Hardness Testing Experiment from Table 5-1 pg. 172
OBS RESPONSE TRMT BLOCK
1 9.3 1 1
2 9.4 2 1
3 9.2 3 1
4 9.7 4 1
5 9.4 1 2
6 9.3 2 2
7 9.4 3 2
8 9.6 4 2
9 9.6 1 3
10 9.8 2 3
11 9.5 3 3
12 10.0 4 3
13 10.0 1 4
14 9.9 2 4
15 9.7 3 4
16 10.2 4 4
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 6 1.21000000 22.69 0.0001
Error 9 0.08000000
Corrected Total 15 1.29000000
R-Square C.V. RESPONSE Mean
0.937984 0.979542 9.62500000
note: For the above F Value, the "full" model contains the overall
mean, treatment effect, and block effect. The "reduced" model
contains just the overall mean.
Source DF Type I SS F Value Pr > F
BLOCK 3 0.82500000 30.94 0.0001
TRMT 3 0.38500000 14.44 0.0009
note: Do not use the Type I F Values to test for significant treatment
and block effects as these are "sequential" hypotheses.
Source DF Type III SS F Value Pr > F
BLOCK 3 0.82500000 30.94 0.0001
TRMT 3 0.38500000 14.44 0.0009
note: Use the Type III F Values to test for significant treatment and
block effects. Here, we see that blocking was helpful and that there
is a significant treatment effect.
Observation Observed Predicted Residual
Value Value
1 9.30000000 9.35000000 -0.05000000
2 9.40000000 9.37500000 0.02500000
3 9.20000000 9.22500000 -0.02500000
4 9.70000000 9.65000000 0.05000000
5 9.40000000 9.37500000 0.02500000
6 9.30000000 9.40000000 -0.10000000
7 9.40000000 9.25000000 0.15000000
8 9.60000000 9.67500000 -0.07500000
9 9.60000000 9.67500000 -0.07500000
10 9.80000000 9.70000000 0.10000000
11 9.50000000 9.55000000 -0.05000000
12 10.00000000 9.97500000 0.02500000
13 10.00000000 9.90000000 0.10000000
14 9.90000000 9.92500000 -0.02500000
15 9.70000000 9.77500000 -0.07500000
16 10.20000000 10.20000000 0.00000000
Sum of Residuals 0.00000000
Sum of Squared Residuals 0.08000000
Sum of Squared Residuals - Error SS -0.00000000
First Order Autocorrelation -0.46093750
Durbin-Watson D 2.89062500
note: The above output simply lists the observed (response), predicted
(fit), and residual (resid) for each observation. Unless you are
going to talk about this output, you do not have to include it.
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= 9 MSE= 0.008889
Critical Value of T= 2.26
Least Significant Difference= 0.1508
Means with the same letter are not significantly different.
T Grouping Mean N TRMT
A 9.87500 4 4
B 9.60000 4 2
B
B 9.57500 4 1
B
B 9.45000 4 3
note: The above output is the Stage II analysis using the Protected
LSD method.
Duncan's Multiple Range Test for variable: RESPONSE
NOTE: This test controls the type I comparisonwise error
rate, not the experimentwise error rate
Alpha= 0.05 df= 9 MSE= 0.008889
Number of Means 2 3 4
Critical Range .1508 .1574 .1612
Means with the same letter are not significantly different.
Duncan Grouping Mean N TRMT
A 9.87500 4 4
B 9.60000 4 2
B
B 9.57500 4 1
B
B 9.45000 4 3
note: The above output is the Stage II analysis using Duncans
procudure.
Plot of EXPECTED*RESID='*'. Plot of STDRESID*TRMT='*'.
-+-----+-----+-----+- -+-----+-----+-----+-
2 + + 2 +------------*------+
| | | |
| * | |* * |
EXPECTED | * | STDRESID | |
| * * | | |
| * | | *|
| * | |* * *|
0 + ** + 0 + *+
| * * | | * * |
| * | |* * |
| * | | |
| * | |* * *|
|* | | * |
| | | |
-2 + + -2 +-------------------+
-+-----+-----+-----+- -+-----+-----+-----+-
-0.1 0.0 0.1 0.2 1 2 3 4
RESID TRMT
NOTE: 2 obs hidden.
Plot of STDRESID*BLOCK='*'. Plot of STDRESID*FIT='*'.
-+-----+-----+-----+- -+-----+-----+-----+-
2 +------*------------+ 2 +---*---------------+
| | | |
| * *| | * * |
STDRESID | | STDRESID | |
| | | |
|* | | * |
|* * * | | * * |
0 + *+ 0 + * +
|* *| | * * |
|* * | | * * |
| | | |
| * * *| | ** |
| * | | * |
| | | |
-2 +-------------------+ -2 +-------------------+
-+-----+-----+-----+- -+-----+-----+-----+-
1 2 3 4 9.0 9.5 10.0 10.5
BLOCK FIT
NOTE: 2 obs hidden.
Variable=RESID
Stem Leaf # Boxplot
1 5 1 |
1 00 2 |
0 5 1 |
0 0222 4 +--+--+
-0 22 2 *-----*
-0 88855 5 +-----+
-1 0 1 |
----+----+----+----+
Multiply Stem.Leaf by 10**-1
Moments
N 16 Sum Wgts 16
Mean 0 Sum 0
Std Dev 0.07303 Variance 0.005333
W:Normal 0.940715 Pr<W 0.3534
note: The above plots makeup what is called the residual analysis.
The first plot is the normal probability plot. It indicates that the
normality assumption is satisfied because it resembles a straight
line. In addition, the stem and leaf and boxplot of the residuals is
fairly symmetric which also indicates normality. Finally, the Shipiro
Wilk test for normality is non-significant. Combining all of this we
can conclude that the normality assumption is satisfied. The next two
plots are used to verify the homogenity of variance assumption as well
as identify any potential outliers. Since the two plots are fairly
random there is no need to suspect that the variance of the errors is
not constant. Also, there appears to be one point that results in a
potential outlier (treatment=3, block=2). In practice it would be a
good idea to go back and examine this point in terms of data entry
errors and or subject matter expertise. We can not do this here
however. Lastly, the plot of the standardized residuals versus the
fit is used check the homogenity of variance assumption and to check
for possible interaction between treatment and block. If there was a
pattern then one would expect some interaction between treatment and
block and would maybe want to consider transforming the data to
account for this. In summary, there is no indication that any of the
underlying assumptions of the model are violated. Hence, our above
analysis is valid.