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Bayesian Decision Making in Clinical Research Dr. Bhaswat S. Chakrabortya a R&D, Cadila Pharmaceuticals Ltd., Ahemdabad, Gujarat 387 810
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Abstract• Objective: To demonstrate various applications of Bayesian statistics in evaluating evidence from clinical and epidemiological research and understanding their implications in conclusions of clinical trials, medical practice, guidelines and policies.• Methods: Conventional (Frequentist) and Bayesian theories of probabilities are compared and contrasted. Inference of the results of a clinical trial comparing treatments A and B was considered. Conventional analysis concluded that treatment A is superior because there is a low probability that such a significant difference would have been observed when the treatments were in fact equal. Bayesian analysis on the other hand looked at the observed difference and induced the likelihood of treatment A being superior to B. A couple of additional case studies were also considered. In all cases, relative merits of the two approaches were analyzed for medical practice, guidelines and policies.• Results: The conventional analysis showed a p value for the difference between treatments A and B is 0.001, which is highly significant at α = 0.05. This means that the chance of observing this difference when A and B are in fact equal is 1in 1000. The Bayesian conditioned probability of A being superior to B was 0.999. Although the same conclusion was reached here, the next two case studies (1. whether cancer can be induced by proximity to high-voltage transmission lines and 2. an increased risk of venous thrombosis with third generation oral contraceptives) showed very different results leading to different conclusions.• Conclusions: Conventional and Bayesian inferences can be similar or different but the key difference between conventional and Bayesian reasoning is that the Bayesian believes that truth is subjective and naturally conditioned by the evidence. Almost all areas of clinical research and medicine now have applications of Bayesian statistics, one of the earliest being diagnostic medicine. From the results of the case studies, we shall also see the application of Bayesian methods clinical trials and epidemiology.
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Rev. Thomas Bayes (1702-1761) • Rev. Thomas Bayes noted that sometimes the probability of a statistical hypothesis is given before event or evidence is observed (Prior); he showed how to compute the probability of the hypothesis after some observations are made (Posterior). • Before Rev. Bayes, no one knew how to measure the probability of statistical hypotheses in the light of data. Only it was known as to how to reject a statistical hypothesis in the light of data.
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Bayes TheoremLet X1, X2, ... , Xn be a set of mutually exclusive events thatform the sample space S. Let Y be any event from the samesample space, such that P(Y) > 0. Then, P( Xk ∩ Y ) P( Xk | Y ) = ….Eq. 1 P( X1 ∩ Y ) + P( X2 ∩ Y ) + P( X3 ∩ Y )Since P( Xk ∩ Y ) = P( Xk )*P( Y | Xk ), Bayes’ theorem canalso be expressed asP( Xk | Y ) = P( Xk )*P( Y | Xk ) P( X1 )*P( Y | X1 ) + P( X2 )*P( Y | X2 ) P( Xn )*P( Y | Xn ) ….Eq. 2
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Case Study 1 – A Superiority Trial: Conventional Statistics• H0 : μA – μB ≤ δ• H1 : μA – μB > δ• The hypotheses were tested at level of significance α =0.05• Experimental data was analyzed to find p = 0.001 – (i.e., the probability of observing this difference by chance is 1 in 1000)• The Null Hypothesis was rejected (conclusion: alternate is true – A is superior to B)
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Case Study 1 – A Superiority Trial: Bayes’ Statistics• Let prior probability Treatment A being superior to Treatment B, P( X A ) = 0.8• and, therefore, prior probability Treatment A NOT being superior to Treatment B, P( XB ) = 0.2• Let experimental (RCTs) probability of concluding A is superior to B, when A is indeed superior, P(Y | XA) = 0.95• and, therefore, experimental probability of concluding A NOT being superior to B, when A is indeed superior, P(Y | X B) = 0.05• What we would like to know is the posterior or conditional probability of A indeed being superior to B when the experimental evidence is superiority of A (following Eq. 2): P( XA | Y ) = P( XA )*P( Y | XA ) P( XA )*P( Y | XA ) + P( XB )*P( Y | XB )
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A Superiority Trial: Bayes’ Statistics contd. P( XA )*P( Y | XA ) P( XA | Y ) = P( XA )*P( Y | XA ) + P( XB )*P( Y | XB ) = 0.8*0.95 0.8*0.95 + 0.2*0.05 = 0.76 0.76 + 0.01 = 0.99That is a 99% probability of the hypothesis that A is Superior to B!
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Case Study 2• Case and the Problem – An elementary school staff was concerned that their high cancer rate could be due to two nearby high voltage transmission lines [1]• Data – there were 8 cases of invasive cancer over a long time among 145 staff members whose average age was between 40 and 44 – based on the national cancer rate among woman this age (approximately 3/100), the expected number of cancers is 4.2• Assumptions – the 145 staff members developed cancer independently of each other – the chance of cancer, θ, was the same for each staff person• Therefore, the number of cancers, X, follows a binomial distribution: – X ~ bin (145, θ) – θ could be 0.03 (national cancer rate) or more, i.e., 0.04, 0.05, 0.06
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Case Study 2 contd.• For each hypothesized θ, we use elementary results about the binomial distribution to calculate: 145 P(X=8 | θ) = θ8 (1 – θ)137 8• Theory A: P(X=8 | θ=0.03 ) ≈ 0.036; Theory B: P(X=8 | θ=0.04 ) ≈ 0.096• Theory C: P(X=8 | θ=0.05 ) ≈ 0.134; Theory D: P(X=8 | θ=0.06 ) ≈ 0.136• This is a ratio of approximately 1:3:4:4. So, Theory B explains the data about 3 times as well as theory A• Likelihood Principle: – Initially, P( X | θ ) is a function of two variables: X and θ. – But once X = 8 has been observed, then P( X | θ ) describes how well each theory, or value of θ explains the data. No other value of X is relevant. – This is an example of the likelihood principal[2]
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Case Study 2 contd..• Conventional Analysis:• Null hypothesis, H0: θ=0.03; Alternate hypothesis, H1: θ=0.03• p-value = P(X=8| θ=0.03)+ P(X=9| θ=0.03)+ +…+ P(X=145| θ=0.03) ≈ 0.07• At α=0.10, the null hypothesis will be rejected and conclusion would be that the high voltage has a significant effect on the cancer rate at the school• Bayesian Analysis:• P(A | X=8 ) = 0.23; P( B | X=8 ) = 0.21• P( C | X=8 ) = 0.28; P( D | X=8 ) = 0.28• The Bayesian P(θ > .03) ≈ 0.77, which would not be sufficient to reject the null hypothesis!!
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Case Study 3• Case and the Problem – Four case-control studies (one nested in a cohort study) of third generation contraceptive pills containing desogestral and gestodene were compared with pills containing other progestagens for the relative risk of venous thromboembolism[3].• Bayesian Analysis: – Showed the posterior distributions are much narrower than the prior distributions, indicating less doubt about the value of the true relative risk (odds ratio of 2.0). The data influenced the posterior distributions more than the prior distributions, such that they are centred on log(1.69) and log(1.76) respectively, and the probability of the true relative risk being greater than 1 is more than 0.999 in both cases.• Why was not this risk picked up by conventional analysis? – As we illustrated in previous examples, conventional analysis does not calculate the probability of a hypothesis being true given the data• The real scientific question could have been: – "What is the probability that the third generation pills increase the risk when compared to the others; what is the probability that they at least double the risk--as measured in the case-control study; and what is the median estimate (as likely to be too small as too large)?" [4]
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Conclusions• Conventional Statistics (Frequentists) make no claims about probabilities – E.g., the 95% CI of mean does not claim there is a 95% probability the mean is in that range – It only states that if the experiment was repeated 100 times, 95 times the estimated mean would lie that range• Likelihood also does not give you probability of a hypothesis – only the likelihood of the data given the hypothesis• Bayesian Statistics does give you the probability of a hypothesis being true given the data – This is closest to intuition and normal process of decision making• Three case studies – The first one shows, if the prior and posterior are equally influenced by data, then the conventional and Bayesian conclusions are nominally the same – However, in the second case, the conditioned (posterior) probability, although influenced by the data, did not yet call for a rejection of the hypothesis – And in the third case, the data changed the posterior probability so much that only the conditioned hypothesis was considered to be true
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References1. Brodeur P, “The Cancer at Slater School”, Annals of radiation, The New Yorker, December 7, 1992, p. 86.2. http://www.home.uchicago.edu/~grynav/bayes/ABSLec1.ppt, access date 06.03.2009.3. McPherson K. Third Generation Oral Contraception and Venous Thromboembolism. BMJ, 1995, 312, 68-9.4. Lilford RJ, Braunholtz D. The Statistical Basis of Public Policy: a Paradigm Shift is Overdue. BMJ, 1996, 313, pp. 603- 7.
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