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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

Results of stage 1 (, ) and stage 2 (, ) analyses of AAA data for the probability of rupture within 3 months, given 50 mm diameter: the estimates are posterior medians with 95% credible intervals; the medians are shown as squares with an area inversely proportional to the posterior variance on the logit scale; I2 is the proportion of total variation due to heterogeneity between studies; the diamond notation is for overall estimates as in Fig. 1
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fig04: Results of stage 1 (, ) and stage 2 (, ) analyses of AAA data for the probability of rupture within 3 months, given 50 mm diameter: the estimates are posterior medians with 95% credible intervals; the medians are shown as squares with an area inversely proportional to the posterior variance on the logit scale; I2 is the proportion of total variation due to heterogeneity between studies; the diamond notation is for overall estimates as in Fig. 1

Mentions: Currently, patients with an AAA diameter between 45 and 54 mm, identified in the National Health Service AAA screening programme, are invited back for re-screening after 3 months. To assess the appropriateness of this monitoring interval, we begin by calculating, for an individual with diameter 50 mm, the study-specific predicted probabilities of rupture and of crossing the surgical intervention threshold (55 mm) within a 3-month period, and respectively. In the second stage, a hierarchical structure is placed on the logit of each predicted probability, by assuming that the study-specific values originate from a common (normal) population distribution. Table 2 shows the overall estimates transformed back to the probability scale. These now have considerably wider credible intervals than those obtained via classical random-effects meta-analysis. Results indicate that the current 3-month screening policy is relatively safe, with point estimates (and 95% credible intervals in parentheses) for the overall/population-median values of the probabilities of rupture before next screen and of crossing the intervention threshold within 3 months being 0.15% (0.076–0.26%) and 1.8% (0.82–3.7%) respectively. There is considerable between-study heterogeneity in these quantities, however, which raises the question of whether there are patient or study level characteristics that may explain this; however, this topic is not pursued here. To illustrate the level of heterogeneity, Fig. 4 shows a forest plot of the stage 1 and stage 2 posterior distributions (medians and 95% credible intervals) for the probability of rupture within 3 months, given a diameter of 50 mm. Note that study-specific estimates are variable, and that those from stage 2, corresponding to the full hierarchical model, are generally more precise than those from stage 1. Considerable shrinkage is also apparent for several studies.


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)

Results of stage 1 (, ) and stage 2 (, ) analyses of AAA data for the probability of rupture within 3 months, given 50 mm diameter: the estimates are posterior medians with 95% credible intervals; the medians are shown as squares with an area inversely proportional to the posterior variance on the logit scale; I2 is the proportion of total variation due to heterogeneity between studies; the diamond notation is for overall estimates as in Fig. 1
© Copyright Policy - open-access
Related In: Results  -  Collection

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

fig04: Results of stage 1 (, ) and stage 2 (, ) analyses of AAA data for the probability of rupture within 3 months, given 50 mm diameter: the estimates are posterior medians with 95% credible intervals; the medians are shown as squares with an area inversely proportional to the posterior variance on the logit scale; I2 is the proportion of total variation due to heterogeneity between studies; the diamond notation is for overall estimates as in Fig. 1
Mentions: Currently, patients with an AAA diameter between 45 and 54 mm, identified in the National Health Service AAA screening programme, are invited back for re-screening after 3 months. To assess the appropriateness of this monitoring interval, we begin by calculating, for an individual with diameter 50 mm, the study-specific predicted probabilities of rupture and of crossing the surgical intervention threshold (55 mm) within a 3-month period, and respectively. In the second stage, a hierarchical structure is placed on the logit of each predicted probability, by assuming that the study-specific values originate from a common (normal) population distribution. Table 2 shows the overall estimates transformed back to the probability scale. These now have considerably wider credible intervals than those obtained via classical random-effects meta-analysis. Results indicate that the current 3-month screening policy is relatively safe, with point estimates (and 95% credible intervals in parentheses) for the overall/population-median values of the probabilities of rupture before next screen and of crossing the intervention threshold within 3 months being 0.15% (0.076–0.26%) and 1.8% (0.82–3.7%) respectively. There is considerable between-study heterogeneity in these quantities, however, which raises the question of whether there are patient or study level characteristics that may explain this; however, this topic is not pursued here. To illustrate the level of heterogeneity, Fig. 4 shows a forest plot of the stage 1 and stage 2 posterior distributions (medians and 95% credible intervals) for the probability of rupture within 3 months, given a diameter of 50 mm. Note that study-specific estimates are variable, and that those from stage 2, corresponding to the full hierarchical model, are generally more precise than those from stage 1. Considerable shrinkage is also apparent for several studies.

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