Bayesian ijupls

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 case
drb.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


INTRODUCTION

Rev. Thomas Bayes (1702 – 1761) 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.

In order to understand what the preceding paragraph means, let us define certain probabilities in
clear ordinary language. Let us say that if two events are mutually exclusive if they have no
sample points in common. An example of such events would be probability of positive diagnosis
of a disease by a kit and the probability of actually developing that disease in a person. Then, let
us define that the probability that event A occurs, given that event B has occurred, is called a
conditional probability. The conditional probability of A, given B, is denoted in statistics by the
symbol P(A|B). Similarly, the probability of event A not occurring is given by P(A'). Further, if
events A and B come from the same sample space, the probability that both A and B occur is the
probability of event A occurring multiplied with the probability of event B occuring, given A has
occurred.

P(A ∩ B) = P(A) * P(B|A)

And finally, probability that either A has occurred or B has occurred or both have occurred is
given by:

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Bayes' Theorem

Let X1, X2, ... , Xn be a set of mutually exclusive events that form a sample space S. Let Y be any
event 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



CASE STUDIES

Case Study 1 – A Superiority Trial

Conventional Statistics

In the first case study, let us consider a randomized clinical trial (RCT) in which a new
therapeutic intervention A is being tested for superiority of its effectiveness over that of B in an
appropriate patient population. Let the null (H0) and alternate (H1) hypotheses be as follows:

H0: μA – μB ≤  and H1: μA – μB >  ….Eq. 3

These hypotheses were tested at level of significance, α = 0.05. Once the trial was over, the
experimental data showed that the mean outcome measure of A was higher in magnitude than
that of B and conventional statistical analysis computed a p = 0.001 i.e., the probability of
observing this difference by chance is 1 in 1000. Consequently, the null hypothesis was rejected
and it was concluded that the alternate is true – A is superior to B.

Bayesian Statistics

We shall understand the basic propositions of Bayesian analysis in Case 1 and not repeat these
for the other two case studies. Let us say that the prior probability of treatment A being superior
to treatment B, P( XA ) is 80% or 0.8. Therefore, as stated above, the prior probability of
treatment A not being superior to Treatment B, P( XB ) = 1 – 0.8 = 0.2. Let the probability of
experimental evidence from the RCT, of concluding A is superior to B, when A is indeed
superior, P(Y | XA) is 95% or 0.95 and, therefore, experimental probability of concluding A not
being 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 indeed
being superior to B when the experimental evidence is superiority of A following Eq. 2. Thus:



Therefore, the Bayesian conclusion, in this case, is that there is a 99% probability of the
hypothesis that treatment effect of A is Superior to the treatment effect of B. This is same as the
one concluded from the conventional analysis by rejecting its null hypothesis.

Case Study 2 – Effect (Cancer) to Cause (High Voltage) Analysis

One of the salient characteristics of Bayesian statistics is, of course, its ability to compute the
probability of hypothesis being true. This allows investigation of effect to cause. In the second
case study, the statistical problem is that an elementary school staff is concerned that their high
cancer rate among the ex- and current employees could be due to two nearby high voltage
transmission lines.[1] The data in support of such suspicion include the fact that there have been
8 cases of invasive cancer over a long time among 145 women staff members whose average age
was between 40 and 44. The national average of incidence of this cancer is 3% in women aged
40-45. Therefore, based on the national cancer rate among woman this age, the expected number
of cancers in this school staff would be 4.2.

What we are assuming in this case study that the staff members developed cancer independently
of 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 as
follows:

X ~ bin (145, ) …. Eq. 4

Where  could be 0.03 (national cancer rate) or more, i.e., 0.04, 0.05, 0.06 which we’ll define as
theories A, B, C, and D, respectively. For each hypothesized, we can use the elementary results
of the binomial distribution to calculate the probabilities:

P(X=8 | θ) = θ8 (1 – θ)(145-8) …. Eq. 5

Thus, 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 of
approximately 1:3:4:4. It is obvious that theory B explains the data about 3 times as well as
theory A. Here, first let us look at the 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 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. 6
And p = P(X = 8 | θ = 0.03)+ P(X = 9 | θ = 0.03) + +…+ P(X = 145 | θ = 0.03)

 0.07


Therefore, at the given level of significance, α = 0.10, the null hypothesis is rejected and and it is
concluded that the high voltage has a significant effect on the cancer rate among the women staff
at the school.

Bayesian Statistics:

For Bayesian analysis, we can look at the probabilities of getting 8 cancers given incidence rates
of 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) = 0.28

If we add the probabilities of X = 8 given  = 0.04,  = 0.05 and  = 0.06, we get 0.77. Thus, the
Bayesian P( > 0.03) = 0.77, which would not be sufficient to reject the null hypothesis. This is
in contrast with the conclusion from the conventional analysis.

Case Study 3 – Another Effect (thromboembolism) to Cause (Contraceptive Treatment)
Analysis

This case study has been taken from many articles published in British Medical Journal.[3-7] Four
case-control studies (including 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.[4-7]

Conventional Statistics

The 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 by
conventional analysis? As we illustrated in previous examples, conventional analysis does not
calculate the probability of a hypothesis being true given the data.

Bayesian Statistics

The details of the Bayesian analysis of this case study may not be warranted as the goal of this
paper is to just introduce the latter to non-statistical readership. However, the

Bayesian analysis showed that 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.
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)?"[8]

CONCLUSIONS

Conventional statistics (also known as Frequentist statistics) make no claims about probabilities
although one get misled that one is establishing a probability of something. For example, the
95% confidence interval (CI) of mean does not claim there is a 95% probability the mean is in
that range. What it states is 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 – it
gives only the likelihood of the data given the hypothesis. Maximum likelihood estimation is
thus only a little more reliable than conventional statistics.

Bayesian statistics, on the other hand, does give you the probability of a hypothesis being true
given the data. This is closest to intuition and normal process of decision making. Everyone
would normally want to know if a hypothesis is given some observed data, such as, given an
effect whether the cause is true. The three case studies that have been looked at in this paper
represent three different scenarios, each of which has its own place in clinical research. The first
one shows, if the prior and posterior are equally influenced by data, then the conventional and
Bayesian 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 the
hypothesis. 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 was
considered to be true.

REFERENCES

1. 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.



6. 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.


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