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<meta property="og:title" content="Functions to transform and backtransform kinetic parameters for fitting — transform_odeparms" />

<meta property="og:description" content="The transformations are intended to map parameters that should only take
  on restricted values to the full scale of real numbers. For kinetic rate
  constants and other paramters that can only take on positive values, a
  simple log transformation is used. For compositional parameters, such as
  the formations fractions that should always sum up to 1 and can not be
  negative, the ilr transformation is used.
The transformation of sets of formation fractions is fragile, as it supposes
  the same ordering of the components in forward and backward transformation.
  This is no problem for the internal use in mkinfit." />
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    <h1>Functions to transform and backtransform kinetic parameters for fitting</h1>
    
    <div class="hidden name"><code>transform_odeparms.Rd</code></div>
    </div>

    <div class="ref-description">
    
    <p>The transformations are intended to map parameters that should only take
  on restricted values to the full scale of real numbers. For kinetic rate
  constants and other paramters that can only take on positive values, a
  simple log transformation is used. For compositional parameters, such as
  the formations fractions that should always sum up to 1 and can not be
  negative, the <code><a href='ilr.html'>ilr</a></code> transformation is used.</p>
<p>The transformation of sets of formation fractions is fragile, as it supposes
  the same ordering of the components in forward and backward transformation.
  This is no problem for the internal use in <code><a href='mkinfit.html'>mkinfit</a></code>.</p>
    
    </div>

    <pre class="usage"><span class='fu'>transform_odeparms</span>(<span class='no'>parms</span>, <span class='no'>mkinmod</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='fu'>backtransform_odeparms</span>(<span class='no'>transparms</span>, <span class='no'>mkinmod</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>)</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>parms</th>
      <td><p>Parameters of kinetic models as used in the differential equations.</p></td>
    </tr>
    <tr>
      <th>transparms</th>
      <td><p>Transformed parameters of kinetic models as used in the fitting procedure.</p></td>
    </tr>
    <tr>
      <th>mkinmod</th>
      <td><p>The kinetic model of class <code><a href='mkinmod.html'>mkinmod</a></code>, containing the names of
    the model variables that are needed for grouping the formation fractions
    before <code><a href='ilr.html'>ilr</a></code> transformation, the parameter names and
    the information if the pathway to sink is included in the model.</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.</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. 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>
    </table>
    
    <h2 class="hasAnchor" id="value"><a class="anchor" href="#value"></a>Value</h2>

    <p>A vector of transformed or backtransformed parameters with the same names
  as the original parameters.</p>
    

    <h2 class="hasAnchor" id="examples"><a class="anchor" href="#examples"></a>Examples</h2>
    <pre class="examples"><div class='input'><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'>list</span>(<span class='kw'>type</span> <span class='kw'>=</span> <span class='st'>"SFO"</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'>list</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</span> <span class='kw'>&lt;-</span> <span class='fu'><a href='mkinfit.html'>mkinfit</a></span>(<span class='no'>SFO_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='fu'>summary</span>(<span class='no'>fit</span>, <span class='kw'>data</span><span class='kw'>=</span><span class='fl'>FALSE</span>) <span class='co'># See transformed and backtransformed parameters</span></div><div class='output co'>#&gt; mkin version used for fitting:    0.9.47.5 
#&gt; R version used for fitting:       3.5.1 
#&gt; Date of fit:     Fri Sep 14 21:00:16 2018 
#&gt; Date of summary: Fri Sep 14 21:00:16 2018 
#&gt; 
#&gt; Equations:
#&gt; d_parent/dt = - k_parent_sink * parent - k_parent_m1 * parent
#&gt; d_m1/dt = + k_parent_m1 * parent - k_m1_sink * m1
#&gt; 
#&gt; Model predictions using solution type deSolve 
#&gt; 
#&gt; Fitted with method Port using 153 model solutions performed in 0.71 s
#&gt; 
#&gt; Weighting: none
#&gt; 
#&gt; Starting values for parameters to be optimised:
#&gt;                  value   type
#&gt; parent_0      100.7500  state
#&gt; k_parent_sink   0.1000 deparm
#&gt; k_parent_m1     0.1001 deparm
#&gt; k_m1_sink       0.1002 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_sink  -2.302585  -Inf   Inf
#&gt; log_k_parent_m1    -2.301586  -Inf   Inf
#&gt; log_k_m1_sink      -2.300587  -Inf   Inf
#&gt; 
#&gt; Fixed parameter values:
#&gt;      value  type
#&gt; m1_0     0 state
#&gt; 
#&gt; Optimised, transformed parameters with symmetric confidence intervals:
#&gt;                   Estimate Std. Error  Lower   Upper
#&gt; parent_0            99.600    1.61400 96.330 102.900
#&gt; log_k_parent_sink   -3.038    0.07826 -3.197  -2.879
#&gt; log_k_parent_m1     -2.980    0.04124 -3.064  -2.897
#&gt; log_k_m1_sink       -5.248    0.13610 -5.523  -4.972
#&gt; 
#&gt; Parameter correlation:
#&gt;                   parent_0 log_k_parent_sink log_k_parent_m1 log_k_m1_sink
#&gt; parent_0           1.00000            0.6075        -0.06625       -0.1701
#&gt; log_k_parent_sink  0.60752            1.0000        -0.08740       -0.6253
#&gt; log_k_parent_m1   -0.06625           -0.0874         1.00000        0.4716
#&gt; log_k_m1_sink     -0.17006           -0.6253         0.47164        1.0000
#&gt; 
#&gt; Residual standard error: 3.211 on 36 degrees of freedom
#&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  61.720 2.024e-38 96.330000 1.029e+02
#&gt; k_parent_sink  0.047920  12.780 3.050e-15  0.040890 5.616e-02
#&gt; k_parent_m1    0.050780  24.250 3.407e-24  0.046700 5.521e-02
#&gt; k_m1_sink      0.005261   7.349 5.758e-09  0.003992 6.933e-03
#&gt; 
#&gt; Chi2 error levels in percent:
#&gt;          err.min n.optim df
#&gt; All data   6.398       4 15
#&gt; parent     6.827       3  6
#&gt; m1         4.490       1  9
#&gt; 
#&gt; Resulting formation fractions:
#&gt;                 ff
#&gt; parent_sink 0.4855
#&gt; parent_m1   0.5145
#&gt; m1_sink     1.0000
#&gt; 
#&gt; Estimated disappearance times:
#&gt;           DT50   DT90
#&gt; parent   7.023  23.33
#&gt; m1     131.761 437.70</div><div class='input'>
</div><div class='input'><span class='no'>fit.2</span> <span class='kw'>&lt;-</span> <span class='fu'><a href='mkinfit.html'>mkinfit</a></span>(<span class='no'>SFO_SFO</span>, <span class='no'>FOCUS_2006_D</span>, <span class='kw'>transform_rates</span> <span class='kw'>=</span> <span class='fl'>FALSE</span>, <span class='kw'>quiet</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>)
<span class='fu'>summary</span>(<span class='no'>fit.2</span>, <span class='kw'>data</span><span class='kw'>=</span><span class='fl'>FALSE</span>)</div><div class='output co'>#&gt; mkin version used for fitting:    0.9.47.5 
#&gt; R version used for fitting:       3.5.1 
#&gt; Date of fit:     Fri Sep 14 21:00:17 2018 
#&gt; Date of summary: Fri Sep 14 21:00:17 2018 
#&gt; 
#&gt; Equations:
#&gt; d_parent/dt = - k_parent_sink * parent - k_parent_m1 * parent
#&gt; d_m1/dt = + k_parent_m1 * parent - k_m1_sink * m1
#&gt; 
#&gt; Model predictions using solution type deSolve 
#&gt; 
#&gt; Fitted with method Port using 350 model solutions performed in 1.417 s
#&gt; 
#&gt; Weighting: none
#&gt; 
#&gt; Starting values for parameters to be optimised:
#&gt;                  value   type
#&gt; parent_0      100.7500  state
#&gt; k_parent_sink   0.1000 deparm
#&gt; k_parent_m1     0.1001 deparm
#&gt; k_m1_sink       0.1002 deparm
#&gt; 
#&gt; Starting values for the transformed parameters actually optimised:
#&gt;                  value lower upper
#&gt; parent_0      100.7500  -Inf   Inf
#&gt; k_parent_sink   0.1000     0   Inf
#&gt; k_parent_m1     0.1001     0   Inf
#&gt; k_m1_sink       0.1002     0   Inf
#&gt; 
#&gt; Fixed parameter values:
#&gt;      value  type
#&gt; m1_0     0 state
#&gt; 
#&gt; Optimised, transformed parameters with symmetric confidence intervals:
#&gt;                Estimate Std. Error     Lower     Upper
#&gt; parent_0      99.600000  1.6140000 96.330000 1.029e+02
#&gt; k_parent_sink  0.047920  0.0037500  0.040310 5.553e-02
#&gt; k_parent_m1    0.050780  0.0020940  0.046530 5.502e-02
#&gt; k_m1_sink      0.005261  0.0007159  0.003809 6.713e-03
#&gt; 
#&gt; Parameter correlation:
#&gt;               parent_0 k_parent_sink k_parent_m1 k_m1_sink
#&gt; parent_0       1.00000        0.6075    -0.06625   -0.1701
#&gt; k_parent_sink  0.60752        1.0000    -0.08740   -0.6253
#&gt; k_parent_m1   -0.06625       -0.0874     1.00000    0.4716
#&gt; k_m1_sink     -0.17006       -0.6253     0.47164    1.0000
#&gt; 
#&gt; Residual standard error: 3.211 on 36 degrees of freedom
#&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  61.720 2.024e-38 96.330000 1.029e+02
#&gt; k_parent_sink  0.047920  12.780 3.050e-15  0.040310 5.553e-02
#&gt; k_parent_m1    0.050780  24.250 3.407e-24  0.046530 5.502e-02
#&gt; k_m1_sink      0.005261   7.349 5.758e-09  0.003809 6.713e-03
#&gt; 
#&gt; Chi2 error levels in percent:
#&gt;          err.min n.optim df
#&gt; All data   6.398       4 15
#&gt; parent     6.827       3  6
#&gt; m1         4.490       1  9
#&gt; 
#&gt; Resulting formation fractions:
#&gt;                 ff
#&gt; parent_sink 0.4855
#&gt; parent_m1   0.5145
#&gt; m1_sink     1.0000
#&gt; 
#&gt; Estimated disappearance times:
#&gt;           DT50   DT90
#&gt; parent   7.023  23.33
#&gt; m1     131.761 437.70</div><div class='input'>
<span class='no'>initials</span> <span class='kw'>&lt;-</span> <span class='no'>fit</span>$<span class='no'>start</span>$<span class='no'>value</span>
<span class='fu'>names</span>(<span class='no'>initials</span>) <span class='kw'>&lt;-</span> <span class='fu'>rownames</span>(<span class='no'>fit</span>$<span class='no'>start</span>)
<span class='no'>transformed</span> <span class='kw'>&lt;-</span> <span class='no'>fit</span>$<span class='no'>start_transformed</span>$<span class='no'>value</span>
<span class='fu'>names</span>(<span class='no'>transformed</span>) <span class='kw'>&lt;-</span> <span class='fu'>rownames</span>(<span class='no'>fit</span>$<span class='no'>start_transformed</span>)
<span class='fu'>transform_odeparms</span>(<span class='no'>initials</span>, <span class='no'>SFO_SFO</span>)</div><div class='output co'>#&gt;          parent_0 log_k_parent_sink   log_k_parent_m1     log_k_m1_sink 
#&gt;        100.750000         -2.302585         -2.301586         -2.300587 </div><div class='input'><span class='fu'>backtransform_odeparms</span>(<span class='no'>transformed</span>, <span class='no'>SFO_SFO</span>)</div><div class='output co'>#&gt;      parent_0 k_parent_sink   k_parent_m1     k_m1_sink 
#&gt;      100.7500        0.1000        0.1001        0.1002 </div><div class='input'>
</div><div class='input'><span class='co'># The case of formation fractions</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'>list</span>(<span class='kw'>type</span> <span class='kw'>=</span> <span class='st'>"SFO"</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'>list</span>(<span class='kw'>type</span> <span class='kw'>=</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'>fit.ff</span> <span class='kw'>&lt;-</span> <span class='fu'><a href='mkinfit.html'>mkinfit</a></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>)
<span class='fu'>summary</span>(<span class='no'>fit.ff</span>, <span class='kw'>data</span> <span class='kw'>=</span> <span class='fl'>FALSE</span>)</div><div class='output co'>#&gt; mkin version used for fitting:    0.9.47.5 
#&gt; R version used for fitting:       3.5.1 
#&gt; Date of fit:     Fri Sep 14 21:00:18 2018 
#&gt; Date of summary: Fri Sep 14 21:00:18 2018 
#&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 deSolve 
#&gt; 
#&gt; Fitted with method Port using 186 model solutions performed in 0.755 s
#&gt; 
#&gt; Weighting: none
#&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; Optimised, transformed parameters with symmetric confidence intervals:
#&gt;                Estimate Std. Error   Lower    Upper
#&gt; parent_0       99.60000    1.61400 96.3300 102.9000
#&gt; log_k_parent   -2.31600    0.04187 -2.4010  -2.2310
#&gt; log_k_m1       -5.24800    0.13610 -5.5230  -4.9720
#&gt; f_parent_ilr_1  0.04096    0.06477 -0.0904   0.1723
#&gt; 
#&gt; Parameter correlation:
#&gt;                parent_0 log_k_parent log_k_m1 f_parent_ilr_1
#&gt; parent_0         1.0000       0.5178  -0.1701        -0.5489
#&gt; log_k_parent     0.5178       1.0000  -0.3285        -0.5451
#&gt; log_k_m1        -0.1701      -0.3285   1.0000         0.7466
#&gt; f_parent_ilr_1  -0.5489      -0.5451   0.7466         1.0000
#&gt; 
#&gt; Residual standard error: 3.211 on 36 degrees of freedom
#&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  61.720 2.024e-38 96.330000 1.029e+02
#&gt; k_parent        0.098700  23.880 5.700e-24  0.090660 1.074e-01
#&gt; k_m1            0.005261   7.349 5.758e-09  0.003992 6.933e-03
#&gt; f_parent_to_m1  0.514500  22.490 4.375e-23  0.468100 5.606e-01
#&gt; 
#&gt; 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</div><div class='input'><span class='no'>initials</span> <span class='kw'>&lt;-</span> <span class='fu'>c</span>(<span class='st'>"f_parent_to_m1"</span> <span class='kw'>=</span> <span class='fl'>0.5</span>)
<span class='no'>transformed</span> <span class='kw'>&lt;-</span> <span class='fu'>transform_odeparms</span>(<span class='no'>initials</span>, <span class='no'>SFO_SFO.ff</span>)
<span class='fu'>backtransform_odeparms</span>(<span class='no'>transformed</span>, <span class='no'>SFO_SFO.ff</span>)</div><div class='output co'>#&gt; f_parent_to_m1 
#&gt;            0.5 </div><div class='input'>
<span class='co'># And without sink</span>
<span class='no'>SFO_SFO.ff.2</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'>list</span>(<span class='kw'>type</span> <span class='kw'>=</span> <span class='st'>"SFO"</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'>FALSE</span>),
  <span class='kw'>m1</span> <span class='kw'>=</span> <span class='fu'>list</span>(<span class='kw'>type</span> <span class='kw'>=</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'>fit.ff.2</span> <span class='kw'>&lt;-</span> <span class='fu'><a href='mkinfit.html'>mkinfit</a></span>(<span class='no'>SFO_SFO.ff.2</span>, <span class='no'>FOCUS_2006_D</span>, <span class='kw'>quiet</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>)
<span class='fu'>summary</span>(<span class='no'>fit.ff.2</span>, <span class='kw'>data</span> <span class='kw'>=</span> <span class='fl'>FALSE</span>)</div><div class='output co'>#&gt; mkin version used for fitting:    0.9.47.5 
#&gt; R version used for fitting:       3.5.1 
#&gt; Date of fit:     Fri Sep 14 21:00:19 2018 
#&gt; Date of summary: Fri Sep 14 21:00:19 2018 
#&gt; 
#&gt; Equations:
#&gt; d_parent/dt = - k_parent * parent
#&gt; d_m1/dt = + k_parent * parent - k_m1 * m1
#&gt; 
#&gt; Model predictions using solution type deSolve 
#&gt; 
#&gt; Fitted with method Port using 104 model solutions performed in 0.416 s
#&gt; 
#&gt; Weighting: none
#&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; 
#&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; 
#&gt; Fixed parameter values:
#&gt;      value  type
#&gt; m1_0     0 state
#&gt; 
#&gt; Optimised, transformed parameters with symmetric confidence intervals:
#&gt;              Estimate Std. Error Lower  Upper
#&gt; parent_0       84.790    2.96500 78.78 90.800
#&gt; log_k_parent   -2.756    0.08088 -2.92 -2.593
#&gt; log_k_m1       -4.214    0.11150 -4.44 -3.988
#&gt; 
#&gt; Parameter correlation:
#&gt;              parent_0 log_k_parent log_k_m1
#&gt; parent_0       1.0000      0.11058  0.46156
#&gt; log_k_parent   0.1106      1.00000  0.06274
#&gt; log_k_m1       0.4616      0.06274  1.00000
#&gt; 
#&gt; Residual standard error: 8.333 on 37 degrees of freedom
#&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 84.79000  28.600 3.939e-27 78.78000 90.80000
#&gt; k_parent  0.06352  12.360 5.237e-15  0.05392  0.07483
#&gt; k_m1      0.01478   8.966 4.114e-11  0.01179  0.01853
#&gt; 
#&gt; Chi2 error levels in percent:
#&gt;          err.min n.optim df
#&gt; All data   19.66       3 16
#&gt; parent     17.56       2  7
#&gt; m1         18.71       1  9
#&gt; 
#&gt; Estimated disappearance times:
#&gt;         DT50   DT90
#&gt; parent 10.91  36.25
#&gt; m1     46.89 155.75</div></pre>
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    <h2>Contents</h2>
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      <li><a href="#arguments">Arguments</a></li>
      
      <li><a href="#value">Value</a></li>
      
      <li><a href="#examples">Examples</a></li>
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    <h2>Author</h2>
    
  Johannes Ranke

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