OBS A B C D RESPONSE
1 -1 -1 -1 -1 45
2 1 -1 -1 -1 71
3 -1 1 -1 -1 48
4 1 1 -1 -1 65
5 -1 -1 1 -1 68
6 1 -1 1 -1 60
7 -1 1 1 -1 80
8 1 1 1 -1 65
9 -1 -1 -1 1 43
10 1 -1 -1 1 100
11 -1 1 -1 1 45
12 1 1 -1 1 104
13 -1 -1 1 1 75
14 1 -1 1 1 86
15 -1 1 1 1 70
16 1 1 1 1 96
note: This is just a listing of the data. This is how you should
enter the data into your .dat file. Since we will need to determine
factor effects you must use -1 and +1 to denote the low and high
levels of a particular factor.
T for H0: Pr > |T| Std Error of
Parameter Estimate Parameter=0 Estimate
INTERCEPT 70.06250000 9999.99 0.0001 0
A 10.81250000 9999.99 0.0001 0
B 1.56250000 9999.99 0.0001 0
A*B 0.06250000 9999.99 0.0001 0
C 4.93750000 9999.99 0.0001 0
A*C -9.06250000 -9999.99 0.0001 0
B*C 1.18750000 9999.99 0.0001 0
A*B*C 0.93750000 9999.99 0.0001 0
D 7.31250000 9999.99 0.0001 0
A*D 8.31250000 9999.99 0.0001 0
B*D -0.18750000 -9999.99 0.0001 0
A*B*D 2.06250000 9999.99 0.0001 0
C*D -0.56250000 -9999.99 0.0001 0
A*C*D -0.81250000 -9999.99 0.0001 0
B*C*D -1.31250000 -9999.99 0.0001 0
A*B*C*D 0.68750000 9999.99 0.0001 0
note: These are the parameter estimates output by SAS. Recall, they
are actually one half the estimates. When analyzing a 2^k factorial
(with only one replicate) you can construct a normal probability plot
of the effects. Enter the effects (not the intercept) into a new data
set in SAS and construct a normal probability plot.
OBS EFFECT BOOKEST
1 A 21.625
2 B 3.125
3 AB 0.125
4 C 9.875
5 AC -18.125
6 BC 2.375
7 ABC 1.875
8 D 14.625
9 AD 16.625
10 BD -0.375
11 ABD 4.125
12 CD -1.125
13 ACD -1.625
14 BCD -2.625
15 ABCD 1.375
Plot of EXPECTED*BOOKEST. Symbol used is '*'.
--+---------+---------+---------+---------+---------+---
2 + +
| |
| * A |
EXPECTED | * AD |
| * D |
| * * C |
| ** |
0 + * +
| ** |
| ** |
| * |
| * |
| * AC |
| |
-2 + +
--+---------+---------+---------+---------+---------+---
-20 -10 0 10 20 30
BOOKEST
note: My SAS program explains how to get this output. We see from the
normal probability plot that effects A, C, D, AC, and AD do not lie
along the straight line determined by the other factors. Thus, we
declare all these other factors non-significant. The "refined" model
will contain only A, C, D, AC, and AD. We next fit the "refined"
model and get residuals so that a complete residual analysis can be
performed.
General Linear Models Procedure
Dependent Variable: RESPONSE
Source DF Sum of Squares F Value Pr > F
Model 5 5535.81250000 56.74 0.0001
Error 10 195.12500000
Corrected Total 15 5730.93750000
R-Square C.V. RESPONSE Mean
0.965952 6.304793 70.0625000
Source DF Type III SS F Value Pr > F
A 1 1870.56250000 95.86 0.0001
C 1 390.06250000 19.99 0.0012
D 1 855.56250000 43.85 0.0001
A*C 1 1314.06250000 67.34 0.0001
A*D 1 1105.56250000 56.66 0.0001
Observation Observed Predicted Residual
Value Value
1 45.00000000 46.25000000 -1.25000000
2 71.00000000 69.37500000 1.62500000
3 48.00000000 46.25000000 1.75000000
4 65.00000000 69.37500000 -4.37500000
5 68.00000000 74.25000000 -6.25000000
6 60.00000000 61.12500000 -1.12500000
7 80.00000000 74.25000000 5.75000000
8 65.00000000 61.12500000 3.87500000
9 43.00000000 44.25000000 -1.25000000
10 100.00000000 100.62500000 -0.62500000
11 45.00000000 44.25000000 0.75000000
12 104.00000000 100.62500000 3.37500000
13 75.00000000 72.25000000 2.75000000
14 86.00000000 92.37500000 -6.37500000
15 70.00000000 72.25000000 -2.25000000
16 96.00000000 92.37500000 3.62500000
note: From the above output we see that all the factors we
subjectively declared significant turn out to be statistically
significant as well. Also, the fits and residuals are given. They
are very close (only round off error) to those given in the table on
page 324.
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= 10 MSE= 19.5125
Critical Value of T= 2.23
Least Significant Difference= 4.9212
Means with the same letter are not significantly different.
T Grouping Mean N A
A 80.875 8 1
B 59.250 8 -1
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= 10 MSE= 19.5125
Critical Value of T= 2.23
Least Significant Difference= 4.9212
Means with the same letter are not significantly different.
T Grouping Mean N C
A 75.000 8 1
B 65.125 8 -1
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= 10 MSE= 19.5125
Critical Value of T= 2.23
Least Significant Difference= 4.9212
Means with the same letter are not significantly different.
T Grouping Mean N D
A 77.375 8 1
B 62.750 8 -1
Least Squares Means
A C RESPONSE Std Err Pr > |T| LSMEAN
LSMEAN LSMEAN H0:LSMEAN=0 Number
1 1 76.7500000 2.2086478 0.0001 1
1 -1 85.0000000 2.2086478 0.0001 2
-1 1 73.2500000 2.2086478 0.0001 3
-1 -1 45.2500000 2.2086478 0.0001 4
Pr > |T| H0: LSMEAN(i)=LSMEAN(j)
i/j 1 2 3 4
1 . 0.0247 0.2887 0.0001
2 0.0247 . 0.0037 0.0001
3 0.2887 0.0037 . 0.0001
4 0.0001 0.0001 0.0001 .
A-C- A-C+ A+C+ A+C-
---- ------------ ----
A D RESPONSE Std Err Pr > |T| LSMEAN
LSMEAN LSMEAN H0:LSMEAN=0 Number
1 1 96.5000000 2.2086478 0.0001 1
1 -1 65.2500000 2.2086478 0.0001 2
-1 1 58.2500000 2.2086478 0.0001 3
-1 -1 60.2500000 2.2086478 0.0001 4
Pr > |T| H0: LSMEAN(i)=LSMEAN(j)
i/j 1 2 3 4
1 . 0.0001 0.0001 0.0001
2 0.0001 . 0.0489 0.1405
3 0.0001 0.0489 . 0.5364
4 0.0001 0.1405 0.5364 .
A-D+ A-D- A+D- A+D+
------------ ----
------------
note: The graphical displays indicate that the combinations A+C- and
A+D+ are statistically different from the rest of the combinations.
Thus, it would seem reasonable to recommend setting factors A and D at
the high levels and factor C at the low level. The same
recommendation was obtained using profile plots in the text book.
Plot of EXPECTED*RESID='*'. Plot of STDRESID*FIT='*'.
-+--------+--------+- -+-----+-----+-----+-
2 + + 2 +-------------------+
| | | * |
| * | | |
EXPECTED | * | STDRESID | * * |
| * | | * *|
| * | | * * |
| ** | | * |
0 + * * + 0 + +
| * | | ** * *|
| * | | * |
| * * | | |
| * | | * |
| * | | |
| | | * * |
-2 + + -2 +-------------------+
-+--------+--------+- -+-----+-----+-----+-
-10 0 10 40 60 80 100
RESID FIT
W:Normal 0.954 Pr<W 0.542
NOTE: 2 obs hidden.
Plot of STDRESID*A='*'. Plot of STDRESID*C='*'.
---+------------+--- ---+------------+---
2 +------------------+ 2 +------------------+
| * | | * |
| | | |
STDRESID | * | STDRESID | * |
| * * | | * * |
| * * | | * |
| * | | * |
0 + + 0 + +
| * * | | * * |
| * | | * |
| | | |
| * | | * |
| | | |
| * * | | * |
-2 +------------------+ -2 +------------------+
---+------------+--- ---+------------+---
-1 1 -1 1
A C
NOTE: 3 obs hidden. NOTE: 5 obs hidden.
Plot of STDRESID*D='*'.
---+------------+---
2 +------------------+
| * |
| |
STDRESID | * * |
| * |
| * |
| * |
0 + +
| * * |
| * |
| |
| * |
| |
| * * |
-2 +------------------+
---+------------+---
-1 1
D
NOTE: 4 obs hidden.