Type I error for smoke detector: probability of alarm given no fire

Bayesian: probability of fire given current air analysis

Bayesian: probability of fire given current air analysis

If I had been taught Bayesian modeling before being taught the frequentist paradigm, I'm sure I would have always been a Bayesian. I started becoming a Bayesian about 1994 because of an influential paper by David Spiegelhalter and because I worked in the same building at Duke University as Don Berry. Two other things strongly contributed to my thinking: difficulties explaining p-values and confidence intervals (especially the latter) to clinical researchers, and difficulty of learning group sequential methods in clinical trials. When I talked with Don and learned about the flexibility of the Bayesian approach to clinical trials, and saw Spiegelhalter's embrace of Bayesian methods because of its problem-solving abilities, I was hooked. [Note: I've heard Don say that he became Bayesian after multiple attempts to teach statistics students the exact definition of a confidence interval. He decided the concept was defective.]

At the time I was working on clinical trials at Duke and started to see that multiplicity adjustments were arbitrary. This started with a clinical trial coordinated by Duke in which low dose and high dose of a new drug were to be compared to placebo, using an alpha cutoff of 0.03 for each comparison to adjust for multiplicity. The comparison of high dose with placebo resulted in a p-value of 0.04 and the trial was labeled completely "negative" which seemed problematic to me. [Note: the p-value was two-sided and thus didn't give any special "credit" for the treatment effect coming out in the right direction.]

I began to see that the hypothesis testing framework wasn't always the best approach to science, and that in biomedical research the typical hypothesis was an artificial construct designed to placate a reviewer who believed that an NIH grant's specific aims must include null hypotheses. I saw the contortions that investigators went through to achieve this, came to see that questions are more relevant than hypotheses, and estimation was even more important than questions. With Bayes, estimation is emphasized. I very much like Bayesian modeling instead of hypothesis testing. I saw that a large number of clinical trials were incorrectly interpreted when p>0.05 because the investigators involved failed to realize that a p-value can only provide evidence against a hypothesis. Investigators are motivated by "we spent a lot of time and money and must have gained something from this experiment." The classic "absence of evidence is not evidence of absence" error results, whereas with Bayes it is easy to estimate the probability of similarity of two treatments. Investigators will be surprised to know how little we have learned from clinical trials that are not huge when p>0.05.

I listened to many discussions of famous clinical trialists debating what should be the primary endpoint in a trial, the co-primary endpoint, the secondary endpoints, co-secondary endpoints, etc. This was all because of their paying attention to alpha-spending. I realized this was all a game.

I came to not believe in the possibility of infinitely many repetitions of identical experiments, as required to be envisioned in the frequentist paradigm. When I looked more thoroughly into the multiplicity problem, and sequential testing, and I looked at Bayesian solutions, I became more of a believer in the approach. I learned that posterior probabilities have a simple interpretation independent of the stopping rule and frequency of data looks. I got involved in working with the FDA and then consulting with pharmaceutical companies, and started observing how multiple clinical endpoints were handled. I saw a closed testing procedures where a company was seeking a superiority claim for a new drug, and if there was insufficient evidence for such a claim, they wanted to seek a non-inferiority claim on another endpoint. They developed a closed testing procedure that when diagrammed truly looked like a train wreck. I felt there had to be a better approach, so I sought to see how far posterior probabilities could be pushed. I found that with MCMC simulation of Bayesian posterior draws I could quite simply compute probabilities such as P(any efficacy), P(efficacy more than trivial), P(non-inferiority), P(efficacy on endpoint A and on either endpoint B or endpoint C), and P(benefit on more than 2 of 5 endpoints). I realized that frequentist multiplicity problems came from the chances you give data to be more extreme, not from the chances you give assertions to be true.

I enjoy the fact that posterior probabilities define their own error probabilities, and that they count not only inefficacy but also harm. If P(efficacy)=0.97, P(no effect or harm)=0.03. This is the "regulator's regret", and type I error is not the error of major interest (is it really even an 'error'?). One minus a p-value is P(data in general are less extreme than that observed if H0 is true) which is the probability of an event I'm not that interested in.

The extreme amount of time I spent analyzing data led me to understand other problems with the frequentist approach. Parameters are either in a model or not in a model. We test for interactions with treatment and hope that the p-value is not between 0.02 and 0.2. We either include the interactions or exclude them, and the power for the interaction test is modest. Bayesians have a prior for the differential treatment effect and can easily have interactions "half in" the model. Dichotomous irrevocable decisions are at the heart of many of the statistical modeling problems we have today. I really like penalized maximum likelihood estimation (which is really empirical Bayes) but once we have a penalized model all of our frequentist inferential framework fails us. No one can interpret a confidence interval for a biased (shrunken; penalized) estimate. On the other hand, the Bayesian posterior probability density function, after shrinkage is accomplished using skeptical priors, is just as easy to interpret as had the prior been flat. For another example, consider a categorical predictor variable that we hope is predicting in an ordinal (monotonic) fashion. We tend to either model it as ordinal or as completely unordered (using k-1 indicator variables for k categories). A Bayesian would say "let's use a prior that favors monotonicity but allows larger sample sizes to override this belief."

Now that adaptive and sequential experiments are becoming more popular, and a formal mechanism is needed to use data from one experiment to inform a later experiment (a good example being the use of adult clinical trial data to inform clinical trials on children when it is difficult to enroll a sufficient number of children for the child data to stand on their own), Bayes is needed more than ever. It took me a while to realize something that is quite profound: A Bayesian solution to a simple problem (e.g., 2-group comparison of means) can be embedded into a complex design (e.g., adaptive clinical trial)

**without modification**. Frequentist solutions require highly complex modifications to work in the adaptive trial setting.
I met likelihoodist Jeffrey Blume in 2008 and started to like the likelihood approach. It is more Bayesian than frequentist. I plan to learn more about this paradigm.

Several readers have asked me how I could believe all this and publish a frequentist-based book such as

*Regression Modeling Strategies*. There are two primary reasons. First, I started writing the book before I knew much about Bayes. Second, I performed a lot of simulation studies that showed that purely empirical model-building had a low chance of capturing clinical phenomena correctly and of validating on new datasets. I worked extensively with cardiologists such as Rob Califf, Dan Mark, Mark Hlatky, David Prior, and Phil Harris who give me the ideas for injecting clinical knowledge into model specification. From that experience I wrote*Regression Modeling Strategies*in the most Bayesian way I could without actually using specific Bayesian methods. I did this by emphasizing subject-matter-guided model specification. The section in the book about specification of interaction terms is perhaps the best example. When I teach the full-semester version of my course I interject Bayesian counterparts to many of the techniques covered.
There are challenges in moving more to a Bayesian approach. The ones I encounter most frequently are:

- Teaching clinical trialists to embrace Bayes when they already do in spirit but not operationally. Unlearning things is much more difficult than learning things.
- How to work with sponsors, regulators, and NIH principal investigators to specify the (usually skeptical) prior up front, and to specify the amount of applicability assumed for previous data.
- What is a Bayesian version of the multiple degree of freedom "chunk test"? Partitioning sums of squares or the log likelihood into components, e.g., combined test of interaction and combined test of nonlinearities, is very easy and natural in the frequentist setting.
- How do we specify priors for complex entities such as the degree of monotonicity of the effect of a continuous predictor in a regression model? The Bayesian approach to this will ultimately be more satisfying, but operationalizing this is not easy.

With new tools such as Stan and well written accessible books such as Kruschke's it's getting to be easier to be Bayesian each day. The R brms package, which uses Stan, makes a large class of regression models even more accessible.