change prior variance to prior sd parametrization in Box

paper/rsabsence.Rnw

@@ -844,7 +844,7 @@ pTOSTa <- function(estimate, se, null = 0, margin) {
 @
 << "pTOST-more-educational-version", eval = FALSE, echo = TRUE, size = "small" >>=
 ## R function to compute TOST p-value based on effect estimate, standard error,
-## null value (default is 0), equivalence margin
+## null value (default is 0), and equivalence margin
 pTOST <- function(estimate, se, null = 0, margin) {
     p1 <- pnorm(q = (estimate - null - margin) / se)
     p2 <- 1 - pnorm(q = (estimate - null + margin) / se)
@@ -866,8 +866,8 @@ pTOST <- function(estimate, se, null = 0, margin) {
         \item Specify a prior distribution for the effect size $\theta$ that
               represents plausible values under the alternative hypothesis that
               there is an effect ($H_{1}\colon \theta \neq \theta_{0})$. For
-              example, specify the mean $m$ and variance $v$ of a normal
-              distribution $\theta \given H_{1} \sim \Nor(m ,v)$.
+              example, specify the mean $m$ and standard deviation $s$ of a normal
+              distribution $\theta \given H_{1} \sim \Nor(m, s^{2})$.
         \item Compute the Bayes factors contrasting
               $H_{0} \colon \theta = \theta_{0}$ to
               $H_{1} \colon \theta \neq \theta_{0}$ for original and replication
@@ -875,25 +875,27 @@ pTOST <- function(estimate, se, null = 0, margin) {
               % $\theta \given H_{1} \sim \Nor(m ,v)$,
               the Bayes factor is
               $$\BF_{01,i}
-              = \sqrt{1 + \frac{v}{\sigma^{2}_{i}}} \, \exp\left[-\frac{1}{2} \left\{\frac{(\hat{\theta}_{i} -
-              \theta_{0})^{2}}{\sigma^{2}_{i}} - \frac{(\hat{\theta}_{i} - m)^{2}}{\sigma^{2}_{i} + v}
+              = \sqrt{1 + \frac{s^{2}}{\sigma^{2}_{i}}} \, \exp\left[-\frac{1}{2} \left\{\frac{(\hat{\theta}_{i} -
+              \theta_{0})^{2}}{\sigma^{2}_{i}} - \frac{(\hat{\theta}_{i} - m)^{2}}{\sigma^{2}_{i} + s^2}
               \right\}\right], ~ i \in \{o, r\}.$$
 \begin{minipage}[c]{0.95\linewidth}
 << "BF01-version-that-we-used", eval = FALSE, echo = FALSE, size = "small" >>=
 ## R function to compute Bayes factor based on effect estimate, standard error,
-## null value (default is 0), prior mean (default is null value), prior variance
-BF01a <- function(estimate, se, null = 0, priormean = null, priorvar) {
+## null value (default is 0), prior mean (default is null value), and prior
+## standard deviation
+BF01a <- function(estimate, se, null = 0, priormean = null, priorsd) {
     f0 <- dnorm(x = estimate, mean = null, sd = se)
-    f1 <- dnorm(x = estimate, mean = priormean, sd = sqrt(se^2 + priorvar))
+    f1 <- dnorm(x = estimate, mean = priormean, sd = sqrt(se^2 + priorsd^2))
     return(f0/f1)
   }
 @
 << "BF01-more-educational-version", eval = FALSE, echo = TRUE, size = "small" >>=
 ## R function to compute Bayes factor based on effect estimate, standard error,
-## null value (default is 0), prior mean (default is null value), prior variance
-BF01 <- function(estimate, se, null = 0, priormean = null, priorvar) {
-    bf <- sqrt(1 + priorvar/se^2) * exp(-0.5 * ((estimate - null)^2 / se^2 -
-            (estimate - priormean)^2 / (se^2 + priorvar)))
+## null value (default is 0), prior mean (default is null value), and prior
+## standard deviation
+BF01 <- function(estimate, se, null = 0, priormean = null, priorsd) {
+    bf <- sqrt(1 + priorsd^2/se^2) * exp(-0.5 * ((estimate - null)^2 / se^2 -
+            (estimate - priormean)^2 / (se^2 + priorsd^2)))
     return(bf)
   }
 @
@@ -1019,55 +1021,27 @@ data from the RPCB were obtained by downloading the files from
 relevant variables as indicated in the R script \texttt{preprocess-rpcb-data.R}
 which is available in our git repository.
 
-\bibliography{bibliography}
 
 
-\section*{Appendix}
 
-\subsection*{Sensitivity analysis}
+\section*{Appendix: Sensitivity analyses}
 
-As discussed before, the post-hoc specification of equivalence margins and the
-choice of the prior distribution for the SMD under the alternative $H_{1}$ is 
-debatable and controversial. Commonly used margins in clinical research are much
-more stringent; for instance, in oncology, a margin of $\Delta = \log(1.3)$ is 
-commonly used for log odds/hazard ratios, whereas in bioequivalence studies a
-margin of \mbox{$\Delta = \log(1.25) % = \Sexpr{round(log(1.25), 2)}
+As discussed before, the post-hoc specification of equivalence margins $\Delta$
+and prior distribution for the SMD under the alternative $H_{1}$ is debatable.
+Commonly used margins in clinical research are much more stringent; for
+instance, in oncology, a margin of $\Delta = \log(1.3)$ is commonly used for log
+odds/hazard ratios, whereas in bioequivalence studies a margin of
+\mbox{$\Delta = \log(1.25) % = \Sexpr{round(log(1.25), 2)}
   $} is the convention \citep[chapter 22]{Senn2008}. These margins would
-translate into margins of $\Delta
-= % \log(1.3)\sqrt{3}/\pi =
+translate into margins of $\Delta = % \log(1.3)\sqrt{3}/\pi =
 \Sexpr{round(log(1.3)*sqrt(3)/pi, 2)}$ and $\Delta = % \log(1.25)\sqrt{3}/\pi =
 \Sexpr{round(log(1.25)*sqrt(3)/pi, 2)}$ on the SMD scale, respectively, using
 the $\text{SMD} = (\surd{3} / \pi) \log\text{OR}$ conversion \citep[p.
-233]{Cooper2019}. As for the Bayesian hypothesis testing we specified a normal
+233]{Cooper2019}. Similarly, for the Bayesian factor we specified a normal
 unit-information prior under the alternative while other normal priors with
-smaller/larger standard deviations could have been considered.
-In the sensitivity analysis in the top plot of Figure~\ref{fig:sensitivity} we 
-show the number of successful replications as a function of the margin $\Delta$ 
-and for different TOST \textit{p}-value thresholds. Such an ``equivalence 
-curve'' approach was first proposed by \citet{Hauck1986}. We see that for
-realistic margins between $0$ and $1$, the proportion of replication successes 
-remains below $50\%$ for the conventional $\alpha = 0.05$ level. To achieve a 
-success rate of $11/15 = \Sexpr{round(11/15*100, 0)}\%$, as was achieved with 
-the non-significance criterion from the RPCB, unrealistic margins of $\Delta > 
-2$ are required, highlighting the paucity of evidence provided by these studies.
-Changing the success criterion to a more lenient level ($\alpha = 0.1$) or a
-more stringent level ($\alpha = 0.01$) hardly changes this conclusion.
-
-In the bottom plot of Figure~\ref{fig:sensitivity} a sensitivity analysis is 
-reported with respect to the choice of the prior standard deviation and the 
-Bayes factor threshold. It is uncommon to specify prior standard
-deviations larger than the unit-information standard deviation of $2$, as this
-corresponds to the assumption of very large effect sizes under the alternatives.
-However, to achieve replication success for a larger proportion of replications
-than the observed $\Sexpr{bfSuccesses}/\Sexpr{ntotal} = 
-\Sexpr{round(bfSuccesses/ntotal*100, 0)}\%$, unreasonably large prior standard
-deviations have to be specified. For instance, a standard deviation of roughly
-$5$ is required to achieve replication success in $50\%$ of the replications at 
-a lenient Bayes factor threshold of $\gamma = 3$. The standard deviation needs 
-to be almost $20$ so that the same success rate $11/15 = \Sexpr{round(11/15*100,
-0)}\%$ as with the non-significance criterion is achieved. The necessary 
-standard deviations are even higher for stricter Bayes factor threshold,
-such as $\gamma = 6$ or $\gamma = 10$.
+smaller/larger standard deviations could have been considered. Here, we
+therefore investigate the sensitivity of our conclusions with respect to these
+parameters.
 
 \begin{figure}[!htb]
 << "sensitivity", fig.height = 6.5 >>=
@@ -1196,6 +1170,39 @@ grid.arrange(plotA, plotB, ncol = 1)
 \label{fig:sensitivity}
 \end{figure}
 
+The top plot of Figure~\ref{fig:sensitivity} shows the number of
+successful replications as a function of the margin $\Delta$ and for different
+TOST \textit{p}-value thresholds. Such an ``equivalence curve'' approach was
+first proposed by \citet{Hauck1986}. We see that for realistic margins between
+$0$ and $1$, the proportion of replication successes remains below $50\%$ for
+the conventional $\alpha = 0.05$ level. To achieve a success rate of
+$11/15 = \Sexpr{round(11/15*100, 0)}\%$, as was achieved with the
+non-significance criterion from the RPCB, unrealistic margins of $\Delta > 2$
+are required, highlighting the paucity of evidence provided by these studies.
+Changing the success criterion to a more lenient level ($\alpha = 0.1$) or a
+more stringent level ($\alpha = 0.01$) hardly changes the conclusion.
+
+The bottom plot of Figure~\ref{fig:sensitivity} shows a sensitivity analysis
+regarding the choice of the prior standard deviation and the Bayes factor
+threshold. It is uncommon to specify prior standard deviations larger than the
+unit-information standard deviation of $2$, as this corresponds to the
+assumption of very large effect sizes under the alternatives. However, to
+achieve replication success for a larger proportion of replications than the
+observed
+$\Sexpr{bfSuccesses}/\Sexpr{ntotal} = \Sexpr{round(bfSuccesses/ntotal*100, 0)}\%$,
+unreasonably large prior standard deviations have to be specified. For instance,
+a standard deviation of roughly $5$ is required to achieve replication success
+in $50\%$ of the replications at a lenient Bayes factor threshold of
+$\gamma = 3$. The standard deviation needs to be almost $20$ so that the same
+success rate $11/15 = \Sexpr{round(11/15*100, 0)}\%$ as with the
+non-significance criterion is achieved. The necessary standard deviations are
+even higher for stricter Bayes factor threshold, such as $\gamma = 6$ or
+$\gamma = 10$.
+
+
+\bibliography{bibliography}
+
+
 << "sessionInfo1", eval = Reproducibility, results = "asis" >>=
 ## print R sessionInfo to see system information and package versions
 ## used to compile the manuscript (set Reproducibility = FALSE, to not do that)