Lecture 35: Residuals – Solitary Confinement
Residuals
Prisoner[Solitary]
The split plot/repeated measures
design has two “Error” terms.
Solitary Factor: Prisoner[Solitary]
Day Factor: Error
Differences among
prisoners in the same
solitary confinement
condition contribute to this
“Error” term.
1
2
Prisoner[Solitary]
Fisher Conditions
“Residual” = Prisoner average
frequency – Solitary Mean.
Equal standard deviation
relative to the factor for that
part of the experiment.
Normal distribution.
Solitary/No; Mean = 15.833 Hz
Solitary/Yes; Mean = 11.766 Hz
3
Solitary “Residuals”
15
10
Average Freq
centered by Solitary
4
There is slightly more variation
for prisoners in solitary
confinement than those not in
solitary confinement.
The Fisher condition of equal
standard deviations is probably
met.
5
0
-5
-10
-15
No
Yes
Solitary
5
6
1
3
.99
2
.95
.90
1
.75
.50
Normal Quantile Plot
Lecture 35: Residuals – Solitary Confinement
Solitary “Residuals”
0
Although the histogram is
slightly skewed right, the box
plot is symmetric and the points
on the Normal plot follow the
Normal model line.
The Fisher condition of Normal
distribution is probably met.
.25
-1
.10
.05
-2
.01
-3
8
7
5
4
Count
6
3
2
1
-15
-10
-5
0
5
10
15
7
8
Day “Residuals”
4
3
The residuals that go into the
“Error” term for evaluating Day are
made up of what is left over after all
the model effects have been
accounted for.
Fit Model – Save Cols – Residuals
automatically calculates and saves
these residuals.
Residual
Frequency (Hz)
2
1
0
-1
-2
-3
-4
1
4
7
Day
10
3
.99
2
Day “Residuals”
.95
.90
1
.75
Normal Quantile Plot
9
0
.50
.25
-1
There is more variation on Day 1
than on Day 4 than on Day 7.
The difference in variation is not
great.
The Fisher condition of equal
standard deviations is probably
met.
.10
.05
-2
.01
-3
10
Count
15
5
11
-4
-3
-2
-1
0
1
2
3
4
12
2
Lecture 35: Residuals – Solitary Confinement
Day “Residuals”
Comment
The histogram is fairly
symmetric, the box plot is
symmetric and the points on the
Normal plot follow the Normal
model line.
The Fisher condition of Normal
distribution is probably met.
Solitary “Residuals” go
from –15 to +15.
Day “Residuals” go from
–4 to +3.
13
14
Comment
Comment
Solitary “Residuals” are more
variable because they have
variation from prisoner to
prisoner, treated the same.
The Day “Residuals” are less
variable because variation from
prisoner to prisoner treated the
same is accounted for separately.
Reusing prisoners (blocking)
in the second part of the
experiment has been effective
in reducing the size of the
mean square error used in
evaluating a factor of interest.
15
16
Question?
Answer?
How could we redesign the
first part of the experiment
to take advantage of the
reduction in mean square
error due to blocking?
Run the first part of the
experiment as a randomized
complete block design
instead of a completely
randomized design.
17
18
3
Lecture 35: Residuals – Solitary Confinement
Blocking
Random Assignment
Form 10 blocks of two prisoners
based on a baseline brainwave
frequency measurement.
Within each block assign
Solitary or No Solitary with a
flip of a coin.
Heads – Prisoner 1 in block gets
no solitary. Prisoner 2 gets
solitary.
Tails – Prisoner 2 in block gets no
solitary. Prisoner 1 gets solitary.
Two highest frequencies – Block 1
Next two highest – Block 2
Etc.
19
Analysis: Solitary
20
“Error” Term
The interaction between
Solitary and Block will serve
as the “Error” term for
evaluating the significance of
the Solitary factor.
Source
df
Solitary
1
Block
9
Solitary*Block&Random 9
21
Source
Solitary
Block
SS DF
MS F Ratio Prob > F
248.067
1 248.067 20.5580 0.0014*
1501.600
9 166.844 13.8269 0.0003*
Block*Solitary&Random
108.600
9 12.067
Day
256.900
2 128.45 45.3353 <.0001*
Day*Solitary
260.433
2 130.217 45.9588 <.0001*
Error
102.000 36
C. Total
4.2588 0.0008*
2.833
2477.600 59
23
22
Effect of Solitary
F = (248.067)/12.067 =
20.558, P-value = 0.0014.
The small P-value indicates
that the difference in Solitary
means is statistically
significant.
24
4
Lecture 35: Residuals – Solitary Confinement
Benefit of Blocking
Benefit of Blocking
By blocking on initial brain
wave frequency, the difference
between prisoners in and out
of solitary confinement is now
statistically significant.
By being able to account
for differences amongst
prisoners’ initial brain wave
frequencies, the “Error”
term is much smaller.
25
Benefit of Blocking
26
Comment
Each part of the Split
Plot/Repeated Measures
design can be run with any
type of design; e.g. completely
randomized, randomized
complete block, Latin square.
Completely Randomized
MS“Error” = 89.455.
Randomized Complete Block
MS“Error” = 12.067.
27
Experiments
28
Experiments
The rest of the semester
will look at different
experiments using different
combinations of factorial
crossing and designs.
Factorial Crossing
Design
CRD, RCBD, RLSq, Split
Plot/Repeated Measures.
29
30
5