Statistical Significance & Frequentist Probability

Deborah Mayo and Richard Morey recently posted a very interesting criticism of the diagnostic screening model of statistical testing (e.g., this, this). Mayo & Morey argue that this approach to criticism of null hypothesis significance testing (NHST) relies on a misguided hybrid of frequentist and Bayesian reasoning. The paper is worth reading in its entirety, but in this post I will focus narrowly on a non-central point that they make.

At the beginning of the section Some Well-Known Fallacies (p. 6), Mayo & Morey write:

From the start, Fisher warned that to use p-values to legitimately indicate incompatibility (between data and a model), we need more than a single isolated low p-value: we must demonstrate an experimental phenomenon.

They then quote Fisher:

[W]e need, not an isolated record, but a reliable method of procedure. In relation to the test of significance, we may say that a phenomenon is experimentally demonstrable when we know how to conduct an experiment which will rarely fail to give us a statistically significant result. (Fisher 1947, p. 14)

It’s worth noting that in the Fisherian approach to NHST, p values provide a continuous measure of discrepancy from a null model (see also, e.g., the beginning of the second section in this commentary by Andrew Gelman). If a continuous measure of discrepancy is dichotomized (e.g., into the categories "statistically significant" and "not statistically significant"), the criterion for dichotomization is typically arbitrary; common p value thresholds like 0.05 and 0.01 are not generally given any kind of rationale, though there is a case for explicit justification of such thresholds.

(As a brief aside, in the Neyman-Pearson approach to NHST, dichotomization of test results is baked in from the start, and p values are not treated as continuous measures of discrepancy between model and data. A priori specification of α sets a hard, though still typically arbitrary, threshold. Here is a tutorial with detailed discussion of the differences between the Fisher and Neyman-Pearson approaches to NHST.)

The point here being that the Fisher quote above could (and probably should) be revised to say that we can describe a phenomenon as experimentally demonstrable when we know how to conduct an experiment that will rarely fail to produce a large discrepancy.

Clearly, in determining what counts as "large", we will run into some of the same problems that we run into in determining cutoffs for statistical significance. But focusing on discrepancies in the space in which our measurements are taken will force us to focus on "clinical significance" rather than "statistical significance". This will make it much easier to argue for or against any particular criterion (assuming you are okay with criteria for "large" vs "not large") or to take costs and benefits directly into account and use statistical decision theory (if you are not okay with such criteria).

To be as clear as I can be, I’m not in favor of "banning" p values (or the statistics that would allow a reader to calculate an otherwise unreported p value, e.g., means and standard errors). If you are concerned with error control, the information provided by p values is important. But the interested reader can decide for him- or herself how to balance false alarms and misses. There is no need for the researcher reporting a study to declare a result "statistically significant" or not.

I wholeheartedly agree with Lakens et al when they write:

[W]e recommend that the label "statistically significant" should no longer be used. Instead, researchers should provide more meaningful interpretations of the theoretical or practical relevance of their results.

In reading and thinking (and writing, e.g., see here, here, here, and here) about statistical significance lately, I feel like some important implications of frequentist statistics have really clicked for me.

A while back, it occurred to me that, at least under certain interpretations of confidence intervals (CIs), it doesn’t make much sense to actually report CI limits. The coverage of CIs is a property of the procedure of constructing CIs, but any particular set of observed CI limits do not tell you anything either probabilistic or useful about the location of a "true" parameter value (scare quotes because it’s not at all obvious to me that the notion of a true parameter value is of much use).

(Under a test-inversion interpretation of CIs, reporting a particular set of CI limits can be useful, since this indicates the range of parameter values that you cannot reject, given a particular test and statistical significance criterion. But, then again, the test-inversion interpretation is not without its own serious problems.)

Anyway, I bring all this up here at the end just to point out an important parallel between CIs and statistical significance. By the logic of frequentist probability – probabilities just are long-run relative frequencies of subsets of events – both CIs and statistical significance are only meaningful (and are only meaningfully probabilistic) across repeated replications. Given this, I am more and more convinced that individual research reports should focus on estimation (and quantification of uncertainty) rather than testing (and uncertainty laundering).

Strangely, and quite possibly incoherently, I think I may have convinced myself that frequentist statistical testing is inherently meta-analytic.

Significant Feelings

This post is a response, of sorts, to the most recent episode of The Black Goat podcast. It is “of sorts” because it’s not just a response, but this post was definitely inspired by the episode. Anyway, the point here being that I stole the title of the episode for this post.

First things first – it’s a good episode, well worth a listen. I listened to (most of) it twice, in fact, taking notes the second time, since I wanted to make sure I was remembering things correctly so that I didn’t misrepresent what they (Srivastava, Tullett, and Vazire; henceforth ST&V) talk about in my discussion here.

There are two main parts: a discussion of a listener question about whether to trust open-science advocates’ pre-rep*-crisis work more than that of non-open-science advocates, and a longer discussion of the hosts’ history with p-values and the various discussions about p-values going on in behavioral and medical sciences (and probably elsewhere). My focus here will be on the latter (which is why the second listen was only to most of the episode – from the 26 minute mark, for what it’s worth).

The p-value discussion starts with each host giving a brief overview of his/her “history with p-values” in classes and in the lab. I’ll do the same here.

A Brief Overview of My History With P Values

The first class I took that was directly relevant to statistical analysis had relatively little statistical analysis in it. It was a second language testing class at the University of Hawai`i. It covered some descriptive stats, but the course focused much more on validity than a standard stats class would. I’m sure my memories aren’t totally accurate, but I think of it now as a course on measurement more than anything else. We probably discussed p values at some point, but statistical testing was not the primary focus of the class.

I took some more standard stats classes from a guy in Educational Psychology at UH, too (he sounded exactly like Donald Sutherland, and kinda-sorta looked like him, which was great). These were more standard, but, I think, not completely like a run of the mill psych stats class. This is maybe illustrated by the fact that the textbook for one course was Measurement, Design, and Analysis: An Integrated Approach. I still own and use it, 16 years later. I’m quite sure p values and NHST came up a lot in the classes I took from Professor Sutherland, but they were secondary to how design and analysis are linked.

At Indiana University, where I went after UH to pursue a PhD in Linguistics and Cognitive Science, I took the two stats courses required for Psychology students. The first was taught by John Kruschke. He was always very careful to define p-values correctly, so this got drilled into my head, early and thoroughly. Also, his class was structured exceptionally well, so the relationships between different statistical tests and the general logic of NHST have stuck with me.

I also took a seminar in Bayesian statistics from Kruschke, the first time he taught a course on this topic. He used the chalkboard and didn’t have prepared slides, which was very different from the other classes I had taken from him. I asked him why at one point, and he said that it was a one-off course for him, so it wasn’t worth the time and energy to prepare elaborate materials he wouldn’t use again. Later on, I was Kruschke’s grader for what became a two-semester sequence on Bayesian stats as he wrote the first edition of his textbook on Bayesian stats. Now he’s on the second edition. Predicting the future is hard. Also, problems with NHST are covered in his book and lectures, so p values were at least a small part of my initial training with Bayesian stats, too.

In addition to those courses, I took a lot of mathematical psychology and cognitive modeling courses at IU, some of which touched on various aspects of statistical testing. A fair amount of this training had nothing to do with p values, though there was a lot of probability theory.

After I was done with my PhD coursework, I also took a couple semesters of classes to fulfill the requirements for the then newly-established MS in Applied Statistics (the rest of the requirements for this degree were fulfilled by the earlier stats and math psych classes I had taken). These post-PhD-coursework courses deepened (and filled some gaps in) my knowledge, and I learned how to do statistical consulting effectively (from the soon-to-be president elect of the ASA). Perhaps not surprisingly, p values came up a lot in some of these classes (e.g., the two-semester sequence on linear models), but less so in others (another course on Bayesian stats, and the statistical theory course, which focused a lot more on things like the central limit theorem, consistency, bias, maximum likelihood estimation, and various other mathematical underpinnings of statistics).

In the research I was doing in the lab(s) I worked in (the Mathematical Psychology Lab, though I ran experiments in the Auditory Perception Lab on the other side of campus; I also did some work in the Linguistic Speech Lab), p values played a relatively small role. Early on, I remember using p values in fairly standard ways (e.g., to pick out what was worth talking about; see my first paper, for example). As I learned more about math psych and Bayesian stats, p values became more irrelevant, though they haven’t disappeared from my research completely.

Finally, I got valuable experience with new (to me) statistical tools in my first job after graduating. I participated in workshops on diagnostic measurement and structural equation modeling at the University of Maryland, and the work I was doing was essentially I/O psychology (as confirmed by an actual I/O psychologist). Plenty of p values here, many of them, in retrospect, likely misused, though not misinterpreted.

Okay, so, well, that was maybe less brief than I envisioned, but for the purposes of discussing my significant feelings, I think it’s important to explain where I’m coming from.

The Discussion

ST&V touch on a wide range of topics in their discussion of p values. As mentioned above, the whole thing is worth listening to, but I’m not going to talk about everything they get into.

Their discussion spurred me to think some more about

  1. how evaluation of statistical analyses is related to evaluation of design and measurement (and how the former is subordinate to the latter);
  2. how individuals evaluate evidence and how this relates to group-level evaluation of evidence;
  3. how we think about (or struggle to think about) continuous vs discrete reports, and how these do or do not relate to how we (struggle to) think about probability;
  4. and how all of this relates (or does not relate) to Bayesian and frequentist statistical philosophies.

Around the 42-minute mark, ST&V start talking about the paper Redefine Statistical Significance (RSS), which Vazire is a co-author of (here is my very negative take on RSS). This leads to a more general discussion about statistical cutoffs and dichotomous vs graded thinking. At around the 53-minute mark, Tullett* brings up differences between evaluation of study designs and evaluation of statistical results, positing that it’s easier to give nuanced evaluation of the former than the latter.

(* I am reasonably confident that it is Tullett talking here and not Vazire. Their voices are different, but I don’t know either voice well enough to be able to map the voices to the names consistently. That is, I would do very well with a discrimination task, but probably substantially less well with a recognition task.)

(1) This is fine as far as it goes – design is, along with measurement, complex and multifaceted. But I want to push this distinction further. It occurred to me a while ago that it is problematic to discuss isolated statistics as “evidence.” I think ST&V mention “statistical evidence” a few times in this discussion, the word “evidence” is used a lot in RSS and the response papers (see links at the top of my RSS post), and I’m pretty sure I’ve seen Bayesians refer to the denominator in Bayes’ rule as “evidence.”

But evidence has to be of something. Without a larger context – a research question, a design, a measurement tool, and a statistical model, at least – a p value is just (tautological) “evidence” of a range of test statistic values being more or less probable. It’s just – per the definition – a statement about the probability of observing a test statistic as or more extreme as what you’ve observed under the assumption that the null hypothesis is true. Which is to say that, in isolation, a p value doesn’t tell us anything at all. Similarly for Bayes factors, sets of *IC fit statistics, or whatever other statistical summary you like.

(2) So, when it comes to evaluation of evidence, an individual must take research questions, design, measurement, and statistics into account. And, as I mentioned in my post on RSS, individuals have now, and always have had, the ability to evaluate evidence as stringently or as laxly as they believe is appropriate for any given situation.

Of course, things get complicated if an individual wants to persuade other individuals about a scientific issue. Shared standards undoubtedly help individuals achieve scientific consensus, but the complications of scientific disagreement and agreement can’t be solved by fixed, arbitrary statistical thresholds. Such thresholds might be a necessary condition of scientific consensus, though I’m skeptical, and they definitely aren’t sufficient.

(3) All that said, when an individual evaluates a scientific report, there can be value in discrete categories. I like the discussion (around the 54-minute mark) of thinking about results in terms of whether you buy them or not and how much you would bet on a replication being successful. I also find the argument for the utility of a three-category fire safety system vs a continuous numerical system compelling.

But it’s worth keeping in mind that, even if a discrete system can better for certain kinds of decisions, a continuous system can be better for other kinds of decisions. “Buy it or not” might be the best way to evaluate a study for some purposes, but from a meta-analytic mind-set, dichotomizing evidence is just throwing information away.

Tangentially related, I agree wholeheartedly with Vazire* that people are bad at probabilistic thinking. Even with all the statistical training and research experience described above, I often have to exert substantial cognitive effort to be careful with probabilistic thinking. That said, I think it’s worth drawing a distinction between, on the one hand, categorical vs continuous thinking, and, on the other, deterministic vs probabilistic thinking.

Obviously, probability plays a role in all this, and the part of the discussion about intuitions about p value distributions under null and alternative hypotheses (~38 minutes) is both interesting and important. But the relative value of dichotomization (or trichotomization) vs indefinitely graded descriptions of statistical analyses is not (only) about probability. It’s also about (mis)interpretation of basic, non-random arithmetical facts (see, e.g., McShane & Gal, 2016, 2017).

(4) Finally, I agree exuberantly with the point that Vazire* makes late in the podcast that these issues are orthogonal to debates about Bayesian and frequentist philosophies of statistics. Srivastava is absolutely right that categorizing Bayes factors as “anecdotal” or “strong evidence” is just as problematic (or not) as categorizing p values as statistically significant, (suggestive,) or not statistically significant. Or maybe I should say that he should have gone further than saying that this kind of categorization “starts to feel a little threshold-y.” It is 100% threshold-y, and so it has all the problems that fixed, arbitrary thresholds have.

If you insist on using thresholds, you should justify your Bayes factor thresholds just as surely as you should Justify Your Alpha. I’m more and more convinced lately that we should be avoiding thresholds (see the McShane & Gal papers linked above, see also the papers linked in the post on the meta-analytic mind-set), but I agree with the larger point of the Justify Your Alpha paper that research decisions (statistical and non) should be described as fully and as transparently as possible.

One last point about metric vs nominal reports and Bayesian vs frequentist statistics: although the former is logically unrelated to the latter, in my experience, Bayesian statistics involves a lot more modeling (model building, parameter estimation, and data and model visualization) than does frequentist statistical testing. Obviously, you can do frequentist model building, parameter estimation, and visualization, but standard psych statistical training typically focuses on (null hypothesis significance) testing, which is inherently, inextricably tied up with categorical reports (e.g., dichotomization). The point being that, while I don’t think there are any deep philosophical reason for Bayesians to be more friendly to continuous, graded reports than are frequentists, there are contingent historical reasons for things to play out this way.


Significant Feelings is a very good episode of The Black Goat podcast, but arbitrary thresholds are bad.


Anti-trolley libertarianism?

This argument against the trolley problem (via) is amusing and makes some interesting points. I don’t totally buy the case that the trolley problem is useless (much less actively harmful), since I think that there are probably some important moral issues related to the action vs inaction distinction that the problem brings up, and that these are probably important for some of the society-level policy-related issues that the authors would prefer we all focus on.

The most amusing bit is a link to a comic making fun of the absurd variations on the trolley problem. Here’s one of the more interesting parts:

By thinking seriously about the trolley problem, i.e. considering what the scenario being described actually involves, we can see why it’s so limited as a moral thought experiment. It’s not just that, as the additional conditions grow, there are not any obvious right answers. It’s that every single answer is horrific, and wild examples like this take us so far afield from ordinary moral choices that they’re close to nonsensical.

I’m not completely convinced by the stretch that follows this, in which the authors argues that pondering these kinds of questions makes us more callous. Maybe it does, maybe it doesn’t. But I do think it’s worth pointing out that many, perhaps most, moral questions involve, on the one hand, uncertainty, and, on the other, both costs and benefits. Uncertainty is relevant because of risk aversion. Costs and benefits are relevant because of asymmetries in approach-avoidance conflicts (e.g., loss aversion). Moral questions involving choices of certain awfulness are inherently limited.

There are some other interesting bits, but the thing that really stood out to me was this, which reads like an argument for pretty hard core libertarianism (bolded emphasis mine):

The “who should have power over lives” question is often completely left out of philosophy lessons, which simply grant you the ability to take others’ lives and then instruct you to weigh them in accordance with your instincts as to who should live or die…. But what about situations where people are making high-level life-or-death decisions from a distance, and thus have the leisure to weigh the value of certain lives against the value of certain other lives? Perhaps the closest real-life parallels to the trolley problem are war-rooms, and areas of policy-making where “cost-benefit” calculuses are performed on lives. But in those situations, what we should often really be asking is “why does that person have that amount of power over others, and should they ever?” (answer: almost certainly not), rather than “given that X is in charge of all human life, whom should X choose to spare?” One of the writers of this article vividly recalls a creepy thought experiment they had to do at a law school orientation, based on the hypothetical that a fatal epidemic was ravaging the human population. The students in the room were required to choose three fictional people out of a possible ten to receive a newly-developed vaccine…. The groups were given biographies of the ten patients: some of them had unusual talents, some of them had dependents, some of them were children, and so on. Unsurprisingly, the exercise immediately descended into eugenics territory, as the participants, feeling that they had to make some kind of argument, began debating the worthiness of each patient and weighing their respective social utilities against each other. (It only occurred to one of the groups to simply draw lots, which would clearly have been the only remotely fair course of action in real life.) This is a pretty good demonstration of why no individual person, or small group of elites, should actually have decision-making authority in extreme situations like this: all examinations of who “deserves” to live rapidly become unsettling, as the decision-maker’s subjective judgments about the value of other people’s lives are given a false veneer of legitimacy through a dispassionate listing of supposedly-objective “criteria.”

Later in the essay, they ask:

Is being rich in a time of poverty justifiable? … Does capitalism unfairly exploit workers?

To which they answer “No” and “Yes,” so it’s clear that they’re not really making a case for libertarianism. Plus, the “about” page for Current Affairs has this unattributed quote placed fairly prominently:

“The Wall Street Journal of surrealistic left-wing policy journals.”

Then again, just below this, it has two more unattributed quotes:

“If Christopher Hitchens and Willy Wonka had edited a magazine together, it might have resembled Current Affairs.”

“The only sensible anarchist thinking coming out of contemporary print media.”

So maybe this apparent inconsistency is just wacky anarchism? Anyway, the whole essay is worth a read.