Null hypothesis

Overview
In statistics, a null hypothesis is a hypothesis set up to be nullified or refuted in order to support an alternate hypothesis. When used, the null hypothesis is presumed true until statistical evidence in the form of a hypothesis test indicates otherwise. In science, the null hypothesis is used to test differences in treatment and control groups, and the assumption at the outset of the experiment is that no difference exists between the two groups for the variable being compared.

Introduction
The null hypothesis proposes something initially presumed true. It is rejected only when it becomes evidently false. That is, when the researcher has a certain degree of confidence, usually 95% to 99%, that the data does not support the null hypothesis.

An example
For example, if we want to compare the test scores of two random samples of men and women, a null hypothesis would be that the mean score of the male population was the same as the mean score of the female population:


 * H0 : &mu;1 = &mu;2

where:


 * H0 = the null hypothesis
 * &mu;1 = the mean of population 1, and
 * &mu;2 = the mean of population 2.

Alternatively, the null hypothesis can postulate that the two samples are drawn from the same population, so that the variance and shape of the distributions are equal, as well as the means.

Formulation of the null hypothesis is a vital step in testing statistical significance. Having formulated such a hypothesis, one can establish the probability of observing the obtained data or data more different from the prediction of the null hypothesis, if the null hypothesis is true. That probability is what is commonly called the "significance level" of the results.

That is, in scientific experimental design, we may predict that a particular factor will produce an effect on our dependent variable — this is our alternative hypothesis. We then consider how often we would expect to observe our experimental results, or results even more extreme, if we were to take many samples from a population where there was no effect (i.e. we test against our null hypothesis). If we find that this happens rarely (up to, say, 5% of the time), we can conclude that our results support our experimental prediction — we accept our alternative hypothesis.

Lack of directionality
The null hypothesis does not have direction. That is, if we formulate a one-tailed alternative hypothesis that application of Drug A will lead to increased growth in patients, the null hypothesis remains that application of Drug A will have no effect on growth in patients. It is not merely the opposite of the alternative hypothesis — that is, it is not that application of Drug A will not lead to increased growth in patients.

To explain why this should be so, it is instructive to consider the nature of the hypotheses outlined above. We are predicting that patients exposed to Drug A will see increased growth compared to a control group who do not receive the drug. That is,


 * H1: &mu;drug > &mu;control

where:


 * &mu; = the patients' mean growth.

The null hypothesis is H0: μdrug = μcontrol

It is not H0 incorrect : μdrug <= μcontrol

This is because, in order to gauge support for the alternative hypothesis, classical hypothesis testing requires us to calculate how often we would have obtained results as extreme as our experimental observations. In order to do this, we need to be able to model the sampling distribution characteristic of the null hypothesis. We are unable to create this model if it is imprecise: as Fisher, who first coined the term "null hypothesis" said, "the null hypothesis must be exact, that is free of vagueness and ambiguity, because it must supply the basis of the 'problem of distribution,' of which the test of significance is the solution."

Thus the null hypothesis must be numerically exact — it must state that a particular quantity or difference is equal to a particular number. In classical science, it is most typically the statement that there is no effect of a particular treatment; in observations, it is typically that there is no difference between the value of a particular measured variable and that of a prediction.

(However, many statisticians believe that it is valid to state direction as a part of null hypothesis (for example see http://davidmlane.com/hyperstat/A73079.html). The logic is quite simple: if the direction is omitted, then if the null hypothesis is not rejected it is quite confusing to interpret the conclusion. Say, the null is that the population mean = 10, and the one-tailed alternative: mean > 10. If the sample evidence obtained through x-bar equals -200 and the corresponding t-test statistic equals -50, what is the conclusion? Not enough evidence to reject the null hypothesis? Surely not! But we cannot accept the one-sided alternative in this case. Therefore, to overcome this ambiguity, it is better to include the direction of the effect if the test is one-sided.)

Limitations
A null hypothesis is only useful if it is possible to calculate the probability of observing a data set with particular parameters from it. In general it is much harder to be precise about how probable the data would be if the alternative hypothesis were true.

If experimental observations contradict the prediction of the null hypothesis, it means that either the null hypothesis is false, or the event under observation occurs very improbably. This gives us high confidence in the falsehood of the null hypothesis, which can be improved in proportion to the number of trials conducted. However, accepting the alternative hypothesis only commits us to a difference in observed parameters; it does not prove that the theory or principles that predicted such a difference is true, since it is always possible that the difference could be due to additional factors not recognized by the theory.

For example, rejecting of a null hypothesis that predicts that the rates of symptom relief in a sample of patients who received a placebo and a sample who received a medicinal drug will be equal allows us to make a non-null statement (that the rates differed); it does not prove that the drug relieved the symptoms, though it gives us more confidence in that hypothesis.

The formulation, testing, and rejection of null hypotheses is methodologically consistent with the falsifiability model of scientific discovery formulated by Karl Popper and widely believed to apply to most kinds of empirical research. However, concerns regarding the high power of statistical tests to detect differences in large samples have led to suggestions for re-defining the null hypothesis, for example as a hypothesis that an effect falls within a range considered negligible. This is an attempt to address the confusion among non-statisticians between significant and substantial, since large enough samples are likely to be able to indicate differences however minor.

The theory underlying the idea of a null hypothesis is closely associated with the frequency theory of probability, in which probabilistic statements can only be made about the relative frequencies of events in arbitrarily large samples. A failure to reject the null hypothesis is meaningful only in relation to an arbitrarily large population from which the observed sample is supposed to be drawn.

Publication bias
In 2002, a group of psychologists launched a new journal dedicated to experimental studies in psychology which support the null hypothesis. The Journal of Articles in Support of the Null Hypothesis (JASNH) was founded to address a scientific publishing bias against such articles. According to the editors,


 * "other journals and reviewers have exhibited a bias against articles that did not reject the null hypothesis. We plan to change that by offering an outlet for experiments that do not reach the traditional significance levels (p < 0.05). Thus, reducing the file drawer problem, and reducing the bias in psychological literature. Without such a resource researchers could be wasting their time examining empirical questions that have already been examined. We collect these articles and provide them to the scientific community free of cost."

The "File Drawer problem" is a problem that exists due to the fact that academics tend not to publish results that indicate the null hypothesis could not be rejected. That is, they got a statistically significant result that indicated the relationship they were looking for did not exist. Even though these papers can often be interesting, they tend to end up unpublished, in "file drawers."

Ioannidis has inventoried factors that should alert readers to risks of publication bias.

Controversy
Null hypothesis testing is controversial when the alternative hypothesis is suspected to be true at the outset of the experiment, making the null hypothesis the reverse of what the experimenter actually believes; it is put forward only to allow the data to contradict it. Many statisticians have pointed out that rejecting the null hypothesis says nothing or very little about the likelihood that the null is true. Under traditional null hypothesis testing, the null is rejected when P(Data | Null) (where P(x|y) denotes the probability of x given y) is very small, say 0.05. However, researchers are really interested in P(Null | Data) which cannot be inferred from a p-value. In some cases, P(Null | Data) approaches 1 while P(Data | Null) approaches 0, in other words, we can reject the null when it's virtually certain to be true. For this and other reasons, Gerd Gigerenzer has called null hypothesis testing "mindless statistics" while Jacob Cohen describes it as a ritual conducted to convince ourselves that we have the evidence needed to confirm our theories.

Bayesian statisticians normally reject the idea of null hypothesis testing. Given a prior probability distribution for one or more parameters, sample evidence can be used to generate an updated posterior distribution. In this framework, but not in the null hypothesis testing framework, it is meaningful to make statements of the general form "the probability that the true value of the parameter is greater than 0 is p".