In contrast, in an adaptive clinical trial, patient outcomes can be used as they become available to adjust the assignment of future patients, assigning more of them to the better treatment. This allows one to improve expected patient outcomes during the experiment, while still being able to reach good statistical decisions in a timely fashion. Thus, adaptive procedures can offer significant ethical and cost advantages over standard fixed procedures. A situation in which adaptive procedures have become particularly relevant are AIDS clinical trials, since many new drugs come to market, yet classical randomized studies may pose ethical dilemmas and time delays. The are also quite important in pre-clinical animal trials, phase I trials, and phase I/II trials. While there had been some opposition to their use quite some time ago, now NIH, FDA, EPA, etc. are encouraging their use.
Adaptive designs also arise naturally in computer-related areas such as computer systems resource allocation or parameter selection for repeated computer simulations, since the computer can both collect data and run a program to decide what to do next. They can be extremely useful in industrial settings as well, such as acceptance testing. Generally speaking, adaptive sampling procedures are of interest to investigators using statistical experimental designs, to those concerned with medical ethics, to people working in stochastic optimization, control theory, decision theory, machine learning (where this subject is sometimes known as "active learning"), as well as to computer scientists.
The experiment starts on the left with a pull on arm 1. (The choice of arm 1 was arbitrary - we could have just as easily started with arm 2). We initially assumed that arm 1 had 1/2 probability of success, and if we observe a success, since we assume that there is some underlying bias, now our estimate of success probability increases 2/3. Similarly, had the first pull been a failure, then our estimate would drop to 1/3. If we have a success then the diagram shows that we try arm 1 again. If we observe failure on the second pull, then we try arm 2, and so forth. The black numbers at the right indicate the number of successes observed if that is the path followed, and the red fractions at the right indicate the probability of that path occurring. This probability is the product of the probabilities along the edges of the path.
There is an important difference between this experiment, in which each arm has some unknown bias, and pulling on arms where it is known what the success rate of the arms are. Suppose, for example, that we know that the success rate for each arm is 1/2 (as opposed to its expected value being 1/2 as it is with the uniform prior). In this case, pulling the arms provides us with no new information, and thus, in 4 pulls the expected number of successes is 2, regardless of which arms are pulled. In our adaptive experiment, however, the outcomes teach us something about the different arms, and by adapting to use the more successful arm, we increase the expected number of successes to more than 2.27, even though initially we only assumed that each arm was equally likely to produce a success or a failure. The more pulls we use, or the more arms there are to choose from, the more the adaptation can help.
The bandit model is extremely general. For example, suppose you have several restaurants in your area, and you want to eat at the best. You can keep going to the same restaurant, or try different ones in the hope that they might turn out to be even better. Going to the same restaurant is sometimes call ``exploitation'', while trying new ones is ``exploration''. In clinical trials these concepts are sometimes called ``learn and confirm'', with the learning coming in early stages of the clinical trial process (such as stage II), and the confirming coming in subsequent stages (such as stage III). Optimal solutions to such bandit problems make tradeoffs between these aspects. Note that if you are soon going to leave the area then you'll emphasize the exploitation part, while if you are new to the area then you'll explore.
The restaurant example provides another possible goal and set of actions: rather than wanting to just identify the best and keep going there, you might instead want to keep a small group that you go to. This is related to what is known as ``multiple plays'', and is useful because it allows one to incorporate other considerations such as variety, or ability to go to a different restaurant if the one you want is full. This is an especially useful strategy if you live near the University of Michigan and you are trying to find a restaurant on a weekend when there is a home football game.
One form that is quite important is when you are trying to find a new streak of gold. You may take rock samples at many different places, and then when you've found some that look promising you'll go back and take more detailed samples in the area. Then you might select a single spot where you'll spend a great deal of time trying to locate the streak.
In this example, geography and location in space are important. You're determining your second round, and perhaps third round, of detailed sampling based on the results of sampling on a much coarser grid. You are assuming that nearby areas are correlated, so that if there is a gold streak then some of the rocks nearby will have at least a little gold in them. It's not exactly arms as in the bandit model, but you'll still balance exploration and exploitation aspects if you want to do well. You should also be prepared for a negative expected outcome.
Another example is when you are trying to estimate the difference between the average income of workers in fast food restaurants versus the average income of waiters in regular restaurants. You survey workers in each population and quickly discover that the fast food workers all get quite similar, low, pay, while the waiters have widely varying income, based on factors such as the quality of the restaurant, the skill of the waiter, etc. Thus if you want to accurately estimate the difference in the average pay, you'll need to sample many waiters to improve your estimate of their average. If you'd known in advance the variance of the waiter's and fast food worker's salaries, then you could have figured out in advance how many to sample from each group, but since you don't know this you first do some sampling to estimate it, and then refine the number of samples you'll take from each group.
Unfortunately, adaptive procedures are more complicated to design and to analyze, and in some settings are more difficult to implement. Most people using formal experimental designs use fixed sampling procedures where they decide in advance how many samples of each alternative they are going to collect. These fixed procedures are easier to design and analyze, because they don't have as many alternatives, but they are less efficient, sometimes significantly so. In many cases, however, users are unaware of the option to use adaptive designs because standard statistical courses and packages do not include them. In medical problems, people may be wary of adaptive designs because of the confusion between a controlled clinical trial using an adaptive design versus allowing the physicians to decide each patient's treatment based on professional experience. The latter can introduce many biases into the learning process, especially because physicians tend to use the exploitation approach for each patient, as opposed to including some exploration.
There are some complicated ethical issues here - which is better the better option?
We have developed a variety of new approaches and computational algorithms that make it much easier to exploit such designs. Here is a more detailed explanation.
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