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\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage{doi}
\usepackage{tikz} % to draw schematics
\usetikzlibrary{decorations.pathreplacing,calligraphy} % for tikz curly braces
\definecolor{darkblue2}{HTML}{273B81}
\definecolor{darkred2}{HTML}{D92102}
\title{Meta-Research: Replication of ``null results'' -- Absence of evidence or
evidence of absence?}
\author[1*\authfn{1}]{Samuel Pawel}
\author[1\authfn{1}]{Rachel Heyard}
\author[1]{Charlotte Micheloud}
\author[1]{Leonhard Held}
\affil[1]{Epidemiology, Biostatistics and Prevention Institute, Center for Reproducible Science, University of Zurich, Switzerland}
\corr{samuel.pawel@uzh.ch}{SP}
\contrib[\authfn{1}]{Contributed equally}
%% custom commands
\input{defs.tex}
\begin{document}
\maketitle
% %% Disclaimer that a preprint
% \vspace{-3em}
% \begin{center}
% {\color{red}This is a preprint which has not yet been peer reviewed.}
% \end{center}
## knitr options
library(knitr)
opts_chunk$set(fig.height = 4,
echo = FALSE,
warning = FALSE,
message = FALSE,
cache = FALSE,
eval = TRUE)
## should sessionInfo be printed at the end?
Reproducibility <- TRUE
## packages
library(ggplot2) # plotting
library(reporttools) # reporting of p-values
## not show scientific notation for small numbers
options("scipen" = 10)
## the replication Bayes factor under normality
BFr <- function(to, tr, so, sr) {
bf <- dnorm(x = tr, mean = 0, sd = so) /
dnorm(x = tr, mean = to, sd = sqrt(so^2 + sr^2))
return(bf)
}
formatBF. <- function(BF) {
if (is.na(BF)) {
BFform <- NA
} else if (BF > 1) {
if (BF > 1000) {
BFform <- "> 1000"
} else {
BFform <- as.character(signif(BF, 2))
}
} else {
if (BF < 1/1000) {
BFform <- "< 1/1000"
} else {
BFform <- paste0("1/", signif(1/BF, 2))
}
}
if (!is.na(BFform) && BFform == "1/1") {
return("1")
} else {
return(BFform)
}
}
formatBF <- Vectorize(FUN = formatBF.)
## Bayes factor under normality with unit-information prior under alternative
BF01 <- function(estimate, se, null = 0, unitvar = 4) {
bf <- dnorm(x = estimate, mean = null, sd = se) /
dnorm(x = estimate, mean = null, sd = sqrt(se^2 + unitvar))
return(bf)
}
\begin{abstract}
In several large-scale replication projects, statistically non-significant
results in both the original and the replication study have been interpreted
as a ``replication success''. Here we discuss the logical problems with this
approach. Non-significance in both studies does not ensure that the studies
provide evidence for the absence of an effect and ``replication success'' can
virtually always be achieved if the sample sizes of the studies are small
enough. In addition, the relevant error rates are not controlled. We show how
methods, such as equivalence testing and Bayes factors, can be used to
adequately quantify the evidence for the absence of an effect and how they can
be applied in the replication setting. Using data from the Reproducibility
Project: Cancer Biology we illustrate that many original and replication
studies with ``null results'' are in fact inconclusive. We conclude that it is
important to also replicate studies with statistically non-significant
results, but that they should be designed, analyzed, and interpreted
appropriately.
\end{abstract}
% \rule{\textwidth}{0.5pt} \emph{Keywords}: Bayesian hypothesis testing,
% equivalence testing, meta-research, null hypothesis, replication success}
% definition from RPCP: null effects - the original authors interpreted their
% data as not showing evidence for a meaningful relationship or impact of an
% intervention.
\textit{Absence of evidence is not evidence of absence} -- the title of the 1995
paper by Douglas Altman and Martin Bland has since become a mantra in the
statistical and medical literature \citep{Altman1995}. Yet, the misconception
that a statistically non-significant result indicates evidence for the absence
of an effect is unfortunately still widespread \citep{Makin2019}. Such a ``null
result'' -- typically characterized by a $p$-value of $p > 0.05$ for the null
hypothesis of an absent effect -- may also occur if an effect is actually
present. For example, if the sample size of a study is chosen to detect an
assumed effect with a power of 80\%, null results will incorrectly occur 20\% of
the time when the assumed effect is actually present. Conversely, if the power
of the study is lower, null results will occur more often. In general, the lower
the power of a study, the greater the ambiguity of a null result. To put a null
result in context, it is therefore critical to know whether the study was
adequately powered and under what assumed effect the power was calculated
\citep{Hoenig2001, Greenland2012}. However, if the goal of a study is to
explicitly quantify the evidence for the absence of an effect, more appropriate
methods designed for this task, such as equivalence testing \citep{Wellek2010}
or Bayes factors \citep{Kass1995}, should be used from the outset.
% two systematic reviews that I found which show that animal studies are very
% much underpowered on average \citep{Jennions2003,Carneiro2018}
The contextualization of null results becomes even more complicated in the
setting of replication studies. In a replication study, researchers attempt to
repeat an original study as closely as possible in order to assess whether
similar results can be obtained with new data \citep{NSF2019}. There have been
various large-scale replication projects in the biomedical and social sciences
in the last decade \citep[among
others]{Prinz2011,Begley2012,Klein2014,Opensc2015,Camerer2016,Camerer2018,Klein2018,Cova2018,Errington2021}.
Most of these projects reported alarmingly low replicability rates across a
broad spectrum of criteria for quantifying replicability. While most of these
projects restricted their focus on original studies with statistically
significant results (``positive results''), the \emph{Reproducibility Project:
Psychology} \citep[RPP,][]{Opensc2015}, the \emph{Reproducibility Project:
Experimental Philosophy} \citep[RPEP,][]{Cova2018}, and the
\emph{Reproducibility Project: Cancer Biology} \citep[RPCB,][]{Errington2021}
also attempted to replicate some original studies with null results.
The RPP excluded the original null results from its overall assessment of
replication success, but the RPCB and the RPEP explicitly defined null results
in both the original and the replication study as a criterion for ``replication
success''. There are several logical problems with this ``non-significance''
criterion. First, if the original study had low statistical power, a
non-significant result is highly inconclusive and does not provide evidence for
the absence of an effect. It is then unclear what exactly the goal of the
replication should be -- to replicate the inconclusiveness of the original
result? On the other hand, if the original study was adequately powered, a
non-significant result may indeed provide some evidence for the absence of an
effect when analyzed with appropriate methods, so that the goal of the
replication is clearer. However, the criterion does not distinguish between
these two cases. Second, with this criterion researchers can virtually always
achieve replication success by conducting two studies with very small sample
sizes, such that the $p$-values are non-significant and the results are
inconclusive. This is because the null hypothesis under which the $p$-values are
computed is misaligned with the goal of inference, which is to quantify the
evidence for the absence of an effect. We will discuss methods that are better
aligned with this inferential goal. % in Section~\ref{sec:methods}.
Third, the criterion does not control the error of falsely claiming the absence
of an effect at some predetermined rate. This is in contrast to the standard
replication success criterion of requiring significance from both studies
\citep[also known as the two-trials rule, see chapter 12.2.8 in][]{Senn2008},
which ensures that the error of falsely claiming the presence of an effect is
controlled at a rate equal to the squared significance level (for example,
$5\% \times 5\% = 0.25\%$ for a $5\%$ significance level). The non-significance
criterion may be intended to complement the two-trials rule for null results,
but it fails to do so in this respect, which may be important to regulators,
funders, and researchers. We will now demonstrate these issues and potential
solutions using the null results from the RPCB.
\section{Null results from the Reproducibility Project: Cancer Biology}
\label{sec:rpcb}
rpcbRaw <- read.csv(file = "../data/rpcb-effect-level.csv")
rpcb <- rpcbRaw %>%
mutate(
## recompute one-sided p-values based on normality
## (in direction of original effect estimate)
zo = smdo/so,
zr = smdr/sr,
po1 = pnorm(q = abs(zo), lower.tail = FALSE),
pr1 = pnorm(q = abs(zr), lower.tail = ifelse(sign(zo) < 0, TRUE, FALSE)),
## compute some other quantities
c = so^2/sr^2, # variance ratio
d = smdr/smdo, # relative effect size
po2 = 2*(1 - pnorm(q = abs(zo))), # two-sided original p-value
pr2 = 2*(1 - pnorm(q = abs(zr))), # two-sided replication p-value
sm = 1/sqrt(1/so^2 + 1/sr^2), # standard error of fixed effect estimate
smdm = (smdo/so^2 + smdr/sr^2)*sm^2, # fixed effect estimate
pm2 = 2*(1 - pnorm(q = abs(smdm/sm))), # two-sided fixed effect p-value
Q = (smdo - smdr)^2/(so^2 + sr^2), # Q-statistic
pQ = pchisq(q = Q, df = 1, lower.tail = FALSE), # p-value from Q-test
BForig = BF01(estimate = smdo, se = so), # unit-information BF for original
BForigformat = formatBF(BF = BForig),
BFrep = BF01(estimate = smdr, se = sr), # unit-information BF for replication
BFrepformat = formatBF(BF = BFrep)
## check the sample sizes
## paper 5 (https://osf.io/q96yj) - 1 Cohen's d - sample size correspond to forest plot
## paper 9 (https://osf.io/yhq4n) - 3 Cohen's w- sample size do not correspond at all
## paper 15 (https://osf.io/ytrx5) - 1 r - sample size correspond to forest plot
## paper 19 (https://osf.io/465r3) - 2 Cohen's dz - sample size correspond to forest plot
## paper 20 (https://osf.io/acg8s) - 1 r and 1 Cliff's delta - sample size correspond to forest plot
## paper 21 (https://osf.io/ycq5g) - 1 Cohen's d - sample size correspond to forest plot
## paper 24 (https://osf.io/pcuhs) - 2 Cohen's d - sample size correspond to forest plot
## paper 28 (https://osf.io/gb7sr/) - 3 Cohen's d - sample size correspond to forest plot
## paper 29 (https://osf.io/8acw4) - 1 Cohen's d - sample size do not correspond, seem to be double
## paper 41 (https://osf.io/qnpxv) - 1 Hazard ratio - sample size correspond to forest plot
## paper 47 (https://osf.io/jhp8z) - 2 r - sample size correspond to forest plot
## paper 48 (https://osf.io/zewrd) - 1 r - sample size do not correspond to forest plot for original study
Figure~\ref{fig:2examples} shows standardized mean difference effect estimates
with confidence intervals from two RPCB study pairs. Both are ``null results''
and meet the non-significance criterion for replication success (the two-sided
$p$-values are greater than 0.05 in both the original and the replication study),
but intuition would suggest that these two pairs are very much different.
\begin{figure}[ht]
## some evidence for absence of effect https://doi.org/10.7554/eLife.45120 I
## can't find the replication effect like reported in the data set :( let's take
## it at face value we are not data detectives
## https://iiif.elifesciences.org/lax/45120%2Felife-45120-fig4-v1.tif/full/1500,/0/default.jpg
## https://iiif.elifesciences.org/lax/25306%2Felife-25306-fig5-v2.tif/full/1500,/0/default.jpg
plotDF1 <- rpcbNull %>%
filter(id %in% c(study1, study2)) %>%
mutate(label = ifelse(id == study1,
"Goetz et al. (2011)\nEvidence of absence",
"Dawson et al. (2011)\nAbsence of evidence"))
## ## RH: this data is really a mess. turns out for Dawson n represents the group
## ## size (n = 6 in https://osf.io/8acw4) while in Goetz it is the sample size of
## ## the whole experiment (n = 34 and 61 in https://osf.io/acg8s). in study 2 the
## ## so multiply by 2 to have the total sample size, see Figure 5A
## ## https://doi.org/10.7554/eLife.25306.012
## plotDF1$no[plotDF1$id == study2] <- plotDF1$no[plotDF1$id == study2]*2
## plotDF1$nr[plotDF1$id == study2] <- plotDF1$nr[plotDF1$id == study2]*2
## create plot showing two example study pairs with null results
conflevel <- 0.95
ggplot(data = plotDF1) +
facet_wrap(~ label) +
geom_hline(yintercept = 0, lty = 2, alpha = 0.3) +
geom_pointrange(aes(x = "Original", y = smdo,
ymin = smdo - qnorm(p = (1 + conflevel)/2)*so,
ymax = smdo + qnorm(p = (1 + conflevel)/2)*so), fatten = 3) +
geom_pointrange(aes(x = "Replication", y = smdr,
ymin = smdr - qnorm(p = (1 + conflevel)/2)*sr,
ymax = smdr + qnorm(p = (1 + conflevel)/2)*sr), fatten = 3) +
geom_text(aes(x = 1.05, y = 2.5,
label = paste("italic(n) ==", no)), col = "darkblue",
parse = TRUE, size = 3.8, hjust = 0) +
label = paste("italic(n) ==", nr)), col = "darkblue",
parse = TRUE, size = 3.8, hjust = 0) +
label = paste("italic(p) ==", formatPval(po))), col = "darkblue",
label = paste("italic(p) ==", formatPval(pr))), col = "darkblue",
labs(x = "", y = "Standardized mean difference (SMD)") +
theme_bw() +
theme(panel.grid.minor = element_blank(),
panel.grid.major.x = element_blank(),
strip.text = element_text(size = 12, margin = margin(4), vjust = 1.5),
strip.background = element_rect(fill = alpha("tan", 0.4)),
axis.text = element_text(size = 12))
@
\caption{\label{fig:2examples} Two examples of original and replication study
pairs which meet the non-significance replication success criterion from the
Reproducibility Project: Cancer Biology \citep{Errington2021}. Shown are
standardized mean difference effect estimates with \Sexpr{round(conflevel*100,
2)}\% confidence intervals, sample sizes, and two-sided $p$-values for the
null hypothesis that the standardized mean difference is zero.}
The original study from \citet{Dawson2011} and its replication both show large
effect estimates in magnitude, but due to the small sample sizes, the
uncertainty of these estimates is very large, too. If the sample sizes of the
studies were larger and the point estimates remained the same, intuitively both
studies would provide evidence for a non-zero effect. However, with the samples
sizes that were actually used, the results seem inconclusive. In contrast, the
effect estimates from \citet{Goetz2011} and its replication are much smaller in
magnitude and their uncertainty is also smaller because the studies used larger
sample sizes. Intuitively, these studies seem to provide some evidence for a
zero (or negligibly small) effect. While these two examples show the qualitative
difference between absence of evidence and evidence of absence, we will now
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\section{Methods for asssessing replicability of null results}
\label{sec:methods}
There are both frequentist and Bayesian methods that can be used for assessing
evidence for the absence of an effect. \citet{Anderson2016} provide an excellent
summary of both approaches in the context of replication studies in psychology.
We now briefly discuss two possible approaches -- frequentist equivalence
testing and Bayesian hypothesis testing -- and their application to the RPCB
data.
\subsection{Equivalence testing}
Equivalence testing was developed in the context of clinical trials to assess
whether a new treatment -- typically cheaper or with fewer side effects than the
established treatment -- is practically equivalent to the established treatment
\citep{Westlake1972,Schuirmann1987}. The method can also be used to assess
whether an effect is practically equivalent to the value of an absent effect,
usually zero. Using equivalence testing as a remedy for non-significant results
has been suggested by several authors \citep{Hauck1986, Campbell2018}. The main
challenge is to specify the margin $\Delta > 0$ that defines an equivalence
range $[-\Delta, +\Delta]$ in which an effect is considered as absent for
practical purposes. The goal is then to reject
the % composite %% maybe too technical?
null hypothesis that the true effect is outside the equivalence range. This is
in contrast to the usual null hypothesis of a superiority test which states that
the effect is zero or smaller than zero, see Figure~\ref{fig:hypotheses} for an
illustration.
\begin{figure}
\begin{center}
\begin{tikzpicture}[ultra thick]
\draw[stealth-stealth] (0,0) -- (6,0);
\node[text width=4.5cm, align=center] at (3,-1) {Effect size};
\draw (2,0.2) -- (2,-0.2) node[below]{$-\Delta$};
\draw (3,0.2) -- (3,-0.2) node[below]{$0$};
\draw (4,0.2) -- (4,-0.2) node[below]{$+\Delta$};
\node[text width=5cm, align=left] at (0,1.5) {\textbf{Equivalence}};
\draw [draw={darkred2},decorate,decoration={brace,amplitude=5pt}]
(2.05,1) -- (3.95,1) node[midway,yshift=1.5em]{\textcolor{darkred2}{$H_1$}};
\draw [draw={darkblue2},decorate,decoration={brace,amplitude=5pt,aspect=0.6}]
(0,1) -- (1.95,1) node[pos=0.6,yshift=1.5em]{\textcolor{darkblue2}{$H_0$}};
\draw [draw={darkblue2},decorate,decoration={brace,amplitude=5pt,aspect=0.4}]
(4.05,1) -- (6,1) node[pos=0.4,yshift=1.5em]{\textcolor{darkblue2}{$H_0$}};
\node[text width=5cm, align=left] at (0,3.5) {\textbf{Superiority \\(two-sided)}};
(3,3) -- (3,3) node[midway,yshift=1.5em]{\textcolor{darkblue2}{$H_0$}};
\draw[darkblue2] (3,2.9) -- (3,3.2);
\draw [draw={darkred2},decorate,decoration={brace,amplitude=5pt,aspect=0.6}]
(0,3) -- (2.95,3) node[pos=0.6,yshift=1.5em]{\textcolor{darkred2}{$H_1$}};
\draw [draw={darkred2},decorate,decoration={brace,amplitude=5pt,aspect=0.4}]
(3.05,3) -- (6,3) node[pos=0.4,yshift=1.5em]{\textcolor{darkred2}{$H_1$}};
\node[text width=5cm, align=left] at (0,5.5) {\textbf{Superiority \\ (one-sided)}};
\draw [draw={darkred2},decorate,decoration={brace,amplitude=5pt,aspect=0.4}]
(3.05,5) -- (6,5) node[pos=0.4,yshift=1.5em]{\textcolor{darkred2}{$H_1$}};
\draw [draw={darkblue2},decorate,decoration={brace,amplitude=5pt,aspect=0.6}]
(0,5) -- (3,5) node[pos=0.6,yshift=1.5em]{\textcolor{darkblue2}{$H_0$}};
\draw [dashed] (2,0) -- (2,1);
\draw [dashed] (4,0) -- (4,1);
\draw [dashed] (3,0) -- (3,1);
\draw [dashed] (3,3.9) -- (3,5);
\end{tikzpicture}
\end{center}
\caption{Null hypothesis ($H_0$) and alternative hypothesis ($H_1$) for
different study designs with equivalence margin $\Delta$.}
\label{fig:hypotheses}
\end{figure}
To ensure that the null hypothesis is falsely rejected at most
$\alpha \times 100\%$ of the time, one either rejects it if the
$(1-2\alpha)\times 100\%$ confidence interval for the effect is contained within
the equivalence range (for example, a 90\% confidence interval for
$\alpha = 5\%$), or if two one-sided tests (TOST) for the effect being
smaller/greater than $+\Delta$ and $-\Delta$ are significant at level $\alpha$,
respectively. A quantitative measure of evidence for the absence of an effect is
then given by the maximum of the two one-sided $p$-values (the TOST $p$-value).
\begin{figure}
\begin{fullwidth}
<< "plot-null-findings-rpcb", fig.height = 8.25, fig.width = "0.95\\linewidth" >>=
rpcbNull$ptosto <- with(rpcbNull, pmax(pnorm(q = smdo, mean = margin, sd = so,
lower.tail = TRUE),
pnorm(q = smdo, mean = -margin, sd = so,
lower.tail = FALSE)))
rpcbNull$ptostr <- with(rpcbNull, pmax(pnorm(q = smdr, mean = margin, sd = sr,
lower.tail = TRUE),
pnorm(q = smdr, mean = -margin, sd = sr,
lower.tail = FALSE)))
rpcbNull$id <- ifelse(rpcbNull$id == "(20, 1, 1)",
"(20, 1, 1) - Goetz et al. (2011)", rpcbNull$id)
rpcbNull$id <- ifelse(rpcbNull$id == "(29, 2, 2)",
"(29, 2, 2) - Dawson et al. (2011)", rpcbNull$id)
## create plots of all study pairs with null results in original study
ggplot(data = rpcbNull) +
facet_wrap(~ id, scales = "free", ncol = 3) +
geom_hline(yintercept = 0, lty = 2, alpha = 0.25) +
## equivalence margin
geom_hline(yintercept = c(-margin, margin), lty = 3, col = 2, alpha = 0.9) +
geom_pointrange(aes(x = "Original", y = smdo,
ymin = smdo - qnorm(p = (1 + conflevel)/2)*so,
ymax = smdo + qnorm(p = (1 + conflevel)/2)*so),
size = 0.25, fatten = 2) +
geom_pointrange(aes(x = "Replication", y = smdr,
ymin = smdr - qnorm(p = (1 + conflevel)/2)*sr,
ymax = smdr + qnorm(p = (1 + conflevel)/2)*sr),
size = 0.25, fatten = 2) +
annotate(geom = "ribbon", x = seq(0, 3, 0.01), ymin = -margin, ymax = margin,
alpha = 0.05, fill = 2) +
labs(x = "", y = "Standardized mean difference (SMD)") +
geom_text(aes(x = 1.05, y = pmax(smdo + 2.5*so, smdr + 2.5*sr, 1.1*margin),
label = paste("italic(n) ==", no)), col = "darkblue",
parse = TRUE, size = 2.3, hjust = 0, vjust = 2) +
geom_text(aes(x = 2.05, y = pmax(smdo + 2.5*so, smdr + 2.5*sr, 1.1*margin),
label = paste("italic(n) ==", nr)), col = "darkblue",
parse = TRUE, size = 2.3, hjust = 0, vjust = 2) +
geom_text(aes(x = 1.05, y = pmax(smdo + 2.5*so, smdr + 2.5*sr, 1.1*margin),
label = paste("italic(p)",
ifelse(po < 0.0001, "", "=="),
formatPval(po))), col = "darkblue",
parse = TRUE, size = 2.3, hjust = 0) +
geom_text(aes(x = 2.05, y = pmax(smdo + 2.5*so, smdr + 2.5*sr, 1.1*margin),
label = paste("italic(p)",
ifelse(pr < 0.0001, "", "=="),
formatPval(pr))), col = "darkblue",
parse = TRUE, size = 2.3, hjust = 0) +
geom_text(aes(x = 1.05, y = pmax(smdo + 2.5*so, smdr + 2.5*sr, 1.1*margin),
label = paste("italic(p)['TOST']",
ifelse(ptosto < 0.0001, "", "=="),
col = "darkblue", parse = TRUE, size = 2.3, hjust = 0, vjust = 3) +
geom_text(aes(x = 2.05, y = pmax(smdo + 2.5*so, smdr + 2.5*sr, 1.1*margin),
label = paste("italic(p)['TOST']",
ifelse(ptostr < 0.0001, "", "=="),
col = "darkblue", parse = TRUE, size = 2.3, hjust = 0, vjust = 3) +
geom_text(aes(x = 1.05, y = pmax(smdo + 2.5*so, smdr + 2.5*sr, 1.1*margin),
label = paste("BF['01']", ifelse(BForig <= 1/1000, "", "=="),
size = 2.3, hjust = 0, vjust = 4.5) +
geom_text(aes(x = 2.05, y = pmax(smdo + 2.5*so, smdr + 2.5*sr, 1.1*margin),
label = paste("BF['01']", ifelse(BFrep <= 1/1000, "", "=="),
theme_bw() +
theme(panel.grid.minor = element_blank(),
panel.grid.major = element_blank(),
strip.text = element_text(size = 8, margin = margin(3), vjust = 2),
strip.background = element_rect(fill = alpha("tan", 0.4)),
\caption{Standardized mean difference (SMD) effect estimates with
\Sexpr{round(conflevel*100, 2)}\% confidence interval for the ``null results''
% (those with original two-sided $p$-value $p > 0.05$)
and their replication studies from the Reproducibility Project: Cancer Biology
\citep{Errington2021}. The identifier above each plot indicates (original
paper number, experiment number, effect number). Two original effect estimates
from original paper 48 were statistically significant at $p < 0.05$, but were
interpreted as null results by the original authors and therefore treated as
null results by the RPCB. The two examples from Figure~\ref{fig:2examples} are
indicated in the plot titles. The dashed gray line represents the value of no
effect ($\text{SMD} = 0$), while the dotted red lines represent the
equivalence range with a margin of $\Delta = \Sexpr{margin}$. The $p$-values
$p_{\text{TOST}}$ are the maximum of the two one-sided $p$-values for the
effect being less than or greater than $+\Delta$ or $-\Delta$, respectively.
The Bayes factors $\BF_{01}$ quantify the evidence for the null hypothesis
$H_{0} \colon \text{SMD} = 0$ against the alternative
$H_{1} \colon \text{SMD} \neq 0$ with normal unit-information prior assigned
to the SMD under $H_{1}$.}
Returning to the RPCB data, Figure~\ref{fig:nullfindings} shows the standardized
mean difference effect estimates with \Sexpr{round(conflevel*100, 2)}\%
confidence intervals for the 20 study pairs with quantitative null results in
the original study ($p > 0.05$). The dotted red lines represent an equivalence
range for the margin $\Delta = \Sexpr{margin}$, for which the shown TOST
$p$-values are computed. This margin is rather lax compared to the margins
typically used in clinical research; we chose it primarily for illustrative
purposes and because effect sizes in preclinical research are typically much
larger than in clinical research. In practice, the margin should be determined
on a case-by-case basis by researchers who are familiar with the subject matter.
However, even with this generous margin, only four of the twenty study pairs --
one of them being the previously discussed example from \citet{Goetz2011} -- are
able to establish equivalence at the 5\% level in the sense that both the
original and the replication 90\% confidence interval fall within the
equivalence range (or equivalently that their TOST $p$-values are smaller than
$0.05$). For the remaining 16 studies -- for instance, the previously discussed
example from \citet{Dawson2011} -- the situation remains inconclusive and there
is neither evidence for the absence nor the presence of the effect.
The distinction between absence of evidence and evidence of absence is naturally
built into the Bayesian approach to hypothesis testing. A central measure of
evidence is the Bayes factor \citep{Kass1995}, which is the updating factor of
the prior odds to the posterior odds of the null hypothesis $H_{0}$ versus the
\begin{align*}
\underbrace{\frac{\Pr(H_{0} \given \mathrm{data})}{\Pr(H_{1} \given
\mathrm{data})}}_{\mathrm{Posterior~odds}}
= \underbrace{\frac{\Pr(H_{0})}{\Pr(H_{1})}}_{\mathrm{Prior~odds}}
\times \underbrace{\frac{p(\mathrm{data} \given H_{0})}{p(\mathrm{data}
\given H_{1})}}_{\mathrm{Bayes~factor}~\BF_{01}}.
\end{align*}
The Bayes factor quantifies how much the observed data have increased or
decreased the probability of the null hypothesis $H_{0}$ relative to the
alternative $H_{1}$. If the null hypothesis states the absence of an effect, a
Bayes factor greater than one (\mbox{$\BF_{01} > 1$}) indicates evidence for the
absence of the effect and a Bayes factor smaller than one indicates evidence for
the presence of the effect (\mbox{$\BF_{01} < 1$}), whereas a Bayes factor not
much different from one indicates absence of evidence for either hypothesis
When the observed data are dichotomized into positive (\mbox{$p < 0.05$}) or null
results (\mbox{$p > 0.05$}), the Bayes factor based on a null result is the
probability of observing \mbox{$p > 0.05$} when the effect is indeed absent
(which is $95\%$) divided by the probability of observing $p > 0.05$ when the
effect is indeed present (which is one minus the power of the study). For
example, if the power is 90\%, we have
\mbox{$\BF_{01} = 95\%/10\% = \Sexpr{round(0.95/0.1, 2)}$} indicating almost ten
times more evidence for the absence of the effect than for its presence. On the
other hand, if the power is only 50\%, we have
\mbox{$\BF_{01} = 95\%/50\% = \Sexpr{round(0.95/0.5,2)}$} indicating only
slightly more evidence for the absence of the effect. This example also
highlights the main challenge with Bayes factors -- the specification of the
alternative hypothesis $H_{1}$. The assumed effect under $H_{1}$ is directly
related to the power of the study, and researchers who assume different effects
under $H_{1}$ will end up with different Bayes factors. Instead of specifying a
single effect, one therefore typically specifies a ``prior distribution'' of
plausible effects. Importantly, the prior distribution, like the equivalence
margin, should be determined by researchers with subject knowledge and before
the data are observed.
In practice, the observed data should not be dichotomized into positive or null
results, as this leads to a loss of information. Therefore, to compute the Bayes
factors for the RPCB null results, we used the observed effect estimates as the
data and assumed a normal sampling distribution for them, as in a meta-analysis.
The Bayes factors $\BF_{01}$ shown in Figure~\ref{fig:nullfindings} then
quantify the evidence for the null hypothesis of no effect
($H_{0} \colon \text{SMD} = 0$) against the alternative hypothesis that there is
an effect ($H_{1} \colon \text{SMD} \neq 0$) using a normal ``unit-information''
prior distribution \citep{Kass1995b} for the effect size under the alternative
$H_{1}$. There are several more advanced prior distributions that could be used
here, and they should ideally be specified for each effect individually based on
domain knowledge. The normal unit-information prior (with a standard deviation
of 2 for SMDs) is only a reasonable default choice, as it implies that small to
large effects are plausible under the alternative. We see that in most cases
there is no substantial evidence for either the absence or the presence of an
effect, as with the equivalence tests. The Bayes factors for the two previously
discussed examples from \citet{Goetz2011} and \citet{Dawson2011} are consistent
with our intuititons -- there is indeed some evidence for the absence of an
effect in \citet{Goetz2011}, while there is even slightly more evidence for the
presence of an effect in \citet{Dawson2011}, though the Bayes factor is very
close to one due to the small sample sizes. With a lenient Bayes factor
threshold of $\BF_{01} > 3$ to define evidence for the absence of the effect,
only one of the twenty study pairs meets this criterion in both the original and
studyInteresting <- filter(rpcbNull, id == "(48, 2, 4)")
noInteresting <- studyInteresting$no
nrInteresting <- studyInteresting$nr
## write.csv(rpcbNull, "rpcb-Null.csv", row.names = FALSE)
Among the twenty RPCB null results, there is one interesting case (the rightmost
plot in the fourth row (48, 2, 4, 1)) where the Bayes factor is qualitatively
different from the equivalence test, revealing a fundamental difference between
the two approaches. The Bayes factor is concerned with testing whether the
effect is \emph{exactly zero}, whereas the equivalence test is concerned with
whether the effect is within an \emph{interval around zero}. Due to the very
large sample size in the original study ($n = \Sexpr{noInteresting}$) and the
replication ($n = \Sexpr{nrInteresting}$), the data are incompatible with an
exactly zero effect, but compatible with effects within the equivalence range.
Apart from this example, however, the approaches lead to the same qualitative
conclusion -- most RPCB null results are highly ambiguous.
We showed that in most of the RPCB studies with ``null results'' (those with
$p > 0.05$), neither the original nor the replication study provided conclusive
evidence for the presence or absence of an effect. It seems logically
questionable to declare an inconclusive replication of an inconclusive original
study as a replication success. While it is important to replicate original
studies with null results, our analysis highlights that they should be analyzed
and interpreted appropriately.
For both the equivalence testing and the Bayes factor approach, it is critical
that the parameters of the procedure (the equivalence margin and the prior
distribution) are specified independently of the data, ideally before the
studies are conducted. Typically, however, the original studies were designed to
find evidence for the presence of an effect, and the goal of replicating the
``null result'' was formulated only after failure to do so. \citet{Campbell2021}
discuss various approaches to post-hoc specification of equivalence margins,
such as motivating it using data from previous studies or using field
conventions. \citet{Hauck1986} propose a sensitivity analysis approach in the
form of plotting the TOST $p$-value against a range of possible margins
(``equivalence curves''). Post-hoc specification of a prior distribution for a
Bayes factor may likewise be based on historical data, field conventions, or
assessed visually with sensitivity analyses.
% As error rate control may no longer be ensured in this case, the TOST
% $p$-values should not be used as dichotomous decision tools, but rather as
% descriptive measures of compatibility between the data and effects outside the
% equivalence region \citep{Amrhein2019, Rafi2020, Greenland2023}.
While the equivalence test and the Bayes factor are two principled methods for
analyzing original and replication studies with null results, they are not the
only possible methods for doing so. For instance, the reverse-Bayes approach
from \citet{Micheloud2022} specifically tailored to equivalence testing in the
replication setting may lead to more appropriate inferences as it also takes
into account the compatibility of the effect estimates from original and
replication studies. In addition, there are various other Bayesian methods which
could potentially improve upon the considered Bayes factor approach. For
example, Bayes factors based on non-local priors \citep{Johnson2010} or based on
interval null hypotheses \citep{Morey2011, Liao2020}, methods for equivalence
testing based on effect size posterior distributions \citep{Kruschke2018}, or
Bayesian procedures that involve utilities of decisions \citep{Lindley1998}.
Finally, the design of replication studies should align with the planned
analysis \citep{Anderson2017, Anderson2022, Micheloud2020, Pawel2022c}.
% The RPCB determined the sample size of their replication studies to achieve at
% least 80\% power for detecting the original effect size which does not seem to
% be aligned with their goal
If the goal of the study is to find evidence for the absence of an effect, the
replication sample size should also be determined so that the study has adequate
power to make conclusive inferences regarding the absence of the effect.
We thank the contributors of the RPCB for their tremendous efforts and for
making their data publicly available. We thank Maya Mathur for helpful advice
with the data preparation. This work was supported by the Swiss National Science
Foundation (grant \href{https://data.snf.ch/grants/grant/189295}{\#189295}).
The code and data to reproduce our analyses is openly available at
\url{https://gitlab.uzh.ch/samuel.pawel/rsAbsence}. A snapshot of the repository
at the time of writing is available at
\url{https://doi.org/10.5281/zenodo.XXXXXX}. We used the statistical programming
language R version \Sexpr{paste(version$major, version$minor, sep = ".")}
\citep{R} for analyses. The R packages \texttt{ggplot2} \citep{Wickham2016},
\texttt{dplyr} \citep{Wickham2022}, \texttt{knitr} \citep{Xie2022}, and
\texttt{reporttools} \citep{Rufibach2009} were used for plotting, data
preparation, dynamic reporting, and formatting, respectively. The data from the
RPCB were obtained by downloading the files from
\url{https://github.com/mayamathur/rpcb} (commit a1e0c63) and extracting the
relevant variables as indicated in the R script \texttt{preprocess-rpcb-data.R}
which is available in our git repository.% The effect estimates and standard
% errors on SMD scale provided in this data set differ in some cases from those in
% the data set available at \url{https://doi.org/10.17605/osf.io/e5nvr}, which is
% cited in \citet{Errington2021}. We used this particular version of the data set
% because it was recommended to us by the RPCB statistician (Maya Mathur) upon
% request.
% For the \citet{Dawson2011} example study and its replication \citep{Shan2017},
% the sample sizes $n = 3$ in th data set seem to correspond to the group sample
% sizes, see Figure 5A in the replication study
% (\url{https://doi.org/10.7554/eLife.25306.012}), which is why we report the
% total sample sizes of $n = 6$ in Figure~\ref{fig:2examples}.
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\appendix
\begin{appendixbox}
% \label{first:sensitivity}
% \section{Sensitivity analyses}
\begin{center}
<< "sensitivity", fig.height = 8 >>=
## compute number of successful replications as a function of the equivalence margin
marginseq <- seq(0.01, 4.5, 0.01)
alphaseq <- c(0.005, 0.05, 0.1)
sensitivityGrid <- expand.grid(m = marginseq, a = alphaseq)
equivalenceDF <- lapply(X = seq(1, nrow(sensitivityGrid)), FUN = function(i) {
m <- sensitivityGrid$m[i]
a <- sensitivityGrid$a[i]
rpcbNull$ptosto <- with(rpcbNull, pmax(pnorm(q = smdo, mean = m, sd = so,
lower.tail = TRUE),
pnorm(q = smdo, mean = -m, sd = so,
lower.tail = FALSE)))
rpcbNull$ptostr <- with(rpcbNull, pmax(pnorm(q = smdr, mean = m, sd = sr,
lower.tail = TRUE),
pnorm(q = smdr, mean = -m, sd = sr,
lower.tail = FALSE)))
successes <- sum(rpcbNull$ptosto <= a & rpcbNull$ptostr <= a)
data.frame(margin = m, alpha = a,
successes = successes, proportion = successes/nrow(rpcbNull))
}) %>%
bind_rows()
## plot number of successes as a function of margin
nmax <- nrow(rpcbNull)
bks <- seq(0, nmax, round(nmax/5))
labs <- paste0(bks, " (", bks/nmax*100, "%)")
plotA <- ggplot(data = equivalenceDF,
aes(x = margin, y = successes,
color = factor(alpha, ordered = TRUE))) +
facet_wrap(~ "Equivalence test") +
geom_vline(xintercept = margin, lty = 2, alpha = 0.4) +
geom_step(alpha = 0.9, linewidth = 0.8) +
scale_y_continuous(breaks = bks, labels = labs) +
## scale_y_continuous(labels = scales::percent) +
guides(color = guide_legend(reverse = TRUE)) +
labs(x = bquote("Equivalence margin" ~ Delta),
y = "Successful replications",
color = bquote(italic("p")["TOST"] ~ "threshold" ~ alpha)) +
theme_bw() +
theme(panel.grid.minor = element_blank(),
panel.grid.major = element_blank(),
strip.background = element_rect(fill = alpha("tan", 0.4)),
strip.text = element_text(size = 12),
legend.position = c(0.85, 0.25),
plot.background = element_rect(fill = "transparent", color = NA),
## axis.text.y = element_text(hjust = 0),
legend.box.background = element_rect(fill = "transparent", colour = NA))
## compute number of successful replications as a function of the prior scale
priorsdseq <- seq(0, 40, 0.1)
bfThreshseq <- c(3, 6, 10)
sensitivityGrid2 <- expand.grid(s = priorsdseq, thresh = bfThreshseq)
bfDF <- lapply(X = seq(1, nrow(sensitivityGrid2)), FUN = function(i) {
priorsd <- sensitivityGrid2$s[i]
thresh <- sensitivityGrid2$thresh[i]
rpcbNull$BForig <- with(rpcbNull, BF01(estimate = smdo, se = so, unitvar = priorsd^2))
rpcbNull$BFrep <- with(rpcbNull, BF01(estimate = smdr, se = sr, unitvar = priorsd^2))
successes <- sum(rpcbNull$BForig >= thresh & rpcbNull$BFrep >= thresh)
data.frame(priorsd = priorsd, thresh = thresh,
successes = successes, proportion = successes/nrow(rpcbNull))
}) %>%
bind_rows()
## plot number of successes as a function of prior sd
plotB <- ggplot(data = bfDF,
aes(x = priorsd, y = successes, color = factor(thresh, ordered = TRUE))) +
facet_wrap(~ "Bayes factor") +
geom_vline(xintercept = 4, lty = 2, alpha = 0.4) +
geom_step(alpha = 0.9, linewidth = 0.8) +
scale_y_continuous(breaks = bks, labels = labs, limits = c(0, nmax)) +
## scale_y_continuous(labels = scales::percent, limits = c(0, 1)) +
guides(color = guide_legend(reverse = TRUE)) +
labs(x = "Prior distribution scale",
y = "Successful replications ",
color = bquote("BF"["01"] ~ "threshold" ~ gamma)) +
theme_bw() +
theme(panel.grid.minor = element_blank(),
panel.grid.major = element_blank(),
strip.background = element_rect(fill = alpha("tan", 0.4)),
strip.text = element_text(size = 12),
legend.position = c(0.85, 0.25),
plot.background = element_rect(fill = "transparent", color = NA),
## axis.text.y = element_text(hjust = 0),
legend.box.background = element_rect(fill = "transparent", colour = NA))
grid.arrange(plotA, plotB, ncol = 1)
@
\captionof{figure}{Number of successful replications of original null results in
the RPCB as a function of the margin $\Delta$ of equivalence test
($p_{\text{TOST}} \leq \alpha$ in both studies) or the scale of the prior
distribution for the effect under the alternative $H_{1}$ of the Bayes
factor ($\BF_{01} \geq \gamma$ in both studies).}
\end{center}
\end{appendixbox}
<< "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)
cat("\\newpage \\section*{Computational details}")
@
<< "sessionInfo2", echo = Reproducibility, results = Reproducibility >>=
cat(paste(Sys.time(), Sys.timezone(), "\n"))