Hypothesis (statistics)

In statistics , a hypothesis is an assumption that is tested using methods of mathematical statistics on the basis of empirical data . A pair of opposites is distinguished between the null hypothesis and the alternative hypothesis (also counter hypothesis or working hypothesis ). The null hypothesis often states that there is no effect or difference or that a certain connection does not exist. This thesis should be rejected, so that the alternative hypothesis remains as a possibility. This indirect approach is intended to keep the probability of an erroneous rejection of the null hypothesis small in a controlled manner. Often, however, confusion arises for the user because this approach suggests the possibility that - provided the null hypothesis cannot be rejected and the alternative hypothesis thus not accepted - the null hypothesis is considered proven. However, this is not the case. A general approach is the testing of so-called general linear hypotheses in the classical linear model of normal regression.

Null hypothesis

In statistics, the null hypothesis is an assumption about the probability distribution of one or more random variables . The alternative hypothesis represents a set of alternative assumptions about the null hypothesis. The task of deciding between the null and alternative hypotheses is called the test problem . If the sample result speaks against the assumption, the hypothesis is rejected; otherwise it is retained. ${\ displaystyle H_ {0}}$

Because an investigation often has the goal of showing that there is a certain difference formulated in the alternative hypothesis, the null hypothesis contains the opposite, i.e. the equality of facts, for example: ${\ displaystyle H_ {0}}$

that there is no difference between groups,
that a certain drug has no effect,
that there is no connection between characteristics.

Examples

statistics

Because one suspects that there is a fundamental difference between men and women with regard to a certain test result, one initially makes the assumption that there is no difference. This assumption is the null hypothesis. One tries to answer the question whether the test result differs statistically significantly between the groups. The null hypothesis in this case would be that the mean results of men and women are the same:

{\ displaystyle H_ {0} \ colon \ mu _ {1} = \ mu _ {2},}

in which:

{\ displaystyle H_ {0}}

the null hypothesis is

{\ displaystyle \ mu _ {1}}

the expected value of the men's test result, and

{\ displaystyle \ mu _ {2}}

the expected value of the women's test result

psychology

In the field of forensic psychology , the null hypothesis was defined by the judgment of the Federal Court of Justice (on the procedure for a credibility assessment): “The basic methodological principle consists in negating a fact to be checked (here: credibility of the specific statement) until this negation is no longer compatible with the facts collected. The expert therefore initially assumes during the assessment that the statement is untrue (so-called null hypothesis). To test this assumption, he has to form further hypotheses. If his test strategy shows that the false hypothesis can no longer be in accordance with the facts collected, it is rejected and the alternative hypothesis that it is a true statement applies. "

Alternative hypothesis

As an alternative hypothesis , or is referred to in empirical science is often a reasoned by observations or considerations assumption or presumption that certain explaining phenomena serves and (namely the null hypothesis) precludes the a potentially widespread acceptance or conjecture. In this respect, the alternative hypothesis can be viewed as innovative . ${\ displaystyle H_ {1}}$ ${\ displaystyle H_ {A}}$

In contrast, there is the null hypothesis. Null and alternative hypotheses must not overlap, i.e. i.e. they must be disjoint . The aim of a statistical test is to reject (reject) the null hypothesis. If this cannot be discarded (e.g. because not enough observations are available), from a statistical point of view there is no reason to assume that the validity of the null hypothesis could be proven (see type 2 error ). A statistical test can only lead to an acceptance of the alternative hypothesis by rejecting the null hypothesis, but not to an (in the narrower sense) acceptance of the null hypothesis. However, this does not mean that a tested null hypothesis cannot also be correct. A repeated failure to refute a null hypothesis then means that the assumption to be tested in the null hypothesis receives additional support.

Types of hypotheses

Directed vs. undirected

If the totality of all possibilities is, then one can formulate a test generally in such a way that the subset represents the null hypothesis and the alternative hypothesis . However, there are no standard tests for this general case. Instead, one looks at the special cases where either only the set that satisfies the null hypothesis is an interval (undirected test) or where both hypotheses can be represented by an interval (directed test). ${\ displaystyle \ Theta}$ ${\ displaystyle \ Theta _ {0} \ subseteq \ Theta}$ ${\ displaystyle \ Theta _ {1} = \ Theta \ backslash \ Theta _ {0}}$

Non-directional alternative hypotheses only assume a difference between the compared parameters. It does not matter whether this difference is directed upwards or downwards. For example: A training is carried out with children. If the hypothesis is undirected, then the alternative hypothesis could be that there is a difference in athletic performance between the population of children with and without training. It does not matter whether the children with training have a higher or lower athletic performance than children without training. Accordingly, the null hypothesis is that there are no differences between the two populations. ${\ displaystyle n = 100}$

Directed hypotheses assume a difference between the examined parameters in a certain direction. In the above example, the alternative hypothesis was either that the children with training have a higher or lower physical fitness than children without training. The null hypothesis then reads that both populations are either equal or differ in the opposite direction.

Specific vs. unspecific

Specific hypotheses make statements about the size of the expected difference or relationship between the examined parameters (e.g. postulate a certain minimum value). Based on the example above, one could e.g. For example, assume that the training population is at least three IQ points better than the starting population.
Non-specific hypotheses, on the other hand, do not make any statements about the size of the expected difference or relationship.

Easy vs. composed

A hypothesis is called simple if it is based on a single distribution. If the hypothesis merely states that the distribution belongs to a family of distributions, one speaks of composite hypotheses. A null hypothesis of the shape for normally distributed quantities with known variance is an example of a simple hypothesis, the associated alternative hypothesis is composed. ${\ displaystyle H_ {0} \ colon \ mu = 0}$ ${\ displaystyle H_ {A} \ colon \ mu \ neq 0}$

Choice of the null and alternative hypothesis

Formally, the null and alternative hypotheses split a parameter space into two disjoint, non-empty subsets and . The null hypothesis includes the statement that the unknown parameter originates from , and the alternative hypothesis that the unknown parameter originates from . ${\ displaystyle \ Theta}$ ${\ displaystyle \ Theta _ {0}}$ ${\ displaystyle \ Theta _ {1}}$ ${\ displaystyle \ theta}$ ${\ displaystyle \ Theta _ {0}}$ ${\ displaystyle \ theta}$ ${\ displaystyle \ Theta _ {1}}$

{\ displaystyle H_ {0} \ colon \ theta \ in \ Theta _ {0}}

vs.

{\ displaystyle H_ {1} \ colon \ theta \ in \ Theta _ {1}}

For two tests, the one - sample t-test (parameter test ) and the Lilliefors test (distribution test ), the following table shows the possible null and alternative hypotheses , etc.

test	Null hypothesis	Alternative hypothesis	Parameter space	${\ displaystyle \ Theta _ {0}}$	${\ displaystyle \ Theta _ {1}}$
One-sample t-test	${\ displaystyle \ mu = \ mu _ {0}}$ (*)	${\ displaystyle \ mu \ neq \ mu _ {0}}$	${\ displaystyle \ mathbb {R}}$	${\ displaystyle \ mu _ {0}}$	${\ displaystyle \ mu}$ With ${\ displaystyle \ mu \ neq \ mu _ {0}}$
	${\ displaystyle \ mu \ leq \ mu _ {0}}$	${\ displaystyle \ mu> \ mu _ {0}}$	${\ displaystyle \ mathbb {R}}$	${\ displaystyle \ mu}$ With ${\ displaystyle \ mu \ leq \ mu _ {0}}$	${\ displaystyle \ mu}$ With ${\ displaystyle \ mu> \ mu _ {0}}$
	${\ displaystyle \ mu \ geq \ mu _ {0}}$	${\ displaystyle \ mu <\ mu _ {0}}$	${\ displaystyle \ mathbb {R}}$	${\ displaystyle \ mu}$ With ${\ displaystyle \ mu \ geq \ mu _ {0}}$	${\ displaystyle \ mu}$ With ${\ displaystyle \ mu <\ mu _ {0}}$
Lilliefors test	${\ displaystyle X}$ is normally distributed (*)	${\ displaystyle X}$ is not normally distributed	All distributions	Normal distribution	all distributions except the normal distribution

The two null hypotheses marked with (*) are simple null hypotheses . In this case, the role of the null hypothesis and the alternative hypothesis cannot be reversed, even if it were desirable from the application point of view.

Only in the case of the other two composite null hypotheses can the role of the null and alternative hypotheses be reversed, i.e. H. one must choose one of the hypothesis pairs. However, it always applies here that the equal sign must be in the null hypothesis.

In the test decision, if the null hypothesis is not rejected, the error of type 2 can be made ( failure to reject the null hypothesis, although the alternative hypothesis applies). However, the likelihood of this is unknown. If the null hypothesis is rejected, the first type of error (rejection of the null hypothesis, although the null hypothesis applies), but the probability of this being less than or equal to the specified level of significance (usually 5%). Hence one is interested in rejecting the null hypothesis.

This leads to the following decision-making scheme in 4 steps:

Does the task indicate whether something should be shown or refuted?

Yes: You formulate what is shown as an alternative hypothesis or what should be refuted as a null hypothesis.

No: Are the consequences of wrong decisions known?
Yes: You turn the mistake with the greatest risk into the type I mistake, because this mistake is fixed.
No: Does the task indicate which interest group the examiner belongs to?
Yes: The alternative hypothesis is formulated in such a way that it is in the interest of the examiner to prove the alternative hypothesis.
No: Then it is not possible to formulate a clear hypothesis.

Example for 1 .: A group of environmentalists and a detergent company are arguing whether the average phosphate content in a detergent is too high (e.g. 18 g per package). ${\ displaystyle \ mu}$

The environmentalists want to prove that the phosphate content is too high. Hence their hypotheses will be: vs. . ${\ displaystyle H_ {0} \ colon \ mu \ leq 18 \; \ mathrm {g}}$ ${\ displaystyle H_ {1} \ colon \ mu> 18 \; \ mathrm {g}}$
The company wants to prove that the phosphate content is okay. Hence their hypotheses will be: vs. . ${\ displaystyle H_ {0} \ colon \ mu \ geq 18 \; \ mathrm {g}}$ ${\ displaystyle H_ {1} \ colon \ mu <18 \; \ mathrm {g}}$

Depending on your interests, you come to different pairs of hypotheses.

Example for 2 .: A bank customer wants a loan of 1,000 euros from his bank. If the banker rejects the loan request and the customer is solvent, he loses the interest paid in the amount of 80 euros. If the banker gives the customer the loan and the customer is insolvent, the banker loses the entire 1,000 euros.

If the hypotheses are formulated with a significance level of 5% in such a way that the 1st type error is precisely the banker rejects the loan request and the customer is solvent , then it is known that the expected loss for the 1st type error is 80 euros × 5% = 4 euros. In the case of the type 2 error, the probability is unknown; i.e. in the worst case it is one and the maximum loss is equal to 1,000 euros. Together, a wrong decision in the test results in an expected loss of 1,004 euros.

If the hypotheses are formulated with a significance level of 5% in such a way that the 1st type error corresponds to the banker just accepting the loan request and the customer is insolvent , it is known that the expected loss in the case of the 1st type error is 1,000 euros × 5% = Is 50 euros. In the case of the type 2 error, the probability is unknown; H. in the worst case it is one and the maximum loss is equal to 80 euros. In total, a wrong decision in the test results in an expected loss of 130 euros.

The hypotheses should therefore be chosen in such a way that the error of the first type corresponds to the banker accepts the loan request and the customer is insolvent , since then the expected loss is lowest.

Individual evidence

↑ BGH, judgment of July 30, 1999 , Az. 1 StR 618/98, full text = BGHSt 45, 164 ff.
↑ George G. Judge, R. Carter Hill, W. Griffiths, Helmut Lütkepohl , TC Lee. Introduction to the Theory and Practice of Econometrics. 2nd Edition. John Wiley & Sons, New York / Chichester / Brisbane / Toronto / Singapore 1988, ISBN 0-471-62414-4 , p. 93
^ H. Rinne: Pocket book of statistics. 2nd Edition. Harri Deutsch Verlag, Frankfurt am Main 1997, ISBN 3-8171-1559-8 , p. 528.

[1] BGH, judgment of July 30, 1999 , Az. 1 StR 618/98, full text = BGHSt 45, 164 ff.

[2] George G. Judge, R. Carter Hill, W. Griffiths, Helmut Lütkepohl , TC Lee. Introduction to the Theory and Practice of Econometrics. 2nd Edition. John Wiley & Sons, New York / Chichester / Brisbane / Toronto / Singapore 1988, ISBN 0-471-62414-4 , p. 93

[3] H. Rinne: Pocket book of statistics. 2nd Edition. Harri Deutsch Verlag, Frankfurt am Main 1997, ISBN 3-8171-1559-8 , p. 528.