ST 524
NCSU - Fall 2007
Incomplete block Design
Incomplete block designs
Example - Balanced Incomplete Block Design
y
6
2
3
5
1
7
2
5
4
1
3
6
trt
t1
t1
t1
t2
t2
t2
t3
t3
t3
t4
t4
t4
B1
*
*
*
T1
T2
T3
T4
block
b1
b2
b3
b1
b2
b4
b1
b3
b4
b2
b3
b4
B2
*
*
*
y
6
2
3
5
1
7
2
5
4
1
3
6
B3
*
*
*
Intercept
1
1
1
1
1
1
1
1
1
1
1
1
T1
1
1
1
0
0
0
0
0
0
0
0
0
B4
*
*
*
T1
T2
T3
T4
T2
0
0
0
1
1
1
0
0
0
0
0
0
B1
6
5
2
13
Source
Treatment (Adjusted)
Block(Adjusted)
Error
Total

T3
0
0
0
0
0
0
1
1
1
0
0
0
T4
0
0
0
0
0
0
0
0
0
1
1
1
B2
2
1
1
4
B1
1
0
0
1
0
0
1
0
0
0
0
0
B3
3
5
3
11
B2
0
1
0
0
1
0
0
0
0
1
0
0
B3
0
0
1
0
0
0
0
1
0
0
1
0
B4
0
0
0
0
0
1
0
0
1
0
0
1
B4
7
4
6
17
11
13
11
10
45
d.f
4–1 =3
4–1=3
Diff= 11 – 3 -3 = 5
4*3 – 1 = 11
Variance between plots in the same block is smaller than the variance between plots in different blocks.
Thursday November 1, 2007
1
t = 4 treatments
k = 3 units per block
r = 3 replications for each treatment
Each treatment occurs together with
each other treatment twice.
ST 524
NCSU - Fall 2007
Incomplete block Design
Linear Model Yijk    i   j   ijk
Adjustment factor
A
 Eb  Ee 
k  r  1 Eb

10.9722  2.35  0.0982
4   3  1 10.9722
Effective error mean square Ee  Ee 1  A  =2.35  (1+0.0982)=2.5808
pooled error Erb 
 SSB  SSE    32.91666667  11.75000000   44.66667  2.7917
16
 k 2  1  r  1
32  1  2  1
% relative precision 
RP 
Erb
100
Ee
2.7917
 100  108.17%
2.5808
Additive Linear Model
Yij     j  b j  ij
Treatment means
Thursday November 1, 2007
2
ST 524
NCSU - Fall 2007
Incomplete block Design
Y11     1  b 1 11
Y 1.     1 
Y12     1  b 2 12
3
3
 b b b        
Y 2.     2  1 2 4  21 22 24
3
3
b 3 b 4   11  12  13   21   22   24 

Y 1.  Y 2.     1   2  

3
3
Y13     1  b 3 13
Y1.          3 1   b 1 b 2 b 3    11  12  13 
Y 1.     1 
 b 1 b 2 b 3   11  12  13 
3
 b 1 b 2 b 3   11  12  13 
3
Block Means
Y .1    b1 
Y11     1  b 1 11
Y21     2  b 1  21
Y .2    b2
Y11     3  b 1  31
Y.1           1  2  3   3b1  11   21   31 
Y .1    b1
Y .3  Y .4   b3  b4  
  b3  b4  
3
3
         
 1 2 4  12 22 42
Y .1  Y .2   b1  b2  
3
 1  2  4 
3
3
3

 1  3  4    2  3  4   13   33   43    24   34   44 
3
 1  2 
3
3

 1  2  4 

11   21   31   12   22   42 
3
3
              
  b1  b2   3 4  11 21 31 12 22 42
3
3
         
 1 2 3  11 21 31
3
 1  2  3  
Treatment differences include block effect
  31 
11   21differences.
3
13   33   43   24   34   44 
3
Block differences include treatment effect
differences.
3
3
Intra-block analysis: obtains treatment information by comparing block-adjusted least squares estimates
Inter-block analysis recovers treatment information contained in the block averages
Solution to uncovering the inter-block information: let block effects to be random, with mean zero, model
turns to be Yij     j  b j  ij ,

 ij ~ N  0,  e2  ,
b j ~ N  0,  b2  , where  ij and b j are independent.
Unconditional comparison of two mean blocks and treatments: BROAD INFERENCE SPACE
E Y 1.  Y 2.    1   2  , and
E Y .3  Y .4  
1   2 
Thursday November 1, 2007
3
3
ST 524
NCSU - Fall 2007
Incomplete block Design

Conditional comparisons of two block or treatment means : NARROW INFERENCE SPACE
b  b 
E Y 1.  Y 2. | b    1   2   3 4 , and
3
E Y .3  Y .4 | b    b3  b4  



1   2 
3
Interblock Analysis: In PROC MIXED – RANDOM Block;
LSMEANS are estimation of treatment means in the broad inference space
Because of non-orthogonality, the estimates of the treatment means as well as their standard errors differ between the fixed
effects (PROC GLM) and mixed effects analysis (PROC MIXED).
PROC MIXED
proc mixed data=b; /*
4 treatments in 4 blocks of size 3 */
class block trt;
model y = trt;
random block;
lsmeans trt / pdiff;
estimate 'I trt 1 mean' intercept 1 trt 1 ;
estimate 'II trt 1 mean' intercept 3 trt 3 |
block 1 1 1 /divisor=3;
estimate 'I trt 2 mean' intercept 1 trt 0 1 ;
estimate 'II trt 2 mean' intercept 3
trt 0 3 |
block 1 1 0 1
estimate 'I trt 3 mean blk1' intercept 1
estimate 'II trt 3 mean'
intercept 3
estimate 'I trt 4 mean blk2' intercept 1
estimate 'II trt 4 mean'
intercept 3
/* inter -analysis
/* intra -analysis - narrow scope */
/* inter -analysis
/divisor=3;
- broad scope*/
- broad scope*/
/* intra -analysis - narrow scope */
trt 0 0 1 ;
trt 0 0 3 |
block 1 0 1 1 /divisor=3;
/* inter -analysis
trt 0 0 0 1 ;
trt 0 0 0 3 |
block 0 1 1 1 /divisor=3;
/* inter -analysis
- broad scope*/
/* intra -analysis - narrow scope */
- broad scope*/
/* intra -analysis - narrow scope */
estimate 'trt1 vs trt3'
trt 1 0 -1;
estimate 'trt1 vs trt3 blk1'
trt 1 0 -1 | block 1;
run;
The Mixed Procedure
Model Information
Data Set
Dependent Variable
Covariance Structure
Estimation Method
Residual Variance Method
Fixed Effects SE Method
Degrees of Freedom Method
Thursday November 1, 2007
WORK.B
y
Variance Components
REML
Profile
Model-Based
Containment
4
ST 524
NCSU - Fall 2007
Incomplete block Design
Class Level Information
Class
Levels
block
trt
4
4
Values
b1 b2 b3 b4
t1 t2 t3 t4
Dimensions
Covariance Parameters
Columns in X
Columns in Z
Subjects
Max Obs Per Subject
2
5
4
1
12
Number of Observations
Number of Observations Read
Number of Observations Used
Number of Observations Not Used
12
12
0
Iteration History
Iteration
Evaluations
-2 Res Log Like
Criterion
0
1
1
1
40.85575344
38.55564281
0.00000000
Convergence criteria met.
Covariance Parameter
Estimates
Cov Parm
block
Residual
Estimate
3.2333
2.3500
Fit Statistics
-2 Res Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
Thursday November 1, 2007
38.6
42.6
45.0
41.3
5
ST 524
NCSU - Fall 2007
Incomplete block Design
Solution for Fixed Effects
Effect
Intercept
trt
trt
trt
trt
trt
Estimate
3.4643
0.7590
0.9018
-0.5180
0
t1
t2
t3
t4
Standard
Error
1.2843
1.3117
1.3117
1.3117
.
DF
3
5
5
5
.
t Value
2.70
0.58
0.69
-0.39
.
Pr > |t|
0.0739
0.5879
0.5224
0.7092
.
Solution for Random Effects
Effect
block
block
block
block
block
b1
b2
b3
b4
Estimate
Std Err
Pred
DF
t Value
Pr > |t|
0.3929
-2.1610
0.09823
1.6699
1.1523
1.1523
1.1523
1.1523
5
5
5
5
0.34
-1.88
0.09
1.45
0.7470
0.1196
0.9354
0.2070
Type 3 Tests of Fixed Effects
Effect
trt
Num
DF
Den
DF
F Value
Pr > F
3
5
0.52
0.6893
Estimates
Label
I trt 1 mean
II trt 1 & Block mean
I trt 2 mean
II trt 2 & Block mean
I trt 3 mean blk1
II trt 3 & Block mean
I trt 4 mean blk2
II trt 4 & Block mean
trt1 vs trt3
trt1 vs trt3 in blk1
II trt 1 in Block 1 pred
II trt 3 in Block 1 pred
Thursday November 1, 2007
Estimate
Standard
Error
DF
t Value
Pr > |t|
4.2233
3.6667
4.3661
4.3333
2.9463
3.6667
3.4643
3.3333
1.2770
1.6699
4.6162
3.3392
1.2843
0.8851
1.2843
0.8851
1.2843
0.8851
1.2843
0.8851
1.3117
1.7459
1.1158
1.1158
5
5
5
5
5
5
5
5
5
5
5
5
3.29
4.14
3.40
4.90
2.29
4.14
2.70
3.77
0.97
0.96
4.14
2.99
0.0218
0.0090
0.0193
0.0045
0.0703
0.0090
0.0429
0.0131
0.3750
0.3828
0.0090
0.0304
6
ST 524
NCSU - Fall 2007
Incomplete block Design
Least Squares Means
Effect
trt
trt
trt
trt
trt
t1
t2
t3
t4
Estimate
Standard
Error
DF
t Value
Pr > |t|
4.2233
4.3661
2.9463
3.4643
1.2843
1.2843
1.2843
1.2843
5
5
5
5
3.29
3.40
2.29
2.70
0.0218
0.0193
0.0703
0.0429
Differences of Least Squares Means
Effect
trt
_trt
trt
trt
trt
trt
trt
trt
t1
t1
t1
t2
t2
t3
t2
t3
t4
t3
t4
t4
Estimate
Standard
Error
DF
t Value
Pr > |t|
-0.1428
1.2770
0.7590
1.4197
0.9018
-0.5180
1.3117
1.3117
1.3117
1.3117
1.3117
1.3117
5
5
5
5
5
5
-0.11
0.97
0.58
1.08
0.69
-0.39
0.9176
0.3750
0.5879
0.3285
0.5224
0.7092
Predicted values
Obs
y
trt
block
Pred
1
2
3
4
5
6
7
8
9
10
11
12
6
2
3
5
1
7
2
5
4
1
3
6
t1
t1
t1
t2
t2
t2
t3
t3
t3
t4
t4
t4
b1
b2
b3
b1
b2
b4
b1
b3
b4
b2
b3
b4
4.61620
2.06228
4.32152
4.75899
2.20506
6.03595
3.33924
3.04456
4.61620
1.30329
3.56253
5.13418
Thursday November 1, 2007
StdErr
Pred
DF
Alpha
1.11580
1.11580
1.11580
1.11580
1.11580
1.11580
1.11580
1.11580
1.11580
1.11580
1.11580
1.11580
5
5
5
5
5
5
5
5
5
5
5
5
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
Lower
Upper
Resid
1.74795
-0.80597
1.45327
1.89074
-0.66319
3.16770
0.47099
0.17631
1.74795
-1.56496
0.69428
2.26593
7.48445
4.93053
7.18977
7.62724
5.07331
8.90420
6.20749
5.91281
7.48445
4.17154
6.43078
8.00243
1.38380
-0.06228
-1.32152
0.24101
-1.20506
0.96405
-1.33924
1.95544
-0.61620
-0.30329
-0.56253
0.86582
7