# Null hypothesis

In statistics, a null hypothesis (H0) is a concept which arises in the context of statistical hypothesis testing. A common convention is to use the symbol H0 to denote the null hypothesis. The null hypothesis describes in a formal way some aspect of the statistical behaviour of a set of data and this description is treated as valid unless the actual behaviour of the data contradicts this assumption. In other words, the null hypothesis is contrasted against another hypothesis. Statistical hypothesis testing is used to make a decision about whether the data does contradict the null hypothesis: this is also called significance testing. A null hypothesis is never proven by such methods, as the absence of evidence against the null hypothesis does not establish the truth of the null hypothesis. In other words, one may either reject, or not reject the null hypothesis; one cannot accept the null hypothesis. Failing to reject H0 says that there is no strong reason to change any decisions or procedures predicated on its truth, but it also allows for the possibility of obtaining further data and then re-examining the same hypothesis.

The term was coined by the English geneticist and statistician Ronald Fisher.

Notionally, the null hypothesis set out for a particular significance test always occurs in conjunction with an alternative hypothesis. Although in some cases it may seem reasonable to consider the alternative hypothesis as simply the negation of the null hypothesis, this would be misleading. In fact, significance testing and statements about hypotheses always take place within the context of a set of assumptions (which may unfortunately be unstated). This provides a way of considering alternative hypotheses which are the negation of the null hypothesis within the context of the overall assumptions. However not all alternative hypotheses are of this "negation type": the simplest cases are directional hypotheses. An important case arises in testing for differences across a number of different groups, where the null hypothesis may be "no difference across groups" with the alternative hypothesis being that the mean values for the groups would be in a certain pre-specified order. In the theory of statistical hypothesis testing, the triple of "assumptions", "null hypothesis" and "alternative hypothesis" provides the basis for choosing an appropriate test statistic.

## Testing for differences

In scientific and medical applications, the null hypothesis plays a major role in testing the significance of differences in treatment and control groups. This use, while widespread, is criticized on a number of grounds.

The assumption at the outset of the experiment is that no difference exists between the two groups (for the variable being compared): this is the null hypothesis in this instance. Examples of other types of null hypotheses are:

• that values in samples from a given population can be modelled using a certain family of statistical distributions.
• that the variability of data in different groups is the same, although they may be centred around different values.

### Example

For example, one may want to compare the test scores of two random samples of men and women, and ask whether or not one group (population) has a mean score (which really is) different from the other. 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 : μ1 = μ2

where:

H0 = the null hypothesis
μ1 = the mean of population 1, and
μ2 = the mean of population 2.

Alternatively, the null hypothesis can postulate (suggest) 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. One can then 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 reject our null hypothesis.

## Directionality

In many statements of null hypotheses there is no appearance that these can have a "directionality", in that the statement says that values are identical. However, null hypotheses can and do have "direction" - in many of these instances statistical theory allows the formulation of the test procedure to be simplified so that the test is equivalent to testing for an exact identity. For instance, if we formulate a one-tailed alternative hypothesis, application of Drug A will lead to increased growth in patients, then the true null hypothesis is the opposite of the alternative hypothesis — that is, application of Drug A will not lead to increased growth in patients. The effective null hypothesis will be application of Drug A will have no effect on growth in patients.

To understand why the effective null hypothesis is valid, 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: μdrug > μcontrol,

where:

μ = the patients' mean growth.

The effective null hypothesis is H0: μdrug = μcontrol.

The true null hypothesis is HT: μdrug ≤ μcontrol.

The reduction occurs 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 or more extreme than our experimental observations. In order to do this, we need first to define the probability of rejecting the null hypothesis for each possibility included in the null hypothesis and second to ensure that these probabilities are all less than or equal to the quoted significance level of the test. For any reasonable test procedure the largest of all these probabilities will occur on the boundary of the region HT, specifically for the cases included in H0 only. Thus the test procedure can be defined (that is the critical values can be defined) for testing the null hypothesis HT exactly as if the null hypothesis of interest was the reduced version H0.

Note that there are some who argue that the null hypothesis cannot be as general as indicated above: 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 according to this view, 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. The usefulness of this viewpoint must be queried - one can note that the majority of null hypotheses test in practice do not meet this criterion of being "exact". For example, consider the usual test that two means are equal where the true values of the variances are unknown - exact values of the variances are not specified.

Most statisticians believe that it is valid to state direction as a part of null hypothesis, or as part of a null hypothesis/alternative hypothesis pair (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. The statistical theory required to deal with the simple cases dealt with here, and more complicated ones, makes use of the concept of an unbiased test.

## Limitations

A test of a null hypothesis is useful because it sets a limit on the probability of observing a data set as or more extreme than that observed if the null hypothesis is true. In general it is much harder to be precise about the corresponding probability if the alternative hypothesis is 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, rejection 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. One way in which a failure to reject the null hypothesis is meaningful is in relation to an arbitrarily large population from which the observed sample is supposed to be drawn. A second way in which it is meaningful is from approach where both an experiment and all details of the statistical analysis are decided before doing the experiment. The significance level of a test is then conceptually identical to the probability of incorrectly rejecting the null hypothesis judged at a pre-experiment stage, where this probability need not be a frequency-based/large-sample one.