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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

Kernel density estimates for all 14 stage 1 posteriors for the log-odds of rupture within 3 months (0.25 years) given a baseline diameter of 50 mm, logit, from independent analyses of study-specific AAA data: , prior distribution, scaled (arbitarily) so that it is visible on the same plot
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fig03: Kernel density estimates for all 14 stage 1 posteriors for the log-odds of rupture within 3 months (0.25 years) given a baseline diameter of 50 mm, logit, from independent analyses of study-specific AAA data: , prior distribution, scaled (arbitarily) so that it is visible on the same plot

Mentions: BUGS code for the stage 1 analysis is given in the on-line appendix A.5. In practice, to ensure that the parameters are less correlated, all time variables (tijk and Tij) are centred at the mean follow-up time for the study, whereas, in the hazard function, is centred at the study mean AAA diameter. Transformations are then required to obtain common parameters across the studies. Such centring is not necessary but can improve convergence substantially. For monitoring convergence, each stage 1 analysis was conducted with two MCMC chains running in parallel; the method of Gelman and Rubin (Gelman and Rubin, 1992; Brooks and Gelman, 1998) was then used. A typical analysis involved a burn-in of 6000 iterations, with 100000 further iterations thinned by 20. In all cases, sufficient iterations for obtaining 10000 approximately independent posterior realizations for each study level parameter of interest were performed. Even with the aforementioned centring to improve convergence, each stage 1 analysis took several hours to perform. Hence a single one-stage analysis of the full hierarchical model would have taken of the order of days to perform. Bearing in mind that there are numerous parameters of interest in this setting, a two-stage approach was considered essential. Prior distributions for all parameters, including derived parameters, were effectively flat within the range of values supported by the corresponding posterior, as illustrated in Fig. 3 for the log-odds of rupture within 3 months (0.25 years) given a baseline diameter of 50 mm, .


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)

Kernel density estimates for all 14 stage 1 posteriors for the log-odds of rupture within 3 months (0.25 years) given a baseline diameter of 50 mm, logit, from independent analyses of study-specific AAA data: , prior distribution, scaled (arbitarily) so that it is visible on the same plot
© Copyright Policy - open-access
Related In: Results  -  Collection

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

fig03: Kernel density estimates for all 14 stage 1 posteriors for the log-odds of rupture within 3 months (0.25 years) given a baseline diameter of 50 mm, logit, from independent analyses of study-specific AAA data: , prior distribution, scaled (arbitarily) so that it is visible on the same plot
Mentions: BUGS code for the stage 1 analysis is given in the on-line appendix A.5. In practice, to ensure that the parameters are less correlated, all time variables (tijk and Tij) are centred at the mean follow-up time for the study, whereas, in the hazard function, is centred at the study mean AAA diameter. Transformations are then required to obtain common parameters across the studies. Such centring is not necessary but can improve convergence substantially. For monitoring convergence, each stage 1 analysis was conducted with two MCMC chains running in parallel; the method of Gelman and Rubin (Gelman and Rubin, 1992; Brooks and Gelman, 1998) was then used. A typical analysis involved a burn-in of 6000 iterations, with 100000 further iterations thinned by 20. In all cases, sufficient iterations for obtaining 10000 approximately independent posterior realizations for each study level parameter of interest were performed. Even with the aforementioned centring to improve convergence, each stage 1 analysis took several hours to perform. Hence a single one-stage analysis of the full hierarchical model would have taken of the order of days to perform. Bearing in mind that there are numerous parameters of interest in this setting, a two-stage approach was considered essential. Prior distributions for all parameters, including derived parameters, were effectively flat within the range of values supported by the corresponding posterior, as illustrated in Fig. 3 for the log-odds of rupture within 3 months (0.25 years) given a baseline diameter of 50 mm, .

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