Examples:
Here we demonstrate the approach by implementation with
metarep, using examples from systematic reviews Cochrane
library. These examples are detailed in the paper as well, along with a
demonstration of a way to incorporate our suggestions in standard
meta-analysis reporting system.
We demonstrate the with an example based on a fixed-effects meta-analysis. This example was included in Jaljuli et. al. 2020, found in review number CD002943 in the Cochrane library. This analysis explores the effect of mammogram invitation on attendance during the following 12 months.
library(metarep)
Loading required package: meta
Loading 'meta' package (version 6.5-0).
Type 'help(meta)' for a brief overview.
Readers of 'Meta-Analysis with R (Use R!)' should install
older version of 'meta' package: https://tinyurl.com/dt4y5drs
library(meta)
data(CD002943_CMP001)
m2943 <- metabin( event.e = N_EVENTS1, n.e = N_TOTAL1,
event.c = N_EVENTS2, n.c = N_TOTAL2,
studlab = STUDY, comb.fixed = T , comb.random = F,
method = 'Peto', sm = CD002943_CMP001$SM[1],
data = CD002943_CMP001)
m2943
Number of studies: k = 5
Number of observations: o = 4166
Number of events: e = 1384
OR 95%-CI z p-value
Common effect model 1.6564 [1.4317; 1.9164] 6.78 < 0.0001
Quantifying heterogeneity:
tau^2 = 0.1314 [0.0073; 1.6698]; tau = 0.3625 [0.0852; 1.2922]
I^2 = 70.3% [24.4%; 88.3%]; H = 1.84 [1.15; 2.93]
Test of heterogeneity:
Q d.f. p-value
13.47 4 0.0092
Details on meta-analytical method:
- Peto method
- Restricted maximum-likelihood estimator for tau^2
- Q-Profile method for confidence interval of tau^2 and tau
summary(m2943)
OR 95%-CI %W(common)
Sutton-1994 1.2836 [0.9933; 1.6589] 32.3
Somkin-1997 1.8739 [1.5372; 2.2844] 54.2
Turnbull-1991 3.5709 [1.9225; 6.6326] 5.5
Mohler-1995 1.8764 [0.5272; 6.6788] 1.3
Bodiya-1999 1.0758 [0.6110; 1.8941] 6.6
Number of studies: k = 5
Number of observations: o = 4166
Number of events: e = 1384
OR 95%-CI z p-value
Common effect model 1.6564 [1.4317; 1.9164] 6.78 < 0.0001
Quantifying heterogeneity:
tau^2 = 0.1314 [0.0073; 1.6698]; tau = 0.3625 [0.0852; 1.2922]
I^2 = 70.3% [24.4%; 88.3%]; H = 1.84 [1.15; 2.93]
Test of heterogeneity:
Q d.f. p-value
13.47 4 0.0092
Details on meta-analytical method:
- Peto method
- Restricted maximum-likelihood estimator for tau^2
- Q-Profile method for confidence interval of tau^2 and tauIn this meta-analysis, the effect of sending invitation letters was examined in five studies. The authors main result is that: “The odds ratio in relation to the outcome, attendance in response to the mammogram invitation during the 12 months after the invitation, was 1.66 (95% CI 1.43 to 1.92)”.
We suggest reporting the replicability-analysis results alongside:
the r-value and lower confidence bounds on the number of studies.
Results of complete replicability analysis can be added to the contents
of a meta object or using the function
metarep(...), as well as to its summary using
summary( metarep(...) ).
To perform assumption-free replicability-analysis requiring
replicability in at least u = 2 (default) studies, we
calculate \(r(2)-value\) using
truncated-Pearson’s’ test with truncation threshold t=0.05
(default):
(m2943.ra <- metarep(x = m2943 , u = 2 , common.effect = F ,t = 0.05 ,report.u.max = T))
Number of studies: k = 5
Number of observations: o = 4166
Number of events: e = 1384
OR 95%-CI z p-value
Common effect model 1.6564 [1.4317; 1.9164] 6.78 < 0.0001
Quantifying heterogeneity:
tau^2 = 0.1314 [0.0073; 1.6698]; tau = 0.3625 [0.0852; 1.2922]
I^2 = 70.3% [24.4%; 88.3%]; H = 1.84 [1.15; 2.93]
Test of heterogeneity:
Q d.f. p-value
13.47 4 0.0092
Details on meta-analytical method:
- Peto method
- Restricted maximum-likelihood estimator for tau^2
- Q-Profile method for confidence interval of tau^2 and tau
- replicability analysis (r-value = 2e-04)The bottom two lines report the \(r(2)-value\), lower bound on the number of
studies with increased effect (\(u^L_{max}\)) and decreased effect (\(u^R_{max}\)) , respectively. The evidence
towards an increased effect was replicable, with \(r(2) − value = 0.0002\). Moreover, with
\(95\%\) confidence, we can conclude
that at least two studies had an increased effect. For higher
replicability requirement, compute \(r(u')-value\) for \(u'>2\) using
metarep(u = u' , ... ).
The two-sided \(r(u)-value\) of the
model can be accessed via r.value:
The replicability-analysis reported was performed with an assumption
free test, based on truncated-Pearson’s’ test with truncation level set
at the nominal hypothesis testing level (i.e., t=0.05,
default). For ordinary Pearson’s’ test, use t=1.
Although the fixed-effect model assumes that all studies are
estimates of the same common effect \(\theta\), we recommend applying
assumption-free replicability-analysis for protection against an
(perhaps) unsupported assumption. Despite that, we extend our suggested
method with the common-effect incorporation in section 7. This analysis
can be performed via
metarep( ... , common.effect = TRUE ).
metarep also allows adding replicability results to the
conventional forest plots by meta. This can be done by
simply applying forest() on a metarep
object.
The lower bounds \(u^L_{max} \, \text{and}
\, u^R_{max}\) are calculated with \(1-\alpha = 95\%\) confidence level (
default), meaning that each of the null hypotheses \[H^{u^L_{max}/n}(L) \;\; \text{and}\;\;
H^{u^L_{max}/n}(R)\] is tested at level \(\alpha /2 = 2.5\%\), resulting in bounds in
overall type error rate \(5\%\). Type I
error rate can be controlled for any desired \(\alpha\) using the argument
confidence = 1 -\(\alpha\).
The calculation of \(u^L_{max} \,
\text{and} \, u^R_{max}\) can also be calculated directly using
the function find_umax() with the option to specify
one-sided alternative, confidence level, truncation threshold and
common-effect assumption. For example, let’s compute \(u^L_{max}\) with the same confidence level
as produced by m2943.ra.bounds.
find_umax(x = m2943 , common.effect = F,alternative = 'less',t = 0.05,confidence = 0.975)
$worst.case
[1] "Sutton-1994" "Somkin-1997" "Turnbull-1991" "Mohler-1995"
[5] "Bodiya-1999"
$side
Direction of the stronger signal
"less"
$u_max
u^L
0
$r.value
r^R
1
$Replicability_Analysis
[1] "out of 5 studies, 0 with decreased effect."Note that this function produces 2 main types of results:
Worst-case scenario studies: A list of \(n-u_{max}^L+1\) studies names yielding the maximum \[\max_{\forall \{i_1,\dots , i_{n-u+1}\} \subset \{1,\dots , n\} } \{\,p^L_{i_1,\dots , i_{n-u+1}}\,\}\]
Replicability-analysis results, including:
\(u^L_{max}\) or \(u^R_{max}\). If
alternative='two-sided', then \(u_{max}=\max\{u^L_{max}\, , \, u^R_{max}\}\) is also reported.\(r(u_{max}^L)-value\) or \(r(u_{max}^R)-value\) if setting
alternative='less'oralternative='greater', respectively. Ifalternative='two-sided', then \[rvalue = r(u_{max}) = 2\cdot\min\{r^R(u_{max}^R) , r^L(u_{max}^L) \}\] is also reported.
For demonstration, see the following example.
find_umax(x = m2943 , common.effect = F,alternative = 'two-sided',t = 0.05,confidence = 0.95)
$worst.case
[1] "Sutton-1994" "Turnbull-1991" "Mohler-1995" "Bodiya-1999"
$side
Direction of the stronger signal
"greater"
$u_max
u_max u^L u^R
2 0 2
$r.value
r.value r^L r^R
2e-04 1e+00 1e-04
$Replicability_Analysis
[1] "out of 5 studies: 0 with decreased effect, and 2 with increased effect."Replicability analysis using a test-statistic based on
The common-effect assumption
find_umax(x = m2943 , common.effect = T,alternative = 'two-sided', confidence = 0.95)
Performing Replicability analysis with the common-effect assumption
$worst.case
[1] "Sutton-1994" "Mohler-1995" "Bodiya-1999"
$u_max
[1] 3
$side
[1] "greater"
$r.value
[1] 0.0468
(m2943.raFE <- metarep(x = m2943 , u = 2 , common.effect = T ,report.u.max = T))
Performing Replicability analysis with the common-effect assumption
Performing Replicability analysis with the common-effect assumption
Number of studies: k = 5
Number of observations: o = 4166
Number of events: e = 1384
OR 95%-CI z p-value
Common effect model 1.6564 [1.4317; 1.9164] 6.78 < 0.0001
Quantifying heterogeneity:
tau^2 = 0.1314 [0.0073; 1.6698]; tau = 0.3625 [0.0852; 1.2922]
I^2 = 70.3% [24.4%; 88.3%]; H = 1.84 [1.15; 2.93]
Test of heterogeneity:
Q d.f. p-value
13.47 4 0.0092
Details on meta-analytical method:
- Peto method
- Restricted maximum-likelihood estimator for tau^2
- Q-Profile method for confidence interval of tau^2 and tau
- replicability analysis (r-value = 0.0011)find_umax(x = m2943 , common.effect = T,alternative = 'two-sided', confidence = 0.95)
Performing Replicability analysis with the common-effect assumption
$worst.case
[1] "Sutton-1994" "Mohler-1995" "Bodiya-1999"
$u_max
[1] 3
$side
[1] "greater"
$r.value
[1] 0.0468
(m2943.raFE <- metarep(x = m2943 , u = 2 , common.effect = T ,report.u.max = T))
Performing Replicability analysis with the common-effect assumption
Performing Replicability analysis with the common-effect assumption
Number of studies: k = 5
Number of observations: o = 4166
Number of events: e = 1384
OR 95%-CI z p-value
Common effect model 1.6564 [1.4317; 1.9164] 6.78 < 0.0001
Quantifying heterogeneity:
tau^2 = 0.1314 [0.0073; 1.6698]; tau = 0.3625 [0.0852; 1.2922]
I^2 = 70.3% [24.4%; 88.3%]; H = 1.84 [1.15; 2.93]
Test of heterogeneity:
Q d.f. p-value
13.47 4 0.0092
Details on meta-analytical method:
- Peto method
- Restricted maximum-likelihood estimator for tau^2
- Q-Profile method for confidence interval of tau^2 and tau
- replicability analysis (r-value = 0.0011)
forest(m2943.raFE, layout='revman5',digits.pval = 2 , test.overall = T )