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authorJohannes Ranke <jranke@uni-bremen.de>2020-05-27 06:06:08 +0200
committerJohannes Ranke <jranke@uni-bremen.de>2020-05-27 06:06:08 +0200
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+<meta property="og:description" content="This function maximises the likelihood of the observed data using the Port
+algorithm stats::nlminb(), and the specified initial or fixed
+parameters and starting values. In each step of the optimisation, the
+kinetic model is solved using the function mkinpredict(), except
+if an analytical solution is implemented, in which case the model is solved
+using the degradation function in the mkinmod object. The
+parameters of the selected error model are fitted simultaneously with the
+degradation model parameters, as both of them are arguments of the
+likelihood function." />
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+ <div class="page-header">
+ <h1>Fit a kinetic model to data with one or more state variables</h1>
+ <small class="dont-index">Source: <a href='http://github.com/jranke/mkin/blob/master/R/mkinfit.R'><code>R/mkinfit.R</code></a></small>
+ <div class="hidden name"><code>mkinfit.Rd</code></div>
+ </div>
+
+ <div class="ref-description">
+ <p>This function maximises the likelihood of the observed data using the Port
+algorithm <code><a href='https://rdrr.io/r/stats/nlminb.html'>stats::nlminb()</a></code>, and the specified initial or fixed
+parameters and starting values. In each step of the optimisation, the
+kinetic model is solved using the function <code><a href='mkinpredict.html'>mkinpredict()</a></code>, except
+if an analytical solution is implemented, in which case the model is solved
+using the degradation function in the <a href='mkinmod.html'>mkinmod</a> object. The
+parameters of the selected error model are fitted simultaneously with the
+degradation model parameters, as both of them are arguments of the
+likelihood function.</p>
+ </div>
+
+ <pre class="usage"><span class='fu'>mkinfit</span>(
+ <span class='no'>mkinmod</span>,
+ <span class='no'>observed</span>,
+ <span class='kw'>parms.ini</span> <span class='kw'>=</span> <span class='st'>"auto"</span>,
+ <span class='kw'>state.ini</span> <span class='kw'>=</span> <span class='st'>"auto"</span>,
+ <span class='kw'>err.ini</span> <span class='kw'>=</span> <span class='st'>"auto"</span>,
+ <span class='kw'>fixed_parms</span> <span class='kw'>=</span> <span class='kw'>NULL</span>,
+ <span class='kw'>fixed_initials</span> <span class='kw'>=</span> <span class='fu'><a href='https://rdrr.io/r/base/names.html'>names</a></span>(<span class='no'>mkinmod</span>$<span class='no'>diffs</span>)[-<span class='fl'>1</span>],
+ <span class='kw'>from_max_mean</span> <span class='kw'>=</span> <span class='fl'>FALSE</span>,
+ <span class='kw'>solution_type</span> <span class='kw'>=</span> <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='st'>"auto"</span>, <span class='st'>"analytical"</span>, <span class='st'>"eigen"</span>, <span class='st'>"deSolve"</span>),
+ <span class='kw'>method.ode</span> <span class='kw'>=</span> <span class='st'>"lsoda"</span>,
+ <span class='kw'>use_compiled</span> <span class='kw'>=</span> <span class='st'>"auto"</span>,
+ <span class='kw'>control</span> <span class='kw'>=</span> <span class='fu'><a href='https://rdrr.io/r/base/list.html'>list</a></span>(<span class='kw'>eval.max</span> <span class='kw'>=</span> <span class='fl'>300</span>, <span class='kw'>iter.max</span> <span class='kw'>=</span> <span class='fl'>200</span>),
+ <span class='kw'>transform_rates</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>,
+ <span class='kw'>transform_fractions</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>,
+ <span class='kw'>quiet</span> <span class='kw'>=</span> <span class='fl'>FALSE</span>,
+ <span class='kw'>atol</span> <span class='kw'>=</span> <span class='fl'>1e-08</span>,
+ <span class='kw'>rtol</span> <span class='kw'>=</span> <span class='fl'>1e-10</span>,
+ <span class='kw'>error_model</span> <span class='kw'>=</span> <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='st'>"const"</span>, <span class='st'>"obs"</span>, <span class='st'>"tc"</span>),
+ <span class='kw'>error_model_algorithm</span> <span class='kw'>=</span> <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='st'>"auto"</span>, <span class='st'>"d_3"</span>, <span class='st'>"direct"</span>, <span class='st'>"twostep"</span>, <span class='st'>"threestep"</span>,
+ <span class='st'>"fourstep"</span>, <span class='st'>"IRLS"</span>, <span class='st'>"OLS"</span>),
+ <span class='kw'>reweight.tol</span> <span class='kw'>=</span> <span class='fl'>1e-08</span>,
+ <span class='kw'>reweight.max.iter</span> <span class='kw'>=</span> <span class='fl'>10</span>,
+ <span class='kw'>trace_parms</span> <span class='kw'>=</span> <span class='fl'>FALSE</span>,
+ <span class='no'>...</span>
+)</pre>
+
+ <h2 class="hasAnchor" id="arguments"><a class="anchor" href="#arguments"></a>Arguments</h2>
+ <table class="ref-arguments">
+ <colgroup><col class="name" /><col class="desc" /></colgroup>
+ <tr>
+ <th>mkinmod</th>
+ <td><p>A list of class <a href='mkinmod.html'>mkinmod</a>, containing the kinetic
+model to be fitted to the data, or one of the shorthand names ("SFO",
+"FOMC", "DFOP", "HS", "SFORB", "IORE"). If a shorthand name is given, a
+parent only degradation model is generated for the variable with the
+highest value in <code>observed</code>.</p></td>
+ </tr>
+ <tr>
+ <th>observed</th>
+ <td><p>A dataframe with the observed data. The first column called
+"name" must contain the name of the observed variable for each data point.
+The second column must contain the times of observation, named "time".
+The third column must be named "value" and contain the observed values.
+Zero values in the "value" column will be removed, with a warning, in
+order to avoid problems with fitting the two-component error model. This
+is not expected to be a problem, because in general, values of zero are
+not observed in degradation data, because there is a lower limit of
+detection.</p></td>
+ </tr>
+ <tr>
+ <th>parms.ini</th>
+ <td><p>A named vector of initial values for the parameters,
+including parameters to be optimised and potentially also fixed parameters
+as indicated by <code>fixed_parms</code>. If set to "auto", initial values for
+rate constants are set to default values. Using parameter names that are
+not in the model gives an error.</p>
+<p>It is possible to only specify a subset of the parameters that the model
+needs. You can use the parameter lists "bparms.ode" from a previously
+fitted model, which contains the differential equation parameters from
+this model. This works nicely if the models are nested. An example is
+given below.</p></td>
+ </tr>
+ <tr>
+ <th>state.ini</th>
+ <td><p>A named vector of initial values for the state variables of
+the model. In case the observed variables are represented by more than one
+model variable, the names will differ from the names of the observed
+variables (see <code>map</code> component of <a href='mkinmod.html'>mkinmod</a>). The default
+is to set the initial value of the first model variable to the mean of the
+time zero values for the variable with the maximum observed value, and all
+others to 0. If this variable has no time zero observations, its initial
+value is set to 100.</p></td>
+ </tr>
+ <tr>
+ <th>err.ini</th>
+ <td><p>A named vector of initial values for the error model
+parameters to be optimised. If set to "auto", initial values are set to
+default values. Otherwise, inital values for all error model parameters
+must be given.</p></td>
+ </tr>
+ <tr>
+ <th>fixed_parms</th>
+ <td><p>The names of parameters that should not be optimised but
+rather kept at the values specified in <code>parms.ini</code>. Alternatively,
+a named numeric vector of parameters to be fixed, regardless of the values
+in parms.ini.</p></td>
+ </tr>
+ <tr>
+ <th>fixed_initials</th>
+ <td><p>The names of model variables for which the initial
+state at time 0 should be excluded from the optimisation. Defaults to all
+state variables except for the first one.</p></td>
+ </tr>
+ <tr>
+ <th>from_max_mean</th>
+ <td><p>If this is set to TRUE, and the model has only one
+observed variable, then data before the time of the maximum observed value
+(after averaging for each sampling time) are discarded, and this time is
+subtracted from all remaining time values, so the time of the maximum
+observed mean value is the new time zero.</p></td>
+ </tr>
+ <tr>
+ <th>solution_type</th>
+ <td><p>If set to "eigen", the solution of the system of
+differential equations is based on the spectral decomposition of the
+coefficient matrix in cases that this is possible. If set to "deSolve", a
+numerical <a href='https://rdrr.io/pkg/deSolve/man/ode.html'>ode solver from package deSolve</a> is used. If
+set to "analytical", an analytical solution of the model is used. This is
+only implemented for relatively simple degradation models. The default is
+"auto", which uses "analytical" if possible, otherwise "deSolve" if a
+compiler is present, and "eigen" if no compiler is present and the model
+can be expressed using eigenvalues and eigenvectors.</p></td>
+ </tr>
+ <tr>
+ <th>method.ode</th>
+ <td><p>The solution method passed via <code><a href='mkinpredict.html'>mkinpredict()</a></code>
+to <code><a href='https://rdrr.io/pkg/deSolve/man/ode.html'>deSolve::ode()</a></code> in case the solution type is "deSolve". The default
+"lsoda" is performant, but sometimes fails to converge.</p></td>
+ </tr>
+ <tr>
+ <th>use_compiled</th>
+ <td><p>If set to <code>FALSE</code>, no compiled version of the
+<a href='mkinmod.html'>mkinmod</a> model is used in the calls to <code><a href='mkinpredict.html'>mkinpredict()</a></code> even if a compiled
+version is present.</p></td>
+ </tr>
+ <tr>
+ <th>control</th>
+ <td><p>A list of control arguments passed to <code><a href='https://rdrr.io/r/stats/nlminb.html'>stats::nlminb()</a></code>.</p></td>
+ </tr>
+ <tr>
+ <th>transform_rates</th>
+ <td><p>Boolean specifying if kinetic rate constants should
+be transformed in the model specification used in the fitting for better
+compliance with the assumption of normal distribution of the estimator. If
+TRUE, also alpha and beta parameters of the FOMC model are
+log-transformed, as well as k1 and k2 rate constants for the DFOP and HS
+models and the break point tb of the HS model. If FALSE, zero is used as
+a lower bound for the rates in the optimisation.</p></td>
+ </tr>
+ <tr>
+ <th>transform_fractions</th>
+ <td><p>Boolean specifying if formation fractions
+constants should be transformed in the model specification used in the
+fitting for better compliance with the assumption of normal distribution
+of the estimator. The default (TRUE) is to do transformations. If TRUE,
+the g parameter of the DFOP and HS models are also transformed, as they
+can also be seen as compositional data. The transformation used for these
+transformations is the <code><a href='ilr.html'>ilr()</a></code> transformation.</p></td>
+ </tr>
+ <tr>
+ <th>quiet</th>
+ <td><p>Suppress printing out the current value of the negative
+log-likelihood after each improvement?</p></td>
+ </tr>
+ <tr>
+ <th>atol</th>
+ <td><p>Absolute error tolerance, passed to <code><a href='https://rdrr.io/pkg/deSolve/man/ode.html'>deSolve::ode()</a></code>. Default
+is 1e-8, which is lower than the default in the <code><a href='https://rdrr.io/pkg/deSolve/man/lsoda.html'>deSolve::lsoda()</a></code>
+function which is used per default.</p></td>
+ </tr>
+ <tr>
+ <th>rtol</th>
+ <td><p>Absolute error tolerance, passed to <code><a href='https://rdrr.io/pkg/deSolve/man/ode.html'>deSolve::ode()</a></code>. Default
+is 1e-10, much lower than in <code><a href='https://rdrr.io/pkg/deSolve/man/lsoda.html'>deSolve::lsoda()</a></code>.</p></td>
+ </tr>
+ <tr>
+ <th>error_model</th>
+ <td><p>If the error model is "const", a constant standard
+deviation is assumed.</p>
+<p>If the error model is "obs", each observed variable is assumed to have its
+own variance.</p>
+<p>If the error model is "tc" (two-component error model), a two component
+error model similar to the one described by Rocke and Lorenzato (1995) is
+used for setting up the likelihood function. Note that this model
+deviates from the model by Rocke and Lorenzato, as their model implies
+that the errors follow a lognormal distribution for large values, not a
+normal distribution as assumed by this method.</p></td>
+ </tr>
+ <tr>
+ <th>error_model_algorithm</th>
+ <td><p>If "auto", the selected algorithm depends on
+the error model. If the error model is "const", unweighted nonlinear
+least squares fitting ("OLS") is selected. If the error model is "obs", or
+"tc", the "d_3" algorithm is selected.</p>
+<p>The algorithm "d_3" will directly minimize the negative log-likelihood
+and independently also use the three step algorithm described below.
+The fit with the higher likelihood is returned.</p>
+<p>The algorithm "direct" will directly minimize the negative log-likelihood.</p>
+<p>The algorithm "twostep" will minimize the negative log-likelihood after an
+initial unweighted least squares optimisation step.</p>
+<p>The algorithm "threestep" starts with unweighted least squares, then
+optimizes only the error model using the degradation model parameters
+found, and then minimizes the negative log-likelihood with free
+degradation and error model parameters.</p>
+<p>The algorithm "fourstep" starts with unweighted least squares, then
+optimizes only the error model using the degradation model parameters
+found, then optimizes the degradation model again with fixed error model
+parameters, and finally minimizes the negative log-likelihood with free
+degradation and error model parameters.</p>
+<p>The algorithm "IRLS" (Iteratively Reweighted Least Squares) starts with
+unweighted least squares, and then iterates optimization of the error
+model parameters and subsequent optimization of the degradation model
+using those error model parameters, until the error model parameters
+converge.</p></td>
+ </tr>
+ <tr>
+ <th>reweight.tol</th>
+ <td><p>Tolerance for the convergence criterion calculated from
+the error model parameters in IRLS fits.</p></td>
+ </tr>
+ <tr>
+ <th>reweight.max.iter</th>
+ <td><p>Maximum number of iterations in IRLS fits.</p></td>
+ </tr>
+ <tr>
+ <th>trace_parms</th>
+ <td><p>Should a trace of the parameter values be listed?</p></td>
+ </tr>
+ <tr>
+ <th>...</th>
+ <td><p>Further arguments that will be passed on to
+<code><a href='https://rdrr.io/pkg/deSolve/man/ode.html'>deSolve::ode()</a></code>.</p></td>
+ </tr>
+ </table>
+
+ <h2 class="hasAnchor" id="value"><a class="anchor" href="#value"></a>Value</h2>
+
+ <p>A list with "mkinfit" in the class attribute.</p>
+ <h2 class="hasAnchor" id="details"><a class="anchor" href="#details"></a>Details</h2>
+
+ <p>Per default, parameters in the kinetic models are internally transformed in
+order to better satisfy the assumption of a normal distribution of their
+estimators.</p>
+ <h2 class="hasAnchor" id="note"><a class="anchor" href="#note"></a>Note</h2>
+
+ <p>When using the "IORE" submodel for metabolites, fitting with
+"transform_rates = TRUE" (the default) often leads to failures of the
+numerical ODE solver. In this situation it may help to switch off the
+internal rate transformation.</p>
+ <h2 class="hasAnchor" id="references"><a class="anchor" href="#references"></a>References</h2>
+
+ <p>Rocke DM and Lorenzato S (1995) A two-component model
+for measurement error in analytical chemistry. <em>Technometrics</em> 37(2), 176-184.</p>
+<p>Ranke J and Meinecke S (2019) Error Models for the Kinetic Evaluation of Chemical
+Degradation Data. <em>Environments</em> 6(12) 124
+<a href='https://doi.org/10.3390/environments6120124'>doi:10.3390/environments6120124</a>.</p>
+ <h2 class="hasAnchor" id="see-also"><a class="anchor" href="#see-also"></a>See also</h2>
+
+ <div class='dont-index'><p><a href='summary.mkinfit.html'>summary.mkinfit</a>, <a href='plot.mkinfit.html'>plot.mkinfit</a>, <a href='parms.html'>parms</a> and <a href='https://rdrr.io/pkg/lmtest/man/lrtest.html'>lrtest</a>.</p>
+<p>Comparisons of models fitted to the same data can be made using
+<code><a href='https://rdrr.io/r/stats/AIC.html'>AIC</a></code> by virtue of the method <code><a href='logLik.mkinfit.html'>logLik.mkinfit</a></code>.</p>
+<p>Fitting of several models to several datasets in a single call to
+<code><a href='mmkin.html'>mmkin</a></code>.</p></div>
+
+ <h2 class="hasAnchor" id="examples"><a class="anchor" href="#examples"></a>Examples</h2>
+ <pre class="examples"><div class='input'>
+<span class='co'># Use shorthand notation for parent only degradation</span>
+<span class='no'>fit</span> <span class='kw'>&lt;-</span> <span class='fu'>mkinfit</span>(<span class='st'>"FOMC"</span>, <span class='no'>FOCUS_2006_C</span>, <span class='kw'>quiet</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>)
+<span class='fu'><a href='https://rdrr.io/r/base/summary.html'>summary</a></span>(<span class='no'>fit</span>)</div><div class='output co'>#&gt; mkin version used for fitting: 0.9.50.3
+#&gt; R version used for fitting: 4.0.0
+#&gt; Date of fit: Wed May 27 05:54:13 2020
+#&gt; Date of summary: Wed May 27 05:54:13 2020
+#&gt;
+#&gt; Equations:
+#&gt; d_parent/dt = - (alpha/beta) * 1/((time/beta) + 1) * parent
+#&gt;
+#&gt; Model predictions using solution type analytical
+#&gt;
+#&gt; Fitted using 222 model solutions performed in 0.043 s
+#&gt;
+#&gt; Error model: Constant variance
+#&gt;
+#&gt; Error model algorithm: OLS
+#&gt;
+#&gt; Starting values for parameters to be optimised:
+#&gt; value type
+#&gt; parent_0 85.1 state
+#&gt; alpha 1.0 deparm
+#&gt; beta 10.0 deparm
+#&gt;
+#&gt; Starting values for the transformed parameters actually optimised:
+#&gt; value lower upper
+#&gt; parent_0 85.100000 -Inf Inf
+#&gt; log_alpha 0.000000 -Inf Inf
+#&gt; log_beta 2.302585 -Inf Inf
+#&gt;
+#&gt; Fixed parameter values:
+#&gt; None
+#&gt;
+#&gt; Results:
+#&gt;
+#&gt; AIC BIC logLik
+#&gt; 44.68652 45.47542 -18.34326
+#&gt;
+#&gt; Optimised, transformed parameters with symmetric confidence intervals:
+#&gt; Estimate Std. Error Lower Upper
+#&gt; parent_0 85.87000 1.8070 81.23000 90.5200
+#&gt; log_alpha 0.05192 0.1353 -0.29580 0.3996
+#&gt; log_beta 0.65100 0.2287 0.06315 1.2390
+#&gt; sigma 1.85700 0.4378 0.73200 2.9830
+#&gt;
+#&gt; Parameter correlation:
+#&gt; parent_0 log_alpha log_beta sigma
+#&gt; parent_0 1.000e+00 -1.565e-01 -3.142e-01 4.770e-08
+#&gt; log_alpha -1.565e-01 1.000e+00 9.564e-01 9.974e-08
+#&gt; log_beta -3.142e-01 9.564e-01 1.000e+00 8.468e-08
+#&gt; sigma 4.770e-08 9.974e-08 8.468e-08 1.000e+00
+#&gt;
+#&gt; Backtransformed parameters:
+#&gt; Confidence intervals for internally transformed parameters are asymmetric.
+#&gt; t-test (unrealistically) based on the assumption of normal distribution
+#&gt; for estimators of untransformed parameters.
+#&gt; Estimate t value Pr(&gt;t) Lower Upper
+#&gt; parent_0 85.870 47.530 3.893e-08 81.2300 90.520
+#&gt; alpha 1.053 7.393 3.562e-04 0.7439 1.491
+#&gt; beta 1.917 4.373 3.601e-03 1.0650 3.451
+#&gt; sigma 1.857 4.243 4.074e-03 0.7320 2.983
+#&gt;
+#&gt; FOCUS Chi2 error levels in percent:
+#&gt; err.min n.optim df
+#&gt; All data 6.657 3 6
+#&gt; parent 6.657 3 6
+#&gt;
+#&gt; Estimated disappearance times:
+#&gt; DT50 DT90 DT50back
+#&gt; parent 1.785 15.15 4.56
+#&gt;
+#&gt; Data:
+#&gt; time variable observed predicted residual
+#&gt; 0 parent 85.1 85.875 -0.7749
+#&gt; 1 parent 57.9 55.191 2.7091
+#&gt; 3 parent 29.9 31.845 -1.9452
+#&gt; 7 parent 14.6 17.012 -2.4124
+#&gt; 14 parent 9.7 9.241 0.4590
+#&gt; 28 parent 6.6 4.754 1.8460
+#&gt; 63 parent 4.0 2.102 1.8977
+#&gt; 91 parent 3.9 1.441 2.4590
+#&gt; 119 parent 0.6 1.092 -0.4919</div><div class='input'>
+<span class='co'># One parent compound, one metabolite, both single first order.</span>
+<span class='co'># Use mkinsub for convenience in model formulation. Pathway to sink included per default.</span>
+<span class='no'>SFO_SFO</span> <span class='kw'>&lt;-</span> <span class='fu'><a href='mkinmod.html'>mkinmod</a></span>(
+ <span class='kw'>parent</span> <span class='kw'>=</span> <span class='fu'><a href='mkinsub.html'>mkinsub</a></span>(<span class='st'>"SFO"</span>, <span class='st'>"m1"</span>),
+ <span class='kw'>m1</span> <span class='kw'>=</span> <span class='fu'><a href='mkinsub.html'>mkinsub</a></span>(<span class='st'>"SFO"</span>))</div><div class='output co'>#&gt; <span class='message'>Successfully compiled differential equation model from auto-generated C code.</span></div><div class='input'><span class='co'># Fit the model to the FOCUS example dataset D using defaults</span>
+<span class='fu'><a href='https://rdrr.io/r/base/print.html'>print</a></span>(<span class='fu'><a href='https://rdrr.io/r/base/system.time.html'>system.time</a></span>(<span class='no'>fit</span> <span class='kw'>&lt;-</span> <span class='fu'>mkinfit</span>(<span class='no'>SFO_SFO</span>, <span class='no'>FOCUS_2006_D</span>,
+ <span class='kw'>solution_type</span> <span class='kw'>=</span> <span class='st'>"eigen"</span>, <span class='kw'>quiet</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>)))</div><div class='output co'>#&gt; <span class='warning'>Warning: Observations with value of zero were removed from the data</span></div><div class='output co'>#&gt; user system elapsed
+#&gt; 0.414 0.000 0.418 </div><div class='input'><span class='fu'><a href='parms.html'>parms</a></span>(<span class='no'>fit</span>)</div><div class='output co'>#&gt; parent_0 k_parent k_m1 f_parent_to_m1 sigma
+#&gt; 99.598481046 0.098697740 0.005260651 0.514475962 3.125503875 </div><div class='input'><span class='fu'><a href='endpoints.html'>endpoints</a></span>(<span class='no'>fit</span>)</div><div class='output co'>#&gt; $ff
+#&gt; parent_m1 parent_sink
+#&gt; 0.514476 0.485524
+#&gt;
+#&gt; $distimes
+#&gt; DT50 DT90
+#&gt; parent 7.022929 23.32966
+#&gt; m1 131.760724 437.69965
+#&gt; </div><div class='input'><span class='co'># \dontrun{</span>
+<span class='co'># deSolve is slower when no C compiler (gcc) was available during model generation</span>
+<span class='fu'><a href='https://rdrr.io/r/base/print.html'>print</a></span>(<span class='fu'><a href='https://rdrr.io/r/base/system.time.html'>system.time</a></span>(<span class='no'>fit.deSolve</span> <span class='kw'>&lt;-</span> <span class='fu'>mkinfit</span>(<span class='no'>SFO_SFO</span>, <span class='no'>FOCUS_2006_D</span>,
+ <span class='kw'>solution_type</span> <span class='kw'>=</span> <span class='st'>"deSolve"</span>)))</div><div class='output co'>#&gt; <span class='warning'>Warning: Observations with value of zero were removed from the data</span></div><div class='output co'>#&gt; <span class='message'>Ordinary least squares optimisation</span></div><div class='output co'>#&gt; Sum of squared residuals at call 1: 15156.12
+#&gt; Sum of squared residuals at call 2: 15156.12
+#&gt; Sum of squared residuals at call 6: 8243.645
+#&gt; Sum of squared residuals at call 12: 6290.712
+#&gt; Sum of squared residuals at call 13: 6290.683
+#&gt; Sum of squared residuals at call 15: 6290.452
+#&gt; Sum of squared residuals at call 18: 1700.749
+#&gt; Sum of squared residuals at call 20: 1700.611
+#&gt; Sum of squared residuals at call 24: 1190.923
+#&gt; Sum of squared residuals at call 26: 1190.922
+#&gt; Sum of squared residuals at call 29: 1017.417
+#&gt; Sum of squared residuals at call 31: 1017.417
+#&gt; Sum of squared residuals at call 33: 1017.416
+#&gt; Sum of squared residuals at call 34: 644.0472
+#&gt; Sum of squared residuals at call 36: 644.047
+#&gt; Sum of squared residuals at call 38: 644.047
+#&gt; Sum of squared residuals at call 39: 590.5025
+#&gt; Sum of squared residuals at call 41: 590.5022
+#&gt; Sum of squared residuals at call 43: 590.5016
+#&gt; Sum of squared residuals at call 44: 543.2196
+#&gt; Sum of squared residuals at call 45: 543.2193
+#&gt; Sum of squared residuals at call 46: 543.2192
+#&gt; Sum of squared residuals at call 50: 391.348
+#&gt; Sum of squared residuals at call 51: 391.3479
+#&gt; Sum of squared residuals at call 56: 386.479
+#&gt; Sum of squared residuals at call 58: 386.479
+#&gt; Sum of squared residuals at call 60: 386.4779
+#&gt; Sum of squared residuals at call 61: 384.0686
+#&gt; Sum of squared residuals at call 63: 384.0686
+#&gt; Sum of squared residuals at call 66: 382.7813
+#&gt; Sum of squared residuals at call 68: 382.7813
+#&gt; Sum of squared residuals at call 70: 382.7813
+#&gt; Sum of squared residuals at call 71: 378.9273
+#&gt; Sum of squared residuals at call 73: 378.9273
+#&gt; Sum of squared residuals at call 75: 378.9272
+#&gt; Sum of squared residuals at call 76: 377.4847
+#&gt; Sum of squared residuals at call 78: 377.4846
+#&gt; Sum of squared residuals at call 81: 375.9738
+#&gt; Sum of squared residuals at call 83: 375.9738
+#&gt; Sum of squared residuals at call 86: 375.3387
+#&gt; Sum of squared residuals at call 88: 375.3387
+#&gt; Sum of squared residuals at call 91: 374.5774
+#&gt; Sum of squared residuals at call 93: 374.5774
+#&gt; Sum of squared residuals at call 95: 374.5774
+#&gt; Sum of squared residuals at call 96: 373.5438
+#&gt; Sum of squared residuals at call 100: 373.5438
+#&gt; Sum of squared residuals at call 102: 373.265
+#&gt; Sum of squared residuals at call 104: 373.265
+#&gt; Sum of squared residuals at call 107: 372.6825
+#&gt; Sum of squared residuals at call 111: 372.6825
+#&gt; Sum of squared residuals at call 114: 372.6356
+#&gt; Sum of squared residuals at call 116: 372.6356
+#&gt; Sum of squared residuals at call 119: 372.6199
+#&gt; Sum of squared residuals at call 121: 372.6199
+#&gt; Sum of squared residuals at call 123: 372.6199
+#&gt; Sum of squared residuals at call 124: 372.5881
+#&gt; Sum of squared residuals at call 126: 372.5881
+#&gt; Sum of squared residuals at call 129: 372.5418
+#&gt; Sum of squared residuals at call 130: 372.4866
+#&gt; Sum of squared residuals at call 131: 372.2242
+#&gt; Sum of squared residuals at call 132: 371.5237
+#&gt; Sum of squared residuals at call 134: 371.5237
+#&gt; Sum of squared residuals at call 137: 371.292
+#&gt; Sum of squared residuals at call 139: 371.292
+#&gt; Sum of squared residuals at call 143: 371.2256
+#&gt; Sum of squared residuals at call 144: 371.2256
+#&gt; Sum of squared residuals at call 146: 371.2256
+#&gt; Sum of squared residuals at call 149: 371.2194
+#&gt; Sum of squared residuals at call 150: 371.2147
+#&gt; Sum of squared residuals at call 153: 371.2147
+#&gt; Sum of squared residuals at call 155: 371.2137
+#&gt; Sum of squared residuals at call 156: 371.2137
+#&gt; Sum of squared residuals at call 157: 371.2137
+#&gt; Sum of squared residuals at call 160: 371.2134
+#&gt; Sum of squared residuals at call 164: 371.2134
+#&gt; Sum of squared residuals at call 165: 371.2134
+#&gt; Sum of squared residuals at call 167: 371.2134
+#&gt; Negative log-likelihood at call 177: 97.22429</div><div class='output co'>#&gt; <span class='message'>Optimisation successfully terminated.</span></div><div class='output co'>#&gt; user system elapsed
+#&gt; 0.371 0.001 0.370 </div><div class='input'><span class='fu'><a href='parms.html'>parms</a></span>(<span class='no'>fit.deSolve</span>)</div><div class='output co'>#&gt; parent_0 k_parent k_m1 f_parent_to_m1 sigma
+#&gt; 99.598480300 0.098697739 0.005260651 0.514475968 3.125503874 </div><div class='input'><span class='fu'><a href='endpoints.html'>endpoints</a></span>(<span class='no'>fit.deSolve</span>)</div><div class='output co'>#&gt; $ff
+#&gt; parent_m1 parent_sink
+#&gt; 0.514476 0.485524
+#&gt;
+#&gt; $distimes
+#&gt; DT50 DT90
+#&gt; parent 7.022929 23.32966
+#&gt; m1 131.760721 437.69964
+#&gt; </div><div class='input'><span class='co'># }</span>
+
+<span class='co'># Use stepwise fitting, using optimised parameters from parent only fit, FOMC</span>
+<span class='co'># \dontrun{</span>
+<span class='no'>FOMC_SFO</span> <span class='kw'>&lt;-</span> <span class='fu'><a href='mkinmod.html'>mkinmod</a></span>(
+ <span class='kw'>parent</span> <span class='kw'>=</span> <span class='fu'><a href='mkinsub.html'>mkinsub</a></span>(<span class='st'>"FOMC"</span>, <span class='st'>"m1"</span>),
+ <span class='kw'>m1</span> <span class='kw'>=</span> <span class='fu'><a href='mkinsub.html'>mkinsub</a></span>(<span class='st'>"SFO"</span>))</div><div class='output co'>#&gt; <span class='message'>Successfully compiled differential equation model from auto-generated C code.</span></div><div class='input'><span class='co'># Fit the model to the FOCUS example dataset D using defaults</span>
+<span class='no'>fit.FOMC_SFO</span> <span class='kw'>&lt;-</span> <span class='fu'>mkinfit</span>(<span class='no'>FOMC_SFO</span>, <span class='no'>FOCUS_2006_D</span>, <span class='kw'>quiet</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>)</div><div class='output co'>#&gt; <span class='warning'>Warning: Observations with value of zero were removed from the data</span></div><div class='input'><span class='co'># Use starting parameters from parent only FOMC fit</span>
+<span class='no'>fit.FOMC</span> <span class='kw'>=</span> <span class='fu'>mkinfit</span>(<span class='st'>"FOMC"</span>, <span class='no'>FOCUS_2006_D</span>, <span class='kw'>quiet</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>)
+<span class='no'>fit.FOMC_SFO</span> <span class='kw'>&lt;-</span> <span class='fu'>mkinfit</span>(<span class='no'>FOMC_SFO</span>, <span class='no'>FOCUS_2006_D</span>, <span class='kw'>quiet</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>,
+ <span class='kw'>parms.ini</span> <span class='kw'>=</span> <span class='no'>fit.FOMC</span>$<span class='no'>bparms.ode</span>)</div><div class='output co'>#&gt; <span class='warning'>Warning: Observations with value of zero were removed from the data</span></div><div class='input'>
+<span class='co'># Use stepwise fitting, using optimised parameters from parent only fit, SFORB</span>
+<span class='no'>SFORB_SFO</span> <span class='kw'>&lt;-</span> <span class='fu'><a href='mkinmod.html'>mkinmod</a></span>(
+ <span class='kw'>parent</span> <span class='kw'>=</span> <span class='fu'><a href='https://rdrr.io/r/base/list.html'>list</a></span>(<span class='kw'>type</span> <span class='kw'>=</span> <span class='st'>"SFORB"</span>, <span class='kw'>to</span> <span class='kw'>=</span> <span class='st'>"m1"</span>, <span class='kw'>sink</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>),
+ <span class='kw'>m1</span> <span class='kw'>=</span> <span class='fu'><a href='https://rdrr.io/r/base/list.html'>list</a></span>(<span class='kw'>type</span> <span class='kw'>=</span> <span class='st'>"SFO"</span>))</div><div class='output co'>#&gt; <span class='message'>Successfully compiled differential equation model from auto-generated C code.</span></div><div class='input'><span class='co'># Fit the model to the FOCUS example dataset D using defaults</span>
+<span class='no'>fit.SFORB_SFO</span> <span class='kw'>&lt;-</span> <span class='fu'>mkinfit</span>(<span class='no'>SFORB_SFO</span>, <span class='no'>FOCUS_2006_D</span>, <span class='kw'>quiet</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>)</div><div class='output co'>#&gt; <span class='warning'>Warning: Observations with value of zero were removed from the data</span></div><div class='input'><span class='no'>fit.SFORB_SFO.deSolve</span> <span class='kw'>&lt;-</span> <span class='fu'>mkinfit</span>(<span class='no'>SFORB_SFO</span>, <span class='no'>FOCUS_2006_D</span>, <span class='kw'>solution_type</span> <span class='kw'>=</span> <span class='st'>"deSolve"</span>,
+ <span class='kw'>quiet</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>)</div><div class='output co'>#&gt; <span class='warning'>Warning: Observations with value of zero were removed from the data</span></div><div class='input'><span class='co'># Use starting parameters from parent only SFORB fit (not really needed in this case)</span>
+<span class='no'>fit.SFORB</span> <span class='kw'>=</span> <span class='fu'>mkinfit</span>(<span class='st'>"SFORB"</span>, <span class='no'>FOCUS_2006_D</span>, <span class='kw'>quiet</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>)
+<span class='no'>fit.SFORB_SFO</span> <span class='kw'>&lt;-</span> <span class='fu'>mkinfit</span>(<span class='no'>SFORB_SFO</span>, <span class='no'>FOCUS_2006_D</span>, <span class='kw'>parms.ini</span> <span class='kw'>=</span> <span class='no'>fit.SFORB</span>$<span class='no'>bparms.ode</span>, <span class='kw'>quiet</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>)</div><div class='output co'>#&gt; <span class='warning'>Warning: Observations with value of zero were removed from the data</span></div><div class='output co'>#&gt; <span class='warning'>Warning: Initial parameter(s) k_parent_free_sink not used in the model</span></div><div class='input'><span class='co'># }</span>
+
+<span class='co'># \dontrun{</span>
+<span class='co'># Weighted fits, including IRLS (error_model = "obs")</span>
+<span class='no'>SFO_SFO.ff</span> <span class='kw'>&lt;-</span> <span class='fu'><a href='mkinmod.html'>mkinmod</a></span>(<span class='kw'>parent</span> <span class='kw'>=</span> <span class='fu'><a href='mkinsub.html'>mkinsub</a></span>(<span class='st'>"SFO"</span>, <span class='st'>"m1"</span>),
+ <span class='kw'>m1</span> <span class='kw'>=</span> <span class='fu'><a href='mkinsub.html'>mkinsub</a></span>(<span class='st'>"SFO"</span>), <span class='kw'>use_of_ff</span> <span class='kw'>=</span> <span class='st'>"max"</span>)</div><div class='output co'>#&gt; <span class='message'>Successfully compiled differential equation model from auto-generated C code.</span></div><div class='input'><span class='no'>f.noweight</span> <span class='kw'>&lt;-</span> <span class='fu'>mkinfit</span>(<span class='no'>SFO_SFO.ff</span>, <span class='no'>FOCUS_2006_D</span>, <span class='kw'>quiet</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>)</div><div class='output co'>#&gt; <span class='warning'>Warning: Observations with value of zero were removed from the data</span></div><div class='input'><span class='fu'><a href='https://rdrr.io/r/base/summary.html'>summary</a></span>(<span class='no'>f.noweight</span>)</div><div class='output co'>#&gt; mkin version used for fitting: 0.9.50.3
+#&gt; R version used for fitting: 4.0.0
+#&gt; Date of fit: Wed May 27 05:54:19 2020
+#&gt; Date of summary: Wed May 27 05:54:19 2020
+#&gt;
+#&gt; Equations:
+#&gt; d_parent/dt = - k_parent * parent
+#&gt; d_m1/dt = + f_parent_to_m1 * k_parent * parent - k_m1 * m1
+#&gt;
+#&gt; Model predictions using solution type analytical
+#&gt;
+#&gt; Fitted using 421 model solutions performed in 0.126 s
+#&gt;
+#&gt; Error model: Constant variance
+#&gt;
+#&gt; Error model algorithm: OLS
+#&gt;
+#&gt; Starting values for parameters to be optimised:
+#&gt; value type
+#&gt; parent_0 100.7500 state
+#&gt; k_parent 0.1000 deparm
+#&gt; k_m1 0.1001 deparm
+#&gt; f_parent_to_m1 0.5000 deparm
+#&gt;
+#&gt; Starting values for the transformed parameters actually optimised:
+#&gt; value lower upper
+#&gt; parent_0 100.750000 -Inf Inf
+#&gt; log_k_parent -2.302585 -Inf Inf
+#&gt; log_k_m1 -2.301586 -Inf Inf
+#&gt; f_parent_ilr_1 0.000000 -Inf Inf
+#&gt;
+#&gt; Fixed parameter values:
+#&gt; value type
+#&gt; m1_0 0 state
+#&gt;
+#&gt; Results:
+#&gt;
+#&gt; AIC BIC logLik
+#&gt; 204.4486 212.6365 -97.22429
+#&gt;
+#&gt; Optimised, transformed parameters with symmetric confidence intervals:
+#&gt; Estimate Std. Error Lower Upper
+#&gt; parent_0 99.60000 1.57000 96.40000 102.8000
+#&gt; log_k_parent -2.31600 0.04087 -2.39900 -2.2330
+#&gt; log_k_m1 -5.24800 0.13320 -5.51800 -4.9770
+#&gt; f_parent_ilr_1 0.04096 0.06312 -0.08746 0.1694
+#&gt; sigma 3.12600 0.35850 2.39600 3.8550
+#&gt;
+#&gt; Parameter correlation:
+#&gt; parent_0 log_k_parent log_k_m1 f_parent_ilr_1 sigma
+#&gt; parent_0 1.000e+00 5.174e-01 -1.688e-01 -5.471e-01 -3.190e-07
+#&gt; log_k_parent 5.174e-01 1.000e+00 -3.263e-01 -5.426e-01 3.168e-07
+#&gt; log_k_m1 -1.688e-01 -3.263e-01 1.000e+00 7.478e-01 -1.406e-07
+#&gt; f_parent_ilr_1 -5.471e-01 -5.426e-01 7.478e-01 1.000e+00 -1.587e-10
+#&gt; sigma -3.190e-07 3.168e-07 -1.406e-07 -1.587e-10 1.000e+00
+#&gt;
+#&gt; Backtransformed parameters:
+#&gt; Confidence intervals for internally transformed parameters are asymmetric.
+#&gt; t-test (unrealistically) based on the assumption of normal distribution
+#&gt; for estimators of untransformed parameters.
+#&gt; Estimate t value Pr(&gt;t) Lower Upper
+#&gt; parent_0 99.600000 63.430 2.298e-36 96.400000 1.028e+02
+#&gt; k_parent 0.098700 24.470 4.955e-23 0.090820 1.073e-01
+#&gt; k_m1 0.005261 7.510 6.165e-09 0.004012 6.898e-03
+#&gt; f_parent_to_m1 0.514500 23.070 3.104e-22 0.469100 5.596e-01
+#&gt; sigma 3.126000 8.718 2.235e-10 2.396000 3.855e+00
+#&gt;
+#&gt; FOCUS Chi2 error levels in percent:
+#&gt; err.min n.optim df
+#&gt; All data 6.398 4 15
+#&gt; parent 6.459 2 7
+#&gt; m1 4.690 2 8
+#&gt;
+#&gt; Resulting formation fractions:
+#&gt; ff
+#&gt; parent_m1 0.5145
+#&gt; parent_sink 0.4855
+#&gt;
+#&gt; Estimated disappearance times:
+#&gt; DT50 DT90
+#&gt; parent 7.023 23.33
+#&gt; m1 131.761 437.70
+#&gt;
+#&gt; Data:
+#&gt; time variable observed predicted residual
+#&gt; 0 parent 99.46 99.59848 -1.385e-01
+#&gt; 0 parent 102.04 99.59848 2.442e+00
+#&gt; 1 parent 93.50 90.23787 3.262e+00
+#&gt; 1 parent 92.50 90.23787 2.262e+00
+#&gt; 3 parent 63.23 74.07319 -1.084e+01
+#&gt; 3 parent 68.99 74.07319 -5.083e+00
+#&gt; 7 parent 52.32 49.91206 2.408e+00
+#&gt; 7 parent 55.13 49.91206 5.218e+00
+#&gt; 14 parent 27.27 25.01257 2.257e+00
+#&gt; 14 parent 26.64 25.01257 1.627e+00
+#&gt; 21 parent 11.50 12.53462 -1.035e+00
+#&gt; 21 parent 11.64 12.53462 -8.946e-01
+#&gt; 35 parent 2.85 3.14787 -2.979e-01
+#&gt; 35 parent 2.91 3.14787 -2.379e-01
+#&gt; 50 parent 0.69 0.71624 -2.624e-02
+#&gt; 50 parent 0.63 0.71624 -8.624e-02
+#&gt; 75 parent 0.05 0.06074 -1.074e-02
+#&gt; 75 parent 0.06 0.06074 -7.381e-04
+#&gt; 1 m1 4.84 4.80296 3.704e-02
+#&gt; 1 m1 5.64 4.80296 8.370e-01
+#&gt; 3 m1 12.91 13.02400 -1.140e-01
+#&gt; 3 m1 12.96 13.02400 -6.400e-02
+#&gt; 7 m1 22.97 25.04476 -2.075e+00
+#&gt; 7 m1 24.47 25.04476 -5.748e-01
+#&gt; 14 m1 41.69 36.69002 5.000e+00
+#&gt; 14 m1 33.21 36.69002 -3.480e+00
+#&gt; 21 m1 44.37 41.65310 2.717e+00
+#&gt; 21 m1 46.44 41.65310 4.787e+00
+#&gt; 35 m1 41.22 43.31312 -2.093e+00
+#&gt; 35 m1 37.95 43.31312 -5.363e+00
+#&gt; 50 m1 41.19 41.21831 -2.831e-02
+#&gt; 50 m1 40.01 41.21831 -1.208e+00
+#&gt; 75 m1 40.09 36.44703 3.643e+00
+#&gt; 75 m1 33.85 36.44703 -2.597e+00
+#&gt; 100 m1 31.04 31.98163 -9.416e-01
+#&gt; 100 m1 33.13 31.98163 1.148e+00
+#&gt; 120 m1 25.15 28.78984 -3.640e+00
+#&gt; 120 m1 33.31 28.78984 4.520e+00</div><div class='input'><span class='no'>f.obs</span> <span class='kw'>&lt;-</span> <span class='fu'>mkinfit</span>(<span class='no'>SFO_SFO.ff</span>, <span class='no'>FOCUS_2006_D</span>, <span class='kw'>error_model</span> <span class='kw'>=</span> <span class='st'>"obs"</span>, <span class='kw'>quiet</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>)</div><div class='output co'>#&gt; <span class='warning'>Warning: Observations with value of zero were removed from the data</span></div><div class='input'><span class='fu'><a href='https://rdrr.io/r/base/summary.html'>summary</a></span>(<span class='no'>f.obs</span>)</div><div class='output co'>#&gt; mkin version used for fitting: 0.9.50.3
+#&gt; R version used for fitting: 4.0.0
+#&gt; Date of fit: Wed May 27 05:54:19 2020
+#&gt; Date of summary: Wed May 27 05:54:19 2020
+#&gt;
+#&gt; Equations:
+#&gt; d_parent/dt = - k_parent * parent
+#&gt; d_m1/dt = + f_parent_to_m1 * k_parent * parent - k_m1 * m1
+#&gt;
+#&gt; Model predictions using solution type analytical
+#&gt;
+#&gt; Fitted using 978 model solutions performed in 0.33 s
+#&gt;
+#&gt; Error model: Variance unique to each observed variable
+#&gt;
+#&gt; Error model algorithm: d_3
+#&gt; Direct fitting and three-step fitting yield approximately the same likelihood
+#&gt;
+#&gt; Starting values for parameters to be optimised:
+#&gt; value type
+#&gt; parent_0 100.7500 state
+#&gt; k_parent 0.1000 deparm
+#&gt; k_m1 0.1001 deparm
+#&gt; f_parent_to_m1 0.5000 deparm
+#&gt; sigma_parent 3.0000 error
+#&gt; sigma_m1 3.0000 error
+#&gt;
+#&gt; Starting values for the transformed parameters actually optimised:
+#&gt; value lower upper
+#&gt; parent_0 100.750000 -Inf Inf
+#&gt; log_k_parent -2.302585 -Inf Inf
+#&gt; log_k_m1 -2.301586 -Inf Inf
+#&gt; f_parent_ilr_1 0.000000 -Inf Inf
+#&gt; sigma_parent 3.000000 0 Inf
+#&gt; sigma_m1 3.000000 0 Inf
+#&gt;
+#&gt; Fixed parameter values:
+#&gt; value type
+#&gt; m1_0 0 state
+#&gt;
+#&gt; Results:
+#&gt;
+#&gt; AIC BIC logLik
+#&gt; 205.8727 215.6982 -96.93634
+#&gt;
+#&gt; Optimised, transformed parameters with symmetric confidence intervals:
+#&gt; Estimate Std. Error Lower Upper
+#&gt; parent_0 99.65000 1.70200 96.19000 103.1000
+#&gt; log_k_parent -2.31300 0.04376 -2.40200 -2.2240
+#&gt; log_k_m1 -5.25000 0.12430 -5.50400 -4.9970
+#&gt; f_parent_ilr_1 0.03861 0.06171 -0.08708 0.1643
+#&gt; sigma_parent 3.40100 0.56820 2.24400 4.5590
+#&gt; sigma_m1 2.85500 0.45240 1.93400 3.7770
+#&gt;
+#&gt; Parameter correlation:
+#&gt; parent_0 log_k_parent log_k_m1 f_parent_ilr_1 sigma_parent
+#&gt; parent_0 1.00000 0.51078 -0.19133 -0.59997 0.035670
+#&gt; log_k_parent 0.51078 1.00000 -0.37458 -0.59239 0.069833
+#&gt; log_k_m1 -0.19133 -0.37458 1.00000 0.74398 -0.026158
+#&gt; f_parent_ilr_1 -0.59997 -0.59239 0.74398 1.00000 -0.041369
+#&gt; sigma_parent 0.03567 0.06983 -0.02616 -0.04137 1.000000
+#&gt; sigma_m1 -0.03385 -0.06627 0.02482 0.03926 -0.004628
+#&gt; sigma_m1
+#&gt; parent_0 -0.033847
+#&gt; log_k_parent -0.066265
+#&gt; log_k_m1 0.024823
+#&gt; f_parent_ilr_1 0.039256
+#&gt; sigma_parent -0.004628
+#&gt; sigma_m1 1.000000
+#&gt;
+#&gt; Backtransformed parameters:
+#&gt; Confidence intervals for internally transformed parameters are asymmetric.
+#&gt; t-test (unrealistically) based on the assumption of normal distribution
+#&gt; for estimators of untransformed parameters.
+#&gt; Estimate t value Pr(&gt;t) Lower Upper
+#&gt; parent_0 99.650000 58.560 2.004e-34 96.190000 1.031e+02
+#&gt; k_parent 0.098970 22.850 1.099e-21 0.090530 1.082e-01
+#&gt; k_m1 0.005245 8.046 1.732e-09 0.004072 6.756e-03
+#&gt; f_parent_to_m1 0.513600 23.560 4.352e-22 0.469300 5.578e-01
+#&gt; sigma_parent 3.401000 5.985 5.662e-07 2.244000 4.559e+00
+#&gt; sigma_m1 2.855000 6.311 2.215e-07 1.934000 3.777e+00
+#&gt;
+#&gt; FOCUS Chi2 error levels in percent:
+#&gt; err.min n.optim df
+#&gt; All data 6.398 4 15
+#&gt; parent 6.464 2 7
+#&gt; m1 4.682 2 8
+#&gt;
+#&gt; Resulting formation fractions:
+#&gt; ff
+#&gt; parent_m1 0.5136
+#&gt; parent_sink 0.4864
+#&gt;
+#&gt; Estimated disappearance times:
+#&gt; DT50 DT90
+#&gt; parent 7.003 23.26
+#&gt; m1 132.154 439.01
+#&gt;
+#&gt; Data:
+#&gt; time variable observed predicted residual
+#&gt; 0 parent 99.46 99.65417 -1.942e-01
+#&gt; 0 parent 102.04 99.65417 2.386e+00
+#&gt; 1 parent 93.50 90.26332 3.237e+00
+#&gt; 1 parent 92.50 90.26332 2.237e+00
+#&gt; 3 parent 63.23 74.05306 -1.082e+01
+#&gt; 3 parent 68.99 74.05306 -5.063e+00
+#&gt; 7 parent 52.32 49.84325 2.477e+00
+#&gt; 7 parent 55.13 49.84325 5.287e+00
+#&gt; 14 parent 27.27 24.92971 2.340e+00
+#&gt; 14 parent 26.64 24.92971 1.710e+00
+#&gt; 21 parent 11.50 12.46890 -9.689e-01
+#&gt; 21 parent 11.64 12.46890 -8.289e-01
+#&gt; 35 parent 2.85 3.11925 -2.692e-01
+#&gt; 35 parent 2.91 3.11925 -2.092e-01
+#&gt; 50 parent 0.69 0.70679 -1.679e-02
+#&gt; 50 parent 0.63 0.70679 -7.679e-02
+#&gt; 75 parent 0.05 0.05952 -9.523e-03
+#&gt; 75 parent 0.06 0.05952 4.772e-04
+#&gt; 1 m1 4.84 4.81075 2.925e-02
+#&gt; 1 m1 5.64 4.81075 8.292e-01
+#&gt; 3 m1 12.91 13.04196 -1.320e-01
+#&gt; 3 m1 12.96 13.04196 -8.196e-02
+#&gt; 7 m1 22.97 25.06847 -2.098e+00
+#&gt; 7 m1 24.47 25.06847 -5.985e-01
+#&gt; 14 m1 41.69 36.70308 4.987e+00
+#&gt; 14 m1 33.21 36.70308 -3.493e+00
+#&gt; 21 m1 44.37 41.65115 2.719e+00
+#&gt; 21 m1 46.44 41.65115 4.789e+00
+#&gt; 35 m1 41.22 43.29465 -2.075e+00
+#&gt; 35 m1 37.95 43.29465 -5.345e+00
+#&gt; 50 m1 41.19 41.19948 -9.479e-03
+#&gt; 50 m1 40.01 41.19948 -1.189e+00
+#&gt; 75 m1 40.09 36.44035 3.650e+00
+#&gt; 75 m1 33.85 36.44035 -2.590e+00
+#&gt; 100 m1 31.04 31.98773 -9.477e-01
+#&gt; 100 m1 33.13 31.98773 1.142e+00
+#&gt; 120 m1 25.15 28.80429 -3.654e+00
+#&gt; 120 m1 33.31 28.80429 4.506e+00</div><div class='input'><span class='no'>f.tc</span> <span class='kw'>&lt;-</span> <span class='fu'>mkinfit</span>(<span class='no'>SFO_SFO.ff</span>, <span class='no'>FOCUS_2006_D</span>, <span class='kw'>error_model</span> <span class='kw'>=</span> <span class='st'>"tc"</span>, <span class='kw'>quiet</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>)</div><div class='output co'>#&gt; <span class='warning'>Warning: Observations with value of zero were removed from the data</span></div><div class='input'><span class='fu'><a href='https://rdrr.io/r/base/summary.html'>summary</a></span>(<span class='no'>f.tc</span>)</div><div class='output co'>#&gt; mkin version used for fitting: 0.9.50.3
+#&gt; R version used for fitting: 4.0.0
+#&gt; Date of fit: Wed May 27 05:54:20 2020
+#&gt; Date of summary: Wed May 27 05:54:20 2020
+#&gt;
+#&gt; Equations:
+#&gt; d_parent/dt = - k_parent * parent
+#&gt; d_m1/dt = + f_parent_to_m1 * k_parent * parent - k_m1 * m1
+#&gt;
+#&gt; Model predictions using solution type analytical
+#&gt;
+#&gt; Fitted using 2088 model solutions performed in 0.714 s
+#&gt;
+#&gt; Error model: Two-component variance function
+#&gt;
+#&gt; Error model algorithm: d_3
+#&gt; Direct fitting and three-step fitting yield approximately the same likelihood
+#&gt;
+#&gt; Starting values for parameters to be optimised:
+#&gt; value type
+#&gt; parent_0 100.7500 state
+#&gt; k_parent 0.1000 deparm
+#&gt; k_m1 0.1001 deparm
+#&gt; f_parent_to_m1 0.5000 deparm
+#&gt; sigma_low 0.1000 error
+#&gt; rsd_high 0.1000 error
+#&gt;
+#&gt; Starting values for the transformed parameters actually optimised:
+#&gt; value lower upper
+#&gt; parent_0 100.750000 -Inf Inf
+#&gt; log_k_parent -2.302585 -Inf Inf
+#&gt; log_k_m1 -2.301586 -Inf Inf
+#&gt; f_parent_ilr_1 0.000000 -Inf Inf
+#&gt; sigma_low 0.100000 0 Inf
+#&gt; rsd_high 0.100000 0 Inf
+#&gt;
+#&gt; Fixed parameter values:
+#&gt; value type
+#&gt; m1_0 0 state
+#&gt;
+#&gt; Results:
+#&gt;
+#&gt; AIC BIC logLik
+#&gt; 141.9656 151.7911 -64.98278
+#&gt;
+#&gt; Optimised, transformed parameters with symmetric confidence intervals:
+#&gt; Estimate Std. Error Lower Upper
+#&gt; parent_0 100.70000 2.621000 95.400000 106.10000
+#&gt; log_k_parent -2.29700 0.008862 -2.315000 -2.27900
+#&gt; log_k_m1 -5.26600 0.091310 -5.452000 -5.08000
+#&gt; f_parent_ilr_1 0.02374 0.055300 -0.088900 0.13640
+#&gt; sigma_low 0.00305 0.004829 -0.006786 0.01289
+#&gt; rsd_high 0.07928 0.009418 0.060100 0.09847
+#&gt;
+#&gt; Parameter correlation:
+#&gt; parent_0 log_k_parent log_k_m1 f_parent_ilr_1 sigma_low rsd_high
+#&gt; parent_0 1.00000 0.67644 -0.10215 -0.76822 0.14294 -0.08783
+#&gt; log_k_parent 0.67644 1.00000 -0.15102 -0.59491 0.34611 -0.08125
+#&gt; log_k_m1 -0.10215 -0.15102 1.00000 0.51808 -0.05236 0.01240
+#&gt; f_parent_ilr_1 -0.76822 -0.59491 0.51808 1.00000 -0.13900 0.03248
+#&gt; sigma_low 0.14294 0.34611 -0.05236 -0.13900 1.00000 -0.16546
+#&gt; rsd_high -0.08783 -0.08125 0.01240 0.03248 -0.16546 1.00000
+#&gt;
+#&gt; Backtransformed parameters:
+#&gt; Confidence intervals for internally transformed parameters are asymmetric.
+#&gt; t-test (unrealistically) based on the assumption of normal distribution
+#&gt; for estimators of untransformed parameters.
+#&gt; Estimate t value Pr(&gt;t) Lower Upper
+#&gt; parent_0 1.007e+02 38.4300 1.180e-28 95.400000 1.061e+02
+#&gt; k_parent 1.006e-01 112.8000 1.718e-43 0.098760 1.024e-01
+#&gt; k_m1 5.167e-03 10.9500 1.171e-12 0.004290 6.223e-03
+#&gt; f_parent_to_m1 5.084e-01 26.0100 2.146e-23 0.468600 5.481e-01
+#&gt; sigma_low 3.050e-03 0.6314 2.661e-01 -0.006786 1.289e-02
+#&gt; rsd_high 7.928e-02 8.4170 6.418e-10 0.060100 9.847e-02
+#&gt;
+#&gt; FOCUS Chi2 error levels in percent:
+#&gt; err.min n.optim df
+#&gt; All data 6.475 4 15
+#&gt; parent 6.573 2 7
+#&gt; m1 4.671 2 8
+#&gt;
+#&gt; Resulting formation fractions:
+#&gt; ff
+#&gt; parent_m1 0.5084
+#&gt; parent_sink 0.4916
+#&gt;
+#&gt; Estimated disappearance times:
+#&gt; DT50 DT90
+#&gt; parent 6.893 22.9
+#&gt; m1 134.156 445.7
+#&gt;
+#&gt; Data:
+#&gt; time variable observed predicted residual
+#&gt; 0 parent 99.46 100.73434 -1.274340
+#&gt; 0 parent 102.04 100.73434 1.305660
+#&gt; 1 parent 93.50 91.09751 2.402486
+#&gt; 1 parent 92.50 91.09751 1.402486
+#&gt; 3 parent 63.23 74.50141 -11.271410
+#&gt; 3 parent 68.99 74.50141 -5.511410
+#&gt; 7 parent 52.32 49.82880 2.491200
+#&gt; 7 parent 55.13 49.82880 5.301200
+#&gt; 14 parent 27.27 24.64809 2.621908
+#&gt; 14 parent 26.64 24.64809 1.991908
+#&gt; 21 parent 11.50 12.19232 -0.692315
+#&gt; 21 parent 11.64 12.19232 -0.552315
+#&gt; 35 parent 2.85 2.98327 -0.133266
+#&gt; 35 parent 2.91 2.98327 -0.073266
+#&gt; 50 parent 0.69 0.66013 0.029874
+#&gt; 50 parent 0.63 0.66013 -0.030126
+#&gt; 75 parent 0.05 0.05344 -0.003438
+#&gt; 75 parent 0.06 0.05344 0.006562
+#&gt; 1 m1 4.84 4.88645 -0.046451
+#&gt; 1 m1 5.64 4.88645 0.753549
+#&gt; 3 m1 12.91 13.22867 -0.318669
+#&gt; 3 m1 12.96 13.22867 -0.268669
+#&gt; 7 m1 22.97 25.36417 -2.394166
+#&gt; 7 m1 24.47 25.36417 -0.894166
+#&gt; 14 m1 41.69 37.00974 4.680263
+#&gt; 14 m1 33.21 37.00974 -3.799737
+#&gt; 21 m1 44.37 41.90133 2.468669
+#&gt; 21 m1 46.44 41.90133 4.538669
+#&gt; 35 m1 41.22 43.45691 -2.236913
+#&gt; 35 m1 37.95 43.45691 -5.506913
+#&gt; 50 m1 41.19 41.34199 -0.151985
+#&gt; 50 m1 40.01 41.34199 -1.331985
+#&gt; 75 m1 40.09 36.61471 3.475295
+#&gt; 75 m1 33.85 36.61471 -2.764705
+#&gt; 100 m1 31.04 32.20082 -1.160823
+#&gt; 100 m1 33.13 32.20082 0.929177
+#&gt; 120 m1 25.15 29.04130 -3.891304
+#&gt; 120 m1 33.31 29.04130 4.268696</div><div class='input'># }
+
+
+</div></pre>
+ </div>
+ <div class="col-md-3 hidden-xs hidden-sm" id="pkgdown-sidebar">
+ <nav id="toc" data-toggle="toc" class="sticky-top">
+ <h2 data-toc-skip>Contents</h2>
+ </nav>
+ </div>
+</div>
+
+
+ <footer>
+ <div class="copyright">
+ <p>Developed by Johannes Ranke.</p>
+</div>
+
+<div class="pkgdown">
+ <p>Site built with <a href="https://pkgdown.r-lib.org/">pkgdown</a> 1.5.1.</p>
+</div>
+
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