Omnibus test

Omnibus test is a term from statistics and describes a special type of statistical test in test theory . The test only checks whether there is a difference between several populations (groups) or not, that is

${\displaystyle H_{0}:}$ There is no difference between all populations (groups).

${\displaystyle H_{1}:}$ There are at least two populations that differ between them.

However, it does not provide any information about which populations are responsible for the difference. So-called post-hoc tests are then carried out for this purpose.

Another fundamental problem in omnibus tests is that the null hypothesis is often composed of several partial hypotheses. One example is the simple analysis of variance , which tests whether the means are the same in normally distributed populations with the same variance. The global hypotheses are ${\displaystyle p}$

${\displaystyle H_{0}:\mu _{1}=\ldots =\mu _{p}}$

${\displaystyle H_{1}:\,}$There are at least two populations (groups) with . ${\displaystyle \mu _{i}\neq \mu _{j}}$

The partial hypotheses here are the paired hypotheses , ..., . If one of the partial hypotheses is rejected, the global null hypothesis should also be rejected. As a result, the significance level for the partial hypotheses must be chosen to be smaller than the significance level for the global null hypothesis . One possibility for this is the Bonferroni method . ${\displaystyle H_{0;1}:\,\mu _{1}=\mu _{2}}$ ${\displaystyle H_{0;k}:\,\mu _{p-1}=\mu _{p}}$ ${\displaystyle H_{0;m}}$ ${\displaystyle H_{0}}$ ${\displaystyle H_{0}}$

Other examples of omnibus tests are

• for differences in location
  • the Kruskal-Wallis test ,
  • the median test ,
• for differences in scatter
  • the Levene test and
• for differences in the probability distribution
  • the chi-square homogeneity test .
• for differences regarding the autocorrelation
  • the Portmanteau tests

literature

• Jürgen Bortz , Christof Schuster: Statistics for human and social scientists . 7th edition. Springer, Berlin 2010, ISBN 978-3-642-12769-4 .