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RANDOMIZED COMPLETE BLOCK DESIGN
1 DESCRIPTION
The simple Randomized Complete Block Design (RCBD) is probably the most frequently used design. The experimental units are divided into homogeneous groups of material (called BLOCKs) each of which constitutes a single replication of the experiment.
At all stages during the experiment, the techniques applied within a block should be as uniform as possible, thus keeping experimental error within blocks as small as possible. Differences between blocks are permitted to be large, but are not of major concern in the analysis, since the comparisons of treatments and the computation of experimental error is within blocks. Blocking will be effective only if the error variance among units within blocks is smaller than the error variance over all units.
The division into blocks needs to be made only at those stages during the study where failure to do so would increase experimental error. For example, blocking may not be required until data are being collected or laboratory analyses are being conducted.
The key to successful blocking is to minimize the variance among units within blocks while allowing the variance among blocks to increase. Precision usually decreases as the number of experiment units (size) per block increases. Therefore block size should be kept as small as possible.
2 ADVANTAGES
2.1 Improves precision (relative to CRD)
2.1.1 reduces MSe, thus reducing the number of replications needed to achieve equal or better sensitivity or power
2.1.2 creates better treatment balance for individual experiments
2.2 Flexible
2.2.1 any number of treatments
2.2.2 any number of block replicates
2.2.3 extra replications for certain treatments may be included (2x, 3x,.. .)
2.3 There are corresponding non-parametric analyses available
2.4 Scope of inference is increased and block means provide an unbiased comparison of the differences among blocks
3.8 Requires some prior knowledge about variability of experimental units for successful blocking.
4 RANDOMIZATION
After the experimental units have been blocked, treatments are assigned at random within each block such that each treatment occurs once in every block for the simple RCBD. During the course of the experiment where the order of processing the material may make a difference, units are processed by block and in a completely random order within each block.
5 WHERE SHOULD THE RCB DESIGN BE USED?
5.1 Where sufficient knowledge exists about the heterogeneity of the experimental units to provide for efficient blocking.
5.1.1 field experiments
5.1.2 experiments with position, location or time effects
5.2 If there is more than one source of heterogeneity in the experimental material, sources may be confounded or multifactor blocking may be required.
5.3 When it is desirable to compare the treatments over a wide range of experimental material or under several different enviromental conditions (i.e. as in variety trials).
6 ANOVA SS (t = number of treatments, b = number of blocks, and bt is the total number of experimental units)
Each observation is identified as a Yij, where i is the treatment identification from i=1 to t, and j is the block identification from j= to b.
8 ANALYSIS OF VARIANCE
8.1 Expected MS Random Model (components in [] are confounded if both exist)
Sources of Components variation MS of variance F Assumptions
Block MSb [Fe^2 + Fbt^2 ] + tFb^2 MSb/MSe none
Treatment MSt [Fe^2 + Fbt^2 ] + bFt^2 MSt/MSe none
Error MSe [Fe^2 + Fbt^2 ]
8.2 Expected MS Fixed Model
Sources of Components variation (^) MS of variance F Assumptions
Block MSb Fe^2 + t (^2) b^2 MSb/MSe (^2) bt^2 = 0
Treatment MSt Fe^2 + b (^2) t^2 MSt/MSe (^2) bt^2 = 0
Error MSe [Fe^2 + (^2) bt^2 ]
8.3 Expected MS Mixed Model
Sources of Components variance MS of variance F Assumptions
Block MSb Fe^2 + tFb^2 MSb/MSe^ Fbt^2 = 0
Treatment MSt [Fe^2 + Fbt^2 ] + b (^2) t^2 MSt/MSe none
Error MSe [Fe^2 + Fbt^2 ]
9 STANDARD ERROR of the ESTIMATES
9.1 Standard error of treatment means (treatments fixed)
9.1.1 Fixed blocks
9.1.2 Random blocks, where Sb^2 is the block variance component
10.3 The log transformed data [LOG 10 (# DEAD)]
T r e a t m e n t Rep A B C D E Y 6 .j
1 1.38 1.83 1.48 2.07 2.16 1. 2 1.43 1.73 1.62 1.89 2.00 1. 3 1.58 2.21 1.94 1.85 2.31 1. 4 1.64 2.11 1.86 2.21 2.26 2. 5 1.45 2.13 2.03 2.33 2.40 2. _ Yi. 1.496 2.002 1.786 2.070 2. _ Y.. 1.
10.4 RCBD ANOVA sums of squares
_ Total SS = E(Yij - Y..)^2
= (1.38-1.916)^2 + (1.43-1.916)^2 +... + (1.43-1.916)^2
= 2.
_ _
Block SS = E(Y.j - Y..)^2
= (1.764-1.916)^2 + (1.734-1.916)^2 +... + (2.068-1.916)^2
=.
_ _
Trt SS = E(Yi. - Y..)^2
= (1.496-1.916)^2 + (2.002-1.916)^2 +... + (2.226-1.916)^2
= 1.
Error SS = E(Yij - Yi. - Y.j + Y..)^2
= Total SS - Block SS - Treatment SS
= 2.2970 - 0.4375 - 1.
=.
10.5 Analysis of variance for log number dead
Sum of Mean Source DF Squares Square F Value Prob <
Block 4 0.4375 0. Treatment 4 1.6026 0.4007 24.95 0.
Error 16 0.2569 0. Total 24 2.
