From 8a3475c59f3d91ce5ce7d980d6de09360617e7fe Mon Sep 17 00:00:00 2001 From: Johannes Ranke Date: Tue, 7 May 2019 08:12:27 +0200 Subject: After the OLS step, use OLS parameter estimates - Fix the respective error in the code - Static documentation rebuilt by pkgdown --- docs/articles/FOCUS_L.html | 329 +++++++++++++++++++++------------------------ 1 file changed, 152 insertions(+), 177 deletions(-) (limited to 'docs/articles/FOCUS_L.html') diff --git a/docs/articles/FOCUS_L.html b/docs/articles/FOCUS_L.html index 9507139c..23fa68c6 100644 --- a/docs/articles/FOCUS_L.html +++ b/docs/articles/FOCUS_L.html @@ -88,7 +88,7 @@

Example evaluation of FOCUS Laboratory Data L1 to L3

Johannes Ranke

-

2019-05-03

+

2019-05-07

@@ -112,17 +112,19 @@

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:     Fri May  3 19:08:47 2019 
-## Date of summary: Fri May  3 19:08:47 2019 
+## Date of fit:     Tue May  7 08:09:13 2019 
+## Date of summary: Tue May  7 08:09:13 2019 
 ## 
 ## Equations:
 ## d_parent/dt = - k_parent_sink * parent
 ## 
 ## Model predictions using solution type analytical 
 ## 
-## Fitted using 133 model solutions performed in 0.281 s
+## Fitted using 41 model solutions performed in 0.088 s
 ## 
 ## Error model:
 ## Constant variance 
@@ -143,25 +145,21 @@
 ## None
 ## 
 ## Optimised, transformed parameters with symmetric confidence intervals:
-##                   Estimate Std. Error  Lower  Upper
-## parent_0            92.470    1.28200 89.740 95.200
-## log_k_parent_sink   -2.347    0.03763 -2.428 -2.267
-## sigma                2.780    0.46330  1.792  3.767
+##                   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
 ## 
 ## Parameter correlation:
-##                     parent_0 log_k_parent_sink      sigma
-## parent_0           1.000e+00         6.186e-01 -1.712e-09
-## log_k_parent_sink  6.186e-01         1.000e+00 -3.237e-09
-## sigma             -1.712e-09        -3.237e-09  1.000e+00
-## 
+## No covariance matrix
 ## Backtransformed parameters:
 ## Confidence intervals for internally transformed parameters are asymmetric.
 ## t-test (unrealistically) based on the assumption of normal distribution
 ## for estimators of untransformed parameters.
-##               Estimate t value    Pr(>t)    Lower   Upper
-## parent_0      92.47000   72.13 8.824e-21 89.74000 95.2000
-## k_parent_sink  0.09561   26.57 2.487e-14  0.08824  0.1036
-## sigma          2.78000    6.00 1.216e-05  1.79200  3.7670
+##               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
 ## 
 ## FOCUS Chi2 error levels in percent:
 ##          err.min n.optim df
@@ -197,26 +195,24 @@
 ##    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 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
+
summary(m.L1.FOMC, data = FALSE)
+
## Warning in summary.mkinfit(m.L1.FOMC, 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:     Fri May  3 19:08:50 2019 
-## Date of summary: Fri May  3 19:08:50 2019 
+## Date of fit:     Tue May  7 08:09:14 2019 
+## Date of summary: Tue May  7 08:09:14 2019 
 ## 
 ## 
 ## Warning: Optimisation did not converge:
@@ -228,7 +224,7 @@
 ## 
 ## Model predictions using solution type analytical 
 ## 
-## Fitted using 899 model solutions performed in 1.885 s
+## Fitted using 743 model solutions performed in 1.558 s
 ## 
 ## Error model:
 ## Constant variance 
@@ -251,28 +247,23 @@
 ## None
 ## 
 ## Optimised, transformed parameters with symmetric confidence intervals:
-##           Estimate Std. Error  Lower  Upper
-## parent_0     92.47     1.2800 89.730 95.220
-## log_alpha    10.58        NaN    NaN    NaN
-## log_beta     12.93        NaN    NaN    NaN
-## sigma         2.78     0.4507  1.813  3.747
+##           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
 ## 
 ## Parameter correlation:
-##           parent_0 log_alpha log_beta   sigma
-## parent_0   1.00000       NaN      NaN 0.01452
-## log_alpha      NaN         1      NaN     NaN
-## log_beta       NaN       NaN        1     NaN
-## sigma      0.01452       NaN      NaN 1.00000
-## 
+## No covariance matrix
 ## Backtransformed parameters:
 ## Confidence intervals for internally transformed parameters are asymmetric.
 ## t-test (unrealistically) based on the assumption of normal distribution
 ## for estimators of untransformed parameters.
-##           Estimate  t value    Pr(>t)  Lower  Upper
-## parent_0     92.47 72.13000 1.052e-19 89.730 95.220
-## alpha     39440.00  0.02397 4.906e-01     NA     NA
-## beta     412500.00  0.02397 4.906e-01     NA     NA
-## sigma         2.78  6.00000 1.628e-05  1.813  3.747
+##           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
 ## 
 ## FOCUS Chi2 error levels in percent:
 ##          err.min n.optim df
@@ -290,19 +281,19 @@
 

 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.

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

 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)

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

+
## Warning in summary.mkinfit(m.L2.FOMC, 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:     Fri May  3 19:08:51 2019 
-## Date of summary: Fri May  3 19:08:51 2019 
+## Date of fit:     Tue May  7 08:09:15 2019 
+## Date of summary: Tue May  7 08:09:15 2019 
 ## 
 ## Equations:
 ## d_parent/dt = - (alpha/beta) * 1/((time/beta) + 1) * parent
 ## 
 ## Model predictions using solution type analytical 
 ## 
-## Fitted using 239 model solutions performed in 0.486 s
+## Fitted using 83 model solutions performed in 0.169 s
 ## 
 ## Error model:
 ## Constant variance 
@@ -350,28 +343,23 @@
 ## None
 ## 
 ## Optimised, transformed parameters with symmetric confidence intervals:
-##           Estimate Std. Error    Lower   Upper
-## parent_0   93.7700     1.6130 90.05000 97.4900
-## log_alpha   0.3180     0.1559 -0.04149  0.6776
-## log_beta    0.2102     0.2493 -0.36460  0.7850
-## sigma       2.2760     0.4645  1.20500  3.3470
+##           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
 ## 
 ## Parameter correlation:
-##             parent_0  log_alpha   log_beta      sigma
-## parent_0   1.000e+00 -1.151e-01 -2.085e-01 -7.637e-09
-## log_alpha -1.151e-01  1.000e+00  9.741e-01 -1.617e-07
-## log_beta  -2.085e-01  9.741e-01  1.000e+00 -1.387e-07
-## sigma     -7.637e-09 -1.617e-07 -1.387e-07  1.000e+00
-## 
+## No covariance matrix
 ## Backtransformed parameters:
 ## Confidence intervals for internally transformed parameters are asymmetric.
 ## t-test (unrealistically) based on the assumption of normal distribution
 ## for estimators of untransformed parameters.
-##          Estimate t value    Pr(>t)   Lower  Upper
-## parent_0   93.770  58.120 4.267e-12 90.0500 97.490
-## alpha       1.374   6.414 1.030e-04  0.9594  1.969
-## beta        1.234   4.012 1.942e-03  0.6945  2.192
-## sigma       2.276   4.899 5.977e-04  1.2050  3.347
+##          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
 ## 
 ## FOCUS Chi2 error levels in percent:
 ##          err.min n.optim df
@@ -392,10 +380,12 @@
      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:     Fri May  3 19:08:52 2019 
-## Date of summary: Fri May  3 19:08:52 2019 
+## Date of fit:     Tue May  7 08:09:16 2019 
+## Date of summary: Tue May  7 08:09:16 2019 
 ## 
 ## Equations:
 ## d_parent/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) *
@@ -404,7 +394,7 @@
 ## 
 ## Model predictions using solution type analytical 
 ## 
-## Fitted using 572 model solutions performed in 1.193 s
+## Fitted using 336 model solutions performed in 0.708 s
 ## 
 ## Error model:
 ## Constant variance 
@@ -429,31 +419,25 @@
 ## None
 ## 
 ## Optimised, transformed parameters with symmetric confidence intervals:
-##          Estimate Std. Error      Lower     Upper
-## parent_0  93.9500  9.998e-01    91.5900   96.3100
-## log_k1     3.1370  2.376e+03 -5616.0000 5622.0000
-## log_k2    -1.0880  6.285e-02    -1.2370   -0.9394
-## g_ilr     -0.2821  7.033e-02    -0.4484   -0.1158
-## sigma      1.4140  2.886e-01     0.7314    2.0960
+##          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
 ## 
 ## Parameter correlation:
-##            parent_0     log_k1     log_k2      g_ilr      sigma
-## parent_0  1.000e+00  5.155e-07  2.371e-09  2.665e-01 -6.849e-09
-## log_k1    5.155e-07  1.000e+00  8.434e-05 -1.659e-04 -7.791e-06
-## log_k2    2.371e-09  8.434e-05  1.000e+00 -7.903e-01 -1.262e-08
-## g_ilr     2.665e-01 -1.659e-04 -7.903e-01  1.000e+00  3.241e-08
-## sigma    -6.849e-09 -7.791e-06 -1.262e-08  3.241e-08  1.000e+00
-## 
+## No covariance matrix
 ## Backtransformed parameters:
 ## Confidence intervals for internally transformed parameters are asymmetric.
 ## t-test (unrealistically) based on the assumption of normal distribution
 ## for estimators of untransformed parameters.
-##          Estimate   t value    Pr(>t)   Lower   Upper
-## parent_0  93.9500 9.397e+01 2.036e-12 91.5900 96.3100
-## k1        23.0400 4.303e-04 4.998e-01  0.0000     Inf
-## k2         0.3369 1.591e+01 4.697e-07  0.2904  0.3909
-## g          0.4016 1.680e+01 3.238e-07  0.3466  0.4591
-## sigma      1.4140 4.899e+00 8.776e-04  0.7314  2.0960
+##          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
 ## 
 ## FOCUS Chi2 error levels in percent:
 ##          err.min n.optim df
@@ -470,18 +454,18 @@
 

 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.

 

@@ -490,11 +474,13 @@
 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]])

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

+
## Warning in summary.mkinfit(mm.L3[["DFOP", 1]]): 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:     Fri May  3 19:08:54 2019 
-## Date of summary: Fri May  3 19:08:54 2019 
+## Date of fit:     Tue May  7 08:09:17 2019 
+## Date of summary: Tue May  7 08:09:17 2019 
 ## 
 ## Equations:
 ## d_parent/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) *
@@ -503,7 +489,7 @@
 ## 
 ## Model predictions using solution type analytical 
 ## 
-## Fitted using 373 model solutions performed in 0.775 s
+## Fitted using 137 model solutions performed in 0.287 s
 ## 
 ## Error model:
 ## Constant variance 
@@ -528,31 +514,25 @@
 ## None
 ## 
 ## Optimised, transformed parameters with symmetric confidence intervals:
-##          Estimate Std. Error   Lower      Upper
-## parent_0  97.7500    1.01900 94.5000 101.000000
-## log_k1    -0.6612    0.10050 -0.9812  -0.341300
-## log_k2    -4.2860    0.04322 -4.4230  -4.148000
-## g_ilr     -0.1229    0.03727 -0.2415  -0.004343
-## sigma      1.0170    0.25430  0.2079   1.827000
+##          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
 ## 
 ## Parameter correlation:
-##            parent_0     log_k1     log_k2      g_ilr      sigma
-## parent_0  1.000e+00  1.732e-01  2.282e-02  4.009e-01 -6.872e-07
-## log_k1    1.732e-01  1.000e+00  4.945e-01 -5.809e-01  3.200e-07
-## log_k2    2.282e-02  4.945e-01  1.000e+00 -6.812e-01  7.673e-07
-## g_ilr     4.009e-01 -5.809e-01 -6.812e-01  1.000e+00 -8.731e-07
-## sigma    -6.872e-07  3.200e-07  7.673e-07 -8.731e-07  1.000e+00
-## 
+## No covariance matrix
 ## Backtransformed parameters:
 ## Confidence intervals for internally transformed parameters are asymmetric.
 ## t-test (unrealistically) based on the assumption of normal distribution
 ## for estimators of untransformed parameters.
-##          Estimate t value    Pr(>t)    Lower     Upper
-## parent_0 97.75000  95.960 1.248e-06 94.50000 101.00000
-## k1        0.51620   9.947 1.081e-03  0.37490   0.71090
-## k2        0.01376  23.140 8.840e-05  0.01199   0.01579
-## g         0.45660  34.920 2.581e-05  0.41540   0.49850
-## sigma     1.01700   4.000 1.400e-02  0.20790   1.82700
+##          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
 ## 
 ## FOCUS Chi2 error levels in percent:
 ##          err.min n.optim df
@@ -573,7 +553,7 @@
 ##    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.

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

 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)

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

+
## Warning in summary.mkinfit(mm.L4[["SFO", 1]], 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:     Fri May  3 19:08:55 2019 
-## Date of summary: Fri May  3 19:08:55 2019 
+## Date of fit:     Tue May  7 08:09:17 2019 
+## Date of summary: Tue May  7 08:09:18 2019 
 ## 
 ## Equations:
 ## d_parent/dt = - k_parent_sink * parent
 ## 
 ## Model predictions using solution type analytical 
 ## 
-## Fitted using 142 model solutions performed in 0.29 s
+## Fitted using 50 model solutions performed in 0.105 s
 ## 
 ## Error model:
 ## Constant variance 
@@ -627,25 +609,21 @@
 ## None
 ## 
 ## Optimised, transformed parameters with symmetric confidence intervals:
-##                   Estimate Std. Error  Lower   Upper
-## parent_0            96.440    1.69900 92.070 100.800
-## log_k_parent_sink   -5.030    0.07059 -5.211  -4.848
-## sigma                3.162    0.79050  1.130   5.194
+##                   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
 ## 
 ## Parameter correlation:
-##                    parent_0 log_k_parent_sink     sigma
-## parent_0          1.000e+00         5.938e-01 3.440e-07
-## log_k_parent_sink 5.938e-01         1.000e+00 5.885e-07
-## sigma             3.440e-07         5.885e-07 1.000e+00
-## 
+## No covariance matrix
 ## Backtransformed parameters:
 ## Confidence intervals for internally transformed parameters are asymmetric.
 ## t-test (unrealistically) based on the assumption of normal distribution
 ## for estimators of untransformed parameters.
-##                Estimate t value    Pr(>t)     Lower     Upper
-## parent_0      96.440000   56.77 1.604e-08 92.070000 1.008e+02
-## k_parent_sink  0.006541   14.17 1.578e-05  0.005455 7.842e-03
-## sigma          3.162000    4.00 5.162e-03  1.130000 5.194e+00
+##                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
 ## 
 ## FOCUS Chi2 error levels in percent:
 ##          err.min n.optim df
@@ -659,18 +637,20 @@
 ## Estimated disappearance times:
 ##        DT50 DT90
 ## parent  106  352

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

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

+
## Warning in summary.mkinfit(mm.L4[["FOMC", 1]], 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:     Fri May  3 19:08:55 2019 
-## Date of summary: Fri May  3 19:08:55 2019 
+## Date of fit:     Tue May  7 08:09:18 2019 
+## Date of summary: Tue May  7 08:09:18 2019 
 ## 
 ## Equations:
 ## d_parent/dt = - (alpha/beta) * 1/((time/beta) + 1) * parent
 ## 
 ## Model predictions using solution type analytical 
 ## 
-## Fitted using 224 model solutions performed in 0.451 s
+## Fitted using 68 model solutions performed in 0.139 s
 ## 
 ## Error model:
 ## Constant variance 
@@ -693,28 +673,23 @@
 ## None
 ## 
 ## Optimised, transformed parameters with symmetric confidence intervals:
-##           Estimate Std. Error   Lower    Upper
-## parent_0   99.1400     1.2670 95.6300 102.7000
-## log_alpha  -0.3506     0.2616 -1.0770   0.3756
-## log_beta    4.1740     0.3938  3.0810   5.2670
-## sigma       1.8300     0.4575  0.5598   3.1000
+##           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
 ## 
 ## Parameter correlation:
-##             parent_0  log_alpha   log_beta      sigma
-## parent_0   1.000e+00 -4.696e-01 -5.543e-01 -2.563e-07
-## log_alpha -4.696e-01  1.000e+00  9.889e-01  4.066e-08
-## log_beta  -5.543e-01  9.889e-01  1.000e+00  6.818e-08
-## sigma     -2.563e-07  4.066e-08  6.818e-08  1.000e+00
-## 
+## No covariance matrix
 ## Backtransformed parameters:
 ## Confidence intervals for internally transformed parameters are asymmetric.
 ## t-test (unrealistically) based on the assumption of normal distribution
 ## for estimators of untransformed parameters.
-##          Estimate t value    Pr(>t)   Lower   Upper
-## parent_0  99.1400  78.250 7.993e-08 95.6300 102.700
-## alpha      0.7042   3.823 9.365e-03  0.3407   1.456
-## beta      64.9800   2.540 3.201e-02 21.7800 193.900
-## sigma      1.8300   4.000 8.065e-03  0.5598   3.100
+##          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
 ## 
 ## FOCUS Chi2 error levels in percent:
 ##          err.min n.optim df
-- 
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