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9.1 Introduction
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tests, or randomizing patients to undergo one of the tests. A within- study comparison of
test performance between two independent but comparable groups of patients could
also be considered for inclusion in such an analysis. Direct comparison has the advantage that each study acts as its own control, thereby dealing with potential confounding
that may occur due to study characteristics. However, the number of comparative
studies available may not be sufficient to allow such an analysis.
9.1.5 Planning theanalysis
Undertaking meta- analyses for a Cochrane Review of diagnostic test accuracy involves
first developing an analysis plan. Some of these decisions can be made at protocol
stage (see Chapter1), others only after the data have been extracted from the reports.
The established and commonly used statistical methods for test accuracy meta-
analysis
use accuracy data for one threshold per study per index test. The following sections and
examples used in Section9.4 follow this approach. More advanced methods for dealing
with data from multiple thresholds per study are discussed in Section9.4.5 and dealt
with in more detail in Chapter10.
The planning stages can be organized as follows.
●
Clearly stating the main questions that need answering, specifying which tests require
estimates of test accuracy, and which tests should be compared with each other.
●
Detailed planning of the way in which comparisons will be made, identifying the
different tests or groups of tests that can be compared, the multiple and pairwise
comparisons that will be made, and the studies and data that will be included in each
analysis. Review authors need to decide whether comparative analyses should
include all studies, or be restricted to studies that evaluate both tests. Covariates for
any heterogeneity analyses similarly need to be specified and coded.
●
From these decisions, a list of the planned main analyses, test comparisons and heterogeneity analyses will be produced. The quantity of data that are available for each
analysis should be determined to guide the choice of analysis method, and to assess
whether adequate data are available for planned heterogeneity analyses.
●
Results can be plotted on forest plots and ROC plots to report study- specific estimates of sensitivity and specificity and the variability between these estimates across
studies.
●
A strategy needs to be developed to deal with the mixed reporting of thresholds that
may occur across studies. A key issue is deciding whether an analysis should be
restricted to studies that share a common threshold value (which allows estimation
of the summary sensitivity and specificity of a test at that threshold) or to include all
studies regardless of threshold value (which allows estimation of SROC curves but
compromises the interpretation of summary points). This will depend on the thresholds at which tests were evaluated in the primary studies, and knowledge of how the
tests are applied in clinical practice. Consideration should also be given to whether
a more complex analysis is feasible that uses multiple thresholds per study
(seeSection9.4.5 and Chapter10, Section10.7).
●
If this analysis plan has been created using RevMan, the data should be exported from
RevMan to the chosen statistics package, and appropriate models fitted. Results
should be collated and tabulated as required, and parameter estimates copied back
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9 Understanding meta- analysis
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into the RevMan graphics function to produce final graphical output showing
summary points or SROC curves as appropriate.
9.2 Graphical andtabular presentation
A Cochrane Review of diagnostic test accuracy uses two main forms of graphical
display: coupled forest plots and SROC plots. Review authors can create these figures
within RevMan for each analysis that is specified.
9.2.1 Coupled forest plots
Forest plots for diagnostic test accuracy report the number of true positives and false
negatives in participants with the target condition (diseased), and true negatives and
false positives in participants who do not have the target condition (non- diseased) in
each study, and the estimated sensitivity and specificity, together with confidence
intervals (CIs). The plots are known as coupled forest plots as they contain two graphical sections: one depicting sensitivity and one specificity (Figure9.2.a). The order of the
studies can be sorted, for example by values of sensitivity, or grouped by test type or
covariate values. While it is possible to observe heterogeneity in sensitivity and specificity individually on such plots, it is not easy to visualize whether there are threshold- like
relationships. Summary statistics computed from meta- analyses are rarely added to
coupled forest plots. In Cochrane Reviews of diagnostic test accuracy, an archive of coupled forest plots for all the tests for which data were entered into RevMan is published
with the review to make the 2×2 tables of the index test result for those with the target
condition (referred to as the diseased) and without the target condition (referred to as
non- diseased) readily accessible.
9.2.2 Summary ROC plots
An SROC plot is a scatterplot of the results of individual studies in ROC space where
each study is plotted as a single (specificity, sensitivity) point. The size of the symbol
(e.g. rectangle, ellipse) used to mark each point can be controlled to depict the precision of the estimate (typically scaled according to their sample sizes). Using sample- size
weights, the height of the symbol is proportional to the number of diseased (and hence
the precision of the sensitivity estimate) and the width is proportional to the number of
non- diseased (and hence the precision of the specificity estimate) (seeFigure9.2.b).
Both the within- study sampling variability and the heterogeneity between studies
contribute to the total variability between studies. Even if the summary plots scale the
symbols to indicate the precision of the estimates from individual studies, it is difficult
to distinguish visually between these two sources of variability.
Two types of meta- analytical summary can be added to an SROC plot: SROC curves
and summary points. Confidence regions for the summary points can be included, as
can prediction regions that give an indication of between- study heterogeneity (see later
Figure9.4.a).
Studies can also be plotted using different symbols and/or colours to indicate different subgroups for investigations of heterogeneity or for test comparisons.
208
Study
A
otsuka 2005
Bas 2003
Bizzar
Bombar
Choi 2005
C
De
Du
Fe
Gar
Gir
Goldbach-Mansk
Gr
Gr
Hitchon 200
J
K
K
Kw
Lee 2003
Lopez-Ho
Nell 2005
Nielen 200
Quinn 2006
Rantapaa-D
Raza 2005
Sa
S
Schellek
Soder
Suzuki 2003
Va
va
va
Vincent 2002
Vit
Zeng 200
1
TP FP FN TN Generation Sensitivity SensitivitySpecificity Specificity
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o 2001
dieri 2004
orrea 2004
Rycke 2004
bucquoi 2004
rnandez-Suarez 2005
cia-Berrocal 2005
elli 2004
einer 2005
ootenboer-Mignot 2004
ansen 2003
amali 2005
umagai 2004
ok 2005
raux 2003
auerland 2005
lin 2004
llbracht 2004
n Gaalen 2005
n Venroooij 2004
tecoq 2004
Figure9.2.a Coupled forest plot of the sensitivity and specificity of Anti- CCP for the diagnosis of rheumatoid arthritis
y 2000
4
yos 2004
5
ahlqvist 2003
ens 2000
3
115
17
16
110
24
40
5
23
0
236
20
74
11
89
4
90
2
31
0
69
8
25
2
43
1
70
5
167
8
26
8
110
3
26
64
71
68
38
42
149
147
47
24
40
171
72
481
190
82
865
139
69
90
1
14
2
14
3
2
7
10
7
3
11
26
14
7
2
23
12
13
79
7
5
7
86
58
88
29
50
22
18
10
63
17
98
15
148
20
15
58
35
60
109
35
20
18
46
60
77
68
105
71
252
101
107
101
73
215
227
7
39
231
8
130
142
129
75
38
40
120
228
88
15
118
56
293
66
132
0
73
96
114
106
375
79
146
443
298
9
51
185
408
301
2218
464
133
313
CCP2
CCP1
CCP1
CCP2
CCP2
CCP2
CCP2
CCP2
CCP2
CCP2
CCP2
CCP1
CCP2
CCP2
CCP2
CCP1
CCP2
CCP2
CCP2
CCP2
CCP2
CCP2
CCP2
CCP2
CCP2
CCP2
CCP1
CCP2
CCP1
CCP2
CCP2
CCP2
CCP2
CCP2
CCP1
CCP2
CCP1
0.88 [0.81, 0.93]
0.56 [0.49, 0.63]
0.41 [0.31, 0.51]
0.77 [0.58, 0.90]
0.73 [0.68, 0.78]
0.90 [0.82, 0.96]
0.75 [0.67, 0.83]
0.64 [0.56, 0.72]
0.58 [0.44, 0.72]
0.79 [0.69, 0.87]
0.71 [0.54, 0.85]
0.41 [0.31, 0.51]
0.80 [0.71, 0.88]
0.63 [0.57 0.69]
0.63 [0.47, 0.78]
0.43 [0.37, 0.49]
0.57 [0.41, 0.71]
0.81 [0.71, 0.89]
0.55 [0.46, 0.64]
0.66 [0.56, 0.75]
1.00 [0.91, 1.00]
0.41 [0.32, 0.51]
0.58 [0.51, 0.64]
0.81 [0.74, 0.86]
0.70 [0.58, 0.81]
0.57 [0.41, 0.72]
0.47 [0.36, 0.58]
0.74 [0.68, 0.80]
0.48 [0.40, 0.57]
0.44 [0.20, 0.70]
0.88 [0.85, 0.90]
0.64 [0.59, 0.70]
0.54 [0.45, 0.62]
0.77 [0.75, 0.80]
0.58 [0.51, 0.64]
0.39 [0.32, 0.47]
0.47 [0.40, 0.54]
0.81 [0.71, 0.89]
0.90 [0.85, 0.93]
0.98 [0.95, 0.99]
1.00 [0.91, 1.00]
0.92 [0.88, 0.95]
0.92 [0.86, 0.96]
0.97 [0.93, 0.99]
0.98 [0.95, 1.00]
1.00 [0.95, 1.00]
0.83 [0.69, 0.92]
0.95 [0.84, 0.99]
0.99 [0.95, 1.00]
0.98 [0.95, 0.99]
0.92 [0.84, 0.96]
0.65 [0.43, 0.84]
0.98 [0.93, 0.99]
0.98 [0.91, 1.00]
0.95 [0.92, 0.97]
0.97 [0.90, 1.00]
0.90 [0.84, 0.95]
0.96 [0.89, 0.99]
0.98 [0.93, 1. 00]
0.94 [0.88, 0.98]
0.91 [0.85, 0.96]
0.98 [0.96, 0.99]
0.96 [0.90, 0.99]
0.93 [0.88, 0.96]
0.94 [0.92, 0.96]
0.96 [0.93, 0.98]
0.96 [0.87, 1.00]
0.89 [0.84, 0.93]
0.97 [0.95, 0.99]
0.96 [0.93, 0.98]
0.97 [0.96, 0.97]
0.99 [0.97, 0.99]
0.96 [0.92, 0.99]
0.98 [0.96, 0.99]
0 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8
9 Understanding meta- analysis
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Sensitivity
Specificity
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0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
Figure9.2.b SROC plot of Anti- CCP for the diagnosis of rheumatoid arthritis
9.2.3 Linked SROC plots
Linked SROC plots are used in analyses of paired test comparisons, where both tests
have been evaluated within a study. The points are plotted as for a routine SROC plot,
but the two estimates (one for each test) from each study are joined by a line. It is thus
possible to get a sense of the difference in accuracy between tests within each study,
and to assess visually the degree of consistency in this difference across studies.
Summary estimates of sensitivity and specificity for each test, as well as SROC curves
obtained from meta- analysis, can be added to these plots (see later Figure9.4.g for an
example plot).
9.2.3.1
Example 1: Anti- CCP forthe diagnosis ofrheumatoid arthritis– descriptive
plots
These data are taken from a review (Nishimura 2007) of anti- cyclic citrullinated peptide
antibody (anti- CCP). The reference standard was based on the 1987 revised American
College of Rheumatology (ACR) criteria for clinical diagnosis. The meta- analysis included
37 studies and their sensitivities and specificities are shown on the forest plot (Figure9.2.a);
these study- specific estimates are also shown in a scatterplot in ROC space.
The forest plot shows the studies in alphabetical order. The figure gives the numbers
for the 2×2 table (true positives (TP), false positives (FP), false negatives (FN), true negatives (TN)) for each study that will form the basis for statistical analyses. Study- specific
estimates of sensitivity and specificity are shown, with their 95% confidence intervals.
These estimates (and confidence intervals) are also represented graphically. The most
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9.3 Meta- analytical summaries
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striking features of this figure are the consistently high specificity and the greater uncertainty (indicated by the confidence interval width) and variability (indicated by the scatter of point estimates) in sensitivity than specificity. The studies can be ordered in
different ways (e.g. in increasing order of sensitivity) to provide a visual representation
of any association between sensitivity and specificity. Ordering studies by year of publication may also be useful to identify trends in test accuracy resulting from changes in
methodology and/or selection criteria over time (Cohen 2016). The figure also includes
information on a covariate, the CCP generation, which may be associated with heterogeneity in test accuracy. (This will be explored in Section9.4.6.2.)
The SROC plot shown in Figure9.2.b also illustrates the greater variability in estimated sensitivity than specificity across studies. Covariate information (e.g. generation
of CCP) could be used to distinguish between studies in different subgroups (e.g. CCP1
vs CCP2). (See Section9.4.6.3 for further exploration of these data.)
Formal statistical analyses are required to obtain a summary estimate of test accuracy
and to explore heterogeneity. These will be covered in Section9.4.6.3. Before proceeding
to these statistical analyses, the review author should decide whether it is appropriate to
focus on a summary point(s) or a summary curve(s) in the statistical analyses that follow.
This will be influenced by the threshold(s) used by the included studies to define a positive test result (see Section9.4).
9.2.4 Tables ofresults
Review authors need to construct additional tables to report results from their metaanalytical models. Review authors might consider creating tables to report the following.
●
The numbers of studies and individuals available for each of the key analyses.
●
Summary estimates of diagnostic accuracy for each test.
●
Statistics for the comparative accuracy and tests of statistical significance for the
pairwise comparisons between tests (a half- matrix display of all possible pairwise
comparisons may be useful). Separate tables based on direct (within- study) compari-
sons and uncontrolled comparisons may be needed (see Section9.4.7).
●
Results of investigations of heterogeneity, including estimates of test accuracy in sub-
groups, summary statistics of comparative accuracy, and tests of statistical signifi-
cance (see Section9.4.6).
●
Results of sensitivity analyses (see Section9.4.9).
This list is not exhaustive, and review authors should explore and identify the best ways
of communicating the results of their analyses.
Cochrane Reviews of diagnostic test accuracy also include ‘Summary of findings’
tables, which are described in Chapter12.
9.3 Meta- analytical summaries
Test accuracy meta- analysis aims to compute the average accuracy of a test, compare
estimates of accuracy between tests, and investigate the heterogeneity between
studies. Review authors should choose which summary statistics to use. In Cochrane
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Reviews that use one 2×2 table per study (per test), the choice is between estimating
average values of sensitivity and specificity for a test at a common threshold (referred
to as the average or summary operating point or summary point), or estimating the
underlying ROC curve for a test across many thresholds (referred to as the summary
ROC curve or SROC curve). In most scenarios estimation of a summary point is preferred, as it facilitates interpretation of the results in terms of the numbers of true
positives, true negatives, false positives and false negatives (see Chapter 11 and
Chapter12). However, there are situations when estimation of an SROC curve is the
only analysis possible, or there are benefits in terms of precision of estimates and
comparisons.
9.3.1 Should Iestimate anSROC curve or asummary point?
In a systematic review of test accuracy studies, it is likely that the collected data on test
performance will have been measured at a mixture of different positivity thresholds.
Athreshold could, for instance, represent a cut- point on a quantitative scale or a judgement on an ordinal scale. A key principle underlying the choice of statistical summary
in meta- analysis of test accuracy is that the sensitivity and specificity of a test will vary
as the positivity thresholds vary, as graphically depicted using an ROC curve. It is
important to note that both hierarchical models recommended for meta- analysis for
Cochrane Reviews of diagnostic test accuracy account for correlation between sensitivity and specificity observed across studies that is due to the functional relationship
between sensitivity and specificity as the criterion for test positivity (threshold) varies.
This correlation is accounted for in the analysis regardless of whether an SROC curve
or a summary point is the output of choice. Separate meta- analysis of sensitivity and
specificity estimates fails to account for the trade- off between sensitivity and specificity, which may lead to under- estimates of test accuracy (Deeks2001). Separate metaanalysis of likelihood ratios also ignores correlations between positive and negative
likelihood ratios, and theoretically can produce estimates that are impossible
(Zwinderman 2008).
While for some tests there is consensus about what value the positivity threshold
should take, often tests are evaluated at different thresholds in different studies.
Presentation of results at multiple thresholds within a single study is also possible, with
some studies presenting estimates of ROC curves that depict the accuracy of the test at
all possible thresholds. In addition, selective reporting of thresholds identified to optimize test accuracy can introduce bias, because sampling variability may influence
which threshold is identified as ‘optimal’ when it is selected based on the observed data
(Leeflang 2008).
Review authors need to decide whether they will use all the studies available and
estimate an SROC curve, or select only studies that use a common threshold to estimate a summary point (an analysis that could be repeated across a set of different common thresholds). Estimating summary sensitivity and specificity by combining studies
that mix thresholds will produce an estimate that relates to some notional unspecified
average of the thresholds that occur in the included studies. This must be avoided,
because it is clinically uninterpretable and not generalizable. More complex methods
do allow use of all data to estimate both curves and summary points, but are not
currently often used due to challenges in their implementation (see Section9.4.5).
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In some contexts, test positivity is based on an ordinal rating scale rather than an
explicit numerical threshold requiring a judgement rather than measurement, which is
likely to lead to variability between individual readers in how they interpret the thresholds on the rating scale. For a study- level meta- analysis of test accuracy, it is generally
reasonable to assume a common threshold across studies if the same point on the rating scale was applied. At the study level, the estimated sensitivity and specificity represent average estimates of accuracy across readers in that study. Even when it is possible
to define a common threshold on the basis of a numerical value or a point on a rating
scale, it must be acknowledged that some variability will remain in the actual threshold
between studies through calibration differences between equipment, differences
between groups of raters or observers in terms of training and experience, as well as
variation in the implementation of tests. The consequence of such variability will be
additional heterogeneity in test results observed at the common threshold. The summary sensitivity and specificity point will reflect the average observed accuracy across
studies, while the prediction region (see Section9.3.2) will reflect the heterogeneity
between studies, as illustrated in the example in Section9.4.2.
Thus, the three main strategies used to handle mixed and variable thresholds in an
analysis are as follows.
●
Estimating summary sensitivity and specificity of the test for a common threshold, or
at each of several different common thresholds if there are sufficient data. Each study
can contribute to one or more analyses depending on what thresholds it reports.
Studies that do not report at any of the selected thresholds are excluded.
●
Estimating the underlying ROC curve that describes how sensitivity and specificity
trade off with each other as thresholds vary. In this case, one threshold per study is
selected to be included in the analysis. A range of thresholds across studies is needed
to inform the shape of the SROC curve.
●
Simultaneous analysis of multiple (one or more) estimates of sensitivity and specific-
ity reported by each study. This approach allows estimation of an SROC curve and
summary points on the curve for particular thresholds of interest.
For analyses that will include one threshold per study, the choice of analytical approach
will be influenced by the variation of thresholds in the available studies. For example,
ifthere is little consistency in the thresholds used, meta- analyses that are restricted
tocommon thresholds will contain very few data, and estimating an SROC curve may
be preferred. If there is little variation in threshold between studies, attempting to
fitan SROC curve will be difficult, as the points are likely to be too tightly clustered
inROC space and there is little information in the data to estimate the shape of the
underlying curve.
It can be reasonable to estimate both SROC curves and summary points in a review,
as they may complement each other in providing clinically useful summaries and powerful ways of detecting effects. For example, separate analyses of test data at different
thresholds may be used to provide clinically informative estimates of sensitivity and
specificity based on the studies that provide data at each threshold. Including all studies to estimate how SROC curves depend on covariates or test type will be the most
powerful way to test hypotheses and investigate heterogeneity when thresholds vary
across studies.
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When some studies provide data for multiple thresholds, more complex models that
estimate an SROC curve and summary points on that curve may be applied (see
Section9.4.5 and Chapter10, Section10.7). However, the potential gain relative to the
more straightforward approaches will be limited if only a small proportion of studies
provide such data.
9.3.2 Heterogeneity
Heterogeneity is to be expected in meta- analyses of diagnostic test accuracy. A consequence of this is that meta- analyses of test accuracy studies tend to focus on computing
average rather than typical effects. In systematic reviews of interventions it is sometimes noted that the estimates of the effect of the intervention in the different studies
are very similar, the differences between them being small enough to be explicable by
chance. In such situations it is appropriate to use a fixed-
effect approach meta- analysis,
which estimates the underlying common effect (and is interpreted as the actual effect
of the intervention). In systematic reviews of test accuracy, large differences are commonly noted between studies, too big to be explained by chance, indicating that actual
test accuracy varies between studies: there is heterogeneity in test accuracy. Randomeffects meta- analysis methods are recommended when effects are heterogeneous.
These methods focus on providing an estimate of the average accuracy of the test and
describing the variability in accuracy between studies. In Cochrane Reviews of diagnostic test accuracy, heterogeneity is presumed to exist and random- effects models are
fitted by default, only simplified to fixed- effect models where there are too few studies
to estimate between- study variability, or analysis and forest or SROC plots demonstrate
that fixed- effect models are appropriate (see Section9.4.8).
Univariate tests for heterogeneity in sensitivity and specificity and estimates of the
I2 statistic (Higgins 2003) are not recommended for systematic reviews of test accuracy, as they do not account for heterogeneity explained by phenomena such as positivity threshold effects, whereby an increasing threshold for defining test positivity
will decrease sensitivity and increase specificity. Problems with the I2statistic that
could lead to misleading conclusions have also been highlighted in the literature
(Rücker 2008, Wetterslev 2009, Zhou 2014); other measures such as the variance
parameter of the random- effects model (Rücker 2008) and also multivariate and test
accuracy- specific I2 statistics (Zhou 2014) have been proposed but are not used
routinely.
The numerical estimates of the random- effects terms in the hierarchical models do
quantify the amount of heterogeneity observed, but they are not easily interpreted as
they represent variation in parameters expressed on log odds scales. Graphical displays
provide a more straightforward means of assessing the magnitude of the observed heterogeneity. For instance, if variation in threshold occurs in a meta- analysis, what matters is the degree to which the observed study results in an SROC plot lie close to the
SROC curve, not how scattered they are in ROC space.
For a summary point estimate, the inclusion of a prediction region around the point
provides a visual assessment of heterogeneity. The region takes account of the variance
of the random effects for logit(sensitivity) and logit(specificity), as well as the correlation between them. The region provides a visual summary of the spread of the true
underlying test accuracy across the studies included in the random- effects model
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(seethe example in Section9.4.2). Hence, a 95% prediction region represents the region
within which one has 95% confidence that the truesensitivity and specificity of any
future study should lie (Harbord 2007). Estimation of a prediction interval or region
relies on the assumption of normal distributions for the effects across studies. This may
be very problematic when the number of studies is small and can lead to spuriously
large (or small) regions (Deeks 2019).
9.4 Fitting hierarchical models
In this section, the bivariate model (Reitsma 2005) and the hierarchical SROC (HSROC)
model of Rutter and Gatsonis (Rutter 2001) are described and discussed for the metaanalysis of studies each of which contributes one threshold to the analysis. These hierarchical models include random study effects that account for the unexplained
heterogeneity between studies that is typical in systematic reviews of test accuracy.
They supersede the earlier, more limited fixed- effect SROC approach of Moses and
Littenberg (Littenberg 1993, Moses 1993).
Both the bivariate and Rutter and Gatsonis HSROC models involve statistical distributions at two levels. At the lower level (level 1), they model the cell counts in the 2×2
tables extracted from each study using binomial distributions and logistic (log odds)
transformations of proportions, thereby taking account of random sampling variability
within studies. At the higher level (level 2), random study effects are assumed to account
for heterogeneity in test accuracy between studies beyond that accounted for by sampling variability at the lower level.
The two models are mathematically equivalent when no covariates are fitted (Harbord
2007, Arends 2008), but differ in their parametrizations. The bivariate parametrization
models sensitivity, specificity and the correlation between them directly, whereas the
Rutter and Gatsonis HSROC parametrization models a function of sensitivity and specificity to define an SROC curve. Given their shared statistical properties, in the absence
of covariates in the models SROC curves can be computed from bivariate models and
summary points from HSROC models.
If review authors are using RevMan, the parameter estimates from either model can
be input to produce an appropriate graphical display in ROC space of the summary estimates of test accuracy for a particular analysis, usually superimposed on the scatterplot of study specific estimates. The options are:
●
the SROC curve; or
●
the summary point (i.e. summary values for sensitivity and specificity);
●
a confidence region around the summary point; and
●
a prediction region around the summary point.
Although it is common to present 95% confidence and 95% prediction regions, RevMan
offers options to use other values. The 95% prediction region around a summary point
illustrates the extent of statistical heterogeneity by depicting a region within which,
assuming the model is correct, we have 95% confidence that the true sensitivity and
specificity of any future study should lie (Harbord 2007). There is also an option of plotting a 50% prediction region, in which the central half of the true values of future studies
would lie (akin to an interquartile range).
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Additional estimates can be derived from the models. Summary estimates for the
positive and negative likelihood ratios and the diagnostic odds ratio, with corresponding confidence intervals, can be computed at the summary point or at any point on the
SROC curve. From the SROC curve the average sensitivity at a given value of specificity
(or the other way round) can also be computed (see Chapter10). Not all of these possible summary measures will be relevant or appropriate for a given analysis.
The motivation for choosing one of these two alternative parameterizations becomes
clear when covariates are added to explore heterogeneity in test accuracy or to compare tests. Ultimately, the choice of method will be determined by the focus one wishes
to adopt, and which of the two models addresses the research question given the
nature of the available data (see Section9.3.1).
Both models require the use of external statistical software. The results given for the
examples included in this chapter have been estimated using frequentist methods.
Although this is the most commonly used approach, Bayesian estimation can also be
used, as illustrated in Chapter10. Publication-
ready graphical output can be created in
RevMan using the model parameter estimates to add model summaries to SROC plots.
Alternative specifications for summary curves based on functions of the bivariate
model parameters have been proposed (Arends 2008, Chappell 2009). This chapter will
focus on the Rutter and Gatsonis model, as it is the most established of the HSROC
specifications.
9.4.1 Bivariate model
The bivariate method models sensitivity and specificity directly. The model can be
regarded as having two levels corresponding to variation within and between studies. At
the lower level, the within- study variability for both sensitivity and specificity is assumed
to follow a binomial distribution. For sensitivity (denoted by A), the number testing
positive yAi~B(nAi, πAi), where nAi and πAi respectively represent the total number of diseased individuals tested and the probability of a positive test result in that group in
study i. Similarly, for specificity (denoted by B), the number testing negative yBi~B(nBi, πBi),
where nBi and πBi respectively represent the total number of non- diseased individuals
tested and the probability of a negative test result in that group in study i. The sensitivity–specificity pair for each study must be modelled jointly within the study at the lower
level of the analysis, because they are linked by shared study characteristics including
the positivity threshold. At the higher (between- study) level, the logit- transformed sensitivities are assumed to have a normal distribution with mean μA and variance σ
while the logit- transformed specificities have a normal distribution with mean μB and
variance σ
2
. Their correlation is included by modelling both at once by a single bivari-
B
ate normal distribution
Ai
Bi
A
N
with
B
B
where σAB is the covariance between logit(sensitivity) and logit(specificity). The model
may also be parametrized using the correlation ρAB= σAB/(σAσB), which may be more
interpretable than the covariance. The bivariate model therefore has five parameters
216
2
,
A
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