Slide set 11 Stat402 (Spring 2016) Last update: March 28, 2016 Stat 402 (Spring 2016): slide set 11 Single Replicate of the 2k Factorial • For even a small k (number of factors), the number of treatment combinations, and therefore the number of runs needed is large. • For example, a 26 factorial has 64 treatment combinations. So if you have 3 replications, 64 × 3 = 192 runs will be needed. • Because of limitations of resources (experimental material, equipment, labor, time or cost) only a single replicate may be possible, unless the experiment is made smaller by eliminating some of the factors from the study. • A major problem with running a single replicate is that there are no degrees of freedom left to estimate the error (or noise) directly. • One approach to alleviate this is to assume that certain higher-order interactions are negligible and combine their mean squares to estimate the error. 1 Stat 402 (Spring 2016): slide set 11 Single Replicate of a 24 Factorial: Example Pilot Plant Filtration Experiment 4 quantitative variables: A - Temperature, B - Pressure, C - Concentration of Formaldehyde, D - Stirring Rate Response - Filtration Rate 2 Stat 402 (Spring 2016): slide set 11 Contrasts for Effects of a 24 The 16 treatment combinations are: (1), a, b, ab, c, ac, ab, abc, d, ad, bd, abd, cd, acd, abd, abcd 3 Stat 402 (Spring 2016): slide set 11 Estimates and SS for Effects of a 24 The 15 factorial effects estmates and sums of squares are: 4 Stat 402 (Spring 2016): slide set 11 Inference from unreplicated factorials Fig. 6.11 shows the normal probabilty plot of the effects. The straight line drawn goes through majority of points and the point (0, 50 percentile), and thus thus capture most of the effects that might be considered non-significant (i.e., close to zero). 5 Stat 402 (Spring 2016): slide set 11 Interpretation of the important effects • From Fig. 6.1, it can be observed that the important effects are the main effects of A, C, and D and the AC and the AD interactions. • The main effects A, C, and D are plotted in the the figure below: • All three effects are positive. If the aim is to maximize filtration rate, we would run the three factors A, C, and D at the high levels. • However, we need to consider any important interactions. The AC and AD interactions are shown in the figure below: 6 Stat 402 (Spring 2016): slide set 11 • From the AC interaction plot: Temperature (A) effect is small when Concentration (C) is at the high level but is very high when Concentration (C) is at the low level. • From the AD interaction plot: Stirring rate (D) effect is small at low Temperature (A) but large and positive at highTemperature (A). • Conclusion: Best filtration rate is obtained when A and D at the high levels and C is at the low level. This will also keep the formaldehyde concentration C to a lower level (another objective). 7 Stat 402 (Spring 2016): slide set 11 Another analysis: design projection • Because Pressure (B) is not important and all interactions involving B are negligible, we may act as if B was not a factor in the experiment. • So the experiment may be regarded as a 23 factorial in A, C, and D with two replicates (i.e., the two responses observed at each level of B). To see this, examine Table 6.10 ignoring the column for factor B. • We say that an unreplicated 24 is projected into a replicated 23. The implication of this is that we now get an estimate of error with 8 degrees of freedom. • See Table 6.13 for the ANOVA table for the prejected experiment. • This method of analysis is also called pseudo-replication. 8 Stat 402 (Spring 2016): slide set 11 9