The Gril Defects Experiment

14 21 The Gril Defects Experiment

  (a) Estimate the factor effects. Based on a normal probability

  plot of the effect estimates, identify a model for the data

  from this experiment.

  (b) Conduct an ANOVA based on the model identified in

  part (a). What are your conclusions? (c) Analyze the residuals and comment on model adequacy.

  (d) Find a regression model to predict yield in terms of the

  actual factor levels.

  (e) Can this design be projected into a 2 3 design with two

  replicates? If so, sketch the design and show the average

  and range of the two yield values at each cube corner. Discuss the practical value of this plot.

  12 12 3.46 14-21. An experiment has run a single replicate of a 2 4 13 3 1.73

  design and calculated the following factor effects:

  A 80.25 AB 53.25 ABC

  B 65.50 AC 11.00 ABD

  C 9.25 AD 9.75 ACD

  D 20.50 BC 18.36 BCD

  7.95 14-23. Consider a 2 factorial experiment with four center BD 15.10 points. The data are ABCD 6.25 112 21, a 125, b 154, ab 352,

  and the responses at the center point are 92, 130, 98, 152. CD 1.25 Compute an ANOVA with the sum of squares for curvature

  (a) Construct a normal probability plot of the effects.

  and conduct an F-test for curvature. Use

  (b) Identify a tentative model, based on the plot of effects in

  14-24. Consider the experiment in Exercise 14-14. Suppose

  part (a).

  that a center point with five replicates is added to the factorial

  (c) Estimate the regression coefficients in this model, assum-

  runs and the responses are 2800, 5600, 4500, 5400, 3600.

  ing that y 400.

  Compute an ANOVA with the sum of squares for curvature

  14-22.

  A two-level factorial experiment in four factors was

  and conduct an F-test for curvature. Use

  conducted by Chrysler and described in the article “Sheet

  14-25. Consider the experiment in Exercise 14-17. Suppose

  Molded Compound Process Improvement” by P. I. Hsieh and

  that a center point with five replicates is added to the factorial

  D. E. Goodwin (Fourth Symposium on Taguchi Methods,

  runs and the responses are 45, 40, 41, 47, and 43.

  JWCL232_c14_551-636.qxd 11910 9:02 PM Page 594

  CHAPTER 14 DESIGN OF EXPERIMENTS WITH SEVERAL FACTORS

  (a) Estimate the experimental error using the center points.

  ⫺1

  1 ⫺1 703.67

  Compare this to the estimate obtained originally in Exercise

  1 ⫺1

  1 ⫺1 642.14

  14-17 by pooling apparently nonsignificant effects.

  ␣ ⫽ 0.05. 692.98

  ⫺1

  ⫺1

  (b) Test for curvature with

  14-26 1

  ⫺1 669.26

  . An article in Talanta (2005, Vol. 65, pp. 895–899)

  presented a 2 3 factorial design to find lead level by using flame

  ⫺1

  1 ⫺1 491.58

  atomic absorption spectrometry (FAAS). The data are shown

  1 ⫺1

  1 ⫺1 475.52

  in the following table.

  ⫺1

  1 ⫺1

  1 ⫺1 478.76

  1 ⫺1

  1 ⫺1 568.23

  ⫺1

  1 ⫺1 444.72

  Factors

  Lead Recovery ()

  39.8 42.1 ⫺1

  ⫺1 428.51

  1 ⫺

  ⫹ 491.47 ⫺

  2 51.3 48 1 ⫺1

  1 ⫺ 607.34 ⫹ ⫺ 57.9 58.1 ⫺1

  78.9 85.9 1 ⫺1

  78.9 84.2 ⫺1

  84.2 1 ⫺1

  94.4 90.9 ⫺1

  The factors and levels are shown in the following table.

  High ( ⫹)

  Reagent concentration (RC)

  Shaking time (ST) (min)

  (a) Construct a normal probability plot of the effect estimates.

  Which effects appear to be large? (b) Conduct an analysis of variance to confirm your findings

  for part (a). (c) Analyze the residuals from this experiment. Are there any

  (a) Estimate the factor effects and use a normal probability

  problems with model adequacy?

  plot of the effects. Identify which effects appear to be

  14-27. An experiment to study the effect of machining fac-

  large.

  tors on ceramic strength was described at http:www.itl.

  (b) Fit an appropriate model using the factors identified in

  nist.govdiv898handbook. Five factors were considered at

  part (a) above.

  two levels each: A ⫽ Table Speed, B ⫽ Down Feed Rate, C ⫽

  (c) Prepare a normal probability plot of the residuals. Also,

  Wheel Grit, D ⫽ Direction, E ⫽ Batch. The response is the av-

  plot the residuals versus the predicted ceramic strength.

  erage of the ceramic strength over 15 repetitions. The following

  Comment on the adequacy of these plots.

  data are from a single replicate of a 2 5 factorial design.

  (d) Identify and interpret any significant interactions. (e) What are your recommendations regarding process oper-

  ating conditions?

  A B C D E Strength

  14-28. Consider the following Minitab output for a 2 3 fac-

  ⫺1

  torial experiment.

  1 ⫺1

  (a) How many replicates were used in the experiment?

  ⫺1 (b) Use Equation 14-17 to calculate the standard error of a 1 ⫺1 702.14

  (c) Calculate the entries marked with “?” in the output.

  14-6 BLOCKING AND CONFOUNDING IN THE 2 k DESIGN

  Factorial Fit: y versus A, B, C

  Estimated Effects and Coefficients for y (coded units)

  Coef SE Coef

  ⫺ 0.1193

  ⫺ ⫹ 0.1196

  A 2.95 1.47 38.46 0.04 0.970

  4 ⫹

  ⫺ 0.1192

  C ⫺37.87 ⫺18.94

  38.46 ⫺0.49 0.636

  6 ⫹

  ⫺

  ⫹ ⫺ 0.1188

  AC ⫺17.11

  ⫺8.55

  38.46 ⫺0.22 0.830

  ⫺ 0.1195

  11 ⫺

  ⫹

  ⫺ ⫹ 0.1196

  S ⫽ 153.832 R⫺Sq ⫽ 5.22 R⫺Sq (adj) ⫽ 0.00

  12 ⫺

  ⫹

  ⫺ 0.1191

  Analysis of Variance for y (coded units)

  ⫹ ⫺ 0.1194

  15 ⫺

  ⫹ 0.1188

  Main Effects

  Interactions 3-Way 1

  The factors and levels are shown in the following table.

  Residual 8 189314 189314 23664.2

  Error

  A RF voltage of the DMS sensor (1200 or 1400 V)

  Pure Error

  B Nitrogen carrier gas flow rate (250 or 500mLmin ⫺1 )

  Total

  C Solid phase microextraction (SPME) filter type (polyacrylate or PDMS–DVB)

  D GC cooling profile (cryogenic and noncryogenic)

  14-29. An article in Analytica Chimica Acta [“Design-

  (a) Estimate the factor effects and use a normal probability plot

  of-Experiment Optimization of Exhaled Breath Condensate

  of the effects. Identify which effects appear to be large, and

  Analysis Using a Miniature Differential Mobility Spectrometer

  identify a model for the data from this experiment.

  (DMS)” (2008, Vol. 628, No. 2, pp. 155–161)] examined four

  (b) Conduct an ANOVA based on the model identified in part

  parameters that affect the sensitivity and detection of the ana-

  (a). What are your conclusions?

  lytical instruments used to measure clinical samples. They op-

  (c) Analyze the residuals from this experiment. Are there any

  timized the sensor function using EBC samples spiked with

  problems with model adequacy?

  acetone, a known clinical biomarker in breath. The following

  (d) Project the design in this problem into a 2 r design for r ⬍4

  table shows the results for a single replicate of a 2 4 factorial

  in the important factors. Sketch the design and show the

  experiment for one of the outputs, the average amplitude of

  average and range of yields at each run. Does this sketch

  acetone peak over three repetitions.

  aid in data representation?

  14-6 BLOCKING AND CONFOUNDING IN THE 2 k DESIGN

  It is often impossible to run all the observations in a 2 k factorial design under homogeneous conditions. Blocking is the design technique that is appropriate for this general situation. However, in many situations the block size is smaller than the number of runs in the complete replicate. In these cases, confounding is a useful procedure for running the 2 k design in 2 p blocks where the number of runs in a block is less than the number of treatment combinations in one complete replicate. The technique causes certain interaction effects to be indistinguishable from

  JWCL232_c14_551-636.qxd 11610 9:56 AM Page 596

  CHAPTER 14 DESIGN OF EXPERIMENTS WITH SEVERAL FACTORS

  b ab +

  = Run in block 1

  = Run in block 2

  ab

  Figure 14-28 A2 2 (1)

  a

  –

  design in two blocks.

  (a) Geometric view.

  A Assignment of the four

  (b) Assignment of the

  Geometric view

  runs to two blocks

  four runs to two blocks.

  (a)

  (b)

  blocks or confounded with blocks. We will illustrate confounding in the 2 k factorial design in