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 br r b = ++ () 13 31 2 2 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. r 2 and b 2 b 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 r 2 and E from the RCBD to estimate , and it may be shown [see Cochran and Cox (1957), pp. 112-114] that b 2 ± r Blocks E bM S b aM S ab 2 11 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 = and 2 (1 ) ) ˆ 1 (5)38.45 6(3)7.33 4(6) 1 14.10 Blocks E r S b S ab −+ = + = = Therefore our estimate of the relative efficiency of the RCBD in this example is
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2 2 (1 ) (3 ) ) ) (15 1)(20 3) 14.10 3)(20 1) 7.33 1.87 br df df R df df r b σ ++ =⋅ = This implies that we would have to use approximately twice times as many replicates with a completely randomized design to obtain the same sensitivity as is obtained by blocking on the metal coupons. Clearly, blocking has paid off handsomely in this experiment. However, suppose that blocking was not really necessary. In such cases, if experimenters choose to block, what do they stand to lose? In general, the randomized complete block design has ( a – 1)( b - 1) error degrees of freedom. If blocking was unnecessary and the experiment was run as a completely randomized design with b replicates we would have had a ( b - 1) degrees of freedom for error. Thus, incorrectly blocking has cost a ( b - 1) ( a - 1 )(b - 1) = b - 1 degrees of freedom for error, and the test on treatment means has been made less sensitive needlessly. However, if block effects really are large, then the experimental error may be so inflated that significant differences in treatment means could possibly remain undetected. (Remember the incorrect analysis of Example 4-1.) As a general rule, when the importance of block effects is in doubt, the experimenter should block and gamble that the block means are different.
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ch04 - Chapter 4 Supplemental Text Material S4-1. Relative...

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