ch04 - Chapter 4 Supplemental Text Material S4-1 Relative...

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Chapter 4 Supplemental Text Material S4-1. Relative Efficiency of the RCBD In Example 4-1 we illustrated the noise-reducing property of the randomized complete block design (RCBD). If we look at the portion of the total sum of squares not accounted for by treatments (302.14; see Table 4-4), about 63 percent (192.25) is the result of differences between blocks. Thus, if we had run a completely randomized design, the mean square for error MS E would have been much larger, and the resulting design would not have been as sensitive as the randomized block design. It is often helpful to estimate the relative efficiency of the RCBD compared to a completely randomized design (CRD). One way to define this relative efficiency is R df df df df b r b r r b = + + + + ( )( ) ( )( ) 1 3 3 1 2 2 σ σ where are the experimental error variances of the completely randomized and randomized block designs, respectively, and are the corresponding error degrees of freedom. This statistic may be viewed as the increase in replications that is required if a CRD is used as compared to a RCBD if the two designs are to have the same sensitivity. The ratio of degrees of freedom in R is an adjustment to reflect the different number of error degrees of freedom in the two designs. σ r 2 and σ b 2 b σ b 2 df df r and To compute the relative efficiency, we must have estimates of . We can use the mean square for error MS σ r 2 and E from the RCBD to estimate , and it may be shown [see Cochran and Cox (1957), pp. 112-114] that σ b 2 ± ( ) ( ) σ r Blocks E b MS b a MS ab 2 1 1 1 = + is an unbiased estimator of the error variance of a the CRD. To illustrate the procedure, consider the data in Example 4-1. Since MS E = 7.33, we have 2 ˆ 7.33 b σ = and 2 ( 1) ( 1) ˆ 1 (5)38.45 6(3)7.33 4(6) 1 14.10 Blocks E r b MS b a MS ab σ + = + = = Therefore our estimate of the relative efficiency of the RCBD in this example is
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