gmeta: Meta-Analysis via a Unified Framework under Confidence...
In gmeta: Meta-Analysis via a Unified Framework of Confidence Distribution
Description
Usage
Arguments
Details
Value
Note
Author(s)
References
See Also
Examples
Description
A unified method for meta-analysis includes combining p-values, fitting meta-analysis fixed-effect and random-effects models, and synthesizing 2x2 tables evidence all under a framework of combining confidence distributions (CDs). The function produces an object of class gmeta with associated functions print, summary, and plot.
Usage
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gmeta(gmi, gmi.type = c('pivot', 'cd', 'pvalue', '2x2'),
method = c('fixed-mle',
'fixed-robust1', 'fixed-robust2', 'fixed-robust2(sqrt12)',
'random-mm', 'random-reml', 'random-tau2',
'random-robust1', 'random-robust2', 'random-robust2(sqrt12)',
'fisher', 'normal', 'stouffer', 'min', 'tippett', 'max', 'sum',
'MH', 'Mantel-Haenszel', 'Peto', 'exact1', 'exact2'),
linkfunc = c('inverse-normal-cdf', 'inverse-laplace-cdf'),
weight = NULL, study.names = NULL, gmo.xgrid = NULL, ci.level = 0.95,
tau2 = NULL, mc.iteration = 10000, eta = 'Inf', verbose = FALSE,
report.error = FALSE)
Arguments
gmi
input, see ‘Details’.
gmi.type
type of input, including 'pivot', 'cd', 'pvalue', and '2x2', see ‘Details’.
method
method used for meta-analysis, including 'fisher', 'normal', 'stouffer', 'min', 'tippett', 'max', and 'sum' for combining p-values; 'fixed-mle', 'fixed-robust1', 'fixed-robust2', 'fixed-robust2(sqrt12)' for fixed-effect meta-analysis models; 'random-mm', 'random-reml', 'random-tau2', 'random-robust1', 'random-robust2', and 'random-robust2(sqrt12)' for random-effects meta-analysis models; and 'MH', 'Mantel-Haenszel', 'Peto', 'exact1', and 'exact2' for synthesizing 2x2 tables, see ‘Details’.
linkfunc
the link function selected for a user specified combination method, see Yang et al. (2013) and ‘Details’.
weight
a vector of user-specified weights for each study. If NULL, the default value depends on the method and linkfunc used for meta-analysis.
study.names
a vector of strings to give a user-specified study name for each study. If NULL, the default will be 'study-1', ..., 'study-k'.
gmo.xgrid
the position to evaluate a combined CD. The output will be reported as empirical cumulative density function (ECDF) on the points specified by gmo.xgrid. If NULL, the default will be gmo.xgrid = seq(from=-1,to=1,by=0.001).
ci.level
the confidence level for confidence intervals.
tau2
a numeric value to provide the heterogeneity estimation or a string to specified the method used to estimate the heterogeneity, see ‘Details’.
mc.iteration
number of iterations to compute the error of coverage probability of the computed confidence interval in 2x2-"exact1" method.
eta
a numeric vector for confidence levels of the one-sided confidence intervals for combining 2x2 tables used in "exact2" method. For example, set eta=Inf to indicate all confidence levels, or set eta=seq(from=0.05,to=0.95,length=23).
verbose
a logical value indicating whether detailed combining information is produced.
report.error
a logical value indicating whether the exact error of coverage probability of the computed confidence interval for 2x2-"exact1" and 2x2-"exact2" method is reported
Details
gmi
The format of gmi depends on the value of gmi.type (see below in this section).
For a classical p-value combination, gmi is a vector of p-values for testing the same hypothesis. For example, gmi=c(0.02,0.03,0.14) with gmi.type='pvalue'.
For model-based meta-analysis, gmi is a list of CDs if gmi.type='cd' with gmi.type='cd' (i.e., x=seq(from=-10,to=10,by=0.001), gmi=rbind(pnorm(x,2,3), pt(x-1,4), pgamma(x-1,1,1))); gmi is a matrix or a data.frame two columns of mean and standard deviations with gmi.type='pivot' (i.e., gmi=data.frame(mean=c(2,1,1),sd=(3,4,1))).
Note that the input confidence distributions should be as complete as possible, which means the distributions should start from almost 0 and end at almost 1, though it is fine if not so. If the distributions are not all specified under different grids, interpolation will be used for interpolating the corresponding probabilities, the closest point values will used for extrapolating the probabilities outside of the original range (see approx). If summary statistics is provided, the corresponding confidence distribution is generated by normal approximation and within 4 standard deviations range symmetric around means.
For 2x2 table-based (log) odds ratio/risk difference combination, gmi is a matrix of Kx4, where K is the number of trials. The first and third column are number of events in case and control group respectively. The second and fourth column are marginal total of case and control group respectively.
gmi.type
gmi.type is a string specifying the type of input data set. The choices are pvalue for classical p-value combination, cd for model-based meta-analysis using a list of CDs, pivot for model-based meta-analysis using summary statistics (means and standard deviations), and pivot for 2x2 table-based (log) odds ratio/risk difference combination.
method
method is a string specifying the method used for meta-analysis.
For classical p-value combination, choices are fisher, normal, stouffer, tippett, min, max, and sum.
For model-based meta-analysis, choices are fixed-mle, fixed-robust1, fixed-robust2,
fixed-robust2(sqrt12), random-mm, random-reml, random-tau2, random-robust1,
random-robust2, and random-robust2(sqrt12).
For 2x2 table-based (log) odds ratio/risk difference combination, choices are exact1, exact2, Mantel-Haenszel, MH, and Peto.
linkfunc
linkfunc is the link function used for combining studies.
The choice of inverse-normal-cdf covers most elementary model-based meta-analysis, and achieves the Fisher efficiency asymptotically.
The choice of inverse-laplace-cdf is more robust and achieves Bahadur efficiency.
The default option of linkfunc is inverse-normal-cdf for model-based meta-analysis, and null for p-value or 2x2 table combination.
tau2
tau2 is either a numeric value for estimating heterogeneity, or a string specifying the method to estimate heterogeneity.
tau2 is only for meta-analysis random-effects models (with method=random-mm, random-reml, random-tau2, random-robust1, random-robust2, or random-robust2(sqrt12))).
Choices for tau2 are DL, HS, SJ, HE, ML, REML and EB for DerSimonian-Laird, Hedges, Sidik-Jonkman, Hunter-Schmidt, Maximum-Likelihood, Restricted-Maximum-Likelihood, and Empirical-Bayesian estimator, respectively.
Value
An object of class gmeta, which has information of the combined inference (summarized in a CD form).
For p-value combination, it is a list of individual.pvalues, method, and combined.pvalue.
For model-based meta-analysis and 2x2 combination, it is a list of x.grids, individual.cds, individual.means, individual.stddevs, individual.medians, individual.cis, combined.cd, combined.density, combined.mean, combined.sd, combined.median, individual.ci, method, linkfunc, weight, tau2, ci.level, etc.
Note
Revised on 2014/12/10.
Author(s)
Guang Yang <gyang.rutgers@gmail.com>, Jerry Q. Cheng <jcheng18@nyit.edu>, and Minge Xie <mxie@stat.rutgers.edu>
References
Efron, B. (1996)
Empirical Bayes Methods for Combining Likelihoods.
Journal of the American Statistical Association, 91 538–550.
Liu, D., Liu, R. and Xie, M. (2014)
Exact meta-analysis approach for discrete data and its application to 2x2 tables with rare events.
Journal of the American Statistical Assocation, 109 1450-1465.
Mantel, N. and Haenszel, W. (1959)
Statistical aspects of the analysis of data from retrospective studies of disease.
Journal of the National Cancer Institute, 22 719-748.
Robins, J., Breslow, N. and Greenland, S. (1986)
Estimators of the Mantel-Haenszel Variance Consistent in Both Sparse Data and Large-Strata Limiting Models.
Biometrics, 42 311-323.
Tian, L., Cai, T., Pfeffer, M. A., Piankov, N., Cremieux, P.-Y., and Wei, L. J. (2009)
Exact and efficient inference procedure for meta-analysis and its application to the analysis of independent 2x2 tables with all available data but without artifficial continuity correction.
Biostatistics, 10 275-281.
Xie, M. and Singh, K. (2013)
Confidence distribution, the frequentist distribution estimator of a parameter (with discussions).
International Statistical Review, 81 3-39.
Xie, M., Singh, K., and Strawderman, W. E. (2011).
Confidencedence distributions and a unifying framework for meta-analysis.
Journal of the American Statistical Association, 106 320-333.
Yang, G., Liu, D., Wang, J. and Xie, M. (2016).
Meta-analysis framework for exact inferences with application to the analysis of rare events.
Biometrics, 72 1378-1386.
Yusuf, S., Peto, R., Lewis, J., Colins, R. and Sleight, P. (1985)
Beta blockade during and after myocardial infarction-An overview of randomized trials.
Progress in Cardiovascular Disease, 27 335-371.
See Also
Examples
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#### gmeta: generalized meta-analysis approach ####
data(ulcer)
ulcer.o <- as.matrix(ulcer)
# p-value combination #
# impute 0.5
ulcer <- ifelse(ulcer.o==0, 0.5, ulcer.o)
# summary statistics
ulcer.theta <- log( (ulcer[,1]*ulcer[,4]) / (ulcer[,2]*ulcer[,3]) )
ulcer.sigma <- sqrt(1/ulcer[,1] + 1/ulcer[,2] + 1/ulcer[,3] + 1/ulcer[,4])
# p-values from individual studies for K0: LOR >=0 vs. Ka: LOR < 0
ulcer.pvalues <- pnorm(ulcer.theta, mean=0, sd=ulcer.sigma)
# p-value combination using gmeta function
gmo.pvalue <- gmeta(ulcer.pvalues, gmi.type='pvalue', method='normal')
gmo.pvalue <- gmeta(ulcer.pvalues, gmi.type='pvalue', method='tippett')
print(gmo.pvalue)
summary(gmo.pvalue)
# model-based meta-analysis #
# data.frame of summary statistics
ulcer.pivots <- data.frame(mns=ulcer.theta, sds=ulcer.sigma)
# fixed-effect model
gmo.mdlfx <- gmeta(ulcer.pivots, method='fixed-mle', gmo.xgrid=seq(from=-10,to=10,by=0.01))
summary(gmo.mdlfx)
# random-effects model, method of moments
gmo.mdlrm <- gmeta(ulcer.pivots, method='random-tau2', weight=rep(1,41), tau2=2,
gmo.xgrid=seq(from=-10,to=10,by=0.01))
summary(gmo.mdlrm)
# plot of the gmeta output - forest plot of CDs
plot(gmo.mdlrm, studies=c(4,8,15,16,23,41)) # default: confidence-distribution-density
plot(gmo.mdlrm, studies=c(4,8,15,16,23,41), plot.option='cv') # using confidence-curve
# 2x2 table-based (log) odds ratio/risk difference combination #
# MH odd-ratio (OR) and Peto's log-odd-ratio (LOR)
ulcer.2x2 <- cbind(ulcer[,1], ulcer[,1]+ulcer[,2], ulcer[,3], ulcer[,3]+ulcer[,4])
# Mantel-Haenszel odd-ratio
gmo.2x2MH <- gmeta(ulcer.2x2, gmi.type='2x2', method='MH', gmo.xgrid=seq(-5,5,by=0.001))
summary(gmo.2x2MH)
plot(gmo.2x2MH, studies=c(4,8,15,16,23,41))
# Peto's log-odd-ratio
gmo.2x2Pt <- gmeta(ulcer.2x2, gmi.type='2x2', method='Peto', gmo.xgrid=seq(-5,5,by=0.001))
summary(gmo.2x2Pt)
plot(gmo.2x2Pt, studies=c(4,8,15,16,23,41))
# Exact meta-analysis on LOR based on Liu et al (2012) and RD based on Tian et al (2009)
ulcer.exact <- cbind(ulcer.o[,1], ulcer.o[,1]+ulcer.o[,2], ulcer.o[,3], ulcer.o[,3]+ulcer.o[,4])
# Exact meta-analysis on log-odd-ratio (LOR) based on Liu et al (2012)
#gmo.exact1 <- gmeta(ulcer.exact, gmi.type='2x2', method='exact1',
# gmo.xgrid=seq(-5,5,by=0.001), report.error=TRUE) # log-odd-ratio
#summary(gmo.exact1)
#plot(gmo.exact1, studies=c(4,8,15,16,23,41))
# Exact meta-analysis on risk difference (RD) based on Tian et al (2009)
#gmo.exact2 <- gmeta(ulcer.exact, gmi.type='2x2', method='exact2',
# gmo.xgrid=seq(-1,1,by=0.001), report.error=TRUE) # risk-difference
#summary(gmo.exact2)
#plot(gmo.exact2, studies=c(4,8,15,16,23,41), plot.option='cv')
Example output
Loading required package: BiasedUrn
Loading required package: binom
P-value combination through CD-Framework
Call:
gmeta.default(gmi = ulcer.pvalues, gmi.type = "pvalue", method = "tippett")
Combine Method: tippett
Combined p-value: 0.0005746768
Individual p-values:
[1] 2.364514e-02 3.204119e-01 7.170408e-01 7.631117e-01 9.045478e-01
[6] 1.435593e-01 2.786246e-02 2.963888e-04 2.499260e-01 1.713687e-03
[11] 3.850228e-02 3.061835e-02 8.296315e-01 8.454856e-01 3.066366e-02
[16] 1.417192e-01 6.571353e-03 7.575993e-02 3.877901e-03 1.602023e-02
[21] 2.165047e-04 8.145715e-01 6.764660e-01 2.568914e-01 8.483522e-03
[26] 1.027274e-01 4.056633e-01 1.735560e-03 1.929439e-02 1.113040e-02
[31] 8.729020e-03 5.737073e-02 1.024082e-01 5.638801e-02 7.908861e-01
[36] 5.038550e-03 6.858163e-01 9.294398e-02 1.960676e-02 1.402044e-05
[41] 6.108883e-01
P-value combination through CD-Framework
Call:
gmeta.default(gmi = ulcer.pvalues, gmi.type = "pvalue", method = "tippett")
Combine Method: tippett
Combined p-value: 0.0005746768
Individual p-values:
[1] 2.364514e-02 3.204119e-01 7.170408e-01 7.631117e-01 9.045478e-01
[6] 1.435593e-01 2.786246e-02 2.963888e-04 2.499260e-01 1.713687e-03
[11] 3.850228e-02 3.061835e-02 8.296315e-01 8.454856e-01 3.066366e-02
[16] 1.417192e-01 6.571353e-03 7.575993e-02 3.877901e-03 1.602023e-02
[21] 2.165047e-04 8.145715e-01 6.764660e-01 2.568914e-01 8.483522e-03
[26] 1.027274e-01 4.056633e-01 1.735560e-03 1.929439e-02 1.113040e-02
[31] 8.729020e-03 5.737073e-02 1.024082e-01 5.638801e-02 7.908861e-01
[36] 5.038550e-03 6.858163e-01 9.294398e-02 1.960676e-02 1.402044e-05
[41] 6.108883e-01
Model-Based Meta-Analysis through CD-Framework
Call:
gmeta.default(gmi = ulcer.pivots, method = "fixed-mle", gmo.xgrid = seq(from = -10,
to = 10, by = 0.01))
Summary of Combined CD:
mean median stddev ci.lower ci.upper
Combined CD -0.8875844 -0.8875834 0.125629 -1.133847 -0.641408
Confidence level = 0.95
Summary of Individual CDs:
mean median stddev ci.lower ci.upper
study-01 -1.8382795 -1.8382795 0.9266964 -3.6545973 -0.02197111
study-02 -0.3184537 -0.3184537 0.6825753 -1.6563103 1.01937778
study-03 0.4111958 0.4111958 0.7162780 -0.9927103 1.81510908
study-04 0.4881568 0.4881568 0.6814521 -0.8474918 1.82381222
study-05 2.0794415 2.0794415 1.5898987 -1.0367162 5.19560086
study-06 -1.7917595 -1.7917595 1.6832508 -5.0908751 1.50736284
study-07 -1.3457091 -1.3457091 0.7033884 -2.7243592 0.03293567
study-08 -4.1743873 -4.1743873 1.2152872 -6.5563253 -1.79245309
study-09 -0.5371429 -0.5371429 0.7960944 -2.0974825 1.02320007
study-10 -2.3795461 -2.3795461 0.8130874 -3.9731943 -0.78589499
study-11 -2.7725887 -2.7725887 1.5679073 -5.8456460 0.30045585
study-12 -2.1690537 -2.1690537 1.1588171 -4.4402960 0.10220062
study-13 0.5996211 0.5996211 0.6293847 -0.6339875 1.83322643
study-14 0.6931472 0.6931472 0.6813851 -0.6423691 2.02865436
study-15 -1.3458030 -1.3458030 0.7192468 -2.7555346 0.06392735
study-16 -0.9650809 -0.9650809 0.8997354 -2.7285435 0.79838292
study-17 -2.7725887 -2.7725887 1.1180340 -4.9639159 -0.58127260
study-18 -1.6519975 -1.6519975 1.1518731 -3.9096303 0.60565314
study-19 -1.7272209 -1.7272209 0.6487167 -2.9986995 -0.45572268
study-20 -1.1143606 -1.1143607 0.5197807 -2.1331526 -0.09556274
study-21 -2.2201089 -2.2201089 0.6308736 -3.4566332 -0.98358353
study-22 0.6641596 0.6641597 0.7421862 -0.7905048 2.11883153
study-23 0.1959978 0.1959977 0.4280930 -0.6430978 1.03510185
study-24 -0.4307829 -0.4307829 0.6597397 -1.7238841 0.86230932
study-25 -4.9698133 -4.9698133 2.0816660 -9.0498046 -0.88982209
study-26 -1.5040774 -1.5040774 1.1879020 -3.8323372 0.82418777
study-27 -0.1823216 -0.1823216 0.7637626 -1.6792774 1.31465758
study-28 -4.7874917 -4.7874917 1.6380883 -7.9980951 -1.57688482
study-29 -3.0910425 -3.0910425 1.4943074 -6.0198322 -0.16224238
study-30 -1.9841314 -1.9841314 0.8679914 -3.6853913 -0.28287627
study-31 -2.7932080 -2.7932080 1.1751393 -5.0964578 -0.48997711
study-32 -0.9162907 -0.9162907 0.5809475 -2.0549691 0.22237587
study-33 -1.2527630 -1.2527630 0.9880235 -3.1892603 0.68375077
study-34 -2.4079456 -2.4079456 1.5184056 -5.3839813 0.56808469
study-35 0.7672552 0.7672551 0.9478141 -1.0904307 2.62496258
study-36 -1.5040774 -1.5040774 0.5845226 -2.6497252 -0.35841189
study-37 0.4795731 0.4795731 0.9908002 -1.4623775 2.42151843
study-38 -0.9852836 -0.9852836 0.7448234 -2.4451435 0.47457608
study-39 -1.1050848 -1.1050848 0.5359444 -2.1555617 -0.05460758
study-40 -8.4390154 -8.4390154 2.0146522 NA -4.49036791
study-41 0.5753641 0.5753641 2.0429418 -3.4287335 4.57945891
Confidence level = 0.95
Model-Based Meta-Analysis through CD-Framework
Call:
gmeta.default(gmi = ulcer.pivots, method = "random-tau2", weight = rep(1,
41), gmo.xgrid = seq(from = -10, to = 10, by = 0.01), tau2 = 2)
Summary of Combined CD:
mean median stddev ci.lower ci.upper
Combined CD -1.294801 -1.294801 0.2709415 -1.825862 -0.7637425
Confidence level = 0.95
Summary of Individual CDs:
mean median stddev ci.lower ci.upper
study-01 -1.8382795 -1.8382795 0.9266964 -5.152174 1.47561963
study-02 -0.3184537 -0.3184537 0.6825753 -3.396242 2.75932347
study-03 0.4111958 0.4111958 0.7162780 -2.695875 3.51826023
study-04 0.4881568 0.4881568 0.6814521 -2.588667 3.56498938
study-05 2.0794415 2.0794415 1.5898987 -2.091087 6.24996587
study-06 -1.7917595 -1.7917595 1.6832508 -6.100713 2.51719963
study-07 -1.3457091 -1.3457091 0.7033884 -4.441439 1.75001379
study-08 -4.1743873 -4.1743873 1.2152872 -7.829040 -0.51973727
study-09 -0.5371429 -0.5371429 0.7960944 -3.717955 2.64367359
study-10 -2.3795461 -2.3795461 0.8130874 -5.576831 0.81773648
study-11 -2.7725887 -2.7725887 1.5679073 -6.911011 1.36584060
study-12 -2.1690537 -2.1690537 1.1588171 -5.752561 1.41445616
study-13 0.5996211 0.5996211 0.6293847 -2.434305 3.63354660
study-14 0.6931472 0.6931472 0.6813851 -2.383627 3.76990775
study-15 -1.3458030 -1.3458030 0.7192468 -4.455507 1.76390071
study-16 -0.9650809 -0.9650809 0.8997354 -4.250303 2.32013984
study-17 -2.7725887 -2.7725887 1.1180340 -6.305977 0.76079055
study-18 -1.6519975 -1.6519975 1.1518731 -5.226895 1.92289950
study-19 -1.7272209 -1.7272209 0.6487167 -4.776748 1.32230353
study-20 -1.1143606 -1.1143606 0.5197807 -4.067468 1.83874187
study-21 -2.2201089 -2.2201089 0.6308736 -5.255223 0.81500564
study-22 0.6641596 0.6641596 0.7421862 -2.466182 3.79450198
study-23 0.1959978 0.1959978 0.4280930 -2.700020 3.09202596
study-24 -0.4307829 -0.4307829 0.6597397 -3.489371 2.62781202
study-25 -4.9698133 -4.9698133 2.0816660 -9.902288 -0.03733784
study-26 -1.5040774 -1.5040774 1.1879020 -5.123986 2.11583093
study-27 -0.1823216 -0.1823216 0.7637626 -3.332534 2.96788997
study-28 -4.7874917 -4.7874917 1.6380883 -9.029056 -0.54592070
study-29 -3.0910425 -3.0910425 1.4943074 -7.123512 0.94142172
study-30 -1.9841314 -1.9841314 0.8679914 -5.236392 1.26812421
study-31 -2.7932080 -2.7932080 1.1751393 -6.397077 0.81065280
study-32 -0.9162907 -0.9162907 0.5809475 -3.912870 2.08027736
study-33 -1.2527630 -1.2527630 0.9880235 -4.634036 2.12850338
study-34 -2.4079456 -2.4079456 1.5184056 -6.474850 1.65895130
study-35 0.7672552 0.7672551 0.9478141 -2.569497 4.10401867
study-36 -1.5040774 -1.5040774 0.5845226 -4.503328 1.49517451
study-37 0.4795731 0.4795731 0.9908002 -2.904820 3.86396548
study-38 -0.9852836 -0.9852836 0.7448234 -4.118026 2.14746049
study-39 -1.1050848 -1.1050848 0.5359444 -4.069263 1.85909405
study-40 -8.4390154 -8.4390154 2.0146522 NA -3.61461736
study-41 0.5753641 0.5753641 2.0429418 -4.294520 5.44524799
Confidence level = 0.95
Exact Meta-Analysis Approach through CD-Framework
Call:
gmeta.default(gmi = ulcer.2x2, gmi.type = "2x2", method = "MH",
gmo.xgrid = seq(-5, 5, by = 0.001))
Summary of Combined CD:
mean median stddev ci.lower ci.upper
Combined CD 0.3452371 0.3452371 0.03818769 0.2703906 0.4200836
Confidence level = 0.95
Summary of Individual CDs:
mean median stddev ci.lower ci.upper
study-01 0.159090909 0.159090909 1.474290e-01 -1.298646e-01 0.448046389
study-02 0.727272727 0.727272727 4.964184e-01 -2.456895e-01 1.700234966
study-03 1.508620690 1.508620690 1.080592e+00 -6.093004e-01 3.626541749
study-04 1.629310345 1.629310345 1.110297e+00 -5.468316e-01 3.805452330
study-05 8.000000000 8.000000000 1.271919e+01 -1.692915e+01 32.929153043
study-06 0.166666667 0.166666667 2.805418e-01 -3.831852e-01 0.716518498
study-07 0.260355030 0.260355030 1.831307e-01 -9.857456e-02 0.619284622
study-08 0.015384615 0.015384615 1.869673e-02 -2.126030e-02 0.052029526
study-09 0.584415584 0.584415584 4.652500e-01 -3.274576e-01 1.496288734
study-10 0.092592593 0.092592593 7.528587e-02 -5.496500e-02 0.240150186
study-11 0.062500000 0.062500000 9.799421e-02 -1.295651e-01 0.254565116
study-12 0.114285714 0.114285714 1.324362e-01 -1.452846e-01 0.373855982
study-13 1.821428571 1.821428571 1.146379e+00 -4.254336e-01 4.068290709
study-14 2.000000000 2.000000000 1.362770e+00 -6.709807e-01 4.670980683
study-15 0.260330579 0.260330579 1.872419e-01 -1.066569e-01 0.627318049
study-16 0.380952381 0.380952381 3.427563e-01 -2.908377e-01 1.052742477
study-17 0.062500000 0.062500000 6.987712e-02 -7.445665e-02 0.199456647
study-18 0.191666667 0.191666667 2.207757e-01 -2.410457e-01 0.624379036
study-19 0.177777778 0.177777778 1.153274e-01 -4.825979e-02 0.403815348
study-20 0.328125000 0.328125000 1.705530e-01 -6.152809e-03 0.662402809
study-21 0.108597285 0.108597285 6.851116e-02 -2.568212e-02 0.242876693
study-22 1.942857143 1.942857143 1.441962e+00 -8.833359e-01 4.769050210
study-23 1.216524217 1.216524217 5.207855e-01 1.958034e-01 2.237245023
study-24 0.650000000 0.650000000 4.288308e-01 -1.904929e-01 1.490492874
study-25 0.006944444 0.006944444 1.445601e-02 -2.138882e-02 0.035277711
study-26 0.222222222 0.222222222 2.639782e-01 -2.951656e-01 0.739610021
study-27 0.833333333 0.833333333 6.364688e-01 -4.141227e-01 2.080789350
study-28 0.008333333 0.008333333 1.365074e-02 -1.842162e-02 0.035088284
study-29 0.045454545 0.045454545 6.792306e-02 -8.767221e-02 0.178581302
study-30 0.137500000 0.137500000 1.193488e-01 -9.641939e-02 0.371419387
study-31 0.061224490 0.061224490 7.194730e-02 -7.978964e-02 0.202238615
study-32 0.400000000 0.400000000 2.323790e-01 -5.545447e-02 0.855454472
study-33 0.285714286 0.285714286 2.822924e-01 -2.675687e-01 0.838997290
study-34 0.090000000 0.090000000 1.366565e-01 -1.778418e-01 0.357841825
study-35 2.153846154 2.153846154 2.041446e+00 -1.847314e+00 6.155006486
study-36 0.222222222 0.222222222 1.298939e-01 -3.236516e-02 0.476809609
study-37 1.615384615 1.615384615 1.600523e+00 -1.521584e+00 4.752352749
study-38 0.373333333 0.373333333 2.780674e-01 -1.716688e-01 0.918335432
study-39 0.331182796 0.331182796 1.774956e-01 -1.670213e-02 0.679067722
study-40 0.000216263 0.000216263 4.356947e-04 -6.376829e-04 0.001070209
study-41 1.777777778 1.777777778 3.631896e+00 -5.340609e+00 8.896164104
Confidence level = 0.95
Exact Meta-Analysis Approach through CD-Framework
Call:
gmeta.default(gmi = ulcer.2x2, gmi.type = "2x2", method = "Peto",
gmo.xgrid = seq(-5, 5, by = 0.001))
Summary of Combined CD:
mean median stddev ci.lower ci.upper
Combined CD -1.103918 -1.103918 0.1087616 -1.317087 -0.8907491
Confidence level = 0.95
Summary of Individual CDs:
mean median stddev ci.lower ci.upper
study-01 -1.5938462 -1.5938462 0.7765803 -3.1159155 -0.07177678
study-02 -0.3090374 -0.3090374 0.6713267 -1.6248135 1.00673875
study-03 0.4060143 0.4060143 0.7087713 -0.9831518 1.79518049
study-04 0.4695661 0.4695661 0.6564843 -0.8171195 1.75625156
study-05 1.6450000 1.6450000 1.1311253 -0.5719649 3.86196485
study-06 -1.3690476 -1.3690476 1.2547529 -3.8283181 1.09022281
study-07 -1.2314975 -1.2314975 0.6355558 -2.4771640 0.01416899
study-08 -2.9523810 -2.9523810 0.7014724 -4.3272415 -1.57752036
study-09 -0.5034965 -0.5034965 0.7526178 -1.9786003 0.97160730
study-10 -1.9860197 -1.9860197 0.5955219 -3.1532213 -0.81881818
study-11 -1.9380252 -1.9380252 0.9396160 -3.7796386 -0.09641177
study-12 -1.8745276 -1.8745276 0.9836148 -3.8023772 0.05332203
study-13 0.5842057 0.5842057 0.6172550 -0.6255918 1.79400317
study-14 0.6764706 0.6764706 0.6667367 -0.6303093 1.98325049
study-15 -1.1863953 -1.1863953 0.6146468 -2.3910810 0.01829026
study-16 -0.9024793 -0.9024793 0.8537714 -2.5758406 0.77088187
study-17 -2.2800000 -2.2800000 0.8717798 -3.9886570 -0.57134301
study-18 -1.4457835 -1.4457835 0.9297790 -3.2681169 0.37654999
study-19 -1.5354596 -1.5354596 0.5532219 -2.6197546 -0.45116465
study-20 -1.0559006 -1.0559006 0.4844013 -2.0053096 -0.10649159
study-21 -2.1512710 -2.1512710 0.5771177 -3.2824009 -1.02014119
study-22 0.6946819 0.6946819 0.7745660 -0.8234396 2.21280336
study-23 0.1947820 0.1947820 0.4265764 -0.6412923 1.03085629
study-24 -0.4183673 -0.4183673 0.6468132 -1.6860980 0.84936328
study-25 -3.1242604 -3.1242604 1.0658774 -5.2133417 -1.03517900
study-26 -1.2280702 -1.2280702 0.9365858 -3.0637446 0.60760428
study-27 -0.1768271 -0.1768271 0.7522278 -1.6511665 1.29751235
study-28 -3.1072222 -3.1072222 0.7998264 -4.6748531 -1.53959134
study-29 -2.0378741 -2.0378741 0.7510356 -3.5098768 -0.56587142
study-30 -1.6699001 -1.6699001 0.6869704 -3.0163372 -0.32346289
study-31 -2.2292308 -2.2292308 0.8236878 -3.8436291 -0.61483241
study-32 -0.8799172 -0.8799172 0.5582449 -1.9740570 0.21422266
study-33 -1.1217949 -1.1217949 0.8861469 -2.8586110 0.61502123
study-34 -1.7072283 -1.7072283 0.9033785 -3.4778175 0.06336096
study-35 0.7189542 0.7189542 0.8892973 -1.0240364 2.46194491
study-36 -1.4820574 -1.4820574 0.5635457 -2.5865867 -0.37752818
study-37 0.4592593 0.4592593 0.9583937 -1.4191579 2.33767643
study-38 -1.0126966 -1.0126966 0.7556378 -2.4937195 0.46832630
study-39 -1.0528573 -1.0528573 0.5038894 -2.0404623 -0.06525227
study-40 -3.8277673 -3.8277673 0.4780412 -4.7647109 -2.89082368
study-41 0.5805423 0.5805423 2.0766800 -3.4896757 4.65076021
Confidence level = 0.95
gmeta documentation built on March 9, 2021, 9:06 a.m.
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Description Usage Arguments Details Value Note Author(s) References See Also Examples
Description
A unified method for meta-analysis includes combining p-values, fitting meta-analysis fixed-effect and random-effects models, and synthesizing 2x2 tables evidence all under a framework of combining confidence distributions (CDs). The function produces an object of class gmeta with associated functions print, summary, and plot.
Usage
1 2 3 4 5 6 7 8 9 10 11 | gmeta(gmi, gmi.type = c('pivot', 'cd', 'pvalue', '2x2'),
method = c('fixed-mle',
'fixed-robust1', 'fixed-robust2', 'fixed-robust2(sqrt12)',
'random-mm', 'random-reml', 'random-tau2',
'random-robust1', 'random-robust2', 'random-robust2(sqrt12)',
'fisher', 'normal', 'stouffer', 'min', 'tippett', 'max', 'sum',
'MH', 'Mantel-Haenszel', 'Peto', 'exact1', 'exact2'),
linkfunc = c('inverse-normal-cdf', 'inverse-laplace-cdf'),
weight = NULL, study.names = NULL, gmo.xgrid = NULL, ci.level = 0.95,
tau2 = NULL, mc.iteration = 10000, eta = 'Inf', verbose = FALSE,
report.error = FALSE)
|
Arguments
gmi |
input, see ‘Details’. |
gmi.type |
type of input, including 'pivot', 'cd', 'pvalue', and '2x2', see ‘Details’. |
method |
method used for meta-analysis, including 'fisher', 'normal', 'stouffer', 'min', 'tippett', 'max', and 'sum' for combining p-values; 'fixed-mle', 'fixed-robust1', 'fixed-robust2', 'fixed-robust2(sqrt12)' for fixed-effect meta-analysis models; 'random-mm', 'random-reml', 'random-tau2', 'random-robust1', 'random-robust2', and 'random-robust2(sqrt12)' for random-effects meta-analysis models; and 'MH', 'Mantel-Haenszel', 'Peto', 'exact1', and 'exact2' for synthesizing 2x2 tables, see ‘Details’. |
linkfunc |
the link function selected for a user specified combination method, see Yang et al. (2013) and ‘Details’. |
weight |
a vector of user-specified weights for each study. If NULL, the default value depends on the |
study.names |
a vector of strings to give a user-specified study name for each study. If NULL, the default will be 'study-1', ..., 'study-k'. |
gmo.xgrid |
the position to evaluate a combined CD. The output will be reported as empirical cumulative density function (ECDF) on the points specified by gmo.xgrid. If NULL, the default will be |
ci.level |
the confidence level for confidence intervals. |
tau2 |
a numeric value to provide the heterogeneity estimation or a string to specified the method used to estimate the heterogeneity, see ‘Details’. |
mc.iteration |
number of iterations to compute the error of coverage probability of the computed confidence interval in 2x2-"exact1" method. |
eta |
a numeric vector for confidence levels of the one-sided confidence intervals for combining 2x2 tables used in "exact2" method. For example, set |
verbose |
a logical value indicating whether detailed combining information is produced. |
report.error |
a logical value indicating whether the exact error of coverage probability of the computed confidence interval for 2x2-"exact1" and 2x2-"exact2" method is reported |
Details
gmi
The format of gmi depends on the value of gmi.type (see below in this section).
For a classical p-value combination, gmi is a vector of p-values for testing the same hypothesis. For example, gmi=c(0.02,0.03,0.14) with gmi.type='pvalue'.
For model-based meta-analysis, gmi is a list of CDs if gmi.type='cd' with gmi.type='cd' (i.e., x=seq(from=-10,to=10,by=0.001), gmi=rbind(pnorm(x,2,3), pt(x-1,4), pgamma(x-1,1,1))); gmi is a matrix or a data.frame two columns of mean and standard deviations with gmi.type='pivot' (i.e., gmi=data.frame(mean=c(2,1,1),sd=(3,4,1))).
Note that the input confidence distributions should be as complete as possible, which means the distributions should start from almost 0 and end at almost 1, though it is fine if not so. If the distributions are not all specified under different grids, interpolation will be used for interpolating the corresponding probabilities, the closest point values will used for extrapolating the probabilities outside of the original range (see approx). If summary statistics is provided, the corresponding confidence distribution is generated by normal approximation and within 4 standard deviations range symmetric around means.
For 2x2 table-based (log) odds ratio/risk difference combination, gmi is a matrix of Kx4, where K is the number of trials. The first and third column are number of events in case and control group respectively. The second and fourth column are marginal total of case and control group respectively.
gmi.type
gmi.type is a string specifying the type of input data set. The choices are pvalue for classical p-value combination, cd for model-based meta-analysis using a list of CDs, pivot for model-based meta-analysis using summary statistics (means and standard deviations), and pivot for 2x2 table-based (log) odds ratio/risk difference combination.
method
method is a string specifying the method used for meta-analysis.
For classical p-value combination, choices are fisher, normal, stouffer, tippett, min, max, and sum.
For model-based meta-analysis, choices are fixed-mle, fixed-robust1, fixed-robust2,
fixed-robust2(sqrt12), random-mm, random-reml, random-tau2, random-robust1,
random-robust2, and random-robust2(sqrt12).
For 2x2 table-based (log) odds ratio/risk difference combination, choices are exact1, exact2, Mantel-Haenszel, MH, and Peto.
linkfunc
linkfunc is the link function used for combining studies.
The choice of inverse-normal-cdf covers most elementary model-based meta-analysis, and achieves the Fisher efficiency asymptotically.
The choice of inverse-laplace-cdf is more robust and achieves Bahadur efficiency.
The default option of linkfunc is inverse-normal-cdf for model-based meta-analysis, and null for p-value or 2x2 table combination.
tau2
tau2 is either a numeric value for estimating heterogeneity, or a string specifying the method to estimate heterogeneity.
tau2 is only for meta-analysis random-effects models (with method=random-mm, random-reml, random-tau2, random-robust1, random-robust2, or random-robust2(sqrt12))).
Choices for tau2 are DL, HS, SJ, HE, ML, REML and EB for DerSimonian-Laird, Hedges, Sidik-Jonkman, Hunter-Schmidt, Maximum-Likelihood, Restricted-Maximum-Likelihood, and Empirical-Bayesian estimator, respectively.
Value
An object of class gmeta, which has information of the combined inference (summarized in a CD form).
For p-value combination, it is a list of individual.pvalues, method, and combined.pvalue.
For model-based meta-analysis and 2x2 combination, it is a list of x.grids, individual.cds, individual.means, individual.stddevs, individual.medians, individual.cis, combined.cd, combined.density, combined.mean, combined.sd, combined.median, individual.ci, method, linkfunc, weight, tau2, ci.level, etc.
Note
Revised on 2014/12/10.
Author(s)
Guang Yang <gyang.rutgers@gmail.com>, Jerry Q. Cheng <jcheng18@nyit.edu>, and Minge Xie <mxie@stat.rutgers.edu>
References
Efron, B. (1996) Empirical Bayes Methods for Combining Likelihoods. Journal of the American Statistical Association, 91 538–550.
Liu, D., Liu, R. and Xie, M. (2014) Exact meta-analysis approach for discrete data and its application to 2x2 tables with rare events. Journal of the American Statistical Assocation, 109 1450-1465.
Mantel, N. and Haenszel, W. (1959) Statistical aspects of the analysis of data from retrospective studies of disease. Journal of the National Cancer Institute, 22 719-748.
Robins, J., Breslow, N. and Greenland, S. (1986) Estimators of the Mantel-Haenszel Variance Consistent in Both Sparse Data and Large-Strata Limiting Models. Biometrics, 42 311-323.
Tian, L., Cai, T., Pfeffer, M. A., Piankov, N., Cremieux, P.-Y., and Wei, L. J. (2009) Exact and efficient inference procedure for meta-analysis and its application to the analysis of independent 2x2 tables with all available data but without artifficial continuity correction. Biostatistics, 10 275-281.
Xie, M. and Singh, K. (2013) Confidence distribution, the frequentist distribution estimator of a parameter (with discussions). International Statistical Review, 81 3-39.
Xie, M., Singh, K., and Strawderman, W. E. (2011). Confidencedence distributions and a unifying framework for meta-analysis. Journal of the American Statistical Association, 106 320-333.
Yang, G., Liu, D., Wang, J. and Xie, M. (2016). Meta-analysis framework for exact inferences with application to the analysis of rare events. Biometrics, 72 1378-1386.
Yusuf, S., Peto, R., Lewis, J., Colins, R. and Sleight, P. (1985) Beta blockade during and after myocardial infarction-An overview of randomized trials. Progress in Cardiovascular Disease, 27 335-371.
See Also
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 | #### gmeta: generalized meta-analysis approach ####
data(ulcer)
ulcer.o <- as.matrix(ulcer)
# p-value combination #
# impute 0.5
ulcer <- ifelse(ulcer.o==0, 0.5, ulcer.o)
# summary statistics
ulcer.theta <- log( (ulcer[,1]*ulcer[,4]) / (ulcer[,2]*ulcer[,3]) )
ulcer.sigma <- sqrt(1/ulcer[,1] + 1/ulcer[,2] + 1/ulcer[,3] + 1/ulcer[,4])
# p-values from individual studies for K0: LOR >=0 vs. Ka: LOR < 0
ulcer.pvalues <- pnorm(ulcer.theta, mean=0, sd=ulcer.sigma)
# p-value combination using gmeta function
gmo.pvalue <- gmeta(ulcer.pvalues, gmi.type='pvalue', method='normal')
gmo.pvalue <- gmeta(ulcer.pvalues, gmi.type='pvalue', method='tippett')
print(gmo.pvalue)
summary(gmo.pvalue)
# model-based meta-analysis #
# data.frame of summary statistics
ulcer.pivots <- data.frame(mns=ulcer.theta, sds=ulcer.sigma)
# fixed-effect model
gmo.mdlfx <- gmeta(ulcer.pivots, method='fixed-mle', gmo.xgrid=seq(from=-10,to=10,by=0.01))
summary(gmo.mdlfx)
# random-effects model, method of moments
gmo.mdlrm <- gmeta(ulcer.pivots, method='random-tau2', weight=rep(1,41), tau2=2,
gmo.xgrid=seq(from=-10,to=10,by=0.01))
summary(gmo.mdlrm)
# plot of the gmeta output - forest plot of CDs
plot(gmo.mdlrm, studies=c(4,8,15,16,23,41)) # default: confidence-distribution-density
plot(gmo.mdlrm, studies=c(4,8,15,16,23,41), plot.option='cv') # using confidence-curve
# 2x2 table-based (log) odds ratio/risk difference combination #
# MH odd-ratio (OR) and Peto's log-odd-ratio (LOR)
ulcer.2x2 <- cbind(ulcer[,1], ulcer[,1]+ulcer[,2], ulcer[,3], ulcer[,3]+ulcer[,4])
# Mantel-Haenszel odd-ratio
gmo.2x2MH <- gmeta(ulcer.2x2, gmi.type='2x2', method='MH', gmo.xgrid=seq(-5,5,by=0.001))
summary(gmo.2x2MH)
plot(gmo.2x2MH, studies=c(4,8,15,16,23,41))
# Peto's log-odd-ratio
gmo.2x2Pt <- gmeta(ulcer.2x2, gmi.type='2x2', method='Peto', gmo.xgrid=seq(-5,5,by=0.001))
summary(gmo.2x2Pt)
plot(gmo.2x2Pt, studies=c(4,8,15,16,23,41))
# Exact meta-analysis on LOR based on Liu et al (2012) and RD based on Tian et al (2009)
ulcer.exact <- cbind(ulcer.o[,1], ulcer.o[,1]+ulcer.o[,2], ulcer.o[,3], ulcer.o[,3]+ulcer.o[,4])
# Exact meta-analysis on log-odd-ratio (LOR) based on Liu et al (2012)
#gmo.exact1 <- gmeta(ulcer.exact, gmi.type='2x2', method='exact1',
# gmo.xgrid=seq(-5,5,by=0.001), report.error=TRUE) # log-odd-ratio
#summary(gmo.exact1)
#plot(gmo.exact1, studies=c(4,8,15,16,23,41))
# Exact meta-analysis on risk difference (RD) based on Tian et al (2009)
#gmo.exact2 <- gmeta(ulcer.exact, gmi.type='2x2', method='exact2',
# gmo.xgrid=seq(-1,1,by=0.001), report.error=TRUE) # risk-difference
#summary(gmo.exact2)
#plot(gmo.exact2, studies=c(4,8,15,16,23,41), plot.option='cv')
|
Example output
Loading required package: BiasedUrn
Loading required package: binom
P-value combination through CD-Framework
Call:
gmeta.default(gmi = ulcer.pvalues, gmi.type = "pvalue", method = "tippett")
Combine Method: tippett
Combined p-value: 0.0005746768
Individual p-values:
[1] 2.364514e-02 3.204119e-01 7.170408e-01 7.631117e-01 9.045478e-01
[6] 1.435593e-01 2.786246e-02 2.963888e-04 2.499260e-01 1.713687e-03
[11] 3.850228e-02 3.061835e-02 8.296315e-01 8.454856e-01 3.066366e-02
[16] 1.417192e-01 6.571353e-03 7.575993e-02 3.877901e-03 1.602023e-02
[21] 2.165047e-04 8.145715e-01 6.764660e-01 2.568914e-01 8.483522e-03
[26] 1.027274e-01 4.056633e-01 1.735560e-03 1.929439e-02 1.113040e-02
[31] 8.729020e-03 5.737073e-02 1.024082e-01 5.638801e-02 7.908861e-01
[36] 5.038550e-03 6.858163e-01 9.294398e-02 1.960676e-02 1.402044e-05
[41] 6.108883e-01
P-value combination through CD-Framework
Call:
gmeta.default(gmi = ulcer.pvalues, gmi.type = "pvalue", method = "tippett")
Combine Method: tippett
Combined p-value: 0.0005746768
Individual p-values:
[1] 2.364514e-02 3.204119e-01 7.170408e-01 7.631117e-01 9.045478e-01
[6] 1.435593e-01 2.786246e-02 2.963888e-04 2.499260e-01 1.713687e-03
[11] 3.850228e-02 3.061835e-02 8.296315e-01 8.454856e-01 3.066366e-02
[16] 1.417192e-01 6.571353e-03 7.575993e-02 3.877901e-03 1.602023e-02
[21] 2.165047e-04 8.145715e-01 6.764660e-01 2.568914e-01 8.483522e-03
[26] 1.027274e-01 4.056633e-01 1.735560e-03 1.929439e-02 1.113040e-02
[31] 8.729020e-03 5.737073e-02 1.024082e-01 5.638801e-02 7.908861e-01
[36] 5.038550e-03 6.858163e-01 9.294398e-02 1.960676e-02 1.402044e-05
[41] 6.108883e-01
Model-Based Meta-Analysis through CD-Framework
Call:
gmeta.default(gmi = ulcer.pivots, method = "fixed-mle", gmo.xgrid = seq(from = -10,
to = 10, by = 0.01))
Summary of Combined CD:
mean median stddev ci.lower ci.upper
Combined CD -0.8875844 -0.8875834 0.125629 -1.133847 -0.641408
Confidence level = 0.95
Summary of Individual CDs:
mean median stddev ci.lower ci.upper
study-01 -1.8382795 -1.8382795 0.9266964 -3.6545973 -0.02197111
study-02 -0.3184537 -0.3184537 0.6825753 -1.6563103 1.01937778
study-03 0.4111958 0.4111958 0.7162780 -0.9927103 1.81510908
study-04 0.4881568 0.4881568 0.6814521 -0.8474918 1.82381222
study-05 2.0794415 2.0794415 1.5898987 -1.0367162 5.19560086
study-06 -1.7917595 -1.7917595 1.6832508 -5.0908751 1.50736284
study-07 -1.3457091 -1.3457091 0.7033884 -2.7243592 0.03293567
study-08 -4.1743873 -4.1743873 1.2152872 -6.5563253 -1.79245309
study-09 -0.5371429 -0.5371429 0.7960944 -2.0974825 1.02320007
study-10 -2.3795461 -2.3795461 0.8130874 -3.9731943 -0.78589499
study-11 -2.7725887 -2.7725887 1.5679073 -5.8456460 0.30045585
study-12 -2.1690537 -2.1690537 1.1588171 -4.4402960 0.10220062
study-13 0.5996211 0.5996211 0.6293847 -0.6339875 1.83322643
study-14 0.6931472 0.6931472 0.6813851 -0.6423691 2.02865436
study-15 -1.3458030 -1.3458030 0.7192468 -2.7555346 0.06392735
study-16 -0.9650809 -0.9650809 0.8997354 -2.7285435 0.79838292
study-17 -2.7725887 -2.7725887 1.1180340 -4.9639159 -0.58127260
study-18 -1.6519975 -1.6519975 1.1518731 -3.9096303 0.60565314
study-19 -1.7272209 -1.7272209 0.6487167 -2.9986995 -0.45572268
study-20 -1.1143606 -1.1143607 0.5197807 -2.1331526 -0.09556274
study-21 -2.2201089 -2.2201089 0.6308736 -3.4566332 -0.98358353
study-22 0.6641596 0.6641597 0.7421862 -0.7905048 2.11883153
study-23 0.1959978 0.1959977 0.4280930 -0.6430978 1.03510185
study-24 -0.4307829 -0.4307829 0.6597397 -1.7238841 0.86230932
study-25 -4.9698133 -4.9698133 2.0816660 -9.0498046 -0.88982209
study-26 -1.5040774 -1.5040774 1.1879020 -3.8323372 0.82418777
study-27 -0.1823216 -0.1823216 0.7637626 -1.6792774 1.31465758
study-28 -4.7874917 -4.7874917 1.6380883 -7.9980951 -1.57688482
study-29 -3.0910425 -3.0910425 1.4943074 -6.0198322 -0.16224238
study-30 -1.9841314 -1.9841314 0.8679914 -3.6853913 -0.28287627
study-31 -2.7932080 -2.7932080 1.1751393 -5.0964578 -0.48997711
study-32 -0.9162907 -0.9162907 0.5809475 -2.0549691 0.22237587
study-33 -1.2527630 -1.2527630 0.9880235 -3.1892603 0.68375077
study-34 -2.4079456 -2.4079456 1.5184056 -5.3839813 0.56808469
study-35 0.7672552 0.7672551 0.9478141 -1.0904307 2.62496258
study-36 -1.5040774 -1.5040774 0.5845226 -2.6497252 -0.35841189
study-37 0.4795731 0.4795731 0.9908002 -1.4623775 2.42151843
study-38 -0.9852836 -0.9852836 0.7448234 -2.4451435 0.47457608
study-39 -1.1050848 -1.1050848 0.5359444 -2.1555617 -0.05460758
study-40 -8.4390154 -8.4390154 2.0146522 NA -4.49036791
study-41 0.5753641 0.5753641 2.0429418 -3.4287335 4.57945891
Confidence level = 0.95
Model-Based Meta-Analysis through CD-Framework
Call:
gmeta.default(gmi = ulcer.pivots, method = "random-tau2", weight = rep(1,
41), gmo.xgrid = seq(from = -10, to = 10, by = 0.01), tau2 = 2)
Summary of Combined CD:
mean median stddev ci.lower ci.upper
Combined CD -1.294801 -1.294801 0.2709415 -1.825862 -0.7637425
Confidence level = 0.95
Summary of Individual CDs:
mean median stddev ci.lower ci.upper
study-01 -1.8382795 -1.8382795 0.9266964 -5.152174 1.47561963
study-02 -0.3184537 -0.3184537 0.6825753 -3.396242 2.75932347
study-03 0.4111958 0.4111958 0.7162780 -2.695875 3.51826023
study-04 0.4881568 0.4881568 0.6814521 -2.588667 3.56498938
study-05 2.0794415 2.0794415 1.5898987 -2.091087 6.24996587
study-06 -1.7917595 -1.7917595 1.6832508 -6.100713 2.51719963
study-07 -1.3457091 -1.3457091 0.7033884 -4.441439 1.75001379
study-08 -4.1743873 -4.1743873 1.2152872 -7.829040 -0.51973727
study-09 -0.5371429 -0.5371429 0.7960944 -3.717955 2.64367359
study-10 -2.3795461 -2.3795461 0.8130874 -5.576831 0.81773648
study-11 -2.7725887 -2.7725887 1.5679073 -6.911011 1.36584060
study-12 -2.1690537 -2.1690537 1.1588171 -5.752561 1.41445616
study-13 0.5996211 0.5996211 0.6293847 -2.434305 3.63354660
study-14 0.6931472 0.6931472 0.6813851 -2.383627 3.76990775
study-15 -1.3458030 -1.3458030 0.7192468 -4.455507 1.76390071
study-16 -0.9650809 -0.9650809 0.8997354 -4.250303 2.32013984
study-17 -2.7725887 -2.7725887 1.1180340 -6.305977 0.76079055
study-18 -1.6519975 -1.6519975 1.1518731 -5.226895 1.92289950
study-19 -1.7272209 -1.7272209 0.6487167 -4.776748 1.32230353
study-20 -1.1143606 -1.1143606 0.5197807 -4.067468 1.83874187
study-21 -2.2201089 -2.2201089 0.6308736 -5.255223 0.81500564
study-22 0.6641596 0.6641596 0.7421862 -2.466182 3.79450198
study-23 0.1959978 0.1959978 0.4280930 -2.700020 3.09202596
study-24 -0.4307829 -0.4307829 0.6597397 -3.489371 2.62781202
study-25 -4.9698133 -4.9698133 2.0816660 -9.902288 -0.03733784
study-26 -1.5040774 -1.5040774 1.1879020 -5.123986 2.11583093
study-27 -0.1823216 -0.1823216 0.7637626 -3.332534 2.96788997
study-28 -4.7874917 -4.7874917 1.6380883 -9.029056 -0.54592070
study-29 -3.0910425 -3.0910425 1.4943074 -7.123512 0.94142172
study-30 -1.9841314 -1.9841314 0.8679914 -5.236392 1.26812421
study-31 -2.7932080 -2.7932080 1.1751393 -6.397077 0.81065280
study-32 -0.9162907 -0.9162907 0.5809475 -3.912870 2.08027736
study-33 -1.2527630 -1.2527630 0.9880235 -4.634036 2.12850338
study-34 -2.4079456 -2.4079456 1.5184056 -6.474850 1.65895130
study-35 0.7672552 0.7672551 0.9478141 -2.569497 4.10401867
study-36 -1.5040774 -1.5040774 0.5845226 -4.503328 1.49517451
study-37 0.4795731 0.4795731 0.9908002 -2.904820 3.86396548
study-38 -0.9852836 -0.9852836 0.7448234 -4.118026 2.14746049
study-39 -1.1050848 -1.1050848 0.5359444 -4.069263 1.85909405
study-40 -8.4390154 -8.4390154 2.0146522 NA -3.61461736
study-41 0.5753641 0.5753641 2.0429418 -4.294520 5.44524799
Confidence level = 0.95
Exact Meta-Analysis Approach through CD-Framework
Call:
gmeta.default(gmi = ulcer.2x2, gmi.type = "2x2", method = "MH",
gmo.xgrid = seq(-5, 5, by = 0.001))
Summary of Combined CD:
mean median stddev ci.lower ci.upper
Combined CD 0.3452371 0.3452371 0.03818769 0.2703906 0.4200836
Confidence level = 0.95
Summary of Individual CDs:
mean median stddev ci.lower ci.upper
study-01 0.159090909 0.159090909 1.474290e-01 -1.298646e-01 0.448046389
study-02 0.727272727 0.727272727 4.964184e-01 -2.456895e-01 1.700234966
study-03 1.508620690 1.508620690 1.080592e+00 -6.093004e-01 3.626541749
study-04 1.629310345 1.629310345 1.110297e+00 -5.468316e-01 3.805452330
study-05 8.000000000 8.000000000 1.271919e+01 -1.692915e+01 32.929153043
study-06 0.166666667 0.166666667 2.805418e-01 -3.831852e-01 0.716518498
study-07 0.260355030 0.260355030 1.831307e-01 -9.857456e-02 0.619284622
study-08 0.015384615 0.015384615 1.869673e-02 -2.126030e-02 0.052029526
study-09 0.584415584 0.584415584 4.652500e-01 -3.274576e-01 1.496288734
study-10 0.092592593 0.092592593 7.528587e-02 -5.496500e-02 0.240150186
study-11 0.062500000 0.062500000 9.799421e-02 -1.295651e-01 0.254565116
study-12 0.114285714 0.114285714 1.324362e-01 -1.452846e-01 0.373855982
study-13 1.821428571 1.821428571 1.146379e+00 -4.254336e-01 4.068290709
study-14 2.000000000 2.000000000 1.362770e+00 -6.709807e-01 4.670980683
study-15 0.260330579 0.260330579 1.872419e-01 -1.066569e-01 0.627318049
study-16 0.380952381 0.380952381 3.427563e-01 -2.908377e-01 1.052742477
study-17 0.062500000 0.062500000 6.987712e-02 -7.445665e-02 0.199456647
study-18 0.191666667 0.191666667 2.207757e-01 -2.410457e-01 0.624379036
study-19 0.177777778 0.177777778 1.153274e-01 -4.825979e-02 0.403815348
study-20 0.328125000 0.328125000 1.705530e-01 -6.152809e-03 0.662402809
study-21 0.108597285 0.108597285 6.851116e-02 -2.568212e-02 0.242876693
study-22 1.942857143 1.942857143 1.441962e+00 -8.833359e-01 4.769050210
study-23 1.216524217 1.216524217 5.207855e-01 1.958034e-01 2.237245023
study-24 0.650000000 0.650000000 4.288308e-01 -1.904929e-01 1.490492874
study-25 0.006944444 0.006944444 1.445601e-02 -2.138882e-02 0.035277711
study-26 0.222222222 0.222222222 2.639782e-01 -2.951656e-01 0.739610021
study-27 0.833333333 0.833333333 6.364688e-01 -4.141227e-01 2.080789350
study-28 0.008333333 0.008333333 1.365074e-02 -1.842162e-02 0.035088284
study-29 0.045454545 0.045454545 6.792306e-02 -8.767221e-02 0.178581302
study-30 0.137500000 0.137500000 1.193488e-01 -9.641939e-02 0.371419387
study-31 0.061224490 0.061224490 7.194730e-02 -7.978964e-02 0.202238615
study-32 0.400000000 0.400000000 2.323790e-01 -5.545447e-02 0.855454472
study-33 0.285714286 0.285714286 2.822924e-01 -2.675687e-01 0.838997290
study-34 0.090000000 0.090000000 1.366565e-01 -1.778418e-01 0.357841825
study-35 2.153846154 2.153846154 2.041446e+00 -1.847314e+00 6.155006486
study-36 0.222222222 0.222222222 1.298939e-01 -3.236516e-02 0.476809609
study-37 1.615384615 1.615384615 1.600523e+00 -1.521584e+00 4.752352749
study-38 0.373333333 0.373333333 2.780674e-01 -1.716688e-01 0.918335432
study-39 0.331182796 0.331182796 1.774956e-01 -1.670213e-02 0.679067722
study-40 0.000216263 0.000216263 4.356947e-04 -6.376829e-04 0.001070209
study-41 1.777777778 1.777777778 3.631896e+00 -5.340609e+00 8.896164104
Confidence level = 0.95
Exact Meta-Analysis Approach through CD-Framework
Call:
gmeta.default(gmi = ulcer.2x2, gmi.type = "2x2", method = "Peto",
gmo.xgrid = seq(-5, 5, by = 0.001))
Summary of Combined CD:
mean median stddev ci.lower ci.upper
Combined CD -1.103918 -1.103918 0.1087616 -1.317087 -0.8907491
Confidence level = 0.95
Summary of Individual CDs:
mean median stddev ci.lower ci.upper
study-01 -1.5938462 -1.5938462 0.7765803 -3.1159155 -0.07177678
study-02 -0.3090374 -0.3090374 0.6713267 -1.6248135 1.00673875
study-03 0.4060143 0.4060143 0.7087713 -0.9831518 1.79518049
study-04 0.4695661 0.4695661 0.6564843 -0.8171195 1.75625156
study-05 1.6450000 1.6450000 1.1311253 -0.5719649 3.86196485
study-06 -1.3690476 -1.3690476 1.2547529 -3.8283181 1.09022281
study-07 -1.2314975 -1.2314975 0.6355558 -2.4771640 0.01416899
study-08 -2.9523810 -2.9523810 0.7014724 -4.3272415 -1.57752036
study-09 -0.5034965 -0.5034965 0.7526178 -1.9786003 0.97160730
study-10 -1.9860197 -1.9860197 0.5955219 -3.1532213 -0.81881818
study-11 -1.9380252 -1.9380252 0.9396160 -3.7796386 -0.09641177
study-12 -1.8745276 -1.8745276 0.9836148 -3.8023772 0.05332203
study-13 0.5842057 0.5842057 0.6172550 -0.6255918 1.79400317
study-14 0.6764706 0.6764706 0.6667367 -0.6303093 1.98325049
study-15 -1.1863953 -1.1863953 0.6146468 -2.3910810 0.01829026
study-16 -0.9024793 -0.9024793 0.8537714 -2.5758406 0.77088187
study-17 -2.2800000 -2.2800000 0.8717798 -3.9886570 -0.57134301
study-18 -1.4457835 -1.4457835 0.9297790 -3.2681169 0.37654999
study-19 -1.5354596 -1.5354596 0.5532219 -2.6197546 -0.45116465
study-20 -1.0559006 -1.0559006 0.4844013 -2.0053096 -0.10649159
study-21 -2.1512710 -2.1512710 0.5771177 -3.2824009 -1.02014119
study-22 0.6946819 0.6946819 0.7745660 -0.8234396 2.21280336
study-23 0.1947820 0.1947820 0.4265764 -0.6412923 1.03085629
study-24 -0.4183673 -0.4183673 0.6468132 -1.6860980 0.84936328
study-25 -3.1242604 -3.1242604 1.0658774 -5.2133417 -1.03517900
study-26 -1.2280702 -1.2280702 0.9365858 -3.0637446 0.60760428
study-27 -0.1768271 -0.1768271 0.7522278 -1.6511665 1.29751235
study-28 -3.1072222 -3.1072222 0.7998264 -4.6748531 -1.53959134
study-29 -2.0378741 -2.0378741 0.7510356 -3.5098768 -0.56587142
study-30 -1.6699001 -1.6699001 0.6869704 -3.0163372 -0.32346289
study-31 -2.2292308 -2.2292308 0.8236878 -3.8436291 -0.61483241
study-32 -0.8799172 -0.8799172 0.5582449 -1.9740570 0.21422266
study-33 -1.1217949 -1.1217949 0.8861469 -2.8586110 0.61502123
study-34 -1.7072283 -1.7072283 0.9033785 -3.4778175 0.06336096
study-35 0.7189542 0.7189542 0.8892973 -1.0240364 2.46194491
study-36 -1.4820574 -1.4820574 0.5635457 -2.5865867 -0.37752818
study-37 0.4592593 0.4592593 0.9583937 -1.4191579 2.33767643
study-38 -1.0126966 -1.0126966 0.7556378 -2.4937195 0.46832630
study-39 -1.0528573 -1.0528573 0.5038894 -2.0404623 -0.06525227
study-40 -3.8277673 -3.8277673 0.4780412 -4.7647109 -2.89082368
study-41 0.5805423 0.5805423 2.0766800 -3.4896757 4.65076021
Confidence level = 0.95
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