Z.3, FOCUS_2006_Z_mkin, quiet = TRUE)

+
m.Z.3 <- mkinfit(Z.3, FOCUS_2006_Z_mkin, quiet = TRUE)

 
## Warning in mkinfit(Z.3, FOCUS_2006_Z_mkin, quiet = TRUE): Observations with
 ## value of zero were removed from the data

-
plot_sep(m.Z.3)

+
plot_sep(m.Z.3)

 

-
summary(m.Z.3, data = FALSE)$bpar

-
## Warning in summary.mkinfit(m.Z.3, data = FALSE): Could not calculate
-## correlation; no covariance matrix

-
##       Estimate se_notrans t value Pr(>t) Lower Upper
-## Z0_0  97.01488         NA      NA     NA    NA    NA
-## k_Z0   2.23601         NA      NA     NA    NA    NA
-## k_Z1   0.48212         NA      NA     NA    NA    NA
-## sigma  4.80411         NA      NA     NA    NA    NA

+
summary(m.Z.3, data = FALSE)$bpar

+
##       Estimate se_notrans t value     Pr(>t)    Lower    Upper
+## Z0_0  97.01488   2.597342  37.352 2.0106e-24 91.67597 102.3538
+## k_Z0   2.23601   0.146904  15.221 9.1477e-15  1.95354   2.5593
+## k_Z1   0.48212   0.041727  11.554 4.8268e-12  0.40355   0.5760
+## sigma  4.80411   0.620208   7.746 1.6110e-08  3.52925   6.0790

 
As there is only one transformation product for Z0 and no pathway to sink, the formation fraction is internally fixed to unity.

 
 

 

 Metabolites Z2 and Z3

 
As suggested in the FOCUS report, the pathway to sink was removed for metabolite Z1 as well in the next step. While this step appears questionable on the basis of the above results, it is followed here for the purpose of comparison. Also, in the FOCUS report, it is assumed that there is additional empirical evidence that Z1 quickly and exclusively hydrolyses to Z2.

-
Z.5 <- mkinmod(Z0 = mkinsub("SFO", "Z1", sink = FALSE),
-               Z1 = mkinsub("SFO", "Z2", sink = FALSE),
-               Z2 = mkinsub("SFO"), use_of_ff = "max")

+
Z.5 <- mkinmod(Z0 = mkinsub("SFO", "Z1", sink = FALSE),
+               Z1 = mkinsub("SFO", "Z2", sink = FALSE),
+               Z2 = mkinsub("SFO"), use_of_ff = "max")

 
## Successfully compiled differential equation model from auto-generated C code.

-
m.Z.5 <- mkinfit(Z.5, FOCUS_2006_Z_mkin, quiet = TRUE)

+
m.Z.5 <- mkinfit(Z.5, FOCUS_2006_Z_mkin, quiet = TRUE)

 
## Warning in mkinfit(Z.5, FOCUS_2006_Z_mkin, quiet = TRUE): Observations with
 ## value of zero were removed from the data

-
plot_sep(m.Z.5)

+
plot_sep(m.Z.5)

 

 
Finally, metabolite Z3 is added to the model. We use the optimised differential equation parameter values from the previous fit in order to accelerate the optimization.

-
Z.FOCUS <- mkinmod(Z0 = mkinsub("SFO", "Z1", sink = FALSE),
-                   Z1 = mkinsub("SFO", "Z2", sink = FALSE),
-                   Z2 = mkinsub("SFO", "Z3"),
-                   Z3 = mkinsub("SFO"),
-                   use_of_ff = "max")

+
Z.FOCUS <- mkinmod(Z0 = mkinsub("SFO", "Z1", sink = FALSE),
+                   Z1 = mkinsub("SFO", "Z2", sink = FALSE),
+                   Z2 = mkinsub("SFO", "Z3"),
+                   Z3 = mkinsub("SFO"),
+                   use_of_ff = "max")

 
## Successfully compiled differential equation model from auto-generated C code.

-
+
 
## Warning in mkinfit(Z.FOCUS, FOCUS_2006_Z_mkin, parms.ini = m.Z.
 ## 5$bparms.ode, : Observations with value of zero were removed from the data

-
plot_sep(m.Z.FOCUS)

+
plot_sep(m.Z.FOCUS)

 

-
summary(m.Z.FOCUS, data = FALSE)$bpar

-
## Warning in summary.mkinfit(m.Z.FOCUS, data = FALSE): Could not calculate
-## correlation; no covariance matrix

-
##             Estimate se_notrans t value Pr(>t) Lower Upper
-## Z0_0       96.838607         NA      NA     NA    NA    NA
-## k_Z0        2.215405         NA      NA     NA    NA    NA
-## k_Z1        0.478300         NA      NA     NA    NA    NA
-## k_Z2        0.451618         NA      NA     NA    NA    NA
-## k_Z3        0.058693         NA      NA     NA    NA    NA
-## f_Z2_to_Z3  0.471508         NA      NA     NA    NA    NA
-## sigma       3.984431         NA      NA     NA    NA    NA

-
endpoints(m.Z.FOCUS)

+
summary(m.Z.FOCUS, data = FALSE)$bpar

+
##             Estimate se_notrans t value     Pr(>t)     Lower      Upper
+## Z0_0       96.838607   1.994273 48.5584 4.0283e-42 92.826626 100.850589
+## k_Z0        2.215405   0.118459 18.7018 1.0415e-23  1.989465   2.467003
+## k_Z1        0.478300   0.028257 16.9267 6.2408e-22  0.424701   0.538662
+## k_Z2        0.451618   0.042138 10.7177 1.6308e-14  0.374328   0.544867
+## k_Z3        0.058693   0.015246  3.8498 1.7806e-04  0.034805   0.098978
+## f_Z2_to_Z3  0.471508   0.058352  8.0804 9.6648e-11  0.357735   0.588320
+## sigma       3.984431   0.383402 10.3923 4.5575e-14  3.213126   4.755736

+
endpoints(m.Z.FOCUS)

 
## $ff
 ##   Z2_Z3 Z2_sink 
 ## 0.47151 0.52849 
@@ -239,77 +231,102 @@
 Using the SFORB model
 
As the FOCUS report states, there is a certain tailing of the time course of metabolite Z3. Also, the time course of the parent compound is not fitted very well using the SFO model, as residues at a certain low level remain.

 
Therefore, an additional model is offered here, using the single first-order reversible binding (SFORB) model for metabolite Z3. As expected, the \(\chi^2\) error level is lower for metabolite Z3 using this model and the graphical fit for Z3 is improved. However, the covariance matrix is not returned.

-
Z.mkin.1 <- mkinmod(Z0 = mkinsub("SFO", "Z1", sink = FALSE),
-                    Z1 = mkinsub("SFO", "Z2", sink = FALSE),
-                    Z2 = mkinsub("SFO", "Z3"),
-                    Z3 = mkinsub("SFORB"))

+
Z.mkin.1 <- mkinmod(Z0 = mkinsub("SFO", "Z1", sink = FALSE),
+                    Z1 = mkinsub("SFO", "Z2", sink = FALSE),
+                    Z2 = mkinsub("SFO", "Z3"),
+                    Z3 = mkinsub("SFORB"))

 
## Successfully compiled differential equation model from auto-generated C code.

-
m.Z.mkin.1 <- mkinfit(Z.mkin.1, FOCUS_2006_Z_mkin, quiet = TRUE)

+
m.Z.mkin.1 <- mkinfit(Z.mkin.1, FOCUS_2006_Z_mkin, quiet = TRUE)

 
## Warning in mkinfit(Z.mkin.1, FOCUS_2006_Z_mkin, quiet = TRUE): Observations
 ## with value of zero were removed from the data

-
plot_sep(m.Z.mkin.1)

+
plot_sep(m.Z.mkin.1)

 

-
summary(m.Z.mkin.1, data = FALSE)$cov.unscaled

-
## Warning in summary.mkinfit(m.Z.mkin.1, data = FALSE): Could not calculate
-## correlation; no covariance matrix

-
## NULL

+
summary(m.Z.mkin.1, data = FALSE)$cov.unscaled

+
##                            Z0_0 log_k_Z0_Z1 log_k_Z1_Z2 log_k_Z2_sink
+## Z0_0                 3.8375e+00  5.4918e-03  3.0584e-02    1.2969e-01
+## log_k_Z0_Z1          5.4918e-03  2.7613e-03 -1.8820e-04    2.6634e-04
+## log_k_Z1_Z2          3.0584e-02 -1.8820e-04  3.3807e-03    3.2177e-03
+## log_k_Z2_sink        1.2969e-01  2.6634e-04  3.2177e-03    3.4256e-02
+## log_k_Z2_Z3_free    -2.4223e-02 -2.6169e-04 -1.1845e-03   -8.1134e-03
+## log_k_Z3_free_sink  -6.5467e-02 -4.0815e-04 -3.2978e-03   -3.6010e-02
+## log_k_Z3_free_bound -6.0659e-02 -4.4768e-04 -3.0588e-03   -3.9074e-02
+## log_k_Z3_bound_free  5.2844e-01  4.5458e-03  7.9800e-03    4.6274e-02
+## sigma                2.0366e-10 -3.4658e-10  8.9910e-11   -2.5946e-10
+##                     log_k_Z2_Z3_free log_k_Z3_free_sink
+## Z0_0                     -2.4223e-02        -6.5467e-02
+## log_k_Z0_Z1              -2.6169e-04        -4.0815e-04
+## log_k_Z1_Z2              -1.1845e-03        -3.2978e-03
+## log_k_Z2_sink            -8.1134e-03        -3.6010e-02
+## log_k_Z2_Z3_free          1.5500e-02         2.1583e-02
+## log_k_Z3_free_sink        2.1583e-02         7.5705e-02
+## log_k_Z3_free_bound       2.5836e-02         1.1964e-01
+## log_k_Z3_bound_free       5.2534e-02         2.9441e-01
+## sigma                     1.3063e-10         3.4170e-10
+##                     log_k_Z3_free_bound log_k_Z3_bound_free       sigma
+## Z0_0                        -6.0659e-02          5.2844e-01  2.0366e-10
+## log_k_Z0_Z1                 -4.4768e-04          4.5458e-03 -3.4658e-10
+## log_k_Z1_Z2                 -3.0588e-03          7.9800e-03  8.9910e-11
+## log_k_Z2_sink               -3.9074e-02          4.6274e-02 -2.5946e-10
+## log_k_Z2_Z3_free             2.5836e-02          5.2534e-02  1.3063e-10
+## log_k_Z3_free_sink           1.1964e-01          2.9441e-01  3.4170e-10
+## log_k_Z3_free_bound          6.5902e-01          5.4737e+00 -6.7704e-10
+## log_k_Z3_bound_free          5.4737e+00          2.8722e+08  7.2421e-02
+## sigma                       -6.7704e-10          7.2421e-02  1.4170e-01

 
Therefore, a further stepwise model building is performed starting from the stage of parent and two metabolites, starting from the assumption that the model fit for the parent compound can be improved by using the SFORB model.

-
Z.mkin.3 <- mkinmod(Z0 = mkinsub("SFORB", "Z1", sink = FALSE),
-                    Z1 = mkinsub("SFO", "Z2", sink = FALSE),
-                    Z2 = mkinsub("SFO"))

+
Z.mkin.3 <- mkinmod(Z0 = mkinsub("SFORB", "Z1", sink = FALSE),
+                    Z1 = mkinsub("SFO", "Z2", sink = FALSE),
+                    Z2 = mkinsub("SFO"))

 
## Successfully compiled differential equation model from auto-generated C code.

-
m.Z.mkin.3 <- mkinfit(Z.mkin.3, FOCUS_2006_Z_mkin, quiet = TRUE)

+
m.Z.mkin.3 <- mkinfit(Z.mkin.3, FOCUS_2006_Z_mkin, quiet = TRUE)

 
## Warning in mkinfit(Z.mkin.3, FOCUS_2006_Z_mkin, quiet = TRUE): Observations
 ## with value of zero were removed from the data

-
plot_sep(m.Z.mkin.3)

+
plot_sep(m.Z.mkin.3)

 

 
This results in a much better representation of the behaviour of the parent compound Z0.

 
Finally, Z3 is added as well. These models appear overparameterised (no covariance matrix returned) if the sink for Z1 is left in the models.

-
Z.mkin.4 <- mkinmod(Z0 = mkinsub("SFORB", "Z1", sink = FALSE),
-                    Z1 = mkinsub("SFO", "Z2", sink = FALSE),
-                    Z2 = mkinsub("SFO", "Z3"),
-                    Z3 = mkinsub("SFO"))

+
Z.mkin.4 <- mkinmod(Z0 = mkinsub("SFORB", "Z1", sink = FALSE),
+                    Z1 = mkinsub("SFO", "Z2", sink = FALSE),
+                    Z2 = mkinsub("SFO", "Z3"),
+                    Z3 = mkinsub("SFO"))

 
## Successfully compiled differential equation model from auto-generated C code.

-
+
 
## Warning in mkinfit(Z.mkin.4, FOCUS_2006_Z_mkin, parms.ini = m.Z.mkin.
 ## 3$bparms.ode, : Observations with value of zero were removed from the data

-
plot_sep(m.Z.mkin.4)

+
plot_sep(m.Z.mkin.4)

 

 
The error level of the fit, but especially of metabolite Z3, can be improved if the SFORB model is chosen for this metabolite, as this model is capable of representing the tailing of the metabolite decline phase.

-
Z.mkin.5 <- mkinmod(Z0 = mkinsub("SFORB", "Z1", sink = FALSE),
-                    Z1 = mkinsub("SFO", "Z2", sink = FALSE),
-                    Z2 = mkinsub("SFO", "Z3"),
-                    Z3 = mkinsub("SFORB"))

+
Z.mkin.5 <- mkinmod(Z0 = mkinsub("SFORB", "Z1", sink = FALSE),
+                    Z1 = mkinsub("SFO", "Z2", sink = FALSE),
+                    Z2 = mkinsub("SFO", "Z3"),
+                    Z3 = mkinsub("SFORB"))

 
## Successfully compiled differential equation model from auto-generated C code.

-
+
 
## Warning in mkinfit(Z.mkin.5, FOCUS_2006_Z_mkin, parms.ini = m.Z.mkin.
 ## 4$bparms.ode[1:4], : Observations with value of zero were removed from the
 ## data

-
plot_sep(m.Z.mkin.5)

+
plot_sep(m.Z.mkin.5)

 

 
The summary view of the backtransformed parameters shows that we get no confidence intervals due to overparameterisation. As the optimized  is excessively small, it seems reasonable to fix it to zero.

-
+
 
## Warning in mkinfit(Z.mkin.5, FOCUS_2006_Z_mkin, parms.ini = c(m.Z.mkin.
 ## 5$bparms.ode[1:7], : Observations with value of zero were removed from the
 ## data

-
plot_sep(m.Z.mkin.5a)

+
plot_sep(m.Z.mkin.5a)

 

 
As expected, the residual plots for Z0 and Z3 are more random than in the case of the all SFO model for which they were shown above. In conclusion, the model  is proposed as the best-fit model for the dataset from Appendix 7 of the FOCUS report.

 
A graphical representation of the confidence intervals can finally be obtained.

-
mkinparplot(m.Z.mkin.5a)

-
## Warning in summary.mkinfit(object): Could not calculate correlation; no
-## covariance matrix

+
mkinparplot(m.Z.mkin.5a)

 

 
The endpoints obtained with this model are

-
endpoints(m.Z.mkin.5a)

+
endpoints(m.Z.mkin.5a)

 
## $ff
 ##   Z0_free_Z1        Z1_Z2      Z2_sink   Z2_Z3_free Z3_free_sink 
 ##      1.00000      1.00000      0.46344      0.53656      1.00000 
diff --git a/docs/articles/web_only/FOCUS_Z_files/figure-html/FOCUS_2006_Z_fits_11b-1.png b/docs/articles/web_only/FOCUS_Z_files/figure-html/FOCUS_2006_Z_fits_11b-1.png
index 4304f6d2..8188c4cf 100644
Binary files a/docs/articles/web_only/FOCUS_Z_files/figure-html/FOCUS_2006_Z_fits_11b-1.png and b/docs/articles/web_only/FOCUS_Z_files/figure-html/FOCUS_2006_Z_fits_11b-1.png differ
diff --git a/docs/articles/web_only/NAFTA_examples.html b/docs/articles/web_only/NAFTA_examples.html
index f6f7c60a..237b506e 100644
--- a/docs/articles/web_only/NAFTA_examples.html
+++ b/docs/articles/web_only/NAFTA_examples.html
@@ -111,19 +111,11 @@
 

 Example on page 5, upper panel

 
p5a <- nafta(NAFTA_SOP_Attachment[["p5a"]])

-
## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix
-
-## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix
-
-## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix

 
## The SFO model is rejected as S_SFO is equal or higher than the critical value S_c

 
## The half-life obtained from the IORE model may be used

-
plot(p5a)

+
plot(p5a)

 

-
print(p5a)

+
print(p5a)

 
## Sums of squares:
 ##       SFO      IORE      DFOP 
 ## 465.21753  56.27506  32.06401 
@@ -133,25 +125,25 @@
 ## 
 ## Parameters:
 ## $SFO
-##               Estimate Pr(>t) Lower Upper
-## parent_0       95.8401     NA    NA    NA
-## k_parent_sink   0.0102     NA    NA    NA
-## sigma           4.8230     NA    NA    NA
+##               Estimate   Pr(>t)  Lower   Upper
+## parent_0       95.8401 4.67e-21 92.245 99.4357
+## k_parent_sink   0.0102 3.92e-12  0.009  0.0117
+## sigma           4.8230 3.81e-06  3.214  6.4318
 ## 
 ## $IORE
-##                     Estimate Pr(>t) Lower Upper
-## parent_0            1.01e+02     NA    NA    NA
-## k__iore_parent_sink 1.54e-05     NA    NA    NA
-## N_parent            2.57e+00     NA    NA    NA
-## sigma               1.68e+00     NA    NA    NA
+##                     Estimate Pr(>t)    Lower    Upper
+## parent_0            1.01e+02     NA 9.91e+01 1.02e+02
+## k__iore_parent_sink 1.54e-05     NA 4.08e-06 5.84e-05
+## N_parent            2.57e+00     NA 2.25e+00 2.89e+00
+## sigma               1.68e+00     NA 1.12e+00 2.24e+00
 ## 
 ## $DFOP
-##          Estimate Pr(>t) Lower Upper
-## parent_0 9.99e+01     NA    NA    NA
-## k1       2.67e-02     NA    NA    NA
-## k2       2.86e-12     NA    NA    NA
-## g        6.47e-01     NA    NA    NA
-## sigma    1.27e+00     NA    NA    NA
+##          Estimate   Pr(>t)   Lower    Upper
+## parent_0 9.99e+01 1.41e-26 98.8116 101.0810
+## k1       2.67e-02 5.05e-06  0.0243   0.0295
+## k2       2.86e-12 5.00e-01  0.0000      Inf
+## g        6.47e-01 3.67e-06  0.6248   0.6677
+## sigma    1.27e+00 8.91e-06  0.8395   1.6929
 ## 
 ## 
 ## DTx values:
@@ -166,20 +158,12 @@
 

 

 Example on page 5, lower panel

-
p5b <- nafta(NAFTA_SOP_Attachment[["p5b"]])

-
## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix
-
-## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix
-
-## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix

+
p5b <- nafta(NAFTA_SOP_Attachment[["p5b"]])

 
## The SFO model is rejected as S_SFO is equal or higher than the critical value S_c

 
## The half-life obtained from the IORE model may be used

-
plot(p5b)

+
plot(p5b)

 

-
print(p5b)

+
print(p5b)

 
## Sums of squares:
 ##      SFO     IORE     DFOP 
 ## 94.81123 10.10936  7.55871 
@@ -189,25 +173,25 @@
 ## 
 ## Parameters:
 ## $SFO
-##               Estimate Pr(>t) Lower Upper
-## parent_0        96.497     NA    NA    NA
-## k_parent_sink    0.008     NA    NA    NA
-## sigma            2.295     NA    NA    NA
+##               Estimate   Pr(>t)    Lower    Upper
+## parent_0        96.497 2.32e-24 94.85271 98.14155
+## k_parent_sink    0.008 3.42e-14  0.00737  0.00869
+## sigma            2.295 1.22e-05  1.47976  3.11036
 ## 
 ## $IORE
-##                     Estimate Pr(>t) Lower Upper
-## parent_0            9.85e+01     NA    NA    NA
-## k__iore_parent_sink 1.53e-04     NA    NA    NA
-## N_parent            1.94e+00     NA    NA    NA
-## sigma               7.49e-01     NA    NA    NA
+##                     Estimate   Pr(>t)    Lower    Upper
+## parent_0            9.85e+01 1.17e-28 9.79e+01 9.92e+01
+## k__iore_parent_sink 1.53e-04 6.50e-03 7.21e-05 3.26e-04
+## N_parent            1.94e+00 5.88e-13 1.76e+00 2.12e+00
+## sigma               7.49e-01 1.63e-05 4.82e-01 1.02e+00
 ## 
 ## $DFOP
-##          Estimate Pr(>t) Lower Upper
-## parent_0 9.84e+01     NA    NA    NA
-## k1       1.55e-02     NA    NA    NA
-## k2       1.16e-11     NA    NA    NA
-## g        6.89e-01     NA    NA    NA
-## sigma    6.48e-01     NA    NA    NA
+##          Estimate   Pr(>t)   Lower   Upper
+## parent_0 9.84e+01 1.24e-27 97.8078 98.9187
+## k1       1.55e-02 4.10e-04  0.0143  0.0167
+## k2       1.16e-11 5.00e-01  0.0000     Inf
+## g        6.89e-01 2.92e-03  0.6626  0.7142
+## sigma    6.48e-01 2.38e-05  0.4147  0.8813
 ## 
 ## 
 ## DTx values:
@@ -222,20 +206,12 @@
 

 

 Example on page 6

-
p6 <- nafta(NAFTA_SOP_Attachment[["p6"]])

-
## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix
-
-## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix
-
-## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix

+
p6 <- nafta(NAFTA_SOP_Attachment[["p6"]])

 
## The SFO model is rejected as S_SFO is equal or higher than the critical value S_c

 
## The half-life obtained from the IORE model may be used

-
plot(p6)

+
plot(p6)

 

-
print(p6)

+
print(p6)

 
## Sums of squares:
 ##       SFO      IORE      DFOP 
 ## 188.45361  51.00699  42.46931 
@@ -245,25 +221,25 @@
 ## 
 ## Parameters:
 ## $SFO
-##               Estimate Pr(>t) Lower Upper
-## parent_0       94.7759     NA    NA    NA
-## k_parent_sink   0.0179     NA    NA    NA
-## sigma           3.0696     NA    NA    NA
+##               Estimate   Pr(>t)   Lower   Upper
+## parent_0       94.7759 7.29e-24 92.3478 97.2039
+## k_parent_sink   0.0179 8.02e-16  0.0166  0.0194
+## sigma           3.0696 3.81e-06  2.0456  4.0936
 ## 
 ## $IORE
-##                     Estimate Pr(>t) Lower Upper
-## parent_0            97.12446     NA    NA    NA
-## k__iore_parent_sink  0.00252     NA    NA    NA
-## N_parent             1.49587     NA    NA    NA
-## sigma                1.59698     NA    NA    NA
+##                     Estimate   Pr(>t)    Lower    Upper
+## parent_0            97.12446 2.63e-26 95.62461 98.62431
+## k__iore_parent_sink  0.00252 1.95e-03  0.00134  0.00472
+## N_parent             1.49587 4.07e-13  1.33896  1.65279
+## sigma                1.59698 5.05e-06  1.06169  2.13227
 ## 
 ## $DFOP
-##          Estimate Pr(>t) Lower Upper
-## parent_0 9.66e+01     NA    NA    NA
-## k1       2.55e-02     NA    NA    NA
-## k2       4.90e-11     NA    NA    NA
-## g        8.61e-01     NA    NA    NA
-## sigma    1.46e+00     NA    NA    NA
+##          Estimate   Pr(>t)   Lower   Upper
+## parent_0 9.66e+01 1.57e-25 95.3476 97.8979
+## k1       2.55e-02 7.33e-06  0.0233  0.0278
+## k2       4.90e-11 5.00e-01  0.0000     Inf
+## g        8.61e-01 7.55e-06  0.8314  0.8867
+## sigma    1.46e+00 6.93e-06  0.9661  1.9483
 ## 
 ## 
 ## DTx values:
@@ -278,20 +254,12 @@
 

 

 Example on page 7

-
p7 <- nafta(NAFTA_SOP_Attachment[["p7"]])

-
## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix
-
-## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix
-
-## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix

+
p7 <- nafta(NAFTA_SOP_Attachment[["p7"]])

 
## The SFO model is rejected as S_SFO is equal or higher than the critical value S_c

 
## The half-life obtained from the IORE model may be used

-
plot(p7)

+
plot(p7)

 

-
print(p7)

+
print(p7)

 
## Sums of squares:
 ##      SFO     IORE     DFOP 
 ## 3661.661 3195.030 3174.145 
@@ -301,25 +269,25 @@
 ## 
 ## Parameters:
 ## $SFO
-##               Estimate Pr(>t) Lower Upper
-## parent_0      96.41796     NA    NA    NA
-## k_parent_sink  0.00735     NA    NA    NA
-## sigma          7.94557     NA    NA    NA
+##               Estimate   Pr(>t)    Lower    Upper
+## parent_0      96.41796 4.80e-53 93.32245 99.51347
+## k_parent_sink  0.00735 7.64e-21  0.00641  0.00843
+## sigma          7.94557 1.83e-15  6.46713  9.42401
 ## 
 ## $IORE
-##                     Estimate Pr(>t) Lower Upper
-## parent_0            9.92e+01     NA    NA    NA
-## k__iore_parent_sink 1.60e-05     NA    NA    NA
-## N_parent            2.45e+00     NA    NA    NA
-## sigma               7.42e+00     NA    NA    NA
+##                     Estimate Pr(>t)    Lower    Upper
+## parent_0            9.92e+01     NA 9.55e+01 1.03e+02
+## k__iore_parent_sink 1.60e-05     NA 1.45e-07 1.77e-03
+## N_parent            2.45e+00     NA 1.35e+00 3.54e+00
+## sigma               7.42e+00     NA 6.04e+00 8.80e+00
 ## 
 ## $DFOP
-##          Estimate Pr(>t) Lower Upper
-## parent_0 9.89e+01     NA    NA    NA
-## k1       1.81e-02     NA    NA    NA
-## k2       1.97e-10     NA    NA    NA
-## g        6.06e-01     NA    NA    NA
-## sigma    7.40e+00     NA    NA    NA
+##          Estimate   Pr(>t)   Lower    Upper
+## parent_0 9.89e+01 9.44e-49 95.4640 102.2573
+## k1       1.81e-02 1.75e-01  0.0116   0.0281
+## k2       1.97e-10 5.00e-01  0.0000      Inf
+## g        6.06e-01 2.19e-01  0.4826   0.7178
+## sigma    7.40e+00 2.97e-15  6.0201   8.7754
 ## 
 ## 
 ## DTx values:
@@ -339,20 +307,17 @@
 

 Example on page 8

 
For this dataset, the IORE fit does not converge when the default starting values used by mkin for the IORE model are used. Therefore, a lower value for the rate constant is used here.

-
p8 <- nafta(NAFTA_SOP_Attachment[["p8"]], parms.ini = c(k__iore_parent_sink = 1e-3))

+
p8 <- nafta(NAFTA_SOP_Attachment[["p8"]], parms.ini = c(k__iore_parent_sink = 1e-3))

 
## Warning in summary.mkinfit(x): Could not calculate correlation; no
 ## covariance matrix
 
-## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix
-
 ## Warning in summary.mkinfit(x): Could not calculate correlation; no
 ## covariance matrix

 
## The SFO model is rejected as S_SFO is equal or higher than the critical value S_c

 
## The half-life obtained from the IORE model may be used

-
plot(p8)

+
plot(p8)

 

-
print(p8)

+
print(p8)

 
## Sums of squares:
 ##       SFO      IORE      DFOP 
 ## 1996.9408  444.9237  547.5616 
@@ -369,11 +334,11 @@
 ## sigma                7.44786     NA    NA    NA
 ## 
 ## $IORE
-##                     Estimate Pr(>t) Lower Upper
-## parent_0            9.77e+01     NA    NA    NA
-## k__iore_parent_sink 6.14e-05     NA    NA    NA
-## N_parent            2.27e+00     NA    NA    NA
-## sigma               3.52e+00     NA    NA    NA
+##                     Estimate   Pr(>t)    Lower    Upper
+## parent_0            9.77e+01 7.03e-35 9.44e+01 1.01e+02
+## k__iore_parent_sink 6.14e-05 3.20e-02 2.12e-05 1.78e-04
+## N_parent            2.27e+00 4.23e-18 2.00e+00 2.54e+00
+## sigma               3.52e+00 5.36e-10 2.67e+00 4.36e+00
 ## 
 ## $DFOP
 ##                     Estimate Pr(>t) Lower Upper
@@ -401,20 +366,12 @@
 

 

 Example on page 9, upper panel

-
p9a <- nafta(NAFTA_SOP_Attachment[["p9a"]])

-
## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix
-
-## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix
-
-## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix

+
p9a <- nafta(NAFTA_SOP_Attachment[["p9a"]])

 
## The SFO model is rejected as S_SFO is equal or higher than the critical value S_c

 
## The half-life obtained from the IORE model may be used

-
plot(p9a)

+
plot(p9a)

 

-
print(p9a)

+
print(p9a)

 
## Sums of squares:
 ##       SFO      IORE      DFOP 
 ## 839.35238  88.57064   9.93363 
@@ -424,25 +381,25 @@
 ## 
 ## Parameters:
 ## $SFO
-##               Estimate Pr(>t) Lower Upper
-## parent_0       88.1933     NA    NA    NA
-## k_parent_sink   0.0409     NA    NA    NA
-## sigma           7.2429     NA    NA    NA
+##               Estimate   Pr(>t)   Lower   Upper
+## parent_0       88.1933 3.06e-12 79.9447 96.4419
+## k_parent_sink   0.0409 2.07e-07  0.0324  0.0516
+## sigma           7.2429 3.92e-05  4.4768 10.0090
 ## 
 ## $IORE
-##                     Estimate Pr(>t) Lower Upper
-## parent_0            9.89e+01     NA    NA    NA
-## k__iore_parent_sink 1.93e-05     NA    NA    NA
-## N_parent            2.91e+00     NA    NA    NA
-## sigma               2.35e+00     NA    NA    NA
+##                     Estimate   Pr(>t)    Lower    Upper
+## parent_0            9.89e+01 1.12e-16 9.54e+01 1.02e+02
+## k__iore_parent_sink 1.93e-05 1.13e-01 3.49e-06 1.06e-04
+## N_parent            2.91e+00 1.45e-09 2.50e+00 3.32e+00
+## sigma               2.35e+00 5.31e-05 1.45e+00 3.26e+00
 ## 
 ## $DFOP
-##          Estimate Pr(>t) Lower Upper
-## parent_0 9.85e+01     NA    NA    NA
-## k1       1.38e-01     NA    NA    NA
-## k2       6.02e-13     NA    NA    NA
-## g        6.52e-01     NA    NA    NA
-## sigma    7.88e-01     NA    NA    NA
+##          Estimate   Pr(>t)  Lower  Upper
+## parent_0 9.85e+01 2.54e-20 97.390 99.672
+## k1       1.38e-01 3.52e-05  0.131  0.146
+## k2       6.02e-13 5.00e-01  0.000    Inf
+## g        6.52e-01 8.13e-06  0.642  0.661
+## sigma    7.88e-01 6.13e-02  0.481  1.095
 ## 
 ## 
 ## DTx values:
@@ -458,20 +415,17 @@
 

 

 Example on page 9, lower panel

-
p9b <- nafta(NAFTA_SOP_Attachment[["p9b"]])

-
## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix
-
-## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix
-
-## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix

+
p9b <- nafta(NAFTA_SOP_Attachment[["p9b"]])

+
## Warning in sqrt(diag(covar)): NaNs wurden erzeugt

+
## Warning in sqrt(diag(covar_notrans)): NaNs wurden erzeugt

+
## Warning in sqrt(1/diag(V)): NaNs wurden erzeugt

+
## Warning in cov2cor(ans$cov.unscaled): diag(.) had 0 or NA entries; non-
+## finite result is doubtful

 
## The SFO model is rejected as S_SFO is equal or higher than the critical value S_c

 
## The half-life obtained from the IORE model may be used

-
plot(p9b)

+
plot(p9b)

 

-
print(p9b)

+
print(p9b)

 
## Sums of squares:
 ##      SFO     IORE     DFOP 
 ## 35.64867 23.22334 35.64867 
@@ -481,25 +435,25 @@
 ## 
 ## Parameters:
 ## $SFO
-##               Estimate Pr(>t) Lower Upper
-## parent_0       94.7123     NA    NA    NA
-## k_parent_sink   0.0389     NA    NA    NA
-## sigma           1.5957     NA    NA    NA
+##               Estimate   Pr(>t)  Lower   Upper
+## parent_0       94.7123 2.15e-19 93.178 96.2464
+## k_parent_sink   0.0389 4.47e-14  0.037  0.0408
+## sigma           1.5957 1.28e-04  0.932  2.2595
 ## 
 ## $IORE
-##                     Estimate Pr(>t) Lower Upper
-## parent_0              93.863     NA    NA    NA
-## k__iore_parent_sink    0.127     NA    NA    NA
-## N_parent               0.711     NA    NA    NA
-## sigma                  1.288     NA    NA    NA
+##                     Estimate   Pr(>t)   Lower  Upper
+## parent_0              93.863 2.32e-18 92.4565 95.269
+## k__iore_parent_sink    0.127 1.85e-02  0.0504  0.321
+## N_parent               0.711 1.88e-05  0.4843  0.937
+## sigma                  1.288 1.76e-04  0.7456  1.830
 ## 
 ## $DFOP
-##          Estimate Pr(>t) Lower Upper
-## parent_0  94.7123     NA    NA    NA
-## k1         0.0389     NA    NA    NA
-## k2         0.0389     NA    NA    NA
-## g          0.7742     NA    NA    NA
-## sigma      1.5957     NA    NA    NA
+##          Estimate   Pr(>t)   Lower   Upper
+## parent_0  94.7123 1.61e-16 93.1355 96.2891
+## k1         0.0389 1.43e-06  0.0312  0.0485
+## k2         0.0389 6.67e-03  0.0186  0.0812
+## g          0.7742      NaN      NA      NA
+## sigma      1.5957 2.50e-04  0.9135  2.2779
 ## 
 ## 
 ## DTx values:
@@ -515,20 +469,12 @@
 

 

 Example on page 10

-
p10 <- nafta(NAFTA_SOP_Attachment[["p10"]])

-
## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix
-
-## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix
-
-## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix

+
p10 <- nafta(NAFTA_SOP_Attachment[["p10"]])

 
## The SFO model is rejected as S_SFO is equal or higher than the critical value S_c

 
## The half-life obtained from the IORE model may be used

-
plot(p10)

+
plot(p10)

 

-
print(p10)

+
print(p10)

 
## Sums of squares:
 ##      SFO     IORE     DFOP 
 ## 899.4089 336.4348 899.4089 
@@ -538,25 +484,25 @@
 ## 
 ## Parameters:
 ## $SFO
-##               Estimate Pr(>t) Lower Upper
-## parent_0      101.7315     NA    NA    NA
-## k_parent_sink   0.0495     NA    NA    NA
-## sigma           8.0152     NA    NA    NA
+##               Estimate   Pr(>t)   Lower    Upper
+## parent_0      101.7315 6.42e-11 91.9259 111.5371
+## k_parent_sink   0.0495 1.70e-07  0.0404   0.0607
+## sigma           8.0152 1.28e-04  4.6813  11.3491
 ## 
 ## $IORE
-##                     Estimate Pr(>t) Lower Upper
-## parent_0               96.86     NA    NA    NA
-## k__iore_parent_sink     2.96     NA    NA    NA
-## N_parent                0.00     NA    NA    NA
-## sigma                   4.90     NA    NA    NA
+##                     Estimate   Pr(>t)  Lower   Upper
+## parent_0               96.86 3.32e-12 90.848 102.863
+## k__iore_parent_sink     2.96 7.91e-02  0.687  12.761
+## N_parent                0.00 5.00e-01 -0.372   0.372
+## sigma                   4.90 1.77e-04  2.837   6.968
 ## 
 ## $DFOP
-##          Estimate Pr(>t) Lower Upper
-## parent_0 101.7315     NA    NA    NA
-## k1         0.0495     NA    NA    NA
-## k2         0.0495     NA    NA    NA
-## g          0.6634     NA    NA    NA
-## sigma      8.0152     NA    NA    NA
+##          Estimate   Pr(>t)   Lower    Upper
+## parent_0 101.7315 1.41e-09 91.6534 111.8097
+## k1         0.0495 6.42e-04  0.0301   0.0814
+## k2         0.0495 1.66e-02  0.0200   0.1225
+## g          0.6634 5.00e-01  0.0000   1.0000
+## sigma      8.0152 2.50e-04  4.5886  11.4418
 ## 
 ## 
 ## DTx values:
@@ -576,20 +522,12 @@
 

 

 Example on page 11

-
p11 <- nafta(NAFTA_SOP_Attachment[["p11"]])

-
## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix
-
-## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix
-
-## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix

+
p11 <- nafta(NAFTA_SOP_Attachment[["p11"]])

 
## The SFO model is rejected as S_SFO is equal or higher than the critical value S_c

 
## The half-life obtained from the IORE model may be used

-
plot(p11)

+
plot(p11)

 

-
print(p11)

+
print(p11)

 
## Sums of squares:
 ##      SFO     IORE     DFOP 
 ## 579.6805 204.7932 144.7783 
@@ -599,25 +537,25 @@
 ## 
 ## Parameters:
 ## $SFO
-##               Estimate Pr(>t) Lower Upper
-## parent_0      96.15820     NA    NA    NA
-## k_parent_sink  0.00321     NA    NA    NA
-## sigma          6.43473     NA    NA    NA
+##               Estimate   Pr(>t)    Lower    Upper
+## parent_0      96.15820 4.83e-13 90.24934 1.02e+02
+## k_parent_sink  0.00321 4.71e-05  0.00222 4.64e-03
+## sigma          6.43473 1.28e-04  3.75822 9.11e+00
 ## 
 ## $IORE
-##                     Estimate Pr(>t) Lower Upper
-## parent_0            1.05e+02     NA    NA    NA
-## k__iore_parent_sink 3.11e-17     NA    NA    NA
-## N_parent            8.36e+00     NA    NA    NA
-## sigma               3.82e+00     NA    NA    NA
+##                     Estimate Pr(>t)    Lower    Upper
+## parent_0            1.05e+02     NA 9.90e+01 1.10e+02
+## k__iore_parent_sink 3.11e-17     NA 1.35e-20 7.18e-14
+## N_parent            8.36e+00     NA 6.62e+00 1.01e+01
+## sigma               3.82e+00     NA 2.21e+00 5.44e+00
 ## 
 ## $DFOP
-##          Estimate Pr(>t) Lower Upper
-## parent_0 1.05e+02     NA    NA    NA
-## k1       4.41e-02     NA    NA    NA
-## k2       7.25e-13     NA    NA    NA
-## g        3.22e-01     NA    NA    NA
-## sigma    3.22e+00     NA    NA    NA
+##          Estimate   Pr(>t)    Lower    Upper
+## parent_0 1.05e+02 9.47e-13  99.9990 109.1224
+## k1       4.41e-02 5.95e-03   0.0296   0.0658
+## k2       7.25e-13 5.00e-01   0.0000      Inf
+## g        3.22e-01 1.45e-03   0.2814   0.3650
+## sigma    3.22e+00 3.52e-04   1.8410   4.5906
 ## 
 ## 
 ## DTx values:
@@ -638,20 +576,14 @@
 

 

 Example on page 12, upper panel

-
p12a <- nafta(NAFTA_SOP_Attachment[["p12a"]])

+
p12a <- nafta(NAFTA_SOP_Attachment[["p12a"]])

 
## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix
-
-## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix
-
-## Warning in summary.mkinfit(x): Could not calculate correlation; no
 ## covariance matrix

 
## The SFO model is rejected as S_SFO is equal or higher than the critical value S_c

 
## The half-life obtained from the IORE model may be used

-
plot(p12a)

+
plot(p12a)

 

-
print(p12a)

+
print(p12a)

 
## Sums of squares:
 ##      SFO     IORE     DFOP 
 ## 695.4440 220.0685 695.4440 
@@ -661,10 +593,10 @@
 ## 
 ## Parameters:
 ## $SFO
-##               Estimate Pr(>t) Lower Upper
-## parent_0       100.521     NA    NA    NA
-## k_parent_sink    0.124     NA    NA    NA
-## sigma            7.048     NA    NA    NA
+##               Estimate   Pr(>t)  Lower   Upper
+## parent_0       100.521 8.75e-12 92.461 108.581
+## k_parent_sink    0.124 3.61e-08  0.104   0.148
+## sigma            7.048 1.28e-04  4.116   9.980
 ## 
 ## $IORE
 ##                     Estimate Pr(>t) Lower Upper
@@ -674,12 +606,12 @@
 ## sigma                  3.965     NA    NA    NA
 ## 
 ## $DFOP
-##          Estimate Pr(>t) Lower Upper
-## parent_0  100.521     NA    NA    NA
-## k1          0.124     NA    NA    NA
-## k2          0.124     NA    NA    NA
-## g           0.877     NA    NA    NA
-## sigma       7.048     NA    NA    NA
+##          Estimate   Pr(>t)   Lower   Upper
+## parent_0  100.521 2.74e-10 92.2366 108.805
+## k1          0.124 5.74e-06  0.0958   0.161
+## k2          0.124 6.61e-02  0.0319   0.484
+## g           0.877 5.00e-01  0.0000   1.000
+## sigma       7.048 2.50e-04  4.0349  10.061
 ## 
 ## 
 ## DTx values:
@@ -694,20 +626,20 @@
 

 

 Example on page 12, lower panel

-
p12b <- nafta(NAFTA_SOP_Attachment[["p12b"]])

-
## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix
-
-## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix
-
-## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix

+
p12b <- nafta(NAFTA_SOP_Attachment[["p12b"]])

+
## Warning in sqrt(diag(covar)): NaNs wurden erzeugt

+
## Warning in qt(alpha/2, rdf): NaNs wurden erzeugt

+
## Warning in qt(1 - alpha/2, rdf): NaNs wurden erzeugt

+
## Warning in sqrt(diag(covar_notrans)): NaNs wurden erzeugt

+
## Warning in pt(abs(tval), rdf, lower.tail = FALSE): NaNs wurden erzeugt

+
## Warning in sqrt(1/diag(V)): NaNs wurden erzeugt

+
## Warning in cov2cor(ans$cov.unscaled): diag(.) had 0 or NA entries; non-
+## finite result is doubtful

 
## The SFO model is rejected as S_SFO is equal or higher than the critical value S_c

 
## The half-life obtained from the IORE model may be used

-
plot(p12b)

+
plot(p12b)

 

-
print(p12b)

+
print(p12b)

 
## Sums of squares:
 ##      SFO     IORE     DFOP 
 ## 58.90242 19.06353 58.90242 
@@ -717,25 +649,25 @@
 ## 
 ## Parameters:
 ## $SFO
-##               Estimate Pr(>t) Lower Upper
-## parent_0       97.6840     NA    NA    NA
-## k_parent_sink   0.0589     NA    NA    NA
-## sigma           3.4323     NA    NA    NA
+##               Estimate  Pr(>t)   Lower    Upper
+## parent_0       97.6840 0.00039 85.9388 109.4292
+## k_parent_sink   0.0589 0.00261  0.0431   0.0805
+## sigma           3.4323 0.04356 -1.2377   8.1023
 ## 
 ## $IORE
-##                     Estimate Pr(>t) Lower Upper
-## parent_0              95.523     NA    NA    NA
-## k__iore_parent_sink    0.333     NA    NA    NA
-## N_parent               0.568     NA    NA    NA
-## sigma                  1.953     NA    NA    NA
+##                     Estimate Pr(>t)     Lower  Upper
+## parent_0              95.523 0.0055 74.539157 116.51
+## k__iore_parent_sink    0.333 0.1433  0.000717 154.57
+## N_parent               0.568 0.0677 -0.989464   2.13
+## sigma                  1.953 0.0975 -5.893100   9.80
 ## 
 ## $DFOP
 ##          Estimate Pr(>t) Lower Upper
-## parent_0  97.6840     NA    NA    NA
-## k1         0.0589     NA    NA    NA
-## k2         0.0589     NA    NA    NA
-## g          0.6902     NA    NA    NA
-## sigma      3.4323     NA    NA    NA
+## parent_0  97.6840    NaN   NaN   NaN
+## k1         0.0589    NaN    NA    NA
+## k2         0.0589    NaN    NA    NA
+## g          0.6902    NaN    NA    NA
+## sigma      3.4323    NaN   NaN   NaN
 ## 
 ## 
 ## DTx values:
@@ -750,20 +682,16 @@
 

 

 Example on page 13

-
p13 <- nafta(NAFTA_SOP_Attachment[["p13"]])

-
## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix
-
-## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix
-
-## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix

+
p13 <- nafta(NAFTA_SOP_Attachment[["p13"]])

+
## Warning in sqrt(diag(covar)): NaNs wurden erzeugt

+
## Warning in sqrt(1/diag(V)): NaNs wurden erzeugt

+
## Warning in cov2cor(ans$cov.unscaled): diag(.) had 0 or NA entries; non-
+## finite result is doubtful

 
## The SFO model is rejected as S_SFO is equal or higher than the critical value S_c

 
## The half-life obtained from the IORE model may be used

-
plot(p13)

+
plot(p13)

 

-
print(p13)

+
print(p13)

 
## Sums of squares:
 ##      SFO     IORE     DFOP 
 ## 174.5971 142.3951 174.5971 
@@ -773,25 +701,25 @@
 ## 
 ## Parameters:
 ## $SFO
-##               Estimate Pr(>t) Lower Upper
-## parent_0      92.73500     NA    NA    NA
-## k_parent_sink  0.00258     NA    NA    NA
-## sigma          3.41172     NA    NA    NA
+##               Estimate   Pr(>t)    Lower    Upper
+## parent_0      92.73500 5.99e-17 89.61936 95.85065
+## k_parent_sink  0.00258 2.42e-09  0.00223  0.00299
+## sigma          3.41172 7.07e-05  2.05455  4.76888
 ## 
 ## $IORE
-##                     Estimate Pr(>t) Lower Upper
-## parent_0             91.6016     NA    NA    NA
-## k__iore_parent_sink   0.0396     NA    NA    NA
-## N_parent              0.3541     NA    NA    NA
-## sigma                 3.0811     NA    NA    NA
+##                     Estimate   Pr(>t)    Lower  Upper
+## parent_0             91.6016 6.34e-16 88.53086 94.672
+## k__iore_parent_sink   0.0396 2.36e-01  0.00207  0.759
+## N_parent              0.3541 1.46e-01 -0.35153  1.060
+## sigma                 3.0811 9.64e-05  1.84296  4.319
 ## 
 ## $DFOP
-##          Estimate Pr(>t) Lower Upper
-## parent_0 92.73500     NA    NA    NA
-## k1        0.00258     NA    NA    NA
-## k2        0.00258     NA    NA    NA
-## g         0.00442     NA    NA    NA
-## sigma     3.41172     NA    NA    NA
+##          Estimate   Pr(>t)    Lower    Upper
+## parent_0 92.73500 9.25e-15 8.95e+01 9.59e+01
+## k1        0.00258 4.28e-01 1.70e-08 3.92e+02
+## k2        0.00258 3.69e-08 2.20e-03 3.03e-03
+## g         0.00442 5.00e-01       NA       NA
+## sigma     3.41172 1.35e-04 2.02e+00 4.80e+00
 ## 
 ## 
 ## DTx values:
@@ -807,20 +735,16 @@
 

 

 DT50 not observed in the study and DFOP problems in PestDF

-
p14 <- nafta(NAFTA_SOP_Attachment[["p14"]])

-
## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix
-
-## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix
-
-## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix

+
p14 <- nafta(NAFTA_SOP_Attachment[["p14"]])

+
## Warning in sqrt(diag(covar)): NaNs wurden erzeugt

+
## Warning in sqrt(1/diag(V)): NaNs wurden erzeugt

+
## Warning in cov2cor(ans$cov.unscaled): diag(.) had 0 or NA entries; non-
+## finite result is doubtful

 
## The SFO model is rejected as S_SFO is equal or higher than the critical value S_c

 
## The half-life obtained from the IORE model may be used

-
plot(p14)

+
plot(p14)

 

-
print(p14)

+
print(p14)

 
## Sums of squares:
 ##      SFO     IORE     DFOP 
 ## 48.43249 28.67746 27.26248 
@@ -830,25 +754,25 @@
 ## 
 ## Parameters:
 ## $SFO
-##               Estimate Pr(>t) Lower Upper
-## parent_0      99.47124     NA    NA    NA
-## k_parent_sink  0.00279     NA    NA    NA
-## sigma          1.55616     NA    NA    NA
+##               Estimate   Pr(>t)    Lower    Upper
+## parent_0      99.47124 2.06e-30 98.42254 1.01e+02
+## k_parent_sink  0.00279 3.75e-15  0.00256 3.04e-03
+## sigma          1.55616 3.81e-06  1.03704 2.08e+00
 ## 
 ## $IORE
 ##                     Estimate Pr(>t) Lower Upper
-## parent_0            1.00e+02     NA    NA    NA
-## k__iore_parent_sink 9.44e-08     NA    NA    NA
-## N_parent            3.31e+00     NA    NA    NA
-## sigma               1.20e+00     NA    NA    NA
+## parent_0            1.00e+02     NA   NaN   NaN
+## k__iore_parent_sink 9.44e-08     NA   NaN   NaN
+## N_parent            3.31e+00     NA   NaN   NaN
+## sigma               1.20e+00     NA 0.796   1.6
 ## 
 ## $DFOP
-##          Estimate Pr(>t) Lower Upper
-## parent_0 1.00e+02     NA    NA    NA
-## k1       9.53e-03     NA    NA    NA
-## k2       7.29e-12     NA    NA    NA
-## g        3.98e-01     NA    NA    NA
-## sigma    1.17e+00     NA    NA    NA
+##          Estimate   Pr(>t)    Lower    Upper
+## parent_0 1.00e+02 2.96e-28 99.40280 101.2768
+## k1       9.53e-03 1.20e-01  0.00638   0.0143
+## k2       7.29e-12 5.00e-01  0.00000      Inf
+## g        3.98e-01 2.19e-01  0.30481   0.4998
+## sigma    1.17e+00 7.68e-06  0.77406   1.5610
 ## 
 ## 
 ## DTx values:
@@ -864,20 +788,17 @@
 

 

 N is less than 1 and DFOP fraction parameter is below zero

-
p15a <- nafta(NAFTA_SOP_Attachment[["p15a"]])

-
## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix
-
-## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix
-
-## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix

+
p15a <- nafta(NAFTA_SOP_Attachment[["p15a"]])

+
## Warning in sqrt(diag(covar)): NaNs wurden erzeugt

+
## Warning in sqrt(diag(covar_notrans)): NaNs wurden erzeugt

+
## Warning in sqrt(1/diag(V)): NaNs wurden erzeugt

+
## Warning in cov2cor(ans$cov.unscaled): diag(.) had 0 or NA entries; non-
+## finite result is doubtful

 
## The SFO model is rejected as S_SFO is equal or higher than the critical value S_c

 
## The half-life obtained from the IORE model may be used

-
plot(p15a)

+
plot(p15a)

 

-
print(p15a)

+
print(p15a)

 
## Sums of squares:
 ##      SFO     IORE     DFOP 
 ## 245.5248 135.0132 245.5248 
@@ -887,25 +808,25 @@
 ## 
 ## Parameters:
 ## $SFO
-##               Estimate Pr(>t) Lower Upper
-## parent_0      97.96751     NA    NA    NA
-## k_parent_sink  0.00952     NA    NA    NA
-## sigma          4.18778     NA    NA    NA
+##               Estimate   Pr(>t)    Lower   Upper
+## parent_0      97.96751 2.00e-15 94.32049 101.615
+## k_parent_sink  0.00952 4.93e-09  0.00824   0.011
+## sigma          4.18778 1.28e-04  2.44588   5.930
 ## 
 ## $IORE
-##                     Estimate Pr(>t) Lower Upper
-## parent_0              95.874     NA    NA    NA
-## k__iore_parent_sink    0.629     NA    NA    NA
-## N_parent               0.000     NA    NA    NA
-## sigma                  3.105     NA    NA    NA
+##                     Estimate   Pr(>t)  Lower  Upper
+## parent_0              95.874 2.94e-15 92.937 98.811
+## k__iore_parent_sink    0.629 2.11e-01  0.044  8.982
+## N_parent               0.000 5.00e-01 -0.642  0.642
+## sigma                  3.105 1.78e-04  1.795  4.416
 ## 
 ## $DFOP
-##          Estimate Pr(>t) Lower Upper
-## parent_0 97.96752     NA    NA    NA
-## k1        0.00952     NA    NA    NA
-## k2        0.00952     NA    NA    NA
-## g         0.17247     NA    NA    NA
-## sigma     4.18778     NA    NA    NA
+##          Estimate   Pr(>t)    Lower    Upper
+## parent_0 97.96752 2.85e-13 94.21914 101.7159
+## k1        0.00952 6.80e-02  0.00277   0.0327
+## k2        0.00952 3.82e-06  0.00902   0.0100
+## g         0.17247      NaN       NA       NA
+## sigma     4.18778 2.50e-04  2.39747   5.9781
 ## 
 ## 
 ## DTx values:
@@ -916,20 +837,16 @@
 ## 
 ## Representative half-life:
 ## [1] 41.33

-
p15b <- nafta(NAFTA_SOP_Attachment[["p15b"]])

-
## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix
-
-## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix
-
-## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix

+
p15b <- nafta(NAFTA_SOP_Attachment[["p15b"]])

+
## Warning in sqrt(diag(covar)): NaNs wurden erzeugt

+
## Warning in sqrt(1/diag(V)): NaNs wurden erzeugt

+
## Warning in cov2cor(ans$cov.unscaled): diag(.) had 0 or NA entries; non-
+## finite result is doubtful

 
## The SFO model is rejected as S_SFO is equal or higher than the critical value S_c

 
## The half-life obtained from the IORE model may be used

-
plot(p15b)

+
plot(p15b)

 

-
print(p15b)

+
print(p15b)

 
## Sums of squares:
 ##       SFO      IORE      DFOP 
 ## 106.91629  68.55574 106.91629 
@@ -939,25 +856,25 @@
 ## 
 ## Parameters:
 ## $SFO
-##               Estimate Pr(>t) Lower Upper
-## parent_0      1.01e+02     NA    NA    NA
-## k_parent_sink 4.86e-03     NA    NA    NA
-## sigma         2.76e+00     NA    NA    NA
+##               Estimate   Pr(>t)    Lower    Upper
+## parent_0      1.01e+02 3.06e-17 98.31594 1.03e+02
+## k_parent_sink 4.86e-03 2.48e-10  0.00435 5.42e-03
+## sigma         2.76e+00 1.28e-04  1.61402 3.91e+00
 ## 
 ## $IORE
-##                     Estimate Pr(>t) Lower Upper
-## parent_0               99.83     NA    NA    NA
-## k__iore_parent_sink     0.38     NA    NA    NA
-## N_parent                0.00     NA    NA    NA
-## sigma                   2.21     NA    NA    NA
+##                     Estimate   Pr(>t)    Lower  Upper
+## parent_0               99.83 1.81e-16 97.51349 102.14
+## k__iore_parent_sink     0.38 3.22e-01  0.00352  41.05
+## N_parent                0.00 5.00e-01 -1.07695   1.08
+## sigma                   2.21 2.57e-04  1.23245   3.19
 ## 
 ## $DFOP
-##          Estimate Pr(>t) Lower Upper
-## parent_0 1.01e+02     NA    NA    NA
-## k1       4.86e-03     NA    NA    NA
-## k2       4.86e-03     NA    NA    NA
-## g        1.50e-01     NA    NA    NA
-## sigma    2.76e+00     NA    NA    NA
+##          Estimate Pr(>t)    Lower    Upper
+## parent_0 1.01e+02     NA 9.82e+01 1.04e+02
+## k1       4.86e-03     NA 6.75e-04 3.49e-02
+## k2       4.86e-03     NA 3.37e-03 6.99e-03
+## g        1.50e-01     NA       NA       NA
+## sigma    2.76e+00     NA 1.58e+00 3.94e+00
 ## 
 ## 
 ## DTx values:
@@ -973,22 +890,14 @@
 

 

 The DFOP fraction parameter is greater than 1

-
p16 <- nafta(NAFTA_SOP_Attachment[["p16"]])

-
## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix
-
-## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix
-
-## Warning in summary.mkinfit(x): Could not calculate correlation; no
-## covariance matrix

+
p16 <- nafta(NAFTA_SOP_Attachment[["p16"]])

 
## The SFO model is rejected as S_SFO is equal or higher than the critical value S_c

 
## The representative half-life of the IORE model is longer than the one corresponding

 
## to the terminal degradation rate found with the DFOP model.

 
## The representative half-life obtained from the DFOP model may be used

-
plot(p16)

+
plot(p16)

 

-
print(p16)

+
print(p16)

 
## Sums of squares:
 ##      SFO     IORE     DFOP 
 ## 3831.804 2062.008 1550.980 
@@ -998,25 +907,25 @@
 ## 
 ## Parameters:
 ## $SFO
-##               Estimate Pr(>t) Lower Upper
-## parent_0        71.953     NA    NA    NA
-## k_parent_sink    0.159     NA    NA    NA
-## sigma           11.302     NA    NA    NA
+##               Estimate   Pr(>t)  Lower Upper
+## parent_0        71.953 2.33e-13 60.509 83.40
+## k_parent_sink    0.159 4.86e-05  0.102  0.25
+## sigma           11.302 1.25e-08  8.308 14.30
 ## 
 ## $IORE
-##                     Estimate Pr(>t) Lower Upper
-## parent_0            8.74e+01     NA    NA    NA
-## k__iore_parent_sink 4.55e-04     NA    NA    NA
-## N_parent            2.70e+00     NA    NA    NA
-## sigma               8.29e+00     NA    NA    NA
+##                     Estimate   Pr(>t)    Lower    Upper
+## parent_0            8.74e+01 2.48e-16 7.72e+01 97.52972
+## k__iore_parent_sink 4.55e-04 2.16e-01 3.48e-05  0.00595
+## N_parent            2.70e+00 1.21e-08 1.99e+00  3.40046
+## sigma               8.29e+00 1.61e-08 6.09e+00 10.49062
 ## 
 ## $DFOP
-##          Estimate Pr(>t) Lower Upper
-## parent_0  88.5333     NA    NA    NA
-## k1        18.5561     NA    NA    NA
-## k2         0.0776     NA    NA    NA
-## g          0.4733     NA    NA    NA
-## sigma      7.1902     NA    NA    NA
+##          Estimate   Pr(>t)   Lower  Upper
+## parent_0  88.5333 7.40e-18 79.9836 97.083
+## k1        18.5561 5.00e-01  0.0000    Inf
+## k2         0.0776 1.41e-05  0.0518  0.116
+## g          0.4733 1.41e-09  0.3674  0.582
+## sigma      7.1902 2.11e-08  5.2785  9.102
 ## 
 ## 
 ## DTx values:
diff --git a/docs/articles/web_only/benchmarks.html b/docs/articles/web_only/benchmarks.html
index 2e83cea9..a250f005 100644
--- a/docs/articles/web_only/benchmarks.html
+++ b/docs/articles/web_only/benchmarks.html
@@ -198,67 +198,67 @@
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.1 8.184
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.2 7.064
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.3 7.296
-## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.4 2.866
+## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.4 5.982
 ##                                                                         t2
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.48.1 11.019
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.1 22.889
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.2 12.558
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.3 21.239
-## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.4 16.533
+## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.4 17.777
 ##                                                                        t3
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.48.1 3.764
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.1 4.649
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.2 4.786
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.3 4.510
-## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.4 2.247
+## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.4 4.545
 ##                                                                         t4
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.48.1 14.347
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.1 13.789
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.2  8.461
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.3 13.805
-## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.4 11.124
+## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.4 16.684
 ##                                                                        t5
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.48.1 9.495
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.1 6.395
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.2 5.675
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.3 7.386
-## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.4 4.887
+## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.4 7.868
 ##                                                                        t6
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.48.1 2.623
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.1 2.542
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.2 2.723
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.3 2.643
-## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.4 1.317
+## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.4  2.66
 ##                                                                        t7
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.48.1 4.587
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.1 4.128
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.2 4.478
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.3 4.374
-## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.4 2.177
+## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.4 4.496
 ##                                                                        t8
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.48.1 7.525
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.1 4.632
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.2 4.862
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.3  7.02
-## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.4 3.991
+## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.4 4.919
 ##                                                                         t9
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.48.1 16.621
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.1  8.171
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.2  7.618
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.3 11.124
-## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.4  7.175
+## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.4 11.683
 ##                                                                       t10
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.48.1 8.576
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.1 3.676
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.2 3.579
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.3 5.388
-## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.4 2.776
+## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.4 5.344
 ##                                                                        t11
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.48.1 31.267
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.1  5.636
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.2  5.574
 ## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.3  7.365
-## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.4  4.658

+## Linux, AMD Ryzen 7 1700 Eight-Core Processor, mkin version 0.9.49.4  7.752

 
save(mkin_benchmarks, file = "~/git/mkin/vignettes/mkin_benchmarks.rda")

 

   

diff --git a/docs/articles/web_only/compiled_models.html b/docs/articles/web_only/compiled_models.html
index ad7716c1..54d3148f 100644
--- a/docs/articles/web_only/compiled_models.html
+++ b/docs/articles/web_only/compiled_models.html
@@ -163,14 +163,14 @@
 ## Warning in mkinfit(SFO_SFO, FOCUS_2006_D, solution_type = "deSolve", quiet
 ## = TRUE): Observations with value of zero were removed from the data

 
##                    test replications elapsed relative user.self sys.self
-## 3     deSolve, compiled            3   1.325    1.000     1.325        0
-## 1 deSolve, not compiled            3  11.175    8.434    11.170        0
-## 2      Eigenvalue based            3   1.799    1.358     1.798        0
+## 3     deSolve, compiled            3   3.067    1.000     3.065        0
+## 1 deSolve, not compiled            3  28.135    9.173    28.122        0
+## 2      Eigenvalue based            3   4.306    1.404     4.303        0
 ##   user.child sys.child
 ## 3          0         0
 ## 1          0         0
 ## 2          0         0

-
We see that using the compiled model is by a factor of around 8 faster than using the R version with the default ode solver, and it is even faster than the Eigenvalue based solution implemented in R which does not need iterative solution of the ODEs.

+
We see that using the compiled model is by a factor of around 9 faster than using the R version with the default ode solver, and it is even faster than the Eigenvalue based solution implemented in R which does not need iterative solution of the ODEs.

 

 

 

@@ -214,8 +214,8 @@
 ## Warning in mkinfit(FOMC_SFO, FOCUS_2006_D, quiet = TRUE): Observations with
 ## value of zero were removed from the data

 
##                    test replications elapsed relative user.self sys.self
-## 2     deSolve, compiled            3   2.322    1.000     2.320        0
-## 1 deSolve, not compiled            3  25.232   10.866    25.208        0
+## 2     deSolve, compiled            3   4.827    1.000     4.825        0
+## 1 deSolve, not compiled            3  52.646   10.907    52.622        0
 ##   user.child sys.child
 ## 2          0         0
 ## 1          0         0

diff --git a/docs/reference/NAFTA_SOP_2015.html b/docs/reference/NAFTA_SOP_2015.html
index 9fbc71c6..84341a17 100644
--- a/docs/reference/NAFTA_SOP_2015.html
+++ b/docs/reference/NAFTA_SOP_2015.html
@@ -155,7 +155,7 @@
     
 
     
Examples

-    
  nafta_evaluation <- nafta(NAFTA_SOP_Appendix_D, cores = 1)
#> Warning: Could not calculate correlation; no covariance matrix
#> Warning: Could not calculate correlation; no covariance matrix
#> Warning: Could not calculate correlation; no covariance matrix
#> The SFO model is rejected as S_SFO is equal or higher than the critical value S_c
#> The representative half-life of the IORE model is longer than the one corresponding
#> to the terminal degradation rate found with the DFOP model.
#> The representative half-life obtained from the DFOP model may be used
  print(nafta_evaluation)
#> Sums of squares:
+    
  nafta_evaluation <- nafta(NAFTA_SOP_Appendix_D, cores = 1)
#> The SFO model is rejected as S_SFO is equal or higher than the critical value S_c
#> The representative half-life of the IORE model is longer than the one corresponding
#> to the terminal degradation rate found with the DFOP model.
#> The representative half-life obtained from the DFOP model may be used
  print(nafta_evaluation)
#> Sums of squares:
 #>       SFO      IORE      DFOP 
 #> 1378.6832  615.7730  517.8836 
 #> 
@@ -164,25 +164,25 @@
 #> 
 #> Parameters:
 #> $SFO
-#>               Estimate Pr(>t) Lower Upper
-#> parent_0       83.7558     NA    NA    NA
-#> k_parent_sink   0.0017     NA    NA    NA
-#> sigma           8.7518     NA    NA    NA
+#>               Estimate   Pr(>t)    Lower   Upper
+#> parent_0       83.7558 1.80e-14 77.18268 90.3288
+#> k_parent_sink   0.0017 7.43e-05  0.00112  0.0026
+#> sigma           8.7518 1.22e-05  5.64278 11.8608
 #> 
 #> $IORE
-#>                     Estimate Pr(>t) Lower Upper
-#> parent_0            9.69e+01     NA    NA    NA
-#> k__iore_parent_sink 8.40e-14     NA    NA    NA
-#> N_parent            6.68e+00     NA    NA    NA
-#> sigma               5.85e+00     NA    NA    NA
+#>                     Estimate Pr(>t)    Lower    Upper
+#> parent_0            9.69e+01     NA 8.88e+01 1.05e+02
+#> k__iore_parent_sink 8.40e-14     NA 1.79e-18 3.94e-09
+#> N_parent            6.68e+00     NA 4.19e+00 9.17e+00
+#> sigma               5.85e+00     NA 3.76e+00 7.94e+00
 #> 
 #> $DFOP
-#>          Estimate Pr(>t) Lower Upper
-#> parent_0 9.76e+01     NA    NA    NA
-#> k1       4.24e-02     NA    NA    NA
-#> k2       8.24e-04     NA    NA    NA
-#> g        2.88e-01     NA    NA    NA
-#> sigma    5.36e+00     NA    NA    NA
+#>          Estimate   Pr(>t)    Lower    Upper
+#> parent_0 9.76e+01 1.94e-13 9.02e+01 1.05e+02
+#> k1       4.24e-02 5.92e-03 2.03e-02 8.88e-02
+#> k2       8.24e-04 6.48e-03 3.89e-04 1.75e-03
+#> g        2.88e-01 2.47e-05 1.95e-01 4.03e-01
+#> sigma    5.36e+00 2.22e-05 3.43e+00 7.30e+00
 #> 
 #> 
 #> DTx values:
diff --git a/docs/reference/NAFTA_SOP_Attachment.html b/docs/reference/NAFTA_SOP_Attachment.html
index 149b4d82..76d91c40 100644
--- a/docs/reference/NAFTA_SOP_Attachment.html
+++ b/docs/reference/NAFTA_SOP_Attachment.html
@@ -152,7 +152,7 @@
     
 
     
Examples

-    
  nafta_att_p5a <- nafta(NAFTA_SOP_Attachment[["p5a"]], cores = 1)
#> Warning: Could not calculate correlation; no covariance matrix
#> Warning: Could not calculate correlation; no covariance matrix
#> Warning: Could not calculate correlation; no covariance matrix
#> The SFO model is rejected as S_SFO is equal or higher than the critical value S_c
#> The half-life obtained from the IORE model may be used
  print(nafta_att_p5a)
#> Sums of squares:
+    
  nafta_att_p5a <- nafta(NAFTA_SOP_Attachment[["p5a"]], cores = 1)
#> The SFO model is rejected as S_SFO is equal or higher than the critical value S_c
#> The half-life obtained from the IORE model may be used
  print(nafta_att_p5a)
#> Sums of squares:
 #>       SFO      IORE      DFOP 
 #> 465.21753  56.27506  32.06401 
 #> 
@@ -161,25 +161,25 @@
 #> 
 #> Parameters:
 #> $SFO
-#>               Estimate Pr(>t) Lower Upper
-#> parent_0       95.8401     NA    NA    NA
-#> k_parent_sink   0.0102     NA    NA    NA
-#> sigma           4.8230     NA    NA    NA
+#>               Estimate   Pr(>t)  Lower   Upper
+#> parent_0       95.8401 4.67e-21 92.245 99.4357
+#> k_parent_sink   0.0102 3.92e-12  0.009  0.0117
+#> sigma           4.8230 3.81e-06  3.214  6.4318
 #> 
 #> $IORE
-#>                     Estimate Pr(>t) Lower Upper
-#> parent_0            1.01e+02     NA    NA    NA
-#> k__iore_parent_sink 1.54e-05     NA    NA    NA
-#> N_parent            2.57e+00     NA    NA    NA
-#> sigma               1.68e+00     NA    NA    NA
+#>                     Estimate Pr(>t)    Lower    Upper
+#> parent_0            1.01e+02     NA 9.91e+01 1.02e+02
+#> k__iore_parent_sink 1.54e-05     NA 4.08e-06 5.84e-05
+#> N_parent            2.57e+00     NA 2.25e+00 2.89e+00
+#> sigma               1.68e+00     NA 1.12e+00 2.24e+00
 #> 
 #> $DFOP
-#>          Estimate Pr(>t) Lower Upper
-#> parent_0 9.99e+01     NA    NA    NA
-#> k1       2.67e-02     NA    NA    NA
-#> k2       2.86e-12     NA    NA    NA
-#> g        6.47e-01     NA    NA    NA
-#> sigma    1.27e+00     NA    NA    NA
+#>          Estimate   Pr(>t)   Lower    Upper
+#> parent_0 9.99e+01 1.41e-26 98.8116 101.0810
+#> k1       2.67e-02 5.05e-06  0.0243   0.0295
+#> k2       2.86e-12 5.00e-01  0.0000      Inf
+#> g        6.47e-01 3.67e-06  0.6248   0.6677
+#> sigma    1.27e+00 8.91e-06  0.8395   1.6929
 #> 
 #> 
 #> DTx values:
diff --git a/docs/reference/logistic.solution.html b/docs/reference/logistic.solution.html
index 082cd4fb..3b45b14f 100644
--- a/docs/reference/logistic.solution.html
+++ b/docs/reference/logistic.solution.html
@@ -205,12 +205,18 @@
     n = 1, reps = 2, digits = 5, LOD = 0.1, seed = 123456)[[1]]
 
   m <- mkinfit("logistic", d_2_1, quiet = TRUE)
-  plot_sep(m)
  summary(m)$bpar
#> Warning: Could not calculate correlation; no covariance matrix
#>              Estimate se_notrans t value Pr(>t) Lower Upper
-#> parent_0 1.057896e+02         NA      NA     NA    NA    NA
-#> kmax     6.398190e-02         NA      NA     NA    NA    NA
-#> k0       1.612775e-04         NA      NA     NA    NA    NA
-#> r        2.263946e-01         NA      NA     NA    NA    NA
-#> sigma    5.332935e+00         NA      NA     NA    NA    NA
  endpoints(m)$distimes
#>            DT50     DT90  DT50_k0 DT50_kmax
+  plot_sep(m)
  summary(m)$bpar
#>              Estimate   se_notrans   t value       Pr(>t)        Lower
+#> parent_0 1.057896e+02 1.9023449649 55.610120 3.768361e-16 1.016451e+02
+#> kmax     6.398190e-02 0.0143201029  4.467978 3.841828e-04 3.929235e-02
+#> k0       1.612775e-04 0.0005866813  0.274898 3.940351e-01 5.846685e-08
+#> r        2.263946e-01 0.1718110773  1.317695 1.061044e-01 4.335843e-02
+#> sigma    5.332935e+00 0.9145907310  5.830952 4.036926e-05 3.340213e+00
+#>                Upper
+#> parent_0 109.9341588
+#> kmax       0.1041853
+#> k0         0.4448750
+#> r          1.1821121
+#> sigma      7.3256566
  endpoints(m)$distimes
#>            DT50     DT90  DT50_k0 DT50_kmax
 #> parent 36.86533 62.41511 4297.854  10.83349