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Teach yourself statistics

Teach yourself statistics

# Randomized Block Designs

This lesson begins our discussion of randomized block experiments. The purpose of this lesson is to provide background knowledge that can help you decide whether a randomized block design is the right design for your study. Specifically, we will answer four questions:

• What is a blocking variable?
• What is blocking?
• What is a randomized block experiment?

We will explain how to analyze data from a randomized block experiment in the next lesson: Randomized Block Experiments: Data Analysis.

Note: The discussion in this lesson is confined to randomized block designs with independent groups. Randomized block designs with repeated measures involve some special issues, so we will discuss the repeated measures design in a future lesson.

## What is a Blocking Variable?

In a randomized block experiment, a good blocking variable has four distinguishing characteristics:

• It is included as a factor in the experiment.
• It is not of primary interest to the experimenter.
• It affects the dependent variable.
• It is unrelated to independent variables in the experiment.

A blocking variable is a potential nuisance variable - a source of undesired variation in the dependent variable. By explicitly including a blocking variable in an experiment, the experimenter can tease out nuisance effects and more clearly test treatment effects of interest.

Warning: If a blocking variable does not affect the dependent variable or if it is strongly related to an independent variable, a randomized block design may not be the best choice. Other designs may be more efficient.

## What is Blocking?

Blocking is the technique used in a randomized block experiment to sort experimental units into homogeneous groups, called blocks. The goal of blocking is to create blocks such that dependent variable scores are more similar within blocks than across blocks.

For example, consider an experiment designed to test the effect of different teaching methods on academic performance. In this experiment, IQ is a potential nuisance variable. That is, even though the experimenter is primarily interested in the effect of teaching methods, academic performance will also be affected by student IQ.

To control for the unwanted effects of IQ, we might include IQ as a blocking variable in a randomized block experiment. We would assign students to blocks, such that students within the same block have the same (or similar) IQ's. By holding IQ constant within blocks, we can attribute within-block differences in academic performance to differences in teaching methods, rather than to differences in IQ.

## What is a Randomized Block Experiment?

A randomized block experiment with independent groups is distinguished by the following attributes:

• The design has one or more factors (i.e., one or more independent variables), each with two or more levels.
• Treatment groups are defined by a unique combination of non-overlapping factor levels.
• Experimental units are randomly selected from a known population.
• Each experimental unit is assigned to one block, such that variability within blocks is less than variability between blocks.
• The number of experimental units within each block is equal to the number of treatment groups.
• Within each block, each experimental unit is randomly assigned to a different treatment group.
• Each experimental unit provides one dependent variable score.

The table below shows the layout for a typical randomized block experiment.

T1 T2 T3 T4
B1 X1,1 X1,2 X1,3 X1,4
B2 X2,1 X2,2 X2,3 X2,4
B3 X3,1 X3,2 X3,3 X3,4
B4 X4,1 X4,2 X4,3 X4,4
B5 X5,1 X5,2 X5,3 X5,4

In this experiment, there are five blocks ( Bi ) and four treatment levels ( Tj ). Dependent variable scores are represented by X i, j , where X i, j is the score for the subject in block i who received treatment j.

With respect to analysis of variance, a randomized block experiment with independent groups has advantages and disadvantages. Advantages include the following:

• With an effective blocking variable - a blocking variable that is strongly related to the dependent variable but not related to the independent variable(s) - the design can provide more precision than other independent groups designs of comparable size.
• The design works with any number of treatments and blocking variables.

• When the experiment has many treatment levels, it can be hard to form homogeneous blocks.
• With an ineffective blocking variable - a blocking variable that is weakly related to the dependent variable or strongly related to one or more independent variables - the design may provide less precision than other independent groups designs of comparable size.
• The design assumes zero interaction between blocks and treatments. If an interaction exists, tests of treatment effects may be biased.

Problem 1

Which, if any, of the following attributes does not describe a good blocking variable?

(A) It is included as a factor in the experiment.
(B) It is not of primary interest to the experimenter.
(C) It affects the dependent variable.
(D) It affects the independent variable.
(E) All of the attributes describe a good blocking variable.

Solution

A good blocking variable is not related to an independent variable. When the blocking variable and treatment variable are related, tests of treatment effects may be biased.

Problem 2

Why would an experimenter choose to use a randomized block design?

(A) To test the effect of a blocking variable on a dependent variable.
(B) To assess the interaction between a blocking variable and an independent variable.
(C) To control unwanted effects of a suspected nuisance variable.
(D) None of the above.
(E) All of the above.

Solution