[UPDATE 2/12/2013: Given a few more weeks to think about it, this post hardly seems worth the electrons. The example I use to justify usage of “accept H0” doesn’t involve distinct H0 and H1 hypotheses: H0 = ~H1. If the only options are A and ~A then, sure, A = ~~A but that’s not exactly a revelation. The real question is “Is it legitimate to say you “accept H0″ when there are distinct H0 and H1 hypotheses?” For that case, I believe “reject H1” is appropriate but “accept H0” is not, i.e., I accept conventional usage. Anyhow, read on if you’re hurting for entertainment.]
Hypothesis testing is a basic problem in statistical analysis. You consider a null hypothesis (H0) in the context of an alternative hypothesis (H1). If your experiment is testing for an effect then H0 is ‘no effect’ and H1 is ‘effect is present’. In statistics parlance, one may “reject H0” or “accept H1” but one never “accepts” H0. You can “reject H1” but you don’t “accept H0”. On the one hand, that makes sense. You could establish with a high degree of confidence that your data isn’t consistent with H1 and therefore reject it but that doesn’t mean that H0 is true. (H1 could be true and you just happened to collect funky data which led you to reject it. Type II error. Oops.)
But on the other hand, maybe it doesn’t make so much sense. Consider a binary decision problem. Let’s take, for example, a fire control decision made in a PATRIOT missile battery. The missiles are launched or they’re not. The decision hypotheses are:
- H0: No threat.
- H1: Threat present. (Incoming missile)
If the system accepts H1 then it launches. If it rejects H1 then it does not. The fire control decision can be (and I presume is) based entirely on accepting or rejecting H1. Now here’s the question of semantics vs substance: How is saying that you* “accept H0” any different from saying that you “reject H1”? They both result in the same action. The consequences of the decision are the same in each case. To be formally correct you’d never say that you “accept H0” even if, in practice, you accept H0. Word choice doesn’t affect Type I or Type II error rates. So is it semantics or substance to insist that one “reject H1” rather than “accept H0”?
UPDATE: Having taken a day to think about it, I believe there is substance to the distinction. I also believe it’s entirely reasonable to say that you “accept H0” in the missile launch case I described – or in any single variable, binary outcome scenario.
When the effect in question isn’t directly observable and/or there is any ambiguity in the data then it seems appropriate to say “reject H1” but not “accept H0”. It could be a weak effect that you’re looking for (low signal-to-noise) and your data may not show it to be statistically significant. If that happens then you (appropriately) reject H1. For example, if your H1 hypothesis were “Children who attend full-day kindergarten achieve higher socio-economic status as adults than do kids who only go for half a day.” that could be very challenging to test.** To begin with, there are a ton of other variables to control for. Even controlling for those, the hypothesis might be true for some populations but not all, definition of socio-economic status is subjective so the criteria for accepting or rejecting H1 would be subjective, etc. In general, when there’s ambiguity in the data or in your definition of what constitutes a positive resul then rejecting H1 doesn’t mean you’re necessarily sold on H0, it just means that you acknowledge that the argument in favor or H1 is dicey.
In contrast, for the missile launch example I described above there’s a single variable and no ambiguity in the outcome.*** The signal-to-clutter ratio of the experiment is essentially infinite. Type I and (particularly) Type II errors are revealed immediately. For any case where you’re dealing with a single variable and a binary outcome, it seems entirely reasonable to use the terms “reject H1” and “accept H0” interchangeably.
* or, more accurately, the PATRIOT battery’s full-automated (I presume) fire control system
** I imagine that hypothesis has been tested. I have no idea whether or not it’s true.
*** What exactly would be your control for the missile launch scenario?