The F distribution will be used to evaluate the significance of the association between an independent variable and an outcome variable in an ANOVA.

*F ***DISTRIBUTION AND APPLICATIONS**

The *F*
distribution will be used to evaluate the significance of the association
between an independent variable and an outcome variable in an ANOVA. The *F* distribution is defined as the
distribution of (*Z*/*n*_{1})/(*W*/*n*_{2}), where *Z* has a chi-square distribution with *n*_{1} degrees of freedom, *W* has a chi-square distribution with *n*_{2} degrees of freedom, and *Z* and *W* are statistically independent. In the one-way analysis of
variance, *Z* = *Q*_{2}/*σ ^{2}*,

The probability density function for this *F* distribution has been derived and is
described in statistical texts [see page 246 in Mood, Graybill, and Boes
(1974)].

The *F*
distribution depends on the two degrees of freedom parameters *n*_{1} and *n*_{2}, called, respectively, the numerator and denominator
degrees of freedom. We in-clude tables of the central *F* distribution based on degree of freedom parameters in Appendix A.
A sample ANOVA is presented in Table 13.1.

Although we do not cover the two-way analysis of
variance, Table 13.2 shows the typical two-way ANOVA table that should help you
see how the ANOVA table generalizes to *N*-way
ANOVAs. Note that as more factors appear, we have more than one *F* test. This appearance of multiple *F* tests is analogous to the several *F* and/or *t* tests in regression that are used to determine the significance
of the regres-sion coefficients. ANOVA Table 13.2 is not the most general
table. A treatment by block effect also can be considered in the model; in this
case, the table would have another row for the interaction term.

We will illustrate the one-way analysis of variance
with a numerical example. Table 13.3 shows some hypothetical data for the
weight gain of pigs fed with three different brands of cereal. A total of 12
pigs are randomly assigned (4 each) to the three cereal brands.

To generate the ANOVA table, we must calculate *SS _{b}* and

Now let us compute *SS _{b}*. We can calculate this directly or calculate

We need to get
the overall or “grand” mean—the weighted average of the group means weighted by
their respective sample sizes. In this case, since all three groups have 4 pigs
each, the result is the same as taking the arithmetic average of the three
group averages. So * _{g}*
= (1.5 + 8 + 15)/3 = 8.1667. For

So by subtraction, *SS _{b}* = 391.0845–23.0 = 368.0845. Now we can fill in the
ANOVA table. Table 13.4 is the ANOVA table of the form of Table 13.1 as applied
to these data.

An *F*
statistic of 72.00 is highly significant. Compare it to values in the *F* distri-bution table with 2 degrees of
freedom in the numerator and 9 degrees of freedom in the denominator (Appendix
A). The critical values are 4.26 at the 5% level and 8.02 at the 1% level. So
we see that the *p*-value is
considerably less than 0.01.

*SS _{b} *can be calculated directly. The
formula is

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