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Fully Bayesian hierarchical modelling in two stages, with application to meta-analysis.

Lunn D, Barrett J, Sweeting M, Thompson S - J R Stat Soc Ser C Appl Stat (2013)

Bottom Line: A Bayesian one-stage approach offers additional advantages, such as the acknowledgement of uncertainty in all parameters and greater flexibility.These are then used as proposal distributions in a computationally efficient second stage.The two-stage Bayesian approach closely reproduces a one-stage analysis when it can be undertaken, but can also be easily carried out when a one-stage approach is difficult or impossible.

View Article: PubMed Central - PubMed

Affiliation: Medical Research Council Biostatistics Unit Cambridge, UK.

ABSTRACT
Meta-analysis is often undertaken in two stages, with each study analysed separately in stage 1 and estimates combined across studies in stage 2. The study-specific estimates are assumed to arise from normal distributions with known variances equal to their corresponding estimates. In contrast, a one-stage analysis estimates all parameters simultaneously. A Bayesian one-stage approach offers additional advantages, such as the acknowledgement of uncertainty in all parameters and greater flexibility. However, there are situations when a two-stage strategy is compelling, e.g. when study-specific analyses are complex and/or time consuming. We present a novel method for fitting the full Bayesian model in two stages, hence benefiting from its advantages while retaining the convenience and flexibility of a two-stage approach. Using Markov chain Monte Carlo methods, posteriors for the parameters of interest are derived separately for each study. These are then used as proposal distributions in a computationally efficient second stage. We illustrate these ideas on a small binomial data set; we also analyse motivating data on the growth and rupture of abdominal aortic aneurysms. The two-stage Bayesian approach closely reproduces a one-stage analysis when it can be undertaken, but can also be easily carried out when a one-stage approach is difficult or impossible.

No MeSH data available.


Related in: MedlinePlus

Posterior density estimates for μ, σ, τ1 and τ2 based on 100000 stage 2 samples from analysis of the pre-eclampsia data: (a)  by using 1000 stage 1 samples; (b)  by using 10000 stage 1 samples; (c)  by using 1000 stage 1 samples; (d)  by using 10000 stage 1 samples; (e)  by using 1000 stage 1 samples; (f)  by using 10000 stage 1 samples; (g)  by using 1000 stage 1 samples; (h)  by using 10000 stage 1 samples
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fig02: Posterior density estimates for μ, σ, τ1 and τ2 based on 100000 stage 2 samples from analysis of the pre-eclampsia data: (a) by using 1000 stage 1 samples; (b) by using 10000 stage 1 samples; (c) by using 1000 stage 1 samples; (d) by using 10000 stage 1 samples; (e) by using 1000 stage 1 samples; (f) by using 10000 stage 1 samples; (g) by using 1000 stage 1 samples; (h) by using 10000 stage 1 samples

Mentions: Fig. 2 compares posterior density estimates from the two two-stage analyses, purposely using the same bandwidth in both cases to emphasize the effect of the stage 1 posterior sample size. Inferences on the overall parameters μ and σ are virtually identical, suggesting that these are not strongly dependent on the number of stage 1 samples collected. Fig. 2 also shows density estimates for τ1 and τ2 as examples of study-specific parameters. In the cases where only 1000 stage 1 samples have been used the density estimates are somewhat ‘granular’ because there are fewer values to choose from, which reduces the resolution with which the target density can be represented.


Fully Bayesian hierarchical modelling in two stages, with application to meta-analysis.

Lunn D, Barrett J, Sweeting M, Thompson S - J R Stat Soc Ser C Appl Stat (2013)

Posterior density estimates for μ, σ, τ1 and τ2 based on 100000 stage 2 samples from analysis of the pre-eclampsia data: (a)  by using 1000 stage 1 samples; (b)  by using 10000 stage 1 samples; (c)  by using 1000 stage 1 samples; (d)  by using 10000 stage 1 samples; (e)  by using 1000 stage 1 samples; (f)  by using 10000 stage 1 samples; (g)  by using 1000 stage 1 samples; (h)  by using 10000 stage 1 samples
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3814003&req=5

fig02: Posterior density estimates for μ, σ, τ1 and τ2 based on 100000 stage 2 samples from analysis of the pre-eclampsia data: (a) by using 1000 stage 1 samples; (b) by using 10000 stage 1 samples; (c) by using 1000 stage 1 samples; (d) by using 10000 stage 1 samples; (e) by using 1000 stage 1 samples; (f) by using 10000 stage 1 samples; (g) by using 1000 stage 1 samples; (h) by using 10000 stage 1 samples
Mentions: Fig. 2 compares posterior density estimates from the two two-stage analyses, purposely using the same bandwidth in both cases to emphasize the effect of the stage 1 posterior sample size. Inferences on the overall parameters μ and σ are virtually identical, suggesting that these are not strongly dependent on the number of stage 1 samples collected. Fig. 2 also shows density estimates for τ1 and τ2 as examples of study-specific parameters. In the cases where only 1000 stage 1 samples have been used the density estimates are somewhat ‘granular’ because there are fewer values to choose from, which reduces the resolution with which the target density can be represented.

Bottom Line: A Bayesian one-stage approach offers additional advantages, such as the acknowledgement of uncertainty in all parameters and greater flexibility.These are then used as proposal distributions in a computationally efficient second stage.The two-stage Bayesian approach closely reproduces a one-stage analysis when it can be undertaken, but can also be easily carried out when a one-stage approach is difficult or impossible.

View Article: PubMed Central - PubMed

Affiliation: Medical Research Council Biostatistics Unit Cambridge, UK.

ABSTRACT
Meta-analysis is often undertaken in two stages, with each study analysed separately in stage 1 and estimates combined across studies in stage 2. The study-specific estimates are assumed to arise from normal distributions with known variances equal to their corresponding estimates. In contrast, a one-stage analysis estimates all parameters simultaneously. A Bayesian one-stage approach offers additional advantages, such as the acknowledgement of uncertainty in all parameters and greater flexibility. However, there are situations when a two-stage strategy is compelling, e.g. when study-specific analyses are complex and/or time consuming. We present a novel method for fitting the full Bayesian model in two stages, hence benefiting from its advantages while retaining the convenience and flexibility of a two-stage approach. Using Markov chain Monte Carlo methods, posteriors for the parameters of interest are derived separately for each study. These are then used as proposal distributions in a computationally efficient second stage. We illustrate these ideas on a small binomial data set; we also analyse motivating data on the growth and rupture of abdominal aortic aneurysms. The two-stage Bayesian approach closely reproduces a one-stage analysis when it can be undertaken, but can also be easily carried out when a one-stage approach is difficult or impossible.

No MeSH data available.


Related in: MedlinePlus