Nonreplicated 2 k Factorial Experiment

15.3 Nonreplicated 2 k Factorial Experiment

  The full 2 k factorial may often involve considerable experimentation, particularly when k is large. As a result, replication of each factor combination is often not feasible. If all effects, including all interactions, are included in the model of the experiment, no degrees of freedom are allowed for error. Often, when k is large, the data analyst will pool sums of squares and corresponding degrees of freedom for high-order interactions that are known or assumed to be negligible. This will produce F-tests for main effects and lower-order interactions.

  Diagnostic Plotting with Nonreplicated 2 k Factorial Experiments

  Normal probability plotting can be a very useful methodology for determining the relative importance of effects in a reasonably large two-level factored experiment when there is no replication. This type of diagnostic plot can be particularly useful when the data analyst is hesitant to pool high-order interactions for fear that some of the effects pooled in the “error” may truly be real effects and not merely random. The reader should bear in mind that all effects that are not real (i.e., they are independent estimates of zero) follow a normal distribution with

  mean near zero and constant variance. For example, in a 2 4 factorial experiment,

  we are reminded that all effects (keep in mind that n = 1) are of the form

  contrast

  AB =

  =¯ y H

  8 − ¯y

  L ,

  15.3 Nonreplicated 2 k Factorial Experiment

  where ¯ y H is the average of eight independent experimental runs at the high, or “+,” level and ¯ y L is the average of eight independent runs at the low, or “

  2 level. −,” Thus, the variance of each contrast is Var(¯ y H

  − ¯y L )=σ 4. For any

  real effects, E(¯ y H − ¯y L ) = 0. Thus, normal probability plotting should reveal

  “significant” effects as those that fall off the straight line that depicts realizations of independent, identically distributed normal random variables.

  The probability plotting can take one of many forms. The reader is referred to Chapter 8, where these plots were first presented. The empirical normal quantile- quantile plot may be used. The plotting procedure that makes use of normal probability paper may also be used. In addition, there are several other types of diagnostic normal probability plots. In summary, the procedure for diagnostic effect plots is as follows.

  Probability Effect

  1. Calculate effects as

  Plots for

  Nonreplicated 2 4 effect = contrast

  2. Construct a normal probability plot of all effects.

  3. Effects that fall off the straight line should be considered real effects. Further comments regarding normal probability plotting of effects are in order.

  First, the data analyst may feel frustrated if he or she uses these plots with a small experiment. On the other hand, the plotting is likely to give satisfying results when there is effect sparsity—many effects that are truly not real. This sparsity will be evident in large experiments where high-order interactions are not likely to be real.

  Case Study 15.1: Injection Molding: Many manufacturing companies in the United States and

  abroad use molded parts as components. Shrinkage is often a major problem. Often, a molded die for a part is built larger than nominal to allow for part shrink- age. In the following experimental situation, a new die is being produced, and ultimately it is important to find the proper process settings to minimize shrink- age. In the following experiment, the response values are deviations from nominal (i.e., shrinkage). The factors and levels are as follows:

  Coded Levels

  −1

  A. Injection velocity (ftsec)

  B. Mold temperature ( ◦ C)

  C. Mold pressure (psi)

  D. Back pressure (psi)

  The purpose of the experiment was to determine what effects (main effects and interaction effects) influence shrinkage. The experiment was considered a prelim- inary screening experiment from which the factors for a more complete analysis might be determined. Also, it was hoped that some insight might be gained into how the important factors impact shrinkage. The data from a nonreplicated 2 4 factorial experiment are given in Table 15.6.

  Chapter 15 2 k Factorial Experiments and Fractions

  Table 15.6: Data for Case Study 15.1

  Combination (cm

  4 ) Combination (cm × 10 4 × 10 )

  72.68 d 73.52

  a 71.74 ad 75.97

  b 76.09 bd 74.28

  ab 93.19 abd

  c 71.25 cd 79.34

  ac 70.59 acd

  bc 70.92 bcd

  Initially, effects were calculated and placed on a normal probability plot. The calculated effects are as follows:

  A = 10.5613,

  BD = −2.2787,

  AD = −1.8238,

  ABD = −1.7813,

  ACD = −3.0438, BCD = −0.4788, ABCD = −1.3063.

  The normal quantile-quantile plot is shown in Figure 15.5. The plot seems to imply that effects A, B, and AB stand out as being important. The signs of the important effects indicate the preliminary conclusions.

  D BC 0 CD AC BCD ABCD ABD

  AD BD

  Theoretical Quantiles −1

  Effects Quantiles

  Figure 15.5: Normal quantile-quantile plot of effects for Case Study 15.1.

  15.3 Nonreplicated 2 k Factorial Experiment

  1. An increase in injection velocity from 1.0 to 2.0 increases shrinkage.

  2. An increase in mold temperature from 100 ◦

  C to 150 ◦

  C increases shrinkage.

  3. There is an interaction between injection velocity and mold temperature; al- though both main effects are important, it is crucial that we understand the impact of the two-factor interaction.

  Interpretation of Two-Factor Interaction

  As one would expect, a two-way table of means provides ease in interpretation of the AB interaction. Consider the two-factor situation in Table 15.7.

  Table 15.7: Illustration of Two-Factor Interaction

  B (temperature)

  A (velocity)

  Notice that the large sample mean at high velocity and high temperature cre- ated the significant interaction. The shrinkage increases in a nonadditive manner . Mold temperature appears to have a positive effect despite the velocity level. But the effect is greatest at high velocity. The velocity effect is very slight at low temperature but clearly is positive at high mold temperature. To control shrinkage at a low level, one should avoid using high injection velocity and high mold temperature simultaneously. All of these results are illustrated graphically in Figure 15.6.

  Figure 15.6: Interaction plot for Case Study 15.1.

  Chapter 15 2 k Factorial Experiments and Fractions

  Analysis with Pooled Mean Square Error: Annotated Computer Printout

  It may be of interest to observe an analysis of variance of the injection molding data with high-order interactions pooled to form a mean square error. Interactions of order three and four are pooled. Figure 15.7 shows a SAS PROC GLM printout. The analysis of variance reveals essentially the same conclusion as that of the normal probability plot.

  The tests and P-values shown in Figure 15.7 require interpretation. A signif- icant P-value suggests that the effect differs significantly from zero. The tests on main effects (which in the presence of interactions may be regarded as the effects averaged over the levels of the other factors) indicate significance for effects A and

  B. The signs of the effects are also important. An increase in the level from low

  The GLM Procedure

  Dependent Variable: y

  Sum of

  Source

  DF Squares Mean Square

  F Value Pr > F

  Corrected Total

  R-Square

  Coeff Var

  Root MSE

  y Mean

  DF Type III SS

  Mean Square

  F Value

  Error t Value

  AD -0.91187500

  BD -1.13937500

  Figure 15.7: SAS printout for data of Case Study 15.1.

  Exercises

  to high of A, injection velocity, results in increased shrinkage. The same is true for

  B. However, because of the significant interaction AB, main effect interpretations may be viewed as trends across the levels of the other factors. The impact of the significant AB interaction is better understood by using a two-way table of means.

  Exercises

  15.1 The following data are obtained from a 2 3 fac- tant, do a complete analysis of the data. Use P-values

  torial experiment replicated three times. Evaluate the in your conclusion. sums of squares for all factorial effects by the contrast method. Draw conclusions.

  15.3 In a metallurgy experiment, it is desired to test

  Treatment

  the effect of four factors and their interactions on the

  Combination Rep 1

  Rep 2 Rep 3

  concentration (percent by weight) of a particular phos-

  12 19 10 phorus compound in casting material. The variables a 15 20 16 are A, percent phosphorus in the refinement; B, per- b 24 16 17 cent remelted material; C, fluxing time; and D, holding

  ab 23 17 27 time. The four factors are varied in a 2 4 factorial exper- c 17 25 21 iment with two castings taken at each factor combina- ac 16 19 tion. The 32 castings were made in random order. The bc 24 23 29 following table shows the data and an ANOVA table is

  abc

  28 25 20 given in Figure 15.8 on page 610. Discuss the effects of the factors and their interactions on the concentration of the phosphorus compound.

  15.2 In an experiment conducted by the Mining Engi- neering Department at Virginia Tech to study a partic-

  Weight

  ular filtering system for coal, a coagulant was added to

  Treatment

  of Phosphorus Compound

  a solution in a tank containing coal and sludge, which

  Combination

  Rep 1

  Rep 2 Total

  was then placed in a recirculation system in order that

  the coal could be washed. Three factors were varied in

  a 28.5 31.4 59.9

  the experimental process:

  b 24.5 25.6 50.1

  Factor A:

  percent solids circulated initially

  ab 25.9 27.2 53.1

  in the overflow

  c 24.8 23.4 48.2

  Factor B:

  flow rate of the polymer

  ac 26.9 23.8 50.7

  Factor C:

  pH of the tank

  The amount of solids in the underflow of the cleans-

  d 31.7 33.5 65.2

  ing system determines how clean the coal has become.

  ad 24.6 26.2 50.8

  Two levels of each factor were used and two experi-

  3 bd 27.6 30.6 mental runs were made for each of the 2 58.2 = 8 combi- abd 26.3 27.8 54.1

  nations. The response measurements in percent solids

  cd 29.9 27.7 57.6

  by weight in the underflow of the circulation system

  acd

  are as specified in the following table:

  Combination Replication 1 Replication 2

  a 21.42 21.35 15.4 A preliminary experiment is conducted to study b 12.66 12.56 the effects of four factors and their interactions on the ab 18.27 16.62 output of a certain machining operation. Two runs are c 7.93 7.88 made at each of the treatment combinations in order to ac 13.18 12.87 supply a measure of pure experimental error. Two lev- bc 6.51 6.26 els of each factor are used, resulting in the data shown

  abc

  18.23 17.83 next page. Make tests on all main effects and interac-

  Chapter 15 2 k Factorial Experiments and Fractions

  Source of

  Sum of Degrees of Mean Computed

  Variation

  Effects

  Squares Freedom Square

  f P-Value

  Main effect :

  A −1.2000

  B −1.2250

  C −2.2250

  Two-factor interaction :

  AD −1.3250

  Three-factor interaction :

  Four-factor interaction :

  Figure 15.8: ANOVA table for Exercise 15.3.

  Treatment

  in an analysis with certain levels of certain processing

  variables. The data are shown below.

  7.9 9.6 Phys. Mixing Blade Nitrogen a 9.1 10.2 Obs. State Time Speed Condition Aluminum

  b 8.6 5.8

  c 10.4 12.0 1 2 16.3 d 7.1 8.3 2 1 2 16.0 ab 11.1 12.3 3 1 16.2 ac 16.4 15.5 4 1 2 1 2 16.1 ad 7.1 8.7 5 1 2 16.0 bc 12.6 15.2 6 1 2 1 16.0 bd 4.7 5.8 7 1 2 1 15.5 cd 7.4 10.9 8 1 2 1 15.9

  15.5 In the study An X-Ray Fluorescence Method for

  Analyzing Polybutadiene-Acrylic Acid (PBAA) Propel- lants (Quarterly Reports, RK-TR-62-1, Army Ord- The variables in the data are given as below. nance Missile Command), an experiment was con-

  A: mixing time

  ducted to determine whether or not there was a signif-

  level 1: 2 hours

  icant difference in the amount of aluminum obtained

  level 2: 4 hours

  Exercises

  B: blade speed

  of interpretation, show two AD interaction plots, one

  level 1: 36 rpm

  for B = −1 and the other for B = +1. From the ap-

  level 2: 78 rpm

  pearance of these, give an interpretation of the ABD

  C: condition of nitrogen passed over propellant

  interaction.

  level 1: dry level 2: 72 relative humidity

  15.9 Consider Exercise 15.6. Use a +1 and −1 scaling

  D: physical state of propellant

  for “high” and “low,” respectively, and do a multiple

  level 1: uncured

  linear regression with the model

  level 2: cured

  Y i =β 0 +β 1 x 1i +β 2 x 2i +β 12 x 1i x 2i + i ,

  Assuming all three- and four-factor interactions to be negligible, analyze the data. Use a 0.05 level of signif- with x 1i = reactant concentration ( −1, +1) and x 2i = icance. Write a brief report summarizing the findings. feed rate ( −1, +1).

  (a) Compute regression coefficients.

  15.6 It is important to study the effect of the concen- (b) How do the coefficients b 1 ,b 2 , and b 12 relate to the

  tration of the reactant and the feed rate on the viscosity

  effects you found in Exercise 15.6(a)?

  of the product from a chemical process. Let the reac-

  tant concentration be factor A, at levels 15 and 25. (c) In your regression analysis, do t-tests on b 1 ,b 2 , and

  Let the feed rate be factor B, with levels 20 lbhr and

  b 12 . How do these test results relate to those in Ex-

  30 lbhr. The experiment involves two experimental

  ercise 15.6(b) and (c)?

  runs at each of the four combinations (L = low and H = high). The viscosity readings are as follows.

  15.10 Consider Exercise 15.5. Compute all 15 effects and do normal probability plots of the effects.

  H 132

  (a) Does it appear as if your assumption of negligible

  three- and four-factor interactions has merit? (b) Are the results of the effect plots consistent with

  B

  what you communicated about the importance of main effects and two-factor interactions in your summary report?

  15.11 In Myers, Montgomery, and Anderson-Cook

  L

  (2009), a data set is discussed in which a 2 H 3 factorial A is used by an engineer to study the effects of cutting

  (a) Assuming a model containing two main effects and speed (A), tool geometry (B), and cutting angle (C)

  an interaction, calculate the three effects. Do you on the life (in hours) of a machine tool. Two levels of each factor are chosen, and duplicates are run at each have any interpretation at this point?

  design point with the order of the runs being random.

  (b) Do an analysis of variance and test for interaction. The data are presented here.

  Give conclusions.

  A B C

  Life

  (c) Test for main effects and give final conclusions re-

  garding the importance of all these effects.

  15.7 Consider Exercise 15.3. It is of interest to the

  researcher to learn not only that AD, BC, and possibly

  ab +

  − 35, 47

  AB are important, but also what they mean scientif-

  c −

  ically. Show two-dimensional interaction plots for all

  three and give an interpretation.

  15.8 Consider Exercise 15.3 once again. Three-factor (a) Calculate all seven effects. Which appear, based interactions are often not significant, and even if they

  on their magnitude, to be important?

  are, they are difficult to interpret. The interaction (b) Do an analysis of variance and observe P -values. ABD appears to be important. To gain some sense

  (c) Do your results in (a) and (b) agree?

  Chapter 15 2 k Factorial Experiments and Fractions

  (d) The engineer felt confident that cutting speed and With only these runs, we have the signs for contrasts

  cutting angle should interact. If this interaction is given by significant, draw an interaction plot and discuss the engineering meaning of the interaction.

  15.12 Consider Exercise 15.11. Suppose there was

  some experimental difficulty in making the runs. In

  fact, the total experiment had to be halted after only 4

  abc

  runs. As a result, the abbreviated experiment is given Comment. In your comments, determine whether or by

  not the contrasts are orthogonal. Which are and which

  Life

  are not? Are main effects orthogonal to each other? In this abbreviated experiment (called a fractional facto-

  a 43 rial), can we study interactions independent of main b 35 effects? Is it a useful experiment if we are convinced c 44 that interactions are negligible? Explain.

  abc