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<meta property="og:title" content="Confidence intervals for parameters of mkinfit objects — confint.mkinfit" />
<meta property="og:description" content="The default method 'quadratic' is based on the quadratic approximation of
the curvature of the likelihood function at the maximum likelihood parameter
estimates.
The alternative method 'profile' is based on the profile likelihood for each
parameter. The method uses two nested optimisations and can take a very long
time, even if parallelized by specifying 'cores' on unixoid platforms. The
speed of the method could likely be improved by using the method of Venzon
and Moolgavkar (1988)." />
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<h1>Confidence intervals for parameters of mkinfit objects</h1>
<div class="hidden name"><code>confint.mkinfit.Rd</code></div>
</div>
<div class="ref-description">
<p>The default method 'quadratic' is based on the quadratic approximation of
the curvature of the likelihood function at the maximum likelihood parameter
estimates.
The alternative method 'profile' is based on the profile likelihood for each
parameter. The method uses two nested optimisations and can take a very long
time, even if parallelized by specifying 'cores' on unixoid platforms. The
speed of the method could likely be improved by using the method of Venzon
and Moolgavkar (1988).</p>
</div>
<pre class="usage"><span class='co'># S3 method for mkinfit</span>
<span class='fu'><a href='https://rdrr.io/r/stats/confint.html'>confint</a></span>(<span class='no'>object</span>, <span class='no'>parm</span>, <span class='kw'>level</span> <span class='kw'>=</span> <span class='fl'>0.95</span>, <span class='kw'>alpha</span> <span class='kw'>=</span> <span class='fl'>1</span> -
<span class='no'>level</span>, <span class='no'>cutoff</span>, <span class='kw'>method</span> <span class='kw'>=</span> <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='st'>"quadratic"</span>, <span class='st'>"profile"</span>),
<span class='kw'>transformed</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>, <span class='kw'>backtransform</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>,
<span class='kw'>cores</span> <span class='kw'>=</span> <span class='fu'><a href='https://rdrr.io/r/base/Round.html'>round</a></span>(<span class='fu'>detectCores</span>()/<span class='fl'>2</span>), <span class='kw'>quiet</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>object</th>
<td><p>An <code><a href='mkinfit.html'>mkinfit</a></code> object</p></td>
</tr>
<tr>
<th>parm</th>
<td><p>A vector of names of the parameters which are to be given
confidence intervals. If missing, all parameters are considered.</p></td>
</tr>
<tr>
<th>level</th>
<td><p>The confidence level required</p></td>
</tr>
<tr>
<th>alpha</th>
<td><p>The allowed error probability, overrides 'level' if specified.</p></td>
</tr>
<tr>
<th>cutoff</th>
<td><p>Possibility to specify an alternative cutoff for the difference
in the log-likelihoods at the confidence boundary. Specifying an explicit
cutoff value overrides arguments 'level' and 'alpha'</p></td>
</tr>
<tr>
<th>method</th>
<td><p>The 'quadratic' method approximates the likelihood function at
the optimised parameters using the second term of the Taylor expansion,
using a second derivative (hessian) contained in the object.
The 'profile' method searches the parameter space for the
cutoff of the confidence intervals by means of a likelihood ratio test.</p></td>
</tr>
<tr>
<th>transformed</th>
<td><p>If the quadratic approximation is used, should it be
applied to the likelihood based on the transformed parameters?</p></td>
</tr>
<tr>
<th>backtransform</th>
<td><p>If we approximate the likelihood in terms of the
transformed parameters, should we backtransform the parameters with
their confidence intervals?</p></td>
</tr>
<tr>
<th>cores</th>
<td><p>The number of cores to be used for multicore processing. This
is only used when the <code>cluster</code> argument is <code>NULL</code>. On Windows
machines, cores > 1 is not supported.</p></td>
</tr>
<tr>
<th>quiet</th>
<td><p>Should we suppress the message "Profiling the likelihood"</p></td>
</tr>
<tr>
<th>...</th>
<td><p>Not used</p></td>
</tr>
</table>
<h2 class="hasAnchor" id="value"><a class="anchor" href="#value"></a>Value</h2>
<p>A matrix with columns giving lower and upper confidence limits for
each parameter.</p>
<h2 class="hasAnchor" id="references"><a class="anchor" href="#references"></a>References</h2>
<p>Bates DM and Watts GW (1988) Nonlinear regression analysis & its applications</p>
<p>Pawitan Y (2013) In all likelihood - Statistical modelling and
inference using likelihood. Clarendon Press, Oxford.</p>
<p>Venzon DJ and Moolgavkar SH (1988) A Method for Computing
Profile-Likelihood Based Confidence Intervals, Applied Statistics, 37,
87–94.</p>
<h2 class="hasAnchor" id="examples"><a class="anchor" href="#examples"></a>Examples</h2>
<pre class="examples"><div class='input'><span class='no'>f</span> <span class='kw'><-</span> <span class='fu'><a href='mkinfit.html'>mkinfit</a></span>(<span class='st'>"SFO"</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/stats/confint.html'>confint</a></span>(<span class='no'>f</span>, <span class='kw'>method</span> <span class='kw'>=</span> <span class='st'>"quadratic"</span>)</div><div class='output co'>#> 2.5% 97.5%
#> parent_0 71.8242430 93.1600766
#> k_parent_sink 0.2109541 0.4440528
#> sigma 1.9778868 7.3681380</div><div class='input'>
<span class='co'># \dontrun{</span>
<span class='fu'><a href='https://rdrr.io/r/stats/confint.html'>confint</a></span>(<span class='no'>f</span>, <span class='kw'>method</span> <span class='kw'>=</span> <span class='st'>"profile"</span>)</div><div class='output co'>#> <span class='message'>Profiling the likelihood</span></div><div class='output co'>#> 2.5% 97.5%
#> parent_0 73.0641834 92.1392181
#> k_parent_sink 0.2170293 0.4235348
#> sigma 3.1307772 8.0628314</div><div class='input'>
<span class='co'># Set the number of cores for the profiling method for further examples</span>
<span class='kw'>if</span> (<span class='fu'><a href='https://rdrr.io/r/base/identical.html'>identical</a></span>(<span class='fu'><a href='https://rdrr.io/r/base/Sys.getenv.html'>Sys.getenv</a></span>(<span class='st'>"NOT_CRAN"</span>), <span class='st'>"true"</span>)) {
<span class='no'>n_cores</span> <span class='kw'><-</span> <span class='kw pkg'>parallel</span><span class='kw ns'>::</span><span class='fu'><a href='https://rdrr.io/r/parallel/detectCores.html'>detectCores</a></span>() - <span class='fl'>1</span>
} <span class='kw'>else</span> {
<span class='no'>n_cores</span> <span class='kw'><-</span> <span class='fl'>1</span>
}
<span class='kw'>if</span> (<span class='fu'><a href='https://rdrr.io/r/base/Sys.getenv.html'>Sys.getenv</a></span>(<span class='st'>"TRAVIS"</span>) <span class='kw'>!=</span> <span class='st'>""</span>) <span class='no'>n_cores</span> <span class='kw'>=</span> <span class='fl'>1</span>
<span class='kw'>if</span> (<span class='fu'><a href='https://rdrr.io/r/base/Sys.info.html'>Sys.info</a></span>()[<span class='st'>"sysname"</span>] <span class='kw'>==</span> <span class='st'>"Windows"</span>) <span class='no'>n_cores</span> <span class='kw'>=</span> <span class='fl'>1</span>
<span class='no'>SFO_SFO</span> <span class='kw'><-</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'>quiet</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>)
<span class='no'>SFO_SFO.ff</span> <span class='kw'><-</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>, <span class='kw'>quiet</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>)
<span class='no'>f_d_1</span> <span class='kw'><-</span> <span class='fu'><a href='mkinfit.html'>mkinfit</a></span>(<span class='no'>SFO_SFO</span>, <span class='fu'><a href='https://rdrr.io/r/base/subset.html'>subset</a></span>(<span class='no'>FOCUS_2006_D</span>, <span class='no'>value</span> <span class='kw'>!=</span> <span class='fl'>0</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/system.time.html'>system.time</a></span>(<span class='no'>ci_profile</span> <span class='kw'><-</span> <span class='fu'><a href='https://rdrr.io/r/stats/confint.html'>confint</a></span>(<span class='no'>f_d_1</span>, <span class='kw'>method</span> <span class='kw'>=</span> <span class='st'>"profile"</span>, <span class='kw'>cores</span> <span class='kw'>=</span> <span class='fl'>1</span>, <span class='kw'>quiet</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>))</div><div class='output co'>#> User System verstrichen
#> 51.297 0.000 51.328 </div><div class='input'><span class='co'># Using more cores does not save much time here, as parent_0 takes up most of the time</span>
<span class='co'># If we additionally exclude parent_0 (the confidence of which is often of</span>
<span class='co'># minor interest), we get a nice performance improvement from about 50</span>
<span class='co'># seconds to about 12 seconds if we use at least four cores</span>
<span class='fu'><a href='https://rdrr.io/r/base/system.time.html'>system.time</a></span>(<span class='no'>ci_profile_no_parent_0</span> <span class='kw'><-</span> <span class='fu'><a href='https://rdrr.io/r/stats/confint.html'>confint</a></span>(<span class='no'>f_d_1</span>, <span class='kw'>method</span> <span class='kw'>=</span> <span class='st'>"profile"</span>,
<span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='st'>"k_parent_sink"</span>, <span class='st'>"k_parent_m1"</span>, <span class='st'>"k_m1_sink"</span>, <span class='st'>"sigma"</span>), <span class='kw'>cores</span> <span class='kw'>=</span> <span class='no'>n_cores</span>))</div><div class='output co'>#> <span class='message'>Profiling the likelihood</span></div><div class='output co'>#> User System verstrichen
#> 0.006 0.003 11.435 </div><div class='input'><span class='no'>ci_profile</span></div><div class='output co'>#> 2.5% 97.5%
#> parent_0 96.456003650 1.027703e+02
#> k_parent_sink 0.040762501 5.549764e-02
#> k_parent_m1 0.046786482 5.500879e-02
#> k_m1_sink 0.003892605 6.702778e-03
#> sigma 2.535612399 3.985263e+00</div><div class='input'><span class='no'>ci_quadratic_transformed</span> <span class='kw'><-</span> <span class='fu'><a href='https://rdrr.io/r/stats/confint.html'>confint</a></span>(<span class='no'>f_d_1</span>, <span class='kw'>method</span> <span class='kw'>=</span> <span class='st'>"quadratic"</span>)
<span class='no'>ci_quadratic_transformed</span></div><div class='output co'>#> 2.5% 97.5%
#> parent_0 96.403841649 1.027931e+02
#> k_parent_sink 0.041033378 5.596269e-02
#> k_parent_m1 0.046777902 5.511931e-02
#> k_m1_sink 0.004012217 6.897547e-03
#> sigma 2.396089689 3.854918e+00</div><div class='input'><span class='no'>ci_quadratic_untransformed</span> <span class='kw'><-</span> <span class='fu'><a href='https://rdrr.io/r/stats/confint.html'>confint</a></span>(<span class='no'>f_d_1</span>, <span class='kw'>method</span> <span class='kw'>=</span> <span class='st'>"quadratic"</span>, <span class='kw'>transformed</span> <span class='kw'>=</span> <span class='fl'>FALSE</span>)
<span class='no'>ci_quadratic_untransformed</span></div><div class='output co'>#> 2.5% 97.5%
#> parent_0 96.403841653 102.79312450
#> k_parent_sink 0.040485331 0.05535491
#> k_parent_m1 0.046611581 0.05494364
#> k_m1_sink 0.003835483 0.00668582
#> sigma 2.396089689 3.85491806</div><div class='input'><span class='co'># Against the expectation based on Bates and Watts (1988), the confidence</span>
<span class='co'># intervals based on the internal parameter transformation are less</span>
<span class='co'># congruent with the likelihood based intervals. Note the superiority of the</span>
<span class='co'># interval based on the untransformed fit for k_m1_sink</span>
<span class='no'>rel_diffs_transformed</span> <span class='kw'><-</span> <span class='fu'><a href='https://rdrr.io/r/base/MathFun.html'>abs</a></span>((<span class='no'>ci_quadratic_transformed</span> - <span class='no'>ci_profile</span>)/<span class='no'>ci_profile</span>)
<span class='no'>rel_diffs_untransformed</span> <span class='kw'><-</span> <span class='fu'><a href='https://rdrr.io/r/base/MathFun.html'>abs</a></span>((<span class='no'>ci_quadratic_untransformed</span> - <span class='no'>ci_profile</span>)/<span class='no'>ci_profile</span>)
<span class='no'>rel_diffs_transformed</span> <span class='kw'><</span> <span class='no'>rel_diffs_untransformed</span></div><div class='output co'>#> 2.5% 97.5%
#> parent_0 FALSE FALSE
#> k_parent_sink TRUE FALSE
#> k_parent_m1 TRUE FALSE
#> k_m1_sink FALSE FALSE
#> sigma FALSE FALSE</div><div class='input'><span class='fu'><a href='https://rdrr.io/r/base/Round.html'>signif</a></span>(<span class='no'>rel_diffs_transformed</span>, <span class='fl'>3</span>)</div><div class='output co'>#> 2.5% 97.5%
#> parent_0 0.000541 0.000222
#> k_parent_sink 0.006650 0.008380
#> k_parent_m1 0.000183 0.002010
#> k_m1_sink 0.030700 0.029100
#> sigma 0.055000 0.032700</div><div class='input'><span class='fu'><a href='https://rdrr.io/r/base/Round.html'>signif</a></span>(<span class='no'>rel_diffs_untransformed</span>, <span class='fl'>3</span>)</div><div class='output co'>#> 2.5% 97.5%
#> parent_0 0.000541 0.000222
#> k_parent_sink 0.006800 0.002570
#> k_parent_m1 0.003740 0.001180
#> k_m1_sink 0.014700 0.002530
#> sigma 0.055000 0.032700</div><div class='input'>
<span class='co'># Investigate a case with formation fractions</span>
<span class='no'>f_d_2</span> <span class='kw'><-</span> <span class='fu'><a href='mkinfit.html'>mkinfit</a></span>(<span class='no'>SFO_SFO.ff</span>, <span class='fu'><a href='https://rdrr.io/r/base/subset.html'>subset</a></span>(<span class='no'>FOCUS_2006_D</span>, <span class='no'>value</span> <span class='kw'>!=</span> <span class='fl'>0</span>), <span class='kw'>quiet</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>)
<span class='no'>ci_profile_ff</span> <span class='kw'><-</span> <span class='fu'><a href='https://rdrr.io/r/stats/confint.html'>confint</a></span>(<span class='no'>f_d_2</span>, <span class='kw'>method</span> <span class='kw'>=</span> <span class='st'>"profile"</span>, <span class='kw'>cores</span> <span class='kw'>=</span> <span class='no'>n_cores</span>)</div><div class='output co'>#> <span class='message'>Profiling the likelihood</span></div><div class='input'><span class='no'>ci_profile_ff</span></div><div class='output co'>#> 2.5% 97.5%
#> parent_0 96.456003650 1.027703e+02
#> k_parent 0.090911032 1.071578e-01
#> k_m1 0.003892605 6.702778e-03
#> f_parent_to_m1 0.471328495 5.611550e-01
#> sigma 2.535612399 3.985263e+00</div><div class='input'><span class='no'>ci_quadratic_transformed_ff</span> <span class='kw'><-</span> <span class='fu'><a href='https://rdrr.io/r/stats/confint.html'>confint</a></span>(<span class='no'>f_d_2</span>, <span class='kw'>method</span> <span class='kw'>=</span> <span class='st'>"quadratic"</span>)
<span class='no'>ci_quadratic_transformed_ff</span></div><div class='output co'>#> 2.5% 97.5%
#> parent_0 96.403840123 1.027931e+02
#> k_parent 0.090823791 1.072543e-01
#> k_m1 0.004012216 6.897547e-03
#> f_parent_to_m1 0.469118710 5.595960e-01
#> sigma 2.396089689 3.854918e+00</div><div class='input'><span class='no'>ci_quadratic_untransformed_ff</span> <span class='kw'><-</span> <span class='fu'><a href='https://rdrr.io/r/stats/confint.html'>confint</a></span>(<span class='no'>f_d_2</span>, <span class='kw'>method</span> <span class='kw'>=</span> <span class='st'>"quadratic"</span>, <span class='kw'>transformed</span> <span class='kw'>=</span> <span class='fl'>FALSE</span>)
<span class='no'>ci_quadratic_untransformed_ff</span></div><div class='output co'>#> 2.5% 97.5%
#> parent_0 96.403840057 1.027931e+02
#> k_parent 0.090491932 1.069035e-01
#> k_m1 0.003835483 6.685819e-03
#> f_parent_to_m1 0.469113361 5.598386e-01
#> sigma 2.396089689 3.854918e+00</div><div class='input'><span class='no'>rel_diffs_transformed_ff</span> <span class='kw'><-</span> <span class='fu'><a href='https://rdrr.io/r/base/MathFun.html'>abs</a></span>((<span class='no'>ci_quadratic_transformed_ff</span> - <span class='no'>ci_profile_ff</span>)/<span class='no'>ci_profile_ff</span>)
<span class='no'>rel_diffs_untransformed_ff</span> <span class='kw'><-</span> <span class='fu'><a href='https://rdrr.io/r/base/MathFun.html'>abs</a></span>((<span class='no'>ci_quadratic_untransformed_ff</span> - <span class='no'>ci_profile_ff</span>)/<span class='no'>ci_profile_ff</span>)
<span class='co'># While the confidence interval for the parent rate constant is closer to</span>
<span class='co'># the profile based interval when using the internal parameter</span>
<span class='co'># transformation, the interval for the metabolite rate constant is 'better</span>
<span class='co'># without internal parameter transformation.</span>
<span class='no'>rel_diffs_transformed_ff</span> <span class='kw'><</span> <span class='no'>rel_diffs_untransformed_ff</span></div><div class='output co'>#> 2.5% 97.5%
#> parent_0 TRUE TRUE
#> k_parent TRUE TRUE
#> k_m1 FALSE FALSE
#> f_parent_to_m1 TRUE FALSE
#> sigma FALSE TRUE</div><div class='input'><span class='no'>rel_diffs_transformed_ff</span></div><div class='output co'>#> 2.5% 97.5%
#> parent_0 0.0005408012 0.0002217857
#> k_parent 0.0009596303 0.0009003981
#> k_m1 0.0307277425 0.0290579163
#> f_parent_to_m1 0.0046884178 0.0027782643
#> sigma 0.0550252516 0.0327066836</div><div class='input'><span class='no'>rel_diffs_untransformed_ff</span></div><div class='output co'>#> 2.5% 97.5%
#> parent_0 0.0005408019 0.0002217863
#> k_parent 0.0046099989 0.0023730118
#> k_m1 0.0146746451 0.0025300990
#> f_parent_to_m1 0.0046997668 0.0023460293
#> sigma 0.0550252516 0.0327066836</div><div class='input'>
<span class='co'># The profiling for the following fit does not finish in a reasonable time,</span>
<span class='co'># therefore we use the quadratic approximation</span>
<span class='no'>m_synth_DFOP_par</span> <span class='kw'><-</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'>"DFOP"</span>, <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='st'>"M1"</span>, <span class='st'>"M2"</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'>M2</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>, <span class='kw'>quiet</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>)
<span class='no'>DFOP_par_c</span> <span class='kw'><-</span> <span class='no'>synthetic_data_for_UBA_2014</span><span class='kw'>[[</span><span class='fl'>12</span>]]$<span class='no'>data</span>
<span class='no'>f_tc_2</span> <span class='kw'><-</span> <span class='fu'><a href='mkinfit.html'>mkinfit</a></span>(<span class='no'>m_synth_DFOP_par</span>, <span class='no'>DFOP_par_c</span>, <span class='kw'>error_model</span> <span class='kw'>=</span> <span class='st'>"tc"</span>,
<span class='kw'>error_model_algorithm</span> <span class='kw'>=</span> <span class='st'>"direct"</span>, <span class='kw'>quiet</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>)
<span class='fu'><a href='https://rdrr.io/r/stats/confint.html'>confint</a></span>(<span class='no'>f_tc_2</span>, <span class='kw'>method</span> <span class='kw'>=</span> <span class='st'>"quadratic"</span>)</div><div class='output co'>#> 2.5% 97.5%
#> parent_0 94.596183241 106.19937044
#> k_M1 0.037605436 0.04490758
#> k_M2 0.008568746 0.01087675
#> f_parent_to_M1 0.021464277 0.62023879
#> f_parent_to_M2 0.015166876 0.37975352
#> k1 0.273897622 0.33388081
#> k2 0.018614564 0.02250380
#> g 0.671943572 0.73583247
#> sigma_low 0.251284138 0.83992136
#> rsd_high 0.040410998 0.07661999</div><div class='input'><span class='fu'><a href='https://rdrr.io/r/stats/confint.html'>confint</a></span>(<span class='no'>f_tc_2</span>, <span class='st'>"parent_0"</span>, <span class='kw'>method</span> <span class='kw'>=</span> <span class='st'>"quadratic"</span>)</div><div class='output co'>#> 2.5% 97.5%
#> parent_0 94.59618 106.1994</div><div class='input'># }
</div></pre>
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