Randomized Block Design

The Definition of Randomized Block Design

When data is grouped based on measurable and known unwanted variables, you say the data has been blocked. This is carried out to prevent undesirable factors from reducing the accuracy of an experiment.

When carrying out an experiment or survey, you should try to reduce errors that may be contributed by various factors. A factor may be known and controllable, so you block (group) the samples based on this factor in a bid to reduce variability caused by this factor. The end goal of this process is to minimize the differences between components in a blocked group as compared to the differences between components of the entire sample. This would help you get more accurate estimates from each block, since the variability of members of each group is low.

In the above example, Femi identified that the floor texture can make a difference. But Femi is interested in which brush is better, so he decided to make three blocks for his experiment: the kitchen, the bedroom, and the sitting room. The factor that led Femi to the decision of making blocks is often regarded as a nuisance factor.

Nuisance factors are not the same thing as lurking variables.

A lurking variable that needs to be accounted for in medical trials is the placebo effect, where people believe the medicine will have an effect so they experience an effect, even if what they are actually getting is a sugar pill instead of real medical treatment.

Let's look at two illustrations of a randomized block design to help clarify how a randomized block design would be constructed.

Fig. 1: Blocking in a randomized block design

From the above figure, you can see how Femi has grouped the experiment into three sections. This is an important idea about the randomized block design.

Randomization in a randomized block design

From the above figure, after blocking into groups, Femi randomly samples each group for the test. After this stage, the analysis of variance is carried out.

Randomized Block Design vs Completely Randomized Design

A completely randomized design is a process of randomly picking samples for an experiment so that all randomly selected items are treated without segregation (grouping). This method is susceptible to an error by chance, since common characteristics are not considered initially, which should minimize variability if they were put in groups. This variability is minimized by the randomized block design through grouping so that a balance is forced between study groups.

You can better understand the difference between a randomized block design versus a completely randomized design with an example.

So which is better, blocking or randomization?

Is Blocking Better Than Randomization or Not?

The randomized block design is more beneficial than complete randomization because it reduces error by creating groups that contain items that are much more similar in comparison to the entire samples.

However, blocking would be preferred only when the sample size is not too large and when the nuisance factor(s) are not too many. When you deal with large samples, there is a higher tendency of numerous nuisance factors, which would require you to increase the grouping as well. The principle is that the more grouping you do, the smaller the sample size in each group. Therefore, when large sample sizes are involved or there are many nuisance factors, then you should approach such cases with a completely randomized design.

Furthermore, as mentioned earlier, when the blocking variable is unknown you should rely on a completely randomized design.

Randomized Block Design vs Matched Pairs Design

A matched pair design deals with the grouping of samples in twos (pairs) based on confounding characteristics (such as age, gender, status, etc.), and members of each pair are randomly assigned treatment conditions. Randomized block designs differ from matched pairs since its there can be more than two groupings. However, when there are just two groups in a randomized block design, then it may appear to be similar to a matched pair design.

Moreover, both the randomized block and matched pair designs are best applied to only small sample sizes.

So what are the advantages of a randomized block design?

What are the Advantages of a Randomized Block Design?

A primary benefit of the randomized block design is the creation of groups which increases similarities between members in the block as compared to the wide variation which may occur when each member is compared with the entire data set. This attribute is very advantageous because:

• It reduces error.

• It increases the statistical reliability of a study.

• It remains a better approach to analyzing smaller sample sizes.

Let's look closer at the model for a randomized block design.

The Statistical Model for a Randomized Block Design

The statistical model for a randomized block design for one blocked nuisance factor is given by:

\[y_{ij}=µ+T_1+B_j+E_{ij}\]

where:

• \(y_{ij}\) is the observation value for treatments in \(j\) and blocks in \(i\);

• \(μ\) is the grand mean;

• \(T_j\) is the \(j\)th treatment effect;

• \(B_i\) is the \(i\)th blocking effect; and

• \(E_{ij}\) is the random error.

The above formula is equivalent to that of ANOVA. You can thus use:

\[SS_T=SS_t+SS_b+SS_e\]

where:

• \(SS_T\) is the total sum of squares;

• \(SS_t\) is the sum of squares of from treatments;

• \(SS_b\) is the sum of squares from blocking; and

• \(SS_e\) is the sum of squares from the error.

The total sum of squares is calculated using:

\[SS_T=\sum_{i=1}^{\alpha} \sum_{j=1}^{\beta}(y_{ij}-\mu)^2\]

The sum of squares from treatments is calculated using:

\[SS_t=\beta \sum_{j=1}^{\alpha}(\bar{y}_{.j}-\mu)^2\]

The sum of squares from blocking is calculated using:

\[SS_b=\alpha \sum_{i=1}^{\beta}(\bar{y}_{i.}-\mu)^2\]

where:

• \(\alpha\) is the number of treatments;

• \(\beta\) is the number of blocks;

• \(\bar{y}_{.j}\) is the mean of the \(j\)th treatment;

• \(\bar{y}_{i.}\) is the mean of the \(i\)th blocking; and

• the total sample size is a product of the number of treatments and blocks, which is \(\alpha \beta\).

The sum of squares of error can be calculated using:

\[SS_e=SS_T-SS_t-SS_b\]

This becomes:

\[SS_e=\sum_{i=1}^{\alpha} \sum_{j=1}^{\beta}(y_{ij}-\mu)^2- \beta \sum_{j=1}^{\alpha}(\bar{y}_{.j}-\mu)^2 -\alpha \sum_{i=1}^{\beta}(\bar{y}_{i.}-\mu)^2\]

However, the value of the test static is obtained by dividing the mean square values of the treatment by that of the error. This is mathematically expressed as:

\[F=\frac{M_t}{M_e}\]

where:

• \(F\) is the test static value.

• \(M_t\) is the mean square value of treatment, which is equivalent to the quotient of the sum of squares from treatments and its degree of freedom, this is expressed as:\[M_t=\frac{SS_t}{\alpha -1}\]

• \(M_e\) is the mean square value of error which is equivalent to the quotient of the sum of squares of error and its degree of freedom, this is expressed as:\[M_e=\frac{SS_e}{(\alpha -1)(\beta -1)}\]

The next section looks at an example to explain the application of these formulas.

Examples of Randomized Block Design

As mentioned at the end of the previous section, you shall have a clearer understanding of the randomized block design with its application in the illustration below.

Try out more examples on your own, while keeping in mind that randomized blocking is essentially ridding off the nuisance factors through blocking (grouping) before randomization. The goal is to create groups that are similar with less variability as compared to the whole samples. Moreover, if variability is more observable within blocks, this is an indication that blocking is not done properly or the nuisance factor is not very good a variable to block. Hoping you'll start blocking afterwards!