OBS FACTORA FACTORB FACTORC RESPONSE
1 10 25 200 -3
2 10 25 200 -1
3 10 25 250 -1
4 10 25 250 0
5 10 30 200 -1
6 10 30 200 0
7 10 30 250 1
8 10 30 250 1
9 12 25 200 0
10 12 25 200 1
11 12 25 250 2
12 12 25 250 1
13 12 30 200 2
14 12 30 200 3
15 12 30 250 6
16 12 30 250 5
17 14 25 200 5
18 14 25 200 4
19 14 25 250 7
20 14 25 250 6
21 14 30 200 7
22 14 30 200 9
23 14 30 250 10
24 14 30 250 11
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 11 328.12500000 42.11 0.0001
Error 12 8.50000000
Corrected Total 23 336.62500000
R-Square C.V. RESPONSE Mean
0.974749 26.93201 3.12500000
note: For the above F Value, the "full" model contains the overall
mean, all main effects, and all possible interaction effects. The
"reduced" model contains just the overall mean.
Source DF Type III SS F Value Pr > F
FACTORA 2 252.75000000 178.41 0.0001
FACTORB 1 45.37500000 64.06 0.0001
FACTORA*FACTORB 2 5.25000000 3.71 0.0558
FACTORC 1 22.04166667 31.12 0.0001
FACTORA*FACTORC 2 0.58333333 0.41 0.6715
FACTORB*FACTORC 1 1.04166667 1.47 0.2486
FACTOR*FACTOR*FACTOR 2 1.08333333 0.76 0.4869
note: The three-way interaction term is not significant. Thus, look
at the two-way interaction terms. Assuming a significance level of
0.05, none of the two-way interaction terms are significant (although
the AB interaction is "borderline"). Since none of the two-way
interaction terms are significant, look at the main effects. The
results indicate that all three factors are significant. Since no
interaction terms were significant, we can perform a stage II analysis
for each of the main effects using the methods of Chapter3 (e.g. LSD).
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= 0.708333
Critical Value of T= 2.18
Least Significant Difference= 0.9169
Means with the same letter are not significantly different.
T Grouping Mean N FACTORA
A 7.3750 8 14
B 2.5000 8 12
C -0.5000 8 10
note: All three means of FactorA are significantly different. Since
we would like the response to be as small as possible, it appears that
10% carbonation yields the smallest fill deviation.
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= 0.708333
Critical Value of T= 2.18
Least Significant Difference= 0.7486
Means with the same letter are not significantly different.
T Grouping Mean N FACTORB
A 4.5000 12 30
B 1.7500 12 25
note: Both levels of FactorB are significantly different.
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= 0.708333
Critical Value of T= 2.18
Least Significant Difference= 0.7486
Means with the same letter are not significantly different.
T Grouping Mean N FACTORC
A 4.0833 12 250
B 2.1667 12 200
note: Both levels of FactorC are significantly different.
General Linear Models Procedure
Least Squares Means
FACTORB RESPONSE Std Err Pr > |T| LSMEAN
LSMEAN LSMEAN H0:LSMEAN=0 Number
FACTORA = 10
25 -1.25000000 0.42081271 0.0117 1
30 0.25000000 0.42081271 0.5635 2
FACTORA = 12
25 1.00000000 0.42081271 0.0350 3
30 4.00000000 0.42081271 0.0001 4
FACTORA = 14
25 5.50000000 0.42081271 0.0001 5
30 9.25000000 0.42081271 0.0001 6
T for H0: LSMEAN(i)=LSMEAN(j) / Pr > |T|
i/j 1 2 3 4 5 6
1 . -2.5205 -3.78076 -8.82176 -11.3423 -17.6435
0.0269 0.0026 0.0001 0.0001 0.0001
2 2.520504 . -1.26025 -6.30126 -8.82176 -15.123
0.0269 0.2315 0.0001 0.0001 0.0001
3 3.780756 1.260252 . -5.04101 -7.56151 -13.8628
0.0026 0.2315 0.0003 0.0001 0.0001
4 8.821765 6.30126 5.041008 . -2.5205 -8.82176
0.0001 0.0001 0.0003 0.0269 0.0001
5 11.34227 8.821765 7.561512 2.520504 . -6.30126
0.0001 0.0001 0.0001 0.0269 0.0001
6 17.64353 15.12302 13.86277 8.821765 6.30126 .
0.0001 0.0001 0.0001 0.0001 0.0001
NOTE: To ensure overall protection level, only probabilities
associated with pre-planned comparisons should be used.
Note: If the AB interaction was significant, then we would use the
above lsmeans output to make conclusions about factorA and factorB
effects. For example, the above LSD type procedure yields the
following summary.
Factor A (lowest-highest)
__ __ __
FACTOR B = 25 10 12 14
__ __ __
FACTOR B = 30 10 12 14
Plot of EXPECTED*RESID='*'. Plot of STDRESID*FIT='*'.
--+-------+-------+-- -+-----+-----+-----+-
2 + + 2 +-------------------+
| * | | * * |
| * | | |
EXPECTED | * | STDRESID | |
| * | | ***** * |
| * | | |
| * | | |
0 + * + 0 + * +
| * | | |
| * | | |
| * | | ***** * |
| * | | |
| * | | |
| * | | * * |
-2 + + -2 +-------------------+
--+-------+-------+-- -+-----+-----+-----+-
-1 0 1 -10 0 10 20
RESID FIT
NOTE: 11 obs hidden. NOTE: 7 obs hidden.
Plot of STDRESID*FACTORA='*'. Plot of STDRESID*FACTORB='*'.
-+--------+--------+- ---+------------+---
2 +-------------------+ 2 +------------------+
|* *| | * * |
| | | |
STDRESID | | STDRESID | |
|* * *| | * * |
| | | |
| | | |
0 +* + 0 + * +
| | | |
| | | |
|* * *| | * * |
| | | |
| | | |
|* *| | * * |
-2 +-------------------+ -2 +------------------+
-+--------+--------+- ---+------------+---
10 12 14 25 30
FACTORA FACTORB
NOTE: 13 obs hidden. NOTE: 15 obs hidden.
Plot of STDRESID*FACTORC='*'.
---+------------+---
2 +------------------+
| * |
| |
STDRESID | |
| * * |
| |
| |
0 + * +
| |
| |
| * * |
| |
| |
| * |
-2 +------------------+
---+------------+---
200 250
FACTORC
NOTE: 17 obs hidden.
Variable=RESID
Stem Leaf # Boxplot
10 00 2 |
8 |
6 |
4 000000000 9 +-----+
2 | |
0 00 2 *--+--*
-0 | |
-2 | |
-4 000000000 9 +-----+
-6 |
-8 |
-10 00 2 |
----+----+----+----+
Multiply Stem.Leaf by 10**-1