Further Meta-analysis

The initial meta-analyses in StatsDirect maybe used to steer further meta-analyses in other software, such as R or WinBUGS, with the guidance of an expert Statistician.

For open-ended modelling and simulation meta-analyses we recommend the R metafor package written by Wolfgang Viechtbauer.

The yi and vi inputs to metafor's advanced functions are equivalent to the "standardised effect" and "variance" outputs from StatsDirect.

You will notice some differences where StatsDirect uses more accurate variance formulae than the escalc function of metafor.

For example, try the odds ratio data from the test worksheet in R metafor as follows:

ttot = c(615,758,832,317,810,2267,8587)

tpos = c(49,44,102,32,85,246,1570)

tneg = ttot - tpos

ctot = c(624,771,850,309,406,2257,8600)

cpos = c(67,64,126,38,52,219,1720)

cneg = ctot-cpos

library(metafor)

dat = escalc(measure="OR", ai=tpos, bi=tneg, ci=cpos, di=cneg)

dat

res = rma(yi, vi, data=dat, method="FE")

The variance estimates in the metafor are made with the logit/Wolf approximation whereas StatsDirect uses an 'exact' alternative called the conditional maximum likelihood calculation, based on convolution of hyper-geometric distributions through polynomial root finding.

When you have yi and vi from either escalc or StatsDirect's standardised effect and variance results you can proceed with a wide range of models, including meta-regression with covariates, via the rma function (the simple fixed effects model is shown above).

You can also perform meta-regression in other software such as Stata. Meta-regression methods (such as the metareg macro for Stata) can be used to explore origins of heterogeneity in meta-analyses. To perform meta-regression you need: the standardised effect and variance for each study as above, plus covariates that might explain the origins of the differences between the studies. There are numerous ways to go wrong in the use and interpretation of meta-regression, therefore please seek the help of a statistician (Thompson and Higgins, 2002).

Some Statisticians might use WinBUGS to perform Bayesian simulations such as the one shown below:

A typical fixed effects odds ratio BUGS model:

model

{

for( i in 1 : Num ) {

rc[i] ~ dbin(pc[i], nc[i])

rt[i] ~ dbin(pt[i], nt[i])

logit(pc[i]) <- mu[i]

logit(pt[i]) <- mu[i] + d

mu[i] ~ dnorm(0.0,1.0E-6)

}

d ~ dnorm(0.0,1.0E-6)

or <- exp(d)

}

A typical random effects odds ratio BUGS model:

{

for( i in 1 : Num ) {

rc[i] ~ dbin(pc[i], nc[i])

rt[i] ~ dbin(pt[i], nt[i])

logit(pc[i]) <- mu[i]

logit(pt[i]) <- mu[i] + delta[i]

mu[i] ~ dnorm(0.0,1.0E-5)

delta[i] ~ dnorm(d, tau)

}

d ~ dnorm(0.0,1.0E-6)

# Choice of priors for random effects variance

#tau ~ dgamma(0.001,0.001)

#sigma <- 1 / sqrt(tau)

tau<-1/(sigma*sigma)

sigma~dunif(0,10)

delta.new ~ dnorm(d, tau)

or <- exp(d)

}