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Prior convictions

bayesian approaches to the analysis and interpretation of clinical megatrials

^* Division of Cardiology, Cedars-Sinai Medical Center, and the School of Medicine, University of California, Los Angeles, California, USA

View larger version (24K): [in a new window]   Figure 1 Alternative prior distributions of log odds ratio (OR) with respect to the null hypothesis, based on a "clinically unimportant" interval of ±5% about the mean of zero (from –0.05 to +0.05). All curves are normalized to the same unit area. An uninformative reference prior (not illustrated) is defined by a uniform distribution for the log OR (p = 10). Smaller standard deviations represent greater degrees of skepticism with respect to the test hypothesis. At a mildly skeptical p = 0.4, 10% of the distribution is contained within the clinically unimportant null interval; at a moderately skeptical p = 0.07, 50% is within the interval; and at a highly skeptical p = 0.03, 90% is within the interval.

View larger version (26K): [in a new window]   Figure 2 Bayesian analysis of a representative clinical megatrial (5). The curve labeled "Prior" represents the operative prior distribution for log odds ratio (OR) (xp = 0, p = 0.07); the curve labeled "Evidence" represents the distribution for log OR based on the empirical data (xe = –0.12, e = 0.06); and the curve labeled "Posterior" represents the distribution for log OR derived from the product of the prior and the evidence (xpe = –0.07, pe = 0.05). The smaller the standard deviation, the narrower is the distribution and the greater is its information content and precision. All curves are normalized to the same unit area. Probabilities for any magnitude of response can be computed directly in terms of the area under the appropriate region of the posterior distribution. The conventional (one-tailed) p value is represented by the proportion of the evidentiary distribution to the right of zero (here, 0.021).

View larger version (18K): [in a new window]   Figure 3 Determination of probabilities from the posterior distribution in Figure 2. The solid area within the "clinically unimportant" null interval for log odds ratio (OR) (ranging from +0.05 to –0.05) represents 34% of the total area under the distribution, and the probability that the log OR lies within this null interval is therefore 0.34. Similarly, the solid area with a log OR <–0.1 (equivalent to >10% improvement in outcome) represents 21% of the total area, and the probability that the log OR is <–0.1 is therefore 0.21.

View larger version (22K): [in a new window]   Figure 4 Relationship between prior and posterior probability for the null hypothesis in the Heart Protection Study (6). The x-axis of this graph represents the prior probability that the log odds ratio (OR) lies within a putative "clinically unimportant" region of equivalence (±5% about a null value of 0, as described in the legend to Figure 1), and the y-axis represents the associated posterior probability. The open circles (and standard deviations) are for the mildly skeptical, moderately skeptical, and highly skeptical priors, as illustrated in Figure 1. An analysis that is insensitive to one's prior degree of skepticism indicates a greater degree of stability in the resultant inferences. See text for further discussion.

View larger version (44K): [in a new window]   Figure 5 Sensitivity analysis with respect to the magnitude of therapeutic benefit (relative risk reduction) for the hypothetical FISH trial, using an uninformative prior. The posterior probability of benefit falls as the threshold of benefit increases.

View larger version (20K): [in a new window]   Figure 6 (Top panel) Sequential Bayesian meta-analysis with respect to "aggressive" versus "conservative" management of acute ischemic syndromes in five clinical trials (A through E). The acronyms and publication dates of the trials are as follows: A = TIMI-IIIB (1994); B = VANQWISH (1998); C = FRISC-II (1999); D = TACTICS TIMI-18 (2001); E = RITA-3 (2002). The y-axis of the graph represents the posterior probability of therapeutic benefit for the hypothesis that the 6- to 12-month risk of death or myocardial infarction exceeded the putative threshold of benefit (>0%, >10%, >20%). The x-axis denotes the sequence of the analysis in parallel with the date of publication: A (given an uninformative prior); B given A; C given A and B; D given A and B and C; E given A and B and C and D. (Bottom panel) A conventional fixed-effects meta-analysis of the same trials. The solid squares represent mean risk ratios derived from the empirical data, and the horizontal lines represent associated 95% confidence intervals (CIs). The solid diamond represents the overall risk ratio (its extremes denoting the associated 95% CI). A chi-square test for heterogeneity reveals significant heterogeneity among the studies (p = 0.017), attributable almost entirely to FRISC-II, and an overall OR of 0.88 in favor of the aggressive approach (95% CI 0.78 to 1.00; p = 0.04).