Traditionally, most trials have used a frequentist approach. However, Bayesian approaches have become more popular. Bayesian statistical inference, is rooted in what we call Bayes theorem. Consider two events, A and B. The probability that A occurs given B, is equal to the probability that both A and B occur, divided by the probability of B, which is proportional to the probability of B given A, times the probability of A. Now, you can see, that we have directly reversed which action we are conditioning on. In the beginning, we were saying A given B, and at the end we're conditioning B given A. In Bayesian statistics, the parameter Theta, you can think of this as your treatment effect, is an unknown quantity with a probability distribution. It is not a fixed value or underlying truth as frequently system. The goal of Bayesian statistics, is to estimate the posterior probability distribution of Theta, given your prior information and observed data. In trials, it is used to formally combine information from previous studies or experience with current data. As Dr. Tom Louis says, "We are all Bayesian at the design phase." The assumptions that are used to design the trial, are based upon expert or prior knowledge. We generate our hypotheses, our sample size, and statistical models based upon these assumptions. How does Bayes theorem apply to statistical inference? There are three components; the prior, the likelihood, and the posterior. The prior, is our initial hypothesized distribution of the parameter Theta, before we collect any data. This is where our data from other studies or expert knowledge comes into play. The likelihood function, which is what frequent is used to analyze their linear regression, or Poisson regression, is a probability function associated with the data Y, conditioned on the parameter Theta. Bayesian statistics combines the prior and the likelihood to create the posterior distribution, and update distribution of the parameter Theta after collecting the data. We have learned from the data, and updated our prior knowledge. Let's look at a motivating example. Suppose we had a 100 mice assigned to one of two treatments, experimental or control designated by E and C. The table below shows the results of our experiment. In the experimental group, 3 out of 47 mice developed a tumor. In the control group, no mice developed a tumor. Scientists in this case, were very excited. This is biologically extremely significant. We've never seen a tumor before. However, the statistician protest it's not statistically significant. A Fisher's exact test, does not achieve the level of significance of less than 0.05. This is where the prior knowledge becomes important. The scientist observed, no tumors out of 450 controls under similar circumstances; same lab, same type of mouse, same exposures. That would update our total number of mice to 0 out of 500 developing a tumor. If we compare that to the 3 out of 50, we are extremely statistically significant. We need to find a way to incorporate that prior knowledge into our test, and not subject extra mice to unnecessary experimentation. How do we choose the prior? We'll the goal of the prior, is to summarize knowledge about the unknown parameter Theta. It can however, be difficult to transfer knowledge into a probability distribution. There are many sources of information for this; data on similar projects, pilot studies, patient registries, trials in different countries. All of them can be used to determine the shape of the prior distribution. However, no matter how much knowledge you use, the choice of your prior can be very controversial. If applying for regulatory approval, you must clear your choice of prior with the regulatory agency before starting the study. Or at the end of the study, they could choose not to approve your intervention, because they disapprove of your prior. Now, priors come in many different shapes. The plot on the right, shows several examples of different shapes of distributions. Informative priors, give more weight to some values than others. Examples of those, would be those curves with a peak, such as the purple line with the peak on the left-hand side around 0.2. A non-informative prior indicates a lack of preference. You are essentially saying, I don't know. In this case, you want to give equal weight to all possible values of Theta. This is often used when you have no prior data. An example of this, is the handlebar type of prior, which is essentially flat in the middle between 0.2 and 0.8. An example of this, is the blue line. It's the line in the middle with a U-shape. It is essentially flat between 0.2 and 0.8, indicating no preference, with a little bit of extra weight at the extreme values. One thing to consider when choosing your prior, is the effective sample size. You don't want the prior to overshadow the actual data from your trial. The goal of calculating the effective sample size, is to quantify the amount of information contained in the prior. How many individuals in your study does this prior equal? It's very useful to give feedback to clinicians on the shape of their chosen prior. For example, if you are planning on 100 patient study, but your prior ESS is 500, then clearly you need to modify the prior, so that it doesn't outweigh your 100 patients. You can confirm that an uninformative prior is actually vague with little weight at any particular place. In an outcome adaptive experiment, you can ensure that the data drives the adaptations, not the prior. If you're reviewing a paper or an application based on a Bayesian design, you can check the designer's assumptions. Let's go through a toxicity example. The dotted blue line is the prior. It is U-shaped and uninformative. We can see that its initial estimate for Theta is at 0.05. We then collect data on a 100 participants. Ten out of the 100 have a toxicity and we assume that they follow a binomial distribution. This allows us to update our prior to the posterior, which is shown by the solid purple line with a peak on the left hand side. You can see that the peak is very close to the 0.1 estimate that we get from the actual data and very little information is being provided by the prior. Our new estimate of the proportion with the toxicity is now 0.104 or 10.4 percent. Let's compare the Bayesian frequentist approaches. For Bayes, the parameter Theta is an unknown quantity with a probability distribution. For frequentists, the parameter is an unknown fixed quantity. There is one single truth. Perhaps one of the biggest differences is in the interpretation of the interval designed to give some idea of the uncertainty or variability of your point estimate. A Bayesian design looks at the credibility interval, for example, a 95 percent credibility interval. The interpretation of this interval is the probability that Theta lies between A and B. The limits of our interval is 0.95. We treat the boundaries, A and B, as fixed and Theta as variable. The interpretation of a confidence interval in a frequentist analysis is very different. Although people often mistakenly describe the credibility interval definition as the confidence interval definition. For a 95 percent confidence interval, we're saying that if we repeated the study multiple times, 95 percent of our intervals will contain the true value of Theta. Theta is fixed, and now our boundaries A and B are variable. You should know that for any specific interval, it will or will not capture the actual true value of Theta. Finally, let's consider the inference that we make. For Bayes, we say, given what we have observed, how likely is it that the null hypothesis is true? Now this directly answers our question. Should we reject the null? In contrast, the inference for the frequentist design is a little more awkward. We say, assuming the null hypothesis is true, what is the chance that we saw this data, or something even more extreme, by which we mean further away from the null. As you can see, the interpretation of a Bayesian analysis is more intuitive than the interpretation of the frequentist analysis, although the two interpretations are often confused. What are the advantages of using a Bayesian design? It's easy to incorporate external and historical information, including multiple sources of uncertainty. It follows the learning paradigm. We update our understanding with new information. We build instead of starting from scratch. Bayesian inferences is more natural than frequentist inference for interpretation and credibility and the results of Bayesian inference is more natural than for frequentist inference. Prediction is also natural as well. What is the probability of the future given the past fits directly into the model? There are some criticisms, however. Probably the most common criticism is in the selection of the prior. Different priors can lead to different conclusions and so this can cause confusion. Also, an influential prior can outweigh the information that you gained from the trial data. However, a large sample size can minimize the influence of a prior. Bayesian studies are more difficult to carry out. Extensive preplanning is needed, and statistical guidance is also necessary to understand the assumptions and interpret the conclusions. Finally, it is computationally intensive, which is less of a concern now with new computing methods. Some final thoughts to consider about Bayesian designs. The interpretation of results and inferences intuitive, which is a big advantage. You do need fast access to the data used to make the decisions. You must consider the risk of the design in terms of bias, type I error, type II error, information leak, distrust of results, which means planning is key. You need a clear protocol and analysis plans. It's important to extensively evaluate the design with simulations under both the scenarios that you expect to occur and those that you do not expect. Then finally, you should be very careful in your choice of prior and way having undue influence. I hope this introduces you to Bayesian trials and why they might be useful.
