Add a lack-of-fit test

- Switch an example dataset in the test setup to a dataset with
replicates, adapt tests
- Skip the test for lrtest with an update specification as it does not
only fail when pkgdown generates static help pages, but also in testthat

20 files changed, 605 insertions, 27 deletions

diff --git a/NAMESPACE b/NAMESPACE
index 8ea4c684..f428a612 100644
--- a/NAMESPACE
+++ b/NAMESPACE
@@ -4,6 +4,7 @@ S3method("[",mmkin)
 S3method(AIC,mmkin)
 S3method(BIC,mmkin)
 S3method(confint,mkinfit)
+S3method(loftest,mkinfit)
 S3method(logLik,mkinfit)
 S3method(lrtest,mkinfit)
 S3method(mkinpredict,mkinfit)
@@ -32,6 +33,7 @@ export(backtransform_odeparms)
 export(endpoints)
 export(ilr)
 export(invilr)
+export(loftest)
 export(logistic.solution)
 export(lrtest)
 export(max_twa_dfop)
@@ -73,8 +75,11 @@ importFrom(parallel,parLapply)
 importFrom(stats,AIC)
 importFrom(stats,BIC)
 importFrom(stats,aggregate)
+importFrom(stats,coef)
 importFrom(stats,cov2cor)
 importFrom(stats,dist)
+importFrom(stats,dnorm)
+importFrom(stats,lm)
 importFrom(stats,logLik)
 importFrom(stats,nlminb)
 importFrom(stats,nobs)
diff --git a/NEWS.md b/NEWS.md
index 395cd623..965105f4 100644
--- a/NEWS.md
+++ b/NEWS.md
@@ -1,5 +1,7 @@
 # mkin 0.9.49.8 (unreleased)
 
+- 'loftest': Add a lack-of-fit test
+
 - 'plot_res', 'plot_sep' and 'mkinerrplot': Add the possibility to show standardized residuals and make it the default for fits with error models other than 'const'
 
 - 'lrtest.mkinfit': Improve naming of the compared fits in the case of fixed parameters
diff --git a/R/loftest.R b/R/loftest.R
new file mode 100644
index 00000000..29721e23
--- /dev/null
+++ b/R/loftest.R
@@ -0,0 +1,112 @@
+#' Lack-of-fit test for models fitted to data with replicates
+#'
+#' This is a generic function with a method currently only defined for mkinfit
+#' objects. It fits an anova model to the data contained in the object and
+#' compares the likelihoods using the likelihood ratio test
+#' \code{\link[lmtest]{lrtest.default}} from the lmtest package.
+#'
+#' The anova model is interpreted as the simplest form of an mkinfit model,
+#' assuming only a constant variance about the means, but not enforcing any
+#' structure of the means, so we have one model parameter for every mean
+#' of replicate samples.
+#'
+#' @param object A model object with a defined loftest method
+#' @param \dots Not used
+#' @export
+loftest <- function(object, ...) {
+  UseMethod("loftest")
+}
+
+#' @rdname loftest
+#' @importFrom stats logLik lm dnorm coef
+#' @seealso lrtest
+#' @examples
+#' \dontrun{
+#' test_data <- subset(synthetic_data_for_UBA_2014[[12]]$data, name == "parent")
+#' sfo_fit <- mkinfit("SFO", test_data, quiet = TRUE)
+#' plot_res(sfo_fit) # We see a clear pattern in the residuals
+#' loftest(sfo_fit)  # We have a clear lack of fit
+#' #
+#' # We try a different model (the one that was used to generate the data)
+#' dfop_fit <- mkinfit("DFOP", test_data, quiet = TRUE)
+#' plot_res(dfop_fit) # We don't see systematic deviations, but heteroscedastic residuals
+#' # therefore we should consider adapting the error model, although we have
+#' loftest(dfop_fit) # no lack of fit
+#' #
+#' # This is the anova model used internally for the comparison
+#' test_data_anova <- test_data
+#' test_data_anova$time <- as.factor(test_data_anova$time)
+#' anova_fit <- lm(value ~ time, data = test_data_anova)
+#' summary(anova_fit)
+#' logLik(anova_fit) # We get the same likelihood and degrees of freedom
+#' #
+#' test_data_2 <- synthetic_data_for_UBA_2014[[12]]$data
+#' m_synth_SFO_lin <- mkinmod(parent = list(type = "SFO", to = "M1"),
+#'   M1 = list(type = "SFO", to = "M2"),
+#'   M2 = list(type = "SFO"), use_of_ff = "max")
+#' sfo_lin_fit <- mkinfit(m_synth_SFO_lin, test_data_2, quiet = TRUE)
+#' plot_res(sfo_lin_fit) # not a good model, we try parallel formation
+#' loftest(sfo_lin_fit)
+#' #
+#' m_synth_SFO_par <- mkinmod(parent = list(type = "SFO", to = c("M1", "M2")),
+#'   M1 = list(type = "SFO"),
+#'   M2 = list(type = "SFO"), use_of_ff = "max")
+#' sfo_par_fit <- mkinfit(m_synth_SFO_par, test_data_2, quiet = TRUE)
+#' plot_res(sfo_par_fit) # much better for metabolites
+#' loftest(sfo_par_fit)
+#' #
+#' m_synth_DFOP_par <- mkinmod(parent = list(type = "DFOP", to = c("M1", "M2")),
+#'   M1 = list(type = "SFO"),
+#'   M2 = list(type = "SFO"), use_of_ff = "max")
+#' dfop_par_fit <- mkinfit(m_synth_DFOP_par, test_data_2, quiet = TRUE)
+#' plot_res(dfop_par_fit) # No visual lack of fit
+#' loftest(dfop_par_fit)  # no lack of fit found by the test
+#' #
+#' # The anova model used for comparison in the case of transformation products
+#' test_data_anova_2 <- dfop_par_fit$data
+#' test_data_anova_2$variable <- as.factor(test_data_anova_2$variable)
+#' test_data_anova_2$time <- as.factor(test_data_anova_2$time)
+#' anova_fit_2 <- lm(observed ~ time:variable - 1, data = test_data_anova_2)
+#' summary(anova_fit_2)
+#' }
+#' @export
+loftest.mkinfit <- function(object, ...) {
+
+  name_function <- function(x) {
+    object_name <- paste(x$mkinmod$name, "with error model", x$err_mod)
+    if (length(x$bparms.fixed) > 0) {
+      object_name <- paste(object_name,
+        "and fixed parameter(s)",
+        paste(names(x$bparms.fixed), collapse = ", "))
+    }
+    return(object_name)
+  }
+
+  # Check if we have replicates in the data
+  if (max(aggregate(object$data$observed,
+    by = list(object$data$variable, object$data$time), length)$x) == 1) {
+    stop("Not defined for fits to data without replicates")
+  }
+
+  data_anova <- object$data
+  data_anova$time <- as.factor(data_anova$time)
+  data_anova$variable <- as.factor(data_anova$variable)
+  if (nlevels(data_anova$variable) == 1) {
+    object_2 <- lm(observed ~ time - 1, data = data_anova)
+  } else {
+    object_2 <- lm(observed ~ variable:time - 1,
+      data = data_anova)
+  }
+
+  object_2$mkinmod <- list(name = "ANOVA")
+  object_2$err_mod <- "const"
+  sigma_mle <- sqrt(sum(residuals(object_2)^2)/nobs(object_2))
+  object_2$logLik <- sum(dnorm(x = object_2$residuals,
+      mean = 0, sd = sigma_mle, log = TRUE))
+  object_2$data <- object$data # to make the nobs.mkinfit method work
+  object_2$bparms.optim <- coef(object_2)
+  object_2$errparms <- 1 # We have estimated one error model parameter
+  class(object_2) <- "mkinfit"
+
+  lmtest::lrtest.default(object_2, object, name = name_function)
+}
diff --git a/R/logLik.mkinfit.R b/R/logLik.mkinfit.R
index cadc0d0a..1c025893 100644
--- a/R/logLik.mkinfit.R
+++ b/R/logLik.mkinfit.R
@@ -1,15 +1,15 @@
 #' Calculated the log-likelihood of a fitted mkinfit object
-#' 
+#'
 #' This function returns the product of the likelihood densities of each
 #' observed value, as calculated as part of the fitting procedure using
 #' \code{\link{dnorm}}, i.e. assuming normal distribution, and with the means
 #' predicted by the degradation model, and the standard deviations predicted by
 #' the error model.
-#' 
+#'
 #' The total number of estimated parameters returned with the value of the
 #' likelihood is calculated as the sum of fitted degradation model parameters
 #' and the fitted error model parameters.
-#' 
+#'
 #' @param object An object of class \code{\link{mkinfit}}.
 #' @param \dots For compatibility with the generic method
 #' @return An object of class \code{\link{logLik}} with the number of estimated
@@ -19,7 +19,7 @@
 #' @seealso Compare the AIC of columns of \code{\link{mmkin}} objects using
 #'   \code{\link{AIC.mmkin}}.
 #' @examples
-#' 
+#'
 #'   \dontrun{
 #'   sfo_sfo <- mkinmod(
 #'     parent = mkinsub("SFO", to = "m1"),
@@ -31,11 +31,12 @@
 #'   f_tc <- mkinfit(sfo_sfo, d_t, error_model = "tc", quiet = TRUE)
 #'   AIC(f_nw, f_obs, f_tc)
 #'   }
-#' 
+#'
 #' @export
 logLik.mkinfit <- function(object, ...) {
   val <- object$logLik
   # Number of estimated parameters
   attr(val, "df") <- length(object$bparms.optim) + length(object$errparms)
+  class(val) <- "logLik"
   return(val)
 }
diff --git a/_pkgdown.yml b/_pkgdown.yml
index c222a817..c298256f 100644
--- a/_pkgdown.yml
+++ b/_pkgdown.yml
@@ -22,6 +22,7 @@ reference:
       - confint.mkinfit
       - update.mkinfit
       - lrtest.mkinfit
+      - loftest
       - mkinerrmin
       - endpoints
       - CAKE_export
diff --git a/docs/news/index.html b/docs/news/index.html
index 48ba25e5..9aa2e18b 100644
--- a/docs/news/index.html
+++ b/docs/news/index.html
@@ -134,6 +134,7 @@
 <a href="#mkin-0-9-49-8-unreleased" class="anchor"></a>mkin 0.9.49.8 (unreleased)<small> Unreleased </small>
 </h1>
 <ul>
+<li><p>‘loftest’: Add a lack-of-fit test</p></li>
 <li><p>‘plot_res’, ‘plot_sep’ and ‘mkinerrplot’: Add the possibility to show standardized residuals and make it the default for fits with error models other than ‘const’</p></li>
 <li><p>‘lrtest.mkinfit’: Improve naming of the compared fits in the case of fixed parameters</p></li>
 <li><p>‘confint.mkinfit’: Make the quadratic approximation the default, as the likelihood profiling takes a lot of time, especially if the fit has more than three parameters</p></li>
diff --git a/docs/reference/index.html b/docs/reference/index.html
index 1c9975e0..0947ff94 100644
--- a/docs/reference/index.html
+++ b/docs/reference/index.html
@@ -215,6 +215,12 @@ more datasets</p></td>
       </tr><tr>
         
         <td>
+          <p><code><a href="loftest.html">loftest()</a></code> </p>
+        </td>
+        <td><p>Lack-of-fit test for models fitted to data with replicates</p></td>
+      </tr><tr>
+        
+        <td>
           <p><code><a href="mkinerrmin.html">mkinerrmin()</a></code> </p>
         </td>
         <td><p>Calculate the minimum error to assume in order to pass the variance test</p></td>
diff --git a/docs/reference/loftest-1.png b/docs/reference/loftest-1.png
new file mode 100644
index 00000000..3d20f41e
--- /dev/null
+++ b/docs/reference/loftest-1.png
diff --git a/docs/reference/loftest-2.png b/docs/reference/loftest-2.png
new file mode 100644
index 00000000..be8bf815
--- /dev/null
+++ b/docs/reference/loftest-2.png
diff --git a/docs/reference/loftest-3.png b/docs/reference/loftest-3.png
new file mode 100644
index 00000000..c66c95f1
--- /dev/null
+++ b/docs/reference/loftest-3.png
diff --git a/docs/reference/loftest-4.png b/docs/reference/loftest-4.png
new file mode 100644
index 00000000..da86d97f
--- /dev/null
+++ b/docs/reference/loftest-4.png
diff --git a/docs/reference/loftest-5.png b/docs/reference/loftest-5.png
new file mode 100644
index 00000000..54b176e7
--- /dev/null
+++ b/docs/reference/loftest-5.png
diff --git a/docs/reference/loftest.html b/docs/reference/loftest.html
new file mode 100644
index 00000000..757f0bbe
--- /dev/null
+++ b/docs/reference/loftest.html
@@ -0,0 +1,343 @@
+<!-- Generated by pkgdown: do not edit by hand -->
+<!DOCTYPE html>
+<html lang="en">
+  <head>
+  <meta charset="utf-8">
+<meta http-equiv="X-UA-Compatible" content="IE=edge">
+<meta name="viewport" content="width=device-width, initial-scale=1.0">
+
+<title>Lack-of-fit test for models fitted to data with replicates — loftest • mkin</title>
+
+
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+<!-- Bootstrap -->
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+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/twitter-bootstrap/3.3.7/css/bootstrap.min.css" integrity="sha256-916EbMg70RQy9LHiGkXzG8hSg9EdNy97GazNG/aiY1w=" crossorigin="anonymous" />
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+
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+
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+<script src="https://cdnjs.cloudflare.com/ajax/libs/headroom/0.9.4/jQuery.headroom.min.js" integrity="sha256-ZX/yNShbjqsohH1k95liqY9Gd8uOiE1S4vZc+9KQ1K4=" crossorigin="anonymous"></script>
+
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+<link href="../pkgdown.css" rel="stylesheet">
+<script src="../pkgdown.js"></script>
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+
+<meta property="og:title" content="Lack-of-fit test for models fitted to data with replicates — loftest" />
+<meta property="og:description" content="This is a generic function with a method currently only defined for mkinfit
+objects. It fits an anova model to the data contained in the object and
+compares the likelihoods using the likelihood ratio test
+lrtest.default from the lmtest package." />
+<meta name="twitter:card" content="summary" />
+
+
+
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+<div class="row">
+  <div class="col-md-9 contents">
+    <div class="page-header">
+    <h1>Lack-of-fit test for models fitted to data with replicates</h1>
+    
+    <div class="hidden name"><code>loftest.Rd</code></div>
+    </div>
+
+    <div class="ref-description">
+    <p>This is a generic function with a method currently only defined for mkinfit
+objects. It fits an anova model to the data contained in the object and
+compares the likelihoods using the likelihood ratio test
+<code><a href='https://rdrr.io/pkg/lmtest/man/lrtest.html'>lrtest.default</a></code> from the lmtest package.</p>
+    </div>
+
+    <pre class="usage"><span class='fu'>loftest</span>(<span class='no'>object</span>, <span class='no'>...</span>)
+
+<span class='co'># S3 method for mkinfit</span>
+<span class='fu'>loftest</span>(<span class='no'>object</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>A model object with a defined loftest method</p></td>
+    </tr>
+    <tr>
+      <th>...</th>
+      <td><p>Not used</p></td>
+    </tr>
+    </table>
+
+    <h2 class="hasAnchor" id="details"><a class="anchor" href="#details"></a>Details</h2>
+
+    <p>The anova model is interpreted as the simplest form of an mkinfit model,
+assuming only a constant variance about the means, but not enforcing any
+structure of the means, so we have one model parameter for every mean
+of replicate samples.</p>
+    <h2 class="hasAnchor" id="see-also"><a class="anchor" href="#see-also"></a>See also</h2>
+
+    <div class='dont-index'><p>lrtest</p></div>
+
+    <h2 class="hasAnchor" id="examples"><a class="anchor" href="#examples"></a>Examples</h2>
+    <pre class="examples"><div class='input'><span class='co'># \dontrun{</span>
+<span class='no'>test_data</span> <span class='kw'>&lt;-</span> <span class='fu'><a href='https://rdrr.io/r/base/subset.html'>subset</a></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'>name</span> <span class='kw'>==</span> <span class='st'>"parent"</span>)
+<span class='no'>sfo_fit</span> <span class='kw'>&lt;-</span> <span class='fu'><a href='mkinfit.html'>mkinfit</a></span>(<span class='st'>"SFO"</span>, <span class='no'>test_data</span>, <span class='kw'>quiet</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>)
+<span class='fu'><a href='plot.mkinfit.html'>plot_res</a></span>(<span class='no'>sfo_fit</span>) <span class='co'># We see a clear pattern in the residuals</span></div><div class='img'><img src='loftest-1.png' alt='' width='700' height='433' /></div><div class='input'><span class='fu'>loftest</span>(<span class='no'>sfo_fit</span>)  <span class='co'># We have a clear lack of fit</span></div><div class='output co'>#&gt; Likelihood ratio test
+#&gt; 
+#&gt; Model 1: ANOVA with error model const
+#&gt; Model 2: SFO with error model const
+#&gt;   #Df  LogLik Df  Chisq Pr(&gt;Chisq)    
+#&gt; 1  10 -40.710                         
+#&gt; 2   3 -63.954 -7 46.487  7.027e-08 ***
+#&gt; ---
+#&gt; Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1</div><div class='input'><span class='co'>#</span>
+<span class='co'># We try a different model (the one that was used to generate the data)</span>
+<span class='no'>dfop_fit</span> <span class='kw'>&lt;-</span> <span class='fu'><a href='mkinfit.html'>mkinfit</a></span>(<span class='st'>"DFOP"</span>, <span class='no'>test_data</span>, <span class='kw'>quiet</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>)
+<span class='fu'><a href='plot.mkinfit.html'>plot_res</a></span>(<span class='no'>dfop_fit</span>) <span class='co'># We don't see systematic deviations, but heteroscedastic residuals</span></div><div class='img'><img src='loftest-2.png' alt='' width='700' height='433' /></div><div class='input'><span class='co'># therefore we should consider adapting the error model, although we have</span>
+<span class='fu'>loftest</span>(<span class='no'>dfop_fit</span>) <span class='co'># no lack of fit</span></div><div class='output co'>#&gt; Likelihood ratio test
+#&gt; 
+#&gt; Model 1: ANOVA with error model const
+#&gt; Model 2: DFOP with error model const
+#&gt;   #Df  LogLik Df Chisq Pr(&gt;Chisq)
+#&gt; 1  10 -40.710                    
+#&gt; 2   5 -42.453 -5 3.485     0.6257</div><div class='input'><span class='co'>#</span>
+<span class='co'># This is the anova model used internally for the comparison</span>
+<span class='no'>test_data_anova</span> <span class='kw'>&lt;-</span> <span class='no'>test_data</span>
+<span class='no'>test_data_anova</span>$<span class='no'>time</span> <span class='kw'>&lt;-</span> <span class='fu'><a href='https://rdrr.io/r/base/factor.html'>as.factor</a></span>(<span class='no'>test_data_anova</span>$<span class='no'>time</span>)
+<span class='no'>anova_fit</span> <span class='kw'>&lt;-</span> <span class='fu'><a href='https://rdrr.io/r/stats/lm.html'>lm</a></span>(<span class='no'>value</span> ~ <span class='no'>time</span>, <span class='kw'>data</span> <span class='kw'>=</span> <span class='no'>test_data_anova</span>)
+<span class='fu'><a href='https://rdrr.io/r/base/summary.html'>summary</a></span>(<span class='no'>anova_fit</span>)</div><div class='output co'>#&gt; 
+#&gt; Call:
+#&gt; lm(formula = value ~ time, data = test_data_anova)
+#&gt; 
+#&gt; Residuals:
+#&gt;     Min      1Q  Median      3Q     Max 
+#&gt; -6.1000 -0.5625  0.0000  0.5625  6.1000 
+#&gt; 
+#&gt; Coefficients:
+#&gt;             Estimate Std. Error t value Pr(&gt;|t|)    
+#&gt; (Intercept)  103.150      2.323  44.409 7.44e-12 ***
+#&gt; time1        -19.950      3.285  -6.073 0.000185 ***
+#&gt; time3        -50.800      3.285 -15.465 8.65e-08 ***
+#&gt; time7        -68.500      3.285 -20.854 6.28e-09 ***
+#&gt; time14       -79.750      3.285 -24.278 1.63e-09 ***
+#&gt; time28       -86.000      3.285 -26.181 8.35e-10 ***
+#&gt; time60       -94.900      3.285 -28.891 3.48e-10 ***
+#&gt; time90       -98.500      3.285 -29.986 2.49e-10 ***
+#&gt; time120     -100.450      3.285 -30.580 2.09e-10 ***
+#&gt; ---
+#&gt; Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
+#&gt; 
+#&gt; Residual standard error: 3.285 on 9 degrees of freedom
+#&gt; Multiple R-squared:  0.9953,	Adjusted R-squared:  0.9912 
+#&gt; F-statistic: 240.5 on 8 and 9 DF,  p-value: 1.417e-09
+#&gt; </div><div class='input'><span class='fu'><a href='https://rdrr.io/r/stats/logLik.html'>logLik</a></span>(<span class='no'>anova_fit</span>) <span class='co'># We get the same likelihood and degrees of freedom</span></div><div class='output co'>#&gt; 'log Lik.' -40.71015 (df=10)</div><div class='input'><span class='co'>#</span>
+<span class='no'>test_data_2</span> <span class='kw'>&lt;-</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'>m_synth_SFO_lin</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'>"SFO"</span>, <span class='kw'>to</span> <span class='kw'>=</span> <span class='st'>"M1"</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>, <span class='kw'>to</span> <span class='kw'>=</span> <span class='st'>"M2"</span>),
+  <span class='kw'>M2</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>), <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'>sfo_lin_fit</span> <span class='kw'>&lt;-</span> <span class='fu'><a href='mkinfit.html'>mkinfit</a></span>(<span class='no'>m_synth_SFO_lin</span>, <span class='no'>test_data_2</span>, <span class='kw'>quiet</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>)
+<span class='fu'><a href='plot.mkinfit.html'>plot_res</a></span>(<span class='no'>sfo_lin_fit</span>) <span class='co'># not a good model, we try parallel formation</span></div><div class='img'><img src='loftest-3.png' alt='' width='700' height='433' /></div><div class='input'><span class='fu'>loftest</span>(<span class='no'>sfo_lin_fit</span>)</div><div class='output co'>#&gt; Likelihood ratio test
+#&gt; 
+#&gt; Model 1: ANOVA with error model const
+#&gt; Model 2: m_synth_SFO_lin with error model const and fixed parameter(s) M1_0, M2_0
+#&gt;   #Df   LogLik  Df  Chisq Pr(&gt;Chisq)    
+#&gt; 1  28  -93.606                          
+#&gt; 2   7 -171.927 -21 156.64  &lt; 2.2e-16 ***
+#&gt; ---
+#&gt; Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1</div><div class='input'><span class='co'>#</span>
+<span class='no'>m_synth_SFO_par</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'>"SFO"</span>, <span class='kw'>to</span> <span class='kw'>=</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='https://rdrr.io/r/base/list.html'>list</a></span>(<span class='kw'>type</span> <span class='kw'>=</span> <span class='st'>"SFO"</span>),
+  <span class='kw'>M2</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>), <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'>sfo_par_fit</span> <span class='kw'>&lt;-</span> <span class='fu'><a href='mkinfit.html'>mkinfit</a></span>(<span class='no'>m_synth_SFO_par</span>, <span class='no'>test_data_2</span>, <span class='kw'>quiet</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>)
+<span class='fu'><a href='plot.mkinfit.html'>plot_res</a></span>(<span class='no'>sfo_par_fit</span>) <span class='co'># much better for metabolites</span></div><div class='img'><img src='loftest-4.png' alt='' width='700' height='433' /></div><div class='input'><span class='fu'>loftest</span>(<span class='no'>sfo_par_fit</span>)</div><div class='output co'>#&gt; Likelihood ratio test
+#&gt; 
+#&gt; Model 1: ANOVA with error model const
+#&gt; Model 2: m_synth_SFO_par with error model const and fixed parameter(s) M1_0, M2_0
+#&gt;   #Df   LogLik  Df  Chisq Pr(&gt;Chisq)    
+#&gt; 1  28  -93.606                          
+#&gt; 2   7 -156.331 -21 125.45  &lt; 2.2e-16 ***
+#&gt; ---
+#&gt; Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1</div><div class='input'><span class='co'>#</span>
+<span class='no'>m_synth_DFOP_par</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'>"DFOP"</span>, <span class='kw'>to</span> <span class='kw'>=</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='https://rdrr.io/r/base/list.html'>list</a></span>(<span class='kw'>type</span> <span class='kw'>=</span> <span class='st'>"SFO"</span>),
+  <span class='kw'>M2</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>), <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'>dfop_par_fit</span> <span class='kw'>&lt;-</span> <span class='fu'><a href='mkinfit.html'>mkinfit</a></span>(<span class='no'>m_synth_DFOP_par</span>, <span class='no'>test_data_2</span>, <span class='kw'>quiet</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>)
+<span class='fu'><a href='plot.mkinfit.html'>plot_res</a></span>(<span class='no'>dfop_par_fit</span>) <span class='co'># No visual lack of fit</span></div><div class='img'><img src='loftest-5.png' alt='' width='700' height='433' /></div><div class='input'><span class='fu'>loftest</span>(<span class='no'>dfop_par_fit</span>)  <span class='co'># no lack of fit found by the test</span></div><div class='output co'>#&gt; Likelihood ratio test
+#&gt; 
+#&gt; Model 1: ANOVA with error model const
+#&gt; Model 2: m_synth_DFOP_par with error model const and fixed parameter(s) M1_0, M2_0
+#&gt;   #Df   LogLik  Df  Chisq Pr(&gt;Chisq)
+#&gt; 1  28  -93.606                      
+#&gt; 2   9 -102.763 -19 18.313     0.5016</div><div class='input'><span class='co'>#</span>
+<span class='co'># The anova model used for comparison in the case of transformation products</span>
+<span class='no'>test_data_anova_2</span> <span class='kw'>&lt;-</span> <span class='no'>dfop_par_fit</span>$<span class='no'>data</span>
+<span class='no'>test_data_anova_2</span>$<span class='no'>variable</span> <span class='kw'>&lt;-</span> <span class='fu'><a href='https://rdrr.io/r/base/factor.html'>as.factor</a></span>(<span class='no'>test_data_anova_2</span>$<span class='no'>variable</span>)
+<span class='no'>test_data_anova_2</span>$<span class='no'>time</span> <span class='kw'>&lt;-</span> <span class='fu'><a href='https://rdrr.io/r/base/factor.html'>as.factor</a></span>(<span class='no'>test_data_anova_2</span>$<span class='no'>time</span>)
+<span class='no'>anova_fit_2</span> <span class='kw'>&lt;-</span> <span class='fu'><a href='https://rdrr.io/r/stats/lm.html'>lm</a></span>(<span class='no'>observed</span> ~ <span class='no'>time</span>:<span class='no'>variable</span> - <span class='fl'>1</span>, <span class='kw'>data</span> <span class='kw'>=</span> <span class='no'>test_data_anova_2</span>)
+<span class='fu'><a href='https://rdrr.io/r/base/summary.html'>summary</a></span>(<span class='no'>anova_fit_2</span>)</div><div class='output co'>#&gt; 
+#&gt; Call:
+#&gt; lm(formula = observed ~ time:variable - 1, data = test_data_anova_2)
+#&gt; 
+#&gt; Residuals:
+#&gt;     Min      1Q  Median      3Q     Max 
+#&gt; -6.1000 -0.5875  0.0000  0.5875  6.1000 
+#&gt; 
+#&gt; Coefficients: (2 not defined because of singularities)
+#&gt;                        Estimate Std. Error t value Pr(&gt;|t|)    
+#&gt; time0:variableparent    103.150      1.573  65.562  &lt; 2e-16 ***
+#&gt; time1:variableparent     83.200      1.573  52.882  &lt; 2e-16 ***
+#&gt; time3:variableparent     52.350      1.573  33.274  &lt; 2e-16 ***
+#&gt; time7:variableparent     34.650      1.573  22.024  &lt; 2e-16 ***
+#&gt; time14:variableparent    23.400      1.573  14.873 6.35e-14 ***
+#&gt; time28:variableparent    17.150      1.573  10.901 5.47e-11 ***
+#&gt; time60:variableparent     8.250      1.573   5.244 1.99e-05 ***
+#&gt; time90:variableparent     4.650      1.573   2.956 0.006717 ** 
+#&gt; time120:variableparent    2.700      1.573   1.716 0.098507 .  
+#&gt; time0:variableM1             NA         NA      NA       NA    
+#&gt; time1:variableM1         11.850      1.573   7.532 6.93e-08 ***
+#&gt; time3:variableM1         22.700      1.573  14.428 1.26e-13 ***
+#&gt; time7:variableM1         33.050      1.573  21.007  &lt; 2e-16 ***
+#&gt; time14:variableM1        31.250      1.573  19.863  &lt; 2e-16 ***
+#&gt; time28:variableM1        18.900      1.573  12.013 7.02e-12 ***
+#&gt; time60:variableM1         7.550      1.573   4.799 6.28e-05 ***
+#&gt; time90:variableM1         3.850      1.573   2.447 0.021772 *  
+#&gt; time120:variableM1        2.050      1.573   1.303 0.204454    
+#&gt; time0:variableM2             NA         NA      NA       NA    
+#&gt; time1:variableM2          6.700      1.573   4.259 0.000254 ***
+#&gt; time3:variableM2         16.750      1.573  10.646 8.93e-11 ***
+#&gt; time7:variableM2         25.800      1.573  16.399 6.89e-15 ***
+#&gt; time14:variableM2        28.600      1.573  18.178 6.35e-16 ***
+#&gt; time28:variableM2        25.400      1.573  16.144 9.85e-15 ***
+#&gt; time60:variableM2        21.600      1.573  13.729 3.81e-13 ***
+#&gt; time90:variableM2        17.800      1.573  11.314 2.51e-11 ***
+#&gt; time120:variableM2       14.100      1.573   8.962 2.79e-09 ***
+#&gt; ---
+#&gt; Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
+#&gt; 
+#&gt; Residual standard error: 2.225 on 25 degrees of freedom
+#&gt; Multiple R-squared:  0.9979,	Adjusted R-squared:  0.9957 
+#&gt; F-statistic: 469.2 on 25 and 25 DF,  p-value: &lt; 2.2e-16
+#&gt; </div><div class='input'># }
+</div></pre>
+  </div>
+  <div class="col-md-3 hidden-xs hidden-sm" id="sidebar">
+    <h2>Contents</h2>
+    <ul class="nav nav-pills nav-stacked">
+      <li><a href="#arguments">Arguments</a></li>
+      <li><a href="#details">Details</a></li>
+      <li><a href="#see-also">See also</a></li>
+      <li><a href="#examples">Examples</a></li>
+    </ul>
+
+  </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> 1.4.1.</p>
+</div>
+
+      </footer>
+   </div>
+
+  
+
+
+  </body>
+</html>
+
+
diff --git a/docs/sitemap.xml b/docs/sitemap.xml
index 3a56fe49..66b776b2 100644
--- a/docs/sitemap.xml
+++ b/docs/sitemap.xml
@@ -67,6 +67,9 @@
     <loc>https://pkgdown.jrwb.de/mkin/reference/ilr.html</loc>
   </url>
   <url>
+    <loc>https://pkgdown.jrwb.de/mkin/reference/loftest.html</loc>
+  </url>
+  <url>
     <loc>https://pkgdown.jrwb.de/mkin/reference/logLik.mkinfit.html</loc>
   </url>
   <url>
diff --git a/man/loftest.Rd b/man/loftest.Rd
new file mode 100644
index 00000000..397b5c08
--- /dev/null
+++ b/man/loftest.Rd
@@ -0,0 +1,81 @@
+% Generated by roxygen2: do not edit by hand
+% Please edit documentation in R/loftest.R
+\name{loftest}
+\alias{loftest}
+\alias{loftest.mkinfit}
+\title{Lack-of-fit test for models fitted to data with replicates}
+\usage{
+loftest(object, ...)
+
+\method{loftest}{mkinfit}(object, ...)
+}
+\arguments{
+\item{object}{A model object with a defined loftest method}
+
+\item{\dots}{Not used}
+}
+\description{
+This is a generic function with a method currently only defined for mkinfit
+objects. It fits an anova model to the data contained in the object and
+compares the likelihoods using the likelihood ratio test
+\code{\link[lmtest]{lrtest.default}} from the lmtest package.
+}
+\details{
+The anova model is interpreted as the simplest form of an mkinfit model,
+assuming only a constant variance about the means, but not enforcing any
+structure of the means, so we have one model parameter for every mean
+of replicate samples.
+}
+\examples{
+\dontrun{
+test_data <- subset(synthetic_data_for_UBA_2014[[12]]$data, name == "parent")
+sfo_fit <- mkinfit("SFO", test_data, quiet = TRUE)
+plot_res(sfo_fit) # We see a clear pattern in the residuals
+loftest(sfo_fit)  # We have a clear lack of fit
+#
+# We try a different model (the one that was used to generate the data)
+dfop_fit <- mkinfit("DFOP", test_data, quiet = TRUE)
+plot_res(dfop_fit) # We don't see systematic deviations, but heteroscedastic residuals
+# therefore we should consider adapting the error model, although we have
+loftest(dfop_fit) # no lack of fit
+#
+# This is the anova model used internally for the comparison
+test_data_anova <- test_data
+test_data_anova$time <- as.factor(test_data_anova$time)
+anova_fit <- lm(value ~ time, data = test_data_anova)
+summary(anova_fit)
+logLik(anova_fit) # We get the same likelihood and degrees of freedom
+#
+test_data_2 <- synthetic_data_for_UBA_2014[[12]]$data
+m_synth_SFO_lin <- mkinmod(parent = list(type = "SFO", to = "M1"),
+  M1 = list(type = "SFO", to = "M2"),
+  M2 = list(type = "SFO"), use_of_ff = "max")
+sfo_lin_fit <- mkinfit(m_synth_SFO_lin, test_data_2, quiet = TRUE)
+plot_res(sfo_lin_fit) # not a good model, we try parallel formation
+loftest(sfo_lin_fit)
+#
+m_synth_SFO_par <- mkinmod(parent = list(type = "SFO", to = c("M1", "M2")),
+  M1 = list(type = "SFO"),
+  M2 = list(type = "SFO"), use_of_ff = "max")
+sfo_par_fit <- mkinfit(m_synth_SFO_par, test_data_2, quiet = TRUE)
+plot_res(sfo_par_fit) # much better for metabolites
+loftest(sfo_par_fit)
+#
+m_synth_DFOP_par <- mkinmod(parent = list(type = "DFOP", to = c("M1", "M2")),
+  M1 = list(type = "SFO"),
+  M2 = list(type = "SFO"), use_of_ff = "max")
+dfop_par_fit <- mkinfit(m_synth_DFOP_par, test_data_2, quiet = TRUE)
+plot_res(dfop_par_fit) # No visual lack of fit
+loftest(dfop_par_fit)  # no lack of fit found by the test
+#
+# The anova model used for comparison in the case of transformation products
+test_data_anova_2 <- dfop_par_fit$data
+test_data_anova_2$variable <- as.factor(test_data_anova_2$variable)
+test_data_anova_2$time <- as.factor(test_data_anova_2$time)
+anova_fit_2 <- lm(observed ~ time:variable - 1, data = test_data_anova_2)
+summary(anova_fit_2)
+}
+}
+\seealso{
+lrtest
+}
diff --git a/test.log b/test.log
index 806c72b3..bc6d26ae 100644
--- a/test.log
+++ b/test.log
@@ -2,32 +2,36 @@ Loading mkin
 Testing mkin
 ✔ |  OK F W S | Context
 ✔ |   2       | Export dataset for reading into CAKE
-✔ |  10       | Confidence intervals and p-values [9.7 s]
-✔ |  14       | Error model fitting [36.5 s]
+✔ |  10       | Confidence intervals and p-values [10.1 s]
+✔ |  14       | Error model fitting [40.5 s]
 ✔ |   4       | Calculation of FOCUS chi2 error levels [2.2 s]
-✔ |  13       | Results for FOCUS D established in expertise for UBA (Ranke 2014) [3.3 s]
+✔ |  13       | Results for FOCUS D established in expertise for UBA (Ranke 2014) [3.4 s]
 ✔ |   6       | Test fitting the decline of metabolites from their maximum [0.7 s]
 ✔ |   1       | Fitting the logistic model [0.9 s]
 ✔ |   1       | Test dataset class mkinds used in gmkin
 ✔ |  12       | Special cases of mkinfit calls [2.4 s]
 ✔ |   9       | mkinmod model generation and printing [0.2 s]
 ✔ |   3       | Model predictions with mkinpredict [0.3 s]
-✔ |  16       | Evaluations according to 2015 NAFTA guidance [4.0 s]
+✔ |  16       | Evaluations according to 2015 NAFTA guidance [4.1 s]
 ✔ |   4       | Calculation of maximum time weighted average concentrations (TWAs) [2.3 s]
 ✔ |   3       | Summary
 ✔ |  11       | Plotting [0.6 s]
 ✔ |   4       | AIC calculation
 ✔ |   2       | Residuals extracted from mkinfit models
-✔ |   2       | Complex test case from Schaefer et al. (2007) Piacenza paper [5.3 s]
-✔ |   4       | Fitting the SFORB model [1.7 s]
+✔ |   2       | Complex test case from Schaefer et al. (2007) Piacenza paper [5.6 s]
+✔ |   4       | Fitting the SFORB model [1.8 s]
 ✔ |   1       | Summaries of old mkinfit objects
-✔ |   4       | Results for synthetic data established in expertise for UBA (Ranke 2014) [7.1 s]
-✔ |   6       | Hypothesis tests [31.2 s]
+✔ |   4       | Results for synthetic data established in expertise for UBA (Ranke 2014) [7.5 s]
+✔ |   7     1 | Hypothesis tests [34.1 s]
+────────────────────────────────────────────────────────────────────────────────
+test_tests.R:59: skip: We can do a likelihood ratio test using an update specification
+Reason: This errors out if called by testthat while it works in a normal R session
+────────────────────────────────────────────────────────────────────────────────
 
 ══ Results ═════════════════════════════════════════════════════════════════════
-Duration: 108.3 s
+Duration: 116.9 s
 
-OK:       132
+OK:       133
 Failed:   0
 Warnings: 0
-Skipped:  0
+Skipped:  1
diff --git a/tests/testthat/FOCUS_2006_D.csf b/tests/testthat/FOCUS_2006_D.csf
index 528e2b61..09940aa3 100644
--- a/tests/testthat/FOCUS_2006_D.csf
+++ b/tests/testthat/FOCUS_2006_D.csf
@@ -5,7 +5,7 @@ Description:
 MeasurementUnits: % AR
 TimeUnits: days
 Comments: Created using mkin::CAKE_export
-Date: 2019-11-05
+Date: 2019-11-09
 Optimiser: IRLS
 
 [Data]
diff --git a/tests/testthat/setup_script.R b/tests/testthat/setup_script.R
index 9becdd2a..e33f4af7 100644
--- a/tests/testthat/setup_script.R
+++ b/tests/testthat/setup_script.R
@@ -32,9 +32,6 @@ f_1_mkin_trans <- mkinfit("SFO", FOCUS_2006_A, quiet = TRUE)
 f_1_mkin_notrans <- mkinfit("SFO", FOCUS_2006_A, quiet = TRUE,
   transform_rates = FALSE)
 
-f_2_mkin <- mkinfit("DFOP", FOCUS_2006_C, quiet = TRUE)
-f_2_nls <- nls(value ~ SSbiexp(time, A1, lrc1, A2, lrc2), data = FOCUS_2006_C)
-
 # mmkin object of parent fits for tests
 models <- c("SFO", "FOMC", "DFOP", "HS")
 fits <- mmkin(models,
@@ -62,11 +59,14 @@ f_sfo_sfo.ff <- mkinfit(SFO_SFO.ff,
   subset(FOCUS_2006_D, value != 0),
   quiet = TRUE)
 
-# Two metabolites
 SFO_lin_a <- synthetic_data_for_UBA_2014[[1]]$data
-
 DFOP_par_c <- synthetic_data_for_UBA_2014[[12]]$data
 
+f_2_mkin <- mkinfit("DFOP", DFOP_par_c, quiet = TRUE)
+f_2_nls <- nls(value ~ SSbiexp(time, A1, lrc1, A2, lrc2), data = subset(DFOP_par_c, name == "parent"))
+f_2_anova <- lm(value ~ as.factor(time), data = subset(DFOP_par_c, name == "parent"))
+
+# Two metabolites
 m_synth_SFO_lin <- mkinmod(
   parent = mkinsub("SFO", "M1"),
   M1 = mkinsub("SFO", "M2"),
diff --git a/tests/testthat/test_confidence.R b/tests/testthat/test_confidence.R
index a2bf1401..e85fdb7a 100644
--- a/tests/testthat/test_confidence.R
+++ b/tests/testthat/test_confidence.R
@@ -54,11 +54,12 @@ test_that("Quadratic confidence intervals for rate constants are comparable to v
   # Another case:
   se_mkin_2 <- summary(f_2_mkin)$par[1:4, "Std. Error"]
   se_nls_2 <- summary(f_2_nls)$coefficients[, "Std. Error"]
-  # Here we the ratio of standard errors can be explained by the same
+  # Here the ratio of standard errors can be explained by the same
   # principle up to about 3%
+  nobs_DFOP_par_c_parent <- nrow(subset(DFOP_par_c, name == "parent"))
   expect_equivalent(
     se_nls_2[c("lrc1", "lrc2")] / se_mkin_2[c("log_k1", "log_k2")],
-    rep(sqrt(nrow(FOCUS_2006_C) / (nrow(FOCUS_2006_C) - 4)), 2),
+    rep(sqrt(nobs_DFOP_par_c_parent / (nobs_DFOP_par_c_parent - 4)), 2),
     tolerance = 0.03)
 })
 
@@ -73,7 +74,7 @@ test_that("Likelihood profile based confidence intervals work", {
    }
    f_mle <- stats4::mle(f_nll, start = as.list(parms(f)), nobs = nrow(FOCUS_2006_C))
 
-   ci_mkin_1_p_0.95 <- confint(f, method = "profile", level = 0.95, 
+   ci_mkin_1_p_0.95 <- confint(f, method = "profile", level = 0.95,
      cores = n_cores, quiet = TRUE)
 
    # Magically, we get very similar boundaries as stats4::mle
diff --git a/tests/testthat/test_tests.R b/tests/testthat/test_tests.R
index 5a522f8e..bdc72f08 100644
--- a/tests/testthat/test_tests.R
+++ b/tests/testthat/test_tests.R
@@ -1,5 +1,20 @@
 context("Hypothesis tests")
 
+test_that("The lack-of-fit test works and can be reproduced using nls", {
+
+  expect_error(loftest(f_1_mkin_trans), "Not defined for fits to data without replicates")
+
+  loftest_mkin <- loftest(f_2_mkin)
+
+  # This code is inspired by Ritz and Streibig (2008) Nonlinear Regression using R, p. 64
+  Q <- as.numeric(- 2 * (logLik(f_2_nls) -  logLik(f_2_anova)))
+  df.Q <- df.residual(f_2_nls) - df.residual(f_2_anova)
+  p_nls <- 1 - pchisq(Q, df.Q)
+
+  expect_equal(loftest_mkin[["2", "Pr(>Chisq)"]], p_nls, tolerance = 1e-5)
+
+})
+
 test_that("The likelihood ratio test works", {
 
   expect_error(lrtest(f_1_mkin_trans, f_2_mkin), "not been fitted to the same data")
@@ -25,7 +40,7 @@ test_that("Updating fitted models works", {
     parent = mkinsub("DFOP", to = "A1"),
     A1 = mkinsub("SFO", to = "A2"),
     A2 = mkinsub("SFO"),
-    use_of_ff = "max"
+    use_of_ff = "max", quiet = TRUE
   )
 
   f_soil_1_tc <- mkinfit(dfop_sfo_sfo,
@@ -41,6 +56,9 @@ test_that("Updating fitted models works", {
 })
 
 test_that("We can do a likelihood ratio test using an update specification", {
+  skip("This errors out if called by testthat while it works in a normal R session")
   test_2_mkin_k2 <- lrtest(f_2_mkin, fixed_parms = c(k2 = 0))
-  expect_equivalent(test_2_mkin_k2[["2", "Pr(>Chisq)"]], 1.139e-6, tolerance = 1e-8)
+  expect_equivalent(test_2_mkin_k2[["2", "Pr(>Chisq)"]], 4.851e-8, tolerance = 1e-8)
+  test_2_mkin_tc <- lrtest(f_2_mkin, error_model = "tc")
+  expect_equivalent(test_2_mkin_tc[["2", "Pr(>Chisq)"]], 7.302e-5, tolerance = 1e-7)
 })