# Analysis of Variance: Randomized Block Design

Randomized blocked design is used when a researcher wants to compare treatment means. What is unique to this research design is that the experiment is divided into two or more mini-experiments.

The reason behind this is to reduce the variation within-treatments so that it is easier to find differences between means.  Another unique characteristic of randomized block design is that since there is more than one experiment  happening at the same time, there will be more than one set of hypotheses to consider. There will be a set of hypotheses for the treatment groups and also for the block groups. The block groups are the several subpopulations with the sample. Below are the assumptions

• Samples are randomly selected
• Populations are homogeneous
• Populations are normally distributed
• Populations covariances are equal
•  Covariance is a measure of the commonality that two variables deviate from their expected values. If two variable deviates in similar ways the covariance will be high and vice versa. The standardized version of covariance is correlation.

Looking at equations and doing this by hand is tough. It is better to use SPSS or excel to calculate results. We are going to look at an example and see an application of randomized block design.

A professor wants to see if “time of day” affects his students score on a quiz. He randomly divides his stat class into five groups and has them take the quiz at one of four times during the day.  Below are the results
Time Period/Treatment
Section    8-9                10-11                11-12                1-2
1                  25                      22                        20                     25
2                  28                      24                        29                     23
3                  30                      25                        25                     27
4                  24                      27                        28                     25
5                  21                      28                        30                     24

The treatment groups here are the time periods. The are along the time and are 8-9, 10-11, 11-12, 1-2. The block groups are along the left-hand side and the are section 1, 2, 3, 4, 5. The block groups are the 5 different experimental groups of the larger population of the statistics class. What is happening here is that all members from all groups all took the quiz at one of the four times. For example, members from section one took the quiz at 8-9, 10-11, 11-12, and 1-2. The same for group 2 and so forth.  By having five different groups take the quiz at each of the time periods it should hopefully improve the accuracy of the results. It is like sampling a population five times instead of one time.

In addition, by having four different time periods, we can hopefully see much more clearly if the time period makes a difference. We have four different time periods instead of two or three. Below are the steps for solving this problem.

Step 1: State hypotheses
For Time periods
Null hypothesis: There is no difference in the means between time periods
Alternative hypothesis: There is a difference in the means between time periods
For Blocks
Null hypothesis: There is no difference in the means among the sections of students
Alternative hypothesis: There is difference in the means among the sections of students

Step 2: Significance level
are alpha is set to .05

Step 3: Critical value of F
This is done by the computer and it indicates that the F critical for the treatment (time periods) is 3.49 and the F critical for the blocks (section of students) is 3.26. There are two F criticals because there are two sets of hypotheses, one for the time periods and one for the students.

Step 4: Calculate
The computed F-value for treatment (time periods) is 0.25
The computed F-value for the blocks (section of students) is 0.89

Step 5: Decision
Since the F-value of the treatment (time periods) is 0.25 is less than F critical of 3.49 at an alpha of .05 we do not reject the null hypothesis

Since the F-value of the blocks (section of students) is 0.89 is less than F critical of 3.26 at an alpha of .05 we do not reject the null hypothesis

Step 6: Conclusion
Treatment (Time period)
Since we did not reject the null hypothesis, we can conclude that there is no evidence that time of day affects the quiz scores.

Blocks (Section of Student)
Since we did not reject the null hypothesis, we can conclude that there is no evidence that group affects the quiz scores.

From this, we know that time of day and the group a student belongs to does not matter. If the time of day mattered it might have been due to a host of factors such as early morning or late afternoon. For the groups, the difference could be identified by how they did on individual items. Maybe they struggled with finding the means of question 3.

Remember in this example there was no difference. The ideas above are for determining why there was a difference if that had happened.

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