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I hope I am making sense with the title. Often, the null hypothesis is formed with the intention of rejecting it. Is there a reason for this, or is it just a convention?

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The purpose of statistical hypothesis testing is largely to impose self-skepticism, making us cautious about promulgating our hypothesis unless there is reasonable evidence to support it. Thus in the usual form of hypothesis testing the null hypothesis provides a "devils advocate", arguing against us, and only promulgate our hypothesis if we can show that the observations mean that it is unlikely that the advocate's argument is sound. So we take $H_0$ to be the thing we don't want to be true and then see if we are able to reject it. If we can reject it, it doesn't mean that our hypothesis is likely to be correct, just that it has passed this basic hurdle and so is worthy of consideration. If we can't, it doesn't mean that our hypothesis is false, it may be that we just don't have enough data to provide suffcient evidence. As @Bahgat rightly suggests (+1) this is very much the idea of Popper's falsificationism idea.

However, it is possible to have a test where $H_0$ is the thing you want to be true, but in order for that to work, you need to show that the test has sufficiently high statistical power in order to be confident of rejecting the null if it actually is false. Computing statistical power is rather more difficult that performing the test, which is why this form of testing is rarely used and the alternative where $H_0$ is what you don't want to be true is normally used instead.

So you don't have to take $H_0$ to oppose your hypothesis, but it does make the testing procedure much easier.

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Karl Popper says "We cannot conclusively affirm a hypothesis, but we can conclusively negate it". So when we do hypothesis testing in statistics, we try to negate (reject) the opposite hypothesis (the null hypothesis) of the hypothesis we are interested in (the alternative hypothesis) and which we can not affirm. Since we can specify the null hypothesis easily, but we don't know what exactly the alternative hypothesis is. We may hypothesize for example that there is a mean difference between the two populations, but we cannot point out how wide the gap would be.

See also Don't believe in the null hypothesis?

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    $\begingroup$ If I understand your question correctly, standard frequentist hypothesis test are non-symmetric (the alternative hypothesis does not even appear in its formulation), so not being able to reject the null hypothesis does not mean that it is true and the alternate hypothesis is false. It could be that the null hypothesis is false, but there is not enough data to provide the evidence to show that it is false. The self-skepticism imposed by the test is due to assuming that $H_0$ is true until "proven" otherwise. $\endgroup$ – Dikran Marsupial Dec 4 '15 at 19:15
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    $\begingroup$ I am not quite sure whether Popper's logic of falsification can be 100% applied to statistical hypothesis testing after Fisher/Neyman. After all Popper says that "We cannot conclusively affirm a hypothesis, but we can conclusively negate it". If I recall correctly, Popper said that a hypothesis is to be framed clearly in order to make it accessible for falsification. As you point out we attempt to reject the null hypothesis. Not sure if Popper intended to falsify the null hypothesis.I think he rather would want to falsify the alternative hypothesis,or the one that actually has a meaning to us. $\endgroup$ – Stefan Dec 4 '15 at 19:18
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    $\begingroup$ @DikranMarsupial, yeah I know his works and also read "Conjectures and Refutations" and after spending some months contemplating what he wrote, I don't think that what he wants is being achieved with hypothesis testing as we do it. He also refuses inductive reasoning over deductive reasoning. Aren't we kind of making inferences all the time? His logic mostly applies to Physics, e.g. his prime example is how Einstein deduced by his calculation that gravity will distort space time and hence bent light. This was then tested on multiple occasions and could not be reject since then. $\endgroup$ – Stefan Dec 4 '15 at 19:30
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    $\begingroup$ @DikranMarsupial but I am open for falsification of my statements :) Great topic though. $\endgroup$ – Stefan Dec 4 '15 at 19:36
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    $\begingroup$ Hypothesis testing does achieve part of his scheme which is that we make our hypotheses explicity testable and falsifiable (at least in probability). In fact statisticans go further, and actually require the test to be performed before proceeding further. I suspect he might approve of Bayesian hypothesis testing rather more than the frequentist equivalent. I don't think falsificationism is fully satisfactory in isolation anyway, so I don't mind Popper not fully approving! ;o) $\endgroup$ – Dikran Marsupial Dec 4 '15 at 19:47
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It is not taken for granted that the null is always to be rejected. In model fit testing, the null is usually that the model fits well, and that is something desirable that we would hate to reject. It is, however, usually true that the sampling distribution of the test statistic is easier to derive under the null, which is usually far more restrictive than the alternative. The null that the mean difference between two groups is zero leads to a $t$-test; the null that the two distributions are the same leading to Kolmogorov-Smirnov test; the null that the linear regression model does not need nonlinear terms via Ramsey RESET test; the null that a latent variable model describes the observed covariance matrix adequately leads to a parametric space of lower dimension than an unrestricted alternative, and an asymptotic chi-square test of the distance to the smooth parametric surface in the space of $p(p+1)/2$ covariances defined by the model. So my take on this is that, as @whuber put it in the comment below, the null is usually a crucial albeit convenient technical assumption. The null is either a point (potentially multivariate) in the parametric space, so that the sampling distribution is fully specified; or a restricted parametric space, with the alternative that can be formulated to be complementary in that space, and the test statistic is based on a distance from the richer set of parameters under the alternative to the set with restrictions under the null; or, in nonparametric rank/order statistics world, the distribution under the null can be derived by the complete enumeration of all possible samples and outcomes (often approximated by something normal in large samples though).

Taking the null as something different (e.g., that the means of the two groups differ by at most 0.01, with the alternative differ by more than 0.01) requires a more complicated set of derivations, e.g., looking at the worst possible situations, which in the above case would still boil down to the point null against a one-sided alternative. The worst case on the right is $H_0: \mu_2 = \mu_1 + 0.01$ vs. $H_1: \mu_2 > \mu_1 +1$, and on the left it is $H_0: \mu_2 = \mu_1 - 0.01$ vs. $H_1: \mu_2 < \mu_1 - 1$.

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    $\begingroup$ +1 Although the references to the philosophy of science are appealing, Fisher and Neyman-Pearson preceded Popper and, I believe, were driven primarily by this crucial technical issue to create the asymmetry between the null and alternate hypothesis. $\endgroup$ – whuber Dec 5 '15 at 0:40
  • $\begingroup$ That (using a simple model unless it is rejected) may be frequent practice, but I am not sure it is necessarily good practice... $\endgroup$ – Björn Dec 5 '15 at 10:48
  • $\begingroup$ @whuber, exactly, but for some reason null hypothesis testing is so often put in context with Popper's philosophy. But the concept of the null hypothesis goes back to Fisher and Neyman-Pearson (as you mentioned). The only thing they had in common though is that they both used/proposed hypotheses to acquire knowledge - and with respect to acquiring knowledge and the scientific method, Popper was clearly more influential. So I think that's the reason why the concept of hypothesis testing (including NHT) in general is linked to Popper... I might be wrong though. $\endgroup$ – Stefan Dec 6 '15 at 0:07
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    $\begingroup$ @whuber and StasK: Would you be able to expand a bit more regarding the "crucial technical issue" which I believe refers to StasK 's comment that the test statistic is easier to derive under the null? Maybe I should ask a new question where this could hopefully be addressed. $\endgroup$ – Stefan Dec 6 '15 at 0:17
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This is a fair and good question. @Tim already gave you all you need to answer your question in a formal way, however if you are not familiar with statistical hypothesis testing you could conceptualize the null hypothesis by thinking about it in a more familiar setting.

Suppose you are being accused of having conducted a crime. Until proven guilty, you are innocent (null hypothesis). The attorney provides evidence that you are guilty (alternative hypothesis), your lawyers try to invalidate this evidence during the trial (the experiment) and in the end the judge rules whether you are innocent given the facts provided by the attorney and the lawyers. If the facts against you are overwhelming, i.e. the probability that you are innocent is very low, the judge (or jury) will conclude that your are guilty given the evidence.

Now with this in mind, you could also conceptualize features of statistical hypothesis testing, for instance why independent measurements (or evidence) are important, since after all your deserve a fair trial.

However, this is example has its limitations and eventually you have to formally understand the concept of the null hypothesis.

So to answer your questions:

  1. Yes there is a reason for the null hypothesis (as described above).

  2. No it's not just a convention, the null hypothesis is the core or statistical hypothesis testing or else it wouldn't work they way it is intended to.

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The law of parsimony (also known as Occam's razor) is a general principle of science. Under that principle, we assume a simple world until it can be shown that the world is more complicated. So, we assume the simpler world of the null hypothesis until it can be falsified. For example:

We assume treatment A and treatment B work the same until we show differently. We assume the weather is the same in San Diego as in Halifax until we show differently, we assume men and women are paid the same until we show differently, etc.

For more, see https://en.wikipedia.org/wiki/Occam%27s_razor

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If I can draw an analogy to logic, a general way to prove something is to assume the opposite and see if that leads to a contradiction. Here the null hypothesis is like the opposite, and rejecting it (i.e. showing that it is very unlikely) is like deriving the contradiction.

You do it that way because it's a way to make an unambiguous statement. Like in my field, it's much easier to say "The statement 'this drug has no benefit' has 5% chance of being right" than it is to say "The statement 'this drug has benefit' has 90% chance of being right". Of course, people want to know how much benefit is being claimed, but first they want to know it isn't zero.

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The null hypothesis is always formed with the intention to reject it that is the basic idea of hypothesis testing. When you are trying to show that something is likely to be true (e.g. a treatment improves or worsens a disease), then the null hypothesis is the default position (e.g. the treatment does not make a difference to the disease). You generate evidence for your desired claim by accumulating data that is (hopefully) so far away from what should have happened under the null hypothesis (in the example patients that are randomized to receive the treatment or a placebo having the same expected outcome) that one concludes that is very unlikely to have arisen under the null hypothesis so that you can reject the null hypothesis. In contrast failure to reject the null hypothesis does not necessarily make it very likely that the null hypothesis is true (just because a clinical trial failed to show a drug works does not mean that the drug really does nothing).

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