gmeta: Meta-Analysis
via a Unified Framework under Confidence Distributiongmeta: Meta-Analysis
via a Unified Framework under Confidence DistributionMeta-analysis combines information from various sources to address a set of related hypotheses. It is a statistical method that is widely used in other scientific fields like biology, chemistry and psychology, as well as in studies that involve clinical trials. The huge amount of google hits, i.e., currently more than 22.6 millions, of "meta analysis" demonstrates its importance to some extent.
Confidence distribution (CD) utilizes a distribution instead of a point (point estimator) or an interval (confidence interval), to estimate the parameter of interest, that it bears a wealth of information for inference and it is a useful tool to combine information from different sources (Xie et al. 2011). We present an R package named "gmeta" that uses CD as a device to perform meta-analysis. As a meta-analysis tool, gmeta not only unifies almost all existing meta analysis procedures under the same framework but it also systemically organizes them and discloses their connections that lead to the development of new approaches.
We start by introducing the gmeta package, the usage of the gmeta function and a plot function associated with it. We then illustrate the CD approaches for meta-analysis through an example where we integrate information from 41 randomized trials of a new surgical treatment for stomach ulcers that were conducted between 1980 and 1989 (Sacks, Charlmers, Blum, Berrier, and Pagano 1990). This data was also used as an example in the illustration of the empirical Bayes approach for meta-analysis by Efron (Efron 1996). We comment on their performace and take this opportunity to bring out two new robust meta-analysis approaches that are developed only under the CD framework. Supporting theories and assumptions of the robust apporaches can be found at Xie et al. 2011.
The advantages of these CD approaches included:
gmeta: Meta-Analysis via a Unified Framework under Confidence Distribution
Unify the p-value combination and various model based meta-analysis methods under the same framework through confidence distribution approaches.
NOTE: The package is still under construction, the current version is only for testing purpose.
Downloads:
gmeta function:
gmeta(gmi, gmi.type = c('pivot','cd','pvalue'),
method =
c('fixed-mle','fixed-bayesian',
'fixed-robust1','fixed-robust2','fixed-robust2(sqrt12)',
'random-mm','random-reml',
'random-robust1','random-robust2','random-robust2(sqrt12)'),
F0 =
c('normal-cdf','double.exponential-cdf'),
tau2 = NULL, weights = NULL, xout =
NULL, n = 512)
plot function:
plot.gmeta.m(gmo, experiments=NULL, option=FALSE, type='l', xlab='x',
ylab='density', xlim=NULL, ...)
Details about the arguments and and result values can be found in reference manual meta.pdf.
1.1 data organization
> data(ulcer)
> ulcer <- as.matrix(ulcer)
> ulcer <- ifelse(ulcer == 0, 0.5, ulcer)
> 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])
# H0: mu < 0 vs. Ha: mu >= 0 #
> ulcer.pvalues <- 1 - pnorm(0, mean=ulcer.theta, sd=ulcer.sigma)
1.2 p-value combinations
# Fisher's method #
> gmeta(ulcer.pvalues, gmi.type='pvalue', method='fisher')
# Tippett's method #
> gmeta(ulcer.pvalues, gmi.type='pvalue', method='tippett')
# current available methods include: fisher, normal (stouffer), tippett
(min), max and sum.
2.1. data organization
# pivot type of input data
> ulcer.data = cbind(ulcer.theta, ulcer.sigma)
# cd type of input data
> ulcer.cdata <- list()
> for (i in 1:41) {
+ li <- ulcer.theta[i] - 4*ulcer.sigma[i]
+ ui <- ulcer.theta[i] + 4*ulcer.sigma[i]
+ xi = seq(from=li, to=ui, length.out = 1024+i)
+ cdi <- pnorm(xi, mean=ulcer.theta[i], sd=ulcer.sigma[i])
+ ulcer.cdata[[i]] = cbind(xi, cdi)
+ }
2.2 model-based meta-analysis
# random effect model - method of moments, using pivot typed data #
> gm1r <- gmeta(ulcer.data, method='random-mm', n=5120)
> summary(gm1r) # summary of the results
# random effect model - method of moments, using cd typed data #
> gm1rc <- gmeta(ulcer.cdata, gmi.type='cd', method='random-mm',
n=5120)
# fixed effect model - MLE method, both pivot typed and cd typed input
data #
> gm1f <- gmeta(ulcer.data, method='fixed-mle', xout=c(-5,4),
n=5120)
> gm1fc <- gmeta(ulcer.cdata, gmi.type='cd', method='fixed-mle',
n=5120)
> plot(gm1r, experiments=c(4,8,15,16,23,41), option=T)
# Figure 1. CD Contains Wealth of Information. #
# an artifical contaminated data set #
> ids <- c(1:41)[(ulcer.theta > 0.5)]
> ulcer.theta.fake <- (ulcer.theta>0.5)*(10*ulcer.theta) +
(ulcer.theta<0.5)*ulcer.theta
> ulcer.data.fake <- cbind(ulcer.theta.fake, ulcer.sigma)
# both fixed effects and random effects models are failed, however,
robust procedures enable the result still acceptable #
# Figure 2. Robust Method under the Unified Framework #
# While taking normal-cdf as the F0 in the CD combination would
usually achieves Fisher Efficiency and gives the shorest confidence
intervals, the CD-Framework is not limited as single choice for F0. In
fact, any monotonic increased function can be taken and result a
different approach. Another F0 to be considered is the double
exponential cdf, which guarantees the Bahadur Efficiency.#
# Figure 3. Robust Method and Using Double Exponential CDF as F0 #
Endoscopic Hemostasis, an Effective Therapy for Bleeding Peptic Ulcers. Sacks, H. S., Chalmers, T. C., Blum, A. L., Berrier, J., and Pagano, D. (1990), Journal of the American Medical Association, 264, 494- 499.
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