🙇‍♂️Randomised Block Design (RBD)

A field trial comparing eight rice varieties faces a problem: soil fertility decreases steadily from the north to the south of the plot. CRD would mix this fertility gradient into the error term, obscuring real varietal differences. By dividing the field into blocks perpendicular to the gradient — each block containing all eight varieties — the Randomised Block Design separates soil variation from treatment effects, giving far more precise results.

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• We have seen that in a completely randomized design no local control measure was adopted excepting that the experimental units should be homogeneous.
• Usually, when experiments require a large number of experimental units, completely randomized designs cannot ensure precision of the estimates of treatment effects.

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• In agricultural field experiments, usually the experimental materials are not homogeneous. Soil fertility, moisture content, slope, and other field conditions naturally vary across the experimental area. In such situations the principle of local control is adopted and the experimental material is grouped into homogeneous sub groups. The subgroup is commonly termed as block. Since each block will consist the entire set of treatments a block is equivalent to a replication. This is a critical concept — in RBD, each block contains one replicate of every treatment, so the number of blocks equals the number of replications.

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• The blocks are formed with units having common characteristics which may influence the response under study.
• In agricultural field experiments the soil fertility is an important character that influences the crop responses. The uniformity trial is used to identify the soil fertility of a field. A uniformity trial involves growing a single crop uniformly across the entire field and then mapping the yield variation to understand the fertility gradient — the direction in which soil fertility changes.
• If the fertility gradient is found to run in one direction (say from north to south) then the blocks are formed in the opposite direction (from east to west). By arranging blocks perpendicular to the fertility gradient, each block captures similar levels of fertility, making plots within a block more uniform.

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• If the number of experimental units within each group is same as the number of treatments and if every treatment appears precisely once in each group, then such an arrangement is called a randomized block design.

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Layout

• Let us consider 5 treatments A, B, C, D and E each replicated 4 times. We divide the whole experimental area into 4 relatively homogeneous blocks and each block into 5 plots. Treatments are then allocated at random to the plots of a block, fresh randomization being done for each block. Fresh randomization for each block means we generate a new random arrangement for every block independently, not using the same order throughout.
• A particular layout as follows:

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Statistical Analysis

• The results from RBD can be arranged in two way table according to the replications (blocks) and treatments, there will be ‘rk’ observations in total. The two-way table structure is what makes RBD a two-way classification — data is organized both by treatments and by blocks. The data can be arranged in the following table.

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Mathematical Model

y_ij = μ + α_i + β_j + ξ_ij

• i = 1, 2,… k
• j = 1, 2,… r
• Where,
  • y_ij is the j^th replication of the i^th treatment
  • μ = general mean effect — the overall average across all treatments and blocks.
  • α_i = the effect due to i^th treatment — the deviation of the i-th treatment mean from the overall mean.
  • β_i = the effect due to j^th block — the deviation of the j-th block mean from the overall mean. By including this term, the block-to-block variation is separated from the error, thereby reducing it.
  • ξ_ij = error effect — the remaining random variation after accounting for treatments and blocks.

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• Source of Variability:
  • Due to treatment
  • Block
  • Error

In RBD, the total variation is partitioned into these three components. The key advantage over CRD is that the block variation is identified and removed from the error, making treatment comparisons more precise.

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Null hypothesis:

• H₀₁: There is no significant difference between the treatment effects.

  i.e. α₁ = α₂ = … = α_k

• H₀₂: There is no significant difference between the block effects

  i.e. β₁ = β₂= … = β_r

Note that while we test both hypotheses, the primary interest is usually in H₀₁ (treatment effects). The block hypothesis H₀₂ is of secondary interest — a significant block effect simply confirms that the blocking was effective and worthwhile.

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• The null hypothesis can be verified by applying the ANOVA procedure. The different steps are in the analysis of data are:

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Advantages and disadvantages of RBD

• Two-way classification (treatment wise and replication/block wise) and one-way control (local control). This means the data is organized in two dimensions (treatments and blocks), and error is controlled in one direction through blocking.
• The principle advantage of RBD is that it increases the precision of the experiment. This is due to the reduction of experimental error by adoption of local control.
• The amount of information obtained in RBD is more as compared to CRD. Hence, RBD is more efficient than CRD.

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• Flexibility is another advantage of RBD. Any number of replications can be included in RBD. If large number of homogeneous units are available, large number of treatments can be included in this design.
• Since the layout of RBD involves equal replication of treatments, statistical analysis is simple. Even when some observations are missing of certain treatments, the data can be analysed by the use of missing plot technique. This technique, also developed by R.A. Fisher, allows the researcher to estimate missing values and proceed with the analysis, although with a slight loss of precision.

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• When the number of treatments is increased, the block size will increase. If the block size is large it may be difficult to maintain homogeneity within blocks. Consequently, the experimental error will be increased. Hence, RBD may not be suitable for large number of treatments. But for this disadvantage, the RBD is a versatile design. It is the most frequently used design in agricultural experiments. This makes RBD the workhorse of agricultural research — used more often than any other design.

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• The optimum block size in field experiments is 21 plots. i.e. we cannot compare treatments which are > 21 in RBD to preserve homogeneity of plots within a block. It is used up to 20 treatments without appreciable loss of efficiency. Beyond 20 treatments, researchers should consider incomplete block designs like Balanced Incomplete Block Design (BIBD) or lattice designs.
• This design appropriate when the fertility gradient in the field is in one direction only. If the fertility varies in two directions simultaneously, Latin Square Design (LSD) would be more appropriate.

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• In RBD the most number of blocks must be equal to the number of replications fixed for coach treatment.
• The number of plots in each block should be equal to the number of treatments.
• It provides more accurate results than CRD due to formation of homogenous blocks.
• This design utilizes all the three basic principles of field experimentation. Unlike CRD (which uses only randomization and replication), RBD employs replication, randomization, and local control together.
• Error degree of freedom in RBD: (r-1) x (k-1). Where r = number of replications (blocks) and k = number of treatments. For example, with 4 blocks and 5 treatments, error d.f. = (4-1) x (5-1) = 3 x 4 = 12.

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Summary Cheat Sheet