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9.4 Fitting hierarchical models
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2
when no covariates are included: μA, μB, σ
2
, σ
and ρAB. (Note: we follow Harbord (2007)
A
B
in using μ where Reitsma (Reitsma 2005) used θ in order to avoid confusion with the
notation from that of the HSROC model that follows.)
The inclusion of a correlation parameter in the model allows for the expected tradeoff in sensitivity and specificity. Where variation between studies arises through a tradeoff resulting from variation in thresholds between studies, this correlation is expected
to be negative. However, the correlation may be positive if there are other sources of
heterogeneity.
Reitsma and colleagues originally proposed fitting these models by approximating
the binomial within- study distributions by normal distributions (Reitsma 2005).
Although this allows the model to be fitted in a slightly larger range of software (e.g. the
MIXED procedure in SAS), Chu (2006) later demonstrated that the approximation can
perform poorly and recommended that software be used that can explicitly model the
binomial within-
study distributions, as is done in the examples that follow and the sam-
ple programs provided in Chapter10.
Meta- analysis of predictive values is possible using the bivariate method (Leeflang
2012), but this is not recommended as it is known that predictive values depend on the
proportion of participants with the target condition, which is likely to vary between
studies. Hence, the average predictive values will relate to use of the test at some average prevalence. Meta- analysis of predictive values may be appropriate when either
sensitivity and specificity is not estimable due to partial verification (i.e. verification of
only test positives or test negatives) or when there is differential verification.
9.4.2 Example 1 continued: anti- CCP forthe diagnosis ofrheumatoid arthritis
We now undertake the first stage of a formal statistical analysis of the data from a
review of anti- cyclic citrullinated peptide antibody (anti- CCP) (Nishimura 2007). If it can
be presumed that the anti- CCP test is deemed positive if any anti- CCP antibody is
detected and that detection can be considered a common threshold, it is appropriate to
focus on summary estimates for sensitivity and specificity.
As noted in the descriptive analyses of these data (Section9.2.3), there appears to be
variability across studies in estimates of test accuracy. This variability appears to be
higher for sensitivity than specificity, which could arise either through heterogeneity or
through estimates of sensitivity being based on smaller samples than estimates of
specificity. Using the bivariate model to estimate a summary point based on the data
for all studies, the parameter estimates from the bivariate model are shown in
Table9.4.a.
The parameter estimates can be input to RevMan to produce the summary point, 95%
confidence region and 95% prediction region shown in Figure9.4.a, superimposed on
the individual study estimates. Computation of confidence and prediction regions also
requires the standard error of the estimates for mean logit(sensitivity), mean
logit(specificity) and the covariance between these estimates, which are 0.1275, 0.1459
and −0.00741, respectively. Note that the covariance shown in Table9.4.a is the covariance between observed estimates of logit(sensitivity) and logit(specificity) across studies, not the covariance between the estimates for mean logit(sensitivity) and mean
logit(specificity). Thelatter can be extracted as demonstrated in Chapter10.
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Table9.4.a Bivariate model parameter estimates foraccuracy ofanti- CCP forthe diagnosis of
rheumatoid arthritis
Label Parameter Estimate Standard error
Mean logit(sensitivity)
Mean logit(specificity) μ
Variance of random effects for logit(sensitivity) σ
Variance of random effects for logit(specificity) σ
Covariance of logit(sensitivity) and logit(specificity) σ
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
μ
A
B
2
A
2
B
AB
0.6534 0.1275
3.1090 0.1459
0.5426 0.1463
0.5717 0.1873
−0.2704 0.1199
0.1
0
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
Figure9.4.a Summary sensitivity and specificity of anti- CCP for the diagnosis of rheumatoid arthritis
The variance coefficients indicate similar heterogeneity in sensitivities and specificities on the logit scale. The magnitude of the heterogeneity in sensitivities and specificities (untransformed) is evident in the size of the prediction region on the SROC plot,
with the variability in specificity being constrained on this scale because the underlying
specificity is high. The summary estimates of sensitivity and specificity are shown by
the solid black dot. The sensitivity and specificity at this point can be computed by
inverse transformation of the logit estimates to give a sensitivity and specificity of 0.66
and 0.96, respectively. Confidence intervals can be computed by inverse transformation
of the confidence intervals computed on the logit scale. (See Chapter 10 for further
details.)
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The plot shows a potential outlier, with a sensitivity of 0.63 and specificity of 0.65. A
sensitivity analysis can be performed by omitting this study to assess its influence on
the summary estimates (see Section9.4.9).
9.4.3 The Rutter andGatsonis HSROC model
The HSROC model proposed by Rutter and Gatsonis (Rutter 1995, Rutter 2001) is based
on a latent scale logistic regression model (McCullagh1980, Tosteson 1988). The HSROC
model assumes that there is an underlying ROC curve in each study with parameters α
and β that characterize the accuracy and asymmetry of the curve.
Accuracy, defined in terms of the lnDOR (natural logarithm of the diagnostic odds
ratio), determines the position of the summary curve relative to the top left corner of
the ROC axes. Each study contributes data at a single threshold to the analysis. The 2×2
table for each study then arises from dichotomizing at a positivity threshold denoted by
θ. Theparameters α and θ are assumed to vary between studies: both are assumed to
have normal distributions, as in conventional random-
effects meta- analysis.
The HSROC model can be regarded as having two levels corresponding to variation
within and between studies. Atthe lower level, the number of diseased individuals
who test positive is denoted by yi1 for the ith study, and the corresponding number of
non- diseased who test positive is denoted by yi2. For each study (i), the number testing positive in each disease group (j) is assumed to follow a binomial distribution such
that yij~B(nij, πij), j=1, 2, where nij and πij, respectively, represent the total number
tested and the probability of a positive test result. The number testing positive in
each diseased and non- diseased pair is analysed jointly within each study at the
lower level in the analysis.
The model takes the form
where disij represents the ‘true’ disease status (coded as −0.5 for the non- diseased and
0.5 for the diseased), therebytaking into account the within- study variability at the
lower level. Using the terminology for this model, θi represents a proxy for positivity
threshold calculated as the mean of the log odds of a positive test result for the diseased and the log odds of a positive test result for the non- diseased groups in study i. αi
(the lnDOR for study i) represents a measure of diagnostic accuracy in the ith study that
incorporates both sensitivity and specificity for that study. The shape (scale) parameter
(β) provides for asymmetry in the SROC curve by allowing accuracy to vary with threshold. Since each study contributes only one estimate of sensitivity and specificity at a
single threshold, it is necessary to assume that the shape of the true underlying ROC
curve in each study is the same, and hence β is fitted as a fixed effect.
The threshold and diagnostic accuracy for each study are specified as random
effects and are assumed to be independent (uncorrelated) and normally distributed.
The accuracy parameter has mean Λ (capital lambda) and variance , while the
positivity (threshold) parameter has mean Θ (capital theta) and variance . The
shape parameter (β) is estimated using data from the studies considered jointly,
assuming normally distributed random effects for test accuracy. When no covariates
are included, the HSROC model has five parameters: Λ, Θ, β, and .
ogit di
ij iiij ij
is
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An SROC curve can be constructed from the HSROC model by choosing a range of
values of 1–specificity and using the estimated average location parameter (Λ) and
scale parameter (β) to compute the corresponding values for sensitivity. The average
sensitivity at a chosen false positive fraction (1–specificity) is given by
ensitivity logitspecificityexp
When β=0, test accuracy can be summarized by Λ, which represents a common average accuracy (lnDOR) across all thresholds, and the resulting summary curve will be
symmetrical.
9.4.4 Example 2: Rheumatoid factor asa marker forrheumatoid arthritis
In this example we will investigate the diagnostic performance of rheumatoid factor
(RF) as a marker for rheumatoid arthritis (RA). The 50 studies included in the analysis
are taken from the same review as Example 1 (Nishimura 2007). The reference standard
was again based on the 1987 revised American College of Rheumatology (ACR) criteria
or clinical diagnosis.
The threshold for test positivity for RF varied between studies and ranged from 3 to
100 U/mL. The variability in threshold used to define test positivity between studies is
reflected in the variability in study- specific estimates of sensitivity and specificity in the
SROC plot shown in Figure9.4.b. Because of the variation in threshold across studies,
asummary point does not have a useful interpretation. An SROC curve is appropriate
to summarize these data and can be estimated by fitting an HSROC model. The
0.9
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Figure9.4.b SROC plot and SROC curve for accuracy of rheumatoid factor
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Table9.4.b HSROC parameter estimates forrheumatoid factor model
Label Parameter Estimate Standard error
Mean accuracy
Mean threshold Θ −0.4370 0.1469
Shape of SROC curve β 0.2267 0.1624
Variance of random effects for accuracy
Variance of random effects for threshold
Λ
2.6016 0.1862
1.3014 0.3046
0.5423 0.1237
parameter estimates shown in Table9.4.b are extracted from the output of example
programs for these data provided in Chapter10.
The parameter estimates can be input to RevMan to draw the summary curve as
shown in Figure9.4.b; 2.6016 estimates the mean of the random effects for accuracy
(Λ), −0.4370 estimates the mean of the random effects for threshold (Θ), 0.2267 estimates the shape parameter (β), 1.3014 estimates the variance of the random effects for
accuracy, and 0.5423 estimates the variance of the random effects for threshold. All of
these estimates are on the logit scale. The resulting curve shows the expected trade- off
between sensitivity and specificity across thresholds. It is recommended that the fitted
curve is displayed across the range of the observed study sensitivities and specificities,
as shown here. The average sensitivity at a chosen specificity, or the other way round,
can be computed from the fitted curve (see Chapter10).
When interpreting the results of the analysis, it is important to note that RF constitutes
part of the ACR criteria. Hence, there is risk of bias in the estimated curve since the index
test is incorporated in the reference standard. This could result in an over- estimation of
the diagnostic accuracy of RF, and consequently a distorted picture of the value of using
RF as a first test for resolving uncertainty in a suspected case of rheumatoid arthritis.
9.4.5 Data reported at multiple thresholds per study
The key approaches that are described and illustrated in this chapter focus on methods
of analysis that use data from one threshold per study. To estimate a summary point,
we use data from a common threshold across studies. To estimate an SROC curve, data
from a range of thresholds are required to inform the shape of the underlying curve
across studies.
Some studies may report sensitivity and specificity at more than one threshold. Using
data from all available thresholds has the potential to provide more accurate estimation of SROC curves and estimates of average sensitivity and specificity values at stated
thresholds. The potential gain will increase as the number of studies that report multiple thresholds increases.
A range of methods has been proposed for the analysis of such data. An early approach
by Dukic (2003) summarizes study- specific ROC curves to obtain an SROC curve within
a Bayesian framework. However, single- threshold studies cannot be included and certain assumptions made in fitting the model can lead to sensitivities and specificities
that are not monotonic (Hamza 2009). An alternative multivariate random- effects metaanalysis approach was proposed (Hamza 2009) that is applicable when there are
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specified thresholds of interest and all studies report sensitivity and specificity at these
thresholds. This method can, in principle, be applied when studies do not provide data
for all thresholds, but model convergence may be problematic in those circumstances.
A subsequent approach based on survival analysis methods (Putter 2010) also requires
that studies provide sensitivity and specificity at the specified thresholds of interest.
Recent approaches, including the methods of Steinhauser (2016) and Jones (2019),
provide a more rigorous and robust approach for dealing with multiple thresholds per
study. These methods can accommodate a variable number of thresholds and a range
of different thresholds across studies. Studies that provide data for only one threshold
can be included, thereby reducing possible bias resulting from the inclusion of an
unrepresentative group of studies. Both methods allow for correlation between sensitivity and specificity across thresholds and also heterogeneity between studies through
the inclusion of random study effects. An SROC curve is estimated and summary sensitivity and specificity can be computed at specified thresholds.
The Steinhauser method models the distribution of test results as a function of the
continuous (or ordinal) thresholds for test positivity within both the diseased and nondiseased groups. Linear mixed- effects modelling is used to estimate the distribution
parameters for a known underlying parametric distribution (usually assumed to be normal logistic) of the test results in the two groups. The more recent method of Jones
(2019) fits multinomial distributions to tables of categorized test results in each of the
diseased and non- diseased groups. The number of categories (equal to the number of
thresholds + 1) is allowed to vary across studies. The Jones model assumes an underlying logistic distribution for some transformation (for example, the natural logarithm) of
the underlying continuous results in each group. The approach is flexible in that the
best- fitting transformation from the set of Box- Cox transformations can be estimated
from the data. Covariates can also be included in the model to investigate sources of
heterogeneity. The methods by Steinhauser and by Jones are described in more detail
in Chapter10, Section10.4, and illustrated using an example.
9.4.6 Investigating heterogeneity
In systematic reviews of test accuracy it is usual to observe variability in test accuracy
between studies that is considerably greater than would be expected from within- study
sampling error alone. This is reflected in the model specifications for the bivariate and
HSROC models, which both allow for random study effects. For the bivariate model, the
summary estimates of sensitivity and specificity represent an average operating point
across studies. Similarly, the estimated SROC curve represents an underlying ROC curve
across studies.
Some of this heterogeneity in test accuracy between studies is likely to arise because
of differences in patient characteristics, test methods, study design and other factors.
Exploratory analyses can be conducted to investigate whether such study characteristics appear to be associated with test accuracy using symbols and colours in SROC plots
to distinguish between studies belonging to different subgroups.
Statistically, it is generally more efficient to make use of all of the data available across
studies when investigating heterogeneity by adding study- level covariates to a hierarchical model to identify factors associated with diagnostic test accuracy. This metaregression approach also allows statistical inferences to be made. It is usually assumed
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that each covariate has a fixed effect when added to the model. This approach is also
applicable to test comparisons, as discussed in Section9.4.7.
The bivariate and HSROC models differ in how study- level covariates are included.
The bivariate method focuses on the estimation of summary estimates of sensitivity
and specificity, and estimating how these values vary with study- level covariates. The
HSROC approach, by contrast, focuses on the estimation of the SROC curve as the basis
for assessing test accuracy, and investigating how the position and shape of the curve
may vary with study- level covariates.
Both models allow the use of categorical and continuous covariates. In practice,
covariates relating to study characteristics are usually categorical and indicator variables are created as in standard regression modelling. For continuous covariates, particular care should be taken to check that the assumption of linear associations is valid.
For the bivariate model, this refers to association with logit(sensitivity) and/or
logit(specificity). For the HSROC model, this refers to association with the accuracy
parameter (lnDOR) and/or the threshold parameter.
The uses and limitations of investigating heterogeneity using subgroup analysis and
regression in Chapter10, Section10.11.5 of the Cochrane Handbook for Systematic
metaReviews of Interventions (Deeks 2019) apply equally to diagnostic studies.
9.4.6.1 Criteria formodel selection
Irrespective of which model is used, review authors should specify what modelling
strategy will be used for adding or removing covariates and what criterion will be used
to decide whether or not a covariate should be included in a model.
The decision as to whether a covariate should be retained in the model may be based in
part on statistical tests. Commonly used software for fitting these models will provide P
values for each estimate in the model based on Wald statistics. A P value based on the
likelihood ratio Chi2 statistic is generally more reliable than the Wald statistic, especially
for small sample sizes (Agresti2007). The likelihood ratio Chi2 statistic is computed as the
change in the −2Log likelihood when a covariate is added (or removed) from a model, with
the degrees of freedom equal to the difference in the number of parameters fitted in these
models. The effect of adding (or removing) covariates on measures of model fit such as
Akaike’s information criterion (AIC) or the Bayesian information criterion (BIC) can also be
used. The deviance information criterion (DIC) is commonly used for models fitted by
Markov chain Monte Carlo (MCMC) simulation. (See Chapter10 for further details.)
Statistical tests can also be used to assess whether allowing for variance of the random effects to vary by test in a comparison of two or more index tests provides a betterfitting model (see Chapter10).
9.4.6.2 Heterogeneity andregression analysis using thebivariate model
The bivariate model allows covariates to affect summary sensitivity or summary specificity, or both. Using the notation of Harbord (2007), and assuming that we have a single
study- level covariate Z that may affect both sensitivity and specificity, then the model
can be extended as follows:
Ai
Bi
AAi
N
vZ
BBi
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As before, ∑ represents the covariance matrix for the random effects for logit(sensitivity)
and logit(specificity). If the covariate does explain some of the heterogeneity in sensitivity
and/or specificity, then we would expect the estimated variance for one or both random
effects to be reduced. The estimated covariance (correlation) parameter may also change.
Assuming that we have a binary study- level covariate (Z) coded as 0 or 1 to represent
the two groups of studies, then μA estimates the logit(sensitivity) at the summary point
for the referent group (Z = 0), and μA+vA estimates the logit(sensitivity) at the summary
point for the other group (Z = 1). Hence, exp(vA) estimates the odds ratio for sensitivity in
group1 relative to the referent group. The average sensitivity is estimated as exp(μA)/
(1+exp(μA)) for the referent group of studies, and as exp(μA+vA)/(1+exp(μA+vA)) for the
other group. Comparisons of specificity between the two groups of studies follow the
same approach as described earlier based on μ
and vB. The fit of the model, with and
B
without the additional parameters vA and vB, can be used to assess whether the covariate
is associated with sensitivity and/or specificity. This joint test will have 2 degrees of
freedom if Z is binary. Separate tests of statistical significance of the covariate with sensitivity and specificity can also be conducted, first to assess whether vAdiffers from 0
(astatistically significant result indicates that there is evidence that sensitivity differs
between the two groups of studies) and secondly whether vBdiffers from 0 (a statistically
significant result indicates that there is evidence that specificity differs between the two
groups of studies). See also Section9.4.6.1 relating to criteria for model selection.
The standard error of a new estimate that is a function of the model parameter estimates
can be obtained using the delta method, on the assumption that the error distribution of
the new estimate is approximately normal (Oehlert1992). The delta method is implemented in standard statistical software packages such as SAS and Stata (see Chapter10).
The bivariate model is easily extended to allow for more than one covariate. However,
this may not be feasible in practice unless the number of studies is large. Typically only
one source of heterogeneity can be investigated at a time. Also, it is important to note
that a covariate may only be associated with sensitivity and not specificity, or the other
way round. It is not required that the same covariates are fitted for both sensitivity and
specificity, although this may commonly be the case. Where a covariate (or covariates)
is allowed to affect both the sensitivity and the specificity, thebivariate model is equivalent to an HSROC model in which the covariate (or covariates) is allowed to affect both
the accuracy and the positivity threshold but not the shape parameter. However, using
the estimates from the bivariate model to test for the effect of covariates on the shape
and position of the SROC curve is not straightforward. Using the HSROC model parametrization allows this to be done in a more direct and straightforward manner.
It is usually assumed that the variance of the random effects (and their correlation in
the case of the bivariate model) is not associated with the covariate. This is probably a
reasonable assumption in most analyses investigating heterogeneity in test accuracy
for a single index test. However, for analyses that compare different index tests, this
assumption is less likely to hold (see Section9.4.7).
9.4.6.3 Example 1 continued: Investigation ofheterogeneity indiagnostic
performance ofanti- CCP
The studies included in the review to assess the diagnostic performance of anti- CCP
used two different generations of the assay: first generation (CCP1, 8 studies) and
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second generation (CCP2, 29 studies). A binary covariate for test version (i.e. CCP1 or
CCP2) with separate coefficients for sensitivity and specificity was added to the model.
The covariate was coded as 0 for CCP1 (the referent group) and 1 for CCP2. Allowing
both sensitivity and specificity to vary by test version in the model resulted in a –2Log
likelihood of 533.4, a reduction of 12.2 compared with the model that contained no
covariates. Hence, there is statistical evidence (Chi2 = 12.2, 2 df, P = 0.002) that test accuracy is associated with the version of test used, but further investigation is required to
ascertain whether this association is for sensitivity, specificity or both. The P values in
Table9.4.c are based on Wald statistics for each parameter estimate, adjusted for the
other variables in the model. Based on these, there is strong evidence that sensitivity is
associated with test version (P = 0.0005), but not specificity (P = 0.21).
The variances of the random effects for logit(sensitivity) and logit(specificity), and
their covariance (
and σAB, respectively), are assumed to be common for both generations of CCP. For the referent group (CCP1in this case) thesummary estimates for
logit(sensitivity), logit(specificity) and the corresponding standard errors are denoted
by μA and μB, respectively). The covariate parameter estimates (νA and νB) give the change
in logit(sensitivity) and logit(specificity) for CCP2 relative to CCP1 – thus the
logit(sensitivity) and logit(specificity) estimates for CCP2 are obtained by adding the
covariate parameter estimates to those of the referent test (μA+νA and μB+νB, respectively). The standard errors for the logit(sensitivity) and logit(specificity) for CCP2 are
most easily computed by refitting the model and defining CCP2 to be the referent group
(covariate coded as 0 for CCP2 and coded as 1 for CCP1). The model parameter estimates can be input to RevMan to display the summary points and corresponding confidence regions shown in Figure9.4.c.
The summary point estimates and corresponding 95% confidence regions shown in
the figure are consistent with the conclusion that the sensitivity varies by test type, but
not specificity. Based on the model parameter estimates in Table9.4.c, the summary
estimates of specificity were 0.97 (95% CI 0.95 to 0.98) for CCP1 and 0.95 (95% CI 0.94 to
0.97) for CCP2. The summary estimates of sensitivity were 0.48 (95% CI 0.37 to 0.58) for
CCP1 and 0.70 (95% CI 0.65 to 0.75) for CCP2. These results indicate an improvement in
Table9.4.c Bivariate parameter estimates forcomparison ofthe accuracy ofCCP1 andCCP2 forthe
diagnosis ofrheumatoid arthritis
Parameter Estimate Standard error P value
μ
A
μ
B
σ
AB
ν
A
ν
B
*These Wald statistics are ignored as they test whether μ
sensitivity = 50% and specificity = 50%, respectively.
−0.09653 0.2203
3.4467 0.2982 < 0.0001*
0.3598 0.1022 0.001
0.5399 0.1802 0.005
−0.1969 0.09836 0.05
0.9626 0.2513 0.0005
−0.4302 0.3377 0.21
= 0 and μB = 0, equivalent to testing hypotheses that
A
0.66
*
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1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
Legend
Generation: CCP1
Generation: CCP2
Figure9.4.c Summary estimates of sensitivity and specificity for CCP1 and CCP2with corresponding
95% confidence regions
Specificity
sensitivity (P < 0.001), without loss of specificity (P = 0.21) for CCP2 compared with
CCP1. (The model could be simplified by removing the covariate for specificity. The
resulting estimate for specificity would then be assumed to be the same for CCP1 and
CCP2.) For final presentation of the results we would wish to estimate the difference in
sensitivity and specificity together with 95% confidence intervals. This is covered in
Chapter10.
Comparing the output from this model with that of the model with no covariates
(see Section 9.4.2), it is clear that the variances of the random effects are smaller,
particularly for sensitivity. Also, checks of the distributions of the random effects
(notshown here) show that adjusting for the use of first- or second- generation anti- CCP
tests results in distributions that more closely follow a normal distribution.
In these analyses, the variances of the random effects were assumed to be the same
for logit(sensitivity) and logit(specificity) for both generations of the test. This assumption can be investigated by fitting additional models, as shown in Chapter10. For the
example here, allowing the variances to differ did not make a substantive change to
theestimates or their interpretation. Chapter10 also demonstrates how to compute
the difference in sensitivity and also the difference in specificity, with corresponding
confidence intervals, using the bivariate model from the example in Section9.4.7.3 for
illustration.
For analyses based on a small number of studies (unlike the example shown here),
confidence intervals around the estimated difference in sensitivity and/or specificity
may be large. Hence, it is important not to rely purely on tests of statistical significance
to conclude that there is a lack of evidence of a difference, as the confidence interval for
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