# Randomized Complete Block Design

In Randomized Complete Design (CRD), there is no restriction on the allocation of the treatments to experimental units. But in practical life there are situations where there is relatively large variability in the experimental material, it is possible to make blocks (in simpler sense groups) of the relatively homogeneous experimental material or units. The design applied in such situations is named as Randomized Complete Block Design (RCBD).

The Randomized Complete Block Design may be defined as the design in which the experimental material is divided into blocks/groups of homogeneous experimental units (experimental units have same characteristics) and each block/group contains a complete set of treatments which are assigned at random to the experimental units.

Actually, RCBD is a one restriction design, used to control a variable which is influence the response variable. The main aim of the restriction is to control the variable causing the variability in response. Efforts of blocking are done to create the situation of homogeneity within the block. Blocking is a source of variability. An example of blocking factor might be the gender of a patient (by blocking on gender), this is a source of variability controlled for, leading to greater accuracy. RCBD is a mixed model in which a factor is fixed and other is random. The main assumption of the design is that there is no contact between the treatment and block effect.

Randomized Complete Block design is said to be complete design because in this design the experimental units and number of treatments are equal. Each treatment occurs in each block.

The general model is defined as

$Y_{ij}=\mu+\eta_i+\xi_j+e_{ij}$

where $i=1,2,3\cdots, t$ and $j=1,2,\cdots, b$ with $t$ treatments and $b$ blocks. $\mu$ is the overall mean based on all observations, $\eta_i$ is the effect of the ith treatment response, $\xi$ is the effect of jth block and $e_{ij}$ is the corresponding error term which is assumed to be independent and normally distributed with mean zero and constant variance.

The main objective of blocking is to reduce the variability among experimental units within a block as much as possible and to maximize the variation among blocks; the design would not contribute to improving the precision in detecting treatment differences.

Randomized Complete Block Design Experimental Layout

Suppose there are $t$ treatments and $r$ blocks in a randomized complete block design, then each block contains homogeneous plots one of each treatment. An experimental layout for such a design using four treatments in three blocks be as follows.

Block 1 Block 2 Block 3
A B C
B C D
C D A
D A B

From RCBD layout we can see that

• The treatments are assigned at random within blocks of adjacent subjects and each of the treatment appears once in a block.
• The number of blocks represents the number of replications
• Any treatment can be adjacent to any other treatment, but not to the same treatment within the block.
• Variation in an experiment is controlled by accounting spatial effects. 