ANOVA - Analysis of Variance - University of Washington

22 Pages·2013·0.2 MB·English

ANOVA - Analysis of Variance
• Extends independent-samples t test
• Compares the means of groups of independent
observations
– Don’t be fooled by the name. ANOVA does not compare
variances.
• Can compare more than two groups
ANOVA – Null and Alternative Hypotheses
Say the sample contains K independent groups
• ANOVA tests the null hypothesis
H : μ = μ = … = μ
0 1 2 K
– That is, “the group means are all equal”
• The alternative hypothesis is
H : μ ≠ μ
for some i, j
1 i j
– or, “the group means are not all equal”
Example:
Accuracy of Implant
Placement
• Implants were placed in a
manikin using metal surgical
guides of varying
occlusogingival heights
• 3 different guide heighths
were evaluated (4mm, 6mm,
and 8mm)
• Distances between a
reference implant and each
placed implant were
measured using a coordinate
measuring machine.
Example:
Accuracy of Implant
Mean Error by Guide Width
Placement
7
2
0.
• 15 implants were placed
m)
m 6
using each guide.
2
(
r 0.
o
r
• The placement error Er

ht
(discrepancy with the g 5
ei 2
H 0.
reference implant) was nt
a
pl
recorded for each implant. m
I 4
n 2
• Mean error for each guide ea 0.
M
are presented in the figure
3
at right. 2
0.
4mm 6mm 8mm
Guide Width
Example:
Accuracy of Implant
Mean Error by Guide Width
Placement 7
2
0.
• The mean of the m)
m 6
combined sample was ( 2
r 0.
o
r
0.248 mm. Er

ht
g 5
• This is called the “grand” ei 2
H 0.

mean, and is often nt
a
pl
m
denoted by (cid:1)(cid:2). I 4
n 2
a 0.
e
• If H were true then we’d M
0
expect the group means
3
2
to be close to the grand 0.
4mm 6mm 8mm
mean.
Guide Width
Example:
Accuracy of Implant
Mean Error by Guide Width
Placement
7
2
0.
• The ANOVA test is based m)
m 6
on the distances of the ( 2
r 0.
o
r
group means from (cid:1)(cid:2) . Er

ht
g 5
• If the combined distances ei 2
H 0.

nt
are large, that indicates
a
pl
m
we should reject H .
I 4
0 n 2
a 0.
e
M
3
2
0.
4mm 6mm 8mm
Guide Width
The Anova Statistic
To combine the differences from the grand mean into one summary statistic we
– Square the differences
– Multiply by the numbers of observations in the groups
– Sum over the groups
( ) ( ) ( )
2 2 2
= - + - + -
SSB 15 X X 15 X X 15 X X
4mm 6mm 8mm
where the (cid:1)(cid:3) are the group means.
∗
S S B
“SSB” = um of quares etween groups
Note: This looks a bit like a variance.
How big is big?
• For the Implant Accuracy Data, SSB = 0.0047
• Is that big enough to reject H ?
0
• As with the t test, we compare the statistic to the variability of the
individual observations.
• In ANOVA the variability is estimated by the Mean Square Error, or
MSE
MSE
Mean Square Error Implant Height Error by Guide Width
5
The Mean Square Error is a
0.
measure of the variability after m)
m
the group effects have been taken 4
r ( 0.
o
into account.
r
Er

ht 3
= 1 ∑ ∑ ( - )2 eig 0.
MSE x X H
N - K ij j nt
j i a
2
mpl 0.
where x is the ith observation in I
ij
the jth group. 1
0.
MSE
= 0.011
4mm 6mm 8mm
Note that the variation of the
Guide Width
means seems quite small
compared to the variance of
observations within groups
Notes on MSE
• If there are only two groups, the MSE is equal to the pooled
estimate of variance used in the equal-variance t test.
• ANOVA assumes that all the group variances are equal.
• Other options should be considered if group variances differ by a
factor of 2 or more.