LeRoy A. Franklin
Rose-Hulman Institute of Technology
Belva J. Cooley
University of Montana
Journal of Statistics Education Volume 10, Number 1 (2002)
Copyright © 2002 by LeRoy A. Franklin and Belva J. Cooley, all rights reserved.
This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the authors and advance notification of the editor.
Key Words: Cell means; Interaction; Parameter estimation; Power; Replication; Sample size.
This paper presents a data set based on an industrial case study using design of experiments. The data set is pedagogically rich because it has a rather large total sample size from an industrial setting that naturally yields a large third order interaction term. The experiment is a 23 design and is initially presented with no replications. The sample size of the data is then doubled and the analysis repeated, comparing these results with previous results. The process is repeated until eight replications are available for each combination of factors and all parameters are estimated. With eight replications, the analysis shows all main effects and all interactions are statistically significant at the a = 0.05 level. With smaller sample sizes, various main effects and interactions are not found to be statistically significant. Through this presentation the instructor can lead students in discussions about the effect of increased sample sizes, power, statistical significance (or insignificance), interaction terms, Type I and Type II errors as well as the importance and the role of the error term. In addition, students can manipulate the data set in a computer laboratory setting to illustrate many of the concepts inherent in the design of experiments and analysis of variance.
Concepts such as power, Type I and Type II errors, model assumptions, statistical significance, statistical non-significance, estimates of the model variance, and the impact these concepts have on interpretation of the data are crucial features for analysis of variance and design of experiments. All too frequently, these concepts are presented in a disjointed fashion without showing their interrelationships.
The authors have found that an "in class discussion" of a data set (based on an industrial case study) which is repetitively presented with progressively larger samples has been a very helpful tool to integrate and synthesize many of these concepts. This discussion occurs after the fundamental concepts, calculations and a few simple examples have been presented to the class. The case is used in an intermediate course in applied statistics with an audience of senior business majors, MBA students and various graduate-level science majors. When it has been used, the reception of the class has been enthusiastic and, although student feedback has been informal, several students have encouraged the authors to continue using such an exercise.
A company engaged in the manufacture of large industrial fans recently acquired a smaller competitor that was also engaged in manufacturing fans. However the two companies had very different manufacturing methods and each company championed its method as superior to the other. The merging of the two companies presented an important opportunity to choose "best practices," standardize operations, and reduce inventories. Since many factors were potentially involved and the decision to standardize on a particular process had substantial long run manufacturing and financial ramifications, the authors were contacted to assist in the design and analysis of the experiments.
After much discussion, the company engineers agreed to look at three important factors in the manufacture of the fans: the type of hole in the fan "spyder" (that is, blades), the type of barrel onto which the spyder was placed, and the type of assembly method for these two components. Each of these factors was qualitative in nature and at two levels. Thus the experiments became a straightforward 23 design that is presented in most design of experiments texts (Cochran and Cox 1957; Box, Hunter, and Hunter 1978; Hicks 1993). Because of industrial and competitive considerations, company representatives have requested that the precise nature of these factor levels not be presented. The two levels of each factor are simply coded -1 and 1. However, Figure 1 "suggests" the type and nature of the factors.
The single most important metric in measuring the quality of a manufactured fan was the torque (in foot-pounds) at which the spyder could be broken from the barrel. When the fan blade broke free of the hub, the shaft of the turning motor would rotate but the blades would not. Such a failure had serious ramifications for the large industrial units being manufactured.
Figure 1. Factors in the manufacture of the fans.
The key technique used in this discussion is the use of one data set that is revisited a total of four times (about all that a single class period will allow). Each time the experiment is revisited the sample size is doubled so that the same experiment is done with 8, then 16, then 32, and finally 64 data points. A powerful approach to teaching this material is to allow students to work through the analysis in a computer laboratory, using prepared diskettes containing the datasets. Students can also get the data from the Internet. If the hands-on approach is not practical, the authors recommend that tables containing the data and the analysis be used as a class handout to focus each student’s attention on interpretation and synthesis of concepts and to deemphasize the hand calculations necessary to achieve those results.
The first experiment is comprised of one observation for each combination of hole type, assembly method, and barrel design. Table 1 shows the MINITAB® results for ANOVA and the parameter estimates for a sample size of eight. Using the printed cell sample means and the standard set of equations, the class can compute the estimates of each of the model terms that are shown in the table. The three-factor interaction term is used to provide an estimate of the error term with one degree of freedom, a course of action sometimes recommended by statisticians (Montgomery 2001). Using the calculated values for F or the p-values, students will find that only the surface of the barrel is statistically significant at the 0.05 significance level. The instructor might note that when MINITAB® reports p-values of 0.000 they are not truly "zero" p-values but rather statements that the actual p-values are less than 0.0005.
The instructor can lead discussion concerning the use of the three-way interaction term as a surrogate for the error term. The analyst is, in fact, assuming that the three-way interaction term is not significant. If the assumption is incorrect and the interaction term is large, main effects of substantial size may be "hidden." Since the fans were destroyed in the testing process, the engineers may have been tempted to test a small number of fans. Had they stopped at eight and used the analysis in Table 1, they may have concluded that only the shape of the barrel had a significant effect on the strength of the fan. They could have concluded that neither the shape of the hole in the spyder nor the assembly method was important. Therefore, inventory and equipment using both types of hole and both types of assembly method could have been routinely used to produce fans. The analysis shows that the average breaking torque of the fans using the best barrel shape was 176.25 foot-pounds.
Analysis of Variance Observed Source DF SS MS F p Power Hole (A) 1 1035.1 1035.1 18.78 0.144 0.266 Assembly (B) 1 1485.1 1485.1 26.94 0.121 0.316 Barrel (C) 1 26565.1 26565.1 481.91 0.029 0.940 Hole*Assembly(A*B) 1 325.1 325.1 5.90 0.249 0.151 Hole*Barrel(A*C) 1 55.1 55.1 1.00 0.500 0.073 Assembly*Barrel(B*C)1 45.1 45.1 0.82 0.532 0.069 Error 1 55.1 55.1 Total 7 29565.9 Hole N Means -1 4 107.25 1 4 130.00 Assembly N Means -1 4 132.25 1 4 105.00 Barrel N Means -1 4 61.00 1 4 176.25 Hole Assembly N Means -1 -1 2 114.50 -1 1 2 100.00 1 -1 2 150.00 1 1 2 110.00 Hole Barrel N Means -1 -1 2 47.00 -1 1 2 167.50 1 -1 2 75.00 1 1 2 185.00 Assembly Barrel N Means -1 -1 2 77.00 -1 1 2 187.50 1 -1 2 45.00 1 1 2 165.00 Estimates of the Model Parameters (all units are in ft-pounds of breaking torque) µ = 118.6 A-1 = -11.4 A1 = 11.4 B-1 = 13.6 B1 = -13.6 C-1 = -57.6 C1 = 57.6 AB-1-1 = -6.4 AB-11 = 6.4 AB1-1 = 6.4 AB11 = -6.4 AC-1-1 = -2.6 AC-11 = 2.6 AC1-1 = 2.6 AC11 = -2.6 BC-1-1 = 2.4 BC-11 = -2.4 BC1-1 = -2.4 BC11 = 2.4
This is a good time for the instructor to review Type I and Type II errors. The hypothesis being evaluated is that the treatment effects are equal to zero. If a conclusion is made that an effect is significant when it is not, the analyst has committed a Type I error. The probability of a Type I error, a , can be controlled by setting a value of a = 0.05, for instance, in conducting the hypothesis test. The p-values, in the table above, give the probability of obtaining more extreme test results by chance if the null hypothesis was true. A Type II error is committed when the analyst concludes that the treatment effects are zero when an effect actually does exist. Students should understand that the probability of committing a Type II error is b, while the probability of identifying a significant effect when there actually is one is 1- b (the power of the test). For a given level of a, the power to detect real differences depends on the size of the sample taken and the magnitude of the difference that must be detected. At this point the instructor should emphasize that factors and terms displaying p-values larger than 0.05 and judged not statistically significant could actually have a significant effect in producing stronger fans. The sample size may simply not be large enough to detect their actual (non-zero) effect.
The last column in the ANOVA portion of Table 1 gives an approximation of the observed power for the hypothesis test, assuming a = 0.05. (SPSS® provides these estimates as optional output.) With the exception of the variable for barrel shape, all power estimates are quite low. For instance, if the observed effect size for the first variable, hole, is a true difference, the null hypothesis would be rejected only 26.6% of the time in repeated experiments of this design. If the power is small, the analyst cannot be confident in a decision not to reject the null hypothesis.
The instructor should next challenge the students to predict what might happen if the sample size is doubled to 16 observations, two at each of the factor combinations. Table 2 gives the results of the MINITAB® analysis for a sample size of 16. Students should note that replicating the experiment allows an estimate of an error term with eight degrees of freedom. The mean squared error is now smaller than in the previous experiment. Doubling the sample size results in an ANOVA showing all main effects to be significant. In addition, the
Analysis of Variance Observed Source DF SS MS F p Power Hole(A) 1 2025.0 2025.0 58.91 0.000 1.000 Assembly(B) 1 3080.3 3080.3 89.61 0.000 1.000 Barrel(C) 1 47089.0 47089.0 1369.86 0.000 1.000 Hole*Assembly(A*B) 1 870.2 870.2 25.32 0.001 0.992 Hole*Barrel(A*C) 1 144.0 144.0 4.19 0.075 0.437 Assembly*Barrel(B*C) 1 56.2 56.2 1.64 0.237 0.204 Hole*Assembly*Barrel 306.3 306.3 8.91 0.017 0.744 (A*B*C) Error 8 275.0 34.4 Total 15 53846.0 Hole N Means -1 8 106.25 1 8 128.75 Assembly N Means -1 8 131.38 1 8 103.62 Barrel N Means -1 8 63.25 1 8 171.75 Hole Assembly N Means -1 -1 4 112.75 -1 1 4 99.75 1 -1 4 150.00 1 1 4 107.50 Hole Barrel N Means -1 -1 4 49.00 -1 1 4 163.50 1 -1 4 77.50 1 1 4 180.00 Assembly Barrel N Means -1 -1 4 79.00 -1 1 4 183.75 1 -1 4 47.50 1 1 4 159.75 Hole Assembly Barrel N Means -1 -1 -1 2 53.00 -1 -1 1 2 172.50 -1 1 -1 2 45.00 -1 1 1 2 154.50 1 -1 -1 2 105.00 1 -1 1 2 195.00 1 1 -1 2 50.00 1 1 1 2 165.00 Estimates of the Model Parameters µ = 117.5 A-1 = -11.3 A1 = 11.3 B-1 = 13.9 B1 = -13.9 C-1 = -54.3 C1 = 54.3 AB-1-1 = -7.4 AB-11 = 7.4 AB1-1 = 7.4 AB11 = -7.4 AC-1-1 = -3.0 AC-11 = 3.0 AC1-1 = 3.0 AC11 = -3.0 BC-1-1 = 1.9 BC-11 = -1.9 BC1-1 = -1.9 BC11 = 1.9 ABC-1-1-1 = -4.4 ABC-1-11 = 4.4 ABC-11-1 = 4.4 ABC-111 = -4.4 ABC1-1-1 = 4.4 ABC1-11 = -4.4 ABC11-1 = -4.4 ABC111 = 4.4
It is helpful to recalculate the estimates of the model parameters from the larger sample and note that these values are close to but not identical to those calculated from the original sample of eight. Prior to experimentation, the company engineers debated how much of a difference in treatment effects would be of material or practical difference. For instance, a person’s fingers can easily exert +/- 1.5 foot-pounds of torque in placing a nut on a bolt. The engineers finally decided that they wanted to detect differences of 10 foot-pounds or more. Each of the main effects in this experiment is estimated at greater than 10 foot-pounds.
The instructor should point out the new estimates for observed power in Table 2 and how they differ from those in Table 1. The
Again, the instructor should ask the students to predict what will happen if the sample
size is doubled to 32 (four observations at each treatment combination).
Table 3 presents the results of the
third experiment. The instructor should point out that the larger sample size results
in more degrees of freedom for the error sum of squares, resulting in an estimate of
mean square error of 30 instead of 34.4. The
Analysis of Variance Observed Source DF SS MS F p Power Hole(A) 1 4028 4028 134.48 0.000 1.000 Assembly(B) 1 6699 6699 223.69 0.000 1.000 Barrel(C) 1 93853 93853 3133.87 0.000 1.000 Hole*Assembly(A*B) 1 1526 1526 50.96 0.000 1.000 Hole*Barrel(A*C) 1 185 185 6.19 0.020 0.665 Assembly*Barrel(B*C) 1 116 116 3.88 0.060 0.473 Hole*Assembly*Barrel 639 639 21.34 0.000 0.993 (A*B*C) Error 24 719 30 Total 31 107765 Hole N Means -1 16 105.81 1 16 128.25 Assembly N Means -1 16 131.50 1 16 102.56 Barrel N Means -1 16 62.88 1 16 171.19 Hole Assembly N Means -1 -1 8 113.37 -1 1 8 98.25 1 -1 8 149.62 1 1 8 106.87 Hole Barrel N Means -1 -1 8 49.25 -1 1 8 162.37 1 -1 8 76.50 1 1 8 180.00 Assembly Barrel N Means -1 -1 8 79.25 -1 1 8 183.75 1 -1 8 46.50 1 1 8 158.63 Hole Assembly Barrel N Means -1 -1 -1 4 54.25 -1 -1 1 4 172.50 -1 1 -1 4 44.25 -1 1 1 4 152.25 1 -1 -1 4 104.25 1 -1 1 4 195.00 1 1 -1 4 48.75 1 1 1 4 165.00 Estimates of the Model Parameters µ = 117.0 A-1 = -11.2 A1 = 11.2 B-1 = 14.5 B1 = -14.5 C-1 = -54.2 C1 = 54.2 AB-1-1 = -6.9 AB-11 = 6.9 AB1-1 = 6.9 AB11 = -6.9 AC-1-1 = -2.4 AC-11 = 2.4 AC1-1 = 2.4 AC11 = -2.4 BC-1-1 = 1.9 BC-11 = -1.9 BC1-1 = -1.9 BC11 = 1.9 ABC-1-1-1 = -4.5 ABC-1-11 = 4.5 ABC-11-1 = 4.5 ABC-111 = -4.5 ABC1-1-1 = 4.5 ABC1-11 = -4.5 ABC11-1 = -4.5 ABC111 = 4.5
Table 4 gives the ANOVA results and parameter estimates for a sample size of 64 (8 observations for each treatment combination). With a sample size of 64, all main effects and all treatment interactions are significant at a = 0.05. This is a good time to introduce the concept of inflating the overall probability of Type I error by making several hypothesis tests. The instructor might suggest using a more conservative value for the significance level, such as 0.01, a Bonferroni-type correction. With seven tests (three main effects and four interactions), the probability of making at least one Type I error would be 1 - (.99)7 or approximately 0.07. The instructor should ask the students to review the results of the four experiments to determine what their conclusions would have been using a = 0.01.
Analysis of Variance Observed Source DF SS MS F p Power Hole(A) 1 8258 8258 266.68 0.000 1.000 Assembly(B) 1 13369 13369 431.73 0.000 1.000 Barrel(C) 1 193050 193050 6234.17 0.000 1.000 Hole*Assembly(A*B) 1 2849 2849 92.00 0.000 1.000 Hole*Barrel(A*C) 1 594 594 19.19 0.000 0.990 Assembly*Barrel(B*C) 1 135 135 4.36 0.041 0.537 Hole*Assembly*Barrel 1397 1397 45.11 0.000 1.000 (A*B*C) Error 56 1734 31 Total 63 221387 Hole N Means -1 32 106.66 1 32 129.38 Assembly N Means -1 32 132.47 1 32 103.56 Barrel N Means -1 32 63.09 1 32 172.94 Hole Assembly N Means -1 -1 16 114.44 -1 1 16 98.87 1 -1 16 150.50 1 1 16 108.25 Hole Barrel N Means -1 -1 16 48.69 -1 1 16 164.63 1 -1 16 77.50 1 1 16 181.25 Assembly Barrel N Means -1 -1 16 79.00 -1 1 16 185.94 1 -1 16 47.19 1 1 16 159.94 Hole Assembly Barrel N Means -1 -1 -1 8 53.25 -1 -1 1 8 175.62 -1 1 -1 8 44.13 -1 1 1 8 153.62 1 -1 -1 8 104.75 1 -1 1 8 196.25 1 1 -1 8 50.25 1 1 1 8 166.25 Estimates of the Model Parameters µ = 118.0 A-1 = -11.4 A1 = 11.4 B-1 = 14.5 B1 = -14.5 C-1 = -54.9 C1 = 54.9 AB-1-1 = -6.7 AB-11 = 6.7 AB1-1 = 6.7 AB11 = -6.7 AC-1-1 = -3.0 AC-11 = 3.0 AC1-1 = 3.0 AC11 = -3.0 BC-1-1 = 1.5 BC-11 = -1.5 BC1-1 = -1.5 BC11 = 1.5 ABC-1-1-1 = -4.7 ABC-1-11 = 4.7 ABC-11-1 = 4.7 ABC-111 = -4.7 ABC1-1-1 = 4.7 ABC1-11 = -4.7 ABC11-1 = -4.7 ABC111 = 4.7
The instructor should again call the students’ attention to the estimated means for the various treatment combinations. With the original sample size of eight, only the barrel shape was significant. If the other factors are ignored and the engineers concentrate only on the best setting for barrel shape, the average breaking torque is estimated to be 176.25. The larger sample size of 64 helps identify significant differences for all factors. If the engineers choose the best settings for all factors, the average breaking torque is estimated at 196.25. Although destructive testing was necessary, the larger sample size helped identify a combination of factor settings that substantially improved the strength of the industrial fans. Figures 2 and 3 show plots of the sample means for the eight different factor combinations.
Figure 2. Breaking Torque (for Hole = -1).
Figure 3. Breaking Torque (for Hole = +1)
Looking back to the beginning of the exercise when only eight fans were tested, the third-order interaction was used as a surrogate for error in the analysis since the experiment was unreplicated. Because the sum of squares for that interaction was large and there was only one degree of freedom, only one main effect (barrel shape) was significant. In fact, using the large interaction term as error hid the impact that the hole shape and assembly method had in producing a stronger fan. A better practice may have been to combine the interactions terms to serve as error and only screen for main effects in the model as seen in Table 5. All main effects are significant at the a = 0.05 level, which is consistent with the findings of the n = 64 experiment.
Analysis of Variance Observed Source DF SS MS F p Power Hole (A) 1 1035 1035 8.617 0.043 0.602 Assembly (B) 1 1485 1485 12.363 0.025 0.749 Barrel (C) 1 26565 26565 221.15 0.000 1.000 Error 4 481 31 Total 7 29566
As a possible homework assignment, the instructor may want each of the students to randomly choose observations from the full dataset to create new datasets of 8, 16, and 32. Requiring this will result in each student’s work being slightly different. However the authors have run several different subsetting patterns. Although the associated p-values do change moderately, it is rare to have a dramatic difference, and the essential results remain the same. However, the instructor may wish the students to explain the cause of any differences that do occur in their analysis as part of the homework assignment.
The instructor may also choose to remind the students that assumptions should be checked when using ANOVA techniques. In this case, a Kolmogorov-Smirnov test of normality on the error terms failed to find a significant non-normality. Tests for homogeneity of variances were mixed. Bartlett's Test resulted in a p-value of 0.116, indicating no significant heterogeneity of variances. However, Levene's Test resulted in a p-value of 0.000, indicating that the variances are not the same at all factor levels. Discussions of these two tests can be found in Montgomery (2001). Box (1954) discusses the robustness of ANOVA to departures from the homogeneity of variances assumption if the sample sizes are equal at all factor levels. In this case, the smallest standard deviation of breaking torques for a factor level is 3.01, while the largest is 8.78. The standard deviations can differ by as much as a factor of three without invalidating the results of the ANOVA analysis.
The instructor could also lead a discussion about other issues, for which information is not available, that may impact the decision. Students might suggest that costs of the various methods be collected and compared. They might suggest that an engineer review the recommended design. Students might ask if breaking torque is the appropriate measure to use and, if so, what value of breaking torque do the customers require. They might explore the existence of other measures of quality that might be appropriate. They should also look at the standard deviation of each of the combinations to determine if the combination of factors that gave the highest breaking strength also resulted in a relatively small standard deviation. A complete discussion of each of these issues will lead to discussion of many more topics in the quality literature and substantially increase the amount of time necessary to complete the exercise. The instructor may wish to assign these topics to small groups of students for further discussion or for short papers.
The repeated analysis of data based on an industrial case study by successively increasing the total sample size has been found to be a useful tool by the authors in summarizing and unifying many analysis of variance concepts. Concepts such as power, statistical significance, statistical non-significance, material differences, Type I and Type II errors, and the effects of sample size are clarified and illustrated progressively.
The file fadata.dat.txt contains the raw data. The file fandata.txt is a documentation file containing a brief description of the dataset.
Data columns 1, 5, 9, and 13 are values for the two types of hole tested (coded -1 and 1)
Data columns 2, 6, 10, and 14 are values for the two types of assembly method used (coded -1 and 1)
Data columns 3, 7, 11, and 15 are values for the two barrel shapes used (coded -1 and 1)
Data columns 4, 8, 12, and 16 are the breaking torques in foot-pounds for the various factor combinations
Data columns 1 - 4 give the results of the experiment with one observation for each factor combination
Data columns 5 - 8 give the results of the experiment with two replications for each factor combination
Data columns 9 - 12 give the results for the experiment with four replications for each factor combination
Data columns 13 - 16 give the results for the experiment with eight replications for each factor combination
The authors wish to thank the referees, the associate editor, and the editor for their constructive comments leading to improvements in this paper.
Box, G. E. P., Hunter, W. G., and Hunter, J. S. (1978), Statistics for Experimenters, New York: John Wiley and Sons.
Box, G. E. P. "Effects of Inequality of Variance and Correlation Between Errors in the Two Way Classification," Annals of Mathematical Statistics, 25, 484-498.
Cochran, W. G., and Cox, G. M. (1957), Experimental Designs, New York: John Wiley and Sons.
Desu, M. M., and Raghavarao, D. (1990), Sample Size Methodology, Boston: Academic Press.
Hicks, C. R. (1993), Fundamental Concepts in the Design of Experiments (4th ed.) New York: Saunders College Publishing.
Montgomery, D. C. (2001), Design and Analysis of Experiments (5th ed.) New York: John Wiley and Sons.
Nelson, L. S. (1985), "Sample Size Tables for Analysis of Variance," Journal of Quality Technology, 17 (3), 167-169.
LeRoy A. Franklin
Rose-Hulman Institute of Technology
5500 Wabash Avenue
Terre Haute, IN 47803
Belva J. Cooley
School of Business Administration
University of Montana
Missoula, MT 59812
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