Test statistic

As a test statistic ( synonymous terms: test size, test statistic, test function ) is called in mathematical statistics a particular sample function that is used in a hypothesis test to test the decision - ie reject the null hypothesis or Nichtablehnen - to meet.

As a test value, the realization of a test statistic is denoted by a random sampling.

Use in a fixed level of significance

Before performing the test, that is, before the drawing of the required for this sample, the test statistic is a random variable whose probability distribution depends on that of the sampling variables, the sample size is. Assuming that the null hypothesis () is correct, a particular distribution model is assumed for the distribution of the test statistic depending on the test method, the distribution parameters resulting from the null hypothesis. Based on this assumed distribution as well as the previously specified significance level of Ablehnbereich is determined at the same time. Now the sample is drawn and calculated from the thereby resulting sample values ​​of the concrete value of the test statistic. To rejection of the null hypothesis occurs if and only if falls within the Ablehnbereich; otherwise is maintained below the level of significance used the null hypothesis. Namely, if the null hypothesis is true, and thus the assumed distribution of the test statistic can be assumed to be correct, corresponds to the probability that the test statistic falls in the Ablehnbereich and thus the null hypothesis is falsely rejected (so-called type 1 error ), exactly the specified significance level. The pitfalls of test size in the Ablehnbereich is tantamount to ( depending on the test problem) exceeds or falls below a certain threshold value, which is also referred to as " Critical value ".

Use with p- value

An alternative, nowadays common in many statistical software applications approach is, instead of previously set to a certain probability of type 1 error to take the test decision by calculating the so-called p- value. In this case, the sample is drawn without prior definition of the significance level, and the value of the test statistic then. Depending on which area of ​​for assumed distribution it falls, there is a value that the more "signals " the rejection of the null hypothesis, the smaller it is (this is why you interpret the p-value as a measure of the " significance " of the null hypothesis ). Thus, in this approach, in contrast to the above described "conventional" method, not even the value of the test statistic used to test decision but it from the determined value.

Examples of test statistics

• The t-test on an expected value of the test statistic is defined as follows:

With = sample mean value = loud null hypothesis assumed exact value or upper limit or lower limit for the expected value, = corrected sample standard deviation. Under the test statistic is the t-test on an expected value t -distributed with degrees of freedom.

• When chi-square goodness of fit test is the test statistic:

With = number of extent Classes of the relevant feature = empirical ( resulting from the sample ) the frequency of occurrence of the feature in the - th class, = theoretical (including imputed ) frequency of occurrence of the feature in the - th class. Under the test statistic is approximately chi-square distributed with degrees of freedom in this test.

• Test theory