Lecture 21: Contrasts
Contrast
Definition
Contrast.
A contrast is a linear
combination of means.
∑
where ∑
0
Estimated Contrast.
Treatment population means.
Factor level population
means.
∑
where ∑
0
1
Recall Experiment
Isolation Time
20 minutes
40 minutes
60 minutes
Mean
21.5
28.0
22.5
Ci
Ci
2
Comparisons
The three sets of Ci values
will give you the three pairwise comparisons of the
Isolation Time means.
Ci
–1 –1 0
+1 0 –1
0 +1 +1
3
Comparisons
4
Sum of Squares
Compare 20 minutes to 40 minutes.
1 21.5
1 28.0
∑
∑
6.5
, 1 df
Where n is the number of
observations in each mean.
estimates the difference in means
.
5
6
1
Lecture 21: Contrasts
Test of Hypothesis
:
0
:
0
Comparisons
 Compare 20 minutes to 40 minutes.
1 21.5
∑
∑
8 6.5
1
169.0
1 28.0
6.5
8 42.25
2
1
7
Test of Hypothesis
:
:
169.0
13.0
8
Another Example
c20= –1, c40= 0, c60= +1
This will compare the mean
for 60 minutes to the mean
)
for 20 minutes (
0
0
13.0
P-value = 0.0020
The small P-value (< 0.05) indicates that the
difference in means is statistically significant.
9
Comparisons
Test of Hypothesis
:
:
 Compare 20 minutes to 60 minutes.
1 21.5
∑
∑
1 22.5
8 1
4.0
1
1
10
0
0
4.0
0.31
13.0
P-value = 0.5859
The P-value is not small (> 0.05) which
indicates that the difference in means is
not statistically significant.
1.0
8 1
2
11
12
2
Lecture 21: Contrasts
Isolation Time Levels
+1
Isolation Time has levels that
are very special.
The levels are equally spaced
values on a numerical scale.
20, 40, and 60.
ci
0
–1
20
40
60
Isolation Time
13
Linear Contrast
14
Test of Hypothesis
: Thereisnolinearrelationshipbetween
RecallandIsolationTime.
: Thereisalinearrelationshipbetween
RecallandIsolationTime.
The values of the ci on a straight line
when plotted against the Isolation
Time levels.
This contrast can also be used to
evaluate the linear relationship
between the mean Recall and
Isolation Time.
4.0
13.0
0.31
P-value = 0.5859
15
Conclusion
16
Comment
The P-value is not small
(> 0.05) which indicates that
there is not a statistically
significant linear relationship
between Recall and Isolation
Time.
We can only make this sort of
conclusion about a
relationship for factors (or
treatments) that have equally
spaced levels on a numerical
scale.
17
18
3
Lecture 21: Contrasts
Quadratic Contrast
Isolation Time
Mean
20 minutes
40 minutes
60 minutes
Ci
21.5
28.0
22.5
Ci
Ci
+1 –1 +½
–2 +2 –1
+1 –1 +½
19
20
Quadratic Contrast
+2
+1
ci
1 21.5
2 28.0
1 22.5
0
12.0
–1
–2
∑
∑
20
40
60
Isolation Time
8
192.0
1
12.0
2
1
8 144.0
6
21
Test of Hypothesis
Conclusion
: Thereisnoquadraticrelationship
betweenRecallandIsolationTime.
: Thereisaquadraticrelationship
betweenRecallandIsolationTime.
192.0
13.0
22
The P-value is small (< 0.05)
which indicates that there is a
statistically significant
quadratic relationship between
Recall and Isolation Time.
14.77
P-value = 0.0012
23
24
4
Lecture 21: Contrasts
JMP
Effect Details
Response Recall
Effect Details
Reinforce
Least Squares Means Table
Analyze – Fit Model
Y: Recall
Construct Model Effects
Level
N
V
Least
Sq Mean
22.000000
26.000000
Std Error
1.0408330
1.0408330
Mean
22.0000
26.0000
IsoTime
Least Squares Means Table
LSMeans Table
LSMeans Plot
LSMeans Contrast
LSMeans Student’s t
LSMeans Tukey HSD
Reinforce, Iso Time,
Reinforce*Iso Time
Level
20
40
60
Least
Sq Mean
21.500000
28.000000
22.500000
Std Error
1.2747549
1.2747549
1.2747549
Mean
21.5000
28.0000
22.5000
Reinforce*IsoTime
Least Squares Means Table
Level
N,20
N,40
N,60
V,20
V,40
V,60
Least
Sq Mean
22.000000
30.500000
13.500000
21.000000
25.500000
31.500000
Std Error
1.8027756
1.8027756
1.8027756
1.8027756
1.8027756
1.8027756
25
Response Recall
26
Interpretations
Effect Details
IsoTime
Least Squares Means Table
Least
Sq Mean
21.500000
28.000000
22.500000
Level
20
40
60
Std Error
1.2747549
1.2747549
1.2747549
Mean
21.5000
28.0000
22.5000
The difference in Recall means
between 20 minutes and 60 minutes
of Isolation Time is not statistically
significant.
There is no statistically significant
linear relationship between Isolation
Time and Recall.
Contrast
Contrast Specification
IsoTime
20
40
60
-1
0
1
Click on + or - to make contrast values.
SS NumDF DenDF
4
1
18
F Ratio Prob > F
0.3077
0.5859
27
Response Recall
28
Interpretations
Effect Details
IsoTime
Least Squares Means Table
Level
20
40
60
Least
Sq Mean
21.500000
28.000000
22.500000
Std Error
1.2747549
1.2747549
1.2747549
Mean
21.5000
28.0000
22.5000
The difference in Recall means
between 40 minutes and the
combination of 20 and 60 minutes of
Isolation Time is statistically
significant.
There is a statistically significant
quadratic relationship between
Isolation Time and Recall.
Contrast
Contrast Specification
IsoTime
20
-0.5
40
1
60
-0.5
Click on + or - to make contrast values.
SS NumDF DenDF
192
1
18
F Ratio Prob > F
14.7692 0.0012*
29
30
5