Example: Treatment Comparisons or Contrasts
In a laboratory experiment, the weight loss in milligrams of a certain machine part due to friction when specimens were used in a wear-tester with four different lubricants SP51, L43, Z90,
and Blu-Tek, were measured. Lower the weight loss, the better the performance of the lubricant. The lubricants were assigned to the test runs completely at random. The model
yij = µi + ij , i = 1, . . . , 4; j = 1, . . . , 8 was used to analyze the resulting data:
i Lubricant
Weight Loss (in milligrams)
ȳi.
1 SP51
12
11
7
10
9
10
12
7
9.75
2 L43
8
10
5
7
10
9
7
8
8.0
3 Z907
10
5
7
8
9
6
7
5
7.125
4 Blu-Tek
8
5
8
7
5
6
6
7
6.5
1. Construct an analysis of variance using numbers from the JMP output. Test H0 : µ1 = µ2 =
µ3 = µ4 vs. Ha : at least one µi is different from the others, using α = .05. Use the p-value
from the JMP output to make a decision.
2. Use the lsd procedure to test all possible differences H0 : µi − µj = 0 vs. Ha : µi − µj 6= 0
using α = .05. Report the results using the underlining display. Summarize your conclusions
from this procedure in sentence or two.
3. Use the Tukey procedure to test all possible differences H0 : µi − µj = 0 vs. Ha : µi − µj 6= 0
using α = .05. Report the results using the underlining display. Point out any conclusions
that are different from those made using the lsd procedure.
4. It would be more useful to test pre-planned comparisons than testing pairwise differences
among the 4 lubricant effects. In this experiment it is known that the 4 factors are
Lubricant
1
2
3
4
Formulation
Petroleum Distillate Base with Standard Silicon Additive
Petroleum Distillate Base with a Hi-tech Detergent Additive
Synthetic Oil Base with Standard Silicon Additive
Synthetic Oil Base with a Hi-tech Detergent Additive
Thus the experimenter could have planned to do the following comparisons.
(a) compare the average effect of the petroleum distillate based lubricants to the average
effect of the synthetic oil based lubricants,
(b) compare the average effect of the standard silicon additive to the average effect of the
hi-tech detergent additive,
(c) compare the effects of the two petroleum distillate based lubricants, one with the silicon
additive to that with the hi-tech detergent additive, and
(d) compare the effects of the two synthetic oil based lubricants, one with the silicon additive
to that with the hi-tech detergent additive.
Compute appropriate t-statistics to test the comparisons given above both by hand and
using JMP controlling the error rate for each comparison at α = .05. Write a statement
in conclusion in each case.
1
Contrasts: Lubricant Example
(a)
Source
d.f .
SS
MS
F
Lubricant
3
47.84375
15.9479
5.5557
Error
28
80.37500
2.8705
C. Total
31
128.21875
Test
H 0 : µ=
µ=
µ=
µ4
1
2
3
H a : at least
at α = .05 .
vs.
Reject the null hypothesis
(b)
The sample means are :
y1 = 9.75 , y2 = 8.0 , y3 = 7.124 , y4 = 6.5
LSD=
2.048 × 2.8705 × 2 / 8 = 1.735
Blu-Tek
Z907
L43
SP51
6.5
7.125
8.0
------------------
9.75
From JMP:
LSMeans Differences Student's t
α=0.050
Level
1
2
3
4
t=2.04841
A
B
B
B
Least Sq Mean
9.7500000
8.0000000
7.1250000
6.5000000
Levels not connected by same letter are significantly different.
(c)
W=
3.87 × 2.8705 × 1/ 8 = 2.32
Blu-Tek
Z907
L43
SP51
6.5
7.125
8.0 9.75
---------------------------
p-value
<.0001
one inequality. Since p-value<.05,
From JMP:
LSMeans Differences Tukey HSD
α= 0.050 Q= 2.73031 <<<<<<<Note :This Q value is different from the value from the table. i.e. 3.87/ 2 =2.736
Level
1
2
3
4
A
A
B
B
B
Least Sq Mean
9.7500000
8.0000000
7.1250000
6.5000000
Levels not connected by same letter are significantly different.
(d)
Hypothesis
H 0 :1/ 2( µ1 + µ2 ) − 1/ 2( µ3 + µ4 ) =
0
Γ1
Contrast : =
1 1 −1 −1
Estimate:
C1 = 9.75 + 8.0 − 7.125 − 6.5 = 4.125
12 + 12 + 12 + 12
4
=
2.8705 = (1.694255)
  1.198
8
8
=
V (ˆ 1 )
Standard Error
=
Thus the t-statistic is:
=
t0
4.125
= 3.4432 and the percentile from the t-table is t.025,28 = 2.048
1.198
Thus R.R. is t > 2.048 ; Thus the null hypothesis is rejected at α = .05 since
3.4434 is in the rejection region.
From JMP output:
t-statistic = 3.4432
p-value = .0018
Reject null hypothesis at α = .05
Hand computations for the other three comparisons are not shown here but those
computations are similar to above and are summarized below.
Contrast
SP51
L43
Z907
Blu-Tek
Est
s.e.
p-value
1.198
𝑡𝑡0
𝐶𝐶1
1
1
-1
-1
4.125
3.4432
.0018
𝐶𝐶2
1
-1
1
-1
2.375
1.198
1.98
.0573
𝐶𝐶3
1
-1
0
0
1.75
0.8471
2.0658
.0482
𝐶𝐶4
0
0
1
-1
0.625
0.8471
0.7378
.4668
The results from JMP are shown below.
Contrast
Test Detail
1
2
3
4
Estimate
Std
Error
t Ratio
Prob>|t|
SS
0.5
0.5
1
0
0.5
-0.5
-1
0
-0.5
0.5
0
1
-0.5
-0.5
0
-1
2.0625 1.1875
1.75 0.625
0.599 0.599 0.8471 0.8471
3.4432 1.9824 2.0658 0.7378
0.0018 0.0573 0.0482 0.4668
34.031 11.281 12.25 1.5625
Summary Statement
The data show that there is a significant difference between the average effects of petroleum distillate
based lubricants from those of synthetic oil based lubricants on the wear of the machine part. The
average wear appear to be lower for the synthetic oil based lubricants. There is no significant difference
between the average effects of silicon additives vs. the hi-tech detergent additives on machine part
wear. There is a significant difference between the average effects of silicon additive vs. the hi-tech
detergent additive when used with petroleum distillate base as opposed to synthetic oil base, when the
two additive effects were not significantly different.