Problem 2.   An engineer is interested in the resonant frequency of a mechanical device as a function of three variables: Pressure, with three levels (Press1,Press2,Press3), Drubness, with two levels (Drub1,Drub2), and Abrasiveness, with three levels (Abr1,Abr2,Abr3). The resonant frequencies of two devices are measured for each set of levels of the three variables. The resulting frequencies are listed in Table 2.

 Table 2. Resonant frequencies of a Device

                Press1                Press2                Press3
            Drub1    Drub2       Drub1     Drub2         Drub1     Drub2
  Abr1   3839 3202  326 117    5950 1254  357 1550     484  227  1915 2924
  Abr2   1313 3202  276 368    1574 8814  530  538    1046 1128  1373 2795
  Abr3   2097 6417  374 429    3614 1293  238 2476     201  886  1803 1647
 

Problem 3. An experimenter wants to test the effect of 6 factors, which she calls A B C D E F, on a response variable YY associated with an industrial process. She can afford to do 12 runs and decides to use the Plackett-Burman PB_12 design, which includes 5 additional columns G H J K L that she does not use. The High/Low settings of the design and the output that she measures are in the following table.

 Table 3. Output of an experiment with High/Low settings for 6 factors
       Rows    A  B  C  D  E  F    G  H  J  K  L   Response
          1.   1 -1  1 -1 -1 -1    1  1  1 -1  1     80.8
          2.   1  1 -1  1 -1 -1   -1  1  1  1 -1     38.8
          3.  -1  1  1 -1  1 -1   -1 -1  1  1  1     31.6
          4.   1 -1  1  1 -1  1   -1 -1 -1  1  1    116.0
          5.   1  1 -1  1  1 -1    1 -1 -1 -1  1     38.1
          6.   1  1  1 -1  1  1   -1  1 -1 -1 -1    102.5
          7.  -1  1  1  1 -1  1    1 -1  1 -1 -1     62.5
          8.  -1 -1  1  1  1 -1    1  1 -1  1 -1     60.3
          9.  -1 -1 -1  1  1  1   -1  1  1 -1  1     68.7
         10.   1 -1 -1 -1  1  1    1 -1  1  1 -1    120.4
         11.  -1  1 -1 -1 -1  1    1  1 -1  1  1     49.7
         12.  -1 -1 -1 -1 -1 -1   -1 -1 -1 -1 -1     47.2 

    (i) Use the Box-Meyer program on the Math420 Web site to find the most likely choice of active factors for the data in Table 3. Analyze all 11 factors in Table 3 together as a control for the 6 factors whose values were actually varied in the data. Recall that one of the advantages of the screens based on the F-statistic and on the Bayesian-model-posterior-probabilities is that they are valid across different model sizes. (That is, 2^1, 2^2, and 2^3 models can be sorted together.) Which submodel has the highest F-statistic? Is it unique? Which submodel has the highest Bayesian posterior probability for submodels? Is it unique? What individual factors (not submodels) have the highest Bayesian posterior probabilities of being active?

    (ii) The Box-Meyer program uses model prior probabilities that depend on two parameters: pi, which is the prior probability that any particular factor is active independently of the other factors, and gamma, which is a function of the estimated selection coefficient of active factors to inert factors. The default settings of the program at pi=0.25 and gamma=2.50. Are your conclusions from part (i) robust with respect to these parameters? Re-run the program with the settings pi=0.10, pi=0.80, gamma=1.0, and gamma=11.0 (four different choices of settings). Are your conclusions similar to part (i)? Are the factors that are most likely to be active the same?

    (iii) Assume that the submodel that you identified in part (i) contains the active factors and that all of the other factors are inert. Do a 2^3-like factorial design analysis of the data in Table 3 using these active factors. Which of the main effects in this analysis are significant? Which of the interactions? Find the P-values of the significant effects. Is this consistent with your answers in part (i)? (Warning: Since these factors and their interactions are not orthogonal in the 12-run Plackett-Burman design, do not do a linear regression on the High/Low settings. Instead, treat the three factors as categorical or class variables and do a 3-factor full factorial analysis.)

Problem 4. Assume that the Low/High settings for the data in Table 3 were given by

 Table 4. Factors and High/Low settings for the data in Table 3
    Factor        Low        High
      A           2.0         4.0
      B          11.0        13.0
      C          15.0        18.0
      D          Light       Dark
      E          Stirred     Not stirred
      F           0.01        0.03  

    (i) Find the parameter estimates for all 11 factors in the design and sort them in decreasing order. Do any of the estimates appear to stick out on the high or low side? Are these the active factors that were predicted in Problem 3?

    (ii) The experimenter wants to carry out additional runs at 5 additional settings in order to increase the value of the response variable. The values that she predicts at new settings will be the estimated regression equation in part (i), with the intercept and the regression coefficients of active factors rounded to two significant figures, and the regression coefficients of inert factors replaced by zero. What is the equation for the response variable at different values of the factors listed in Table 4 that she predicts? (See Section 12.1 in the text for a similar analysis.)

    (iii) In general, the direction of fastest ascent of a function w=f(x,y,z) is given by the gradient of f(x,y,z) at (x,y,z). For example, if w=10+2*x+3*y+4*z, the direction of fastest ascent is the vector (2,3,4). Which is the vector direction of fastest ascent for the linear function in part (ii)? If the value of the first active variable is increased by one, by how much should the second and third active variables be changed to stay on the line of fastest ascent?