STATISTICAL APPROACHES TO EXPERIMENTAL DATA
CO M PA R ING SCREENING DESIGNS a
When applying screening designs to many-varia ble processes, proceed with caution. Though better than the univairiate approach, the designs still may be inadequate K. R. W I L L I A M S
s chemical processes become more complex, research experimental situations in which large numbers of variables may be important. The traditional way to attack such problems has been to restrict experiments to a few primary variables, selected from prior information or theory, and disregard the others which are fixed at arbitrary levels. For many problems this approach is inadequate. I n the early stages of the work, an experimental design is needed so that a large number of variables can be examined to determine which affect the results and in which direction. During the last 20 years statisticians have come to recognize the technological importance of screening designs and a number have been developed. Some designs (5,6)were published in the forties, but utility of such methods has not been recognized until recently. One recent trend has been the development of more efficient screening designs; that is, where the number of experiments is only slightly greater than the number of variables studied. Box ( I ) has used the term “saturated” to describe a design in which all of the available degrees of freedom are used to determine main effects. Chemical research workers now have available a wide variety of screening designs, differing in experimental layout and degree of saturation, for any particular problem. Recognizing that each problem is different and that no “best” design covers all cases, the question may be asked, “Are there preferences in selection?” I n a n attempt to establish guide lines for design selection, Brooks (2) used simulated problems to investigate various alternative maximum-seeking methods. He concluded that experiments should be chosen at random in factor space when the number of factors is large. However, simulated problems are unsatisfactory because a factor-response relationship and distribution and size of the error terms must be assumed. These assumptions, particularly regarding error characteristics, may not be in accord with Nature. A different approach is discussed in this article. A real experimental problem, containing 24 recognized
A workers are confronted with
variables, was studied using three separate screening designs to compare factor-response information obtained with each pattern. Aims of this work were different from those of Brooks, which were to compare maximum response values. But practically, they were similar. The specific problem studied concerned development of an epoxide adhesive system for bonding polyester cords to a natural rubber elastomer. This problem is typical of much current industrial research on products such as synthetic fibers, elastomers, adhesives, and films where many variables may have important influences. Results from each design were analyzed separately and additional experimental work was used to check validity of the predictions obtained with the three designs. I hree near-saturatrd, two-level designs were used : a fractional factorial ( 4 ) in 32 experiments; a PlackettBurman (6) in 28 experiments; and a random balance (7) in 28 experiments. r
>
Data Analysis Techniques
Analysis of data from a two-level fractional factorial or Plackett-Burman design is straightforward. The contribution of each factor is determined by adding values of the response from all experiments run at one level of a factor and substracting the response values of all experiments run at the other level. This difference, divided by the total number of experiments run, indicates influence of the factor on response. Three procedures were used to analyze the random balance design. The first was simple addition and subtraction of results as in the two orthogonal desicps. With a random balance design, lack of orthogonality can significantly bias some of the effects. Thus, if one effect has a large influence on a response, then the fact that it does not appear at its two levels an equal number of times a t the two levels of a second variable can lead to a n erroneous conclusion regarding the importapce of the second factor. (Continued on next Page) VOL.55
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This biasing of factor significance can be partially offset by correcting the results. Budne (3) recommends selecting the three most important effects, finding their average influence, then adjusting the data based on the levels of the three factors appearing in each experiment. Significance of the remaining factors is then determined by examining the adjusted data. Two recommended (3) techniques to determine the three most significant effects use scatter plots where response (adhesion level) is plotted on the ordinate and the two levels of each factor are plotted on the abscissa. One correction technique consists of choosing the three factors which show the greatest difference in median value between their two levels and then adjusting each response for these three factors. New scatter plots are made, based on the corrected data, and the three most important effects are again selected and the data adjusted a second time. This procedure continues until no more significant effects are uncovered. The complete correction procedure is described by Budne (3). Another way to select the three most important factors from the scatter plots is to use the outside count test developed by Tukey (8). This consists of counting the points in one column which exceed all points in the second column and adding the number of points in the second column which are lower than any point in the first column. If this sum is seven or more, the effect is significant at the 95% level. However, with these two selection procedures, the same variables in this problem were not chosen for correction. Thus, data adjustment followed two different paths, and different sets of predicted significant effects were obtained. The Adhesive System
Polyester fiber has a number of desirable characteristics as a reinforcement in rubber articles; however, its bond strength to natural and SBR-type elastomers is low when common adhesives of the resorcinol-formaldehyde-latex type are used. Therefore, a number of other adhesive systems was studied, and one which
A, AfD ROLL
showed promise was an epoxide-based adhesive. In the procedure used (see below), polyester cord was passed through a preheating oven and then a solution containing an epoxide and a surfactant was applied. Next, the cord was stretched and dried before application of a second solution composed of a mixture of two vinylpyridine terpolymer latices and an epoxide curing agent. Finally, the cord was dried at constant length and wound onto a spool. A single-end strip adhesion test was used as the principal means of determining adhesion performance. Single cords were cured under tension into the surface of a layer of rubber inch thick. The average force to peel the cord from the strip was determined a t room temperature.
,
Experimantd Factors
The 24 variables, shown in Table I, may be considered under four broad categories: the cord, the adhesive formulation, the processing conditions, and the adhesive testing procedure. Sixteen of the variables were quantitative and the specific numerical values used are given. The two levels of the remaining qualitative variables are and - . shown as Item 6 in Table I refers to a ratio, fixed a t two levels, of two commercially available vinylpyridine terpolymer latices used in the adhesive formulations. Item 23 refers to the procedure for selecting dipped cord from a spool for use in the strip adhesion test. The standard in the designs) was sclection procedure (denoted by a to cut off consecutive 26-inch pieces from a spool. The procedure denoted by - was to discard about 3 feet of cord between each cord segment placed in the strip adhesion mold.
+
+
Experimental Proc.dun
All experiments in each design were run and tested in order, starting with the fractional factorial followed by the Plackett-Burman and then the random balance. Two variations in procedure occurred during this work. An insufficient amount of one cord lot was obtained so I,10. 1 DRAW ROLL
C, NO. 2 DRAW ROLL
NO. 1 OVEN
h
6MY CORD SMLl 30
fFOXlDf A ~ U T I O N
INDUSTRIAL AND ENGINEERING C H E M I S T R Y
no. 2 OYfW
that lots A and B were used in the fractional factorial program, but lots A and C were used with the other designs and in the subsequent follow-up program. Also, an insufficient quantity of old rubber stock was retained in the fractional factorial program so that this factor was not studied with this design. Following completion of the three screening design programs, further studies were made. Two path-ofsteepest-ascent programs were run, aimed at locating the preferred levels for some of the significant variables uncovered in the screening studies. A third test series was run to confirm the significance of additional factors from the screening studies; therefore, these factors were studied at the same levels as in the original work. I n this latter test, several factors, whose importance to adhesion level was already established, were fixed at one of the levels used in the original screening work. The first path-of-steepest-ascent test was a 12-experiment Plackett-Burman design in which six factors were investigated : epoxide concentration, latex ratio, and second-oven temperature mixing time, latex concentration, and surfactant concentration. The second test series was a factorial design in eight experiments where the number of variables was reduced to five: epoxide concentration, surfactant concentration, latex concentration, latex ratio, and second-oven temperature. The final test series was a 16-experiment fractional factorial design in which cord lot, mixing method, curing agent concentration, preliminary drying, first-oven temperature, applied stretch, rubber curing temperature, and rubber curing time were investigated. Discussion
From the follow-up studies it was obvious that secondoven temperature, mixing tirne, rubber cure temperature, preliminary drying, rubber cure tension, latex ratio, surfactant concentration, and curing agent concentration were significant. At the levels considered in the original work, epoxide concentration, mixing method, and latex solids level do not influence adhesion. Cord lot, first-oven temperature, and applied stretch affect adhesion but not significantly at the 9Oy0level. However, these effects are real because follow-up tests of these variables showed the same direction of departure as the original designs. Table I1 shows those effects which the screening designs indicated were significant at the 90% level together with a list, in estimated order of importance, of factors found siynificant in the follow-up work. T o compare designs, we used a 90% confidence limit in the orthogonal designs. However, an 80% limit probably would be better if additional confirmatory work were a part of the program. AUTHOR K . R. Williams is a Research Associate, Textile Fibers Department, Chestnut R u n Laboratory, E. I. du Pont de Nemours &? Go., Inc., Wilmington, Del. T . C. Mayberry i s responsible for iesearch on the epoxide adhesive system discussed here.
TABLE I. 1.
2. 3. 4. 5.
6. 7. 8.
EXPERIMENTAL FACTORS Cord Level Two lots ( and -)
Cord lot Adhesive System Epoxide concn., yo Surfactant concn., % Curing agent concn., % Latex concn., % Ratio of vinylpyridine terpolymer elastomers Mixing time (epoxide surfactant), min. Mixing method (epoxide surfactant)
+
+
9.
Adhesive age at time of application
IO.
Processing Conditions Preliminary drying
11. 12. 13. 14. 15.
Cord tension before adhesive application, lb. Applied stretch in ovens, 70 First-oven temp., O F. Time in first oven, sec. Cord travel in first and second ovens
16. Second-oven temp., O F. 17. Time in second oven, sec. 18. Quantity of adhesive in dip troughs
19. 20. 21. 22. 23.
Adhesion Testing Cure time, min. Cure temp., O F. Quantity of rubber in curing mold Cord tension during curing, Ib. Selection of dip cord for testing
24.
Age of rubber stock
+
0.3, 0.5
4, 6
1.8, 2 . 3 14, 18 0.6, 0 . 8
4, 6 2 types of stirrers ( + and -) New, old Yes, no ( and -) 0.2, 1 . 0
+
3.0, 4 . 5 275, 325 50, 70 Long, short ( + a n d -) 425, 475 50, 70 Large, small ( + and - ) 55, 65 293, 310
+, -
0.5, 1 . 0 2 procedures ( + a n d -1 New, old
Three generalizations can be made about the results: Several factors influenced adhesion level ; all designs located the most important variable, second-oven temperature; and no design uncovered most or all of the significant factors. The unadjusted random balance data show a large number of significant factors because nonorthogonality of the design inflates significance. Thus, these data contain appreciable misinformation which must be screened out. When adjusted, however, according to the two procedures outlined previously, the number of significant factors was reduced to three when medians were used, and to four when the outside count test was used. Except for second-oven temperature, the two adjustment procedures predicted different sets of significant factors. Although the outside count data adjustment procedure indicated that amount of applied stretch was significant, it predicted the wrong direction for adjustment. This is undesirable because follow-up studies are centered in a n undesirable region of factor space and in subsequent work considerable back-tracking must be done. Also random balance designs require a long time for data analysis-roughly 10 times longer than with orthogonal designs. Several complete sets of scatter plots YOL. 5 5
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TABLE 1 1 .
SIGNIFICANT EFFECTS FROM SCREENING DESIGNS AND SUBSEQUENT STUDIES
90% significance level for orthogonal designs, 0.13 lb. Random Balance I
Fractional Factorial
1
Placke f t - B u r m a n
2nd-oven temp. 2nd-oven time
I‘
Using M e d i a n s
Unadjusted Data
Using Outside Count
Effect, 1b.
Facto?
Effect, Ib.
0.46
2nd-oven temp.
0.42
0.17
Cure temp. Adh. quant. Mix time Cure tension Applied stretch Latex ratio Cord lot
Piedicted direction t o improve adhen
0.25 0.20 0.18 0.17 0.16 0.14 0.14
Factor
2nd.o~-entemp. Surfactant concn. Preliminary drying 1st-oven temp. Pretension Dip age Applied stretch“ Mix time 2nd-oven time Cord selection
1
I
I I ~
effect, lb.
0 28 0 23 0 23
0 21 0 21 0 17
0 14
Subsequent Studies
Factor
2nd-oven temp. Preliminary drying 1st-oven temp.
’
0 33 2 n d - o ~ e n t e m p . 0 31 1 Pretension 1
Sui factant
i
10
10
2nd-oven temp. Preliminary drying Curing temp. Mixing timr Curing agent concn. Curing tensio:i Latex ratio Surfact ant concn. 1st-oven temp. Applicd rtretch Cord lot
war incorrect in faridom balance desz
must be made along with one or or more data adjustment calculations. The fractional factorial design showed only t\vo factors (second-oven temperature and second-oven time) to be significant at the 90Yc level. Sec.ond-oven time was not checked in subsequent work so its influence on adhesion level was not defined. Three other factors, shown in Table I (surfactant concentration, cord selection, and curing agent concentration), had effects close to the 90Yc significance level, and if this design alone had been used in a screening program, these three factors would probably have been selected for further study. The small number of significant factors found with the fractional factorial design may have resulted partly because two-factor interactions confounded main effects. The Plackett-Burman design indicated that eight factors affected adhesion, of which seven were shown correct by subsequent work. The eighth factor, adhesive quantity, was not studied further. Of the three designs studied, the most information was obtained from the Plackett-Burman design. The two orthogonal designs were combined to see if prediction could be improved. However, two factors, cord lot and rubber age, could not be considered in the pooled data because the same cord lots were not used in both tests and rubber age was not studied in the factorial design. The 90% confidence level for the pooled data is narrower (=k0.095), because the standard deviation for the effect is smaller (Seffect= 0.03 lb.) than for each indilTidua1 design; also, the factor in the t distribution is lower because of the larger sample size (60 experiments). The pooled data agreed with both subsequent work and the Plackett-Burman desiq-i.e., the three tech32
1
~
I Factor
.4djusted D a t a
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
niques indicated that the same seven factors \vex significant. Cord selection method and surfactant concentration were close to thc 90y0 significance level for the pooled data. If the fractional factorial design failed to uncover significant effects because of confounding with two-factor interactions, then probably one PlackettBurman design of 60 experiments would have yielded more information than the pooled data. The results of this work, where 45Tc of the variables significantly affected the response, clear1)- chionstrate the value of screening designs in the investigation of multivariable, high variancc systems as opposed to thc traditional univariate experimental approach. Even the use of near-saturated screening designs is not suficient when most or all of the significant factors in a system need to be determined. The best near-saturated design used failed to point out 4 of the 11 significant variables. Random designs were ineffecti1.e because their lack of orthogonality confounded main efi‘ects and led t o determination of only a few significant variables. The fractional factorial design was no better, because main effects and two-factor interactions were completely confounded. The Plackett-Burman design with sufficient number of degrees of freedom is best suited to an eyperimental situation with many variables and high variance. LITERATURE CITED ( 1 ) Box, G. E. P., Tec/tnometric.r 1, 174 (1959). ( 2 ) Brooks, S. H., Operations Res. 7, 430 (1959). ( 3 ) Budne, T. A . , Technometrics 1, 139 (1959). ( 4 ) Davies, 0. L., “Design and Analysis of Industrial Experiments,” Hafner, 1954. ( 5 ) Finnev, D. J., Annals of Eugenics 12, 291 (1945). ( 6 ) Plackett, R. L., Burman, .J. P.: Biometrika 33, 305 (1946). ( 7 ) Satterrhwaite, F. E., Technometrics 1, 111 (1359). ( 8 ) Tukey, J. W.? Ibzd., 1, p. 32.
