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10 Undertaking meta- analysis
i. Parameter estimates
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Box 10.4.k R output of parameter and covariance estimates for unequal variance model
summary(E)
## Generalized linear mixed model fit by maximum likelihood (Laplace
## Approximation) [glmerMod]
## Family: binomial ( logit )
## Formula: cbind(true, n - true) ~ 0 + seCT + seMRI + spCT + spMRI + (0 +
## seMRI + spMRI | Study_ID) + (0 + seCT + spCT | Study_ID)
## Data: Y
##
## AIC BIC logLik deviance df.resid
## 963.3 997.1 -471.7 943.3 206
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -2.09753 -0.33455 0.04077 0.53006 1.43650
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## Study_ID seMRI 0.1127 0.3357
## spMRI 0.7122 0.8439 -0.52
## Study_ID.1 seCT 1.1070 1.0521
## spCT 0.8801 0.9381 0.31
## Number of obs: 216, groups: Study_ID, 103
##
## Fixed effects:
## Estimate Std. Error z value Pr(>|z|)
## seCT 3.5594 0.1739 20.464 < 2e-16 ***
## seMRI 1.9671 0.1624 12.109 < 2e-16 ***
## spCT 1.9317 0.1225 15.773 < 2e-16 ***
## spMRI 0.8410 0.2406 3.495 0.000474 ***
ii. Covariance between the mean logits
(vcovE = (summary(E))$vcov)
## 4 × 4 Matrix of class "dpoMatrix"
## seCT seMRI spCT spMRI
## seCT 3.025237e-02 2.055647e-09 3.550535e-03 -2.960073e-09
## seMRI 2.055647e-09 2.638712e-02 8.912427e-10 -8.289464e-03
## spCT 3.550535e-03 8.912427e-10 1.499821e-02 -3.943513e-09
## spMRI -2.960073e-09 -8.289464e-03 -3.943513e-09 5.791074e-02
●
Comparing summary points: To compare summary points, we add a few lines to the
rjags model to calculate the difference between the summary sensitivity of CCP2 and
summary sensitivity of CCP1 (Difference_Se) and difference between the summary
specificity of CCP2 and summary specificity of CCP1 (Difference_Sp). Using the step()
288
10.4 Comparison ofsummary points
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function in rjags, we can calculate the probability that the summary sensitivity of
CCP2 is greater than the summary sensitivity of CCP1 (prob_Se), and similarly for
specificity (prob_Sp).
10.4.4.1 Summary statistics
After ensuring that the MCMC algorithm successfully converged, the summary statistics for the parameters of interest can be calculated based on a sample from the posterior distribution. In the output from the rjags program in Box 10.4.l, the logit
transformed summary sensitivity and specificity for CCP1 (mu[1] and mu[2]) and the
between- study standard deviations and correlation are reported as well as the difference in logit mean sensitivities and specificities between CCP2 and CCP1 (nu[1] and
nu[2]). These results are similar to the ones reported in Section10.4.1. The difference
in mean sensitivities and specificities between CCP2 and CCP1 is also reported on
Box 10.4.l Rjags output from Bayesian bivariate meta- regression of CCP generation
Iterations = 26001:46000
Thinning interval = 1
Number of chains = 3
Sample size per chain = 20000
1. Empirical mean and standard deviation for each
variable,
plus standard error of the mean:
Mean SD Naive SE Time- series SE
Summary_Sp_CCP1 0.96791 0.009514 3.884e- 05 1.586e- 04
Summary_Sp_CCP2 0.95258 0.007405 3.023e- 05 7.236e- 05
Summary_Se_CCP1 0.47494 0.056278 2.298e- 04 8.337e- 04
Summary_Se_CCP2 0.70383 0.025918 1.058e- 04 1.860e- 04
mu[1] -
0.10161 0.228722 9.338e- 04 3.388e- 03
mu[2] 3.44889 0.301046 1.229e- 03 5.130e- 03
nu[1] 0.97037 0.260469 1.063e- 03 3.936e- 03
nu[2] - 0.43651 0.341737 1.395e- 03 5.917e- 03
tau.sq[1] 0.39251 0.112203 4.581e- 04 1.170e- 03
tau.sq[2] 0.56943 0.194723 7.950e- 04 2.933e- 03
rho - 0.38057 0.174221 7.113e- 04 2.391e- 0
Difference_Se 0.22888 0.061937 2.529e- 04 9.445e- 04
Difference_Sp - 0.01533 0.012038 4.914e- 05 1.971e- 04
prob_Se 0.99980 0.014141 5.773e- 05 1.900e- 04
prob_Sp 0.09882 0.298418 1.218e- 03 3.755e- 03
289
10 Undertaking meta- analysis
Density
(a)
(b)
prob_Sp
Iteration
0
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2.5% 25% 50% 75% 97.5%
Summary_Sp_CCP1 0.94600 0.96264 0.96914 0.974560 0.98292
Summary_Sp_CCP2 0.93696 0.94789 0.95299 0.957715 0.96596
Summary_Se_CCP1 0.36477 0.43696 0.47499 0.512617 0.58609
Summary_Se_CCP2 0.65157 0.68684 0.70431 0.721308 0.75340
mu[1] -
0.55471 - 0.25350 - 0.10013 0.050479 0.34781
mu[2] 2.86329 3.24917 3.44701 3.645675 4.05251
nu[1] 0.46336 0.79715 0.96846 1.142235 1.49064
nu[2]
- 1.10806 - 0.66086 - 0.43822 - 0.214684 0.24391
tau.sq[1] 0.22526 0.31331 0.37531 0.452647 0.659967
tau.sq[2] 0.28326 0.43244 0.53753 0.671007 1.037957
rho - 0.68655 - 0.50609 - 0.39200 - 0.267176 - 0.01105
prob_Se 1.00000 1.00000 1.00000 1.000000 1.00000
prob_Sp 0.00000 0.00000 0.00000 0.000000 1.00000
Difference_Se 0.10751 0.18769 0.22887 0.270338 0.35087
Difference_Sp - 0.03735 - 0.02328 - 0.01598 - 0.008228 0.01051
the probability scale. The median of the difference in summary sensitivities of CCP2
versus CCP1 is about 22.9 percentage points (95% credible interval ranging from 10.8
to 35.1 percentage points, see red boxes). The median of the difference in summary
specificities (see green boxes) is −1.6 percentage points (with a 95% credible interval
of −3.7 to 1.1 percentage points). The blue box highlights that there is a 0.999 probability
that the summary sensitivity of CCP2will be greater than the summary sensitivity of
CCP1. However, there is only a 0.099 probability that the summary specificity of
CCP2will be higher than that of CCP1. Figure10.4.c shows the diagnostics plots of the
0 5000 10000 15000
0.0
Figure10.4.c MCMC diagnostics plots for a parameter (prob_Sp) from the Bayesian bivariate
meta- regression model
290
0.2
0.4 1. 00.80.6
0.30.20.10.0
Running mean
05000 1000015000
2000
22
~ , and ~ , .
ii
NN
ii
sZ
22
22
~, and~, .
ii
ii
NN
04
1. 00.80.6
prob_Sp
(c)
Iteration
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0.40.00.2
10.5 Comparison ofsummary curves
30000 350004000
Figure10.4.c (Continued)
5000
probability that the summary specificity of CCP2will be higher than that of CCP1. We
can see that at any given iteration this probability is either 0 or 1. Its posterior mean
is the desired summary statistic.
10.5 Comparison ofsummary curves
The Rutter and Gatsonis HSROC model, like the bivariate model, can be extended to a
meta- regression model for investigating heterogeneity (Chapter9, Section9.4.6.4) or
comparing test accuracy (Chapter9, Section 9.4.7.4). HSROC meta- regression models
were illustrated in Chapter9, Section9.4.6.4, using the RF data to investigate whether
the laboratory technique (nephelometry, latex agglutination (LA) and ELISA) used to
measure RF is associated with diagnostic performance. Covariate terms were included
in the HSROC model to assess whether accuracy, threshold or the shape of the SROC
curve varied with technique. The variances of the random effects for threshold (θ) and
accuracy (α) were assumed to be equal for all three techniques (see Part I of Appendix
13 of the online supplementary material (10.S1 Code for undertaking metaSAS code) and modelled as follows:
analysis) for
Similar to the bivariate model, the variance parameters of the HSROC model can be
allowed to differ between subgroups or tests. For example, if Zi represents a vector of
study- level covariates (i= 1, .. ., I) from I studies included in a meta- analysis, allowing
the difference in threshold and accuracy between tests to vary gives
t
ij iiiiii ij
ZZdi
(10.4)
(10.5)
disexp
j
(10.6)
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10 Undertaking meta- analysis
22
2
22
2
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Covariance terms are needed between the random effects for accuracy (i.e. αi and ξi)
and between the random effects for threshold (i.e. θi and γi), otherwise the variances for
the referent test will be larger than those of the comparator test(s) (Macaskill2003).
Therefore, σασξ is the covariance between the random effects for accuracy and σθσγ is
the covariance between the random effects for threshold. The variances of the random
effects for the comparator can be obtained using the expression for the variance of a
sum that is
for accuracy and
for threshold. The compu-
tation of these variances will be illustrated in Section10.5.1.
To investigate the effect of laboratory technique on the diagnostic performance of
RF, HSROC meta- regression models will be fitted using SAS in Section 10.5.1 and
Bayesian estimation will be illustrated in Section10.5.2. Following the notation of
equation 10.5, since laboratory technique has three categories, we define two
dichotomousvectors of data Z1i and Z2i, where Z1i takes the value 1 if the laboratory
technique is nephelometry and0 otherwise, while Z2i takes the value 1 if the laboratory
technique is ELISA and 0 otherwise. The laboratory technique LA is arbitrarily chosen
to be the reference group that corresponds to the situation where Z1i and Z2i are simultaneously equal to 0.
10.5.1 Fitting the HSROC model in SAS to compare summary curves
The SAS code for this analysis is in Appendix 13 of the online supplementary material
(10.S1 Code for undertaking meta- analysis). The output of the model that included
covariate terms for the shape, accuracy and threshold parameters but assumed equal
variances is shown in Box10.5.a. The parameter estimates for LA, the reference group,
are shown in the red box in the parameter estimates table, while the estimates for the
accuracy, threshold and shape parameters for ELISA and nephelometry are shown in
the blue box in the additional estimates table. Since the variances were assumed to be
equal for the three techniques, the variance estimates in the red box (s2ua = 1.2817
and s2ut = 0.4780) are also the estimates for ELISA and nephelometry.
Further modelling was undertaken by removing the covariate terms for shape from
the model, i.e. assuming a common shape for the three curves. The increase in the
−2Log likelihood was negligible (Chi2 = 753.1−752.9 = 0.2, 2 df, P= 0.90). Parameter estimates for the model that assumes a common shape are given in Chapter9, Table9.4.d,
with the corresponding HSROC curves shown in Chapter9, Figure 9.4.e. Since there
appeared to be no difference in shape, covariate terms for accuracy were removed from
the common shape model to assess the effect on the accuracy parameter. The output
from fitting this model is shown in Box10.5.b. The removal of the covariate terms for
accuracy has a negligible effect on the fit of the model (Chi2 = 753.7−753.1 = 0.6 on 2 df,
P = 0.74), indicating no statistical evidence of a difference in diagnostic accuracy of RF
according to technique (see Chapter9, Section9.4.6.4).
Fitting HSROC meta- regression models with a more complex variance–covariance
structure (equation10.6) is computationally intensive and prone to convergence issues.
Such models may require simplification, e.g. assumption of same or no shape parameter (see Section10.6.2). For an example of an HSROC meta- regression with unequal
variances and the accompanying code, see Takwoingi (2016).
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10.5 Comparison ofsummary curves
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Box 10.5.a SAS output of HSROC meta- regression of RF technique: differences in
shape, accuracy and threshold parameters, equal variances
Fit Statistics
<.0001
0.0463
0.3006
0.0002
0.0001
0.5674
0.4137
0.4919
0.2766
0.8132
0.6872
752.9
774.9
778.1
795.2
1.7559
–0.9952
–0.2582
0.6565
0.2486
–0.7393
–0.5375
–1.0038
–0.3287
–0.9625
–0.9637
3.0907
–0.00853
0.8174
1.9069
0.7074
1.3316
1.2836
0.4899
1.1217
0.7593
0.6408
–0.00026
–0.00033
–0.00028
–0.00083
–0.00062
–0.00028
0.000267
–0.00037
–0.00027
2.043E–6
0.000412
Parameter
alpha
theta
beta
s2ua
s2ut
a1
a2
t1
t2
b1
b2
Estimate
2.4233
–0.5019
0.2796
1.2817
0.4780
0.2962
0.3730
–0.2569
0.3965
–0.1016
–0.1614
-2 Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
Parameter Estimates
Standard
Error DF t Value Pr > |t| 95% Confidence Limits Gradient
0.3314
0.2449
0.2670
0.3104
0.1139
0.5141
0.4521
0.3708
0.3601
0.4274
0.3983
45
45
45
45
45
45
45
45
45
45
45
7.31
–2.05
1.05
4.13
4.20
0.58
0.83
–0.69
1.10
–0.24
–0.41
Label
alpha ELISA
theta ELISA
beta ELISA
alpha Nephelometry
theta Nephelometry
beta Nephelometry
Estimate
2.7194
–0.7588
0.1780
2.7963
–0.1054
0.1182
Additional Estimates
Standard
Error DF t Value Pr > |t| Alpha Lower Upper
0.3935
0.2785
0.3338
0.3079
0.2637
0.2955
45
45
45
45
45
45
6.91
–2.72
0.53
9.08
–0.40
0.40
<.0001
0.0091
0.5964
<.0001
0.6914
0.6912
0.05
0.05
0.05
0.05
0.05
0.05
1.9269
–1.3198
–0.4943
2.1761
–0.6364
–0.4771
3.5120
–0.1978
0.8503
3.4165
0.4257
0.7134
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10 Undertaking meta- analysis
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Box 10.5.b SAS output of HSROC meta- regression of RF technique: differences in
threshold parameter, equal variances
Fit Statistics
Parameter
alpha
theta
beta
s2ua
s2ut
t1
t2
Estimate
2.6574
–0.5565
0.1912
1.2994
0.4784
–0.1939
0.4971
-2 Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
Parameter Estimates
Standard
Error DF t Value Pr > |t| 95% Confidence Limits Gradient
0.1923
0.2133
0.1651
0.3148
0.1139
0.2614
0.2626
45
45
45
45
45
45
45
13.82
–2.61
1.16
4.13
4.20
–0.74
1.89
<.0001
0.0123
0.2529
0.0002
0.0001
0.4621
0.0648
753.7
767.7
769.0
780.6
–0.03171
2.2702
–0.9860
–0.1413
0.6653
0.2490
–0.7203
3.0447
–0.1269
0.5236
1.9335
0.7079
0.3325
1.0259
0.000052
0.000080
–0.00005
0.000011
–0.00009
0.000174
–0.00005
10.5.2 Bayesian estimation of the HSROC model for comparing summary curves
See Appendix 6 of the online supplementary material (10.S1 Code for undertaking
meta- analysis), Section A6.4, for the complete Bayesian program. The rjags model
involves specification of the following components.
●
Likelihood: The likelihood of the observed data (TP and FP cells) follows a binomial
distribution, with the unknown parameters being the TPR and FPR in each study.
●
Prior distributions: The logit TPR and the logit FPR are expressed as functions of the
positivity threshold (theta) for each study (i), the lnDOR (alpha) for each study and a
regression equation of the shape parameter (beta, which is assumed to be common
across studies) and the covariates Z1i and Z2i (delta[1] and delta[2]). At the first level of
the prior distribution, theta and alpha parameters in the included studies are assumed
to follow normal distributions with precision (i.e. 1/variance) prec[2] and prec[1],
respectively, assumed to be equal for all three techniques. The mean of the prior distri-
bution is expressed as a function of the covariates Z1i and Z2i via a regression equation.
The mean of theta is expressed as a function of THETA, gamma[1] and gamma[2], where
THETA, THETA+gamma[1] and THETA+gamma[2] are the mean positivity threshold
parameters of the laboratory techniques LA, nephelometry and ELISA, respectively.
Similarly, the mean of alpha is a function of LAMBDA, epsilon[1] and epsilon[2], where
LAMBDA, LAMBDA+epsilon[1] and LAMBDA+epsilon[2] are the mean lnDOR parameters
of LA, nephelometry and ELISA, respectively.
294
10.5 Comparison ofsummary curves
Density
LAMBDA
(a)
(b)
(c)
Iteration
20000
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●
At the second level of the prior distribution, vague prior distributions are provided for
THETA, gamma[1], gamma[2], prec[2], LAMBDA, epsilon[1], epsilon[2], prec[1], beta,
delta[1] and delta[2].
10.5.2.1 Monitoring convergence
The rjags program and the detailed script on how to run it are provided in Appendix 6
of the online supplementary material (10.S1 Code for undertaking meta- analysis).
Before interpreting the results, the MCMC algorithm should be checked for convergence. As an illustrative example, Figure10.5.a represents the convergence behaviour
of LAMBDA, the mean lnDOR parameter of nephelometry. The overlapping results from
three independent MCMC chains in each of the panels indicate that convergence was
achieved.
10.5.2.2
Summary statistics
With convergence successfully achieved, we display the rjags output of the summary
statistics of the parameters of interest based on their posterior distributions in
Box10.5.c. For comparison with the results obtained with the frequentist analysis in
Section10.5.1, the output returns posterior estimates of the variances of the random
effects for threshold (tau.sq[1]) and accuracy (tau.sq[2]), THETA, LAMBDA and beta
parameters for LA; THETA_nephelometry (THETA+gamma[1]), LAMBDA_ nephelometry
(LAMBDA+epsilon[1]) and beta_ nephelometry (beta+delta[1]) parameters for nephelometry; and THETA_ELISA (THETA+gamma[2]), LAMBDA_ELISA (LAMBDA+epsilon[2])
and beta_ELISA (beta+delta[2]) parameters for ELISA.
3.02.82.62.2 2.4
Running mean
0.0 0.2 0.4 0.6 0.8 1. 0
1. 0
1. 5
2.0 3.5 4.03.02.5
LAMBDA
4.03.02.0
0 5000 10000 15000
Iteration
1. 0
Figure10.5.a MCMC diagnostics plots for a parameter (LAMBDA) from the HSROC meta- regression model
30000 35000 40000 45000
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10 Undertaking meta- analysis
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Box 10.5.c Rjags output from Bayesian HSROC meta- regression of RF laboratory
techniques
Iterations = 26001:46000
Thinning interval = 1
Number of chains = 3
Sample size per chain = 20000
1. Empirical mean and standard deviation for each variable,
plus standard error of the mean:
Mean SD Naive SE Time-
LAMBDA
LAMBDA_ELISA 2.77382 0.42765 1.746e- 03 0.0153399
LAMBDA_Nephelometry
THETA - 0.52722 0.26313 1.074e- 03 0.0084638
THETA_ELISA - 0.77078 0.30704 1.253e- 03 0.0128415
THETA_Nephelometry - 0.12863 0.28407 1.160e- 03 0.0081274
beta 0.25933 0.27488 1.122e- 03 0.0168935
beta_ELISA 0.17310 0.35237 1.439e- 03 0.0195180
beta_Nephelometry 0.08596 0.30435 1.243e- 03 0.0116891
tau.sq[1] 0.59907 0.14225 5.807e- 04 0.0008611
tau.sq[2] 1.45742 0.36373 1.485e- 03 0.0026122
2. Quantiles for each variable:
2.5% 25% 50% 75% 97.5%
LAMBDA 1.759833 2.22213 2.45811 2.695339 3.169631
LAMBDA_ELISA 1.984570 2.48382 2.75725 3.041644 3.661214
LAMBDA_Nephelometry 2.198794 2.62092 2.83738 3.058208 3.497508
THETA
THETA_ELISA - 1.392830 - 0.97026 - 0.76279 - 0.563522 - 0.193243
THETA_Nephelometry
beta - 0.266138 0.07160 0.25477 0.442268 0.813086
beta_ELISA
beta_Nephelometry - 0.517020 - 0.11818 0.08346 0.291679 0.684564
tau.sq[1] 0.378056 0.49776 0.57973 0.679221 0.929534
tau.sq[2] 0.896167 1.19925 1.40845 1.658469 2.306908
2.46056 0.35761 1.460e- 03 0.0053571
2.83999 0.32917 1.344e- 03 0.0023498
- 1.047216 - 0.70485 - 0.52802 - 0.351234 - 0.007843
- 0.688319 - 0.31738 - 0.12788 0.061554 0.425279
- 0.497171 - 0.06738 0.17259 0.414194 0.855344
series SE
10.6 Meta- analysis ofsparse data anda typical data sets
When fitting hierarchical meta- analytical models within a frequentist framework using
a maximum likelihood approach, estimation problems such as unreliable parameter
estimates or lack of convergence often occur due to small number of studies or sparse
data. Bayesian analyses of such data can also be challenging and checking convergence
296
10.6 Meta- analysis ofsparse data anda typical data sets
0 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 1
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Study
Li 2009
Fenton 1989
Montariol 1998
Silverstein 1998
Wu 2005
FN
FP
TP
1
21
6
19
3
41
0
14
0
23
Sensitivity (95% CI)
TN
7
74
0
131
0
171
0
76
0
67
0.75 [0.55, 0.89]
1.00 [0.82, 1. 00]
1.00 [0.91, 1. 00]
1. 00 [0.77, 1. 00]
1.00 [0.85, 1. 00]
Specificity (95% Cl)
0.99 [0.93, 1.00]
0.96 [0.91, 0.98]
0.98 [0.95, 1.00]
1.00 [0.95, 1. 00]
1.00 [0.95, 1. 00]
Sensitivity (95% CI) Specificity (95% Cl)
Figure10.6.a Forest plot of intraoperative cholangiography for diagnosis of common bile duct
stones. Studies are ordered by sensitivity and study identifier. Source: Adapted from Gurusamy 2015
of the MCMC algorithm is critical. In addition, priors for variance parameters can have
more influence than intended when the number of studies is small (Lambert 2005).
Sparse data are a common occurrence in meta-
analysis of very accurate tests where
most of the studies have 100% sensitivity and/or specificity. For example, in a Cochrane
Review of intraoperative cholangiography (IOC) for diagnosis of common bile duct
stones, four of the five IOC studies had a sensitivity of 100% (Figure10.6.a) (Gurusamy
2015)).
There are also atypical situations where data are only available in a 2×1 (e.g. only sensitivity or only specificity estimated) rather than a 2×2 table format, or where false positives
are impossible and thus are structural zeros. Anexample of the latter is a Cochrane Review
of the accuracy of laparoscopy following CT scanning for assessing resectability with
curative intent in pancreatic and periampullary cancer (Allen 2016). Since laparotomy
(the reference standard) will be performed only if histopathology of the biopsy of the
suspicious lesion on diagnostic laparoscopy shows no evidence of cancer, false positives
are not possible, i.e. specificity is always 1. In the Cochrane Review of serum screening
tests used in the first trimester of pregnancy, although studies reported results at different
thresholds, it is common in this clinical field for studies to report sensitivity (detection
rate) at a fixed specificity (usually a 5% FPR). The chosen FPR level is determined as the
FPR deemed acceptable in a particular screening programme and so all studies report the
same specificity. In both of these examples there is no need to account for correlation
between sensitivity and specificity across studies in a hierarchical meta- analytical model.
Approaches for dealing with sparse data were introduced in Chapter9, Section9.4.8,
and will be explained further and illustrated in this section with a focus on frequentist
analyses.
10.6.1 Facilitating convergence
Irrespective of the model fitted or the software program used for frequentist estimation, adaptive Gaussian quadrature is commonly used for the maximum likelihood estimation. In sparse data situations, the variance–covariance parameters are often on the
boundary of the parameter space (Riley 2007, Chung 2013); the maximum likelihood
estimate on the boundary will have at least one of the variances in an HSROC model
equal to 0 or the correlation parameter in a bivariate model equal to +1 or –1.
The direct comparison of CT versus MRI for CAD included five comparative accuracy
studies (see Chapter9, Section9.4.7.5). The forest plot of CT and MRI shows that except
for one study, the studies reported the same sensitivity for CT (Figure10.6.b). This lack
of heterogeneity in the sensitivity of CT, in addition to the small number of studies, may
lead to estimation problems when a bivariate model is fitted to the data.
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