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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_2767_Библиотеки_им_академика_М_И_Перельмана
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Figure10.2.a Screenshot of MetaDTA (version 2.0) showing anti- CCP data uploaded
https://t.me/medicina_free
Figure10.2.b Screenshot of MetaDTA (version 2.0) showing the SROC plot for anti- CCP
https://t.me/medicina_free
10 Undertaking meta- analysis
## Generalized linear mixed model fit by maximum likelihood (Laplace
## Approximation) [glmerMod]
## Family:
## Formula: cbind(true, n - true) ~ 0 + sens + spec + (0 + sens + spec |
##
##
##
##
##
##
## Scaled residuals:
##
##
##
## Random effects:
##
##
##
## Number of obs: 74, groups:
##
## Fixed effects:
##
## sens
## spec
## --
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
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Box 10.2.f R output of bivariate model parameters using glmer
binomial (logit)
Study.ID)
Data: Y
AIC BIC logLik deviance df.resid
555.6 567.1 -272.8 545.6 69
Min 1Q Median 3Q Max
-1.9629 -0.2391 0.0097 0.2396 1.9864
Groups Name Variance Std.Dev. Corr
Study.ID sens 0.5426 0.7366
spec 0.5712 0.7558 -0.49
Study.ID, 37
Estimate Std. Error z value Pr(>|z|)
260
The code provided in Appendix 5 of the online supplementary material (10.S1 Code for
undertaking metaglmer, and obtain the summary estimates. Box10.2.f shows the output obtained after
summarizing the glmer model output. The five parameters (mean logits, variances and
correlation) of the bivariate model are shown in the red boxes. This correlation estimate
is the correlation of the logits across studies. In the glmer function, the nAGQ argument
controls the number of quadrature points. The default isnAGQ = 1, which equates to the
Laplace approximation (see green box). Values greater than 1 produce greater accuracy
in the evaluation of the log- likelihood. However, when there is more than one randomeffects parameter in the model, the number of quadrature points used by glmer cannot
be increased. This is a limitation of using glmer to fit the bivariate model, since the
standard model without a covariate has two random- effects parameters.
The variance- covariance matrix can be displayed to obtain the covariance between
the summary estimates of logit(sensitivity) and logit(specificity), as shown in the blue
box in Box10.2.g. This covariance and the parameter estimates are needed to generate
the SROC plot in RevMan.
0.6534 0.1274 5.127 2.95e-07 ***
3.1084 0.1451 21.420 < 2e-16 ***
-
analysis) shows how to load the data, fit the bivariate model using
10.2 Estimation ofa summary point
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Box 10.2.g R output of variance–covariance matrix
(summary(MA_Y))$vcov
## 2 × 2 Matrix of class "dpoMatrix"
## sens spec
## sens 0.016241813 -0.007404185
## spec -0.007404185 0.021058854
10.2.4 Bayesian estimation ofthe bivariate model
10.2.4.1 Specification ofthe bivariate model inrjags
Both levels of the bivariate model introduced in Chapter9, Section9.4.1, must be specified in rjags. This section covers the different components of this rjags model, which is
the core of the code (see Box10.2.h). The complete program is in Appendix 6 of the
online supplementary material (10.S1 Code for undertaking meta- analysis), Section
A6.1.
●
Likelihood: The likelihood specifies that the observed data (TP and TN cells) in each
study follow a binomial distribution, with the unknown parameters being the sensitivity (se) and specificity (sp) in each study.
●
Prior distribution: To carry out Bayesian estimation, a prior distribution function
must be specified for each unknown parameter in the likelihood. As this is a prior distribution for a meta- analysis model, it has a hierarchical structure. At the first level of
the hierarchy, sensitivity and specificity parameters in the individual studies are
assumed to have the same prior distribution. This is specified by assuming the logit
sensitivities and logit specificities of the included studies follow a bivariate normal
distribution whose mean is the vector (mean logit sensitivity (mu[1]), mean logit
specificity (mu[2])) and whose variance–covariance matrix is parametrized by the
between- study standard deviation in the logit sensitivities (tau[1]), the betweenstudy standard deviation in the logit specificities (tau[2]) and the correlation between
the logit sensitivities and the logit specificities (rho). Specifying the prior distribution
for the logit(sensitivity) (l[,1]) and logit(specificity) (l[,2]) rather than for the sensitivity
and specificity parameters directly makes it easier to expand the meta-
analysis to a
meta- regression model in Section10.4.
At the second level of the hierarchy, prior distributions are provided for the parameters
mu[1], mu[2], prec[1], prec[2] and rho, where prec[1] and prec[2] are the between- study
precision in the logit sensitivities and logit specificities, respectively. The between- study
precision is the inverse of the between- study variance, e.g. prec[1] = 1/tau.sq[1]. These
prior distributions are typically defined to be vague, i.e. their parameter values are
selected so they have a negligible influence on the results of the meta- analysis.
●
Summary sensitivity and specificity: The mean logit(sensitivity) and mean logit(specificity)
are transformed to the probability scale to obtain the summary sensitivity (Summary_
Se) and summary specificity (Summary_Sp) across studies.
261
Box 10.2.h Specification of the bivariate model for Bayesian estimation in rjags
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model {
#=== LIKELIHOOD ===#
for(i in 1:n) {
TP[i] ~ dbin(se[i],pos[i])
TN[i] ~ dbin(sp[i],neg[i])
# === PRIOR DISTRIBUTIONS FOR INDIVIDUAL LOGIT SENSITIVITY,
SPECIFICITY === #
logit(se[i]) <- l[i,1]
logit(sp[i]) <- l[i,2]
l[i,1:2] ~ dmnorm(mu[], T[,])
}
#=== HYPER PRIOR DISTRIBUTIONS MEAN LOGIT SENSITIVITY AND
SPECIFICITY === #
mu[1] ~ dnorm(0,0.25)
mu[2] ~ dnorm(0,0.25)
# Between- study variance- covariance matrix
T[1:2,1:2]<- inverse(TAU[1:2,1:2])
TAU[1,1] <- tau[1]*tau[1]
TAU[2,2] <- tau[2]*tau[2]
TAU[1,2] <- rho*tau[1]*tau[2]
TAU[2,1] <- rho*tau[1]*tau[2]
#=== HYPER PRIOR DISTRIBUTIONS FOR PRECISION OF LOGIT
SENSITIVITY ===#
#=== AND LOGIT SPECIFICITY, AND CORRELATION BETWEEN THEM === #
prec[1] ~ dgamma(2,0.5)
prec[2] ~ dgamma(2,0.5)
rho ~ dunif(- 1,1)
# === PARAMETERS OF INTEREST === #
# BETWEENtau.sq[1]<- pow(prec[1],- 1)
tau.sq[2]<- pow(prec[2],- 1)
# BETWEEN-STUDY STANDARD DEVIATION OF LOGIT SENSITIVITY
AND SPECIFICITY
tau[1]<-pow(prec[1],-0.5)
tau[2]<-pow(prec[2],-0.5)
# SUMMARY SENSITIVITY AND SPECIFICITY
Summary_Se <- 1/(1+exp(- mu[1]))
Summary_Sp <- 1/(1+exp(- mu[2]))
# PREDICTED SENSITIVITY AND SPECIFICITY IN A NEW STUDY
l.predicted[1:2] ~ dmnorm(mu[],T[,])
Predicted_Se <- 1/(1+exp(- l.predicted[1]))
Predicted_Sp <- 1/(1+exp(- l.predicted[2]))
STUDY VARIANCE OF LOGIT SENSITIVITY AND SPECIFICITY
}
10.2 Estimation ofa summary point
Density
Summary_Se
(a)
(b)
(c)
Iteration
000
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●
Predicted sensitivity and specificity: Calculation of the predicted sensitivity and spec-
ificity parameters in a future study can be done by adding a line within the rjags model
specifying the distribution of logit(sensitivity) and logit(specificity) (l.predicted) in a
single, future study. This would be the same bivariate normal distribution as the
observed studies in the meta- analysis. Transforming back to the probability scale
gives posterior distributions for the predicted sensitivity (Predicted_Se) and pre-
dicted specificity (Predicted_Sp) in a future study.
10.2.4.2 Monitoring convergence
Convergence of the MCMC algorithm can be examined by running multiple MCMC chains
with different starting values and examining the results using graphs such as those in
Figure10.2.c, created using the mcmcplots package in R (see Appendix 6 of the online
supplementary material (10.S1 Code for undertaking meta-
analysis) for the complete R
script used to generate Figure10.2.c). In Figure10.2.c, results from three chains (identified by different colours) are superimposed. Panel (a) shows posterior density plots for
Summary_Se; (b) the running posterior mean value; and (c) the history plots for
Summary_Se. The very similar results from all three chains suggest that the algorithm
converged to the same solution in each case.
12 14
0246810
0.50
0.55
0.60 0.65
Summary_Se
0.750.700.65
0.600.55
6000 800010000 1200014000 16000
0.70 0.7502000 4000 6000
0.670.650.630.61
Running mean
8000 10
Iteration
Figure10.2.c MCMC diagnostics plots for a parameter (Summary_Se) from the bivariate model
263
10 Undertaking meta- analysis
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Besides these graphs, many statistics for assessing convergence have been described.
A well- known statistic for assessing convergence is the Gelman- Rubin statistic, which
can be calculated separately for each parameter in the model. When this statistic
diverges from 1 there is concern that the MCMC algorithm has not converged.
10.2.4.3 Summary statistics
Once convergence of the MCMC algorithm is achieved, summary statistics of the parameters of interest may be extracted from their posterior distributions. Output from the
rjags program is given in Box10.2.i. The focus here is on the mean and predicted values
Box 10.2.i Rjags output from Bayesian estimation of the bivariate model
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_Se 0.6575 0.028885 1.179e–04 1.924e–04
Summary_Sp 0.9568 0.005987 2.444e–05 5.896e–05
Predicted_Se 0.6427 0.155745 6.358e– 04 6.469e–04
Predicted_Sp 0.9453 0.043130 1.761e– 04 1.815e–04
mu[1] 0.6550 0.128844 5.260e–04 8.592e–04
mu[2] 3.1062 0.145372 5.935e–04 1.456e–03
rho - 0.4253 0.156699 6.397e–04 1.817e–03
tau.sq[1] 0.5546 0.151128 8.725e–04 2.040e–03
tau.sq[2] 0.5699 0.186720 1.078e–03 3.853e–03
2. Quantiles for each variable:
2.5% 25% 50% 75% 97.5%
Summary_Se 0.5996 0.6386 0.6578 0.6770 0.71300
Summary_Sp 0.9442 0.9530 0.9570 0.9609 0.96780
Predicted_Se 0.3045 0.5404 0.6590 0.7608 0.89549
Predicted_Sp 0.8303 0.9310 0.9569 0.9733 0.99033
mu[1] 0.4037 0.5692 0.6536 0.7401 0.90998
mu[2] 2.8282 3.0088 3.1028 3.2007 3.40312
rho –0.6993 –0.5378 –0.4362 –0.3248 –0.09108
tau.sq[1] 0.3276 0.4482 0.5318 0.6367 0.91251
tau.sq[2] 0.2907 0.4373 0.5400 0.6714 1.01380
264
10.2 Estimation ofa summary point
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of sensitivity and specificity, but other parameters, such as the sensitivities and specificities in individual studies, may also be extracted. The logit- transformed mean sensitivity and specificity and the between- study variances and correlation are also reported
(red box) in Box10.2.i. Note that the posterior standard deviation (SD) estimated (blue
box) is the standard error in frequentist terms. The results obtained using a Bayesian
approach are similar to those obtained using a frequentist approach (see Box10.2.b
and Box10.2.f).
It is common to report the posterior median values of the summary sensitivity and
summary specificity, which would be 0.6578 and 0.9570, respectively, in this example, to
cover the possibility that the posterior distribution is skewed. In the current example, it
so happens that the posterior mean and posterior median values of the summary sensitivity and summary specificity are in fact quite similar. Obtaining a 95% credible interval
is straightforward and does not require gathering values of the standard errors or the
covariance parameters. To obtain a 95% equal-
tailed credible interval for a parameter,
use the 2.5% and 97.5% quantiles of its posterior distribution. These quantiles and the
median are shown in the green boxes in Box10.2.i. The 95% credible interval for the summary sensitivity is (0.5996, 0.7130), while the 95% credible interval for the summary
specificity is (0.9442, 0.9678). Each of these intervals can be interpreted as having 95%
probability of including the true value of the parameter given the observed data.
There is considerable between- study variability in both sensitivity and specificity. This
is reflected in the much wider 95% prediction intervals (i.e. the 95% credible interval
around the predicted values) compared to the 95% credible intervals for the mean values.
10.2.4.4 Generating an SROC plot
In this section we describe two ways of obtaining the SROC plot following Bayesian
estimation with rjags.
●
SROC plot with the DTAplots package: The package DTAplots within R was used
to produce the plot in Figure10.2.d with a 95% credible region (red line) and 95%
prediction region (black dotted line) around the summary estimates of sensitivity and
specificity (solid black circle). The posterior samples and the number of studies were
provided as arguments to the SROC_rjags function (see Appendix 6 of the online
supplementary material (10.S1 Code for undertaking meta- analysis), Section A6.1).
The points on the plot represent sensitivity and specificity estimates in individual
studies scaled according to sample size. The plot is very similar to Chapter 9,
Figure9.4.a, obtained with a frequentist approach. As already observed in Figure9.4.a,
the prediction region is much larger than the credible region due to considerable het-
erogeneity in the sensitivity and specificity estimates across studies. The DTAplots
package uses the point estimates from the rjags output (Box10.2.i) in the equations
from Harbord (2007) to obtain the SROC plot.
●
SROC plot in RevMan: In order to create a similar figure in RevMan, in addition to the
point estimates of mu[1], mu[2], rho, tau.sq[1] and tau.sq[2] provided in the red and
blue boxes in Box10.2.i, the covariance between the posterior samples of the mean
logit- transformed sensitivity and mean logit- transformed specificity must be pro-
vided. This term can be computed by adding an extra line to the R script (see Appendix
6 of the online supplementary material (10.S1 Code for undertaking meta- analysis),
Section A6.1).
265
10 Undertaking meta- analysis
2
;
0
Sensitivity
Specificity
0.0
https://t.me/medicina_free
1.
0.80.60.4
0.20.0
1. 0 0.8 0.6 0.4 0.2
Figure10.2.d SROC plot of anti- CCP for rheumatoid arthritis showing 95% credible and 95% prediction
regions
10.2.4.5 Sensitivity analyses
As already noted, Bayesian estimation of the bivariate 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. It has been shown that when the number of studies is small (fewer
than 10), results can be highly sensitive to the choice of prior distribution over the
between- study variability parameters (Spiegelhalter 2004). In Box10.2.h 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, tau[1] and tau[2].
10.3 Estimation ofa summary curve
As described in Chapter9, Section9.4.3, the focus of the HSROC model is the estimation
of a summary curve. When no covariates are included, the HSROC model has five
parameters: the accuracy parameter with mean Λ (capital lambda) and variance
the positivity (threshold) parameter with mean Θ (capital theta) and variance
the shape parameter β. Rheumatoid factor (RF) for diagnosis of rheumatoid arthritis
was used to illustrate the Rutter and Gatsonis HSROC model in Chapter9, Section9.4.4,
and is the example used in this section. The sensitivities and specificities of the 50
included studies are shown on a forest plot (Figure10.3.a) along with their thresholds
for defining test positivity and RF measurement method.
2
and
;
266
Study
Da
vis 1989
Sa
Banchuin 1
Gomes-D
Bas 2003
v
Aho 1
W
Bizzar
Gr
Nell 200
Schellek
de Bois 1
De
Anuradha 2005
Visser 1
Choi 2005
V
Kw
Suzuki 2003
Bombar
Vit
D
V
Du
Rantapaa-D
Soder
K
Thammanichanond 2005
Goldbach-Mans
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Hitchon 2004
Spir
Sw
Gir
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Lopez-Ho
Raza 200
Ja
Jonsson 1
Sa
Co
Quinn 2006
Y
Fe
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Lee 2003
Ca
Ber
Despr
TP
FP
FN
TN
ThresholdSensitivity (95% CI)Specificity (95% CI)
Method
Sensitivity (95% CI)
Specificity (95% CI)
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raux 2003
992
audrix 1994
an Leeuwen 1988
999
inkles 1989
o 2001
einer 2005
5
ens 2000
996
Rycke 2004
996
allbracht 2004
ok 2005
dieri 2004
tecoq 2004
Figure10.3.a Coupled forest plot of the sensitivity and specificity of rheumatoid factor for the diagnosis of rheumatoid arthritis. Studies on the
plot are sorted by threshold, method of measurement and sensitivity. ELISA, enzymerheumatoid arthritis. Source: Data taken from Nishimura 2007.
as 2004
asiliauskiene 2001
amali 2005
oung 1991
bucquoi 2004
ahlqvist 2003
lin 2004
ky 2000
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itus 2004
edler
elli 2004
uerland 2005
yos 2004
5
nsen 2003
998
raux 1995
rdonnier 1996
rnandez-Suarez 2005
tecoq 2001
rpenter 1989
thelot 1995
es 1994
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linked immunosorbent assay; LA,latex agglutination; RA,
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