# Wald F statistics

## Testing of Fixed Effects with Wald F statistics

The table for testing fixed effects has 4 forms:

Source DF F_{i}c
Source DF F_{i}c F_{c}n M
Source DF DDF_{i}c F_{i}c P_{i}c
Source DF DDF_{c}n F_{i}c F_{c}n M P_{c}n

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,

F_{i}c
is the F statistic for the incremental sum of squares,

F_{c}n
is the F statistic for the conditional sum of squares,

M
is an alpha code indicating conditioning level in F_{c}n
,

P_{i}c, P_{c}n
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 ( F_{i}c) 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.
F_{c}n 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
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