OBS TRMT BLOCK RESPONSE
1 1 1 73
2 1 2 74
3 1 3 .
4 1 4 71
5 2 1 .
6 2 2 75
7 2 3 67
8 2 4 72
9 3 1 73
10 3 2 75
11 3 3 68
12 3 4 .
13 4 1 75
14 4 2 .
15 4 3 72
16 4 4 75
note: This is just a listing of the data. This is how you should
enter the data into your .dat file. The "." in the response column
represents a missing value in SAS. Thus, this will be analyzed as a
balanced incomplete block design.
General Linear Models Procedure
Dependent Variable: RESPONSE
Source DF Sum of Squares F Value Pr > F
Model 6 77.75000000 19.94 0.002
Error 5 3.25000000
Corrected Total 11 81.00000000
R-Square C.V. RESPONSE Mean
0.959877 1.112036 72.5000000
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 55.00000000 28.21 0.0015
TRMT 3 22.75000000 11.67 0.0107
note: Type I SS for BLOCK is SS[Blocks] on page 156 of the text book.
If you want SS[Treatments] (pg. 158) you can change the model line in
the sas program to model response = trmt block / p;.
Source DF Type III SS F Value Pr > F
BLOCK 3 66.08333333 33.89 0.0010
TRMT 3 22.75000000 11.67 0.0107
note: Type III SS for BlOCK and TRMT are SS[Blocks(adjusted)] and
SS[Treatments(adjusted)] respectively (see pages 155-158 of the
textbook). Use the corresponding F Values to test for no BLOCK and no
TRMT effect.
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= 5 MSE= 0.65
Number of Means 2 3 4
Critical Range 1.692 1.745 1.767
Means with the same letter are not significantly different.
Duncan Grouping Mean N TRMT
A 74.0000 3 4
A
B A 72.6667 3 1
B
B 72.0000 3 3
B
B 71.3333 3 2
note: The conclusions here do not correspond to the conclusions given
in the textbook (pg. 213). This is because SAS is running Duncan's
procedure on the "unadjusted" means and the textbook is running
Duncan's procedure on the "adjusted" treatment effects. Thus, in order
to do a proper state II analysis, we need to run the LSD procedure on
the "adjusted" (or least squares) means. This is done next.
General Linear Models Procedure
Least Squares Means
TRMT RESPONSE Std Err Pr > |T| LSMEAN
LSMEAN LSMEAN H0:LSMEAN=0 Number
1 71.3750000 0.4868051 0.0001 1
2 71.6250000 0.4868051 0.0001 2
3 72.0000000 0.4868051 0.0001 3
4 75.0000000 0.4868051 0.0001 4
note: We can use this information to get confidence intervals for the
true treatment population means.
T for H0: LSMEAN(i)=LSMEAN(j) / Pr > |T|
i/j 1 2 3 4
1 . -0.35806 -0.89514 -5.19183
0.7349 0.4117 0.0035
2 0.358057 . -0.53709 -4.83378
0.7349 0.6142 0.0047
3 0.895144 0.537086 . -4.29669
0.4117 0.6142 0.0077
4 5.191833 4.833775 4.296689 .
0.0035 0.0047 0.0077
note: We can use this information to get confidence intervals for the
difference between two treatment population means by solving for the
appropriate standard error.
note: The results of the LSD procedure (on the "adjusted" means) is as
follows.
TRMT1 TRMT2 TRMT3 TRMT4
--------------------- -----
This agrees with the conclusions found on page 160 of the textbook.
Plot of STDRESID*FIT='*'. Plot of STDRESID*TRMT='*'.
-+-----+-----+-----+- -+-----+-----+-----+-
2 +-------------------+ 2 +-------------------+
| * | | *|
| * | |* |
STDRESID | | STDRESID | |
| * * | | * |
| | | |
| * | | * |
0 + * + 0 + * *+
| * | | * |
| | | |
| * * | |* |
| | | |
| * | | * |
| * | | *|
-2 +-------------------+ -2 +-------------------+
-+-----+-----+-----+- -+-----+-----+-----+-
65 70 75 80 1 2 3 4
FIT TRMT
NOTE: 4 missing. 1 hidden. NOTE: 4 missing. 2 hidden.
Plot of STDRESID*BLOCK='*'. Plot of EXPECTED*RESID='*'.
-+-----+-----+-----+- -+--------+--------+-
2 +-------------------+ 2 + +
| * | | |
|* | | * |
STDRESID | | EXPECTED | * |
| * *| | |
| | | * |
|* | | * |
0 + * *+ 0 + * +
| * | | * |
| | | * |
| * *| | |
| | | * |
| * | | * |
|* | | |
-2 +-------------------+ -2 + +
-+-----+-----+-----+- -+--------+--------+-
1 2 3 4 -1 0 1
BLOCK RESID
NOTE: 4 missing. NOTE: 4 missing. 3 hidden.