barcode font reporting services Analysis of Variance in Software

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CHAPTER 9 Analysis of Variance
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The F ratios in the last column of Table 9-6 can be used to test the null hypotheses H(1): All treatment (row) means are equal, i.e., aj 0 H(2): 0 All block (column) means are equal, i.e., bk 0 0 0
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H(3): There are no interactions between treatments and blocks, i.e., gjk 0 H(3) 0
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From a practical point of view we should first decide whether or not can be rejected at an appropriate level of significance using the F ratio ^2 >s 2 of Table 9-6. Two possible cases then arise. s i ^e Case I H(3) Cannot Be Rejected: In this case we can conclude that the interactions are not too large. We can then 0 test H(1) and H(2) by using the F ratios ^2 >s 2 and ^2 >s 2, respectively, as shown in Table 9-6. Some stats r ^e s c ^e 0 0 isticians recommend pooling the variations in this case by taking the total vi ve and dividing it by the total corresponding degrees of freedom, (a 1)(b 1) ab(c 1), and using this value to replace se the denominator ^2 in the F test.
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Case II H(3) Can Be Rejected: In this case we can conclude that the interactions are significantly large. Differ0 ences in factors would then be of importance only if they were large compared with such interactions. s r ^i For this reason many statisticians recommend that H(1) and H(2) be tested using the F ratios ^2 >s 2 and 0 0 ^2 ^2 s c >s i rather than those given in Table 9-6. We shall use this alternate procedure also. The analysis of variance with replication is most easily performed by first totaling replication values that correspond to particular treatments (rows) and blocks (columns). This produces a two-factor table with single entries, which can be analyzed as in Table 9-5. The procedure is illustrated in Problem 9.13.
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Experimental Design
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The techniques of analysis of variance discussed above are employed after the results of an experiment have been obtained. However, in order to gain as much information as possible, the details of an experiment must be carefully planned in advance. This is often referred to as the design of the experiment. In the following we give some important examples of experimental design. 1. COMPLETE RANDOMIZATION. Suppose that we have an agricultural experiment as in Example 9.1, page 314. To design such an experiment, we could divide the land into 4 4 16 plots (indicated in Fig. 9-1 by squares, although physically any shape can be used) and assign each treatment, indicated by A, B, C, D, to four blocks chosen completely at random. The purpose of the randomization is to eliminate various sources of error such as soil fertility.
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Fig. 9-1
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Fig. 9-2
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Fig. 9-3
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Fig. 9-4
2. RANDOMIZED BLOCKS. When, as in Example 9.2, it is necessary to have a complete set of treatments for each block, the treatments A, B, C, D are introduced in random order within each block I, I, III, IV (see Fig. 9-2) and for this reason the blocks are referred to as randomized blocks. This type of design is used when it is desired to control one source of error or variability, namely, the difference in blocks (rows in Fig. 9-2). 3. LATIN SQUARES. For some purposes it is necessary to control two sources of error or variability at the same time, such as the difference in rows and the difference in columns. In the experiment of Example 9.1, for instance, errors in different rows and columns could be due to changes in soil fertility in different parts of the land. In that case it is desirable that each treatment should occur once in each row and once in each column, as in Fig. 9-3. The arrangement is called a Latin square from the fact that Latin letters A, B, C, D are used.
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