diff --git a/biblio.bib b/biblio.bib
index 5c826f1748d159eb50ba260ff729b9f010c7e218..660d912b750e78de71afc9cac6fb9adf7b6909bc 100644
--- a/biblio.bib
+++ b/biblio.bib
@@ -185,7 +185,7 @@ year = {2010}
@book{fisher:34,
author = {Fisher, R. A.},
- year = {1934},
+ year = {1932},
title = {{S}tatistical {M}ethods for {R}esearch {W}orkers},
publisher = {{O}liver \& {B}oyd},
address = {Edinburgh},
@@ -228,3 +228,63 @@ year = {2010}
pages = {1--51},
file = {Full Text:/home/felix/Zotero/storage/5RPPKK2B/Röver - 2020 - Bayesian Random-Effects Meta-Analysis Using the ba.pdf:application/pdf},
}
+
+@article{lil:2023,
+ title = {Bayesian random-effects meta-analysis with empirical heterogeneity priors for application in health technology assessment with very few studies},
+ author = {Lilienthal, Jona and Sturtz, Sibylle and Schürmann, Christoph and Maiworm, Matthias and Röver, Christian and Friede, Tim and Bender, Ralf},
+ year = {2023},
+ note = {submitted for publication},
+}
+
+@article{langan_comparison_2019,
+ title = {A comparison of heterogeneity variance estimators in simulated random-effects meta-analyses},
+ volume = {10},
+ issn = {1759-2887},
+ doi = {10.1002/jrsm.1316},
+ abstract = {Studies combined in a meta-analysis often have differences in their design and conduct that can lead to heterogeneous results. A random-effects model accounts for these differences in the underlying study effects, which includes a heterogeneity variance parameter. The DerSimonian-Laird method is often used to estimate the heterogeneity variance, but simulation studies have found the method can be biased and other methods are available. This paper compares the properties of nine different heterogeneity variance estimators using simulated meta-analysis data. Simulated scenarios include studies of equal size and of moderate and large differences in size. Results confirm that the DerSimonian-Laird estimator is negatively biased in scenarios with small studies and in scenarios with a rare binary outcome. Results also show the Paule-Mandel method has considerable positive bias in meta-analyses with large differences in study size. We recommend the method of restricted maximum likelihood (REML) to estimate the heterogeneity variance over other methods. However, considering that meta-analyses of health studies typically contain few studies, the heterogeneity variance estimate should not be used as a reliable gauge for the extent of heterogeneity in a meta-analysis. The estimated summary effect of the meta-analysis and its confidence interval derived from the Hartung-Knapp-Sidik-Jonkman method are more robust to changes in the heterogeneity variance estimate and show minimal deviation from the nominal coverage of 95\% under most of our simulated scenarios.},
+ language = {eng},
+ number = {1},
+ journal = {Research Synthesis Methods},
+ author = {Langan, Dean and Higgins, Julian P. T. and Jackson, Dan and Bowden, Jack and Veroniki, Areti Angeliki and Kontopantelis, Evangelos and Viechtbauer, Wolfgang and Simmonds, Mark},
+ month = mar,
+ year = {2019},
+ pmid = {30067315},
+ keywords = {Algorithms, Analysis of Variance, Computer Simulation, Data Interpretation, Statistical, DerSimonian-Laird, heterogeneity, Humans, Likelihood Functions, Meta-Analysis as Topic, Models, Statistical, Odds Ratio, Probability, random-effects, REML, Reproducibility of Results, Research Design, Selection Bias, simulation, Software, Systematic Reviews as Topic},
+ pages = {83--98},
+ file = {Full Text:/home/felix/Zotero/storage/TIR2XXPU/Langan et al. - 2019 - A comparison of heterogeneity variance estimators .pdf:application/pdf},
+}
+
+@article{paul:man:82,
+ title = {Consensus {Values} and {Weighting} {Factors}},
+ volume = {87},
+ issn = {0160-1741},
+ doi = {10.6028/jres.087.022},
+ abstract = {A method is presented for the statistical analysis of sets of data which are assembled from multiple experiments. The analysis recognizes the existence of both within group and between group variabilities, and calculates appropriate weighting factors based on the observed variability for each group. The weighting factors are used to calculate a "best" consensus value from the overall experiment. The technique for obtaining the consensus value is applicable to either the determination of the weighted average value, or to the parameters associated with a weighted least squares regression problem. The calculations are made by using an iterative technique with a truncated Taylor series expansion. The calculations are straightforward, and are easily programmed on a desktop computer. An examination of the observed variabilities, both within groups and between groups, leads to considerable insight into the overall experiment and greatly aids in the design of future experiments.},
+ language = {eng},
+ number = {5},
+ journal = {Journal of Research of the National Bureau of Standards (1977)},
+ author = {Paule, Robert C. and Mandel, John},
+ year = {1982},
+ pmid = {34566088},
+ pmcid = {PMC6768160},
+ keywords = {ANOVA (within-between), components of variance, consensus values, design of experiments, pooling of variance, weighted average, weighted least squares regression},
+ pages = {377--385},
+ file = {Full Text:/home/felix/Zotero/storage/N6JIMAM3/Paule and Mandel - 1982 - Consensus Values and Weighting Factors.pdf:application/pdf},
+}
+
+@article{harv:77,
+ title = {Maximum {Likelihood} {Approaches} to {Variance} {Component} {Estimation} and to {Related} {Problems}},
+ volume = {72},
+ issn = {0162-1459},
+ url = {https://www.jstor.org/stable/2286796},
+ doi = {10.2307/2286796},
+ abstract = {Recent developments promise to increase greatly the popularity of maximum likelihood (ML) as a technique for estimating variance components. Patterson and Thompson (1971) proposed a restricted maximum likelihood (REML) approach which takes into account the loss in degrees of freedom resulting from estimating fixed effects. Miller (1973) developed a satisfactory asymptotic theory for ML estimators of variance components. There are many iterative algorithms that can be considered for computing the ML or REML estimates. The computations on each iteration of these algorithms are those associated with computing estimates of fixed and random effects for given values of the variance components.},
+ number = {358},
+ urldate = {2023-11-08},
+ journal = {Journal of the American Statistical Association},
+ author = {Harville, David A.},
+ year = {1977},
+ note = {Publisher: [American Statistical Association, Taylor \& Francis, Ltd.]},
+ pages = {320--338},
+ file = {JSTOR Full Text PDF:/home/felix/Zotero/storage/KDTIT8CF/Harville - 1977 - Maximum Likelihood Approaches to Variance Componen.pdf:application/pdf},
+}
diff --git a/makefile b/makefile
index 12cb9ee8330cd04c9b3f9bcfda54f2cd0f182c98..a422ab722af67b8704d993e36a79e17190ba8493 100755
--- a/makefile
+++ b/makefile
@@ -1,9 +1,9 @@
all: protocol.pdf
-protocol.pdf: protocol.tex
+protocol.pdf: protocol.tex biblio.bib
latexmk -pdf $<
-preview: protocol.tex
+preview: protocol.tex biblio.bib
latexmk -pdf -pvc $<
clean:
diff --git a/protocol.pdf b/protocol.pdf
index f9b70d5637339a8a6cda5e3f8e2b6099b5532c6f..f44c102af1c0469f64bd3f40117d7142f7ece9c2 100644
Binary files a/protocol.pdf and b/protocol.pdf differ
diff --git a/protocol.tex b/protocol.tex
index 9f046dfa28f31e678ed75358edb9c78f9ff6ad21..67a0c049053cfa215f560cbad656074cc07e4596 100644
--- a/protocol.tex
+++ b/protocol.tex
@@ -69,8 +69,8 @@
\begin{document}
\begin{center}
{\noindent \LARGE \bf Simulation protocol:\\[2mm]
- Comparison of confidence intervals summarizing the\\[2mm]
- uncertainty of the combined estimate of a meta-analysis
+ Comparison of confidence intervals summarizing\\[2mm]
+ the uncertainty of the combined estimate of a meta-analysis
}\\
\bigskip
{\noindent \Large Leonhard Held, Felix Hofmann
@@ -90,12 +90,12 @@ The simulation is implemented in \texttt{simulate\_all.R}.
The aim of this simulation study is the comparison of confidence intervals
(CIs) summarizing the uncertainty of the combined estimate of a meta-analysis.
-Specifically, we focus on CIs constructed using p-value functions that
-implement the methods from \citet{edgington:72} and \citet{fisher:34}. The
-underlying data sets are simulated as described in Section~\ref{sec:simproc}
-and Section~\ref{sec:scenario}. The resulting intervals are then compared to CIs
-constructed using the other methods listed in Section~\ref{sec:method} using the
-measures defined in Section~\ref{sec:meas}.
+Specifically, we focus on CIs constructed using $p$-value functions that
+implement the $p$-value combination methods from \citet{edgington:72} and
+\citet{fisher:34}. The underlying data sets are simulated as described in
+Section~\ref{sec:simproc}. In Section~\ref{sec:analysis} we describe which
+CI construction methods we compare in this simulation study and what criteria
+we use to evaluate them.
\section{Simulation of the data sets} \label{sec:simproc}
@@ -211,14 +211,10 @@ and $\text{se}_i$ by newly simulated values, which are then again accepted
with the given probability above. This procedure is repeated until the required
number of studies is simulated.
-To obtain a similar scenario as in \citet{henm:copa:10} we set
-$$
-\theta / \sqrt{2/n_i} \overset{!}{=} 1 \Rightarrow \theta = \sqrt{2/n_i}
-\vadjust{\todo{Where does this come from?}}
-$$
However, we assume that only small studies with $n_i = 50$ are subject to
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()}.
+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
@@ -241,7 +237,7 @@ For each scenario in Section~\ref{sec:scenario} we
Section~\ref{sec:meas}
\end{enumerate}
-\section{Analysis of the confidence intervals}
+\section{Analysis of the confidence intervals} \label{sec:analysis}
This section contains an overview over the construction methods for CIs
that we consider in this simulation. Moreover, we explain what measures we
@@ -252,12 +248,12 @@ use in order to compare the different CIs with each other.
For this project, we will calculate 95\% CIs according to the following methods.
\begin{enumerate}
- \item Hartung Knapp Sidik Jonkman (HK) \citep{IntHoutIoannidis}.
+ \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 $\sigma = 0.3$ \citep{rov:20, }.
- \todo{Insert citation for Lilienthal et al. for $\sigma = 0.3$?}
+ 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}
@@ -273,10 +269,9 @@ mentioned methods.
\begin{enumerate}
\item No heterogeneity, \ie $\tau^2 = 0$.
\item DerSimonian-Laird \citep{ders:lair:86}.
- \item Paule-Mandel.
- \item REML.
+ \item Paule-Mandel \citep{paul:man:82}.
+ \item REML \citep{harv:77}.
\end{enumerate}
-\todo{Add citation for these estimators?}
The calculation of the estimates in the simulation will be done using the
\texttt{metagen} function from the \texttt{R} package ``\emph{meta}''.