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Ch 18 Analysis of a Randomized-blocks Experiment
18-3 Principle of Blocking
-- A block is a group of homogeneous units (sex, city, school district, site).
-- A number of units per block = a number of treatments
-- “a” treatments are assigned randomly to “a” units within each block
(block randomization)
The Randomized-blocks Experiment with one observation in each cell.
Data Let the data be in the following format;
Blocks
Levels 1 2 3 b Sumi Mean i
1 Y
Y11 Y12 Y13 ..... 1b Y1. Y1.
2 Y
Y21 Y22 Y23 ..... 2 b Y2. Y2.
:
a Y
Ya1 Ya 2 Ya 3 ..... ab Ya . Ya .
_________________________
Sum j Y.1 Y.2 Y.3 ..... Y.b Y..
Mean j Y.1 Y.2 Y.3 ..... Y.b Y..
b Yi.
where Yi. Yij Yi.
j1 b
a Y a b Y..
Y. j Yij Yi. i. Y.. Yij Y.. N = ab
i 1 a i 1 j1 N
Assumptions
1. The observations are statistically independent of one another.
2. Each observation is selected from a normally distributed population with 2
3. There is no block-treatment interaction effect.
Model
Yij i j E ij
where i = 1,...,a , j = 1,..., b ,
i is the mean of the ith level,
b
j is the jth block effect, j 0
j1
and E ij is the random error with NID (0, 2 )
Hypotheses H 0 : 1 2 ... a vs H 1 : Not H 0 .
Table 3. The ANOVA Table for a Randomized Complete Block Design
with an observation.
Source of Degrees of Sum of Squares Mean F
Variation Freedom Squares
a 2
1 y ..
SST y i. -
2
Treatment a-1 b i 1 N MST MST/MSE
Blocks b-1 1 b 2 y 2 .. MSB MSB/MSE
SSB y . j
a j1 N
Error (a – 1)(b – 1) SSE = TSS SST - SSB MSE
2
a b y ..
TSS y ij -
2
Total N–1 N
i 1 j1
Estimation of parameters
i Yi.
ˆ
ˆ
j Y. j Y..
2 MSE
ˆ
Why Block?
-- Blocking decreases unexplained variability, that is, SSE.
-- We anticipate block-to-block variation
-- We are interested in the treatment effect.
TSS = SST + SSB + SSE
Regression Model
Model
y ij i j E ij
where i = 1,...,a, j = 1,.., b,
a
is an overall mean, i is the effect of the ith treatment, i 0
i 1
b
j is the effect of the jth block, j 0
j1
and E ij is the random error with NID (0, 2 ) .
Hypotheses H 0 : 1 2 ... a 0 vs H 1 : Not H 0 .
Parameter Estimates
Y..
ˆ
i Yi. Y..
ˆ
If H 0 is not rejected, conclude that there is no difference among the means of
different levels. If H 0 is rejected, then perform the follow-up test of multiple
comparison in order to determine which means are different from others. Based on the
result of multiple comparison, estimate the differences between the means using the
option of estimate.
Random block effect Model
Often blocks are considered random
Yij i B j E ij
B j ~ N (0, B )
2
where
E ij ~ N (0, 2 )
Hypotheses H 0 : 1 2 ... a 0 vs H 1 : Not H 0 .
ANOVA Table and F-tests are the same as the Fixed block effect Model
Example
Title 'Randomized Complete Block Design';
proc plan seed=38477; /* first, plan the randomized design*/
factors block=3 ordered tx=4;
output out=RCBD;
run;
proc print data=RCBD; /* with the plan, collect the data */
run;
data one; /* after collecting the data, add y */
set RCBD;
input y @@;
cards;
3 5 4 6 4 5 5 4 5 5 4 4
run;
proc tabulate;
class block tx;
var y;
table block, tx*(y);
run;
proc glm data=one;
class block tx;
model y = block tx;
lsmeans tx /pdiff;
means tx /tukey cldiff;
run;
Output
Randomized Complete Block Design
Procedure PLAN
Factor Select Levels Order
------ ------ ------ -------
BLOCK 3 3 Ordered
TX 4 4 Random
BLOCK TX
-------- -+-+-+-+
1 4 2 3 1
2 3 4 1 2
3 4 2 1 3
Randomized Complete Block Design
OBS BLOCK TX
1 1 4
2 1 2
3 1 3
4 1 1
5 2 3
6 2 4
7 2 1
8 2 2
9 3 4
10 3 2
11 3 1
12 3 3
Randomized Complete Block Design
„ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ…ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ†
‚ ‚ TX ‚
‚ ‡ƒƒƒƒƒƒƒƒƒƒƒƒ…ƒƒƒƒƒƒƒƒƒƒƒƒ…ƒƒƒƒƒƒƒƒƒƒƒƒ…ƒƒƒƒƒƒƒƒƒƒƒƒ‰
‚ ‚ 1 ‚ 2 ‚ 3 ‚ 4 ‚
‚ ‡ƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒ‰
‚ ‚ Y ‚ Y ‚ Y ‚ Y ‚
‚ ‡ƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒ‰
‚ ‚ SUM ‚ SUM ‚ SUM ‚ SUM ‚
‡ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒ‰
‚BLOCK ‚ ‚ ‚ ‚ ‚
‡ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ‰ ‚ ‚ ‚ ‚
‚1 ‚ 6.00‚ 5.00‚ 4.00‚ 3.00‚
‡ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒ‰
‚2 ‚ 5.00‚ 4.00‚ 4.00‚ 5.00‚
‡ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒ‰
‚3 ‚ 4.00‚ 5.00‚ 4.00‚ 5.00‚
Šƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ‹ƒƒƒƒƒƒƒƒƒƒƒƒ‹ƒƒƒƒƒƒƒƒƒƒƒƒ‹ƒƒƒƒƒƒƒƒƒƒƒƒ‹ƒƒƒƒƒƒƒƒƒƒƒƒŒ
Randomized Complete Block Design
General Linear Models Procedure
Class Level Information
Class Levels Values
BLOCK 3 1 2 3
TX 4 1 2 3 4
Number of observations in data set = 12
General Linear Models Procedure
Dependent Variable: Y
Source DF Sum of Squares Mean Square F Value Pr > F
Model 5 1.66666667 0.33333333 0.38 0.8494
Error 6 5.33333333 0.88888889
Corrected Total 11 7.00000000
R-Square C.V. Root MSE Y Mean
0.238095 20.95131 0.94280904 4.50000000
Source DF Type I SS Mean Square F Value Pr > F
BLOCK 2 0.00000000 0.00000000 0.00 1.0000
TX 3 1.66666667 0.55555556 0.63 0.6246
Source DF Type III SS Mean Square F Value Pr > F
BLOCK 2 0.00000000 0.00000000 0.00 1.0000
TX 3 1.66666667 0.55555556 0.63 0.6246
Randomized Complete Block Design
General Linear Models Procedure
Least Squares Means
TX Y Pr > |T| H0: LSMEAN(i)=LSMEAN(j)
LSMEAN i/j 1 2 3 4
1 5.00000000 1 . 0.6801 0.2416 0.4198
2 4.66666667 2 0.6801 . 0.4198 0.6801
3 4.00000000 3 0.2416 0.4198 . 0.6801
4 4.33333333 4 0.4198 0.6801 0.6801 .
NOTE: To ensure overall protection level, only probabilities associated with pre-planned
comparisons should be used.
Randomized Complete Block Design
General Linear Models Procedure
Tukey's Studentized Range (HSD) Test for variable: Y
NOTE: This test controls the type I experimentwise error rate.
Alpha= 0.05 Confidence= 0.95 df= 6 MSE= 0.888889
Critical Value of Studentized Range= 4.896
Minimum Significant Difference= 2.6648
Comparisons significant at the 0.05 level are indicated by '***'.
Simultaneous Simultaneous
Lower Difference Upper
TX Confidence Between Confidence
Comparison Limit Means Limit
1 - 2 -2.3315 0.3333 2.9982
1 - 4 -1.9982 0.6667 3.3315
1 - 3 -1.6648 1.0000 3.6648
2 - 1 -2.9982 -0.3333 2.3315
2 - 4 -2.3315 0.3333 2.9982
2 - 3 -1.9982 0.6667 3.3315
4 - 1 -3.3315 -0.6667 1.9982
4 - 2 -2.9982 -0.3333 2.3315
4 - 3 -2.3315 0.3333 2.9982
3 - 1 -3.6648 -1.0000 1.6648
3 - 2 -3.3315 -0.6667 1.9982
3 - 4 -2.9982 -0.3333 2.3315
Since the F value for TX is not significant, we conclude that there is no difference
between the means of different treatments.
HW problems for ch18 are #7,9,10
In order to read the data set in #7, follow this data step
Data ch1807;
input genotype $ @@;
do site=1 to 5;
input bodysize @@;
output;
end;
cards;
hom 29.87 28.16 32.08 30.84 29.44
het 32.51 30.82 34.17 33.46 32.99
wld 35.76 33.14 36.29 34.95 35.89
run;
For #10, use the following data step.
Data ch1810;
input obs school $ grade $ score;
cards;
.......as shown in the book...
run;
