# Wald F statistics

## Testing of Fixed Effects with Wald F statistics

The table for testing fixed effects has 4 forms:
``` Source   DF             Fic
Source   DF             Fic  Fcn M
Source   DF  DDFic    Fic            Pic
Source   DF  DDFcn    Fic  Fcn M   Pcn
```
depending on whether conditional F-statistics are reported (requested by the !FCON qualifier ) and whether the denominator degrees of freedom are reported.
Source is the model term,
DF is the numerator degrees of freedom ,
DDF is the denominator degrees of freedom for either the incremental or conditional F statistic,
Fic is the F statistic for the incremental sum of squares,
Fcn is the F statistic for the conditional sum of squares,
M is an alpha code indicating conditioning level in Fcn ,
Pic, Pcn is probability associated with the corresponding F statistic.
Note that the probability is only available when the denominator degrees of freedom is also present.

ASReml always reports incremental F-statistics ( Fic) for the fixed model terms (in the DENSE partition) conditional on the order the terms were nominated in the model. Users should consult the Theory chapter of the ASReml User Guide (PDF document) for details of this table.

The 'conditional maximum' model used as the basis for the conditional F-statistic is spelt out in the .aov file.

The numerator degrees of freedom for each term is easily determined as the number of non-singular equations involved in the term. However, in general, calculation of the denominator degrees of freedom is not trivial. ASReml will by default attempt the calculation for small analyses, by one of two methods. In larger analyses, users can request the calculation be attempted using the !DDF qualifier. Use !DDF -1 to prevent the calculation.

#### Sample of Wald F statistics table

```     Source of Variation   NumDF     DenDF    F-inc             Prob
7 mu                        1       5.0   245.14            <.001
4 variety                   2      10.0     1.49            0.272
2 nitrogen                  3      45.0    37.69            <.001
8 variety.nitrogen          6      45.0     0.30            0.932
```

## !FOWN directive

!FOWN terms to test ; background terms

allows the user to specify the test in the 'conditional' F-statistic column of the Wald F statistics table.

It is placed on a separate line immediately before or after the model line. Multiple !FOWN statements should appear together. It generates an F-test statistic for each model term in terms to test which tests its contribution after all after terms in terms to test and background terms, conditional on all terms that appear in the SPARSE equations, and on not changing the degrees of freedom associated with a term. It should only include terms which will appear in the ANOVA table.

#### For example,

``` !FOWN  A B C ; mu
!FOWN  A.B B.C A.C ; mu A B C
!FOWN  A.B.C  ; mu A B C A.B  B.C A.C
```
would request the tests
``` F( A ; mu B C  sparse),
F( B ; mu A C  sparse),
F( C ; mu A B  sparse),
F( A.B ; mu A B C B.C A.C  sparse),
F( B.C ; mu A B C A.B A.C  sparse),
F( A.C ; mu A B C A.B B.C  sparse) and
F( A.B.C ; mu A B C A.B A.C B.C  sparse).
```

#### Warnings:

This qualifier is provided for advanced users who have a good understanding of marginality issues in ASReml. ASReml does not verify the tests requested satisfy marginality considerations which are normally relevant. Any model terms in the !FOWN lists which do not appear in the actual model, are ignored without flagging an error. Any model terms which are omitted from !FOWN statements are tested with the usual conditional test. If any model terms are listed twice, only the first test is performed. Fcn tests specified in !FOWN statements are given model codes O, P, ....

The !FOWN statements are parsed by the same routine that parses the model line and so accepts the same model syntax options. Care should be taken to ensure term names are consistently spelt. If the !FOWN statements appear before the model line, model terms that are not previously defined may not be abbreviated (truncated) in !FOWN statements relative to their form in the model line because they are defined on their first appearance.
• Explanation of the ASReml ANOVA table