Banishing “Black/White Thinking”: A Trio of Teaching Tricks

Literally hundreds of statisticians have rightly called for an end to statistical significance testing (Amrhein et al., 2019; Wasserstein et al., 2019). But the practice of arbitrarily thresholding p values is not only deeply embedded in statistical practice, it is also congenial to the human mind.


Introduction
Humans are natural born categorizers. We instinctively take continuous variables and draw (often) arbitrary boundaries that allow us to put names to groups. For example, we divide the continuous visible spectrum up into discrete colors like "red," "yellow," and "blue." And the body mass index (BMI) is a continuous measure of a person's weight-to-height ratio, yet a brief scan of the Internet turns up repeated examples of the classification shown in Table 1.
In some cases, such as for color, certain categories appear to be "natural," as if they were baked into our brains (Rosch, 1973). In other cases, categorization is related to the need to make decisions, as is the case for many medical classifications. And the fact that we communicate our ideas using language-words being discrete entities-surely contributes to this tendency.
Nowhere is the tendency more dramatic-and more pernicious-than in the practice of null hypothesis significance testing (NHST), based on p values, where an arbitrary cutoff of 0.05 is used to separate "truth" from "falsehood." Let us set aside the first obvious problem that in NHST we never accept the null (i.e., proclaim falsehood) but rather only fail to reject it. And let us also ignore the debate about whether we should change the cutoff to something more stringent, say 0.005 (Benjamin et al., 2018), and instead focus on what I consider to be Significance Statement I offer specific teaching examples to help students properly think about p values and interval statistics. the real problem: the cutoff itself. This is the problem I refer to as "black/white thinking." Because this tendency to categorize using p values is (1) natural and (2) abundantly reinforced in many statistics courses, we must do more than simply tell our students that it is wrong. We must show them why it is wrong and offer better ways of thinking about statistics. What follows are some practical methods I have found useful in classroom discussions with graduate students and postdoctoral fellows in neuroscience.

Example 1
In class, I start with an example of a statistical error that is known to be extremely common in the neuroscience literature (Nieuwenhuis et al., 2011). I took the numbers directly from the classic article by Gelman and Stern (2006) that has one of my all-time favorite titles: "The Difference Between 'Significant' and 'Not Significant' is not Itself Statistically Significant." In this made-up example ( Fig. 1), we compare two drugs being tested for their efficacy in increasing the time a genetic mouse model of amyotrophic lateral sclerosis can remain on a rotating rod. Drug A, on average, increases performance by 25 s with an SE of 10 s-what most people would categorize as a "statistically significant effect" (p ϭ 0.012). Drug B, on the other hand, is not even close (effect size, 10 Ϯ 10 s; p ϭ 0.32).
Before showing the data for the direct contrast in Figure  1 (A-B), I ask the class the simple question of whether, based on the data shown on the left part of Figure 1, we can conclude that drug A is "significantly better" than drug B at increasing performance. I encourage them to first discuss the question with their immediate neighbors, and then I ask them to vote "yes" or "no," either with a show of hands or, if available, some form of a clicker response. In general, the vast majority of the class votes yes, but there is always some visible trepidation (e.g., sheepish hand raising), since the students figure that there must be a trick if I am asking them something so apparently obvious. I then display the contrast shown on the right side of Figure 1 as "A-B." A direct comparison of the two drugs reveals a difference of 15 s with an SE of 14, and a corresponding p value of 0.29. So I ask them, "How can this be? One drug 'works' and the other drug does not, so there must be a difference, right?" This generates some murmuring among the class, and this is an excellent opportunity for a discussion of "What's going on here?" At some point, I usually interject that, given any two drugs with any two nonequal p values, I can set a criterion that makes one of the drugs "work" and the other not. This helps point out the arbitrary nature of any p value cutoff and the major error of interpreting the failure to reject the null hypothesis (H0) for one of the drugs as not working, as well as the important idea that we want to make our inference about differences between the drugs based on the difference A-B. I close off this example by displaying the title of the article by Gelman and Stern (2006), and I encourage the students to repeat the title as a mantra each night before they go to bed and each morning when they awake for the next 2 months. And I add in a favorite quote from Rosnow and Rosenthal (1989): "That is, we want to underscore that, surely, God loves the 0.06 nearly as much as the 0.05. Can there be any doubt that God views the strength of evidence for or against the null as a fairly continuous function of the magnitude of p?" (emphasis added). As this quote is bang on and moderately funny, it puts a memorable cap on the exercise.

Example 2
From Figure 1, there is an easy segue to the second concept I find useful, that of the "counternull," first described by Rosenthal and Rubin (1994). To do this, I simply replace the SE bars with 95% confidence intervals (CIs; Fig. 2), which clearly shows why the null was not rejected for drug B: the 95% CI contains the null value of zero. But Rosenthal and Rubin (1994) would also have us consider the point that is equally distant from the mean but on the opposite side: this is the value they refer to as the counternull. Simply put, it is that value of the effect size (in our example, a performance increase of just Ͼ20 s) that is supported by exactly the same amount of evidence as the null value.
I like this statistic, because it gets the students thinking about more of the confidence interval than just whether or   Figure 1, except that SE bars have been replaced with 95% confidence intervals. The counternull is the value of the effect size that is equidistant from the mean (vs the null), but on the opposite side. See Rosenthal and Rubin (1994).
Commentary not it contains the null value (i.e., NHST). I ask them, would an effect size of this magnitude be of behavioral significance? If so, we might not be so quick to give up on drug B. We can certainly see that there is a broad range of plausible effect sizes that would be beneficial (as well as some that would be detrimental).
This also presents a good opportunity to start a discussion about how one might decide which of two drugs to take. Is statistical significance a good criterion? By simply rescaling the axes, one could show a strongly significant effect for a change in performance that would be negligible in terms of behavioral benefit-so the actual effect size, and not just its p value, matter. And what if we were told that drug A has a high incidence of toxic side effects? Or that it needed to be taken by a twice-daily intravenous injection?
One additional issue that can be brought up here is how we write statistical results in our articles. If we banish from our students' lexicons the phrase "statistically significant," what do we give them as a replacement? My practice is to encourage them to always include a point estimate of the effect size, generally the mean, along with a 95% confidence interval-a practice that is increasingly recommended by neuroscience journals (Calin-Jageman and Cumming, 2019), including eNeuro. In certain cases where the null value is included within the 95% CI, it might be useful to include the counternull, particularly when its magnitude would represent an important biological effect.

Example 3
My third example is taken from an interesting article by Johnson (2013a), and it is designed to nudge the students toward a Bayesian perspective (Fig. 3).
In this example, we are given a single datum, namely that X ϭ 2 and are asked to make an inference about the distribution from which it was drawn. A good Frequentist (Fig. 3A) would look at the probabilities under H0 and determine that she should reject H0 for X ϭ 2 or 3, as this would give a p value of Ͻ0.01. However, a Bayesian would compare the values in the red box of Figure 3B and realize that, for X ϭ 2, H0 is eight times more likely than HA (a Bayes factor). In fact, a simple calculation using Bayes' rule (which I do on the whiteboard) and assuming that the two hypotheses are a priori equally likely, reveals that the posterior probability that H0 is true is 0.89, although our Frequentist has confidently rejected it at p Ͻ 0.01 ("highly significant!"). This effectively creates a tension between what common sense tells us is the better approach and what the students have long held to be the right way to think.
Apart from this heavily rigged example, why is a Bayesian perspective helpful in combating black/white thinking? Well, the spirit of Bayesian data analysis is exactly what we want to inculcate in our students: using our experimental data, via the likelihood, to inform us how much we should change our beliefs. It encourages the better interpretation that the results of experiments should change our beliefs about hypotheses in a continuous way and not be used to draw sharp lines between truth and falsehood. This is not to say that Bayesian thinking is a panaceaone can create thresholds with Bayes factors as surely as one can with p values, and it is the threshold setting that is the problem. So what I try to communicate to my students is that we will continue to publish and perish in a largely Frequentist world for some time (Efron, 2013), but it behooves us all to be more Bayesian in spirit. And it even appears that Bayesian analyses may be on an upward trend in the neurosciences (Boekel et al., 2015).
Finally, the introduction of Bayes' rule allows us to address another critical shortcoming of NHST by considering priors. While this is a thorny topic when approached broadly, a narrower consideration of the prior probability of H0 is useful when considering, for example, "groundbreaking experiments" that are proffered with no more evidence than "p Ͻ 0.05." I introduce this problem with my favorite xkcd cartoon [Fig. 4 (see https://xkcd.com/ 1132/)].
If we start with the belief that it is extremely unlikely for the sun to explode in any small interval of time, then we will not be persuaded by such flimsy evidence as p Ͻ 0.03. This is a nice illustration of the LaPlacian notion that extraordinary claims require extraordinary evidence. From this perspective, the exercise of converting p values to minimum Bayes' factors (Goodman, 2001) and then ap- Figure 3. Each table shows the probabilities with which a random variable, X, can take on values of 1, 2, or 3 under two different hypotheses. A, The red box shows the Frequentist perspective in which only the H0 probabilities are considered and the most powerful test is to reject H0 for X ϭ 2 or 3. B, A more Bayesian perspective is shown. Example is from Johnson (2013a), who apparently borrowed it from Berger and Wolpert (1988). plying Bayes' rule to different scenarios of prior probability (Nuzzo, 2014) can be eye opening for the students. A p value just under 0.05 does not push us as far away from H0 as we would like to think it does (Johnson, 2013b).
And in the spirit of closing with a memorable quote, I share a favorite exclamation of one of my early mentors, David Hubel, whenever I approached him with some claim that struck him as highly implausible: "That's the kind of result you wouldn't believe even if it were true!" For some time, this statement bothered me a lot-was this great scientist scoffing at evidence?-until I realized that it reflected a Bayesian perspective combined with a characteristically deep awareness of the brittleness of a p value criterion for "truth."

Discussion
There is a broad consensus among statisticians that significance testing based on p values is bad statistical practice. Moreover, this consensus has existed for many years (Wasserstein et al., 2019). So why does the practice persist so stubbornly? I have argued that it is not just inertia in the teaching and practice of statistics, but that it also stems from our natural proclivity to sort continuous data into clean classes to which we can give names-what I have called black/white thinking. Because of this tendency, we need to work harder and be more creative in teaching our students better ways of thinking. I have offered here several specific teaching examples (including PowerPoint slides; Extended Data Fig. 1-1) that I have found useful in this regard. I hope they will be added to by others.