diff --git a/protocol.pdf b/protocol.pdf
index 0ed0bcde97f0c9f5418ef35c57e1e1bef11f8aed..3746d8558fa9f86933fd2e7cb4e7a880b35780de 100644
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diff --git a/protocol.tex b/protocol.tex
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--- a/protocol.tex
+++ b/protocol.tex
@@ -111,9 +111,9 @@ We save the output of \texttt{sessionInfo()} giving information on the used
version of R, packages, and platform with the simulation results.
\subsection{Random number generator}
-We use the package \pkg{doRNG} with its default random number generator to
-ensure that random numbers generated inside parallel for loops are independent
-and reproducible.
+We use the package \pkg{doRNG} \todo{add citation} with its default random
+number generator to ensure that random numbers generated inside parallel
+for loops are independent and reproducible.
\subsection{Scenarios to be investigated} \label{sec:scenario}
@@ -121,13 +121,10 @@ The $720$ simulated scenarios consist of all combinations
of the following parameters:
\begin{itemize}
\item Higgin's $I^2$ heterogeneity measure $\in \{0, 0.3, 0.6, 0.9\}$.
-% \item We always use an additive heterogeneity model. \todo{Maybe remove this entirely?}
\item Number of studies summarized by the meta-analysis $k \in \{3, 5, 10, 20, 50\}$.
\item Publication bias is $\in \{\text{'none'}, \text{'moderate'}, \text{'strong'}\}$
following the terminology of \citet{henm:copa:10}.
\item The average study effect $\theta \in \{0.2, 0.5\}$.
- %, and we set it to $\theta = 0.2$ to
- %obtain a similar scenario as used in \citet{henm:copa:10}.
\item The distribution to draw the true study values $\delta_i$ is either
'Gaussian' or 't' with 4 degrees of freedom. The latter still has finite mean
and variance, but leads to more 'outliers'.
@@ -140,8 +137,6 @@ Note that \citet{IntHoutIoannidis} use a similar setup.
\subsection{Simulation details}
-% For the \textbf{Additive heterogeneity model without publication bias}, the
-% simulation of one meta-analysis dataset is performed as follows:
The simulation of one meta-analysis data set is performed as follows:
\begin{enumerate}
@@ -170,34 +165,6 @@ The simulation of one meta-analysis data set is performed as follows:
The marginal variance of this simulation procedure is
$\tau^2 + 2/n_i$, so follows the additive heterogeneity model as intended.
-% For the \textbf{Multiplicative model without publication bias}, the simulation
-% of one meta-analysis dataset is performed as follows:
-% \begin{enumerate}
-% \item Compute the within-study variance
-% $\epsilon^2 = \frac{2}{k} \sum\limits_{i=1}^k \frac{1}{n_i}$.
-% \item Compute the multiplicative heterogeneity factor
-% $\phi = \frac{1}{1-I^2}$. Compute the corresponding
-% \begin{equation}\label{eq:eq2}
-% \tau^2 = \epsilon^2 \, (\phi-1) .
-% \end{equation}
-% \item For a trial $i$ of the meta-analysis with $k$ trials, $i = 1, \dots, k$:
-% \begin{enumerate}
-% \item Simulate the true effect size using the Gaussian model:
-% $\delta_i \sim \N(\theta, \tau^2)$ or using a Student-$t$ distribution
-% such that the samples have mean $\theta$ and variance $\tau^2$.
-% \item Simulate the effect estimates of each trial
-% $y_i \sim \N(\delta_i, \frac{2}{n_i})$.
-% \item Simulate the standard errors of the trial outcomes:
-% $\text{se}_i \sim \sqrt{\frac{\chi^2(2n_i-2)}{(n_i-1)n_i}}$.
-% \end{enumerate}
-% \end{enumerate}
-%
-%
-% \paragraph{Note: The marginal variance}\mbox{}\\
-% The marginal variance of this simulation procedure is
-% $\frac{2}{n} \, (\phi-1) + \frac{2}{n} = \frac{2\phi}{n} = \phi \, \epsilon^2$,
-% so follows the multiplicative model as intended.
-
\paragraph{Note: Publication bias}\mbox{}\\
To simulate studies under \textbf{publication bias}, we follow the suggestion
of \citet{henm:copa:10} and accept each simulated study with probability
@@ -216,18 +183,6 @@ publication bias. Thus, larger studies with $n_i = 500$ are always accepted.
As described in Section~\ref{sec:scenario}, we set $\theta \in \{0.2, 0.5\}$.
See the R function \texttt{simREbias()}.
-% The mean study effect $\theta$ and the sample size $n_i$ have an influence
-% on the acceptance probability
-
-% \paragraph{Note: Unbalanced sample sizes}\mbox{}\\
-% To study the effect of unbalanced sample sizes $n_1, \ldots, n_k$ we consider
-% the following setup:
-% \begin{enumerate}
-% \item Increase the sample size of \textbf{one} of the $k$ by a factor 10.
-% \item Increase the sample size of \textbf{two} of the $k$ by a factor 10.
-% \end{enumerate}
-%% See the argument \texttt{large} of \texttt{simREbias()}.
-
\subsection{Simulation procedure}
For each scenario in Section~\ref{sec:scenario} we
\begin{enumerate}
@@ -251,9 +206,6 @@ For this project, we will calculate 95\% CIs according to the following methods.
\item Hartung-Knapp-Sidik-Jonkman (HK) \citep{IntHoutIoannidis}.
\item Random effects model.
\item Henmi and Copas (HC) \citep{henm:copa:10}.
- \item Bayesian random effects meta analysis (Bayesmeta) with half-normal prior
- distribution with scale parameter $\sigma = 0.3$ for $\tau$, the square root
- between-study heterogeneity \citep{rov:20, lil:2023}.
\item Edgington's method \citep{edgington:72}.
\item Fisher's method \citep{fisher:34}.
\end{enumerate}
@@ -279,18 +231,6 @@ The calculation of the estimates in the simulation will be done using the
The adjusted study-specific standard errors are then given by
$\text{se}_{\text{adj}}(\hat{\theta_i}) = \sqrt{\text{se}(\hat{\theta_i})^2 + \tau^2}$.
-% As stated in Subsection~\ref{sec:method}, the harmonic mean and $k$-trials
-% methods can be extended such that heterogeneity between the individual studies
-% is taken into account. In scenarios where the additive variance adjustment is
-% used, we estimate the between study variance $\tau^2$ using the REML method
-% implemented in the \texttt{metagen} \texttt{R}-package ``meta'' and adjust
-% the study-specific standard errors such that
-% $\text{se}_{\text{adj}}(\hat{\theta_i}) = \sqrt{\text{se}(\hat{\theta_i})^2 + \tau^2}$.
-% In case of the multiplicative variance adjustment, we estimate the
-% multiplicative parameter $\phi$ as described in \citet{mawd:etal:17} and adjust
-% the study-specific standard errors such that
-% $\text{se}_{\text{adj}}(\hat{\theta_i}) = \text{se}(\hat{\theta_i}) \cdot \sqrt{\phi}$.
-
\subsection{Measures considered} \label{sec:meas}
We assess the CIs using the following criteria
@@ -298,19 +238,6 @@ We assess the CIs using the following criteria
\item CI coverage of combined effect, \ie, the proportion of intervals
containing the true effect. If the CI does not exist given a specific
simulated data set, we treat the coverage as as missing (\texttt{NA}).
- % coverage_true
- % \item CI coverage of study effects, \ie, the proportion of intervals
- % containing the true study-specific effects % coverage_effects
- % \item CI coverage of all study effects, \ie, whether or not the CI covers
- % all of the study effects %coverage_all
- % \item CI coverage of at least one of the study effects, \ie, whether or not
- % the CI covers at least one of the study effects % coverage_effects_min1
- % \item Prediction Interval (PI) coverage, \ie, the proportion of intervals
- % containing the treatment effect of a newly simulated study. The newly
- % simulated study has $n = 50$ and is not subject to publication bias. All
- % other simulation parameters stay the same as for the simulation of the
- % original studies (only for Harmonic mean, $k$-trials, REML, and HK methods)
- % % coverage_prediction
\item CI width. If there is more than one interval, the width is the sum of
the lengths of the individual intervals. If the interval does not exist for
a simulated data set, the width will be recorded as missing (\texttt{NA}).
@@ -319,8 +246,8 @@ We assess the CIs using the following criteria
a simulated data set, the score will be recorded as missing (\texttt{NA}).
% score
\item Number of CIs (only for Fisher and Edgington methods). If the interval
- does not exist for a simulated data set, the number of CIs will be recorded as
- 0. % n
+ does not exist for a simulated data set, the number of CIs will be recorded as
+ 0. % n
\end{enumerate}
Furthermore, we calculate the following measure related to the point estimates