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# The Difference Between the GRBD and the RCBD with by sja20118

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```									          The Difference Between the GRBD and the RCBD with Subsampling

There are important differences between a RCBD where each experimental unit is subsampled and a
block design with genuine replications of the treatments within each block (a GRBD). These differences
are important to recognize, since you can usually not tell from the data structure alone what the design
may be.
Consider the following data set on three treatments in three blocks. There are a total of 18 observations.

OBS       BLOCK     TX     N           Y

1         1         1    1      -1.09314
2         1         1    2      -0.46962
3         1         2    1      -2.41522
4         1         2    2      -1.65248
5         1         3    1      -5.04992
6         1         3    2      -4.01389
7         2         1    1      -0.30574
8         2         1    2       0.53438
9         2         2    1      -2.62680
10         2         2    2      -0.23635
11         2         3    1      -3.39847
12         2         3    2      -5.22330
13         3         1    1       1.01157
14         3         1    2       1.24340
15         3         2    1      -1.00363
16         3         2    2      -0.36448
17         3         3    1      -2.70124
18         3         3    2      -1.72609

It is not obvious whether the variable N denotes 2 subsamples on each of 9 EUs (three per block) or 2
genuine replications of the treatments within each block (6 EUs per block). If you encounter data such as
these you must ask the right questions first:
• How was the design randomized?
• How many EUs per block?
• How many subsamples per EU?
By the way, I simulated these data with the SAS statements

data test;
do block = 1 to 3; do tx = 1 to 3; do n=1 to 2;
y = rannor(124) + block - 2*tx;
output;
end; end; end;
run;
options nocenter ls=60;
proc print; run;

Let's look at the two designs side-by-side.
GRBDa> é \$Ç , é \$Ç ? é #b                               RCBDa> é \$Ç , é \$b with 2 subsamples

Randomization/Size                                      Randomization/Size
Here we have 6 EUs per block and the treatments         Just like in a regular RCBD we have 3 EUs per
in each block are randomized as if the block were       block and the treatments in each block are
a CRDa> é \$Ç ? é #b.                                    randomized as if the block were a
CRDa> é \$Ç ? é "b. Then take two subsamples
from each EU

Layout                                                  Layout
2       1                                       2    1        3
1           3                       2        3

3    2        1
2       3        1       3      3        1

1   2         3
2           1        1       3      2        3

Linear Model (formula (8.8), p. 265)                    Linear Model (formula (8.9), p. 265)

C345 é .  73  34  a73b34  /345                    C345 é .  73  34  /34  .345

3 é "Ç ÆÆÆÇ > é \$
3 é "Ç ÆÆÆÇ > é \$
4 é "Ç ÆÆÆÇ , é \$
4 é "Ç ÆÆÆÇ , é \$
5 é "Ç ÆÆÆÇ 8 é #
5 é "Ç ÆÆÆÇ ? é #                                                    #
#                                         /34  a!Ç 5/ b
/345  a!Ç 5/ b                                                        #
.345  a!Ç 5. b

Noteworthy                                              Noteworthy
• Only one error term (/345 é experimental error)       • Two error terms (/34 é EE, .345 é OE)
• Block*Tx interaction                                  • No Block*Tx interaction in model

ANOVA                                                   ANOVA
Source            df                                    Source     df
Block             ,"              #                    Block      ,"              #
Tx                >"              #                    Tx         >"              #
Block ô Tx        a,  "ba>  "b   %                    EError     a,  "ba>  "b   %
EError            ,>a?  "b        *                    OError     ,>a?  "b        *
Total             ,>?  "          "&                   Total      ,>?  "          "&

Noteworthy                                              Noteworthy
• Block ô Tx interaction is testable                    • Block ô Tx interaction is not permissible,
• The df that go into OError in the RCBD w/             otherwise there is no term for experimental error
subsampling are assigned to EError here                 • The df that go into EError in the GRBD are
• Both designs have same number of total df and         assigned to OError here.
the same number of sources of variability               • If blocks and treatments interact, this design is
biased
Hypothesis Tests                                         Hypothesis Tests

L! : no treatment effects                                L! : no treatment effects
J9,= é QW aX Bb¶QW aII b                                 J9,= é QW aX Bb¶QW aII b
CV: J!Ç>"Ç,>a?"b                                       CV: J!Ç>"Ça,"ba>"b
Here: J!Æ!&Ç#Ç*                                          Here: J!Æ!&Ç#Ç%

L! : no Block ô Tx interaction                           L! : no Block ô Tx interaction
J9,= é QW aBlock ô X Bb¶QW aII b                                          No test available
CV: J!Ça,"ba>"bÇ,>a?"b                                  Design only valid in the absence of Block ô Tx
Here: J!Æ!&Ç%Ç*

Standard Errors of Treatment Means                       Standard Errors of Treatment Means

/=/ of C3Æ é °QW aII b¶a,ù?b                             /=/ of C3Æ é °QW aII b¶a,ù8b

LSD                                                      LSD
#
#
LSD! é > ! Ç,>a?"b ±QW aII b ,?                         LSD! é > ! Ça,"ba>"b ±QW aII b ,8
#
#

In SAS                                                   In SAS
proc glm data=whatever;                                  proc glm data=whatever;
class block tx;                                          class block tx;
model y = block tx block*tx;                             model y = block tx block*tx;
means tx / lsd;                                          test h=tx e=block*tx;
run; quit;                                                  means tx / lsd e=block*tx;
run; quit;

Noteworthy                                               Noteworthy
Both designs have the same class and model               block*tx term in RCBD(SS) is a random variable
statements, since both must generate the same            (EE) and must be used as the denominator in all
ANOVA df decomposition.                                  tests (e= option of test and means statement)

The key differences between the two designs are the ability to estimate the Block ô Tx interaction and the
degrees of freedom for experimental error. Assume that both designs yield the same estimate of
experimental error, QW aII b. Also assume that the number of subsamples a8b equals the number of
replicates within each block a?b in the GRBD. Which design has more power? The standard errors of the
treatment means will now be the same in the two designs, but because of the error degrees of freedom
we have
.0 IIKVFH é ,>a?  "b  .0 IIVG FH¸WW¹ é a,  "ba>  "b

The GRBD will be more powerful.

When am I going to choose a RCBD with subsampling, then?
1. If the assumption of the RCBD of no treatment ô block interaction is met.
2. If observational error (sampling variability within EU, measurement error) is substantial and eliminating
this source reduces QW aII bVG FH in comparison to QW aII bKVFH sufficiently to offset the smaller
experimental error degrees of freedom in the subsampling design.

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