Consider the following two statements about an estimand \(\beta\):
- The null hypothesis, \(\mathbb H_0: \beta = c\).
- The alternative hypothesis, \(\mathbb H_1: \beta \neq c\).
This is the most common hypothesis test in the world. A statistical test uses data to choose between the two statements. Almost every researcher uses the terminology “reject the null hypothesis” or “fail to reject the null” to describe the results of this test. Note they never “accept the null hypothesis”. Say “accept the null” at a conference and someone will promptly correct you. You just made their week. They’ve waited years for that moment.
Most textbooks also insist on that terminology. And the internet agrees. My quick Google search started with an AI lecture about how I can never be “100% certain” and a “courtroom analogy”: a verdict of “not guilty” does not mean the person is proven innocent. The first link I got was a Reddit discussion where every user agrees it’s wrong to say “accept the null”. It might be the only instance of complete agreement I’ve ever witnessed online.
I regard this convention as silly. A test is just a decision rule. The rule splits the sample space into two regions. One where we reject \(\mathbb H_0\) and one where we do not. The data will land in only one of them. There is no third option. (At least not in the test I’m considering!)
Then I can just use the law of the excluded middle from classical propositional logic: a proposition is either true or false, with nothing in between. Either \(\mathbb H_0\) holds or \(\mathbb H_1\) does. Not rejecting \(\mathbb H_0\) is deciding against \(\mathbb H_1\), which is accepting \(\mathbb H_0\). Can we stop with the word play now?
The main argument of the other camp is that you may never know the truth. By saying “accept” researchers claim to have found the truth. They haven’t, so they are wrong.
This argument makes no sense, for three reasons. First, it’s a double-edged sword. Granted, you can never know the null is true. But you can never know it is false either. I can play the same game and say that a researcher who rejects the null claims to have found that the truth is not \(c\). From a technical standpoint, a rejection at the 5% level is just as fallible a decision as accepting. When the null is true, the test still rejects it 5% of the time. Type I and Type II errors are symmetric objects.
Second, we (my camp) never claimed to know the truth. Neither did the two founders of this framework. Jerzy Neyman and Egon Pearson never claimed a test reveals the truth. They proposed rules of behaviour instead. As they wrote in 1933:
Without hoping to know whether each separate hypothesis is true or false, we may search for rules to govern our behaviour with regard to them, in following which we insure that, in the long run of experience, we shall not be too often wrong. Here, for example, would be such a “rule of behaviour”: to decide whether a hypothesis, H, of a given type be rejected or not, calculate a specified character, \(x\), of the observed facts; if \(x > x_0\) reject H, if \(x \leq x_0\) accept H. Such a rule tells us nothing as to whether in a particular case H is true when \(x \leq x_0\) or false when \(x > x_0\).
Neyman later named this “inductive behaviour” and defended it as the proper meaning of accepting a hypothesis. The word survives today in the term acceptance region, which ironically appears in the same textbooks that forbid the word “accept”.
Third, I can switch the content of the null and the alternative! Take a one-sided test, like the one in the Neyman and Pearson quote: \(\mathbb H_0: \beta \leq c\) against \(\mathbb H_1: \beta > c\). The purists only let me say I “fail to reject” that \(\beta \leq c\). Now swap the labels. Test \(\mathbb H_0: \beta > c\) against \(\mathbb H_1: \beta \leq c\). If the data rejects this null, everyone happily concludes that \(\beta \leq c\), the very statement I was forbidden to accept a moment ago.2
However, the purists have a counterargument to my last argument. It goes like the argument-from-ignorance fallacy: deciding to “accept” is equivalent to “not proven false, therefore true”. So the statement you want to establish must go in the alternative, and the null must be the default you try to knock down. Their underlying assumption is that a scientist must prove that an effect exists while holding a prior of no effect. There is no reason why that should be the case in science. (I grant that it’s a useful rule for our daily lives; see Hitchens’s razor.) And I’m getting off topic: even if that were a good prior (it’s not), rejecting the null means accepting the alternative! We end up in my initial argument.
Besides, accepting the null is often the more substantive result. Alberto Abadie argued in a 2020 paper published in AER: Insights that “rejection of a point null often carries very little information”, whereas a non-rejection can be highly informative. Because few economists ever believed an effect was exactly \(c\), learning that it isn’t teaches them little, while learning that it is close to \(c\) rules out a whole range of effects they did take seriously.
So say it. Accept the null. Mean by it what Neyman meant: a decision taken with known error rates, not a revelation. And accept only when the test had the power to detect an effect worth caring about. Do both and you have nothing to apologize for.