Understanding Confounding in Two-Level Factorial Experiments
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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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Statistics
Date
Dec 10, 2024
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2
Uploaded by niumei85167
13.3 Confounding Using Contrasts 429 Y. oul Jj =0, Labels: Y au Jj =1, Labels: | s+ 0. | 55 50 + 1.-""3._0 50 0 451 45 . 01 | 40 6 AR ! 0 1 2 0 Lk Figure 13.2 BCD interaction plot for the field experiment Yis. Labels: k 51+ 9 A 49 + ‘ ' (i ., i st : 0 1 i Figure 13.3 AC interaction plot for the field experiment with the earlier observation that the contrast estimate for B is negative, suggesting that the low level is better. If B is set at its low level, the left-hand graph of Figure 13.2 suggests that both C and D should be at their low levels. The AC interaction plot in Figure 13.3 shows that either both C and A should be at their low levels or both C and A should be at their high levels. The contrast estimators for A and D are both negative, suggesting that on average the low level is better, although the difference in yield is minor. More importantly, the low levels in this experiment are cheaper. Therefore, all the evidence points towards not adding any fertilizer ingredients in the quantities studied in the experiment. A followup experiment could be run with the same four factors but with an increased “high” level of C and lower “high” levels of A, B, and D. Since it is possible that the response is quadratic for each of the factors, a 3 experiment could be run. Suppose the experimenters had known ahead of time that factors A and D do not interact, so that interactions AD, ABD and ACD could have been assumed negligible; then the corresponding terms would have been omitted from the model. There would then have been 3 degrees of freedom for estimating the error variance. The error sum of squares would have been the total of the sums of squares for AD, ABD, and AC D listed in Table 13.5, so that 67 = msE = }(ss(4D) + ss(ABD) + ss(ACD)) = 1(36.25) = 12.0833. The analysis of variance table would then have been as shown in Table 13.6, and we see that at an overall significance level of at most @ = 11(0.005) = 0.055, none of the contrasts would have been judged as significantly different from zero, since Fi.3.0005 = 55.55.
430 Chapter 13 Confounded Two-Level Factorial Experiments Table 13.6 Analysis of variance for the field experiment Source of Degrees of Sum of Mean Variation Freedom Squares Square Ratio p-value Block 1 2.25 2.25 - - A 1 2.25 225 0.186 0.6952 B 1 256.00 256.00 21.186 0.0193 C 1 0.25 0.25 0.021 0.8947 D 1 20.25 20.25 1.676 0.2861 AB 1 6.25 6.25 0.517 0.5240 AC 1 81.00 81.00 6.703 0.0811 B8C 1 20.25 20.25 1.676 0.2861 BD 1 12.25 12.25 1.014 0.3882 cD 1 1.00 1.00 0.083 0.7923 ABC 1 16.00 16.00 1.324 0.3332 BCD 1 121.00 121.00 10.014 0.0507 Error 3 36.25 12.0833 Total 15 575.00 Confidence intervals could be calculated for each contrast at an overall confidence level of at least 94.5%, using the Bonferroni method (formula (4.4.21)) with error degrees of freedom df = 3 and withr; = 8 being the number of observations averaged over to obtain the estimate. For example, a confidence interval for the difference in the high and low levels of B is Bl — B € (%Ecijuyhiju + 13,0.0025/ MSE (16/64)) = (-8+7.4532x1.7381) = (=20.954,4.954). We remind the reader that if both of the above analyses are done, that is, if the interactions AD, ABD, and ACD are dropped from the model after examining the normal probability plot, then it is no longer meaningful to talk about the significance levels of the tests or the confidence levels of the intervals (see Section 6.5.6). O 13.3.3 Experiments in Four Blocks We can extend the method of confounding that we used for two blocks to obtain b = 4 — 22 blocks. We then need to use two contrasts to divide up the treatment combinations. For example, suppose that in a 2* experiment, all interactions except for the two-factor interactions are thought to be negligible. We can select one of the negligible interactions to produce two blocks of size 8 and then select a second interaction to subdivide each of these two blocks into two smaller blocks, giving a total of 4 blocks of size 4. Now, b—1 = 3 degrees of freedom are used to measure blocks, which means that a third treatment contrast must also be confounded. Since we require this third contrast to be among the negligible contrasts, care must be taken as to which pair of contrasts is initially selected for confounding. The choice of ABCD and ABC, for example, is a very poor choice even if these high-order interactions may be thought to be negligible. We can see this by examining the design and the third confounded contrast as follows.