Null hypothesis


The null hypothesis is the claim in scientific research that the effect being studied does not exist. The null hypothesis can also be described as the hypothesis in which no relationship exists between two sets of data or variables being analyzed. If the null hypothesis is true, any experimentally observed effect is due to chance alone, hence the term "null". In contrast with the null hypothesis, an alternative hypothesis is developed, which claims that a relationship does exist between two variables.

Basic definitions

The null hypothesis and the alternative hypothesis are types of conjectures used in statistical tests to make statistical inferences, which are formal methods of reaching conclusions and separating scientific claims from statistical noise.
The statement being tested in a test of statistical significance is called the null hypothesis. The test of significance is designed to assess the strength of the evidence against the null hypothesis, or a statement of 'no effect' or 'no difference'. It is often symbolized as.
The statement that is being tested against the null hypothesis is the alternative hypothesis. Symbols may include and.
A statistical significance test starts with a random sample from a population. If the sample data are consistent with the null hypothesis, then you do not reject the null hypothesis; if the sample data are inconsistent with the null hypothesis, then you reject the null hypothesis and conclude that the alternative hypothesis is true.
Consider the following example. Given the test scores of two random samples, one of men and one of women, does one group score better than the other? A possible null hypothesis is that the mean male score is the same as the mean female score:
where
the null hypothesis,
the mean of population 1, and
the mean of population 2.
A stronger null hypothesis is that the two samples come from populations with equal variances and shapes of their respective distributions. This is known as a pooled variance.

Terminology

; Simple hypothesis : Any hypothesis that specifies the population distribution completely. For such a hypothesis the sampling distribution of any statistic is a function of the sample size alone.
; Composite hypothesis : Any hypothesis that does not specify the population distribution completely. Example: A hypothesis specifying a normal distribution with a specified mean and an unspecified variance.
The simple/composite distinction was made by Neyman and Pearson.
; Exact hypothesis: Any hypothesis that specifies an exact parameter value is an exact hypothesis, also known as a point hypothesis. Example:.
; Inexact hypothesis : Those specifying a parameter range or interval. Examples: ;.
Fisher required an exact null hypothesis for testing.
A one-tailed hypothesis is an inexact hypothesis in which the value of a parameter is specified as being either:
  • above or equal to a certain value, or
  • below or equal to a certain value.
A one-tailed hypothesis is said to have directionality.
Fisher's original example was a one-tailed test. The null hypothesis was asymmetric. The probability of guessing all cups correctly was the same as guessing all cups incorrectly, but Fisher noted that only guessing correctly was compatible with the lady's claim.

Technical description

The null hypothesis is a default hypothesis that a quantity to be measured is zero. Typically, the quantity to be measured is the difference between two situations. For instance, trying to determine if there is a positive proof that an effect has occurred or that samples derive from different batches.
The null hypothesis is generally assumed to remain possibly true. Multiple analyses can be performed to show how the hypothesis should either be rejected or excluded e.g. having a high confidence level, thus demonstrating a statistically significant difference. This is demonstrated by showing that zero is outside of the specified confidence interval of the measurement on either side, typically within the real numbers. Failure to exclude the null hypothesis does not logically confirm or support the null hypothesis. So failure of an exclusion of a null hypothesis amounts to a "don't know" at the specified confidence level; it does not immediately imply null somehow, as the data may already show a indication for a non-null. The used confidence level does absolutely certainly not correspond to the likelihood of null at failing to exclude; in fact in this case a high used confidence level expands the still plausible range.
A non-null hypothesis can have the following meanings, depending on the author a) a value other than zero is used, b) some margin other than zero is used and c) the "alternative" hypothesis.
Testing the null hypothesis provides evidence that there are statistically sufficient grounds to believe there is a relationship between two phenomena. Testing the null hypothesis is a central task in statistical hypothesis testing in the modern practice of science. There are precise criteria for excluding or not excluding a null hypothesis at a certain confidence level. The confidence level should indicate the likelihood that much more and better data would still be able to exclude the null hypothesis on the same side.
The concept of a null hypothesis is used differently in two approaches to statistical inference. In the significance testing approach of Ronald Fisher, a null hypothesis is rejected if the observed data are significantly unlikely to have occurred if the null hypothesis were true. In this case, the null hypothesis is rejected and an alternative hypothesis is accepted in its place. If the data are consistent with the null hypothesis statistically possibly true, then the null hypothesis is not rejected. In neither case is the null hypothesis or its alternative proven; with better or more data, the null may still be rejected. This is analogous to the legal principle of presumption of innocence, in which a suspect or defendant is assumed to be innocent until proven guilty beyond a reasonable doubt.
In the hypothesis testing approach of Jerzy Neyman and Egon Pearson, a null hypothesis is contrasted with an alternative hypothesis, and the two hypotheses are distinguished on the basis of data, with certain error rates. It is used in formulating answers in research.
Statistical inference can be done without a null hypothesis, by specifying a statistical model corresponding to each candidate hypothesis, and by using model selection techniques to choose the most appropriate model..

Principle

Hypothesis testing requires constructing a statistical model of what the data would look like if chance or random processes alone were responsible for the results. The hypothesis that chance alone is responsible for the results is called the null hypothesis. The model of the result of the random process is called the distribution under the null hypothesis. The obtained results are compared with the distribution under the null hypothesis, and the likelihood of finding the obtained results is thereby determined.
Hypothesis testing works by collecting data and measuring how likely the particular set of data is, when the study is on a randomly selected representative sample. The null hypothesis assumes no relationship between variables in the population from which the sample is selected.
If the data-set of a randomly selected representative sample is very unlikely relative to the null hypothesis, the experimenter rejects the null hypothesis, concluding it is false. This class of data-sets is usually specified via a test statistic, which is designed to measure the extent of apparent departure from the null hypothesis. The procedure works by assessing whether the observed departure, measured by the test statistic, is larger than a value defined, so that the probability of occurrence of a more extreme value is small under the null hypothesis.
If the data do not contradict the null hypothesis, then only a weak conclusion can be made: namely, that the observed data set provides insufficient evidence against the null hypothesis. In this case, because the null hypothesis could be true or false, in some contexts this is interpreted as meaning that the data give insufficient evidence to make any conclusion, while in other contexts, it is interpreted as meaning that there is not sufficient evidence to support changing from a currently useful regime to a different one. Nevertheless, if at this point the effect appears likely and/or large enough, there may be an incentive to further investigate, such as running a bigger sample.
For instance, a certain drug may reduce the risk of having a heart attack. Possible null hypotheses are "this drug does not reduce the risk of having a heart attack" or "this drug has no effect on the risk of having a heart attack". The test of the hypothesis consists of administering the drug to half of the people in a study group as a controlled experiment. If the data show a statistically significant change in the people receiving the drug, the null hypothesis is rejected.

Goals of null hypothesis tests

There are many types of significance tests for one, two or more samples, for means, variances and proportions, paired or unpaired data, for different distributions, for large and small samples; all have null hypotheses. There are also at least four goals of null hypotheses for significance tests:
  • Technical null hypotheses are used to verify statistical assumptions. For example, the residuals between the data and a statistical model cannot be distinguished from random noise. If true, there is no justification for complicating the model.
  • Scientific null assumptions are used to directly advance a theory. For example, the angular momentum of the universe is zero. If not true, the theory of the early universe may need revision.
  • Null hypotheses of homogeneity are used to verify that multiple experiments are producing consistent results. For example, the effect of a medication on the elderly is consistent with that of the general adult population. If true, this strengthens the general effectiveness conclusion and simplifies recommendations for use.
  • Null hypotheses that assert the equality of effect of two or more alternative treatments, for example, a drug and a placebo, are used to reduce scientific claims based on statistical noise. This is the most popular null hypothesis; It is so popular that many statements about significant testing assume such null hypotheses.
Rejection of the null hypothesis is not necessarily the real goal of a significance tester. An adequate statistical model may be associated with a failure to reject the null; the model is adjusted until the null is not rejected. The numerous uses of significance testing were well known to Fisher who discussed many in his book written a decade before defining the null hypothesis.
A statistical significance test shares much mathematics with a confidence interval. They are mutually illuminating. A result is often significant when there is confidence in the sign of a relationship. Whenever the sign of a relationship is important, statistical significance is a worthy goal. This also reveals weaknesses of significance testing: A result can be significant without a good estimate of the strength of a relationship; significance can be a modest goal. A weak relationship can also achieve significance with enough data. Reporting both significance and confidence intervals is commonly recommended.
The varied uses of significance tests reduce the number of generalizations that can be made about all applications.