Bayesian ijupls

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Bayesian ijupls

  1. 1. International Standard Serial Number (ISSN): 2249-6793 International Journal of Universal Pharmacy and Life Sciences 2(1): January-February 2012 INTERNATIONAL JOURNAL OF UNIVERSAL PHARMACY AND LIFE SCIENCES Pharmaceutical Sciences Original Article……!!! Received: 02-01-2012; Accepted: 08-01-2012 BAYESIAN DECISION MAKING IN CLINICAL RESEARCH: A PRIMER FOR NON- STATISTICIANS Bhaswat S. Chakraborty* Cadila Pharmaceuticals, Ahmedabad, Gujarat 387810, India. ABSTRACT Keywords: The objective of this article is to demonstrate a few fundamental applications of Bayesian statistics in evaluating evidence from clinical and epidemiological Conventional (Frequentist) research and understanding their implications in conclusions of clinical trials, statistics – Bayesian medical practice, guidelines and policies. The methodology mainly compares statistics – posterior and contrasts the conventional (Frequentist) and Bayesian theories of probabilities through examples. First, inference of the results of a clinical trial (conditioned) probability – comparing treatments A and B was considered. Conventional analysis clinical trial – inference concluded that treatment A is superior because there is a low probability that such a significant difference would have been observed when the treatments For Correspondence: 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. Two Dr. Bhaswat S. Chakraborty additional case studies (case study 2 concerning that their high cancer rate could be due to two nearby high voltage transmission lines and case study 3 Senior Vice President, concerning third generation contraceptive pills containing desogestral and Research & Development, gestodene causing venous thromboembolism) were also analyzed using Bayes’ rule. In all cases, relative merits of the two approaches were analyzed Cadila Pharmaceuticals for medical practice, guidelines and policies. As results of the case study 1, the Limited, 1389, Trasad Road, conventional analysis showed a p value for the difference between treatments Dholka 387810, Ahmedabad, A and B is 0.001, which is highly significant at α = 0.05. This means that the Gujarat, India chance of observing this difference when A and B are in fact equal is 1 in a 1000. The Bayesian conditioned probability of A being superior to B was E-mail: 0.999. Although the same conclusion was reached here, the next two casedrb.chakraborty@cadilapharma.co.in studies (case study 2: whether cancer can be induced by proximity to high- voltage transmission lines and case study3: an increased risk of venous thrombosis with third generation oral contraceptives) showed very different results leading to different conclusions. It is concluded that conclusions from both conventional and Bayesian inferences can be similar 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. 48 Full Text Available On www.ijupls.com
  2. 2. International Standard Serial Number (ISSN): 2249-6793INTRODUCTIONRev. Thomas Bayes (1702 – 1761) noted that sometimes the probability of a statisticalhypothesis is given before event or evidence is observed (Prior). He showed how to compute theprobability of the hypothesis after some observations are made (Posterior).Before Rev. Bayes, no one knew how to measure the probability of statistical hypotheses in thelight of data. Only it was known as to how to reject a statistical hypothesis in the light of data.In order to understand what the preceding paragraph means, let us define certain probabilities inclear ordinary language. Let us say that if two events are mutually exclusive if they have nosample points in common. An example of such events would be probability of positive diagnosisof a disease by a kit and the probability of actually developing that disease in a person. Then, letus define that the probability that event A occurs, given that event B has occurred, is called aconditional probability. The conditional probability of A, given B, is denoted in statistics by thesymbol P(A|B). Similarly, the probability of event A not occurring is given by P(A). Further, ifevents A and B come from the same sample space, the probability that both A and B occur is theprobability of event A occurring multiplied with the probability of event B occuring, given A hasoccurred.P(A ∩ B) = P(A) * P(B|A)And finally, probability that either A has occurred or B has occurred or both have occurred isgiven by:P(A ∪ B) = P(A) + P(B) - P(A ∩ B)Bayes TheoremLet X1, X2, ... , Xn be a set of mutually exclusive events that form a sample space S. Let Y be anyevent from a same sample space, such that P(Y) > 0. Then,Since P( Xk ∩ Y ) = P( Xk )*P( Y | Xk ), Bayes’ theorem can also be expressed as 49 Full Text Available On www.ijupls.com
  3. 3. International Standard Serial Number (ISSN): 2249-6793CASE STUDIESCase Study 1 – A Superiority TrialConventional StatisticsIn the first case study, let us consider a randomized clinical trial (RCT) in which a newtherapeutic intervention A is being tested for superiority of its effectiveness over that of B in anappropriate patient population. Let the null (H0) and alternate (H1) hypotheses be as follows: H0: μA – μB ≤  and H1: μA – μB >  ….Eq. 3These hypotheses were tested at level of significance, α = 0.05. Once the trial was over, theexperimental data showed that the mean outcome measure of A was higher in magnitude thanthat of B and conventional statistical analysis computed a p = 0.001 i.e., the probability ofobserving this difference by chance is 1 in 1000. Consequently, the null hypothesis was rejectedand it was concluded that the alternate is true – A is superior to B.Bayesian StatisticsWe shall understand the basic propositions of Bayesian analysis in Case 1 and not repeat thesefor the other two case studies. Let us say that the prior probability of treatment A being superiorto treatment B, P( XA ) is 80% or 0.8. Therefore, as stated above, the prior probability oftreatment A not being superior to Treatment B, P( XB ) = 1 – 0.8 = 0.2. Let the probability ofexperimental evidence from the RCT, of concluding A is superior to B, when A is indeedsuperior, P(Y | XA) is 95% or 0.95 and, therefore, experimental probability of concluding A notbeing superior to B, when A is indeed superior, P(Y | XB) = 1 – 0.95 = 0.05.What we would like to know is whether the posterior or conditional probability of A indeedbeing superior to B when the experimental evidence is superiority of A following Eq. 2. Thus: 50 Full Text Available On www.ijupls.com
  4. 4. International Standard Serial Number (ISSN): 2249-6793Therefore, the Bayesian conclusion, in this case, is that there is a 99% probability of thehypothesis that treatment effect of A is Superior to the treatment effect of B. This is same as theone concluded from the conventional analysis by rejecting its null hypothesis.Case Study 2 – Effect (Cancer) to Cause (High Voltage) AnalysisOne of the salient characteristics of Bayesian statistics is, of course, its ability to compute theprobability of hypothesis being true. This allows investigation of effect to cause. In the secondcase study, the statistical problem is that an elementary school staff is concerned that their highcancer rate among the ex- and current employees could be due to two nearby high voltagetransmission lines.[1] The data in support of such suspicion include the fact that there have been8 cases of invasive cancer over a long time among 145 women staff members whose average agewas between 40 and 44. The national average of incidence of this cancer is 3% in women aged40-45. Therefore, based on the national cancer rate among woman this age, the expected numberof cancers in this school staff would be 4.2.What we are assuming in this case study that the staff members developed cancer independentlyof each other and the rate of developing cancer, , was the same for each woman staff member.Therefore, the number of cancers, X, which follows a binomial distribution can be given asfollows:X ~ bin (145, ) …. Eq. 4Where  could be 0.03 (national cancer rate) or more, i.e., 0.04, 0.05, 0.06 which we’ll define astheories A, B, C, and D, respectively. For each hypothesized, we can use the elementary resultsof the binomial distribution to calculate the probabilities:P(X=8 | θ) = θ8 (1 – θ)(145-8) …. Eq. 5Thus, theory A gives 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; and theory D: P(X=8 | θ=0.06 ) = 0.136. This is a ratio ofapproximately 1:3:4:4. It is obvious that theory B explains the data about 3 times as well astheory A. Here, first let us look at the likelihood principle. Initially, P(X | θ) is a function of twovariables: X and θ. But once X = 8 has been observed, then P(X | θ) describes how well eachtheory, or value of θ explains the data. No other value of X is relevant and we should treat Pr(X |) simply as Pr( X = X | ).[2]Conventional Statistics:Once again, let us state the null (H0) and alternate (H1) hypotheses as follows:H0: θ = 0.03 and H1: θ ≠ 0.03 …. Eq. 6And p = P(X = 8 | θ = 0.03)+ P(X = 9 | θ = 0.03) + +…+ P(X = 145 | θ = 0.03)  0.07 51 Full Text Available On www.ijupls.com
  5. 5. International Standard Serial Number (ISSN): 2249-6793Therefore, at the given level of significance, α = 0.10, the null hypothesis is rejected and and it isconcluded that the high voltage has a significant effect on the cancer rate among the women staffat the school.Bayesian Statistics:For Bayesian analysis, we can look at the probabilities of getting 8 cancers given incidence ratesof θ = 0.03 or more (define as theories A, B, C, and D).Thus, P(A | X = 8) = 0.23; P(B | X = 8) = 0.21; P(C | X = 8) = 0.28; and P(D | X = 8) = 0.28If we add the probabilities of X = 8 given  = 0.04,  = 0.05 and  = 0.06, we get 0.77. Thus, theBayesian P( > 0.03) = 0.77, which would not be sufficient to reject the null hypothesis. This isin contrast with the conclusion from the conventional analysis.Case Study 3 – Another Effect (thromboembolism) to Cause (Contraceptive Treatment)AnalysisThis case study has been taken from many articles published in British Medical Journal.[3-7] Fourcase-control studies (including one nested in a cohort study) of third generation contraceptivepills containing desogestral and gestodene were compared with pills containing otherprogestagens for the relative risk of venous thromboembolism.[4-7]Conventional StatisticsThe pills we are talking about were all declared “safe and effective” by conventional analysis.Now, the question arises as to why was not the risk of increased thromboembloism picked up byconventional analysis? As we illustrated in previous examples, conventional analysis does notcalculate the probability of a hypothesis being true given the data.Bayesian StatisticsThe details of the Bayesian analysis of this case study may not be warranted as the goal of thispaper is to just introduce the latter to non-statistical readership. However, theBayesian analysis showed that the posterior distributions are much narrower than the priordistributions, indicating less doubt about the value of the true relative risk (odds ratio of 2.0). 52 Full Text Available On www.ijupls.com
  6. 6. International Standard Serial Number (ISSN): 2249-6793The data influenced the posterior distributions more than the prior distributions, such that theyare centred on log(1.69) and log(1.76) respectively, and the probability of the true relative riskbeing greater than 1 is more than 0.999 in both cases.The real scientific question could have been – "what is the probability that the third generationpills increase the risk when compared to the others; what is the probability that they at leastdouble the risk – as measured in the case-control study; and what is the median estimate (aslikely to be too small as too large)?"[8]CONCLUSIONSConventional statistics (also known as Frequentist statistics) make no claims about probabilitiesalthough one get misled that one is establishing a probability of something. For example, the95% confidence interval (CI) of mean does not claim there is a 95% probability the mean is inthat range. What it states is that if the experiment was repeated 100 times, 95 times the estimatedmean would lie that range. Likelihood also does not give you probability of a hypothesis – itgives only the likelihood of the data given the hypothesis. Maximum likelihood estimation isthus only a little more reliable than conventional statistics.Bayesian statistics, on the other hand, does give you the probability of a hypothesis being truegiven the data. This is closest to intuition and normal process of decision making. Everyonewould normally want to know if a hypothesis is given some observed data, such as, given aneffect whether the cause is true. The three case studies that have been looked at in this paperrepresent three different scenarios, each of which has its own place in clinical research. The firstone shows, if the prior and posterior are equally influenced by data, then the conventional andBayesian conclusions are nominally the same. In the second case, however, the conditioned(posterior) probability, although influenced by the data, did not yet call for a rejection of thehypothesis. And in the third case, the most complicated and challenging for the decision makers,the data changed the posterior probability so much that only the conditioned hypothesis wasconsidered to be true.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, 68-69.4. Poulter NR,Chang CL, Farley TMM, Meirik O, Marmot MG. Venous thromboembolic disease and combined oral contraceptives: results of international multicentre case-control study. Lancet, 1995, 346, 1575–1582.5. Farley TMM, Meirik O, Chang CL, Marmot MG, Poulter NR. Effect of different progestagens in low oestrogen oral contraceptives on venous thromboembolic disease. Lancet, 1995, 346, 1582–1588. 53 Full Text Available On www.ijupls.com
  7. 7. International Standard Serial Number (ISSN): 2249-67936. Jick H, Jick SS, Gurewich V, Myers MW, Vasilakis C. Risk of idiopathic cardiovascular death and non-fatal venous thromboembolism in women using oral contraceptives with differing progestagen components. Lancet, 1995, 346, 1589–1593.7. Bloemenkamp KWM, Rosendaal FR, Helmerhorst FM, Buller HR, Vandenbroucke JP. Enhancement by factor V Leiden mutation of risk of deep-vein thrombosis associated with oral contraceptives containing third-generation progestage. Lancet, 1995, 346, 1593–1596.8. Lilford RJ, Braunholtz D. The Statistical Basis of Public Policy: a Paradigm Shift is Overdue. BMJ, 1996, 313, pp. 603-607. 54 Full Text Available On www.ijupls.com

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