designOfExperimentsMontgomeryRunger

designOfExperimentsMontgomeryRunger - Design of Engineering...

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Unformatted text preview: Design of Engineering Experiments CHAPTER OUTLINE .wnmmeowmwmu ~. .rr. 7-1 THE STRATEGY CF 76.2. Blocking and Confounding . EXPERMNTATION 7-7 FRACTIONAL REPLICATION OF 7-2 FACTORIAL EXPERIMENTS A 2" DESIGN 7-3 A 2" FACTORIAL DESIGN 7—7.1 One—Half Fraction of a 2" Design ‘ 7—3.1 2.2 Example 717.2 Smaller Fractions: 2"—p ‘752 statistical Analysis Fractional Factorial Designs 7-3.3 Residual Analysis and Model 7’8 RESPONSE SURFACE METHODS Checking AND DESIGNS , 7,4. 2" DESIGN FOR k a 3 FACTORS 7-8.1 Method of Steepest Ascent SlNGLE REPLICATE OF A 21: 7-8.2 Analysis of a Second—Order DESIGN Response Surface 7-6 CENTER 130st AND 7-9 FACTORIAL EXPERIMENTS WITH BLOCKING IN 2* DESIGNS MORE THAN TWO LEVELS ' 7—6.1 Addition of Center Points 7-1 THE STRATEGY OF EXPERIMENTATION Recall from Chapter 1 that engineers conduct tests or experiments as a natural part of their work. Statistically based experimental design techniques are particularly useful in the engi- neering world for improving the performance of a manufacturing process. They also have extensive application in the development of new processes. Most processes can be described in terms of several controllable variables, such as temperature, pressure, and feed rate. By us— ing designed experiments, engineers can'determine which subset of the process variables have the mast influence on process performance. The results of such an experiment can lead to 1. Improved process yield 2. Reduced variability in the process and closer conformance to nominal or target requirements 317 3 18 CHAPTER 7 DESIGN OF ENorNEEruNo EXPERIMENTS 3. Reduced design and development time 4. Reduced cost of operation Experimental design methods are also usefiil in engineering design activities, where new products are developed and existing ones improved. Some typical applications of statistically designed experiments in engineering design include 1. Evaluation and comparison of basic design configurations 2. Evaluation of different materials 3. Selection of design parameters so that the product will work well under a wide vari- ety of field conditions (or so that the design will be robust) , 4. Determination of key product design parameters that affect product performance The use of experimental design in the engineering design process can result in products that are easier to manufacture, have better field performance and reliability than their competitors, and can be designed, developed, and produced in less time. Designed experiments are usually employed sequentially, hence the term sequential ex- perimentation. That is, the first experiment with a complex system (perhaps a manufacturing process) that has many controllable variables is ofien a screening experiment designed to de— termine which variables are most important. Subsequent experiments are then used to refine this information and determine which adjustments to these critical variables are required to improve the process. Finally, the objective of the experimenter is optimization—that is, to de- termine which levels of the critical variables result in the best process performance. This is the KISS principle: “keep it small and sequential”. When small steps are completed, the knowl- edge gained can improve the subsequent experiments. Every experiment involves a sequence of activities: Conjecture—the original hypothesis that motivates the experiment. Experiment—the test performed to investigate the, conjecture. Analysis—the statistical analysis of the data from the experiment. Conclusionflwhat has been learned about the original conjecture from the experi— ment will lead to a revised conjecture, a new experiment, and so forth. Statistical methods are essential to good experimentation. All experiments are designed experiments; some of them are poorly designed, and as a result, valuable resources are used ineffectively. Statistically designed experiments permit efficiency and economy in the exper- imental process, and the use of statistical methods in examining the data results in scientific objectivity when drawing conclusions. In this chapter we focus on experiments that include two or more factors that the experi— menter thinks may be important. The factorial experimental design will be introduced as a powerful technique for this type of problem. Generally,-in a factorial experimental design,‘ experimental trials (or runs) are performed at all combinations of factor levels. For example, if a chemical engineer is interested in investigating the effects of reaction time and reaction temperature on the yield of a process, and if two levels of time (1 and 1.5 hr) and two levels of temperature (125 and lSO°F) are considered important, a factorial experiment would consist of making the experimental runs at each of the four possible combinations of these levels of reaction time and reaction temperature. Most of the statistical concepts introduced previously can be extended to the factorial experiments of this chapter. We will also introduce several graphical methods that are useful in analyzing the data from designed experiments. 7—2 FACTORiAL EXPERIMENTS 3 1 9 7-2 FACTORIAL EXPERIMENTS When several factors are of interest in an experiment, a factorial experiment should be used. As noted previously, in these experiments factors are varied together. replicate of .expéri'sf if" I fthe atlsareinvestigated. Thus, if there are two factors A and B with a levels of factor A and [2 levels of factor B, each replicate contains all ab treatment combinations. I The effect of a factoris defined as the change in response produced by a change in the level of the factor. It is called a main effect because it refers to the primary factors in the study. For example, consider the data in Table 7-1. This is a factorial experiment with two factors, A and B, each at two levels (Am, Amgh, and Blow, Bhigh). The main efiectgfifactoniisthe differ- ence between the average response at the high level of if andhthe'average response at the low level of A, or !r M” g m+fi m+m * A— 2 — 2 .«m That is, changing factorA from the low level to the high level causes an average response in- crease of 20 units. Similarly, the main effect of B is + B=m+m_m m=m 2 2 In some experiments, the difi‘erence in response between the levels of one factor is not the same at all levels of the other factors. When this occurs, there is an interaction between the fac- tors. For example, consider the data in Table 7-2. At the low level of factor B, the A effect is A = 30 — 10 = 20 and at the high level of factor B, the A effect is A=0-20=-20 Because the effect‘of A e ends on the lev there is interaction between A and B. Table 7—1 A Factorial Experiment Table 7-2. A Factorial Experiment without Interaction with Interaction CHAPTER 7 DESIGN OF ENGINEERING EXPERIMENTS When an interaction is large, the corresponding main effects have very little practical meaning. For example, by using the data in Table 7~2, we find the main effect of A as 2 and we would be tempted to conclude that there is not factor A effect. However, when we ex~ amined the effects of A at dzfifzrent levels offirctor B, we saw that this was not the case. The ef- fect of factor A depends on the levels of factor B. Thus, knowledge of the AB interaction is more useful than knowledge of the main effect. A significant interaction can mask the signifi- cance of main effects. Consequently, when interaction is present, the main effects of the fac— tors involved in the interaction may not have much meaning. It is easy to estimate the interaction effect in factorial experiments such as those illus— trated in Tables 7-1 and 7-2. In this type of experiment, when both factors have two levels, the AB interaction efiect is the difference in the diagonal averages. This represents one—half the difference between the A effects at the two levels of B. For example, in Table 7-1, we find the AB interaction effect to be innxmmwmt‘mVAumtnw 2 As we noted before, the interaction effect in these data is very large. The concept of interaction can be illustrated graphically in several ways. Figure 7-1 is a plot of the data in Table 7-1 against the levels of A for both levels of B. Note that the Blow and Bligh lines are approximately parallel, indicating that factors A and 13 do not interact signifi- cantly. Figure 7-2 presents a similar plot for the data in Table 7-2. In this graph, the BMW and Bug}, lines are not parallel, indicating the interaction between factors A and B. Such graphical displays are called two—factor interaction plots. They are often useful in presenting the re- sults of experiments, and many computer sofiware programs used for analyzing data from de— signed experiments will construct these graphs automatically. Figures 7—3 and 7-4 present another graphical illustration of the data fi‘om Tables 7-1 and 7-2. In Fig. 7-3 we have shown a three-dimensional surface plot of the data fi'om Table "H, where the low and high levels are set at — I and 1, respectively, for both A and B. The equations Observation H N C) Observation N on O H O Alow Ahigh Aiow Factor A FactorA Figure 7'1 An interaction plot of a factorial Figure 7—2 An interaction plot of a factorial experiment, no interaction. experiment, with interaction. i. E t :5 7—2 Figure 7-3 Three-dimensional surface plot for the data from Table 7-1, showing main effects of the two factors A and B. only way to discover interactions between variables. a-time procedure, suppose that we are interested in finding i ._._i..?i!i Mir—i-.. mugs... .. 140 150 160 170 18 Temperature (°F) 1.0 1.5 2.0'2'5' Time {hr)_ Figure 75 Yield versus reaction time with temperature constant at 155 DF. Figure 76 Yield versus temperature with reaction time constant at 1.7 hours. Figure 7-4 Three—dimensional surface plot for the data from Table 7-2, showing the effect of the A and B interaction. for these surfaces are discussed later in the chapter. These data contain no interaction, and the surface plot is a plane lying above the A—B space. The slope of the plane in the A and B direc— tions is proportional to the main effects of factors A and B, respectively. Figure 7-4 is a surface plot for the data from Table 7—2. 'Note that the effect of the interaction in these data is to “twist” the plane so that there is curvature in the response function. Factorial An alternative to the factorial design that is (unfortunately) used in practice is to change the factors one at a time rather than to vary them simultaneously. To illustrate this one-factor—at— that maximize the yield of a chemical process. Suppose that we fix temperature at 155° F (the cur- rent "operating level) and perform five runs at different levels of time—say, 0.5, 1.0, 1.5, 2.0, and ' 2.5 hours. The results of this series cfruns are shown in Fig. 7-5. This figure indicates that maxi- mum yield is achieved at about 1.7 hours of reaction time. then fixes time at 1.7 hours (the apparent optimum) and performs five runs at different tempera— turesm—say, 140, 150, 160, 170, and 180°F. The results of this set of runs are plotted in Fig. 7—6. Maximum yield occurs at about 155°F. Therefore, we would conclude that running the process at , 155°F and 1.7 hours is the best set of operating conditions, resulting in yields of around 75%. Figure 7-7 displays the contour plot of yield as a function of temperature and time with the one-factor—at—a—timc experiments superimposed on the contours. Clearly, this one-factor at-a—time approach has failed dramatically here, because the true optimum is at least 20 yield points higher and occurs at much lower reaction times and higher temperatures. The failure to FACTORIAL EXPERlMENTS 321 experiments are the the values of temperature and pressure To optimize temperature, the engineer Temperature (°F) >—- i—I I—l |-‘ I—- .3: m m \I oo o o o o 0 0.53.0 1.5 2.0 2.5 Time(hr} Figure 7-7 Contour plot of a yield function and an optimization experiment using the one-factor—at-a—time method. mm“: 322 ._ CHAPTER? DESIGN OF ENGINEERING EXPERIMENTS discover the importance of the shorter reaction times is particularly important because this could have significant impact on production volume or capacity, production planning, manu- facturing cost, and total productivity. The one—factor—at—a—time approach has failed here because it cannot detect the interaction between temperature and time. Factorial experiments are the only way to detect interactions. Furthermore, the one-factor-at—a—time method is inefficient. It will require more experimenta— tion than a factorial, and as We have just seen, there is no assurance that it will produce the cor- rect results. 7—3 2" FACTORIAL DESiGN Factorial designs are frequently used in experiments involving several factors where it is nec— essary to study the joint effect of the factors on a response. However, several special cases of the general factorial design are important because they are widely employed in research work and because they form the basis of other designs of considerable practical value. The most important of these special cases is that of 1: factors, each at only two levels. These levels may be quantitative, such as two values of temperature, pressure, or time; or they may be qualitative, such as two machines, two operators, the “high” and “low” levels of a fac- tor, or perhaps the presence and absence of a factor. A complete replicate of such a design re- quires 2 X 2 X X 2 = 2" observations and is called a 2" factorial design. The 2" design is particularly usefiil in the early stages of experimental work, when many factors are likely to be investigated. It provides the smallest number of runs for which 1: factors can be studied in a complete factorial design. Because there are only two levels for each factor, we must assume that the response is approximately linear over the range of the factor levels chosen. The 2" design is a basic building block that is used to begin the study of a system. 7-3.1 22 Example The simplest type of 2" design is the 22—tl1at is, two factors A and B, each at two levels. We usually think of these levels as the low and high levels of the factor. The 22 design is shown in Fig. 7-8. Note that the design can be represented geometrically as a square with the 22 = 4 runs, or treatment combinations, forming the corners of the square (Fig. 7-8a). In the 22 design High I: (+) (-) Law A High (a) Geometric view (b) Design or test matrix for the 22 factorial design Figure 7,8 The 22 factorial design. 7—3 2k FACTORIAL DESIGN 323 it is customary to denote the low and high levels of the factors A andB by the signs d and +, respectively. This is sometimes called the geometric notation for the design. Figure 7—81) shows the test, or design, matrix for the 22 design. Each row of the matrix is a run in the design and the —,+ signs in each row identify the factor settings for that run. A special notation is used to label the treatment combinations. In general, a treatment combination is represented by a series of lowercase letters. If a letter is present, the corre- sponding factor is run at the high level in that treatment combination; if it is absent, the factor is run at its low level. For example, treatment combination a indicates that factor A is at the high level and factor B is at the low level. The treatment combination with both factors at the low level is represented by (1). This notation is used throughout the 2" design series. For ex— ample, the treatment combination in a 24 with A and C at the high level and B and D at the low level is denoted by ac. The effects of interest in the 22 teraction AB. Let the letters (1), a, b, and ab also I at each of these design points. It is easy to estimate the e main effect of A, we would average the observations on the ri where A is at the high level, and subtract from this the average 0 side of the square, where A is at the low level, or design are the main effects A and B and the two-factor in— epresent the totals of all n observations taken ffects of these factors. To estimate the ght side of the square in Fig. 7-8 f the observations on the left Main Effect cf A Similarly, the main effect of B is found by averaging the observations on the top of the square, the bottom of where B is at the high level, and subtracting the average of the observations on the square, where B is at the low level: Main Effect of B Finally, the AB interaction is estimated by taking the difference in the diagonal averages in Fig. 7-8, or AB __ __ _. _. I it t' I Iab+1 +5 .1 ' ' ' -' " . neracion 7 _ . .-._AB=H+L2_LM:_"-[ab+£1.)_a_b]g: (76) Effect - _ I I. _' '_2n 2?: 2n The quantities in brackets in equations 7-1, 7—2, and 7-3 are called contrasts. For exam— ple, the A contrast is ContrastA = a + ab — b — (1) nts are always either +1 or — l. A table of plus and mi- In these equations, the contrast coefficie ombination nus signs, such as Table 7-3, can be used to determine the sign on each treatment 0 WWW-flex: f CHAPTER 7 DESIGN OF ENGINEERING EXPERIMENTS Table 7-3 Signs for Effects in the 21 Design for a particular contrast. The column headings for Table 7—3 are the main effects A and B, the AB interaction, and I, which represents the total. The row headings are the treatment combi- nations. Note that the signs in the AB column are the product of signs from columns A and B. To generate a contrast from this table, multiply the signs in the appropriate column of Table 7—3 by the treatment combinations listed in the rows and add. For example, centrastAB : {(1)} + [—a] + [—b] + [ab] = ab + (1) — a — b ’ Contrasts are used in calculating both the effect estimates and the sums of squares for A, B, and the AB interaction. The sums of squares formulas are The analysis of variance is completed by computing the total sum of squares SST (with 412 — 1 degrees of freedom) as usual and obtaining the error sum of squares SSE [with 401. — 1) de— grees of freedom] by subtraction. An article in the AT&T Technical Journal (Vol. 65, March/April 1986, pp. 39-50) describes the application of two-level factorial designs to integrated circuit manufacturing. A basic pro— cessing step in this industry is to grow an epitaxial layer on polished silicon wafers. The wafers are mounted on a susceptor and positioned inside a bell jar. Chemical vapors are introduced through nozzles near the top of the jar. The susceptor is rotated, and heat is applied. These con— ditions are maintained until the epitaxial layer is thick enough; Table 7-4 presents the results of a 22 factorial design with n = 4 replicates using the factors A = deposition time and B = arsenic flow rate. The two levels of deposition time "»1L'.'1”J>fixs‘kaxww~t¥§fli m- trams. "sch" um Table 7-4 The 22 Design for the Epitaxial Process Experiment Tim: same-mummnmwmmsnowman-unzm 11'de5shawlVSfiiv‘f'miivrsfl:tavern»mewmx/wmw-wlmut:vnrmxmtwa‘mmmme r l3 7-3 2* FACTORIAL DESIGN 325 are — short and + = long, and the two levels of arsenic flow rate are — 2 55% and + = 59%. The response variable is epitaxial layer thickness (pm). We may find the estimates of the effects using equations 7—1, 7-2, and 7—3 as follows: 1 213—— 2n[u!+ab—b--(1)] 2(4) 1 B=w~w [59.299 + 59.156 “- 55.686 - 56.081] = 0.836 [b+ab—a—(l)] [55.686 + 59.156 ~— 59299 — 56.081] = —{).067 l rm 1—45 {59.156 + 56.081 — 59.299 — 55.686] = 0.032 The numerical estimates of the effects indicate that the eifect of deposition time is large and has a positive direction (increasing deposition time increases thickness), because changing deposition time from low to high changes the mean epitaxial layer thickness by 0.836 11m. The eifects of arsenic flow rate (B) and the AB interaction appear small. The importance of these effects may be confirmed with the analysis of variance. The sums of squares for A, B, and AB are computed as follows: [a + ab — b — (1)]2 6.6882 _ 16 16 — 2.7956 {b + ab — a e (1)]2 _ —0.5382 16 _ 16 _ [ab + (1) — a — 612 0.2522 _ SSAB — 16 16 w 0.0040 (56.081 + + 59.156)2 16 SS4 2 SS3 E = 0.0181 SST = 14.0372 + + 14.9322 — = 3.0672 The analysis of variance is summarized in Table 7—5 and confirms our conclusions obtained by examining the magnitude and direction of the effects. Deposition time is the only factor that significantly affects epitaxial layer thickness, and from the direction of the effect estimates we know that longer deposition times lead to thicker epitaxial layers. 7.13.2 Statistical Analysis We present two related methods for determining which effects are significantly different from zero. In the first method, the magnitude of an effect is compared to its estimated standard error. In the second method, a regression model is used in which each effect is 326 CHAPTER? DESIGN OF ENGINEERING EXPERIMENTS Table 7-5 Analysis of Variance for the Epitaxial Process Experiment A (deposition time) . 134.40 B (arsenic flow) . 0.87 AB . 0.19 Error Total associated with a regression coefficient. Then the regression results developed in Chapter 6 can be used to conduct the analysis. The two methods produce identical results for two-level designs. One might choose the method that is easiest to interpret or the one that is used by the available computer software. A third method that uses normal probability plots is dis~ cussed later in this chapter. Standard Errors of the Effects _ The magnitude of the eifects in Example 7-1 can be judged by comparing each eifect to its es- timated standard error. In a 2" design with :1 replicates, there are a total of N = I22" measure- ments. An effect estimate is the difference between two means and each mean is calculated from half the measurements. Consequently, the variance of an effect estimate is 02 2 2 2 V(Efi‘ect) = — + a — 20 G N/2 N72 — WE = n2k_2 (7'5) To obtain the estimated standard error of an effect, replace 02 by an estimate 6-2 and take the square root of equation 7—5. If there are :1 replicates at each of the 2" runs in the design, and if y“, ya, . . . , y," are the observations at the ith run, 2; (J’s—ft? 6,-2=J_—(n_—1)*— i=l,2,...,2k is an estimate of the variance at the ith run. The 2" variance estimates can be pooled (averaged) to give an overall variance estimate 2" 6.2 62 = (7—6) Each 6;?“ is associated with n. — 1 degrees of freedom, and so 6’2 is associated with 2% — 1) degrees of freedom. To illustrate this approach for the epitaxial process experiment, we find that 2 = 0.0121 + 0.0026 + 0.0059 + 0.0625 4 = 0.0208 6 \{yt/VM/V 7-3 2" FACTORIAL DESIGN 327 and the estimated standard error of each effect is - se(Effect) = V [event—2)] = V [0.0208/(4 - 22 ‘2)] = 0.072 In Table 7—6, each effect is divided by this estimated standard error and the resulting t ratio is compared to a t distribution with 22 - 3 = 12 degrees of fieedom. Recall that the 1‘ ratio is used to judge whether the effect is significantly different from zero. The significant effects are the important ones in the experiment. Two standard error limits on the effect estimates are also 3 shown in Table 7—6. These intervals are approximate 95% CIs. 317:; I The magnitude and direction of the effects were examined previously, and the analysis in " i ' Table 7-6 confirms these earlier, tentative conclusions. Deposition time is the only factor that significantly affects epitaxial layer thickness, and flour the direction of the effect estimates we know that longer depOSition times lead to thicker epitaxial layers. Regression Analysis In any designed experiment, it is important to examine a model for predicting responses. Furthermore, there is a close relationship between the analysis of a designed experiment and a regression model that can be used easily to obtain predictions from a 2" experiment. For the epitaxial process experiment, an initial regression model is mum-mm Y: 130 + Bixl + [32352 + 312351352 + ‘5 The deposition time and arsenic flow are represented by coded variables x1 and x2, respectively. The low and high levels of deposition time are assigned values it, = —l and x, = +1, respectively, and the low and high levels of arsenic flow are assigned values x2 = — I and x2 = +1, respectively. The cross-product term xlxz represents the effect of the interaction between these variables. The least squares fitted model is . — . . 2 5’ = + x] ‘i' x2 ‘1' )1?le where the intercept 30 is the grand average of ail 16 observations, The estimated coefficient of x, is one-half the effect estimate for deposition time. The estimated coefficient is one—half the effect estimate because regression coefficients measure the effect ofa unit change in x] on the mean of‘ Y, and the effect estimate is based on a two—unit change from -1 to +1. Similarly, the estimated coefficient of 1:2 is one-half of the effect of arsenic flow and the estimated coefficient of the cross-product term is one-half of the interaction effect. Table 7—6 t—Tests of the Effects for Example 7'1 A 0.836 0.072 11.61 0.00 0.836 t 0.144 B —0.067 0.072 —0.93 0.38 "0.067 t 0.144 AB 0.032 0.072 0.44 0.67 0.032 1‘ 0.144 Degrees offreedom = 2*(n — l) = 22(4 - l) = 12. : m|ur21r .A m CHAPTER 7 DESIGN OF ENGINEERING EXPERIMENTS The regression analysis is shown in Table 7—7. Note that mean square error equals the es- timate of 02 calculated previously. Because the P—value associated with the F-test for the model in the analysis of variance (or ANOVA) portion of the display is small (less than 0.05), we conclude that one or more of the effects are important. The t—test for the hypothesis H0: B,- = 0 versus H1: 8, 95 0 (for each coefficient Bl, [32, and [33 in the regression analysis) is iden- tical to the one computed from the standard error of the effects in Table 7-6. Consequently, the results in Table 7-7 can be interpreted as r-tests of regression coefficients. Because each esti— mated regression coefficient is one-half of the effect estimate, the standard errors in Table 7-7 are one-half of those in Table 7-6. The r—test from a regression analysis is identical to the twtesl obtained from the standard error of an efiect in a 2" design whenever the estimate 32 is the same in both analyses. Similar to a regression analysis, a simpler model that uses only the important effects is the preferred choice to predict the response. Because the r—tests for the main effect of B and the AB interaction effect are not significant, these terms are removed from the model. The model then becomes 0. 6 f2=14.389+( 33 )x1’ That is, the estimated regression coefficient for any effect is the same, regardless of the model considered. Although this is not true in general for a regression analysis, an estimated regres- sion coefficient does not depend on the model in a factorial experiment. Consequently, it is easy to assess model changes when data are collected in one of these experiments. One may also revise the estimate of 0'2 by using the mean square error obtained from the ANOVA table for the simpler model. These analysis methods are summarized as follows. . We reassess " ' 7-3.3 Residual Analysis and Model Checking The analysis of a 2* design assumes that the observations are normally and independently distributed with the same variance in each treatment or factor level. These assumptions should .. . .A . .u...‘x.~n-n...mun—amendwmmlvauwirhv‘fi-NsfiipawnrflvrflIMMWW-‘Wrfiwlfiflffu 7.3 2" FACTORIAL DESIGN 329 Table '7'? Regression Analysis for Example 7'1. The regression equation is Thickness 14.4 + 0.418x1 - 0.0336162 + 0.0158x3x2 Model 2.81764 3 0.93921 45.18 0.000 Error 0.24948 12 0.02079 Total 3.06712 15 0.0360 399.17 0.000 Intercept 14.3889 A oer 0.41800 0.03605 11.60 0.000 B oer —0.03363 0.03605 —0.93 0.369 AB or )1le 0.01575 0.03605 0.44 0.670 be checked by examining the residuals. Residuals are calculated the same as in regression analysis. A residual is the difference between an observation y and its estimated (or fitted) value from the statistical model being studied, denoted as )7. Each residual is e=y-fi The normality assumption can be checked by constructing a normal probability plot of the residuals. To check the assumption of equal variances at each factor level, plot the residuals against the factor levels and compare the spread in the residuals. It is also useful to plot the residuals against 3'}; the variability in the residuals should not depend in any way on the value of j}. When a pattern appears in these plots, it usually suggests the need for transformation—that is, analyzing the data in a different metric. For example, if the vari- ability in the residuals increases with j», a transformation such as log y or V)? should be _ considered. In some problems, the dependence of residual scatter on the fitted value 52 is very important information. It may be desirable to select the factor level that results in maximum response; however, this level may also cause more variation in response from run to run. , The independence assumption can be checked by plotting the residuals against the time or run order in which the experiment was performed. A pattern in this plot, such as sequences of positive and negative residuals, may indicate that the observations are not independent. This suggests that time or run order is important or that variables that change over time are impor— tant and have not been included in the experimental design. This phenomenon should be stud- ied in a new experiment. It is easy to obtain residuals from a 2"“ design by fitting a regression model to the data. For the epitaxial process experiment in Example 7—1, the regression model is 0.836 51 = 14.389 +( 2 )x1 because the only active variable is deposition time. 330 CHAPTER 7 DESIGN OF ENGINEERING EXPERIMENTS This model can be used to obtain the predicted values at the four points that form the corners of the square in the design. For example, consider the point with low deposition time (x; = — 1) and low arsenic flow rate. The predicted value is 0.836 5; = 14.389 +(T)(—1)= 13.9711tm and the residuals are e; = 14.037 — 13.971 x 0.066 e; = 14.165 — 13.971 = 0.194 83 = 13.972 w 13.971 = 0.001 64 x 13.907 - 13.971: "-0.064 It is easy to verify that the remaining predicted values and residuals are, for low deposition time (x: = — l) and high arsenic flow rate, 52 = 14.389 + (0.836/2)(—1) = 13.971 um e5 = 13.880 - 13.971 = —0.091 36 = 13.860 - 13.971 = —0.111 e, = 14.032 "j 13.971 = 0.061 93 = 13.914 —~ 13.971 = “0.057 for high deposition time (x1 +1) and low arsenic flow rate, 51 = 14.389 + (0.836/2)(+1) «1 14.807 um e9 = 14.821 — 14.807 = 0.014 em = 14.757 — 14.807 = —0.050 en = 14.843 - 14.807 = 0.036 312 = 14.878 — 14.807 = 0.071 and for high deposition time (x1 = +1) and high arsenic flow rate, 52 = 14.389 + (O.836/2)(+1) = 14.807 am .215 = 14.415 - 14.807 =' —0.392 315 = 14.932 — 14.807 = 0.125 (213 = 14.888 *— 14.807 = 0.081 e14 = 14.921 — 14.807 = 0.114 A normal probability plot of these residuals is shown in Fig. 7~9. This plot indicates that one residual 65 = —0.392 is an outlier. Examining the four runs with high deposition time and high arsenic flow rate reveals that observation yls = 14.415 is considerably smaller than the other three observations at that treatment combination. This adds some additional evidence to the tentative conclusion that observation 15 is an outlier. Another possibility is that some process variables affect the variability in epitaxial layer thickness. If we could discover which variables produce this effect, we could perhaps adjust these variables to levels that would min- imize the variability in epitaxial layer thickness. This could have important implications in subsequent manufacturing stages. Figures 7-10 and 7-11 are plots of residuals versus deposi- tion time and arsenic flow rate, respectively. Apart from that unusually large residual associ- ated with yls, there is no strong evidence that either deposition time or arsenic flow rate influences the variability in epitaxial layer thickness. Figure 7-12 shows the standard deviation of epitaxial layer thickness at all four runs in the 22 design. These standard deviations were calculated using the data in Table 7-4. Note that the standard deviation of the four observations with A and B at the high level is considerably larger than the standard deviations at any of the other three design points: Most of this difference is attributable to the unusually low thickness measurement associated with yls. The standard 7—3 2*FACTORIAL DESIGN 331 g.. .. .g; Low I High Deposition time. A O CID'G-O-O-O— 6-0 L_._. I ‘ .L._._.___ i —0.392 —.0.294 —0.197 -0.099 —0.001 0.096 0.194 9 Residual —0.5 Figure 7—9 Normal probability plot of residuals for the Figure 7-10 Plot of residuals versus deposition epitaxial process experiment. time. ‘IIU Low gmHigh Arsenic flow rater B a O —-—-———-—-—-——-——1—-—-—-———-—-m—mm~1~i —0.5 '— Figure 7-11 Plot of residuals versus arsenic flow rate. 0.251 ab (1) 0.110 A 0.055 *- + Figure 7—12 The standard deviation of epitaxial layer thickness at the four runs in the 22 design. deviation of the four observations with A and B at the low level is also somewhat larger than the standard deviations at the remaining two runs. This could indicate that other process vari— ables not included in this experiment may affect the variability in epitaxial layer thickness. Another experiment to study this possibility, involving other process variables, could be de— signed and conducted. (The original paper in the AT&T Technical Journal shows that two additional factors, not considered in this example, affect process variability) EXERCISES FOR SECTION 7—3 M For each of the fol-lowing designs in Exercises 7-1 through (d) 7-8, answer the following questions. (3) Compute the estimates of the effects and their standard er— (3) rors for this design. (b) Construct two—factor interaction plots and comment on the interaction of the factors. (c) Use the t ratio to determine the significance of each effect with or = 0.05. Comment on your findings. Compute an approximate 95% CI for each effect. Compare your results with those in (c) and comment. Perform an analysis of variance of the appropriate regression model for this design. Include in your analy- sis hypothesis tests for each coefficient, as well as resid— ual analysis. State your final conclusions about the adequacy of the model. Compare your results to part (c) and comment. i i g E 5 i % 332 781. An experiment involves a storage battery used in the launching mechanism of a shoulder-fired ground-to-air mis- sile. Two material types can be used to make the battery plates. The objective is to design a battery that is relatively unaffected by the ambient temperature. The output response from the bat— tery is effective life in hours. Two temperature levels are se- lected, and a factorial experiment with four replicates is run. The data are as follows. 130 155 20 70 74 180 82 58 138 110 168 160 82 60 7-2. An engineer suspects that the surface finish of metal parts is influenced by the type of paint used and the drying time. She selects two drying times—20 and 30 minutes-wand uses two types of paint. Three parts are tested with each com- bination of paint type and drying time. The data are as follows. 78 64 85 50 92 92 66 '86 . 45 68 85 73. An experiment was designed to identify a better ultra- filtration membrane for separating proteins and peptide drugs from fermentation broth. Two levels of an additive PVP (% wt) and time duration (hours) were investigated to determine the better membrane. The separation values (measured in %) resulting from these experimental runs are as follows. CHAPTER 7 DESIGN OF ENGINEERING EXPERIMENTS 74. An experiment was conducted to determine whether either firing temperature or furnace position affects the baked density of a carbon anode. The data are as follows. 570 ' 1063 565 1080 583 1043 528 988 547 l026 521 1004 7-5. Johnson and Leone (Statistics and Experimental Design in Engineering and the Physical Sciences, John Wiley, 1977) describe an experiment conducted to investigate warp- ing of copper plates. The two factors studied were temperature and the copper content of the plates. The response variable is the amount of warping. Some of the data are as follows. 24, 22 25, 23 17, 20 100 16, 12 76. An article in the Journal of Testing and Evaluation (Vol. 16, no. 6, 1988, pp. 508—5 15) investigated the effects of cyclic loading frequency and environment conditions on fatigue crack growth at a constant 22 MPa streSS for a particu- lar material. Some of the data from the experiment are shown here. The response variable is fatigue crack growth rate. Frequency 7-7. A article in the IEEE Transactions on Electron Devices (November 1986, p. 1754) describes a study on the effects of two variables—polysilicon doping and anneal con- ditions (time and temperature)—on the base current of a bipolar transistor. Some of the data from this experiment are as follows. i i 74 2k DESIGN FOR k 2 3 FACTORS 333 cleaning method (spin rinse dry, or SRD, and spin dry, or SD) and the position on the wafer where the charge was measured. The surface charge (X 10” q/cm3) response data are as shown. 1 x 102" Polysilicon Doping 5:; 2 X 1020 Cleaning it} 7—8. An article in the IEEE Transactions on Semiconductor Method Manufacturing (Vol. 5, no. 3, 1992, pp. 214—222) describes an experiment to investigate the surface charge on a silicon wafer. The factors thought to influence induced surface charge are 7-4 2" DESIGN FOR k 2 3 FACTORS The methods presented in the previous section for factorial designs with k = 2 factors each at two levels can be easily extended to more than two factors. For example, consider I: = 3 I factors, each at two levels. This design is a 23 factorial design, and it has eight runs or treat- ment combinations. Geometrically, the design is a cube as shown in Fig. 7—1351, with the eight runs forming the corners of the cube. The test matrix or design matrix is shown in Fig. 7—1319. This design allows three main effectsto be estimated (A, B, and C) along with three two-factor interactions (AB, AC, and BC) and a three-factor interaction (ABC). The main effects can easily be estimated. Remember that the lowercase letters (1), a, I), ab, 6. ac, be, and abc represent the total of all 12 replicates at each of the eight runs in the de- sign. As seen in Fig. 7-14a, note that the main effect of A can be estimated by averaging the four treatment combinations on the right~hand side of the cube, where A is at the high level, and by subtracting from this quantity the average of the four treatment combinations on the left-hand side of the cube where A is at the low level. This gives A 3i“ _ .1721— _a+ab+ac+abc_(1)+b+c+bc _ 4n 4n B C Label A 1 2 _ 3 + 4 + — ab 5 — + c 6 + — + no 7 — + + be 8 + + + abc \ (a) Geometric View (5) The test matrix Figure 7—13 The 23 design. _m_ mm. ._ 0 = + runs 0 =— runs Three-factor interaction (c) Figure 7-14 Geometric presentation of contrasts corresponding to the main effects and interaction in the 23 design. (a) Main effects. (b) Two-factor interactions. (c) Three-factor interaction This equation can be rearranged as In a similar manner, the efl’ect of B is the difference in averages between the four treat- ment combinations in the back face of the cube (Fig. 7—14a), and the four in the front. This yields The effect of C is the difference in average response between the four treatment combinations in the top face of the cube (Fig. 7-14a) and the four in the bottom; that is, The two—factor interaction efi'ects may be computed easiiy. A measure of the AB interaction is ' _ the difference between the average A effects at the two levels of B. By convention, one-half of this difference is called the AB interaction. Symbolicaliy, [(abc — be) + (ab — an 2n {(516 r c) + [a - (1)3} 2n . [abcwbc+ab—b—ac+c—a+(1)] Difference 2” High (+) Low (—) Because the AB interaction is one—half of this difference, We could write equation 7-10 as follows: _abc+ab+c+(1) bc+b+ac+a AB 4n 4n In this form, the AB interaction is easily seen to be the difference in averages between runs on two diagonal planes in the cube in Fig. 7—1419. Using similar logic and referring to Fig. 7—1413, we find that the AC and BC interactions are AC _= + b *- ab -.Ic'-+. ac —. .bc_+--qb__c] I I 1 BC=“4; [(1)-+ a — b -* ab — c —' ac +bc abc] iii-(7‘13) The ABC interaction is defined as the average difference between the AB interactions for the two different levels of C. Thus, ABC = Z1}; [abc — be] — [ac — c] — [ab — b] + [a — (1)1} EXAMPLE 7'2 i i ABC %."E_'[ubé — be — ac + c. — ab + a + a — (1)] (7—14) I;. . I" ; As before, we can think of the ABC interaction as the difference in two averages. If the runs in the two averages are isolated, they define the vertices of the two tetrahedra that comprise the cube in Fig. 7- 14:. In equations 7—8 through 7—14, the quantities in brackets are contrasts in the treatment com— binations. A table of plus and minus signs can be developed from the contrasts and is shown in Table 7-8. Signs for the main effects are determined by associating a plus with the high level and a minus with the low level. Once the signs for the main effects have been established, the signs for the remaining columns can be obtained by multiplying the appropriate preceding columns, row by row. For example, the signs in the AB column are the products of the A and B column signs in each row. The contrast for any effect can easily be obtained from this table. Table 7-8 has several interesting properties: 1. Except for the identity column I, each column has an equal number of plus and minus signs. 2. The sum of products of signs in any two columns is zero; that is, the columns in the table are orthogonal. Multiplying any column by column I leaves the column unchanged; that is, I is an identity element. The product of any two columns yields a column in the table, for example A X B = AB, and AB >< ABC = AZBZC = C, because any column multiplied by itself is the identity column. The estimate of any main efiect or interaction in a 2" design is determined by multiplying the treatment combinations in the first column of the table by the signs in the corresponding main effect or interaction column, adding the result to produce a contrast, and then dividing the contrast by one-half the total number of runs in the experiment. A mechanical engineer is studying the surface roughness of a part produced in a metal cutting operation. A 23 factorial design in the factors feed rate (A), depth of cut (B), and tool angle (C ), with n = 2 replicates, is run. The levels for the three factors are: low A = 20 in./min, Table 7-8 Algebraic Signs for Calculating Effects in the 23 Design ++++++++ 7-4 2" DESIGN FOR 1: 2 3 FACTORS Table 7-9 Surface Roughness Data for Example 72 2.0 2.0 2.0 4.5 0.5 4.5 2.0 2.0 2.4375 abc Average highA = 30 in./min; lowB = 0.025 in., highB = 0.040 in; low C 7: 15°, high C = 25°. Table . 7-9 presents the observed surface roughness data. -- The main effects may be estimated using equations 7-8 through 7—14. The effect of A, for example. is A=-—4:‘[a+ab+ac+abcf(1)—b—c—bc] 1 =— + + — — —— — 4(2)[22 27 23+30 16 20 21 18] = $27] = 3.375 It is easy to verify that the other effects are i 2; 5'5 E E2 i . is g 53 f i) g: g g; i‘ E 1': e 8 i 1'2 1.1 t3. .3! B = 1.625 C = 0.875 AB = 1.375 AC = 0.125 BC: —0.625 ABC = 1.125 Hua—yn'mxz‘: Examining the magnitude of the effects clearly shows that feed rate (factor A) is dominant, fol— lowed by depth of cut (B) and the AB interaction, although the interaction effect is relatively Small. v mas: 3a":4l;'::-.:rvn€:i'<a‘< 5.4" For the surface roughness experiment, we find from pooling the variances at each of the eight treatments as in equation 7-2 that 62 = 2.4375 and the estimated standard error of each efl‘ect is A2 2.4375 se(effect) = 1 {RSI-P2 = 1 {W 2 0.78 Therefore, two standard error limits on the effect estimates are A: 3.375 i 1.56 B: 1.625 i 1.56 C: 0.875 1' 1.56 AB: 1.375 t 1.56 AC: 0.125 i 1.56 BC: -0.625 : 1.56 ABC: 1.125 $1.56 338 CHAPTER 7 DESIGN OF ENGINEERING EXPERIMENTS These intervals are approximate 95% confidence intervals. They indicate that the two main effects A and B are important, but that the other effects are not, because the intervals for all effects except A and B include zero. This CI approach is a nice method of analysis. With relatively simple modifications, it can be used in situations where only a few of the design points have been replicated. Normal prob- ability plots can also be used to judge the significance of effects. We will illustrate that method in the next section. Regression Model and Residual Analysis We may obtain the residuals from a 2" design by using the method demonstrated earlier for the 22 design. As an example, consider the surface roughness experiment. The three largest effects are A, B, and the AB interaction. The regression model used to obtain the predicted values is Y: 50 + 31361 + 52352 + 312051362 + 6 where x1 represents factor A, x2 represents factor B, and 3cle represents the AB interaction. The regression coefficients [31, [32, and [312 are estimated by one-half the corresponding effect esti- mates, and Bo is the grand average. Thus . 5 1.625 . j} = + )x1 +( 2 )x2 “I” (Egé)x1x2 and the predicted values would be obtained by substituting the low and high levels of A and B into this equation. To illustrate this, at the treatment combination where A, B, and C are all at the low level, the predicted value is 11.065 + (iiiywl) + (-1) + (—991) a 9.25 J9 Because the observed values at this run are 9 and 7, the residuals are 9 — 9.25 = .—0.25 and 7 — 9.25 = —2.25. Residuals for the other 14 runs are obtained similarly. A normal probability plot of the residuals is Shown in Fig. 7—15. Because the residuals lie approximately along a straight line, we do not suspect any problem with normality in the data. There are no indications of severe outliers. It would also be helpful to plot the residuals versus the predicted values and against each of the factors A, B, and C. Projection of a 2k Design Any 2" design will collapse or project into another 2" design in fewer variables if one or more of the original factors are dropped. Sometimes this can provide additional insight into the remaining factors. For example, consider the surface roughness experiment. Because factor C and all its interactions are negligible, we could eliminate factor C from the design. The result is to collapse the cube in Fig. 7—13 into a square in the A—B plane; therefore, each of the four runs in the new design has four replicates. In general, if we delete 1: factors so that r = k — [1 factors remain, the original 2" design with :1 replicates will project into a 2" design with :22” replicates. ...
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