Null hypothesis

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In statistical hypothesis testing, the null hypothesis (H₀) formally describes some aspect of the statistical "behaviour" of a set of data. This description is assumed to be valid unless the actual behaviour of the data contradicts this assumption. Thus, the null hypothesis is contrasted against another or alternative hypothesis. Statistical hypothesis testing, which involves a number of steps, is used to decide whether the data contradicts the null hypothesis. This is called significance testing. A null hypothesis is never proven by such methods, as the absence of evidence against the null hypothesis does not establish its truth. In other words, one may either reject, or not reject the null hypothesis; one cannot accept it. This means that one cannot make decisions or draw conclusions that assume the truth of the null hypothesis. Just as failing to reject it does not "prove" the null hypothesis, one does not conclude that the alternative hypothesis is dis-proven or rejected, even though this seems reasonable. One simply concludes that the null hypothesis is not rejected.^{[clarification needed]} Not rejecting the null hypothesis still allows for getting new data to test the alternative hypothesis again. On the other hand, rejecting the null hypothesis only means that the alternative hypothesis may be true, pending further testing.

For example, imagine flipping a coin three times for three heads and then forming the opinion that we have used a two-headed trick coin. Clearly this opinion is based on the premise that such a sequence is unlikely to have arisen using a normal coin. In fact, such sequences (three consecutive heads or three consecutive tails) occur a quarter of the time on average when using normal unbiased coins. Therefore, the opinion that this coin is two-headed has little support. Formally, the hypothesis to be tested in this example is "this is a two-headed coin." One tests it by assessing whether the data contradict the null hypothesis that "this is a normal, unbiased coin." Since the observed data arise reasonably often by chance under the null hypothesis, we cannot reject the null hypothesis as an explanation for the data, and we conclude that we cannot assert our hypothesis on the basis of the observed sequence.

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 (see straw man, Bayesian criticism and publication bias). 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 wish to compare the test scores of two random samples of men and women, and ask whether the mean score of one population-group differs 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:

where:

H₀ = the null hypothesis

μ₁ = the mean of population 1, and

μ₂ = the mean of population 2.

Alternatively, the null hypothesis may postulate (suggest) that the two samples are drawn from the same population, thus the variance and shape of the distributions would be equal, likewise the mean values.

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 commonly named the "significance level" of the results.

That is, in scientific experimental design, one may predict that a particular factor will produce an effect on our dependent variable — this is the 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 in which 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

Quite often statements of null hypotheses may appear not to have a "directionality," namely it is stated 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 thus the test is equivalent to testing for an exact identity. For instance, when formulating 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 i.e. 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.

In order to understand why the effective null hypothesis is valid, it is instructive to consider the nature of the hypotheses outlined above. It is predicted that patients exposed to Drug A will see increased growth compared to a control group who do not receive the drug. That is,

H₁: μ_drug > μ_control,

where:

μ = the patients' mean growth.

The effective null hypothesis is H₀: μ_drug = μ_control.

The true null hypothesis is H_T: μ_drug ≤ μ_control.

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 H_T, 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 H_T 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."^[1] 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.

See also

Statistics portal

• Counternull
• Statistical hypothesis testing

References

1. ^ Fisher, R.A. (1966). The design of experiments. 8th edition. Hafner:Edinburgh.

External links

• HyperStat Online: Null hypothesis