Post-hoc analysis

Post- hoc tests are tests of significance of mathematical statistics. With the one-factor ANOVA, the Kruskal -Wallis test or the median test is only found that there are significant differences in a group of mean values. The post-hoc tests indicate either pairwise mean comparisons or comparisons with subgroup information, which means differ significantly from each other.

  • 4.1 Tukey test
  • 4.2 Student-Newman- Keuls test
  • 4.3 Duncan test
  • 5.1 Pairwise Comparisons
  • 5.2 Group -wise comparisons

Overview of post-hoc tests

The post-hoc tests differ in various criteria, such as are the sample sizes in all treatment groups ( balanced case ) or not ( unbalanced case) or the variance in all groups ( homogeneity of variance ) or not ( heterogeneity of variance ). The variance homogeneity can be checked with the Levene test.

The tests can be partially ordered, depending on how conservative they are:

Prerequisites and Notation

It is assumed that the alternative hypothesis was accepted at the mean comparisons in groups and with a significance level, ie there are differences between at least two group means. The hypotheses for all of the following tests are

It should also be the number of observations in the group and the number of all observations. The tests are divided into tests for the balanced case () and for the unbalanced case ( the sample sizes in the groups may be different).

Tests for the unbalanced case

Least Significant Difference test

In the Least Significant Difference test is the test statistic:

With

And the group variance of the group.

The Least Significant Difference test is based on the two-sample t-test, however, the variance is calculated with the help of all groups.

Least Significant Difference Bonferroni test

The Least Significant Difference Bonferroni test, the test statistic is identical to the test statistic in the Least Significant Difference test. However, the significance level is corrected according to the Bonferroni method. ANOVA is performed with the level of significance, the significance level corrected for the pairwise comparisons of mean value is used:

The critical values ​​for the corrected level of significance can be found in special tables or can use the approximation

Be determined. is the quantile of the standard normal distribution.

The test should not be applied to large only, otherwise the corrected significance level is too small and overlap non- rejection regions of the t-test. If, for example, and, then.

Scheffé test

The Scheffé test requires actually the homogeneity of variances in the groups, but it is insensitive to the violation of this condition.

Easy Scheffé test

The simple Scheffé test checks vs. using the test statistic

The simple Scheffé test is a special case of the general Scheffe test for a linear contrast for two means.

Linear contrast

A linear contrast of one or more mean values ​​is defined as

For the simple Scheffé test is the linear contrast:

Two contrasts and are called orthogonal if and only if

General Scheffé test

The hypotheses for all ( orthogonal ) contrasts vs. For the general Scheffé test. for at least one contrast. The test statistic is given by

The idea is based on the variance decomposition of the estimated contrast

Since under the null hypothesis is true.

Tests for the balanced case

This test is intended for the balanced case, ie the sample size in each group is the same. SPSS performs the test by unequal sample sizes in each group, however, is then calculated as the harmonic mean of sample sizes.

The test statistic is for the following tests always the same

The critical values ​​are only tabulated ( usually for or ). In this case, between the mean values ​​, and further averages.

Tukey test

In the Tukey test, the critical values ​​are obtained from

That is, There is no Bonferroni correction and the number of over -loaded means is not considered.

Student-Newman- Keuls test

In the Student-Newman- Keuls test, the critical values ​​are obtained from

That is, There is no Bonferroni correction and the number of over -loaded means is taken into account.

Duncan test

In the Duncan test, the critical values ​​are obtained from

That is, There is a Bonferroni correction and the number of over -loaded means is taken into account.

Example

For the Mietbelastungsquote ( = share of gross rental income on net household income ), taken from the CAMPUS Files for the 2002 micro-census of the Federal Statistical Office, result, both the non-parametric median test as well as parametric one-way ANOVA highly significant differences in the medians or averages of the federal states. That there are differences between the provinces in the middle on rent ( relative to income ).

Since the Levene test rejects the null hypothesis of homogeneity of variance and the numbers of observations differ significantly in the sample, only the following test procedure to determine difference remain:

  • Least significant difference
  • Least significant difference, Bonferroni
  • Scheffé

Since the Scheffé test is carried out in SPSS both pairwise comparisons as outputs also homogeneous subgroups, we look at the results.

Pairwise comparisons

In the respective pairwise comparisons are output for each combination of two states:

  • The difference,
  • The standard error,
  • The p-value (column: significance), which means a rejection of the equality of means falls below the predetermined level of significance, and
  • A 95 % confidence interval for the difference of the mean value. If the confidence interval is not the null is rejected the null hypothesis at the significance level of 5 %.

For a given level of significance of 5 %, only the mean values ​​of Schleswig- Holstein and Saxony significant ( p- value equal to 2.1 %), and not at other compare with Schleswig -Holstein.

Group -wise comparisons

It is an iterative process carried out in order to find homogeneous subgroups, ie, groups in which the null hypothesis of equality of means is not rejected. For this purpose, the observed mean values ​​are sorted according to size and it is carried out a series of tests.

In the first step, the null hypothesis is tested and rejected; we already know that the means are different. Then first

  • The state away with the largest mean and tested the null hypothesis and
  • The state away with the smallest mean and tested the null hypothesis.

In both tests, only groups with 15 federal states are tested. If the null hypothesis is rejected for one of the tests ( in the table red), the state with the largest mean value and the state are removed from the group with the smallest mean and retested. So that a sequence of null hypotheses to be tested is created with a smaller and smaller number of averages.

The method is aborted when

  • Either the null hypothesis in any of the tests can not be rejected ( in the table green) or
  • The considered null hypothesis is not rejected part of a null hypothesis is already ( yellow in the table ) or
  • Only one state is left.

The "green" sub-groups are output from SPSS.

For example, two homogeneous subgroups arise, each with 14 federal states. That is, here could not be rejected the null hypothesis of equality of means. For a better interpretation would be preferred or only a few overlapping homogeneous subgroups, but this is here with 12 states not the case and an interpretation of this result is correspondingly difficult.

Levene's test Digital Object Identifier
658471
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