# Example evaluation of FOCUS Example Dataset D

#### Johannes Ranke

#### 2016-06-28

This is just a very simple vignette showing how to fit a degradation model for a parent compound with one transformation product using `mkin`. After loading the library we look a the data. We have observed concentrations in the column named `value` at the times specified in column `time` for the two observed variables named `parent` and `m1`.

    library("mkin")

    ## Loading required package: minpack.lm

    ## Loading required package: rootSolve

    ## Loading required package: inline

    ## Loading required package: parallel

    print(FOCUS_2006_D)

    ##      name time  value
    ## 1  parent    0  99.46
    ## 2  parent    0 102.04
    ## 3  parent    1  93.50
    ## 4  parent    1  92.50
    ## 5  parent    3  63.23
    ## 6  parent    3  68.99
    ## 7  parent    7  52.32
    ## 8  parent    7  55.13
    ## 9  parent   14  27.27
    ## 10 parent   14  26.64
    ## 11 parent   21  11.50
    ## 12 parent   21  11.64
    ## 13 parent   35   2.85
    ## 14 parent   35   2.91
    ## 15 parent   50   0.69
    ## 16 parent   50   0.63
    ## 17 parent   75   0.05
    ## 18 parent   75   0.06
    ## 19 parent  100     NA
    ## 20 parent  100     NA
    ## 21 parent  120     NA
    ## 22 parent  120     NA
    ## 23     m1    0   0.00
    ## 24     m1    0   0.00
    ## 25     m1    1   4.84
    ## 26     m1    1   5.64
    ## 27     m1    3  12.91
    ## 28     m1    3  12.96
    ## 29     m1    7  22.97
    ## 30     m1    7  24.47
    ## 31     m1   14  41.69
    ## 32     m1   14  33.21
    ## 33     m1   21  44.37
    ## 34     m1   21  46.44
    ## 35     m1   35  41.22
    ## 36     m1   35  37.95
    ## 37     m1   50  41.19
    ## 38     m1   50  40.01
    ## 39     m1   75  40.09
    ## 40     m1   75  33.85
    ## 41     m1  100  31.04
    ## 42     m1  100  33.13
    ## 43     m1  120  25.15
    ## 44     m1  120  33.31

Next we specify the degradation model: The parent compound degrades with simple first-order kinetics (SFO) to one metabolite named m1, which also degrades with SFO kinetics.

The call to mkinmod returns a degradation model. The differential equations represented in R code can be found in the character vector `$diffs` of the `mkinmod` object. If a C compiler (gcc) is installed and functional, the differential equation model will be compiled from auto-generated C code.

    SFO_SFO <- mkinmod(parent = mkinsub("SFO", "m1"), m1 = mkinsub("SFO"))

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

    print(SFO_SFO$diffs)

    ##                                                       parent
    ## "d_parent = - k_parent_sink * parent - k_parent_m1 * parent"
    ##                                                           m1
    ##             "d_m1 = + k_parent_m1 * parent - k_m1_sink * m1"

We do the fitting without progress report (`quiet = TRUE`).

    fit <- mkinfit(SFO_SFO, FOCUS_2006_D, quiet = TRUE)

A plot of the fit including a residual plot for both observed variables is obtained using the `plot` method for `mkinfit` objects.

    plot(fit, show_residuals = TRUE)

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

    mkinparplot(fit)

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

    summary(fit)

    ## mkin version:    0.9.44
    ## R version:       3.3.1
    ## Date of fit:     Tue Jun 28 10:42:08 2016
    ## Date of summary: Tue Jun 28 10:42:09 2016
    ##
    ## Equations:
    ## d_parent = - k_parent_sink * parent - k_parent_m1 * parent
    ## d_m1 = + k_parent_m1 * parent - k_m1_sink * m1
    ##
    ## Model predictions using solution type deSolve
    ##
    ## Fitted with method Port using 153 model solutions performed in 1.654 s
    ##
    ## Weighting: none
    ##
    ## Starting values for parameters to be optimised:
    ##                  value   type
    ## parent_0      100.7500  state
    ## k_parent_sink   0.1000 deparm
    ## k_parent_m1     0.1001 deparm
    ## k_m1_sink       0.1002 deparm
    ##
    ## Starting values for the transformed parameters actually optimised:
    ##                        value lower upper
    ## parent_0          100.750000  -Inf   Inf
    ## log_k_parent_sink  -2.302585  -Inf   Inf
    ## log_k_parent_m1    -2.301586  -Inf   Inf
    ## log_k_m1_sink      -2.300587  -Inf   Inf
    ##
    ## Fixed parameter values:
    ##      value  type
    ## m1_0     0 state
    ##
    ## Optimised, transformed parameters with symmetric confidence intervals:
    ##                   Estimate Std. Error  Lower   Upper
    ## parent_0            99.600    1.61400 96.330 102.900
    ## log_k_parent_sink   -3.038    0.07826 -3.197  -2.879
    ## log_k_parent_m1     -2.980    0.04124 -3.064  -2.897
    ## log_k_m1_sink       -5.248    0.13610 -5.523  -4.972
    ##
    ## Parameter correlation:
    ##                   parent_0 log_k_parent_sink log_k_parent_m1 log_k_m1_sink
    ## parent_0           1.00000            0.6075        -0.06625       -0.1701
    ## log_k_parent_sink  0.60752            1.0000        -0.08740       -0.6253
    ## log_k_parent_m1   -0.06625           -0.0874         1.00000        0.4716
    ## log_k_m1_sink     -0.17006           -0.6253         0.47163        1.0000
    ##
    ## Residual standard error: 3.211 on 36 degrees of freedom
    ##
    ## 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  61.720 2.024e-38 96.330000 1.029e+02
    ## k_parent_sink  0.047920  12.780 3.050e-15  0.040890 5.616e-02
    ## k_parent_m1    0.050780  24.250 3.407e-24  0.046700 5.521e-02
    ## k_m1_sink      0.005261   7.349 5.758e-09  0.003992 6.933e-03
    ##
    ## Chi2 error levels in percent:
    ##          err.min n.optim df
    ## All data   6.398       4 15
    ## parent     6.827       3  6
    ## m1         4.490       1  9
    ##
    ## Resulting formation fractions:
    ##                 ff
    ## parent_sink 0.4855
    ## parent_m1   0.5145
    ## m1_sink     1.0000
    ##
    ## Estimated disappearance times:
    ##           DT50   DT90
    ## parent   7.023  23.33
    ## m1     131.761 437.70
    ##
    ## Data:
    ##  time variable observed predicted   residual
    ##     0   parent    99.46 9.960e+01 -1.385e-01
    ##     0   parent   102.04 9.960e+01  2.442e+00
    ##     1   parent    93.50 9.024e+01  3.262e+00
    ##     1   parent    92.50 9.024e+01  2.262e+00
    ##     3   parent    63.23 7.407e+01 -1.084e+01
    ##     3   parent    68.99 7.407e+01 -5.083e+00
    ##     7   parent    52.32 4.991e+01  2.408e+00
    ##     7   parent    55.13 4.991e+01  5.218e+00
    ##    14   parent    27.27 2.501e+01  2.257e+00
    ##    14   parent    26.64 2.501e+01  1.627e+00
    ##    21   parent    11.50 1.253e+01 -1.035e+00
    ##    21   parent    11.64 1.253e+01 -8.946e-01
    ##    35   parent     2.85 3.148e+00 -2.979e-01
    ##    35   parent     2.91 3.148e+00 -2.379e-01
    ##    50   parent     0.69 7.162e-01 -2.624e-02
    ##    50   parent     0.63 7.162e-01 -8.624e-02
    ##    75   parent     0.05 6.074e-02 -1.074e-02
    ##    75   parent     0.06 6.074e-02 -7.382e-04
    ##   100   parent       NA 5.151e-03         NA
    ##   100   parent       NA 5.151e-03         NA
    ##   120   parent       NA 7.155e-04         NA
    ##   120   parent       NA 7.155e-04         NA
    ##     0       m1     0.00 0.000e+00  0.000e+00
    ##     0       m1     0.00 0.000e+00  0.000e+00
    ##     1       m1     4.84 4.803e+00  3.704e-02
    ##     1       m1     5.64 4.803e+00  8.370e-01
    ##     3       m1    12.91 1.302e+01 -1.140e-01
    ##     3       m1    12.96 1.302e+01 -6.400e-02
    ##     7       m1    22.97 2.504e+01 -2.075e+00
    ##     7       m1    24.47 2.504e+01 -5.748e-01
    ##    14       m1    41.69 3.669e+01  5.000e+00
    ##    14       m1    33.21 3.669e+01 -3.480e+00
    ##    21       m1    44.37 4.165e+01  2.717e+00
    ##    21       m1    46.44 4.165e+01  4.787e+00
    ##    35       m1    41.22 4.331e+01 -2.093e+00
    ##    35       m1    37.95 4.331e+01 -5.363e+00
    ##    50       m1    41.19 4.122e+01 -2.831e-02
    ##    50       m1    40.01 4.122e+01 -1.208e+00
    ##    75       m1    40.09 3.645e+01  3.643e+00
    ##    75       m1    33.85 3.645e+01 -2.597e+00
    ##   100       m1    31.04 3.198e+01 -9.416e-01
    ##   100       m1    33.13 3.198e+01  1.148e+00
    ##   120       m1    25.15 2.879e+01 -3.640e+00
    ##   120       m1    33.31 2.879e+01  4.520e+00