18 An experiment was carried out to investigate shrinkage in the plastic casing material used

Example 11.18 An experiment was carried out to investigate shrinkage in the plastic casing material used

  for speedometer cables (“An Explanation and Critique of Taguchi’s Contribution to Quality Engineering,” Quality and Reliability Engr. Intl., 1988: 123–131). The engineers started with 15 factors: liner outside diameter, liner die, liner material, liner line speed, wire braid type, braiding tension, wire diameter, liner tension, liner temperature, coating material, coating die type, melt temperature, screen pack, cooling method, and line speed. It was suspected that only a few of these factors were important, so a screening experi-

  ment in the form of a 2 15211 factorial (a 12 11 fraction of a 2 15 factorial experiment) was

  carried out. The resulting alias structure is quite complicated; in particular, every main effect is confounded with two-factor interactions. The response variable was the percent- age of shrinkage for a cable specimen produced at designated levels of the factors.

  Figure 11.15 displays a normal probability plot of the effect contrasts. All but two of the points fall quite close to a straight line. The discrepant points correspond to effects

  E 5 wire braid type and

  G 5 wire diameter , suggesting that these two

  factors are the only ones that affect the amount of shrinkage.

  Contrast

  .8 G Wire diameter

  1.6 E Wire-braid type

  z percentile

  ■ The subjects of factorial experimentation, confounding, and fractional replica-

  Figure 11.15 Normal probability plot of contrasts from Example 11.18

  tion encompass many models and techniques we have not discussed. Please consult the chapter references for more information.

  CHAPTER 11 Multifactor Analysis of Variance

  EXERCISES Section 11.4 (38–49)

  38. The accompanying data resulted from an experiment to

  40. In a study of processes used to remove impurities from cel-

  study the nature of dependence of welding current on

  lulose goods (“Optimization of Rope-Range Bleaching of

  three factors: welding voltage, wire feed speed, and tip-

  Cellulosic Fabrics,” Textile Research J., 1976: 493–496),

  to-workpiece distance. There were two levels of each fac-

  the following data resulted from a 2 4 experiment involving

  tor (a 2 3 experiment) with two replications per

  the desizing process. The four factors were enzyme con-

  combination of levels (the averages across replications

  centration (A), pH (B), temperature (C), and time (D).

  agree with values given in the article “A Study on Prediction of Welding Current in Gas Metal Arc

  Starch

  Welding,” J. Engr. Manuf., 1991: 64–69). The first two

  En-

  by Weight

  given numbers are for the treatment (1), the next two for

  Treat

  zyme

  Temp. Time

  1st 2nd

  a, and so on in standard order: 200.0, 204.2, 215.5, 219.5,

  a. Verify that the sums of squares are as given in the accom-

  a .75

  panying ANOVA table from Minitab.

  b .50

  b. Which effects appear to be important, and why?

  Analysis of Variance for current

  39. The accompanying data resulted from a 2 3 experiment with

  three replications per combination of treatments designed to

  a. Use Yates’s algorithm to obtain sums of squares and the

  study the effects of concentration of detergent (A), concen-

  ANOVA table.

  tration of sodium carbonate (B), and concentration of

  b. Do there appear to be any second-, third-, or fourth-order

  sodium carboxymethyl cellulose (C) on the cleaning ability

  interaction effects present? Explain your reasoning.

  of a solution in washing tests (a larger number indicates bet-

  Which main effects appear to be significant?

  ter cleaning ability than a smaller number).

  41. In Exercise 39, suppose a low water temperature has been used to obtain the data. The entire experiment is then

  Factor Levels

  repeated with a higher water temperature to obtain the fol-

  A B C Condition

  Observations

  lowing data. Use Yates’s algorithm on the entire set of 48 observations to obtain the sums of squares and ANOVA

  table, and then test appropriate hypotheses at level .05.

  2 1 a 198, 200, 214 1 2 1 b 197, 202, 185

  Condition

  Observations

  2 1 ab 329, 331, 307 1 2 c 149, 169, 135

  a. After obtaining cell totals x ijk , compute estimates of b 1 ,

  b. Use the cell totals along with Yates’s method to compute

  abcd

  the effect contrasts and sums of squares. Then construct

  42. The following data on power consumption in electric-

  an ANOVA table and test all appropriate hypotheses

  furnace heats (kW consumed per ton of melted product)

  using . a 5 .05

  resulted from a 2 4 factorial experiment with three replicates

  11.4 2 p Factorial Experiments

  (“Studies on a 10-cwt Arc Furnace,” J. of the Iron and Steel

  ab 94 abd 79

  Institute, 1956: 22). The factors were nature of roof (A,

  c 86 cd 69

  lowhigh), power setting (B, lowhigh), scrap used (C,

  ac 83 acd 75

  tubeplate), and charge (D, 700 lb1000 lb).

  x ijklm

  x ijklm

  c. Assume that all three-way interaction effects are absent, so that the associated sums of squares can be combined

  to yield an estimate of s 2 , and carry out all appropriate

  a 946, 800, 840

  ad 966, 976, 876

  tests at level .05.

  45. a. An experiment was carried out to investigate the effects on

  c 1017, 990, 954

  cd 922, 808, 868

  audio sensitivity of varying resistance (A), two capacitances

  (B, C), and inductance of a coil (D) in part of a television

  circuit. If four blocks were used with four treatments per

  block and the defining effects for confounding were AB and CD, which treatments appeared in each block?

  Construct the ANOVA table, and test all hypotheses of

  b. Suppose two replications of the experiment described in

  interest using a 5 .01 .

  part (a) were performed, resulting in the accompanying

  43. The article “Statistical Design and Analysis of Qualification

  data. Obtain the ANOVA table, and test all relevant

  Test Program for a Small Rocket Engine” (Industrial

  hypotheses at level .01.

  Quality Control, 1964: 14–18) presents data from an exper-

  Treat-

  iment to assess the effects of vibration (A), temperature

  ment

  x ijkl1

  x ijkl2

  x

  cycling (B), altitude cycling (C), and temperature for alti-

  ijkl1

  x ijkl2

  tude cycling and firing (D) on thrust duration. A subset of

  the data is given here. (In the article, there were four levels

  of D rather than just two.) Use the Yates method to obtain

  sums of squares and the ANOVA table. Then assume that

  three- and four-factor interactions are absent, pool the cor-

  responding sums of squares to obtain an estimate of s 2 , and

  test all appropriate hypotheses at level .05.

  abcd 449 455 C 1 C 2 C 1 C 2 46. In an experiment involving four factors (A, B, C, and D) and

  four blocks, show that at least one main effect or two-factor

  B A 1 1 21.60 11.54 11.50 B interaction effect must be confounded with the block effect.

  B 1 21.60 21.86 11.75 9.82 47. a. In a seven-factor experiment (A, c, G) , suppose a

  A 2 B

  2 19.57 21.85 11.69 11.18 quarter-replicate is actually carried out. If the defining

  effects are ABCDE and CDEFG, what is the third nones-

  44. a. In a 2 4

  experiment, suppose two blocks are to be used,

  timable effect, and what treatments are in the group

  and it is decided to confound the ABCD interaction with

  containing (1)? What are the alias groups of the seven

  the block effect. Which treatments should be carried out

  main effects?

  in the first block [the one containing the treatment (1)],

  b. If the quarter-replicate is to be carried out using four blocks

  and which treatments are allocated to the second block?

  (with eight treatments per block), what are the blocks if the

  b. In an experiment to investigate niacin retention in veg-

  chosen confounding effects are ACF and BDG?

  etables as a function of cooking temperature (A), sieve size (B), type of processing (C), and cooking time (D),

  48. The article “Applying Design of Experiments to Improve a

  each factor was held at two levels. Two blocks were used,

  Laser Welding Process” (J. of Engr. Manufacture, 2008:

  with the allocation of blocks as given in part (a) to con- 4 1035–1042) included the results of a half replicate of a 2 found only the ABCD interaction with blocks. Use

  experiment. The four factors were: A. Power (2900 W, 30

  Yates’s procedure to obtain the ANOVA table for the

  W), B. Current (2400 mV, 3600 mV), C. Laterals cleaning

  accompanying data.

  (No, Yes), and D. Roof cleaning (No, Yes). a. If the effect ABCD is chosen as the defining effect for the

  Treatment

  x ijkl

  x ijkl

  replicate and the group of eight treatments for which data is obtained includes treatment (1), what other treatments

  91 d 72 are in the observed group, and what are the alias pairs? a 85 ad 78 b. The cited article presented data on two different response b 92 bd 68 variables, the percentage of defective joints for both the

  CHAPTER 11 Multifactor Analysis of Variance

  right laser welding cord and the left welding cord. Here

  and higher-order interactions to be negligible) test at level

  we consider just the latter response. Observations are

  .01 for the presence of main effects. Also construct a normal

  listed here in standard order after deleting the half not

  probability plot.

  observed. Assuming that two- and three-factor interac- tions are negligible, test at level .05 for the presence of

  Treat-

  main effects. Also construct a normal probability plot.

  a 70.4 acd

  b 72.1 ace

  c 70.4 ade

  49. A half-replicate of a 2 5 experiment to investigate the effects

  d 67.4 bcd

  of heating time (A), quenching time (B), drawing time (C),

  e 68.0 bce

  position of heating coils (D), and measurement position (E)

  on the hardness of steel castings resulted in the accompany-

  ing data. Construct the ANOVA table, and (assuming second

  abe

  67.8 abcde