Rocket Propellant Problem from Table 5-9 pg. 194
OBS BLOCK1 BLOCK2 TRMT RESPONSE
1 1 1 A 24
2 1 2 B 20
3 1 3 C 19
4 1 4 D 24
5 1 5 E 24
6 2 1 B 17
7 2 2 C 24
8 2 3 D 30
9 2 4 E 27
10 2 5 A 36
11 3 1 C 18
12 3 2 D 38
13 3 3 E 26
14 3 4 A 27
15 3 5 B 21
16 4 1 D 26
17 4 2 E 31
18 4 3 A 26
19 4 4 B 23
20 4 5 C 22
21 5 1 E 22
22 5 2 A 30
23 5 3 B 20
24 5 4 C 29
25 5 5 D 31
note: This is just a listing of the data. This is how you should
enter the data into your .dat file.
General Linear Models Procedure
Dependent Variable: RESPONSE
Source DF Sum of Squares F Value Pr > F
Model 12 548.00000000 4.28 0.0089
Error 12 128.00000000
Corrected Total 24 676.00000000
R-Square C.V. RESPONSE Mean
0.810651 12.85821 25.4000000
note: For the above F Value, the "full" model contains the overall
mean, treatment effect, block1 effect, and block2 effect. The
"reduced" model contains just the overall mean.
Source DF Type III SS F Value Pr > F
BLOCK1 4 68.00000000 1.59 0.2391
BLOCK2 4 150.00000000 3.52 0.0404
TRMT 4 330.00000000 7.73 0.0025
note: Use the Type III F Values to test for significant treatment and
blocking effects. Here, we see that blocking was helpful for block2
(i.e. operator) while it appears that blocking on batch of raw
material was not necessary. However, it is common practice to block
on batches of raw material when possible. Furthermore, note that
there is a significant treatment effect (p-value = 0.0025). Thus, we
will proceed to the stage II analysis.
T tests (LSD) for variable: RESPONSE
NOTE: This test controls the type I comparisonwise error rate
not the experimentwise error rate.
Alpha= 0.05 df= 12 MSE= 10.66667
Critical Value of T= 2.18
Least Significant Difference= 4.5005
Means with the same letter are not significantly different.
T Grouping Mean N TRMT
A 29.800 5 D
A
A 28.600 5 A
A
B A 26.000 5 E
B
B C 22.400 5 C
C
C 20.200 5 B
mote: Using the Protected LSD method, we see that treatment
(formulation) B gives a statistically "better" burning rate than the
other formulations, except treatment C. However, since treatment C is
not different from treatment E I would be inclined to choose treatment
B as the "best" formulation.
OBS RESPONSE FIT RESID STDRESID EXPECTED
1 24 21.4 2.60000 1.14905 1.02008
2 20 20.2 -0.20000 -0.08839 0.09656
3 19 18.0 1.00000 0.44194 0.61514
4 24 27.2 -3.20000 -1.41421 -1.76883
5 24 24.2 -0.20000 -0.08839 0.00000
6 17 17.6 -0.60000 -0.26517 -0.09656
7 24 27.0 -3.00000 -1.32583 -1.42608
8 30 30.0 0.00000 0.00000 0.39573
9 27 28.0 -1.00000 -0.44194 -0.29338
10 36* 31.4 4.60000 2.03293 1.42608
11 18 19.0 -1.00000 -0.44194 -0.19403
12 38* 33.6 4.40000 1.94454 1.19838
13 26 25.4 0.60000 0.26517 0.50240
14 27 29.8 -2.80000 -1.23744 -1.02008
15 21 22.2 -1.20000 -0.53033 -0.50240
16 26 26.0 0.00000 0.00000 0.29338
17 31 29.4 1.60000 0.70711 0.73632
18 26 27.6 -1.60000 -0.70711 -0.73632
19 23 21.0 2.00000 0.88388 0.86942
20 22 24.0 -2.00000 -0.88388 -0.86942
21 22 23.0 -1.00000 -0.44194 -0.39573
22 30 32.8 -2.80000 -1.23744 -1.19838
23 20 20.0 0.00000 0.00000 0.19403
24 29* 24.0 5.00000 2.20971 1.76883
25 31 32.2 -1.20000 -0.53033 -0.61514
note: The above output lists the residual analysis variables from the
data set checkass in the .sas file. I have "starred" the three
potential outliers appearing in the plots below. However, these
observations probably are not extreme enough to influence the
analysis.
Plot of STDRESID*TRMT='*'. Plot of STDRESID*BLOCK1='*'.
---+--+--+--+--+--- --+---+---+---+---+--
STDRESID | | STDRESID | |
4 + + 4 + +
| | | |
| | | |
| | | |
2 +--*-----*--*-----+ 2 +-----*---*-------*-+
| | | |
| * * | | * * |
| * * | | * * * |
0 + * * * + 0 + * * * * +
| * * * * * | | * * * * |
| * * | | * * * |
| * * | | * * |
-2 +-----------------+ -2 +-------------------+
| | | |
---+--+--+--+--+--- --+---+---+---+---+--
A B C D E 1 2 3 4 5
TRMT BLOCK1
NOTE: 6 obs hidden. NOTE: 4 obs hidden.
Plot of STDRESID*BLOCK2='*'. Plot of STDRESID*FIT='*'.
--+---+---+---+---+-- -+-----+-----+-----+-
STDRESID | | STDRESID | |
4 + + 4 + +
| | | |
| | | |
| | | |
2 +-----*-------*---*-+ 2 +--------*----**----+
| | | |
| * * | | * |
| * * | | * * * |
0 + * * * * + 0 + * ** * +
| * * * * | | * ** * * |
| * * * | | * * * |
| * * | | * |
-2 +-------------------+ -2 +-------------------+
| | | |
--+---+---+---+---+-- -+-----+-----+-----+-
1 2 3 4 5 10 20 30 40
BLOCK2 FIT
NOTE: 5 obs hidden. NOTE: 5 obs hidden.
Plot of EXPECTED*RESID='*'. Plot of RESPONSE*TRMT='*'.
-+--------+--------+- ---+--+--+--+--+---
2 + + RESPONSE | |
| *| 40 + +
| * | | * |
EXPECTED | * * | | * |
| ** | | |
| ** | 32 + * * +
| * | | * * * |
0 + ** + | * * |
| * | | * * * |
| * | 24 + * * * * * +
| ** | | * * * |
| * | | * * |
| * | | * * |
| * | 16 + +
-2 + + | |
-+--------+--------+- ---+--+--+--+--+---
-5 0 5 A B C D E
RESID TRMT
NOTE: 7 obs hidden. NOTE: 1 obs hidden.
W:Normal 0.921064
Pr<W 0.0559
Variable=RESID
Stem Leaf # Boxplot
5 0 1 0
4 46 2 0
3
2 06 2 |
1 06 2 +-----+
0 0006 4 | + |
-0 622 3 *-----*
-1 622000 6 +-----+
-2 880 3 |
-3 20 2 |
----+----+----+----+
note: The above plots indicate that the homogeneity of variance
assumption and the normality assumption are tenable although there is
some evidence that suggests normality may be suspect. Furthermore,
there does not appear to be any interaction between the blocks and
treatments since there is no patterns in the plots.