Retrospective sampling, on the other hand is powerful in the context of infinite dimensional. The computation when using this method is inexpensive and exact. A few simulation examples illustrated in the talk are for unnormalized probability density function, alternating series and retrospective MCMC.

Simulation of diffusions is a Markov process described by stochastic differential equation. It involves Brownian motion and Euler approximation. An interest lies in simulating without discretisation error and obtaining realisations of the whole path.

1. Apply Bayesian adaptive to a case study

2. Demonstrate the benefit of it and provide advance understanding of using this approach in regulatory environment.

The authors focused on phase III study, in a trial for patients who had DSM-IV diagnosis of schizophrenia and related disease. The methods advocated here follow all aspects of trial protocol except that a Bayesian approach is employed in the analyses, decision criteria and treatment allocation which is based on predictive probability of treatment response without the side effect. The prior was obtained by combining historical pharmacokinetics data and expert opinon. Patients were randomized first according to the ratio 2:2:1 and then Bayesian adaptive is applied to assign treatment to patients. To assign this treatment, it depends on the calculated joint predictive probability of treatment safe and patient response. This method will reduce patients’ exposure to ineffective treatments or treatments with adverse side effects. MCMC was used to draw since the posterior distribution did not have a closed form.

The authors reached the same conclusion to the analysis previously done, except that they could stop early for efficacy. They required only a half number of patients from the original frequentist study. Since the trial completed earlier, they estimated that 40% of original time and 1 million dollar were saved. Sensitivity of the design with respect to prior distributions was investigated by simulation. If informative priors were used with respect to simulated data, then power is 100% for rejecting false null hypothesis. On the other hand if noninformative priors with respect to simulated data were used, power can be achieved up to 79%.

The authors believed that adaptive design is a favorable approach to ensure that the new entity reach the market sooner and people can obtain the medicine at a lower cost. At the end of this paper, Spann et al. (2008 ) wrote a bit on the difference of frequentist and Bayesian ideas.

The purpose of this paper written by Daimon (2008 ) was to demonstrate the use of Bayesian approach to calculate the sample size, by assuming normal prior for the “sum of the true difference and the non-inferiority margin”. Note that in the frequentist approach, uncertainty surrounds the value given to the true difference and the margin. However, when one construct the equation for the sample size, it can be seen that those two values can be taken as one unknown value, which is the “sum of the true difference and the non-inferiority margin”.

Daimon first described the idea of Neyman-Pearson power before giving a detailed account on Bayesian probabilities. In the context of Bayesian approach, he discussed three possible methods to find the necessary sample size:

Hybrid Neyman-Pearson-Bayesian  probability (hNPB): This is also known as the expected power or strength. It takes account of available prior and is used to critically evaluate Neyman-Pearson power formula. It utilizies Neyman-Pearson critical region.

Conditional Bayesian probability (CB): This is a fully Bayesian design where priors are incorporated explicitly. This method allows one to find the probability of obtaining a significant Bayesian result, for any given data.

Unconditional Bayesian probability (UB): Probability of obtaining a significant Bayesian result is not conditional on the pre-specification of the sum of alternative difference and noninferiority margin.

Sample size are calculated and compared in these four approaches: Neyman-Pearson, hNPB, CB dan UB. It is found that sample size from CB is smaller to what has been obtained when using Neyman-Pearson. Sometimes, when using Bayesian approach, there is a possibility of obtaining only 1 subject for the trial, and this is not realistic. So in this situation, Bayesian method is not applicable.

Daimon proposed that this method could possibly be applied to hazard ratios, odds ratios and continuous variables because it uses a large sample aprroximation and the conjugate normal analysis. Daimon suggested that it is worth to check all the sample sizes from those methods and choose the appropriate sample size. It is also a good practice to check the validity of prior specification; for example normal prior here if not reliable can be substituted with beta prior or log normal prior.

Sample size calculation is an essential step in any designed trial.A conventional single-stage design for testing both objectives will choose the sample size based on the primary objective. In this situation, power for the second objective is going to be affected and the potential to reproduce the result is lowered as well. Given these problems, the athours of this paper, Shao et al. (2008 ) proposed a hybrid Bayesian-frequentist method. The essence of this approach is by specifying a probability distribution to the experimental treatment as a way to express uncertainty surrounding the efficacy.

The authors discussed this idea in the context of binomial data of 2-arm trial, involving the experimental and the reference treatment.  Testing for both objectives is done based on closed-testing principle. Based on this principle, there is no inflation of type I error because the null hypotheses are nested (Morikawa and Yoshida, 1995) and (Dunnett and Tamhane, 1997).  However, type II error is inflated and this leads to a severe loss of resources. Shao and Chow (2002) stated that result of the trial in phase III must be able to reproduce and so there will be an issue here when the power of the trial is low.  Kayama and Westfall (2005) favoured this design rather than the test-one-only design, based on their study conducted from the perspective of decision theory.

The authors of this paper applied Bayes formula in tackling the issue of power and reproducibility in this two-tests design. The uncertainty of either experimental or reference treatment is expressed via probability distribution. They considered two situations: first situation is when experimental is uncertain but reference is a known constant and the second situation is when both are uncertain. Power is evaluated with respect to the uncertainty of experimental. There are two types of power: adjusted power and average power.  The sensitivity of the result to the choice of distribution is studied as well. The final analysis however does not rely on any distribution but uses the classical frequentist methods.

There is a section in this article, where the author described the basic of Bayesian idea, called “Bayesics“. One must be familiar with the terms such as: probability model for the data, prior probability, posterior probability, predictive distribution for future data, marginal posterior density, bivariate marginal density, posterior moments and highest posterior density.There are many areas of clinical trials where Bayesian has come into the picture. One of them is bioequivalence.

The aim of bioequivalence study is to determine whether two formulations of a drug are equivalent. Westlake (1972) claimed that hypothesis testing is not relevant. His claim was supported by Metzler (1972). Westlake argued that clinical pharmacologists work with ratios rather than difference and so confidence interval is more appropriate in that sense. Bayesian approach started to develop in the bioequivalence area in 1979 by three independent groups: Merck, Sharp and Dohme, Ciba-Geigy (US) and Ciba-Geigy (Swiss). In 1983, the ratio of the area under the curve (AUC) was commonly accepted as 0.8 and 1.2 and that 95% confidence interval is required for that ratio. Bayesian approach advocates the use of credible interval or posterior probability.

In the situation where a small single bioequivalence study is being conducted, proving that two drugs are similar in efficacy would be difficult. So, a 2-stage Bayesian approach is adopted. The first stage is called the screening stage and the sample size chosen here is small. Sample size in the second stage is chosen based on the result of the screening stage. This is an early example of adaptive design and making use of predictive distribution.

The standard approach to bioequivalence today is by conducting two “one-sided t-test”and so simultaneous rejection will indicate that two treatments are equivalent. Bauer and Bauer (1994) raised a question where the means of those two could possibly be close but the variances on the other hand, are large. They proposed testing an extra two “one-sided F test”; so there will be four null hypotheses altogether to be rejected to be able to claim bioequivalence. In the Bayesian perspective, this can be done by determining the joint posterior of say, the difference in means and the ratio of variances.

Grieve did not review the implementation of Bayesian method in non-inferiority trial. The non-inferiority term was first introduced by Blackwelder (1982). However, he quoted a statement from Susan Ellenberg:  “area of non-inferiority is one that really lends itself to the Bayesian approach”, which pretty much summarized up Grieve’s idea.

In the late 1970s, the crossover design received many attention. The basic design is by having 2 treatments with 2 periods. The FDA however, said that it is not a good design when “unequivocal evidence” of a treatment effect was required. In the 1980s, Grieve started to work on Bayesian approach in the crossover design. Bayes factor is used to determine whether carryover effect should be included or not. In this section, Behrens-Fisher distribution came up as well. The problem of Behrens-Fisher is about interval estimation and hypothesis testing betwen the means of two normally distributed population, when the population variances are assumed unequal and come from independent samples.

In the early to mid 1980s, analytic approximation to posterior quantities of interest were being studied. There are four approaches to analytic approximation:

Specific expansion for ratios of integral to determine posterior moments and predictive distributions: This method requires calculation up to third order derivatives of log likelihood function, which is cumbersome. It also provides no correction to variance and covariance.

Laplace approximation to integrals: The advantage of this compared to the first one, is it only evaluates up to second derivative only. It is also possible to find marginal distribution by integrating over a subset of the parameters in the numerator.

Importance sampling: The idea is to choose a density of the parameter that approximates the probability model of the data. The efficiency depends on how well the approximation is.

Iterated Gaussian quadrature: Basic assumption of this method is posterior can be approximated by the multiplication of normal density and low order polynomial. In practice, we would not know the posterior mean and the covariance matrix and so Gaussian quadrature will update the estimates till it converges.

All the approaches described above behave well in small-dimensional problems; that is up to 6-10 parameters. If the parameters increased, the best way is to use Gibbs sampling or MCMC.

In the early 1980s, time was spent to studying the analysis of animal toxicology. The objective of this study is to estimate a dose-response curve and to determine LD50, a dose that kills 50% of the animals exposed to it in a fixed time interval. For this kind of design, logit and probit model are used and parameters of the model are estimated using the maximum likelihood estimator, weighted least squares and Fieller’s theorem. Fieller’s theorem is used to find the confidence interval for the log of LD50. Technical issues are noted to arise when using Fieller’s method: the interval consists of whole real line and as a result, little is learnt about LD50. This is said due to a small number of experimental units being studied. Grieve hoped that Bayesian could circumvent this issue.

Systematic investigation regarding the use of Bayesian in the drug development was carried out in 1982. The work was published in Racine et al. (1986). The obstacles identified in incorporating Bayesian approach are related to numerical implementation, utilizing priors in regulated environment and conservatory attitute among pharmaceutical statisticians and respective colleagues. Compared to the same paper read by John Lewis in 1983, Racine et al. (1986) was received in defeaning silence by the Royal Statistical Society.

The MCMC revolution in the 1990s enable multi-parameter problems to be tackled without having to simplify the models or the assumptions. The general form of MCMC is based on Metropolis-Hastings algorithm which is a special case of Gibbs sampler. The idea of Gibbs sampler is like this: say we have k parameters of interest and we want to sample from the posterior density. Starting values for the parameters are given and sampling is done in sequence for a large number of iterations, say 10 000. In practice, the first say, 1000 iterations are to be discarded, also called as “burn-in”. For a more complex problems the number of discarded iterations could be more than 1000. The mean of values from the sample is the estimate of the posterior mean. The same goes for the posterior variance. The kernel estimate is used to construct the graph of a posterior.

Other than the issue of efficacy and safety, one question that need to be answered is whether the drug can be manufactured consistently in terms of content, uniformity and level of contaminants. So, batches of drugs have to be analyzed and Bayesian approach can be implemented here. Historical data will be taken into consideration and a prior is formed. For example, one could assume that proportion of defectives in a particular batch has a beta distribution. The number to be sampled from the batch will be based on these calculation.

Adaptive dose-finding studies make an appearance in the 21st century, though the idea has already been discussed in 1933 by William Thompson. William described this design as “play-the-winner” design. There are two classes of design: one is to determine maximum tolerated dose in oncology studies and the other is to learn about dose-response curve when it can be non-monotonic.

Pharmacoeconomics is another area where concern arises due to escalating health-care costs. So, demonstration value for money is required and the method is by evaluating the cost data. From the Bayesian point of view, probability statements can be made about populations incremental mean values. John Stevens and Tony O’ Hagan are among the famous statisticians working in this area.

To conclude, Bayesian approach enable direct probability statements about parameters to be made and easier to be interpreted. Full accounting of uncertainty can be seen for example in using Bayes factor in the crossover design. Other than estimating and testing hypothesis, predicting is also of interest. Grieve suggested using predictive ideas. Predictive powers on the other hand is when it is conditioned on the observed data. Grieve quoted a statement from Stephen Senn: “nowhere will Bayesian statistics to be carried out with greater discipline than in the pharmaceutical industry”. Grieve predicted that Bayesian will be heading next to drug safety, playing the role in not just the development but the post-marketing as well.

Pham-Gia and Turkkan (1992) remarked that sample size determination is rarely discussed in the Bayesian context because of the nature use of posterior as a prior for the next step and information will be updated and so on until a conclusion is reached. This is also called a sequential approach. However, in real life sequential approach is not always feasible and so the fixed sample size has to be determined.

Adcock (1988 ) studied the sample size in the normally distributed data with known variance and unknown variance.  A closed formulae for each case is provided and is done by averaging the fixed length posterior credible sets over predictive distribution. This is a similar technique advocated by Joseph and Belisle (1997). For binomial sampling, there are plenty of papers that can be referred [Adcock (1987, 1992, 1995), Joseph et al. 91995) and Pezeshk and Gittins (2002)]. Joseph et al. (1997) studied the sample size for the difference in binomial sampling.

Pezeshk divides the approaches used in determining the sample size to four major categories:-

1. Bayesian and mixed Bayesian/likelihood criteria

2. Bayes factor or Weiss’ method

3. Methods of moments

4. Decision theoretic/fully Bayesian

The author defines that the posterior is the normalized likelihood when the prior used in predictive density is uniform. He also reserves the use of fully Bayesian approach when one implements utility or loss function only. This differs from Joseph et al. (1997), stating that when true prior is used in predictive and posterior, then it is called fully Bayesian. Method named Average Coverage Criterion (ACC) is also known as tolerance intervals in Adcock (1987). Fraser and Guttmann (1956) named it as tolerance region approach while Raiffa and Schlaifer (1961) referred to it as pre-posterior analysis. Pezeshk also discussed the idea of average length criterion, which was first introduced in Joseph et al. (1995) and the worst outcome criterion (WOC).

Bayes factor is a standard way to test two hypothesis. What we need to do is,  compare the posteriors of these two hypotheses. Sometimes when the priors are not available, it is sufficient to just look at Bayes factor, which is the likelihhod ratio. Since the likelihood ratio yields a non-negative value, it is convenient to take the log of it. If a positive value is obtained, that means the null hypothesis is favorable than the alternative. The interpretation of Bayes factor can be found in Kass and Raftery (1995). Weiss’s method take the idea of Bayes factor and determine the sample size by first specifying the type I and type II error, followed by solving 2 equations simultaneously to obtain the cut-off point and the sample size.

Methods of moments avoid difficulties in the previous methods discussed. Here, we need moments of posterior distribution (eg: posterior variance) and average over predictive distribution. An upper bound of the posterior variance is used to determine the sample size. In this section, Pezeshk listed some important definitions:-

Loss function:describes the discrepancy between value of the parameter and its estimate. It is defined to be greater or equal to 0.

Expected value of perfect information (EVPI): also known as expected cost of uncertainty surrounding the decision problem. It means expected loss of action that is optimal under the current state of information, prior to sampling.

Expected value of sample information (EVSI): describes an amount that is paid for information provided by the sample. This is a function in terms of sample size.

Expected net gain of sampling (ENGS): this is also a function in terms of sample size and an optimal sample size is determined by maximizing this function. This is also equal to EVSI – sampling cost.

Pezeshk gave a short review of fully Bayesian approach done by Raiffa and Schlaifer (1961). They made use of utility function and their method is also known as maximization of expected utility (MEU). Other work in this area include O’Hagans and Seteves who studied the cost effective analysis from clinical trial data, Statlard et al. who concentrated on sample size for phase III trial and Pezeshk and Gittins who provided a simplified version of Raiffa and Schlaifer (1961) in the form of algebraic terms.

Pezeshk gave an example where Bayesian approach is applied to normal distribution. In the case of known variance, Joseph and Belisle pointed out average coverage criterion (ACC), average length criterion(ALC) and worst outcome criterion (WOC) give the same sample simple. The result from using frequentist method and Bayesian is found to be approximately similar. However, the similarity ends in the case of unknown variance.  To handle the situation of unknown variance, Bayesian approach will have to consider a prior for the variance. The use of Bayes factor is possible but will be more complicated. Frequentist, on the other hand may need to sample twice or get the independent estimate of variance.

Another example given is related to the use of decision theory in the actual clinical trial with binary response. It is normal to specify a beta prior for binomial likelihood. In this example, one needs to consider the total cost per patient and the expected net benefit. One thing noted when Bayesian approach is to be applied in the analysis is, regulator might choose the same beta prior but with different parameters to reflect scepticism.

To compare two treatments; warfarin and heparin for binomial data in the clinical trial for patients with major knee surgery, what we need is the estimates of the proportions in these two groups. However, in practice we don’t have point estimates. Instead, we have a distribution over a range of values.  Therefore, the solution would be to use a prior distribution.

Many papers related to finding the necessary sample size have been published, although in different context. Dudewicz (1972)  studied confidence interval for power for normal density. Goldstein (1981) derived the sample size in estimating the mean of an arbitrary distribution. Gould (1983) considered both frequentist and Bayesian to see the relationship between power and sample size. In 1985, Berger adopted decision theoretic approach to find optimal sample size for fixed and sequential sampling. Spiegelhalter and Freedman (1986) produced a paper about predictive approach based on power considerations.

Adcock (1987, 1988 ) and Pham-Gia and Turkkan (1992) determined the sample size in multinomial, normal and single binomial expressions.  Gould (1993) had the sample size for difference in binomial proportions for equivalence trials. Joseph et al. (1995) used HPD (highest posterior density) from exact posterior distribution to find the sample size for single binomial parameter. This approach is said to be optimal because the sample size is minimum for any given coverage. To get the sample size for complex survival studies, a large number of data sets were simulated in Yateman and Skene (1993).

To use Bayesian approach, we need to be familiar with predictive distribution which is also known as pre posterior marginal distribution of data and the posterior distribution. The method is termed as fully Bayesian when true prior is used in predictive and posterior equation. Else, it is called mixed Bayesian/likelihood when true prior is used in predictive but a uniform density if used to replace the prior in the posterior equation. The mixed Bayesian/likelihood could be expressed as using the prior for predictive and only likelihood for the inference. This fits the situation where investigators recognize the importance of Bayesian approach in the planning stage and final analysis is then based on data set.

The aim of this research is to have highest posterior density (HPD) or posterior credible interval of length l to cover the parameter at (1-alpha) coverage probability. Note that in the planning stage, posterior distribution depends on data that has not been observed yet. So, uncertainty comes in and this is not good. There are four ways to eliminate this uncertainty:-

1. Average Coverage Criterion (ACC) : where HPD is fixed at l but coverage probability varies depending on the data.

2. Average Length Coverage (ALC): where probability coverage is fixed but HPD varies depending on the data.

3. Worst Outcome Criterion (WOC): because ACC and ALC depends on data, it can have large/small coverages/lengths depend on which sample is being observed. This property is not desirable. A conservative approach is to fix a maximum length and a minimum coverage probability and the sample size is chosen based on these two constraints.

4. Modified Worst Outcome Criterion (MWOC)

In this research, the parameter of interest is the difference of two binomial proportions. Sample size is found by implementing ACC, ALC ,WOC and MWOC.  When the sample size is large relative to prior information, the estimation obtained from posterior distribution is found robust against prior specifications (either independent or dependent prior). To illustrate the idea, three examples are given:

1. Data on patients with deep vein thrombosis (DVT)

2. Rare events

3. Dependent priors

For ACC and ALC, the sample size obtained is based on the average of 10 repetitions of MC simulation. MC simulation is used when it is difficult to calculate the posterior. However, MC simulation is not needed for WOC and MWOC. In the first example of DVT, WOC method gave a slightly less sample size than the frequentist method. For the rare event, like the rates of myocardial infarction patients with acute angina pectoris, the difference in sample size between Bayesian and frequentist is more pronounced. The last example compared the sample size calculated for independent and dependent priors.

Result showed that fully Bayesian or mixed Bayesian/likelihood give lower sample sizes than the frequentist due to the use of prior. A caution has to be exercised when employing mixed Bayesian/likelihood as it may lead to paradoxes. The methods discussed in determining the sample size (ACC, ALC, WOC, MWOC) may be applicable to multivariate analysis, relative risks or odds ratios. Choosing ALC or ACC is arbitrary though given certain situations they don’t lead to the same sample size.  However, ACC is said to be more conventional because reports are always based on coverage. Note that criteria based on average (ACC and ALC) attain target values approximately half of the time. Therefore, the authors suggest that to get a good sample size, it is best to consider all criteria.

“Bayesian paradigm has little to add and moderate amount to lose”. Fisher suggests that:-

1. Data is not being optimally processed by humans, given the situation is stable.

2. Humans tend to weigh more heavily on recent experience.

3. Humans weigh more heavily evidence from direct experience.

Therefore, it is difficult for a person to judge a problem and state the prior rationally. More reasons are provided by Fisher to support his argument.

1. Humans do not behave in a Bayesian manner.

2. Individual prior distributions must of necessity be distorted because of a lack of an adequate initial distribution.

3. Whose prior to be used when there is a large range of prior distributions?

4. Lack of acceptance of result because of dispute over prior beliefs.

5. Misdirection of scientific effort from the debating of particular science issues to work on eliciting prior distribution.

6. Ethical difficulties because of differing types of belief.

In 1993, the data from GUSTO trial is reanalyzed using stylized Bayesian (Diamond et al., 1993). GUSTO stands for global utilization of streptokinase and tissue plasminogan activator for occulded coronary arteries. In this analysis, individual priors are not elicited. Instead, priors are specified based on Jeffreys (1939). A true Bayesian would object this approach. This example given by Fisher is to demonstrate the effect of prior on the interpretation.

In the end, Fisher concludes that Bayesian approach should not be used in randomized clinical trials. He believes that frequentist method or stylized Bayesian is more suitable to ensure the analysis is robust against particular beliefs.

Goldstein discussed the following criteria in determining the sample size:-

1) Stop if further sampling does not give much difference (in terms of the estimate)

2) A stringent criteria would be to require a high probability that future observation satisfy criteria (1)

The author suggested that once the smallest n is obtained by fulfilling the inequality constraint posed, then an upper bound of that quantity can be derived. In order to do this, we need the variance of a prior measure for parameter of interest and prior expected variance of the data.