The distribution used for the hypothesis test is a new one. It is called the
FF distribution,
named after Sir Ronald Fisher, an English statistician. The
FF statistic is a ratio (a
fraction). There are two sets of degrees of freedom; one for the numerator and one for
the denominator.
For example, if
F
F follows an
F
F distribution and the degrees of freedom for the
numerator are 4 and the degrees of freedom for the denominator are 10, then
FF ~
F
4
,
10
F
4
,
10
.
The
FF distribution is derived from the Student's-t distribution. One-Way ANOVA expands the tt-test for comparing more than two groups. The scope of that derivation is beyond the level of this course.
To calculate the
F
F
ratio, two estimates of the variance are made.
- Variance between samples: An estimate of
σ
2
σ
2
that is the variance of the sample
means multiplied by n (when there is equal n). If the samples are different sizes, the variance between samples is weighted to
account for the different sample sizes. The variance is also called variation due to treatment or
explained
variation.
- Variance within samples: An estimate of
σ
2
σ
2
that is the average of the sample
variances (also known as a pooled variance). When the sample sizes are different, the
variance within samples is weighted. The variance is also called the variation due to error or
unexplained variation.
-
SS
between
=
SS
between
= the sum of squares that represents the variation among the different
samples.
-
SS
within
=
SS
within
= the sum of squares that represents the variation within samples that is
due to chance.
To find a "sum of squares" means to add together squared quantities which, in some
cases, may be weighted. We used sum of squares to calculate the sample variance and
the sample standard deviation in Descriptive Statistics.
MSMS means "mean square."
MS
between
MS
between
is the variance between groups and
MS
within
MS
within
is the variance within groups.
- kk size 12{k} {} = the number of different groups
- njnj size 12{n rSub { size 8{j} } } {} = the size of the
jthjth size 12{ ital "jth"} {} group
- sjsj size 12{s rSub { size 8{j} } } {}= the sum of the values in the
jthjth size 12{ ital "jth"} {} group
- nn size 12{n} {} = total number of all the values combined. (total sample size: ∑nj∑nj size 12{ Sum {n rSub { size 8{j} } } } {})
- xx = one value:
∑x=∑sj∑x=∑sj size 12{ Sum {x} = Sum {s rSub { size 8{j} } } } {}
- Sum of squares of all values from every group combined: ∑x2∑x2 size 12{ Sum {x rSup { size 8{2} } } ={}} {}
- Between group variability:
SS
total
=
∑
x
2
−
∑
x
2
n
SS
total
=
∑
x
2
−
∑
x
2
n
size 12{ ital "SS" rSub { size 8{ ital "total"} } = Sum {x rSup { size 8{2} } } - { { left ( Sum {x} right ) rSup { size 8{2} } } over {n} } } {}
- Total sum of squares:
∑
x
2
−
(
∑
x
)
2
n
∑
x
2
−
(
∑
x
)
2
n
size 12{ Sum {x rSup { size 8{2} } } - { { \( Sum {x} \) rSup { size 8{2} } } over {n} } } {}
- Explained variation- sum of squares representing variation among the different samples
SS
between
=
∑
[
(
sj
)
2
n
j
]
−
(
∑
s
j
)
2
n
SS
between
=
∑
[
(
sj
)
2
n
j
]
−
(
∑
s
j
)
2
n
size 12{ ital "SS" rSub { size 8{ ital "between"} } = Sum { \[ { { \( ital "sj" \) rSup { size 8{2} } } over {n rSub { size 8{j} } } } \] } - { { \( Sum {s rSub { size 8{j} } \) rSup { size 8{2} } } } over {n} } } {}
- Unexplained variation- sum of squares representing variation within samples due to chance:
SS
within
=
SS
total
−
SS
between
SS
within
=
SS
total
−
SS
between
size 12{ ital "SS" rSub { size 8{ ital "within"} } = ital "SS" rSub { size 8{ ital "total"} } - ital "SS" rSub { size 8{ ital "between"} } } {}
- df's for different groups (df's for the numerator):
df
between
=
k
−
1
df
between
=
k
−
1
size 12{ ital "df" rSub { size 8{ ital "between"} } =k - 1} {}
-
Equation for errors within samples (df's for the denominator):
df
within
=
n
−
k
df
within
=
n
−
k
size 12{ ital "df" rSub { size 8{ ital "within"} } = n - k} {}
- Mean square (variance estimate) explained by the different groups:
MS
between
=
SS
between
df
between
MS
between
=
SS
between
df
between
size 12{ ital "MS" rSub { size 8{ ital "between"} } = { { ital "SS" rSub { size 8{ ital "between"} } } over { ital "df" rSub { size 8{ ital "between"} } } } } {}
-
Mean square (variance estimate) that is due to chance (unexplained):
MS
within
=
SS
within
df
within
MS
within
=
SS
within
df
within
size 12{ ital "MS" rSub { size 8{ ital "within"} } = { { ital "SS" rSub { size 8{ ital "within"} } } over { ital "df" rSub { size 8{ ital "within"} } } } } {}
MS
between
MS
between
and
MS
within
MS
within
can be written as follows:
-
MS
between
=
SS
between
df
between
=
SS
between
k
−
1
MS
between
=
SS
between
df
between
=
SS
between
k
−
1
-
MS
within
=
SS
within
df
within
=
SS
within
n
−
k
MS
within
=
SS
within
df
within
=
SS
within
n
−
k
The One-Way ANOVA test depends on the fact that
MS
between
MS
between
can be influenced by population
differences among means of the several groups. Since
MS
within
MS
within
compares values of
each group to its own group mean, the fact that group means might be different does
not affect
MS
within
MS
within
.
The null hypothesis says that all groups are samples from populations having the same
normal distribution. The alternate hypothesis says that at least two of the sample
groups come from populations with different normal distributions. If the null hypothesis
is true,
MS
between
MS
between
and
MS
within
MS
within
should both estimate the same value.
The null hypothesis says that all the group population means are equal. The hypothesis of equal means implies that the populations have the same normal distribution because it is assumed that the populations are normal and that they have equal variances.
F
=
MS
between
MS
within
F=
MS
between
MS
within
(1)If
MS
between
MS
between
and
MS
within
MS
within
estimate the same value (following the belief that
H
o
H
o
is
true), then the F-ratio should be approximately equal to 1. Mostly just sampling errors
would contribute to variations away from 1. As it turns out,
MS
between
MS
between
consists of
the population variance plus a variance produced from the differences between the
samples.
MS
within
MS
within
is an estimate of the population variance. Since variances are
always positive, if the null hypothesis is false,
MS
between
MS
between
will generally be larger than
MS
within
MS
within
.
Then the F-ratio will be larger than 1.
However, if the population effect size is small it is not unlikely that
MS
within
MS
within
will be larger in a give sample.
The above calculations were done with groups of different sizes. If the groups are the same size, the calculations simplify somewhat and the F ratio can be written as:
F
=
n
⋅
s
x_
2
s
2
pooled
F
=
n
⋅
s
x_
2
s
2
pooled
(2)- n=n= size 12{n={}} {}the sample size
-
df
numerator
=
k
−
1
df
numerator
=
k
−
1
size 12{ ital "df" rSub { size 8{ ital "numerator"} } =k - 1} {}
-
df
denominator
=
n
−
k
df
denominator
=
n
−
k
size 12{ ital "df" rSub { size 8{ ital "denominator"={}} } k \( n - 1 \) =N - k} {}
-
s
2
pooled
=
s
2
pooled
=
the mean of the sample variances (pooled variance)
-
sx¯
2
=
s
x
2
=
the variance of the sample means
The data is typically put into a table for easy viewing. One-Way ANOVA results are often displayed in this manner by computer software.
Table 1
| Source of Variation |
Sum of Squares (SS) |
Degrees of Freedom (df) |
Mean Square (MS) |
F |
Factor
(Between) |
SS(Factor) |
k - 1 |
MS(Factor) = SS(Factor)/(k-1) |
F = MS(Factor)/MS(Error) |
Error
(Within) |
SS(Error) |
n - k |
MS(Error) = SS(Error)/(n-k) |
|
| Total |
SS(Total) |
n - 1 |
|
|
Three different diet plans are to be tested for mean weight loss. The entries in the table are the weight losses for the different plans. The One-Way ANOVA table is shown below.
Table 2
| Plan 1 |
Plan 2 |
Plan 3 |
| 5 |
3.5 |
8 |
| 4.5 |
7 |
4 |
| 4 |
|
3.5 |
| 3 |
4.5 |
|
One-Way ANOVA Table: The formulas for SS(Total), SS(Factor) = SS(Between) and SS(Error) = SS(Within) are shown above. This same information is provided by the TI calculator hypothesis test function ANOVA in STAT TESTS (syntax is ANOVA(L1, L2, L3) where L1, L2, L3 have the data from Plan 1, Plan 2, Plan 3 respectively).
Table 3
| Source of Variation |
Sum of Squares (SS) |
Degrees of Freedom (df) |
Mean Square (MS) |
F |
Factor
(Between) |
SS(Factor)
= SS(Between)
=2.2458 |
k - 1
= 3 groups - 1
= 2 |
MS(Factor)
= SS(Factor)/(k-1)
= 2.2458/2
= 1.1229 |
F =
MS(Factor)/MS(Error)
= 1.1229/2.9792
= 0.3769 |
Error
(Within) |
SS(Error)
= SS(Within)
= 20.8542 |
n - k
= 10 total data - 3 groups
= 7 |
MS(Error)
= SS(Error)/(n-k)
= 20.8542/7
= 2.9792 |
|
| Total |
SS(Total)
= 2.9792 + 20.8542
=23.1 |
n - 1
= 10 total data - 1
= 9 |
|
|
The One-Way ANOVA hypothesis test is always right-tailed because larger F-values are
way out in the right tail of the F-distribution curve and tend to make us reject
H
o
H
o
.
The notation for the F distribution is
FF ~
F
df(num)
,
df(denom)
F
df(num)
,
df(denom)
where
df(num)
=
df
between
df(num)=
df
between
and
df(denom)
=
df
within
df(denom) =
df
within
The mean for the F distribution is
μ
=
df(num)
df(denom)
−
1
μ=
df(num)
df(denom)
−
1
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