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10 Undertaking meta- analysis
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.
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Box 10.3.a SAS output of HSROC model parameters
Parameter Estimates
Standard
Parameter
alpha
theta
beta
s2ua
s2ut
Estimate
2.6016
–0.4370
0.2267
1.3014
0.5423
Error DF t ValuePr > |t| 95% Confidence Limits Gradient
0.1862
0.1469
0.1624
0.3046
0.1237
48
48
48
48
48
13.97
–2.98
1.40
4.27
4.39
<.0001
0.0046
0.1691
<.0001
<.0001
2.2273
–0.7323
–0.09978
0.6890
0.2936
2.9759
–0.1417
0.5532
1.9138
0.7909
0.000069
0.000103
0.000039
0.000020
–0.00012
10.3.1 Fitting the HSROC model using SAS
The HSROC model was used to estimate a summary curve using Proc NLMIXED in
SAS (see code in Appendix 7 of the online supplementary material (10.S1 Code for
undertaking meta- analysis)) to obtain the output shown in Box10.3.a.
The parameter estimates in the red box can be entered into RevMan to draw the summary curve shown in Chapter9, Figure9.4.b; the estimate of the mean for accuracy (Λ,
lambda) is 2.6016, −0.4370 for the mean for threshold (Θ, theta), 0.2267 for the shape
parameter (β, beta), 1.3014 for the variance of the random effects for accuracy
0.5423 for the variance of the random effects for threshold
2
2
and
Estimation of a summary sensitivity and specificity are not clinically meaningful estimates for RF, since the 50 studies used different thresholds for RF. However, the expected
sensitivity at a chosen specificity (or vice versa) can be computed from the fitted curve
by using the equation given by
logit sensitivity logit 1 specificity .
0.5
The equation can be included in an ESTIMATE statement in NLMIXED (see code in
Appendix 7 of the online supplementary material (10.S1 Code for undertaking metaanalysis)). The ESTIMATE statement computes additional estimates as a function of
parameter values and produces standard errors and confidence intervals using the
delta method. For RF, the median (interquartile range) of specificities from the 50 studies was 0.87 (0.80 to 0.93). These three values of specificity were used in ESTIMATE
statements to obtain the corresponding values of sensitivity and their 95% CIs. The estimates of logit(sensitivity) with their 95% CIs are presented in the additional estimates
table in the SAS output, as shown in the red boxes in Box10.3.b. Inverse transformations of the logit estimates (see equations in Section10.2.1) give the estimates of sensitivity and their 95% CIs at the fixed values of specificity.
10.3.2 Bayesian estimation of the HSROC model
10.3.2.1 Specification of the HSROC model in rjags
The HSROC model equations introduced in Chapter9, Section9.4.3, are written out
directly in rjags. This section covers the different components of this rjags model, which
is the core of the program (comment lines separate the components in Box10.3.c).
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10.3 Estimation ofa summary curve
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Box 10.3.b SAS output of additional estimates produced using HSROC model parameters
Additional Estimates
Label Estimate
E(logitSe_sp80)
E(logitSe_sp87)
E(logitSe_sp93)
1.2177
0.8074
0.2608 0.1780 48 1.46 0.1494 0.05
Standard
ErrorDFt Value Pr > |t| Alpha Lower Upper
0.1697 48 7.18
0.1553 48 5.20
<.0001
<.0001
0.8765 1.5589
0.05
0.4952 1.1197
0.05
–0.09713 0.6187
Specificity Sensitivity (95% CI
0.80 0.77 (0.71 to 0.83)
0.87 0.69 (0.62 to 0.75)
0.93 0.56 (0.48 to 0.65)
Box 10.3.c Specification of the HSROC model for Bayesian estimation in rjags
model {
# === LIKELIHOOD === #
for(i in 1:n) {
TP[i] ~ dbin(TPR[i],pos[i])
FP[i] ~ dbin(FPR[i],neg[i])
# === PRIOR DISTRIBUTIONS FOR TPR AND FPR === #
logit(TPR[i]) <- (theta[i] + 0.5*alpha[i])/exp(beta/2)
logit(FPR[i]) <- (theta[i] - 0.5*alpha[i])*exp(beta/2)
theta[i] ~ dnorm(THETA,prec[2])
alpha[i] ~ dnorm(LAMBDA,prec[1])
}
### === HYPER PRIOR DISTRIBUTIONS === ###
THETA ~ dunif(- 10,10)
LAMBDA ~ dunif(- 2,20)
beta ~ dunif(- 5,5)
for(i in 1:2) {
prec[i] ~ dgamma(2.1,2)
tau.sq[i] <-
1/prec[i]
tau[i] <- pow(tau.sq[i],0.5)
}
}
●
Likelihood: The likelihood specifies that the observed data in each study, the TP and
FP cells, follow a binomial distribution with probability TPR (true positive rate) and
FPR (false positive rate), respectively (Box10.3.c).
●
Prior distributions: The parameters logit TPR and logit FPR are expressed as functions
of three additional parameters: the proxy for positivity threshold (theta) for each
study, the lnDOR (alpha) for each study and the shape parameter (beta) (β in Chapter9,
Section9.4.3), which is assumed to be common across studies. The parameters theta
and alpha are assumed to follow hierarchical prior distributions, thus allowing for
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,
1
,
Density
LAMBDA
(a)
(b)
(c)
Iteration
0
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both within- study and between- study variability. The theta parameters are assumed
to have a normal prior distribution with mean THETA (Θ) and precision prec[2] (
where 2 is referred to as tau.sq[2] in Box10.3.c and is the variance for the theta
parameters). The alpha parameters are assumed to have a normal prior distribution
with mean LAMBDA (Λ) and precision prec[1] (
where 2 is referred to as tau.
2
sq[1] in Box10.3.c and is the variance for the alpha parameters). The parameters
THETA, prec[2], LAMBDA, prec[1] and beta are provided with vague prior distribu-
tions, with the intention that they have a negligible impact on the results.
2
10.3.2.2
Monitoring convergence
Detailed information on how to run the rjags program in Box10.3.c is given in Appendix
6 of the online supplementary material (10.S1 Code for undertaking meta- analysis),
Section A6.2. Here, the focus is on interpreting the results of the program when applied
to the rheumatoid factor data of Nishimura (2007) introduced in Chapter9. As in the
case of the bivariate model (see Section10.2.4.2), begin by examining whether the
MCMC algorithm converged. Figure10.3.b shows the diagnostics plots for LAMBDA, but
similar plots can be obtained for all parameters. These plots, created using the mcmcplots
3.02.82.72.6 2.92.52.4
0.0 0.5 1. 01.5 2.0
2.0
3.53.0
2.5
LAMBDA
3.0 3.5
Running mean
0500010000 15000
Iteration
2000
270
2.52.0
Figure10.3.b MCMC diagnostics plots for a parameter (LAMBDA) from the HSROC model
30000 3500040000 45000
10.3 Estimation ofa summary curve
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Box 10.3.d Rjags output from Bayesian estimation of the HSROC model
Iterations = 6001:16000
Thinning interval = 1
Number of chains = 3
Sample size per chain = 10000
1. Empirical mean and standard deviation for each
variable,
plus standard error of the mean:
Mean SD Naive SE Time-
series SE
LAMBDA 2.6187 0.191119 1.103e- 03 3.787e- 03
THETA - 0.4490 0.152213 8.788e- 04 5.690e- 03
beta 0.2155 0.163026 9.412e- 04 8.912e- 03
tau.sq[1] 1.3781 0.321925 1.859e- 03 2.875e- 03
tau.sq[2] 0.6241 0.139763 8.069e- 04 1.133e- 03
2. Quantiles for each variable:
2.5% 25% 50% 75% 97.5%
LAMBDA 2.2434 2.4922 2.6167 2.7448 2.9985
THETA - 0.7518 - 0.5516 - 0.4476 - 0.3460 - 0.1534
beta - 0.1066 0.1069 0.2189 0.3256 0.5348
tau.sq[1] 0.8750 1.1481 1.3378 1.5589 2.1281
tau.sq[2] 0.4034 0.5243 0.6065 0.7036 0.9472
package in R, suggest that the MCMC algorithm has converged, as the results from the
three independent MCMC chains (identified by different colours) with different initial
values overlap nearly perfectly. The running mean plot in panel (b) shows some discordance between the chains when the number of iterations is small, but the values on the
y- axis indicate that the apparent differences are in fact very small.
10.3.2.3 Summary statistics and SROC plot
Following successful convergence of the MCMC algorithm, the summary statistics of the
parameters of interest can be calculated using a sample from the posterior distribution.
Output from the rjags program (see red box in Box10.3.d) gives the estimates for
LAMBDA, THETA and beta, and the between- study variance of alpha (tau.sq[1]) and
theta (tau.sq[2]). For comparison, see the results in Section10.3.1 that were obtained
using a frequentist approach in SAS.
The summary curve on the SROC plot (Figure10.3.c) from the Bayesian estimation
using the DTAplots package is very similar to the curve from the frequentist estimation
(Chapter9, Figure9.4.b). Study points on the plot were scaled according to sample size.
This plot can also be obtained in RevMan by providing the parameter estimates in the
red box in Box10.3.d.
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Specificity
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1.
0.80.60.4
0.20.0
1. 0 0.8 0.6 0.4 0.2
Figure10.3.c SROC plot of rheumatoid factor for rheumatoid arthritis
10.3.2.4 Sensitivity analyses
As already noted, Bayesian estimation of the HSROC model typically relies on vague
prior distribution functions. Since there is no unique way to specify a vague prior distribution, it is important to verify the impact of using an alternative vague prior distribution, as indicated in Section10.2.4.5. In Box10.3.c Gamma priors were used over the
precision parameters. An alternative approach would be to use a uniform or half- normal
prior distribution over the standard deviation parameters.
10.4 Comparison ofsummary points
The bivariate model is a regression model that can be extended to incorporate covariates
(i.e. meta- regression) to compare summary points (see Chapter9, Section 9.4.6.2 and
Section9.4.7.2). In Chapter9, Section9.4.6.3, a bivariate meta- regression was used to
investigate heterogeneity by assessing differences in the sensitivity and specificity of two
different generations of the anti- CCP test for diagnosis of rheumatoid arthritis. Similarly,
a bivariate meta- regression was used in Chapter9, Section 9.4.7.3, to compare the
accuracy of two imaging modalities – multislice computed tomography (CT) and
magnetic resonance imaging (MRI)– for diagnosis of coronary artery disease (CAD). For
both analyses, the variances of the random effects were assumed to be equal for the logit
sensitivities and the logit specificities.
The assumption of equal variances for the random effects of the logit sensitivities and
the logit specificities of different subgroups may be reasonable in many situations when
investigating heterogeneity in the accuracy of a single test, but less so when comparing
the accuracy of different tests. For ease of reference, subgroups or tests will simply be
272
10.4 Comparison ofsummary points
,,
2
AK
2
Bk
00
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referred to as ‘group’ in the model specifications. The model that allows for unequal variances for each group can be written as
2
ABk Bk
(10.1)
2
where μ
and μ
Aik
Aik A A k AK ABk
N
Bik B B k
are the logit(sensitivity) and logit(specificity) for the kth group in the
Bik
vZ
vZ
ith study; Zk represents the study- level covariate; μA estimates the mean logit(sensitivity)
for the referent group (note that for test comparisons this is not the reference standard
but another index test); μB estimates the mean logit(specificity) for the referent group;
μA+vAZk estimates the mean logit(sensitivity) for the kth group; and μB+vBZk estimates
the mean logit(specificity) for the kth group. Hence, exp(v
) and exp(vB) estimate the
A
odds ratio for sensitivity and specificity in the kth group relative to the referent group.
The parameters
cities for the kth group; and σ
and
are the variances for the logit sensitivities and logit specifi-
is the covariance between the logits across studies
ABk
evaluating the group (Takwoingi 2016). The variance–covariance structure in equation10.1 is typically modelled assuming independence between groups. For a binary
covariate (e.g. two tests), this variance–covariance matrix can be expressed as
2
A1 A1B1
2
0
2
B1
(10.2)
0
2
B2
The means
Ai
1
N
and
A
B
12B
B
are column vectors of the means of logit sensi-
B
Ai
2 A2 A2B2
~ , with .
1
Bi
2
Bi
12A
A
A
tivities and logit specificities for the two groups. Since test comparisons may include
only studies that used a paired design in which individuals received all index tests, the
bivariate model can allow for correlation in test performance between tests by estimating all between- study and between- test variability using the following unstructured
variance- covariance matrix:
Ai
Ai
Bi
Bi
Note that potential within- study correlation between tests is not taken into account
in equation10.3– this would require individual participant data or aggregate data in
the form of 2×4 tables of the results of two index tests cross- classified for individuals
with and for those without the target condition. Such tables are seldom reported in
primary studies. There may be biological/clinical justification for other variants of the
variance–covariance matrix in equation10.3, such as assuming independence between
the sensitivity of one test and the specificity of another test (i.e.σ
2
1
2 A2 A2B1 A2B2
~ , with .
1
2
A
N
B
A1 A1A2 A1B1 A1B2
2
2
B1 B1B 2
2
B2
A1B2
(10.3)
=0 and σ
A2B1
=0).
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Sections10.4.1,10.4.2 and10.4.3 revisit and extend the analyses presented in sections
9.4.6.3 and 9.4.7.3 to show how to fit models that allow for unequal variances using the
variance–covariance matrix expressed in equation 10.2. Section 10.4.4 presents
Bayesian estimation of the meta- regression analyses.
10.4.1 Fitting the bivariate model in SAS to compare summary points
The SAS code for the investigation of heterogeneity presented in Chapter 9,
Section9.4.6.3, is in Part I of Appendix 8 of the online supplementary material (10.S1
Code for undertaking meta- analysis). This appendix also contains code for metaregression using MetaDAS. The parameter estimates required to draw the summary
points and regions in RevMan can be extracted from the Proc NLMIXED output in
Box10.4.a. The variances of the random effects for logit(sensitivity) and logit(specificity),
and their covariance, are common for both generations of CCP (see blue box in output). For the referent group (CCP1in this example) the mean logit(sensitivity), mean
logit(specificity), the corresponding standard errors and covariance are shown in the
red boxes in the output. The covariate parameter estimates, se2 and sp2 (i.e. νA and
νB), give the expected change in logit(sensitivity) and logit(specificity) for CCP2 relative to CCP1. The mean logit(sensitivity) for CCP2 is thus estimated by msens+se2, and
the mean logit(specificity) is estimated by mspec+sp2. These additional estimates and
their standard errors and covariance can be obtained as described in Chapter 9,
Section9.4.6.3, or using the ESTIMATE statement in Proc NLMIXED, as shown in the
Box 10.4.a SAS output of bivariate meta- regression of CCP generation: equal
variances (model 1)
Fit Statistics
274
Parameter Gradient95% Confidence LimitsPr > |t|t ValueDFEstimate
msens
mspec
s2usens
s2uspec
covsesp
se2
sp2
–0.09653
3.4467
0.3598
0.5399
–0.1969
0.9626
–0.4302
-2 Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
Parameter Estimates
Standard
Error
0.2203
0.2982
0.1022
0.1802
0.09836
0.2513
0.3377
35
35
35
35
35
35
35
–0.44
11.56
3.52
3.00
–2.00
3.83
–1.27
0.6640
<.0001
0.0012
0.0050
0.0532
0.0005
0.2111
533.4
547.4
549.1
558.6
–0.5438
2.8412
0.1524
0.1742
0.4523
–1.1158
0.3507
4.0522
0.5673
0.9057
0.002825
1.4728
0.2554
0.000317
–0.00005
8.325E-6
0.000159
–0.00004
0.000319
–0.00004
msens
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mspec
s2usens
s2uspec
covsesp
se2
sp2
Label
logitsens CCP2
logitspec CCP2
logLR+ CCP1
logLR-CCP1
logLR+ CCP2
logLR-CCP2
–0.02464
–0.00012
–0.00001
–0.00003
–0.04855
Covariance Matrix of Parameter Estimates
0.04854
0.02465
0.004772
Estimate
0.8660
3.0165
2.7355
–0.6147
2.7132
–1.1693
–0.02464
0.08895
–0.00002
–0.00065
0.02463
–0.08834
–0.00012
–0.00002
0.01044
0.002118
–0.00440
0.000693
–0.00005
Additional Estimates
Standard
Error DF t Value Pr > |t| Alpha Lower
0.1209
0.1622
0.2681
0.1018
0.1458
0.08270
35
35
35
35
35
35
–0.00001
0.004772
0.002118
0.03246
–0.00860
–0.00006
–0.00039
7.16
18.59
10.21
–6.04
18.60
–14.14
10.4 Comparison ofsummary points
–0.00003
–0.00065
–0.00440
–0.00860
0.009674
0.000100
–0.00091
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
–0.04855
0.02463
0.000693
–0.00006
0.000100
0.06317
–0.03160
0.05
0.05
0.05
0.05
0.05
0.05
–0.08834
–0.00005
–0.00039
–0.00091
–0.03160
0.6207
2.6871
2.1913
–0.8213
2.4171
–1.3372
sp2se2covsesps2uspecs2usensmspecmsens
0.02465
0.1141
Upper
1.1114
3.3459
3.2797
–0.4081
3.0093
–1.0014
Covariance Matrix of Additional Estimates
Label
logitsens CCP2
logitspec CCP2
logLR+ CCP1
logLR-CCP1
logLR+ CCP2
logLR-CCP2
Cov1 Cov2 Cov3 Cov4 Cov5 Cov6
0.01461
–0.00697
–0.00001
5.023E-6
–0.00231
–0.00996
–0.00697
0.02632
0.000598
–0.00002
0.02303
0.003674
–0.00001
0.000598
0.07185
–0.00301
0.000566
–0.00002
5.023E-6
–0.00002
–0.00301
0.01035
–0.00002
–2.45E-6
–0.00231
0.02303
0.000566
–0.00002
0.02127
0.000554
–0.00996
0.003674
–0.00002
–2.45E-6
0.000554
0.006840
program in Appendix 8 of the online supplementary material (10.S1 Code for undertaking meta- analysis). The additional estimates and their covariance matrix are shown in
the green boxes in Box10.4.a. The resulting SROC plot from entering parameter estimates, standard errors and covariances into RevMan is shown in Chapter9, Figure9.4.c,
and panel (a) of Figure10.4.a.
Part II of Appendix 8 of the online supplementary material (10.S1 Code for undertaking meta- analysis) fits the more complex model that allows for unequal variances for
generations of anti- CCP (model 1). The regression equation specified in Proc NLMIXED
is not in the same format as for model 1 because dummy variables have been created
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(a)(b)
1
0.9
0.8
0.7
0.6
0.5
Sensitivity
0.4
0.3
0.2
0.1
0
10.9
0.8 0.70.6 0.5
Legend
Generation: CCP1 Generation: CCP2
Generation Summary
CCP1 0.48 (0.37 to 0.58) 0.97 (0.95 to 0.98)CCP1 0.48 (0.43 to 0.53)0.97 (0.95 to 0.98)
CCP2 0.70 (0.65 to 0.75) 0.95 (0.94 to 0.97)CCP2 0.71 (0.64 to 0.76)0.95 (0.94 to 0.97)
These results indicate an improvement in sensitivity
(P < 0.001), without loss of specificity (P = 0.21).
sensitivity
(95% CI)
0.40.3 0.2 0.10
Specificity
Summary
specificity
(95% CI)
1
0.9
0.8
0.7
0.6
0.5
Sensitivity
0.4
0.3
0.2
0.1
0
0.8 0.70.6 0.5
10.9
Legend
Generation: CCP1 Generation: CCP2
Generation Summary
These results indicate an improvement in sensitivity
(P < 0.0001), without loss of specificity (P = 0.19).
sensitivity
(95% CI)
0.4 0.3 0.2 0.10
Specificity
Summary
specificity
(95% CI)
Figure10.4.a SROC plots produced using estimates from (a) model 1: equal variances and
(b)model 2: unequal variances. The dotted region around each summary point (solid circle) is the
95% confidence region while the dashed region is the 95% prediction region
and included for both CCP1 and CCP2. This approach is more straightforward for directly
obtaining all parameter estimates and for specifying the variance–covariance structure.
The statistical significance of differences between the model that assumed equal variances (model 1) and the model that allowed for unequal variances (model 2) can be
assessed using a likelihood ratio test (see Chapter9, Section9.4.6.1). The likelihood
ratio Chi2 statistic is the difference in the −2Log likelihood when covariate terms are
added or removed from a model. The degrees of freedom used along with the Chi
statistic to obtain a P value are the difference in the number of parameters fitted in the
two models being compared.
Prior to fitting model 1, it is useful to perform separate meta- analyses for CCP1 and
CCP2 to explore whether or not the values of the variances are close or differ substantially between the two CCP generations. The output of these subgroup analyses are
shown in Box10.4.b. The variances of the random effects for the logit sensitivities
(s2usens) are 0.04277 and 0.4830 for CCP1 and CCP2, respectively (see red boxes). This
suggests much greater variation in sensitivity across CCP2 studies compared to CCP1
studies, which is also apparent from the scatter of the study points in Figure10.4.a. The
variances of the random effects for the logit specificities (s2uspec) also differ, but perhaps not to the extent that alone would justify fitting a model with unequal variances.
It was evident from Box10.4.a that the variances of the random effects have reduced,
particularly for sensitivity, compared to the analysis without the covariate in Box10.2.a.
Nevertheless, it is worth investigating the assumption of equal variances, especially as
2
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Meta-analysis of CCP1
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10.4 Comparison ofsummary points
Box 10.4.b Subgroup analyses of CCP1 and CCP2 generations of anti- CCP test for
rheumatoid arthritis
Parameter Estimates
Standard
Parameter
msens
mspec
s2usens
s2uspec
covsesp
Meta-analysis of CCP2
Parameter
msens
mspec
s2usens
s2uspec
covsesp
Estimate
–0.07773
3.4586
0.04277
0.4793
–0.06258
Estimate
0.8725
3.0147
0.4830
0.5661
–0.2326
Error DF t Value Pr > |t| 95% Confidence Limits Gradient
–0.83
0.09375
0.2910
0.03305
0.3201
0.07902
Standard
Error DF t Value Pr > |t| 95% Confidence Limits Gradient
0.1377
0.1655
0.1530
0.2185
0.1303
6
6
6
6
6
Parameter Estimates
27
27
27
27
27
11.89
1.29
1.50
–0.79
6.33
18.21
3.16
2.59
–1.79
0.4387
<.0001
0.2432
0.1850
0.4585
<.0001
<.0001
0.0039
0.0153
0.0854
–0.3071
2.7466
–0.03811
–0.3040
–0.2559
0.5899
2.6750
0.1691
0.1177
–0.4999
0.1517
4.1706
0.1237
1.2626
0.1308
1.1551
3.3543
0.7968
1.0145
0.03468
0.000033
0.000043
–0.00019
–0.00002
–0.00008
0.000025
0.000489
–0.00004
0.000055
0.000114
there are many included studies and so model overfitting is not a concern. In addition
to providing insight into whether or not an assumption of equal variances for the two
generations is justifiable, the subgroup analyses can guide the choice of starting values.
It is advisable not to use the exact values of the parameter estimates from the analyses,
but rather values close to the estimates, e.g. in Box10.4.b the estimate for ‘msens’ for
CCP1 is −0.07773 and was used to inform a starting value of 0.1 for ‘msens’ in the
meta-
regression analysis (see Part II in Appendix 8 of the online supplementary material
(10.S1 Code for undertaking meta- analysis)).
Box10.4.c shows the parameter estimates obtained for CCP1 (red box) and for
CCP2 (blue box) and their covariances (green boxes) from fitting model 2. The fit
statistics table gives a –2Log likelihood of 524.6, a reduction of 8.8 (533.4−524.6) compared with model 1 that assumed equal variances. Hence, there is statistical evidence
(Chi2 = 8.8, 3df, P = 0.032) that the variances are unequal for CCP1 and CCP2. The differences in mean logit sensitivity and mean logit specificity (i.e. νA and νB) are shown in
Box10.4.d (see red box). The P values based on Wald statistics indicate an improvement
in sensitivity (P < 0.0001), without loss of specificity (P = 0.19); there is a similar conclusion based on the results of model 1 (Figure10.4.a). Comparing the summary estimates
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