Bayesian statistics is an approach to statistics based on a different philosophy from that which underlies significance tests and confidence intervals. It is essentially about updating of evidence. In a Bayesian analysis, initial uncertainty is expressed through a prior distribution about the quantities of interest. Current data and assumptions concerning how they were generated are summarized in the likelihood. The posterior distribution for the quantities of interest can then be obtained by combining the prior distribution and the likelihood. The posterior distribution may be summarized by point estimates and credible intervals, which look much like classical estimates and confidence intervals. Bayesian analysis cannot be carried out in RevMan, but may be performed using WinBUGS software (Smith 1995, Lunn 2000).
In the context of a meta-analysis, the prior distribution will describe uncertainty regarding the particular effect measure being analysed, such as the odds ratio or the mean difference. This may be an expression of subjective belief about the size of the effect, or it may be from sources of evidence not included in the meta-analysis, such as information from non-randomized studies. The width of the prior distribution reflects the degree of uncertainty about the quantity. When there is little or no information, a ‘non-informative’ prior can be used, in which all values across the possible range are equally likely. The likelihood summarizes both the data from studies included in the meta-analysis (for example, 2×2 tables from randomized trials) and the meta-analysis model (for example, assuming a fixed effect or random effects).
The choice of prior distribution is a source of controversy in Bayesian statistics. Although it is possible to represent beliefs about effects as a prior distribution, it may seem strange to combine objective trial data with subjective opinion. A common practice in meta-analysis is therefore to use non-informative prior distributions to reflect a position of prior ignorance. This is particularly true for the main comparison. However, prior distributions may also be placed on other quantities in a meta-analysis, such as the extent of among-study variation in a random-effects analysis. It may be useful to bring in judgement, or external evidence, on some of these other parameters, particularly when there are few studies in the meta-analysis. It is important to carry out sensitivity analyses to investigate how the results depend on any assumptions made.
A difference between Bayesian analysis and classical meta-analysis is that the interpretation is directly in terms of belief: a 95% credible interval for an odds ratio is that region in which we believe the odds ratio to lie with probability 95%. This is how many practitioners actually interpret a classical confidence interval, but strictly in the classical framework the 95% refers to the long-term frequency with which 95% intervals contain the true value. The Bayesian framework also allows a review author to calculate the probability that the odds ratio has a particular range of values, which cannot be done in the classical framework. For example, we can determine the probability that the odds ratio is less than 1 (which might indicate a beneficial effect of an experimental intervention), or that it is no larger than 0.8 (which might indicate a clinically important effect). It should be noted that these probabilities are specific to the choice of the prior distribution. Different meta-analysts may analyse the same data using different prior distributions and obtain different results.
Bayesian methods offer some potential advantages over many classical methods for meta-analyses. For example, they can be used to:
incorporate external evidence, such as on the effects of interventions or the likely extent of among-study variation;
extend a meta-analysis to decision-making contexts, by incorporating the notion of the utility of various clinical outcome states;
allow naturally for the imprecision in the estimated between-study variance estimate (see Chapter 9, Section 9.5.4);
investigate the relationship between underlying risk and treatment benefit (see Chapter 9, Section 9.6.7);
perform complex analyses (e.g. multiple-treatments meta-analysis), due to the flexibility of the WinBUGS software; and
examine the extent to which data would change people’s beliefs (Higgins 2002).
Statistical expertise is strongly recommended for review authors wishing to carry out Bayesian analyses. There are several good texts (Sutton 2000, Sutton 2001, Spiegelhalter 2004).