The F-Ratio

The F-Ratio
Supose we have two linear models:
• The general model Ω with sum of squared residuals
SS Ω and degrees of freedom for error df Ω
• The benchmark model ω with sum of squared residuals
SS ω and degrees of freedom for error df ω
and the benchmark model ω is a special case of the general
model Ω obtainable by setting some of the Ω parameters to
constants, or by imposing some linear restrictions on their
parameters.
The F-ratio is
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F = ( SS ω − SS Ω
)/( SS Ω
)
df ω − df Ω df Ω
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Interpreting the F-ratio
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• The numerator of the F-ratio is the decrease in error
sum of squares per degree of freedom spent by moving
from the more specific and less powerful model ω to the
more general and powerful model Ω.
• The denominator of the F-ratio is the remaining error
sum of squares per parameter left to spend to drive it to
0.
• Each ratio can be considered as a “return on investment”,
degrees of freedom being invested, and decrease
in error sum of squares being returned.
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• We want to keep the move from the benchmark model
ω to the general model Ω if its “return on investment” is
large relative to what remains to be achieved.
• We prefer the benchmark model ω if the “return on investment”
ratio represented by F is near one.
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F-ratio and the F-distribution
The F-ratio has an F-distribution over repeated samples if,
for any combination of independent variable values, the distribution
of the residuals have
• mean 0 (i. e. the benchmark model is correct)
• standard deviation σ (i. e. is constant)
• a normal distribution
• and are independent of the residuals for any other set
of independent variable values
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• If the F-ratio turns out to be surprisingly large relative
to tabled critical values, we should conclude that something
is wrong with one or more of the above assumptions.
• It may not necessarily be the benchmark model that is
wrong.
• Also, in accepting the general model Ω, we only take it
as being better than the benchmark model ω, and not
as necessarily the right model.
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