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<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 'profile' 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>
<small class="dont-index">Source: <a href='http://github.com/jranke/mkin/blob/master/R/confint.mkinfit.R'><code>R/confint.mkinfit.R</code></a></small>
<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 'profile' 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='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='kw'>rel_tol</span> <span class='kw'>=</span> <span class='fl'>0.01</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.
On Windows machines, cores > 1 is currently not supported.</p></td>
</tr>
<tr>
<th>rel_tol</th>
<td><p>If the method is 'profile', what should be the accuracy
of the lower and upper bounds, relative to the estimate obtained from
the quadratic method?</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 elapsed
#> 3.689 0.991 3.361 </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'>#> <span class='warning'>Warning: scheduled cores 2, 1, 3 encountered errors in user code, all values of the jobs will be affected</span></div><div class='output co'>#> <span class='error'>Error in dimnames(x) <- dn: length of 'dimnames' [2] not equal to array extent</span></div><div class='output co'>#> <span class='message'>Timing stopped at: 0.007 0.042 0.193</span></div><div class='input'><span class='no'>ci_profile</span></div><div class='output co'>#> 2.5% 97.5%
#> parent_0 96.456003640 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</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 96.403839460 1.027931e+02
#> k_parent 0.090823790 1.072543e-01
#> k_m1 0.004012216 6.897547e-03
#> f_parent_to_m1 0.469118713 5.595960e-01
#> 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 96.403839413 1.027931e+02
#> k_parent 0.090491931 1.069035e-01
#> k_m1 0.003835483 6.685819e-03
#> f_parent_to_m1 0.469113365 5.598386e-01
#> sigma 2.396089689 3.854918e+00</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 TRUE TRUE
#> k_parent TRUE TRUE
#> k_m1 FALSE FALSE
#> f_parent_to_m1 TRUE 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 0.000960 0.000900
#> k_m1 0.030700 0.029100
#> f_parent_to_m1 0.004690 0.002780
#> 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 0.004610 0.002370
#> k_m1 0.014700 0.002530
#> f_parent_to_m1 0.004700 0.002350
#> 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 96.456003640 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 96.403839460 1.027931e+02
#> k_parent 0.090823790 1.072543e-01
#> k_m1 0.004012216 6.897547e-03
#> f_parent_to_m1 0.469118713 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 96.403839413 1.027931e+02
#> k_parent 0.090491931 1.069035e-01
#> k_m1 0.003835483 6.685819e-03
#> f_parent_to_m1 0.469113365 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 FALSE</div><div class='input'><span class='no'>rel_diffs_transformed_ff</span></div><div class='output co'>#> 2.5% 97.5%
#> parent_0 0.0005408080 0.0002217794
#> k_parent 0.0009596417 0.0009003876
#> k_m1 0.0307277370 0.0290579182
#> f_parent_to_m1 0.0046884130 0.0027782556
#> 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 0.0005408085 0.0002217799
#> k_parent 0.0046100096 0.0023730229
#> k_m1 0.0146746469 0.0025301011
#> f_parent_to_m1 0.0046997599 0.0023460223
#> 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 94.59613833 106.19939215
#> k_M1 0.03760542 0.04490759
#> k_M2 0.00856874 0.01087675
#> f_parent_to_M1 0.02146166 0.62023888
#> f_parent_to_M2 0.01516502 0.37975343
#> k1 0.27389751 0.33388078
#> k2 0.01861456 0.02250379
#> g 0.67194349 0.73583256
#> sigma_low 0.25128383 0.83992146
#> rsd_high 0.04041100 0.07662001</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 94.59614 106.1994</div><div class='input'># }
</div></pre>
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