Confounding Contrasts in 2^3 Factorial Experiments Explained
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Tennessee Colleges of Applied Technology, Murfreesb**We aren't endorsed by this school
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PSYCHOLOGY 101386TYFY
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Mathematics
Date
Dec 10, 2024
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2
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13.3 Confounding Using Contrasts 425 Table 13.2 Table 13.3 Contrasts for a 23 experiment A B C AB AC BC ABC 000 -1 -1 -1 1 1 1 -1 001 -1 -1 1 1T -1 -1 1 010 -1 1T =1 = 1T -1 1 011 -1 1 1T =1 -1 1 -1 100 1 -1 -1 -1 - 1 1 101 1 -1 1 =1 TR -1 10 1 1T -1 1T -1 -1 -1 11 1 1 1 1 1 1 1 the interaction contrasts are also divided by v/2, the contrast estimators all have variance Y c?0? = 40? /v, and the A B interaction, for example, then compares the average response when factors A and B are at the same level with the average response when they are at different levels. We shall use either v/2 or 1 for the divisor for all contrasts in 27 experiments both here and in Chapter 15. The full set of factorial treatment contrasts (without divisors) for a 23 experiment is shown in Table 13.2 written as columns. The row headings are the treatment combinations (inlexicographical order) whose observations are to be multiplied by the contrast coefficients when estimating the contrast. The contrasts in Table 13.2 are orthogonal. This can be verified by multiplying together corresponding digits in any two columns and showing that the sum of the products is zero. A table of orthogonal contrasts, such as Table 13.2, is sometimes called an orthogonal array. 13.3.2 Experiments in Two Blocks We start with an example. Suppose that a single-replicate 2° experiment is to be run in two blocks of size four. Suppose also that the experimenter knows that one of the factors, say factor A, does not interact with either of the other two factors. This means that the interactions AB, AC, and ABC may be assumed to be negligible and that the contrasts labeled A, B, C, and BC in Table 13.2 are the only contrasts to be measured. Since there will be b = 2 blocks, it follows that b — 1 = 1 degree of freedom will be used to measure block differences and one treatment contrast will be confounded with blocks. Without too much difficulty, we can ensure that the confounded contrast is one of the negligible contrasts. For example, we can confound the negligible A BC contrast by placing in one block those treatment combinations corresponding to —1 in the ABC contrast, and placing in the second block those treatment combinations corresponding to +1 in the same contrast. Referring to Table 13.2, we can see that the design in Table 13.3 results. The ABC contrast is now identical to a block contrast that compares Block I with Block II, and 23 experiment in 2 blocks of 4, confounding ABC Blockl 000 011 101 110 BlockIl 001 010 100 111
426 Chapter 13 Confounded Two-Level Factorial Experiments Table 13.4 Data for the field experiment (ABCD is Example 13.3.1 confounded) Block | Block Il TC Response TC Response 0000 58 0001 55 0011 51 0010 45 0101 44 0100 42 0110 50 0111 36 1001 43 1000 53 1010 50 1011 55 1100 41 1101 41 1111 44 1110 48 Source: Experimental Designs, Second Edition, by W. G. Cochran and G. M. Cox, Copyright © 1957, John Wiley & Sons, New York. Adapted by permission of John Wiley & Sons, Inc. consequently, the contrast is confounded with blocks. The other two negligible contrasts, AB and AC, provide two degrees of freedom to estimate o'2. Since all the nonnegligible factorial contrasts are orthogonal to AB, AC, and ABC , they can be measured as though there were no blocks present. Block design randomization (see Section 11.2.2) needs to be carried out before the design in Table 13.3 can be used in practice. A similar method of confounding can be used for any 27 experiment in b = 2 blocks of size k = 27~!. All factorial contrasts except for the one confounded contrast can be estimated. Field experiment The data shown in Table 13.4 form part of the results of a field experiment on the yield of beans using various types of fertilization. The experiment was conducted at Rothamsted Experimental Station in 1936 and was reported by W. G. Cochran and G. M. Cox in their book Experimental Designs. There were four treatment factors each at two levels. Factor A was the amount of dung (0 or 10 tons) spread per acre, factors B, C, and D were the amounts of nitrochalk (0 and 45 Ib), superphosphate (0 and 67 1b), and muriate of potash (0 and 112 Ib), respectively, per acre. The experimental area was divided into two possibly dissimilar blocks of land, each of which was subdivided into eight plots (experimental units). Since this was a single-replicate experiment with 2¢ = 16 treatment combinations (TC) divided into b = 2 blocks of size k = 8, one treatment contrast had to be confounded. The experimenters chose to confound the A BC D contrast, since the four-factor interaction was of least interest. The A BC D contrast is shown below, and it can be verified that the treatment combinations corresponding to contrast coefficient — 1 appear in Block I of Table 13.4, while those corresponding to coefficient +1 appear in Block II. All the other factorial contrasts are orthogonal to the ABC D contrast, so they can all be estimated without adjusting for the block effects. We take as examples the B and BC contrasts shown below.