Avoiding unnecessary bias in robust design analysis

Computational North-Holland

Statistics

& Data Analysis

Avoiding unnecessary design analysis Kwok-Leung

535

18 (1994) 535-546

bias in robust

Tsui

School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, USA Received October 1992 Revised July 1993 Abstract: The objective of robust design method is to identify control factor combinations which make product and process designs insensitive (i.e., robust) to hard-to-control variations (noise). Robustness is quantified by some loss measures such as signal-to-noise ratios. In many applications, however, directly modeling the loss measures creates unnecessary biases for the factorial effect estimates and may lead to non-optimal solutions. The response model approach, an alternative approach proposed in Welch et al. (19901, Box and Jones (1990), Lucas (19901, and Shoemaker et al. (1991), does not have this shortcoming and may give more reliable results in some situations. As pointed out by Box and Jones (1990) and others, the variance of response due to noise may be quadratic in the control factors even if the response is linear in the control factors. This paper shows that, when the design for the control factors (control array) is highly fractionated, the main effect estimates from the loss model approach are often biased with the control-by-control this bias problem and how it may lead to interactions. A real example is used to illustrate non-optimal control factor combinations. Keywords: Taguchi array.

method;

Loss model;

Response

model;

Control

array;

Noise

array;

Product

1. Introduction Robust design is an important method for improving product quality, manufacturability, and reliability at low cost. Taguchi’s introduction of this method (Taguchi, 1986) in 1980 to several major American industries resulted in examples of significant quality improvement in product and manufacturing process design. In planning robust design experiments, Taguchi proposed construction of two separate experiment plans for the control and noise factors. We call that the experiment plan for the control factors is “control array”, and that for the noise Correspondence to: K.-L. Tsui, Ph.D., School of Industrial Institute of Technology, Atlanta, GA 30332-0205, USA. 0167-9473/94/$07.00 0 1994 - Elsevier SSDZ 0167-9473(93)E0031-X

Science

and Systems

B.V. All rights reserved

Engineering,

Georgia

536

K.-L. Tsui / Avoiding unnecessary bias

factors is “noise array”, and the experiment set-up is “product array” format. In the experiment, the control factors are varied according to the combinations of the control array; and for each row of the control array, the noise factors are varied according to the combinations of the noise array. In analyzing robust design experiments, Taguchi proposed to first calculate the SN ratios for each row of the control array (design matrix for control factors), model the SN ratios as function of control factors and then use the model to identify the “optimal” factor combinations to maximize the SN ratios. Shoemaker et al. (1991) classify this analysis approach as a special case of the loss model approach in which estimates of performance measures are first computed and then a model is fitted to these performance measures to determine the “optimal” factor combinations. Leon et al. (1987) and Box (1988) discussed the choice of performance measures for different underlying models. In this paper, we will not address the problem of choosing appropriate performance measures. We will focus on the objective of reducing variance and comment on other performance measures in the discussion. As pointed out in Shoemaker and Tsui (19931, variance reduction is a central part of most robust design problems. They also explained the relationship between variance reduction and other robust design objectives. An alternative approach for analyzing data is to model the observed response first, and then determine the “optimal” factor settings from the fitted response model. This approach is called the “response model” approach and was first proposed in Welch et al. (1990) and further developed by Shoemaker et al. (1991). Informal examples of this approach can be found in Pignatiello and Ramberg (1985) and some early case studies in American Supplier Institute (1984, 1985). Related approaches have also been discussed by Box and Jones (19901, Freeny and Nair (1992), Lucas (19891, Montgomery (19911, and Myers et al. (1992). The loss model approach is appealing because it is conceptually simple and provides direct estimates of the loss. As a result, the loss model approach has been used frequently among industrial practitioners. American Suppliers Institute (1984-1992) documented many successful case studies in which the product or process design had been improved by the application of robust design methods. Among these case studies, most of the experiments are highly fractionated (resolution III) and the analysis methods used are often main effect analyses of the SN ratios (the loss model approach). As shown in this paper, the use of the loss model approach in highly fractionated experiments may create unnecessary biases for the factorial effect estimates, and thus may lead to non-optimal solutions. On the other hand, the response model approach does not have this bias problem and may give more reliable results in some situations. As pointed out in Box and Jones (19901, Myers et al. (19921, and Shoemaker and Tsui (1993), the variance of response due to noise can be quadratic in the control factors even if the response is linear in the control factors. This paper studies the consequences of this fact and explains the major problem of the loss model approach - the unnecessary biases of factorial effect estimates and thus

K.-L. Tsui / Al:oiding unnecessary bias

537

may lead to non-optimal control factor combinations. Section 2 discusses the source of bias and shows that, when the experiment plan of the control factors (i.e., the control array) is highly fractionated (e.g. resolution III>, the loss model approach often produces biased effect estimates even though the effect estimates from the response model are unbiased. Section 3 illustrates with a real example that how serious the biases could be and how the bias problem would lead to non-optimal results. We conclude with a summary in Section 4.

2. Source of bias in the loss model approach In this section we investigate the source of bias under an additive model with control-by-noise interactions. The more general situation will be discussed in Section 4. Assume that the true model of the response (Y) is linear in p control factors (C,, C,, ***>C,) and linear in 4 noise factors (N,, N2,. . . , N,) but each control factor Ci interacts with each noise factor Nj, i.e.,

Y=p

+crTC+

5 (Y,+@)Nj+q

j=l

(2.1)

where C = CC,, . . . ,CpIT, (Y= ((Y,, . . . , apIT, Pj = (Pjl,. . . , pj,>T, N’jS are independently distributed with mean zero and variance u,,,,~, E distributed with mean zero and variance a’, and Nj and E are independent. Suppose the experiment is a saturated or a resolution III design for the control factors (control array) crossed with a similar design for the noise factors. This implies that the control factor main effects are confounded with the control-by-control interactions and the noise factor main effects are confounded with the noise-by-noise interactions. Since there are no control-by-control and noise-by-noise interactions in model (2.11, if the experimenter models the response (Y) directly, he/she can obtain unbiased estimates of all effects in the response model in (2.1). In contrast, as noted by Box and Jones (1990) and others, the variance of response due to noise and random error under model (2.1) is quadratic in the control factors (Cl even though the original response is linear in C, i.e.,

Since there are control-by-control interaction terms in (2.21, if one models the variance (var(Y)> in (2.2) instead of the response (Y), the control factor main effects will be confounded with the control-by-control interactions. Obviously, if these interactions are large, the main effect estimates of the variance model will

538

K.-L. Tsui / Avoiding unnecessary bias

be seriously biased. As illustrated later, these biases could lead to non-optimal factor combinations in robust design problems. In general, under model (2.1), if the experimental design allows confounding between main effects and 2-factor interactions in the control array (i.e., resolution III), there will be a bias problem in the loss model approach when the confounded effects are large (since variance is quadratic in C,‘s). As shown in equation (2.21, the coefficients of the Ci main effect and Ci X C, interaction are equal to Cq= ,a; rjpji and Cg= ius PjiPjk respectively. Clearly these coefficients equal zero if the variances of all noise factors (a:~> are zero. Also, since rjpji and PjiPjk can be negative, these coefficients can drop to zero due to cancellation of positive and negative terms. In practice, however, it is unlikely that all uzs are (close to) zero or the sum drops to zero due to cancellation. ‘As pointed out in Shoemaker and Tsui (1993), the interaction coefficients would drop to zero if there is separability in model (2.1) (i.e., each noise factor only interacts with at most one control factor). That is, in the coefficient of the interaction, uipjipjk, if either pji or pjk is zero for each j. On the other hand, if the noise factor interacts with more than one control factor, the interactions between these control factors in the variance model will be non zero. For example, if there are six control factors that interact with the same noise factor, this will contribute “6 choose 2” (15) control-by-control interactions to the variance model. When there are multiple noise factors, there is a high chance to have non-zero interaction coefficients. Thus, the effect estimates calculated from the loss model approach under a highly fractionated (resolution III) control array are almost always biased. The next example, based on an experiment reported in Pfaff (1987), illustrates how the loss model approach results in some seriously biased effect estimates, and thus leads to non-optimal control factor combinations.

3. A power supply transformer

example

This experiment addressed a transformer processing problem involving inductance changes. Due to inductance falling out of the required ranges, much time and cost had been involved in dealing with the problem. The primary objective of the experiment was to minimize variation of inductance from transformer to transformer after processing. The process was described in Pfaff (1987). There are nine control factors, A, B, C, D, E, F, G, H, and L in the experiment. Each factor was tested at two levels, except for factor L for which eight levels were tested. The control array was constructed by customizing a 215-” saturated fractional factorial design. Seven columns of the saturated design were collapsed to generate an eight-level column for factor L and the remaining eight factors were assigned to the other eight columns. For the noise array, three two-level noise factors, r, s, and t, were assigned to a 23-1 saturated fractional factorial design. As shown in Table 3.1, the resulting

539

K.-L. Tsui / Avoiding unnecessary bias Table 3.1 Design and data of the transformer example Run

Noise array

Control array r S

t LABCDEFGH 1 2 3 4 5 6 7 8 9 0 1 2 .3 14 15 16

1 2 3 4 1 2 3 4 5 6 7 8 5 6 7 8

-1 -1 -1 -1

1 1 1 1 1 1 1 1

-1 -1 -1 -1 1 1 1 1

-1 -1 -1 -1 -1 -1 -1 -1

1 1 -1 -1 -1 -1

1

-1 -1 -1 -1

-1 -1 1 1

1

1 1 1

-1

1 1

-1

-1

1 1

-1

-1 -1

-1 1

1

-1 -1

1

-1

1

-1 1

-1 -1

-1 -1

-1

-1 1 1

-1 1

1 1

1

1 1 1

-1 1

-1

-1 -1 -1 -1 1 1

-1

-1 1 1 1 1

1 1 1 1

1 1 1 1

-1 -1

1

-1 -1 1

1 -1 --1 1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1

-1 -1 -1 Data

-1

9.44 9.07 8.41 10.20 9.56 9.08 9.30 9.72 9.10 9.63 9.94 9.63 10.11 9.89 10.20 8.72

10.21 9.68 7.23 10.48 8.39 9.18 8.11 9.83 8.88 9.77 9.17 7.85 8.52 10.65 9.87 8.94

1 1

1 1 1

-1 1 9.54 8.82 8.87 10.62 8.85 9.30 9.43 9.92 9.43 9.90 10.40 9.52 9.84 10.19 10.87 9.14

-1 9.73 8.84 8.17 11.08 7.87 8.94 9.04 9.85 10.08 9.73 9.15 7.87 7.83 10.71 10.73 8.91

experiment plan is a product array of two saturated designs for control factors and noise factors. Following the loss model approach, the variance over the noise runs for each control combination was first calculated. To calculate the effect estimates, we first decomposed factor L to seven orthogonal contrasts CL,, . . . , L,) which correspond to the seven columns of the original 215-” saturated design. The main effect estimates of all nine control factors were computed and shown in Table 3.2. Note that each main effect is confounded with seven 2-factor interactions even though third or higher order interactions are assumed to be zero. As pointed out earlier, the main effect estimates are likely to be biased because the 2-factor interactions are unlikely to be zero. Below we will study these possible biases by applying the response model approach to the example. Following the response model approach, we first modeled the response as a function of both the control and noise factors. The following fitted model was obtained using the method in Lenth (1989) to identify significant effects at cr = 0.20 under the assumption that main effects are more important than 2-factor interactions. y^= 9.41 + 0.14B + O.llC - 0.26G + 0.08H + O.l3L, - O.llL,

+ O.l5L,

- 0.24L, - 0.47L, + 0.07r - 0.18s + 0.08Dr - 0.09Gr + O.l2L,r - O.O8L,r + O.lOL?s+ 0.18C.r + 0.06Ds + 0.13Es + 0.09Fs - O.llGs - O.O7L,S - O.O9L,s + O.l5L,S - O.l3L,S.

(3.1)

K.-L. Tsui / Avoiding unnecessary bias

540 Table 3.2 Effect estimates

under

loss model

approach

A B c D E F G H L, L, L, L, L5 L, L,

-

0.004 0.005 0.166 0.103 0.131 0.063 0.031 - 0.035 0.089 0.010 - 0.010 -0.125 0.017 0.091 0.122

In order to study approach in Box process variance variables Y and s the variance of 9

the biases of the effect estimates on Table 3.2, we follow the and Jones (1990) and Myers et al. (1992) to estimate the (as suggested by a referee). By assuming that the noise in (3.1) are uncorrelated random variables with unit variances, over the noise can be estimated by

var( y^) = (0.07 + 0.080

- 0.09G + O.l2L,

+ 0.18C + 0.060 +o.l5L,

- 0.08L3)2 + ( -0.18

+ 0.13E + 0.09F - O.llG

- o.13L,)2,

+ O.lOB

- O.O7L, - O.O9L, (3.2)

which is clearly a quadratic function of the control factors. Note that this variance estimate is equivalent to the variance estimate computed by the “imputation” method used in Welch et al. (1990). Following the example of Welch et al., the variance estimate can be calculated from the data imputed from the fitted model (3.1) based on a full factorial design on the significant control factors crossed with a 4-run full factorial design on the two significant noise factors r and s. This “imputation” variance estimate is the same as the estimate in (3.2) because the use of a full factorial design of (- 1, 1) to Y and s is equivalent to assuming that Y and s follow uncorrelated discrete uniform distributions (- 1, 1) with means zero and variances one. The only possible difference between the two estimates is that they will be off by a multiplication constant of t if the variance estimate of the “imputation” method is calculated by the sample variance formula, i.e., divided by n - 1 (3) instead of 12(4). Since the two methods are equivalent, we preferred to use the former method for its simplicity. In addition, the variance estimate in (3.2) can be interpreted as an estimate of the process variance if the assumption on the noise distributions is correct.

0.045 CL,)

-0.062(C) -0.011(D) -0.047(E) - 0.033 (F) 0.025(G) 0.0(H) 0.024(L,) 0.049(L,) -0.012CL,) -0.051(L,) O.O(L,) 0.0CL,)

0.0 (A) 0.037 (B)

M.e. est.

Table 3.3 Effect estimates

-0.024(FL,) - 0.019 (BL,) -0.001(GL,) O.O3O(BL,) -0.026(BL,) -0.017(FL,) 0.025EF) -0.028(EG) 0.037( BC) 0.020(BF) 0.028( BE) O.O47(CE)

- 0.022 (BG)

and their aliases

- O.O20(FG) - O.O37(CG) - 0.015 (DG) -0.027(L,L,) 0.033 (CF)

-0.013(FL,) -0.044(CL,) 0.051(CL,) 0.014(GL ,) -0.019(L,L,)

- 0.014 (Lx,) 0.027 (FL,) -0.031(GL,) -0.033(EL,)

effect est.

variance

- 0.032 (CL,) - 0.019 (DL,)

Confounded

of predicted

0.038 (EL,)

0.017 (L,L,)

0.023(L,L,)

-0.037(L,L,)

-O.O24(EL,)

0.015 (GL,)

0.027(GL,) -0.009(DL,) - 0.024(CL,)

(Ix,)

-0.020(L,L,)

0.013 (L,L,)

- 0.018(EL,)

-0.014

0.068 0.091

0.037 0.056

0.030 0.034 -0.032 -0.068 0.016

- 0.003

0.026

- 0.050

-0.004 -0.125 -0.077 -0.098 - 0.047 0.023 -0.027 0.067 0.007 -0.008 -0.094 0.013

- 0.003

Biased m.e. est. (raw data)

0.002 -0.126 - 0.087 -0.061

- 0.022

Biased m.e. est. (sum)

z w E $ ST E

iii

s 3

0

??

3 E. \

?? h

542

K.-L. Tsui / Avoiding unnecessary bias

By expanding equation (3.21, there are forty-two nonzero 2-factor interactions (2-f.i.1 estimates. These estimates. are the sources of bias for the main effect estimates in Table 3.2. Table 3.3 summarizes the confoundings of all main effects and the forty-two nonzero 2-f.i. estimates from equation (3.2) for each of the main effect estimates in Table 3.2. The effect estimates in this table are unbiased if equation (3.1) is the true model. The second-to-last column, which is the sum of the confounded effect estimates in the first five columns, represents the biased main effect estimates that would have been obtained if the data were imputed based on (3.1) using the design in Table 3.1. The last column, which is the biased main effect estimates in Table 3.2 multiplied by i, represents the biased main effect estimates obtained from the raw data. The differences between these two columns of biased effect estimates are due to random error if the fitted model (3.1) is believed to be the correct model. By comparing these two columns, it was found that the two sets of biased effect estimates match qualitatively well for the seven largest effects, C, D, E, L,, L,, L,, L,. By examining the confounded effect estimates for these seven effects, many of them were found to be seriously biased. For example, the largest effect estimate of C was inflated by 100%. The magnitudes of the effect estimates of D and L, were increased from 0.011 and 0 to 0.087 and 0.037. Other estimates were not as seriously biased due to cancellation of non-zero interaction estimates. Note that the overall mean estimate of the loss model approach is also biased with the sum of the main effect quadratic components of the control factors. This bias would not affect the result if all control factors are qualitative with only two levels. However, if some control factors are quantitative, the optimal control factor combinations will be affected by these biases. Lorenzen’s discussion in Nair (1992) discussed this problem in more detail. Below we investigate if the serious bias of the loss model approach would lead to different results from the response model approach. According to Table 3.2, the four largest effects (C, E, L,, L7) were identified to be significant using the method in Lenth (19891, and the best control factor setting is C = 1, E = 1, L,= 1, and L,= -1. For the response model approach, the variance estimate in (3.2) was optimized over the discrete choices of (- 1, 1) for the eleven significant control factors (totally 2 l1 = 2 064 choices). Table 3.4 lists the best twenty combinations and their correspondmg variance estimates. It was found that, out of these twenty combinations, none of the factor settings for C, E, L,, and L, overlap with the best setting identified from the loss model approach, i.e., C = 1, E = 1, L, = 1, and L,.= - 1. This means that the best setting of the loss model approach does not make the first twenty best combinations of the response model approach. To see the performance of the loss model approach under the fitted response model (3.11, we calculated the minimum and the average over the variance estimates of the 128 combinations whose factor settings overlap with the loss model best setting (C = 1, E = 1, L, = 1, and L, = - 1). They are 0.005 and 0.249 respectively. Thus, if model (3.1) is true, the best possible

543

K.-L. Tsui / Avoiding unnecessary bias Table 3.4 Best combinations of response model approach Comb. 1

8 9 10 11 12 13 14 1.5 16 17 18 19 20

B -1. 1 1 1 1 -1 1 1 -1 1 -1 1 1 -1 1 -1 1 -1 -1 -1

C

D 1 1 1

1 1 1 1 1 1 1

-1 1 -1 -1 -1 1 1 1 -1 -1 -1

E

-1 -1 -1

-1 -1 -1

1 1 1 1

-1 -1 -1

-1

-1 -1 -1

1 1 1 1 1

-1

1

1 1 1

-1

1 -1

1 1 -1 -1 -1

1

-1 -1

1

-1

1

1 1

-1 1

-1

-1 1 1

-1 -1

G 1

1

1

1

F

1 1 -1 -1

1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 -1 1 -1 -1

L,

L,

L,

L,

-1 -1 -1

-1 -1 -1 -1-l -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1

-1 -1 -1

-1

1 -1 1 1 -1 1 -1 -1 -1 1 -1 1 -1 1 -1 1 -1

1

1 1 1 1

1 1

-1 -1

-1 1 1 1 1

1 -1 -1 -1 1

1 1 1

1 -1 -1 -1

-1 1 1

-1 -1 1

-1 -1 1 -1 -1

-1 -1

-1

Var. est.

L,

-1 -1 1 1

1 1 1 -1 1 -1 -1 1 1 1 1

0.0009 0.0010 0.0011 0.0013 0.0014 0.0014 0.0016 0.0016 0.0016 0.0016 0.0017 0.0017 0.0017 0.0017 0.0017 0.0017 0.0018 0.0019 0.0019 0.0019

variance estimate (0.005) and the average variance estimate (0.249) from the loss model approach are five times and 249 times the best variance estimate (0.001) from the response model approach. Therefore, in this example, the bias problem of the loss model approach led to “non-optimal” control factor combinations which are far away from the “optimal” combination determined from the response model approach. Note that the comparison above assumed that the fitted model for the response (3.1) is correct. In reality, the results of the response model approach may critically depend on the goodness-of-fit of the response model. As suggested in Shoemaker et al. (19911, additional analysis techniques, such as normal probability plot, stepwise regression, CP, prediction sum of squares statistics, model diagnostics, and data transformation (see Box et al., 1978; Daniel, 1976; Draper and Smith, 1981; Carroll and Ruppert, 19881, are needed to identify the best model for the response in practice. Also, in our response model approach, we only computed the variance estimates of the fitted model for comparison and optimization purpose. In practice, other analysis methods of the approach, such as the graphical methods proposed in Shoemaker et al. (1991) and Shoemaker and Tsui (19931, should be used in conjunction to gain additional information about the process. One may suspect that quadratic effects on the variance measure can be linearized by applying the log transformation to the variance. As shown in the Appendix, a second-order Taylor series approximation indicates that the

544

K.-L. Tsui / Avoiding unnecessary bias

quadratic terms of the log variance are negligible only when the magnitude of the control-by-noise interaction is much smaller than that of the noise main effect. In the example above noise main effects and control-by-noise interactions are of the same magnitude, so the log transformation would be ineffective for linearizing the variance model.

4. Discussion The variance of response due to noise is quadratic in the control factors even if the response is linear in the control factors. When the design for the control factors (control array) is highly fractionated (e.g. resolution III) and the loss model approach is used to analyze the data, the effect estimates on variance are often biased. As illustrated in Section 3, if the experimenter conducts a main effect analysis and ignores the alias structure of the design, these biased estimates may easily lead to non-optimal control factor combinations. If the experimenter conducts a more careful analysis and considers the potential aliases of main effects and 2-factor interactions, he/she may conclude that additional experiments are needed to de-alias these confoundings. On the other hand, the response model analysis does not have this problem and may not require additional experiments. If a simple model is adequate for describing the relationship between the response and control and noise factors, the experimenter can obtain unbiased effect estimates by fitting a model to the response. However, such a simple response model is not always available in practice. Since the response model approach highly depends on the adequacy of the fitted response model, the results may be unreliable when the fitted model is inadequate. In practice, as suggested in Wu’s discussion in Nair (1992) and Leon et al. (1993), a good overall strategy might be to try both the loss model and the response model approaches, if possible. The two approaches should give similar best control combinations if the effect estimates of the fitted loss model are not seriously biased and the fitted response model is adequate. When the control array is of resolution IV or V, and if model (2.3) is the true response model, there is no confounding between main effects and 2-factor interactions and thus the loss model approach may give unbiased effect estimates. However, if the true response model involves some higher-order terms such as control-by-control interactions, the variance of the response due to noise will contain some third-order interaction terms. Since a resolution V control array would confound second-order interactions with third-order interactions, in this case, the loss model approach may still result in biased effect estimates. In addition to modeling variance, Taguchi also recommends modeling other SN ratios, such as the square of coefficient of variation or the mean squared error. Nevertheless, all these performance measures suffer from the same problem as the variance in that they create unnecessary biases for factorial effect estimates and thus may lead to non-optimal results.

K.-L.

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545

Acknowledgements

The author would like to thank C.F.J. Wu and two referees for their helpful comments which significantly improve the quality of this paper. This research was supported by the National Science Foundation under Grants DDM-9114554 and DDM-9257918.

Appendix

Suppose there is only one noise factor in model (2.21, i.e., var(Y) = (y +/?‘C)‘ci By a second-order

+ ff2.

Taylor series approximation,

log var(Y) = log(y20i + (TV)+

-

2rfl;

[

2

2u;

(2Y@;)2

y2u;

PTC

y q$+u2

2+ +(

we have

+ al)

cTppTc. y2u;

+ a2 1

For robust design problems, the variation due to noise factors are often much larger than the random error, i.e., y20i > u2. Thus, log var( Y) = log( y ST;) + 2pTc/y

- 2cTppTc/y2.

Therefore, the second-order term is close to zero when the magnitude of the control-by-noise interaction c/3> is much smaller than that of the noise main effect (y). The derivation above assumes that there is one noise variable and the cases of more than one noise factor are more complicated. Nevertheless, it is generally true that the second-order term of the log variance due to noise is close to zero when the magnitude of the control-by-noise interaction is much smaller than that of the noise main effect for each noise factor.

References American Supplier Institute, Proceedings of symposia on Tuguchi Methods (American Supplier Institute, Dearborn, MI, 1984-1992). Box, G.E.P., Signal to noise ratios, performance criteria and transformation, Technometrics, 30 (1988),

1-31.

Box, G.E.P., W.G. Hunter and J.S. Hunter, Stntistics for experimenters (Wiley, New York, 1978). Box, G.E.P. and S. Jones, Designs for minimizing the effects of environmental variables, Technical report (University of Wisconsin, Madison, WI, 1990). Carroll, R.J. and D. Ruppert, Transformation and weighting in regression (Chapman and Hall, New York, 1988).

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