Graphical Analysis of Variance (Graphical ANOVA) This set of slides ...

Graphical Analysis of Variance (Graphical ANOVA)
This set of slides introduces a graphical method
for analyzing multiple samples which makes the
main idea of ANOVA ... accessible .... ... when we ask if a set of
sample means gives evidence for differences among the
population means, what matters is not how far apart the sample
means are but how far apart they are relative to the variability of
individual observations.*
*David Moore, The Basic Practice of Statistics. 2010, p. 642.
1

The Typical One Factor Layout
The data consist of m samples: sets of n
independent measurements taken under various
conditions or treatments:
Treatment 1 Treatment 2 · · · Treatment m
X 11 X 21 · · · X m1
X 12 X 22 · · · X m2
· · · · · · · · · · · ·
X 1n X 2n · · · X mn
2

A Paper Airplane Experiment
Does the distance travelled by a paper
airplane after being thrown depend on the
weight of the paper used in its construction?
3

A Paper Airplane Experiment
12 paper airplanes of a single design were
constructed using
• 20 × 27 cm sheets of paper
• 3 different weights of paper (light, medium,
heavy)
m = 3 treatment groups with n = 4
• Response: distance travelled (in m)
• Factor of interest: weight of paper
4

The Paper Airplane Experiment Data
> airplane

Paper Airplane Data
> head(airplane, n=6)
paper distance
1 light 3.1
2 light 3.3
3 light 2.1
4 light 1.9
5 medium 4.0
6 medium 3.5
6

Paper Airplane Data
> tail(airplane, n=2)
paper distance
11 heavy 4.7
12 heavy 5.3
7

Displaying the Data Differently
> unstack(airplane, distance ~ paper)
heavy light medium
1 5.1 3.1 4.0
2 3.1 3.3 3.5
3 4.7 2.1 4.5
4 5.3 1.9 6.1
8

Compare Averages for the 3 Groups
First, split the data frame up by paper type:
> airplane.split

Split Data Frame
> airplane.split
$heavy
[1] 5.1 3.1 4.7 5.3
$light
[1] 3.1 3.3 2.1 1.9
$medium
[1] 4.0 3.5 4.5 6.1
10

Compare Averages for the 3 Groups
Now, compute the averages:
> means means
heavy
light medium
4.550 2.600 4.525
11

Discussion
Flight distances are different for different
treatments, but they are also different within
treatments
What has caused the variation within each
treatment?
Factors other than paper weight must be having
an effect: unexplained variation or error.
12

Discussion
Some of the possibilities are: individual airplane
construction, initial throwing height, and initial
thrust and direction
These unmeasured factors probably vary slightly
from throw to throw and thus could account for
some or all of the variation observed within each
treatment
Is there confounding? (What is confounding?)
13

Discussion
The presence of unmeasured factors makes it
difficult to tell if differences in paper type have an
effect on flight distance
Are the differences in the treatment averages due
only to the unmeasured factors?
14

Identifying Within Group Variation
Subtract the within-group averages from each of
the distance values to remove any possible
effects due to different types of paper.
15

Identifying Within Group Variation
First step: Create a vector of distances.
> unlist(airplane.split)
heavy1 heavy2 heavy3 heavy4 light1
5.1 3.1 4.7 5.3 3.1
light2 light3 light4 medium1 medium2
3.3 2.1 1.9 4.0 3.5
medium3 medium4
4.5 6.1
16

Identifying Within Group Variation
Second step: Create vector containing
corresponding averages.
Do this using the rep() function. We need the
sample sizes first:
> n n # sample sizes
heavy
light medium
4 4 4
17

Identifying Within Group Variation
The vector of averages:
> rep(means, n)
heavy heavy heavy heavy light
4.550 4.550 4.550 4.550 2.600
light light light medium medium
2.600 2.600 2.600 4.525 4.525
medium medium
4.525 4.525
18

Identifying Within Group Variation
Subtract the averages from the original
measurements.
> errors

Identifying Within Group Variation
Variation due to factors other than paper type:
> errors
heavy1 heavy2 heavy3 heavy4 light1
0.550 -1.450 0.150 0.750 0.500
light2 light3 light4 medium1 medium2
0.700 -0.500 -0.700 -0.525 -1.025
medium3 medium4
-0.025 1.575
20

Does paper type have an effect on mean distance?
Averages of the samples vary, but maybe only
because of the unmeasured factors.
If this is true (and paper type does not affect mean
distance), then
Var( ¯X i ) = σ2
n
where ¯X i is the average distance in the ith
sample, n is the sample size and
σ 2 is the variance of the error.
21

Does paper type have an effect on mean distance?
Observe that, for the ith sample,
Var( √ n ¯X i ) = σ 2
if variability is due only to unmeasured factors.
the variability of (scaled) sample averages ∼
the variability in the errors, when paper type has
no effect.
22

Does paper type have an effect on mean distance?
Let ¯X = the grand mean of the flight distances.
If paper type has no effect on distance, then
√ n( ¯X i − ¯X)
and the errors will both have mean 0 and the same
variance.
(Why do we need to subtract the grand mean?)
23

Graphically testing whether paper type has an
effect on mean distance
A histogram of the errors shows the variability in
the errors.
Since the scaled averages are supposed to have
the same amount of variability when paper type
has no effect, we can plot the locations of the
scaled averages on the error histogram and look
for any surprises.
24

Graphically testing whether paper type has an
effect on mean distance
A surprising result represents evidence that paper
type has an effect.
In other words, look for outliers relative to the
histogram.
25

The Graphical Test
> hist(errors, xlim=c(-3,3))
> scaled.means points(scaled.means, rep(0,3),
+ pch=16, col=2, cex=2.5)
26

The Graphical Test
Histogram of errors
Frequency
0.0 0.5 1.0 1.5 2.0 2.5 3.0
●
●
−3 −2 −1 0 1 2 3
errors
27

The Graphical Test Result
One of the scaled means is an outlier.
We have strong evidence that paper type affects
flight distance.
28

The Graphical ANOVA Procedure
1. Compute averages ¯X 1 , ¯X 2 , . . . , ¯X m and the grand average, ¯X.
2. Remove treatment effects from each sample by computing errors:
e ij = X ij − ¯X i ,
for j = 1, . . . , n, and i = 1, . . . , m.
3. Center and scale the ith treatment average by subtracting the grand
average and multiplying by the sample size:
√ n( ¯X i − ¯X).
4. Compare the m scaled averages with a histogram of the errors.
29

Graphical ANOVA Function
> graphicalANOVA

Application of the function to the Airplane Data
> graphicalANOVA(airplane)
Frequency
0.0 0.5 1.0 1.5 2.0 2.5 3.0
●
average heavy
average light
average medium
●
−3 −2 −1 0 1 2
errors
31

The Graphical ANOVA Plot
The light paper average is an outlier relative to the
histogram
There is strong evidence that the treatment
means are not all the same
Otherwise, the treatment averages should be
located in regions corresponding to higher
histogram density
32

Motor Vibration Example
5 different brands of bearings are compared in
terms of the amount of vibration they generate in
an electric motor*
*The motor vibration data can be found in the Devore5 library (Bates, 2004)
33

Motor Vibration Data
> motor head(motor)
V1 V2 V3 V4 V5
1 13.1 16.3 13.7 15.7 13.5
2 15.0 15.7 13.9 13.7 13.4
3 14.0 17.2 12.4 14.4 13.2
4 14.4 14.9 13.8 16.0 12.7
5 14.0 14.4 14.9 13.9 13.4
6 11.6 17.2 13.3 14.7 12.3
34

Motor Vibration Data
> names(motor) motors head(motors)
values ind
1 13.1 Brand 1
2 15.0 Brand 1
3 14.0 Brand 1
4 14.4 Brand 1
5 14.0 Brand 1
6 11.6 Brand 1
35

Motor Vibration Data
> tail(motors, n=2)
values ind
29 13.4 Brand 5
30 12.3 Brand 5
36

Motor Vibration Data
> names(motors) head(motors, n=3)
vibration bearing brand
1 13.1 Brand 1
2 15.0 Brand 1
3 14.0 Brand 1
37

Motor Vibration Data
> graphicalANOVA(motors[,c(2,1)])
38

Motor Vibration Data
Frequency
0 2 4 6 8 10
●
●
average Brand 1
average Brand 2
average Brand 3
average Brand 4
average Brand 5
●
●
−2 0 2 4
errors
39

Motor Vibration
The second brand average appears in the extreme
right tail.
There is clear evidence that the amount of
vibration depends on the type of bearing.
40

An Example Using Simulated Data
What does this ANOVA plot look like if there really
is no difference between the samples?
To check, we simulate 40 independent
observations from a normal distribution with
mean 0 and variance 1.
We pretend that the first 10 observations are from
1 sample, the second set of 10 are from another
sample, and so on.
41

Simulated Data
> set.seed(12030); pretenddata samplenumber pretend.df rm(pretenddata, samplenumber)
> head(pretend.df, n=3)
samplenumber pretenddata
1 1 0.3752435
2 1 -0.6727016
3 1 0.8656027
42

Simulated Data
> graphicalANOVA(pretend.df)
43

Simulated Data
Frequency
0 5 10 15
●
average 1
average 2
average 3
average 4
●
−3 −2 −1 0 1 2 3 4
errors
44