# Multiple Comparisons (Post Hoc Testing)

## The basic idea

Whenever a statistical test concludes that a relationship is significant, when, in reality, there is no relationship, a false discovery has been made. When multiple tests are conducted this leads to a problem known as the multiple testing problem (also known as the multiple comparisons problem, or the post hoc testing problem, data dredging and, sometimes, data mining), whereby the more tests that are conducted, the more false discoveries that are made. Multiple comparison corrections attempt to fix this problem. The basic way that they work is that they require results to have smaller p-Values in order to be classified as significant.

Refer to the Multiple Comparisons (Post Hoc Testing) page on Displayr for more information about the theory and practice of correcting for multiple comparisons.

## How multiple comparison corrections are performed within Q

Multiple comparison corrections are, by default, applied in the following situations:

By default, Q uses the False Discovery Rate correction but other corrections can be selected in Statistical Assumptions.
(Beginning in Q5.14.1.0, the False Discovery Rate and other corrections can be selected in Statistical Assumptions > Column Comparisons)

Q does not correct for multiple comparisons when:

## Multiple comparison corrections available within Q

In Q, the false discovery rate is applied by first computing the p-values for all the cells in the table. These can all be viewed by selecting Statistics - Cells and p. Then, all the cells that are NETs or copies of other cells are discarded and the remaining p-values are sorted and the cutoff computed. This cutoff is then used to determine which cells are marked as significant and which are not; this cutoff is also applied to any NETs and copies of other cells. A corrected p-value, which is computed by multiplying the actual p-value by the correction factor is available by selecting Corrected p from Statistics - Cells.

## Multiple comparison corrections

Multiple comparisons correction Description
None A significance test is employed for each cell using the selected value of the Overall significance level.
Fisher LSD Uses the Multiple Comparisons t-Test (Fisher LSD), which makes no correction for multiple comparisons. Traditionally, an F-Test (ANOVA) is conducted initially and the t-tests are only conducted if this test is significant (this is done by selecting ANOVA-Type Test in Statistical Assumptions.

Note that this test has stringent requirements about the relationships between the columns - see How to Specify Comparisons for ANOVA-Based Tests.

Duncan Duncan’s New Multiple Range Test
The familywise error rate is determined using the Statistical Assumptions setting of Overall significance level.
Tukey HSD Tukey HSD
The familywise error rate is determined using the Statistical Assumptions setting of Overall significance level
Newman-Keuls
(S-N-K)
Newman-Keuls (S-N-K)
The familywise error rate is determined using the Statistical Assumptions setting of Overall significance level
False Discovery Rate (FDR) Significance tests are conducted in accordance with the specifications for Statistical tests for categorical and numeric data in Statistical Assumptions and whether Within Row and Span is or is not selected. The False Discovery Rate Correction is used to compute a Corrected p which is then evaluated using the specified Overall significance level is used as the false discovery rate (i.e., q).
False Discovery Rate (pooled t-test) Multiple Comparisons t-Test with False Discovery Rate Correction is used to compute a Corrected p. It is evaluated using the specified Overall significance level is used as the false discovery rate (i.e., q).
Bonferroni Significance tests are conducted in accordance with the specifications for Statistical tests for categorical and numeric data in Statistical Assumptions. The Bonferroni Correction is used to compute a Corrected p which is then evaluated using the specified Overall significance level is used as the false discovery rate (i.e., q).
Bonferroni (pooled t-test) Multiple Comparisons t-Test with Bonferroni Correction is used to compute a Corrected p which is then evaluated using the specified Overall significance level is used as the false discovery rate (i.e., q).
Dunnett Dunnett’s Pairwise Multiple Comparison
The familywise error rate is determined using the Statistical Assumptions setting of Overall significance level

## Specifying multiple comparison corrections

(Beginning in Q5.14.1.0, multiple comparison corrections can be selected in Statistical Assumptions > Column Comparisons)

### Cell comparisons

The correction used when testing the cells on tables is specified using Multiple comparison correction in Cell comparisons in Statistical Assumptions.

There is a choice of None and False Discovery Rate (FDR).

### Column comparisons

The correction used with Column Comparisons is specified using Multiple comparison correction in Column comparisons in Statistical Assumptions.

### ANOVA-Type Tests

The corrections used in ANOVA-Type Tests is determined by the Multiple comparison correction specified for Column comparisons in Statistical Assumptions.

### Smart Tables

Smart Tables uses the Multiple comparison correction specified for Cell comparisons in Statistical Assumptions.