Section author: Danielle J. Navarro and David R. Foxcroft
Why be a Bayesian?¶
Up to this point I’ve focused exclusively on the logic underpinning Bayesian statistics. We’ve talked about the idea of “probability as a degree of belief”, and what it implies about how a rational agent should reason about the world. The question that you have to answer for yourself is this: how do you want to do your statistics? Do you want to be an orthodox statistician, relying on sampling distributions and p-values to guide your decisions? Or do you want to be a Bayesian, relying on things like prior beliefs, Bayes factors and the rules for rational belief revision? And to be perfectly honest, I can’t answer this question for you. Ultimately it depends on what you think is right. It’s your call and your call alone. That being said, I can talk a little about why I prefer the Bayesian approach.
Statistics that mean what you think they mean¶
You keep using that word. I do not think it means what you think it means
To me, one of the biggest advantages to the Bayesian approach is that it answers the right questions. Within the Bayesian framework, it is perfectly sensible and allowable to refer to “the probability that a hypothesis is true”. You can even try to calculate this probability. Ultimately, isn’t that what you want your statistical tests to tell you? To an actual human being, this would seem to be the whole point of doing statistics, i.e., to determine what is true and what isn’t. Any time that you aren’t exactly sure about what the truth is, you should use the language of probability theory to say things like “there is an 80% chance that Theory A is true, but a 20% chance that Theory B is true instead”.
This seems so obvious to a human, yet it is explicitly forbidden within the orthodox framework. To a frequentist, such statements are a nonsense because “the theory is true” is not a repeatable event. A theory is true or it is not, and no probabilistic statements are allowed, no matter how much you might want to make them. There’s a reason why, back in section The p-value of a test, I repeatedly warned you not to interpret the p-value as the probability that the null hypothesis is true. There’s a reason why almost every textbook on statstics is forced to repeat that warning. It’s because people desperately want that to be the correct interpretation. Frequentist dogma notwithstanding, a lifetime of experience of teaching undergraduates and of doing data analysis on a daily basis suggests to me that most actual humans think that “the probability that the hypothesis is true” is not only meaningful, it’s the thing we care most about. It’s such an appealing idea that even trained statisticians fall prey to the mistake of trying to interpret a p-value this way. For example, here is a quote from an official Newspoll report in 2013, explaining how to interpret their (frequentist) data analysis:
Throughout the report, where relevant, statistically significant changes have been noted. All significance tests have been based on the 95 percent level of confidence. This means that if a change is noted as being statistically significant, there is a 95 percent probability that a real change has occurred, and is not simply due to chance variation. (emphasis added)
Nope! That’s not what p < 0.05 means. That’s not what 95% confidence means to a frequentist statistician. The bolded section is just plain wrong. Orthodox methods cannot tell you that “there is a 95% chance that a real change has occurred”, because this is not the kind of event to which frequentist probabilities may be assigned. To an ideological frequentist, this sentence should be meaningless. Even if you’re a more pragmatic frequentist, it’s still the wrong definition of a p-value. It is simply not an allowed or correct thing to say if you want to rely on orthodox statistical tools.
On the other hand, let’s suppose you are a Bayesian. Although the bolded passage is the wrong definition of a p-value, it’s pretty much exactly what a Bayesian means when they say that the posterior probability of the alternative hypothesis is greater than 95%. And here’s the thing. If the Bayesian posterior is actually the thing you want to report, why are you even trying to use orthodox methods? If you want to make Bayesian claims, all you have to do is be a Bayesian and use Bayesian tools.
Speaking for myself, I found this to be the most liberating thing about switching to the Bayesian view. Once you’ve made the jump, you no longer have to wrap your head around counter-intuitive definitions of p-values. You don’t have to bother remembering why you can’t say that you’re 95% confident that the true mean lies within some interval. All you have to do is be honest about what you believed before you ran the study and then report what you learned from doing it. Sounds nice, doesn’t it? To me, this is the big promise of the Bayesian approach. You do the analysis you really want to do, and express what you really believe the data are telling you.
Evidentiary standards you can believe¶
If [p] is below 0.02 it is strongly indicated that the [null] hypothesis fails to account for the whole of the facts. We shall not often be astray if we draw a conventional line at 0.05 and consider that [smaller values of p] indicate a real discrepancy.
Consider the quote above by Sir Ronald Fisher, one of the founders of what has become the orthodox approach to statistics. If anyone has ever been entitled to express an opinion about the intended function of p-values, it’s Fisher. In this passage, taken from his classic guide Statistical Methods for Research Workers, he’s pretty clear about what it means to reject a null hypothesis at p < 0.05. In his opinion, if we take p < 0.05 to mean there is “a real effect”, then “we shall not often be astray”. This view is hardly unusual. In my experience, most practitioners express views very similar to Fisher’s. In essence, the p < 0.05 convention is assumed to represent a fairly stringent evidential standard.
Well, how true is that? One way to approach this question is to try to convert p-values to Bayes factors, and see how the two compare. It’s not an easy thing to do because a p-value is a fundamentally different kind of calculation to a Bayes factor, and they don’t measure the same thing. However, there have been some attempts to work out the relationship between the two, and it’s somewhat surprising. For example, Johnson (2013) presents a pretty compelling case that (for t-tests at least) the p < 0.05 threshold corresponds roughly to a Bayes factor of somewhere between 3:1 and 5:1 in favour of the alternative. If that’s right, then Fisher’s claim is a bit of a stretch. Let’s suppose that the null hypothesis is true about half the time (i.e., the prior probability of H0 is 0.5), and we use those numbers to work out the posterior probability of the null hypothesis given that it has been rejected at p < 0.05. Using the data from Johnson (2013), we see that if you reject the null at p < 0.05, you’ll be correct about 80% of the time. I don’t know about you but, in my opinion, an evidential standard that ensures you’ll be wrong on 20% of your decisions isn’t good enough. The fact remains that, quite contrary to Fisher’s claim, if you reject at p < 0.05 you shall quite often go astray. It’s not a very stringent evidential threshold at all.
The p-value is a lie.¶
The cake is a lie.The cake is a lie.The cake is a lie.The cake is a lie.
Okay, at this point you might be thinking that the real problem is not with orthodox statistics, just the p < 0.05 standard. In one sense, that’s true. The recommendation that Johnson (2013) gives is not that “everyone must be a Bayesian now”. Instead, the suggestion is that it would be wiser to shift the conventional standard to something like a p < 0.01 level. That’s not an unreasonable view to take, but in my view the problem is a little more severe than that. In my opinion, there’s a fairly big problem built into the way most (but not all) orthodox hypothesis tests are constructed. They are grossly naive about how humans actually do research, and because of this most p-values are wrong.
Sounds like an absurd claim, right? Well, consider the following scenario. You’ve come up with a really exciting research hypothesis and you design a study to test it. You’re very diligent, so you run a power analysis to work out what your sample size should be, and you run the study. You run your hypothesis test and out pops a p-value of 0.072. Really bloody annoying, right?
What should you do? Here are some possibilities:
- You conclude that there is no effect and try to publish it as a null result
- You guess that there might be an effect and try to publish it as a “borderline significant” result
- You give up and try a new study
- You collect some more data to see if the p-value goes up or (preferably!) drops below the “magic” criterion of p < 0.05
Which would you choose? Before reading any further, I urge you to take some time to think about it. Be honest with yourself. But don’t stress about it too much, because you’re screwed no matter what you choose. Based on my own experiences as an author, reviewer and editor, as well as stories I’ve heard from others, here’s what will happen in each case:
- Let’s start with option 1. If you try to publish it as a null result, the paper will struggle to be published. Some reviewers will think that p = 0.072 is not really a null result. They’ll argue it’s borderline significant. Other reviewers will agree it’s a null result but will claim that even though some null results are publishable, yours isn’t. One or two reviewers might even be on your side, but you’ll be fighting an uphill battle to get it through.
- Okay, let’s think about option number 2. Suppose you try to publish it as a borderline significant result. Some reviewers will claim that it’s a null result and should not be published. Others will claim that the evidence is ambiguous, and that you should collect more data until you get a clear significant result. Again, the publication process does not favour you.
- Given the difficulties in publishing an “ambiguous” result like p = 0.072, option number 3 might seem tempting: give up and do something else. But that’s a recipe for career suicide. If you give up and try a new project every time you find yourself faced with ambiguity, your work will never be published. And if you’re in academia without a publication record you can lose your job. So that option is out.
- It looks like you’re stuck with option 4. You don’t have conclusive results, so you decide to collect some more data and re-run the analysis. Seems sensible, but unfortunately for you, if you do this all of your p-values are now incorrect. All of them. Not just the p-values that you calculated for this study. All of them. All the p-values you calculated in the past and all the p-values you will calculate in the future. Fortunately, no-one will notice. You’ll get published, and you’ll have lied.
Wait, what? How can that last part be true? I mean, it sounds like a perfectly reasonable strategy doesn’t it? You collected some data, the results weren’t conclusive, so now what you want to do is collect more data until the the results are conclusive. What’s wrong with that?
Honestly, there’s nothing wrong with it. It’s a reasonable, sensible and rational thing to do. In real life, this is exactly what every researcher does. Unfortunately, the theory of null hypothesis testing as I described it in chapter Hypothesis testing forbids you from doing this.[2] The reason is that the theory assumes that the experiment is finished and all the data are in. And because it assumes the experiment is over, it only considers two possible decisions. If you’re using the conventional p < 0.05 threshold, those decisions are:
| Outcome | Action |
|---|---|
| p less than 0.05 | Reject the null |
| p greater than 0.05 | Retain the null |
What you’re doing is adding a third possible action to the decision making problem. Specifically, what you’re doing is using the p-value itself as a reason to justify continuing the experiment. And as a consequence you’ve transformed the decision-making procedure into one that looks more like this:
| Outcome | Action |
|---|---|
| p less than 0.05 | Stop the experiment and reject the null |
| p between 0.05 and 0.1 | Continue the experiment |
| p greater than 0.1 | Stop the experiment and retain the null |
The “basic” theory of null hypothesis testing isn’t built to handle this sort of thing, not in the form I described back in chapter Hypothesis testing. If you’re the kind of person who would choose to “collect more data” in real life, it implies that you are not making decisions in accordance with the rules of null hypothesis testing. Even if you happen to arrive at the same decision as the hypothesis test, you aren’t following the decision process it implies, and it’s this failure to follow the process that is causing the problem (a related problem). Your p-values are a lie.
Worse yet, they’re a lie in a dangerous way, because they’re all too small. To give you a sense of just how bad it can be, consider the following (worst case) scenario. Imagine you’re a really super-enthusiastic researcher on a tight budget who didn’t pay any attention to my warnings above. You design a study comparing two groups. You desperately want to see a significant result at the p < 0.05 level, but you really don’t want to collect any more data than you have to (because it’s expensive). In order to cut costs you start collecting data but every time a new observation arrives you run a t-test on your data. If the t-tests says p < 0.05 then you stop the experiment and report a significant result. If not, you keep collecting data. You keep doing this until you reach your pre-defined spending limit for this experiment. Let’s say that limit kicks in at N = 1000 observations. As it turns out, the truth of the matter is that there is no real effect to be found: the null hypothesis is true. So, what’s the chance that you’ll make it to the end of the experiment and (correctly) conclude that there is no effect? In an ideal world, the answer here should be 95%. After all, the whole point of the p < 0.05 criterion is to control the Type I error rate at 5%, so what we’d hope is that there’s only a 5% chance of falsely rejecting the null hypothesis in this situation. However, there’s no guarantee that will be true. You’re breaking the rules. Because you’re running tests repeatedly, “peeking” at your data to see if you’ve gotten a significant result, all bets are off.
Fig. 200 How badly can things go wrong if you re-run your tests every time new data arrive? If you are a frequentist, the answer is “very wrong”.
So how bad is it? The answer is shown as the solid black line in Fig. 200, and it’s astoundingly bad. If you peek at your data after every single observation, there is a 49% chance that you will make a Type I error. That’s, um, quite a bit bigger than the 5% that it’s supposed to be. By way of comparison, imagine that you had used the following strategy. Start collecting data. Every single time an observation arrives, run a Bayesian t-tests and look at the Bayes factor. I’ll assume that Johnson (2013) is right, and I’ll treat a Bayes factor of 3:1 as roughly equivalent to a p-value of 0.05.[3] This time around, our trigger happy researcher uses the following procedure. If the Bayes factor is 3:1 or more in favour of the null, stop the experiment and retain the null. If it is 3:1 or more in favour of the alternative, stop the experiment and reject the null. Otherwise continue testing. Now, just like last time, let’s assume that the null hypothesis is true. What happens? As it happens, I ran the simulations for this scenario too, and the results are shown as the dashed line in Fig. 200. It turns out that the Type I error rate is much much lower than the 49% rate that we were getting by using the orthodox t-test.
In some ways, this is remarkable. The entire point of orthodox null hypothesis testing is to control the Type I error rate. Bayesian methods aren’t actually designed to do this at all. Yet, as it turns out, when faced with a “trigger happy” researcher who keeps running hypothesis tests as the data come in, the Bayesian approach is much more effective. Even the 3:1 standard, which most Bayesians would consider unacceptably lax, is much safer than the p < 0.05 rule.
Is it really this bad?¶
The example I gave in the previous section is a pretty extreme situation. In real life, people don’t run hypothesis tests every time a new observation arrives. So it’s not fair to say that the p < 0.05 threshold “really” corresponds to a 49% Type I error rate (i.e., p = 0.49). But the fact remains that if you want your p-values to be honest then you either have to switch to a completely different way of doing hypothesis tests or enforce a strict rule of no peeking. You are not allowed to use the data to decide when to terminate the experiment. You are not allowed to look at a “borderline” p-value and decide to collect more data. You aren’t even allowed to change your data analysis strategy after looking at data. You are strictly required to follow these rules, otherwise the p-values you calculate will be nonsense.
And yes, these rules are surprisingly strict. As a class exercise a couple of years back, I asked students to think about this scenario. Suppose you started running your study with the intention of collecting N = 80 people. When the study starts out you follow the rules, refusing to look at the data or run any tests. But when you reach N = 50 your willpower gives in… and you take a peek. Guess what? You’ve got a significant result! Now, sure, you know you said that you’d keep running the study out to a sample size of N = 80, but it seems sort of pointless now, right? The result is significant with a sample size of N = 50, so wouldn’t it be wasteful and inefficient to keep collecting data? Aren’t you tempted to stop? Just a little? Well, keep in mind that if you do, your Type I error rate at p < 0.05 just ballooned out to 8%. When you report p < 0.05 in your paper, what you’re really saying is p < 0.08. That’s how bad the consequences of “just one peek” can be.
Now consider this. The scientific literature is filled with t-tests, ANOVAs, regressions and χ²-tests. When I wrote this book I didn’t pick these tests arbitrarily. The reason why these four tools appear in most introductory statistics texts is that these are the bread and butter tools of science. None of these tools include a correction to deal with “data peeking”: they all assume that you’re not doing it. But how realistic is that assumption? In real life, how many people do you think have “peeked” at their data before the experiment was finished and adapted their subsequent behaviour after seeing what the data looked like? Except when the sampling procedure is fixed by an external constraint, I’m guessing the answer is “most people have done it”. If that has happened, you can infer that the reported p-values are wrong. Worse yet, because we don’t know what decision process they actually followed, we have no way to know what the p-values should have been. You can’t compute a p-value when you don’t know the decision making procedure that the researcher used. And so the reported p-value remains a lie.
Given all of the above, what is the take home message? It’s not that Bayesian methods are foolproof. If a researcher is determined to cheat, they can always do so. Bayes’ rule cannot stop people from lying, nor can it stop them from rigging an experiment. That’s not my point here. My point is the same one I made at the very beginning of the book in section On the psychology of statistics: the reason why we run statistical tests is to protect us from ourselves. And the reason why “data peeking” is such a concern is that it’s so tempting, even for honest researchers. A theory for statistical inference has to acknowledge this. Yes, you might try to defend p-values by saying that it’s the fault of the researcher for not using them properly, but to my mind that misses the point. A theory of statistical inference that is so completely naive about humans that it doesn’t even consider the possibility that the researcher might look at their own data isn’t a theory worth having. In essence, my point is this:
Good laws have their origins in bad morals.
Good rules for statistical testing have to acknowledge human frailty. None of us are without sin. None of us are beyond temptation. A good system for statistical inference should still work even when it is used by actual humans. Orthodox null hypothesis testing does not.[4]
| [1] | I should note in passing that I’m not the first person to use this quote to complain about frequentist methods. Rich Morey and colleagues had the idea first. I’m shamelessly stealing it because it’s such an awesome pull quote to use in this context and I refuse to miss any opportunity to quote The Princess Bride. |
| [2] | In the interests of being completely honest, I should acknowledge that not all orthodox statistical tests rely on this silly assumption. There are a number of sequential analysis tools that are sometimes used in clinical trials and the like. These methods are built on the assumption that data are analysed as they arrive, and these tests aren’t horribly broken in the way I’m complaining about here. However, sequential analysis methods are constructed in a very different fashion to the “standard” version of null hypothesis testing. They don’t make it into any introductory textbooks, and they’re not very widely used in the psychological literature. The concern I’m raising here is valid for every single orthodox test I’ve presented so far and for almost every test I’ve seen reported in the papers I read. |
| [3] | Some readers might wonder why I picked 3:1 rather than 5:1, given that Johnson (2013) suggests that p = 0.05 lies somewhere in that range. I did so in order to be charitable to the p-value. If I’d chosen a 5:1 Bayes factor instead, the results would look even better for the Bayesian approach. |
| [4] | Okay, I just know that some knowledgeable frequentists will read this and start complaining about this section. Look, I’m not dumb. I absolutely know that if you adopt a sequential analysis perspective you can avoid these errors within the orthodox framework. I also know that you can explictly design studies with interim analyses in mind. So yes, in one sense I’m attacking a “straw man” version of orthodox methods. However, the straw man that I’m attacking is the one that is used by almost every single practitioner. If it ever reaches the point where sequential methods become the norm among experimental psychologists and I’m no longer forced to read 20 extremely dubious ANOVAs a day, I promise I’ll rewrite this section and dial down the vitriol. But until that day arrives, I stand by my claim that default Bayes factor methods are much more robust in the face of data analysis practices as they exist in the real world. Default orthodox methods suck, and we all know it. |