6.3 - Restriction on Randomization: RCBD

The Randomized Complete Block Design (RCBD) Section

  See Text Chapter 21

In using the CRD we have to assume that the positions on the bench are equivalent. In reality, this is rarely the case, even when investigators try to minimize any differences. In the greenhouse example, we might expect that there is some micro-environmental variation due to the Glass wall on one end, and the Open walkway at the other end of the bench.

A powerful alternative to the CRD is to restrict the randomization process to form blocks. Blocks are usually set up at right angles to suspected gradients in variation. Within a block we want all the treatment levels or treatment combinations to be represented. With 6 replications, we then would have 6 blocks of 4 positions.

To assign treatments to experimental units here using our original data (Lesson1 Data), we work within each block separately. For Block 1, The following assignment is made using Minitab: Calc > Random Data > Sample From Columns.

sample from columns diablog box in Minitab

Here we specify 4 rows - the treatment levels, 4 to fill a complete block.

We get:

Minitab worksheet

Then, for each block, repeat the process. The key element is that each treatment level or treatment combination appears in each block (forming complete blocks), and were assigned at random within each block.

Fertilizer 1 (Blue) Fertilizer 2 (Red) Fertilizer 3 (Black) No Fertilizer (White) Wall Open Walkway 4 3 2 1 5 6 7 8 9 1 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Block I Block II Block III Block IV Block V Block VI
 

Blocks are usually treated as random effects, as they would represent the population of all possible blocks. In other words, we are usually not concerned with mean comparisons among specific blocks, but are using blocks to represent the population of all possible blocks.

The variation among blocks gets partitioned out of the Error MS of the CRD, and can yield a smaller MSE for testing hypotheses about treatments.

Here is Dr. Shumway stepping through this experimental design in the greenhouse.

Now, let's take a look at the results that were obtained for this experiment.

For the Greenhouse single factor example, using the SAS code below, consider a modified version of our original greenhouse dataset (RCBD Data) with blocks identified:

data RCBD_oneway;
input block Fert $ Height;
datalines;
1      Control      19.5
2      Control      20.5
3      Control      21
4      Control      21
5      Control      21.5
6      Control      22.5
1      F1      25
2      F1      27.5
3      F1      28
4      F1      28.6
5      F1      30.5
6      F1      32
1      F2      22.5
2      F2      25.2
3      F2      26
4      F2      26.5
5      F2      27
6      F2      28
1      F3      27.5
2      F3      28
3      F3      29.2
4      F3      29.5
5      F3      30
6      F3      31
;

/* CRD for comparison */
proc mixed data=RCBD_oneway method=type3;
class fert;
model height=fert;
run;

/* RCBD */
proc mixed data=RCBD_oneway method=type3;
class block fert;
model height=fert;
random block;
run;

Output:

block Fert _TYPE_ _FREQ_ mean se
-   0 24 26.1667 0.75238
- Control 1 6 21.0000 0.40825
- F1 1 6 28.6000 0.99499
- F2 1 6 25.8667 0.77531
- F3 1 6 29.2000 0.52599
1   2 4 23.6250 1.71239
2   2 4 25.3000 1.71221
3   2 4 26.0500 1.80808
4   2 4 26.4000 1.90657
5   2 4 27.2500 2.06660
6   2 4 28.3750 2.13478
1 Control 3 1 19.5000 -
1 F1 3 1 25.0000 -
1 F2 3 1 22.5000 -
1 F3 3 1 27.5000 -
2 Control 3 1 20.5000 -
2 F1 3 1 27.5000 -
2 F2 3 1 25.2000 -
2 F3 3 1 28.0000 -
3 Control 3 1 21.0000 -
3 F1 3 1 28.0000 -
3 F2 3 1 26.0000 -
3 F3 3 1 29.2000 -
4 Control 3 1 21.0000 -
4 F1 3 1 28.6000 -
4 F2 3 1 26.5000 -
4 F3 3 1 29.5000 -
5 Control 3 1 21.5000 -
5 F1 3 1 30.5000 -
5 F2 3 1 27.0000 -
5 F3 3 1 30.0000 -
6 Control 3 1 22.5000 -
6 F1 3 1 32.0000 -
6 F2 3 1 28.0000 -
6 F3 3 1 31.0000 -
CRD
Type 3 Analysis of Variance
Source DF Sum of Squares Mean Square Expected Mean Square F Value Pr > F
Fert 3 251.440000 83.813333 Var(Residual) + Q(Fert) 27.46 <.0001
Residual 20 61.033333 3.051667 Var(Residual)    
RCBD
Type 3 Analysis of Variance
Source DF Sum of Squares Mean Square Expected Mean Square F Value Pr > F
Fert 3 251.440000 83.813333 Var(Residual) + Q(Fert) 162.96 <.0001
block 5 53.318333 10.663667 Var(Residual) + 4 Var(block) 20.73 <.0001
Residual 15 7.715000 0.514333 Var(Residual)    

Notice that the F statistic for the treatment has increased from 27.46 to 162.96. Although not affecting significance in this example, this sort of improvement in the F statistic often means the difference between rejecting or failing to reject a Null Hypothesis. The reduction in MSE by using blocks is seen as the partition in SSE for the CRD (61.033) into SSBlock (53.32) + SSE (7.715). The potential reduction in SSE by blocking is to some degree offset by losing degrees of freedom for the blocks, but more often than not, is worth it in terms of the improvement in the calculated F statistic.