EXAMPLE 12 Demonstrating how RBd works

Demonstrate how randomized block design works, using the results from Examples 10 and 11.

Solution

Figure 30 from Example 10 provides us with

The RBD from Example 11 (Figure 32) separates into the sum of the following two quantities:

The new sum of squares error is small enough that the resulting new statistic is large, leading to a rejection of the null hypothesis.

Let and represent the number of treatments and the number of blocks, respectively. Then, the ANOVA table for the randomized block design is as shown in Table 8.

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Table 12.33: Table 8 ANOVA table for randomized block design
Source Sum of
squares		 Degrees of
freedom		 Mean square
Treatments SSTR
Blocks SSB
Error SSE
Total SST

Note the following facts about the ANOVA table for randomized block design:

  • Notice that SSTR, its degrees of freedom , and MSTR are all the same quantities as in the one-way ANOVA table (Table 3) on page 672.
  • is denoted simply as SSE.
  • Quantities in the “Mean square” column equal the ratio of the quantities in the “Sum of squares” column divided by their respective degrees of freedom.
  • We have , and the three degrees of freedom values sum to .
  • We are not interested in the blocks and thus the mean square blocks MSB, so there is no statistic for blocks.
  • In one-way ANOVA, the degrees of freedom for is . In RBD, this error degrees of freedom is partitioned into the degrees of freedom for SSB, , and the degrees of freedom for the new SSE, . An exercise in this section asks the student to show that these two degrees of freedom sum to .

NOW YOU CAN DO

Exercises 11–13.