diff --git a/vignettes/modelling.Snw b/vignettes/modelling.Rmd
similarity index 73%
rename from vignettes/modelling.Snw
rename to vignettes/modelling.Rmd
index eec92e2eb124dd1992f2f1ea1584daa4fcc28a26..97cf4998f59492eabba7da5399dfdee642e92d07 100644
--- a/vignettes/modelling.Snw
+++ b/vignettes/modelling.Rmd
@@ -1,21 +1,25 @@
-\documentclass{article}
-
-\usepackage{fullpage}
-\usepackage{amsmath}
-\usepackage{amsfonts}
-
-%\VignetteIndexEntry{Models}
-\title{Models in 'morse' package}
-
-\begin{document}
-\SweaveOpts{concordance=TRUE}
+---
+title: "Models in 'morse' package"
+output: rmarkdown::html_vignette
+vignette: >
+ %\VignetteIndexEntry{Models in 'morse' package}
+ %\VignetteEngine{knitr::rmarkdown}
+ %\VignetteEncoding{UTF-8}
+bibliography: biblio.bib
+---
+
+\newcommand*{\INT}{${\tiny INT}$}
+\newcommand*{\diffdchar}{\mathrm{d}}
+\newcommand*{\dd}{\mathop{\diffdchar\!}}
+\newcommand*{\Cint}{C^{\mbox{\INT}}}
+\newcommand*{\z}{z_w}
+\newcommand*{\tz}{t^z}
-\maketitle
-This document describes the statistical models used in 'morse' to
+This document describes the statistical models used in **morse** to
analyze survival and reproduction data, and as such serves as a
mathematical specification of the package. For a more practical
-introduction, please consult the ``Tutorial'' vignette ; for
+introduction, please consult the **Tutorial** vignette ; for
information on the structure and contents of the library, please
consult the reference manual.
@@ -24,25 +28,26 @@ where posterior distributions are computed from the
likelihood of observed data combined with prior distributions on the
parameters. These priors are specified after each model description.
-\section{Survival toxicity tests}
+# Survival toxicity tests
+
In a survival toxicity test, subjects are exposed to a measured concentration of a
contaminant over a given period of time and the number of surviving
organisms is measured at certain time points during exposure.
In most standard toxicity tests, the concentration is held constant throughout the
-whole experiment, which we will assume for \emph{\ref{subsect:TT}~Analysis of target
-time survival toxicity tests}, but not for \emph{\ref{subsect:TKTD}~Toxicokinetic-Toxicodynamic modeling}
+whole experiment, which we will assume for
+**Analysis of target time survival toxicity tests**,
+but not for **Toxicokinetic-Toxicodynamic modeling**
which can handled time variable exposure.
In the case of constant exposure, an experiment is generally replicated
several times and also repeated for various levels of the contaminant.
For time-variable exposure, a profile of exposure is usually unique,
and the experiment is repeated with several profiles of exposures.
-In so-called \emph{target time} toxicity tests, the mortality is usually analyzed
+In so-called *target time* toxicity tests, the mortality is usually analyzed
at the end of the experiment. The chosen time point for this analysis is called
-\emph{target time}. Let us see how this particular case is handled in 'morse'.
+*target time*. Let us see how this particular case is handled in 'morse'.
-\subsection{Analysis of target time survival toxicity tests}
-\label{subsect:TT}
+## Analysis of target time survival toxicity tests
A dataset from a target time survival toxicity test is a collection $D = \{
(c_i, n_i^{init}, n_i) \}_i$ of experiments, where $c_i$ is the tested
@@ -50,10 +55,11 @@ concentration, $n_i^{init}$ the initial number of organisms and
$n_i$ the number of organisms at the chosen target time. Triplets such that
$c_i = 0$ correspond to control experiments.
-\subsubsection{Modelling}
+### Modelling
+
In the particular case of target time analysis, the
model used in 'morse' is defined as follows. Let $t$ be the target time
-in days. We suppose the \emph{mean survival rate after $t$ days} is
+in days. We suppose the *mean survival rate after $t$ days* is
given by a function $f$ of the contaminant level $c$. We also suppose
that the death of two organisms are two independent events. Hence,
given an initial number $n^{init}_i$ of organisms in the toxicity test at
@@ -74,30 +80,27 @@ corresponds to the survival rate in absence of contaminant and $e$
corresponds to the $LC_{50}$.
Parameter $b$ is related to the effect intensity of the contaminant.
-\subsubsection{Inference}
+### Inference
+
Posterior distributions for parameters $b$,
-$d$ and $e$ are estimated using the JAGS software~\cite{rjags2016} with the following priors:
-\begin{itemize}
-\item we assume the range of tested concentrations in an
+$d$ and $e$ are estimated using the JAGS software [@rjags2016] with the following priors:
+
+* we assume the range of tested concentrations in an
experiment is chosen to contain the $LC_{50}$ with high
probability. More formally, we choose:
$$\log_{10} e \sim \mathcal{N}\left(\frac{\log_{10} (\min_i c_i) + \log_{10} (\max_i c_i)}{2}, \frac{\log_{10} (\max_i c_i) - \log_{10} (\min_i c_i)}{4} \right)$$
which implies $e$ has a probability slightly higher than 0.95 to lie
between the minimum and the maximum tested concentrations.
-\item we choose a quasi non-informative prior distribution for the
+* we choose a quasi non-informative prior distribution for the
shape parameter $b$:
$$\log_{10} b \sim \mathcal{U}(-2,2)$$
-\end{itemize}
-\begin{align*}
-\end{align*}
The prior on $d$ is chosen as follows: if we observe
no mortality in control experiments then we set $d = 1$, otherwise we
assume a uniform prior for $d$ between 0 and 1.
-\subsection{Toxicokinetic-Toxicodynamic modeling}
-\label{subsect:TKTD}
+## Toxicokinetic-Toxicodynamic modeling
For datasets featuring time series measurements, more complete models
can be used to estimate the effect of a contaminant on survival.
@@ -108,43 +111,26 @@ t_m$ and $t_0 = 0$), thus providing a collection $D = {(c_i, t_k,
n_i^k)}_{i,k}$ of experiments.
In 'morse', we propose two
Toxicokinetic-Toxicodynamic (TKTD) models belonging to
-the General Unified Threshold model for Survival (GUTS)~\cite{jager2011, Jager2018GUTSbook}.
-One is known as the \emph{reduced stochastic death} model~\cite{nyman2012} or
-GUTS-SD and the other is the \emph{reduced organism tolerance} model or GUTS-IT, which we
+the General Unified Threshold model for Survival (GUTS) [@jager2011; @Jager2018GUTSbook].
+One is known as the *reduced stochastic death* model [@nyman2012] or
+GUTS-SD and the other is the *reduced organism tolerance* model or GUTS-IT, which we
describe now.
-\newcommand*{\diffdchar}{\mathrm{d}}
-\newcommand*{\dd}{\mathop{\diffdchar\!}}
-\newcommand*{\Cint}{C^{\mbox{\tiny INT}}}
-\newcommand*{\z}{z_w}
-\newcommand*{\tz}{t^z}
+| Parameters | Symbols | Alternative symbols | Units | Models |
+|:--------------------------------------------|:--------|---------------------|--------------------|-----------|
+| Background hazard rate | $h_b$ | $m_0$ | $\text{time}^{-1}$ | SD and IT |
+| Dominant toxicokinetic rate constant | $k_d$ | $\mbox{NEC}$ | $\text{time}^{-1}$ | SD and IT |
+| Threshold for effects | $\z$ | $m_0$ | $[D]$ | SD |
+| Killing rate constant | $b_w$ | $k_k$ | $[D]^{-1}$ | SD |
+| Median of the threshold effect distribution | $m_w$ | $\alpha$ | $[D]$ | IT |
+| Shape of the threshold effect distribution | $\beta$ | $-$ | $n.d.$ | IT |
+: *Table: Parameters and symbols used for GUTS-SD and GUTS-IT models. Alternative symbols are used within pubications (see for instance [@jager2011; @delignette2017; @Jager2018GUTSbook]. The unit $[D]$ refers to unit of actual damage, $n.d$ for non dimensional. For GUTS-IT model, we assume a log-logistic distributions, but other distributions are occasionally used [@albert2016].*
-\begin{table}
-\caption{Parameters and symbols used for GUTS-SD and GUTS-IT models. Alternative symbols are used within pubications (see for instance \cite{jager2011, delignette2017, Jager2018GUTSbook}). The unit $[D]$ refers to unit of actual damage, $n.d$ for non dimensional. For GUTS-IT model, we assume a log-logistic distributions, but other distributions are occasionally used \cite{albert2016}.}
-\label{tab:parameters}
-\begin{tabular}{p{6.5cm} c c c c}
-Parameters & Symbols & Alternative symbols & Units & Models \\
-\hline
-Background hazard rate & $h_b$ & $m_0$ & $\text{time}^{-1}$ & SD and IT\\
-%
-Dominant toxicokinetic rate constant & $k_d$ & $k_e$ & $\text{time}^{-1}$ & SD and IT \\
-%
-Threshold for effects & $\z$ & $\mbox{\it NEC}$, $z$ & $[D]$ & SD \\
-%
-Killing rate constant & $b_w$ & $k_s$, $k_k$ & $[D]^{-1}.\text{time}^{-1}$ & SD \\
-%
-Median of the threshold effect distribution & $m_w$ & $\alpha$ & $[D]$ & IT \\
-%
-Shape of the threshold effect distribution & $\beta$ & - & n.d. & IT \\
-%
-\hline
-\end{tabular}
-\end{table}
-
-
-\paragraph{GUTS Modelling} The number of survivors at time $t_k$ given the
+#### GUTS Modelling
+
+The number of survivors at time $t_k$ given the
number of survivors at time $t_{k-1}$ is assumed to follow a binomial
distribution:
$$
@@ -157,12 +143,12 @@ $$
f_i(t_{k-1}, t_k) = \frac{S_i(t_k)}{S_i(t_{k-1})}
$$
-The formulation of the survival probability $S_i(t)$ in GUTS~\cite{jager2011} is given by integrating the \emph{instantaneous mortality
-rate} $h_i$:
-\begin{equation}
- \label{eq:survivaldef}
- S_i(t) = \exp \left( \int_0^t - h_i(u)\mbox{d}u \right)
-\end{equation}
+The formulation of the survival probability $S_i(t)$ in GUTS [@jager2011] is
+given by integrating the *instantaneous mortality rate* $h_i$:
+$$
+S_i(t) = \exp \left( \int_0^t - h_i(u)\mbox{d}u \right)
+\tag{2}
+$$
In the model, function $h_i$ is expressed using the internal
concentration of contaminant (that is, the concentration inside an
@@ -170,46 +156,56 @@ organism) $\Cint_i(t)$. More precisely:
$$
h_i(t) = b_w \max(\Cint_i(t) - \z, 0) + h_b
$$
-where (see Table \ref{tab:parameters}):
-\begin{itemize}
-\item $b_w$ is the \emph{killing rate} and expressed in
+where (see Table of parameters):
+
+
+* $b_w$ is the \emph{killing rate} and expressed in
concentration$^{-1}$.time$^{-1}$ ;
-\item $\z$ is the so-called \emph{no effect concentration} and
+* $\z$ is the so-called \emph{no effect concentration} and
represents a concentration threshold under which the contaminant has
no effect on organisms ;
-\item $h_b$ is the \emph{background mortality} (mortality in absence of
+* $h_b$ is the \emph{background mortality} (mortality in absence of
contaminant), expressed in time$^{-1}$.
\end{itemize}
-\noindent The internal concentration is assumed to be driven by the external
+The internal concentration is assumed to be driven by the external
concentration, following:
-\begin{equation}
+
+$$
\frac{\dd \Cint_i}{\dd t}(t) = k_d (c_i(t) - \Cint_i(t))
-\label{eq:dynInternalConc}
-\end{equation}
+\tag{1}
+$$
-We call parameter $k_d$ of Eq.~\eqref{eq:dynInternalConc} the
-``dominant rate constant'' (expressed in time$^{-1}$). It
+
+We call parameter $k_d$ of Eq.(1) the
+*dominant rate constant* (expressed in time$^{-1}$). It
represents the speed at which the internal concentration in
contaminant converges to the external concentration. The model could
be equivalently written using an internal damage instead of an
-internal concentration as a dose metric~\cite{jager2011}.
+internal concentration as a dose metric [@jager2011].
+$$
\bigskip
+$$
If we denote $f_z(\z)$ the probability distribution of the no effect concentration threshold, $\z$, then the survival function is given by:
-\begin{equation}
+
+$$
S(t) = \int_0^t S_i(t) f_z(\z) \mbox{d} \z = \int \exp \left( \int_0^t - h_i(u)\mbox{d} u \right) f_z(\z) \mbox{d} \z
-\end{equation}
+$$
+
+Then, the calculation of $S(t)$ depends on the model of survival, GUTS-SD or GUTS-IT [@jager2011].
-Then, the calculation of $S(t)$ depends on the model of survival, GUTS-SD or GUTS-IT~\cite{jager2011}:
+#### GUTS-SD
-\paragraph{GUTS-SD} In GUTS-SD, all organisms are assumed to have the same internal concentration
+In GUTS-SD, all organisms are assumed to have the same internal concentration
threshold (denoted $\z$), and, once exceeded, the instantaneous probability
to die increases linearly with the internal concentration.
-In this situation, $f_z(\z)$ is a Dirac delta distribution, and the survival rate is given by Eq.~\eqref{eq:survivaldef}.
+In this situation, $f_z(\z)$ is a Dirac delta distribution, and the survival rate is given by Eq.(2).
-\paragraph{GUTS-IT} In GUTS-IT, the threshold concentration is distributed among all the organisms, and once
+#### GUTS-IT
+
+In GUTS-IT, the threshold concentration is distributed among all the organisms, and once
exceeded for one organism, this organism dies immediately.
In other words, the killing rate is infinitely high (e.g. $k_k = + \infty$), and the survival rate is given by:
$$
@@ -217,34 +213,36 @@ S_i(t) = e^{-h_b t} \int_{\max\limits_{0<\tau <t}(\Cint_i(\tau))}^{+\infty} f_z(
$$
where $F_z$ denotes the cumulative distribution function of $f_z$.
-\bigskip
Here, the exposure concentration $c_i(t)$ is not supposed constant.
In the case of time variable exposure concentration, we use an midpoint ODE integrator (also known as modified Euler, or Runge-Kutta 2) to solve models GUTS-SD and GUTS-IT.
When the exposure concentration is constant, then, explicit formulation of integrated equations are used. We present them in the next subsection.
-\subsubsection{For constant concentration exposure}
+### For constant concentration exposure
+
+If $c_i(t)$ is constant, and assuming $\Cint_i(0) = 0$, then we can integrate the previous equation (1) to obtain:
-If $c_i(t)$ is constant, and assuming $\Cint_i(0) = 0$, then we can integrate the previous equation \eqref{eq:dynInternalConc} to obtain:
-\begin{equation}
-\label{eq:cint}
+$$
\Cint_i(t) = c_i(1 - e^{-k_d t})
-\end{equation}
+\tag{4}
+$$
-\paragraph{GUTS-SD}
+#### GUTS-SD
In the case $c_i < \z$, the organisms are never affected by the contaminant:
-\begin{equation}
- \label{eq:bgsurvival}
- S_i(t) = \exp( - h_b t )
-\end{equation}
+
+$$
+S_i(t) = \exp( - h_b t )
+\tag{3}
+$$
+
When $c_i > \z$, it takes time $\tz_i$ before the internal
concentration reaches $\z$, where:
$$
\tz_i = - \frac{1}{k_d} \log \left(1 - \frac{\z}{c_i} \right).
$$
-Before that happens, Eq.~\eqref{eq:bgsurvival} applies, while for $t >
-\tz_i$, integrating Eq.~\eqref{eq:survivaldef} results in:
+Before that happens, Eq.(3) applies, while for $t >
+\tz_i$, integrating Eq.(2) results in:
$$
S_i(t) = \exp \left(- h_b t - b_w(c_i - \z) (t - \tz_i) - \frac{b_w c_i}{k_d} \left(e^{- k_d t} - e^{-k_d \tz_i} \right) \right)
$$
@@ -254,9 +252,9 @@ $\z$, we can simulate trajectories by using $S_i(t)$ to compute
conditional survival probabilities. In 'morse', those parameters are
estimated using Bayesian inference. The choice of priors is defined hereafter.
-\paragraph{GUTS-IT}
+#### GUTS-IT
-With constant concentration, Eq.~\ref{eq:cint} provides that $\Cint_i(t)$ is an increasing function, meaning that:
+With constant concentration, Eq.(4) provides that $\Cint_i(t)$ is an increasing function, meaning that:
$$
\max\limits_{0 < \tau < t} (\Cint_i(\tau)) = c_i(1 - e^{-k_d t})
@@ -270,10 +268,10 @@ $$
where $m_w>0$ is the scale parameter (and also the median) and $\beta>0$ is the shape parameter of the log-logistic distribution.
-\subsubsection{Inference}
+### Inference
Posterior distributions for all parameters $h_b$, $b_w$, $k_d$, $\z$,
-$m_w$ and $\beta$ are computed with JAGS~\cite{rjags2016}.
+$m_w$ and $\beta$ are computed with JAGS [@rjags2016].
We set prior distributions on those
parameters based on the actual experimental design used in a
toxicity test.
@@ -282,7 +280,7 @@ the range of tested concentrations.
For each parameter $\theta$, we derive in a similar
manner a minimum ($\theta^{\min}$) and a maximum ($\theta^{\max}$)
value and state that the prior on $\theta$ is a log-normal
-distribution \cite{delignette2017}.
+distribution [@delignette2017].
More precisely:
$$
\log_{10} \theta \sim \mathcal{N}\left(\frac{\log_{10} \theta^{\min} + \log_{10} \theta^{\max}}{2} \, , \,
@@ -290,12 +288,13 @@ $$
$$
With this choice, $\theta^{\min}$ and $\theta^{\max}$ correspond to the
2.5 and 97.5 percentiles of the prior distribution on $\theta$. For each parameter, this gives:
-\begin{itemize}
-\item $\z^{\min} = \min_{i, c_i \neq 0} c_i$ and $\z^{\max} = \max_i c_i$,
+
+
+* $\z^{\min} = \min_{i, c_i \neq 0} c_i$ and $\z^{\max} = \max_i c_i$,
which amounts to say that $\z$ is most probably contained in the
range of experimentally tested concentrations ;
-\item similarly, $m_w^{\min} = \min_{i, c_i \neq 0} c_i$ and $m_w^{\max} = \max_i c_i$ ;
-\item for background mortality rate $h_b$, we assume a maximum value
+* similarly, $m_w^{\min} = \min_{i, c_i \neq 0} c_i$ and $m_w^{\max} = \max_i c_i$ ;
+* for background mortality rate $h_b$, we assume a maximum value
corresponding to situations where half the indivuals are lost at the
first observation time in the control (time $t_1$), that is:
$$
@@ -309,7 +308,7 @@ With this choice, $\theta^{\min}$ and $\theta^{\max}$ correspond to the
$$
e^{- h_b^{\min} t_m} = 0.999 \Leftrightarrow h_b^{\min} = - \frac{1}{t_m} \log 0.999
$$
-\item $k_d$ is the parameter describing the speed at which the
+* $k_d$ is the parameter describing the speed at which the
internal concentration of contaminant equilibrates with the external
concentration. We suppose its value is such that the internal
concentration can at most reach 99.9\% of the external concentration
@@ -324,7 +323,7 @@ With this choice, $\theta^{\min}$ and $\theta^{\max}$ correspond to the
$$
1 - e^{- k_d^{\min} t_m} = 0.001 \Leftrightarrow k_d^{\min} = - \frac{1}{t_m} \log 0.999
$$
-\item $b_w$ is the parameter relating the internal concentration of
+* $b_w$ is the parameter relating the internal concentration of
contaminant to the instantaneous mortality. To fix a maximum value,
we state that between the closest two tested concentrations, the
survival probability at the first time point should not be
@@ -346,11 +345,10 @@ With this choice, $\theta^{\min}$ and $\theta^{\max}$ correspond to the
$$
e^{- b_w^{\min} \Delta^{\max} t_m} = 0.001 \Leftrightarrow b_w^{\min} = - \frac{1}{\Delta^{\max} t_m} \log 0.999
$$
-\item for the shape parameter $\beta$, we used a quasi non-informative log-uniform distribution: $$\log_{10} \beta \sim \mathcal{U}(-2,2)$$
-\end{itemize}
+* for the shape parameter $\beta$, we used a quasi non-informative log-uniform distribution: $$\log_{10} \beta \sim \mathcal{U}(-2,2)$$
-\section{Reproduction toxicity tests}
+# Reproduction toxicity tests
In a reproduction toxicity test, we observe the number of offspring
produced by a sample of adult organisms exposed to a certain
@@ -367,8 +365,8 @@ As already mentionned, it is often the case that part of the
organisms die during the observation period. In
previous approaches, it was proposed to consider the cumulated number
of reproduction outputs without accounting for
-mortality~\cite{OECD2004,OECD2008}, or to exclude replicates where
-mortality occurred~\cite{OECD2012}. However, organisms may have
+mortality [@OECD2004; @OECD2008], or to exclude replicates where
+mortality occurred [@OECD2012]. However, organisms may have
reproduced before dying and thus contributed to the observed
response. In addition, organisms dying the first are probably the most
sensitive, so the information on reproduction of these prematurely
@@ -376,7 +374,7 @@ dead organisms is valuable ; ignoring it is likely to bias the
results in a non-conservative way. This is particularly critical at
high concentrations, when mortality may be very high.
-In a toxicity test, mortality is usually regularly recorded, \textit{i.e}.
+In a toxicity test, mortality is usually regularly recorded, *i.e*.
at each time point when reproduction outputs are counted. Using these
data, we can approximately estimate for each organism the period it
has stayed alive (which we assume coincides with the period it may
@@ -390,7 +388,7 @@ number of outputs per organism-day.
In the following, we denote $M_{ijk}$ the observed number of surviving
organisms at concentration $c_i$, replicate $j$ and time $t_k$.
-\subsection{Estimation of the effective observation period}
+## Estimation of the effective observation period
We define the effective observation period as the sum for all
organisms of the time they spent alive in the experiment.
@@ -410,13 +408,14 @@ NID_{ij} = \sum_k M_{ij(k+1)} (t_{k+1} - t_k)
+ (M_{ijk} - M_{ij(k+1)})\left( \frac{t_{k+1}+t_k}{2} - t_k \right)
$$
-\subsection{Target time analysis}
-In this paragraph, we describe our so-called ``target time analysis'',
+## Target time analysis
+
+In this paragraph, we describe our so-called *target time analysis*,
where we model the cumulated number of offspring up to a target time
as a function of contaminant concentration and effective observation
time in this period (cumulated life times of all organisms in the
experiment, as described above). A more detailed presentation can be
-found in \cite{delignette2014}.
+found in [@delignette2014].
We keep the convention that index $i$ is used for concentration
levels and $j$ for replicates. The data will therefore correspond to a
@@ -426,19 +425,21 @@ output. These observations are supposed to be drawn independently from
a distribution that is a function of the level of contaminant $c_i$.
-\subsubsection{Modelling}
+### Modelling
+
We assume here that the effect of the considered
-contaminant on the reproduction rate\footnote{that is, the number of
- reproduction outputs during the experiment per organism-day} does
+contaminant on the reproduction rate ^[that is, the number of
+ reproduction outputs during the experiment per organism-day] does
not depend on the exposure period, but only on the concentration of the
contaminant.
More precisely, the reproduction rate in an
experiment at concentration $c_i$ of contaminant is modelled by a
three-parameters log-logistic model, that writes as follows:
-\[
+
+$$
f(c;\theta)=\frac{d}{1+(\frac{c}{e})^b} \quad \textrm{with} \quad
\theta=(e,b,d)
-\]
+$$
Here $d$ corresponds to the reproduction rate in absence of
contaminant (control condition) and $e$ to the value of the
$EC_{50}$, that is the concentration dividing the average number of
@@ -448,10 +449,10 @@ replicate $j$ can be modelled using a Poisson distribution:
$$
N_{ij} \sim Poisson(f(c_i ; \theta) \times NID_{ij})
$$
-This model is later referred to as ``Poisson model''. If there
+This model is later referred to as *Poisson model*. If there
happens to be a non-negligible variability of the reproduction rate
between replicates at some fixed concentrations, we propose a second
-model, named ``gamma-Poisson model'', stating that:
+model, named *gamma-Poisson model*, stating that:
$$
N_{ij} \sim Poisson(F_{ij} \times NID_{ij})
$$
@@ -469,38 +470,36 @@ parameter (the greater its value, the greater the inter-replicate
variability)
-\subsubsection{Inference}
+### Inference
+
Posterior distributions for parameters $b$,
-$d$ and $e$ are estimated using JAGS~\cite{rjags2016} with the following priors:
-\begin{itemize}
-\item we assume the range of tested concentrations in an
+$d$ and $e$ are estimated using JAGS [@rjags2016] with the following priors:
+
+* we assume the range of tested concentrations in an
experiment is chosen to contain the $EC_{50}$ with high
probability. More formally, we choose:
$$\log_{10} e \sim \mathcal{N} \left(\frac{\log_{10} (\min_i c_i) + \log_{10} (\max_i c_i)}{2}, \frac{\log_{10} (\max_i c_i) - \log_{10} (\min_i c_i)}{4} \right)$$
which implies $e$ has a probability slightly higher than 0.95 to lie
between the minimum and the maximum tested concentrations.
-\item we choose a quasi non-informative prior distribution for the
+* we choose a quasi non-informative prior distribution for the
shape parameter $b$:
$$\log_{10} b \sim \mathcal{U}(-2,2)$$
-\item
- as $d$ corresponds to the reproduction rate without contaminant, we
+* as $d$ corresponds to the reproduction rate without contaminant, we
set a normal prior $\mathcal{N}(\mu_d,\sigma_d)$ using the control:
+ $$
\begin{align*}
\mu_d & = \frac{1}{r_0} \sum_j \frac{N_{0j}}{NID_{0j}}\\
\sigma_d & = \sqrt{\frac{\sum_j \left( \frac{N_{0j}}{NID_{0j}} - \mu_d\right)^2}{r_0(r_0 - 1)}}\\
\end{align*}
+ $$
where $r_0$ is the number of replicates in the control
condition. Note that since they are used to estimate the prior
distribution, the data from the control condition are not used in the
fitting phase.
-
-\item we choose a quasi non-informative prior distribution for the
+* we choose a quasi non-informative prior distribution for the
$\omega$ parameter of the gamma-Poisson model:
$$log_{10}(\omega) \sim \mathcal{U}(-4,4)$$
-\end{itemize}
-\begin{align*}
-\end{align*}
For a given dataset, the procedure implemented in 'morse' will fit both
models (Poisson and gamma-Poisson) and use an information criterion
@@ -511,6 +510,4 @@ will provide more reliable estimates. That is why a Poisson model is
preferred unless the gamma-Poisson model has a sufficiently lower DIC
(in practice we require a difference of 10).
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+# References
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