The Bayesian approach - Estimating the Heterogeneity Variance in a Random-Effects Meta-Analysis

I described the general framework for Bayesian meta-analysis in the introduction chapter (section 1.5.4). Bayesian heterogeneity variance estimators are based on this model and allow for prior beliefs of model parameters to be combined with meta- analysis data. This is the dening dierence from the frequentist estimators outlined in the rest of this chapter. I introduce the full Bayesian approach in section 2.6.1. Following this, I introduce a series of semi-Bayesian τ2 _{estimators that are more}

simple to compute including approximate Bayes (AB), empirical Bayes (EB), Bayes modal (BM) and estimators proposed by Rukhin [93].

2.6.1 Full Bayesian

The full Bayesian approach estimates the heterogeneity variance simultaneously with all other parameters of interest in the model. In doing so, it can account for un- certainty of these parameters [106]. In a Bayesian random-eect model with no covariates, we can dene prior distributions for τ2 _{and θ:}

τ2 ∼ p1(ϕ1)

θ ∼ p2(ϕ2)

where p1 and p2are the chosen probability distributions with xed parameter vectors

ϕ1 and ϕ2.

Prior distributions and their xed parameters vary between meta-analyses in practice and are chosen based on external evidence, expert opinion or they are vague to reect a lack of prior knowledge [111]. Therefore, it is not possible to dene a distinct full Bayesian method. Possible assumed distributions for τ2 include the inverse gamma,

uniform or normal [66, 88]. A normal distribution for θi is often assumed and is

therefore the chosen prior distribution for θ [111].

The aim of this approach is to calculate a joint posterior distribution for τ2 _{and θ by}

combining prior distributions with meta-analysis data. The posterior distribution is derived by Markov Chain Monte Carlo (MCMC) methods such as Gibb's sampler [106]. This requires specialist software such as WinBUGS [72]. From the joint posterior distribution, expected values and credibility intervals for τ2 _{and θ can be}

extracted.

2.6.2 Approximate Bayes Estimator (AB)

The approximate Bayes estimator (AB) was originally proposed within the context of sequential meta-analysis [50]. It does not require any form of Gibbs sampling or process of iteration and is therefore more simple to compute than full Bayes.

The prior for τ2 follows the inverse-gamma distribution Γ−1_{(η, λ)}_{, with parameters}

η and λ dening the shape and spread of the distribution respectively and zero probability of τ2 _{< 0}. The inverse-gamma distribution has the following p.d.f:

p τ2; η, λ = λ η Γ (η) τ 2−η−1 exp −λ τ2

where Γ represents the gamma function.

The underlying eects of each study i are assumed known, i.e. σ2

i = 0. Higgins

et al. [50] suggest this has minimal impact on ˆτ2. By making this assumption, it

follows that the posterior distribution will also be an inverse-gamma distribution with parameters η = η0+ (k/2) and λ = λ0+ (kτ2/2) [35] . τ2 in this case represents the

heterogeneity variance from the data, for which Higgins et al. [50] suggests using the DL estimate (section 2.2.1). A posterior estimate of τ2can be derived by substituting

the formulas for the posterior parameters η and λ into the formula for the mean of an inverse-gamma distribution: ˆ τ_AB2 = λ η − 1 = 2λ0+ kˆτDL2 2 (η0− 1) + k

The prior distribution for τ2 has mean λ

0/ (η0− 1), implying the expected value of

λ0 is ˆτ02(η0− 1). This can be substituted into the formula for ˆτAB2 above:

τ_AB2 = 2τ

0 (η0− 1) + kˆτDL2

2 (η0− 1) + k

This last step is carried out because it is often easier to dene a prior value for τ2

0 than the spread parameter λ0. To calculate ˆτAB2 , we must provide two of three

prior values for τ2

0, η0 and λ0. In the context of sequential meta-analysis in the

original publication [50], τ2

0 is the estimate of τ2 from the previous update to the

meta-analysis. Outside of this context, τ2

0 can represent our best estimate from prior

2.6.3 Bayes modal (BM)

Bayes Modal (BM) can estimate τ2 _{numerically without the need for MCMC meth-}

ods. It imposes a gamma prior distribution for τ [16, 17]:

p (τ ; η, λ) = 1 Γ (α)τ

α−1_e−λτ

α and λ are the shape and scale parameters dened from prior information. Chung et al. [16] suggest using α = 2 and λ ≈ 0 for a vague prior. The gamma distribution is chosen because it has the property p (τ = 0; η, λ) = 0 for any α or λ, thus avoid- ing zero estimates of τ2 _{from the posterior. The density function of the posterior}

distribution can be derived if we assume an improper uniform prior for θ:

p (τ, θ) = lM L θ, τ2 + (α − 1) logτ − λτ + c

where c is an undened constant and lM L(θ, τ2)is the log-likelihood function (2.10).

Software packages are available to nding estimates of (τ, θ) that maximise the above equation such as lmer in R and gllamm in Stata [17]. The BM estimator can al- ternatively be considered a maximum likelihood approach with a penalty imposed to avoid zero estimates; the above log-likelihood is that of lM L(θ, τ2)with added terms

[17].

2.6.4 Rukhin's approach

Rukhin [93] proposed two semi-Bayesian heterogeneity variance estimators. These dier from estimators derived from a more typical Bayesian approach, which involves specifying prior distributions for the unknown parameters and requires MCMC methods t observed data to the model. Rukhin's estimators are more simple to compute and only require a xed prior estimate of τ2, denoted ˆτ2

Rukhin's estimators are derived from a generalised version of method of moments from section 2.2. He rst explicitly derives the formula for V ar (ˆτ2₎under this unied

approach. Then, his general formula nds ˆτ2 _{such that V ar (ˆτ}2₎_{is locally minimised}

around the prior estimate of τ2. I refer the reader to the original paper for a detailed

derivation of this approach [93]. The general formula for Rukhin's heterogeneity variance estimators is:

ˆ τ_RB2 = Pk i=1 ˆθi− ˆθ 2 k + 1 + Pk i=1(n1i+ n2i) − k 2kˆτ2 0 − (k − 1) Pk i=1σˆ 2 i Pk i=1(n1i+ n2i) − k + 2 k (k + 1) (2.13)

where n1i and n2i are the sample sizes in intervention groups one and two, ˆθ =

Pk i=1 ˆθiwˆi /Pk i=1wˆi and ˆwi = (ˆσ 2 i + ˆτ02) −1 . Rukhin [93] proposed two formulae for ˆτ2

1. ˆτ2

0 = 0, which leads to following heterogeneity variance estimator (B0):

ˆ τ_B02 = Pk i=1 ˆθi− ˆθ 2 k + 1 + Pk i=1(n1i+ n2i) − k (k − 1)Pk i=1σˆ 2 i Pk i=1(n1i+ n2i) − k + 2 k (k + 1) 2. ˆτ2 0 = 0.5 (k − 1) Pk i=1(ˆσ 2 i) /k, which leads to (BP): ˆ τ_BP2 = Pk i=1 ˆθi − ˆθ 2 k + 1 For a given k, ˆτ2

BP is a xed proportion of the total sample variance V ar ˆθ

= (k − 1)−1Pk

i=1 ˆθi− ˆθ

. There is little logic for this relationship, so BP is unlikely to have good properties. Rukhin [93] suggested this prior for τ2 only to simplify the

Another estimator, which forms a part of the same approach, assumes σ2

i to be

known. This estimator is given by the formula:

ˆ τ_SB2 = Pk i=1 ˆθi− ˆθ 2 k + 1 + 2ˆτ2 0k − (k − 1) Pk i=1σˆ 2 i k (k + 1) Prior estimates ˆτ2

0 must be specied to calculate ˆτSB2 , but Rukhin [93] made no

specic suggestions.

In document Estimating the Heterogeneity Variance in a Random-Effects Meta-Analysis (Page 61-66)