# One-factor Completely Randomized Design

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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. 
j1               b
a               Y                     a   b                   Y..
Y. j   Yij       Yi.  i.          Y..    Yij            Y..             N = ab
i 1              a                 i 1 j1                   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
j1

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 j1      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 j1

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
j1

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;

```
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