Randomized Block Designs

Why Group Units Before Random Assignment?

Suppose an agricultural researcher wants to compare two fertilizers using seedlings. Seedlings that are already larger may grow more, regardless of which fertilizer they receive. If the researcher assigns all seedlings to treatments completely at random, the larger seedlings could, by chance, be unevenly distributed between the two groups. Random assignment protects the comparison from systematic selection, but it does not guarantee that every chance imbalance will be small.

A randomized block design addresses a known source of variation by first grouping experimental units that are similar in a relevant way, then randomly assigning treatments within each group. For the seedling study, initial size could define the groups. The researcher still uses random assignment, but comparisons can now be made among seedlings that began in similar size ranges.

The block characteristic is not itself the treatment. It is a feature of the units, such as initial size, production shift, or baseline severity, that is known before treatment assignment and is expected to influence the outcome. A block should contain units similar to one another with respect to that characteristic. Random assignment within blocks determines which units receive which treatments.

Planning a Randomized Block Design

A useful design starts by stating the experimental question, naming the units that will receive treatments, and identifying the response to be measured. Then the researcher chooses a relevant blocking variable. That variable should be measured before treatments begin and should plausibly explain variation in the response. Blocking by an irrelevant characteristic adds complexity without addressing the source of variability that matters.

After forming the blocks, the researcher decides how many units in each block will receive each treatment. Random assignment must take place separately within each block. The assignment process might use a random-number generator or a lottery; the particular method matters less than using a genuine random mechanism and following it as planned.

A valid plan needs enough units in each block to assign the treatments being compared. If a block contains no units assigned to one of the treatments, it cannot contribute a within-block comparison for that treatment. The design also relies on units receiving their assigned treatments as planned and on the response being measured consistently. Random assignment supports a causal comparison of the treatments for the experimental units, while the blocking variable helps organize that comparison.

What Blocking Accomplishes

Blocking makes the treatment comparison within groups of units that are similar on the chosen characteristic. If that characteristic strongly affects the response, comparisons within blocks can be less affected by variation between those groups. The design does not remove all variability: units in the same block can still differ, and other sources of response variation remain.

The basic logic can be shown with a simple model for two treatments. Suppose the response for unit \(i\) in block \(j\), receiving treatment \(t\), can be written as a common level, a block contribution, a treatment contribution, and random error:

Here, \(\mu\) is a common level, \(\beta_j\) is the contribution associated with block \(j\), \(\tau_t\) is the contribution associated with treatment \(t\), and \(\varepsilon_{ijt}\) is the remaining unit-level variation. To see the benefit, compare the average response under Treatment A with the average under Treatment B within the same block. If there are \(r\) units per treatment in that block, the difference is:

The block contribution \(\beta_j\) cancels because both treatments are compared within the same block. This cancellation is the central reason to block on a characteristic that creates differences between groups. The algebra assumes that the treatment contribution is the same across blocks, as in the model above. In real applications, treatment effects may differ by block; blocking is not a guarantee that every treatment effect is identical across all kinds of units.

If the remaining errors are independent, have mean zero, and have variance \(\sigma^2\), then the error in each treatment average for one block has variance \(\sigma^2/r\). The two treatment groups contain different units, so their errors are independent; consequently, the variance of the within-block difference is:

This calculation is not a guarantee that a particular blocked experiment will be more precise. It shows why, under the stated model, a shared block contribution does not add variation to the within-block treatment contrast. Blocking is most useful when the chosen characteristic predicts response differences between blocks and units can still be randomly assigned to each treatment within them.

Interpreting Results Across Blocks

After the experiment, a treatment comparison should respect the design. One simple approach is to calculate a treatment difference within every block, then combine those differences using weights that reflect the intended comparison. If blocks have equal sizes and equal allocations, an unweighted average of block differences may be appropriate for an average across blocks. If block sizes or allocations differ, the weighting needs to be chosen deliberately; a simple average of block averages may not represent the intended average effect across all units.

For example, in a two-treatment study, let \(d_j\) be the mean response for Treatment A minus the mean response for Treatment B in block \(j\). With three equally sized blocks, an equally weighted summary is:

Choosing Blocks Carefully

A useful blocking variable has two features: it is available before treatment assignment, and it is expected to predict the response. The aim is to group similar units with respect to a known source of outcome variation, not to create as many groups as possible. Too many narrow blocks can leave too few units in each one to assign treatments reliably.

The design must also be practical. A blocking variable should be measured consistently, and the categories should be defined before outcomes are observed. If units cannot be assigned to all treatment conditions within a block, the researcher should reconsider the grouping plan or the available treatments. A well-designed block is not a substitute for random assignment: the units still need random treatment allocation within each block.

Blocking also differs from controlling a variable by giving a treatment to one group and withholding it from another. That would risk making treatment assignment depend systematically on the blocking characteristic. In a randomized block design, the characteristic organizes random assignment; it does not determine which treatment a unit receives.

Common Mistake

• Wrong reasoning: “The smaller seedlings should get Fertilizer A and the larger seedlings should get Fertilizer B, because that balances the groups.” This is not random assignment within blocks. Treatment and initial size become linked, so an outcome difference may reflect either factor.
• Correction: Form size blocks first, then use random assignment within each block so every fertilizer is represented among small and large seedlings. Blocking handles the known size differences while random assignment protects the treatment comparison.