### F-test

```The F-test
why? later in gory detail
now? brief explanation of logic of F-test
for now -- intuitive level: F is big (i.e., reject Ho: μ1 = μ2 = ... = μa) when
MSbetw is large relative to MSw/in
e.g., F would be big here, where group diffs are clear:
F would be smaller here where diffs are less clear:
F-test Intuitively:
Variance Between vs. Variance Within
x1
x2
price: high mdm
x3
1
F= Var Betw Grps
------------------Var W/in Grps
2
3
4
low
sales,
F= Var Betw Grps
------------------Var W/in Grps
F-test Intuitively:
Variance Between vs. Variance Within
sales,
\$Price: low medium high
A)
low medium high
B)
low medium high
C)
low medium high
D)
ANOVA: Model
Group:
x1
III
x2
II
I
x3
Grand Mean
Model: Yij    i   ij
Yij
Another take on intuition follows, more mathy, less visual
Brief Explanation of Logic of F-test
For the simple design we've been working with (l factor, complete
randomization; subjects randomly assigned to l of a group--no blocking
or repeated measures factors, etc.),
model is: Yij = μ + αi + εij
where μ & αi (of greatest interest) are structural components, and the εij's
are random components.
assumptions on ε ij's (& in effect on Yij's):
l) εij's mutually indep (i.e., randomly assign subjects to groups & one
subject's score doesn't affect another's)
2) εij's normally distributed with mean=0 (i.e., errors cancel each other) in
each population.
3) homogeneity of variances: σ21=σ22=...=σ2ε <--error variance
In particular -- test statistic F
Later - general rules to generate F tests in diff designs
Logic of F-test
• Yij's -- population of scores - vary around group mean because of εij's:
• draw sample size n, compute stats like μ's & MSA's repeatedly draw
such samples, compile distribution of stats (Keppel pp.94-96):
• means of the corresponding theoretical distributions are the "expected
values“
• E(MSS/A) = σ2ε
UE of error variance
• E(MSA) = σ2 ε + [nΣ(αi)2]/(a-1) not UE of error variance, but
also in combo w. treatment effects
• F = MSA/MSS/A
compare their E'd values:
2
n

(
)

i
2

+
E( MS A )
a -1
=
E( MS S/A )
 2
Logic of F-test, cont’d
• if Ho : μ1=μ2=...=μa were true, then nΣ(αi)2/(a-1)=0
• Ho above states no group diffs. This is equivalent to
Ho: α1=α2=...=αa=0, again stating no group diffs.
• all groups would have mean μ+αi=μ+0=μ; no treatment effects.
• if (Ho were true and therefore) all αi's=0, then (αi)2=0,
so nΣ(αi)2/(a-1) would equal nΣ0/(a-1)=0.
• SO! under H0 then, F would be:
2
+0
ˆ error
2
ˆ error
• that is, F would be a ratio of 2 independent estimates of error variance,
so F should be "near" 1.
Logic of F-test, cont’d
• When F is large, reject Ho as not plausible, because:
• when Ho is not true, each (αi)2 will be > or = 0,
n(  i )2
 will be >0
a -1
• and F will be :
2
+ som ething
ˆ error
much >1
2
ˆ error
(for more on the intuition underlying the F-test, see Keppel pp.26-28; and for
more on expected mean squares, see Keppel p.95.)
```