piqc.com.pk Abdul H. Chowdhury School of Business North South University, Dhaka, Bangladesh

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1 Pakistan s 10th International Convention on Quality Improvement November 27~28, 2006 Lahore, Pakistan DESIGN OF EXPERIMENTS IN LIFE-DATA ANALYSIS WITH CENSORED DATA Abdul H. Chowdhury School of Business North South University, Dhaka, Bangladesh hannan@northsouth.edu Nasser Fard School of Engineering Northeastern University, Boston, USA n.fard@neu.edu piqc.com.pk

2 DESIGN OF EXPERIMENTS IN LIFE-DATA ANALYSIS THE CENSORED DATA AUTHORS Abdul H. Chowdhury School of Business North South University, Dhaka, Bangladesh Nasser Fard School of Engineering Northeastern University, Boston, USA ABSTRACT Traditionally, design of experiments (DOE) has widely been used for product quality improvement. Analysts are often challenged in analyzing life-testing experiments with censored data because it encounters many unforeseen situations and the test could be terminated before the predetermined testing time. In life-testing experiment a typical analysis of variance (ANOVA) assumes that the observed responses (life-times) are normally distributed and they are not subject to censoring. This paper presents a robust design experimental technique to analyze right censored data. The procedure also introduced an imputation technique for censored data in DOE problem for quality improvement. This paper provides description of DOE terminologies and different classes of experimental plans for quality improvement for a multi-factor/multi-level experiment. To demonstrate the application of the proposed method numerical example has been given. Optimal parameter designs for reducing variation in the lifetime and maximizing the mean lifetime are identified The options that support a cultural change include leadership activities, using stories and setting agendas, involving employees, quality initiatives and process redesign. For any TQM implementation program to succeed in an organization, the understanding, measuring and changing of organizational culture is a prerequisite. TQM implementation programs should be accompanied with the infusion and reinforcement of a cultural change. Keywords: Total Quality Management, Organizational Culture, Behavior, Socialability, Solidarity, Myths, Planning, Implementation, Institutionalization, Quality Initiatives INTRODUCTION In life testing experimentation, reduction of variation in a manufactured product is an integral part of quality improvement. Recent development in quality improvement methods have led to considerable interest in the design of experiment (DOE). In the traditional DOE, the expectation of the response is a primary concern, rather than the variability, and the variability is only considered with respect to inference on the expectation (Box, Hunter and Hunter, 1978). On the other hand, in a robust design of experiment (RDE), the variability of the response is of as much interest as the mean. Ongoing Pakistan Institute of Quality Control 2

3 researchers have emphases on applying RDE to reduce variation, improve product/process quality and to achieve robust reliability. Kacker (1985), Leon, Shoemaker & Kacker (1987), Phadke (1989), Quinlan (1989), Taguchi (1991), Nair (1992), Freeny and Nair (1992), and Hamada (1993) studied RDE methodology. Taguchi s RDE reduces the effect of uncontrollable variations, by exploiting the relationships between the response variable and the control and noise variables. In life testing experiments, tests are usually terminated before all units fail, which would lead to censored data. This paper analyzed right-censored data from a designed experiment using RDE technique. In life testing experiment a typical ANOVA assumes that the observed responses are normally distributed and they are not subject to censoring. The paper provides description of DOE terminologies and different classes of experimental plans for quality improvement in studying multi factor/ multi levels experiment with censored response data. This paper also introduced an imputation technique for censored data in DOE problem for quality improvement. The paper also describes an application of RDE method to improve and to achieve robust reliability. An industrial experiment is revisited and reanalyzed to obtain the optimal parameter design through imputation of right-censored data. DOE VERSUS RDE Michaels (1964) worked on robustness and robust design and discussed how robustness is improved by exploiting interactions between design and environmental variables. Taguchi (1987, 1991) developed RDE methods to optimize the industrial process experiments using the DOE philosophy. His philosophy provided a new dimension in the quality improvement field that differs from traditional DOE practices. Taguchi considered three stages in a product/ process improvement: system, parameter and tolerance design and designed experiments by applying orthogonal array (OA). The traditional DOE uses OA to study parameter space that contains a large number of design factors, whereas the RDE uses OA, to study the parameter space that contains a large number of decision variables, with a small number of experiments. However, OA reduces the number of experimental configurations significantly, since Taguchi simplified their use by providing tabulated sets of standard OA and corresponding linear graphs to fit a specific project. Using OA, the robust design approach improved the efficiency of generating the necessary information to design systems which are robust to variations in manufacturing processes and operating conditions. As a result, development time and research and development costs is reduced considerably. The RDE is a systematic method for producing high quality product by reducing the cost with a low operating cost. Taguchi (1987) defined two types of application for RDE which he referred to as static and dynamic characteristics. The applications of static characteristics involve situations where obtaining the value of a quality characteristic (y), close to a specified target value is the main objective. Nevertheless, the applications of dynamic characteristics involve situation where the performance of the system is determined by relationship between a signal factor, and an observe response. For more detail about static characteristics, consult Nair (1992) and Box (1988) and for dynamic characteristics, consult Phadke (1989) and Miller & Wu (1995). Freeny and Nair (1992) proposed a method to analyze a situation where noise variables are uncontrolled but observable. Their method involved treating the noise variable as covariates and modeling both the location parameters and the regression coefficients (equivalent to signal-to-noise ratio) as functions of the design factors. Bryne and Taguchi (1986), Phadke (1989), Taguchi (1986) and Logothetis and Salmon (1988) presented several applications of RDE to optimize quality characteristics. RDE TERMINOLOGY S Several terminologies are used in RDE method, for instance, loss function, control factors, noise factors and signal to noise ratio (SNR). Loss function is a measure of cost associated with the deviation of product characteristic from its target Pakistan Institute of Quality Control 3

4 value. Taguchi recommends using the quadratic loss function in equation 1 to approximate the loss (Leon, Shoemaker, and Kacker, 1987; Phadke, 1989). The equation for the loss function is of the second order in terms of deviation of quality characteristic, Lyt (, ) = ky ( t) 2, (1) where y be the quality characteristic of a product, t be the target value for y and k is a constant (quality loss co-efficient) that depends on the cost structure of a manufacturing process. Constant k is determined by using a simple economic argument (Taguchi and Wu, 1980 and Phadke, 1989). The term (y - t) represents the deviation of quality characteristic y from the target value t. By taking expectation of equation 1 with respect to y, we obtain the expected loss ELyt [ (, )] = ke[( y t) 2 ]. (2) If y is nonnegative and t = 0, then the loss function reduces to ELyt [ (, = 0)] = key [ 2 ]. (3) Note that it is possible to estimate the expected loss if a random sample from the distribution of y is obtained. In RDE two types quality systems are of great importance, they are off-line and on-line quality system. The off-line quality system is all activities, which take place before production begins (activities in product and process design). These activities usually include market research, product and process development. The off-line quality control system reduces performance variation and hence the product s lifetime cost. The on-line quality system is all activities, which take place after production begins (activities in manufacturing). These activities usually include manufacturing, quality assurance, and customer support. Another important aspect in RDE is noise. Noise is any source of variation that affects product output. Noise factors are those parameters that are uncontrollable or too expensive to control. There are three types of noise: Outer (environmental), Inner (deteriorative), and Product-to-product (manufacturing imperfection) noise. Taguchi s RDE uses a statistical measure of performance called signal-to-noise ratio (SNR). The SNR is a measure of how sensitive a system is to noise (or variance). An insensitive (robust) system will have a high SNR. The SNR is a performance measure to choose control levels that best cope with noise and it takes into account both the mean and the variability. In its simplest form, the SNR is the ratio of the mean (signal) to the standard deviation (noise). The SNR depends on the criterion for the quality characteristic to be optimized. Although there are many different possible SNR, three of them are considered standard and are generally applicable in situations given below: Smallest-is-best quality characteristic: The measure of performance characteristic y has a nonnegative distribution, the target t is set to zero, and the loss L(y) increases as y increases from zero. In this case, the expected loss in equation 3 is proportional to 2 MSE( x) = E[ y ] (4) and Taguchi (1976, 1977) recommends using the performance measure SNR = 10log 10 MSE( x). (5) The larger the performances measure, the smaller is the mean squared error Pakistan Institute of Quality Control 4

5 (MSE). The performance statistic can then be estimated as SNR S 1 2 = 10log 10[ yi ]. (6) n This is employed when the objective is to make the response as small as possible. Biggest-is-best quality characteristic: The measure of performance characteristic y has a nonnegative distribution, the target t is set to infinity, and the loss L(y) decreases as y increases from zero. Recall that this is a particular application of the smallest is best case, where 1 y is the performance characteristic. The target t of 1 y is set to zero. In this case, the expected loss in equation 3 is proportional to 2 MSE( x) = E[( 1/ y) ] (7) and the performance measure is SNR = 10log 10 MSE( x). (8) The larger the performances measure, the smaller is the mean squared error (MSE). The performance statistic reduces to 1 1 SNRB = 10log 10[ 2 ]. (9) n yi This is employed when the objective is to make the response as large as possible. Nominal-is-best quality characteristic: The performance characteristics y has a specific target t = t 0 and the loss function L(y) increases as y deviates from t 0 in either direction. In this case, the expected loss in equation 2 is proportional to MSE( x) = E[( y t) 2 ]. (10) The performance statistic can then be estimated as 2 y SNRT = 10log 10[ 2 ]. (11) s This is employed when the objective is closeness to target. Regardless of the type of quality characteristic, the transformations are such that the SNR is always interpreted the same, the larger the SNR ratio the better. A high value of the SNR represents a small variance and therefore a small quality loss. DOE AND RDE PLAN In practice, three factors such as signal (m), noise (n), and control (x) factors affect the quality characteristics of a product/ processes. The signal factors are selected by the experimenters based on the experience and are set by the users or operator to express the intended value for the response of the product. Noise factors are those factors whose settings (levels) are uncontrollable or expensive to control by the designer. On the other hand, control factors are those factors specified by the designer and it is the designer s duty to determine the best Pakistan Institute of Quality Control 5

6 values for these factors. Figure 1 shows a typical block diagram of a product/process, which is used to represent a process or a system, usually referred to as p-diagram (Phadke, 1989). Noise factors (n) Signal factors (m) Product /Process Response (y)/ Quality characteristics Control factors (x) Figure 1: Block diagram of a product/ process in RDE Suppose that in a sample of (X i, Y i ), where independent variable (X i ) are completely observed and in the response variable (Y i ) some values are censored. This structure is expressed by a monotone pattern of censored data expressed in figure 2. This pattern helps to classify the censored data strategy according to whether the probability of the response: depends on response variable Y and X variable as well; depends on X variable but not on Y variable; and is independent of both X and Y variable. X Y x 11 x k1 y 1 x 12 x k2 y 2 : y com : y m-1 : : y m : : y m+1 : : y m+2 x 1(n-1) x k(n-1) : y cen x 1n x kn y n-1 y n Figure 2: Pattern for censored data in the DOE In RDE planning, it is important to know which control factor change manufacturing cost and which do not. To solve the RDE problem, Taguchi suggests a two-part experimenting strategy, and formulates the RDE problem to choose the levels of the control factors (x) to minimize the expected loss caused by the noise factors (n) illustrated in figure 3. In RDE, three quality characteristics such as the mean ( y ), the variance (s 2 ) and the SNR are used as the response variables. Estimates of y and s 2 are obtained from the repeated product of each setting and the estimates of SNR for different cases are given in equation 6, 9 and 11. Whenever an experiment involves repeated observations at each of the trial conditions, the SNR provides a practical way to measure and control combined influence of deviation of the population mean from the target and the variation around the mean. Pakistan Institute of Quality Control 6

7 Control Factor (x) Noise Factor (n) Response (Y) Run D E A BC F G H I J Y1 Y2 Y y1,1 ycen y1, : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : ycen y27,2 y27, Figure 3: Pattern for Censored data in RDE METHODOLOGY The analysis of censored response data in a designed experiment requires some modifications in the usual procedures since the actual failure time would be greater than the censored time. Suppose that the response Y is written as (y 1, y 2,,y m, y m+1,,y n ) and the response column is divided into two sub groups, such as Y com = (y 1, y 2,,y m ) for the completely observed response and Y cen = (y m+1,,y n ) for the censored response variables. Let, the response variable Y be arranged according to Ycom Yi =, and the design matrix (consists only control factor) be expressed by Ycen X, then confine the structure of the data set as follows: Ycom = Xβ + ε. (12) Ycen Where, Y is an n-vector of possible observations, X is an (n k) design matrix (n > k), β is a k-factor of unknown parameters, while ε is a vector of random errors. It is assumed that errors are independently normally distributed with means zero and a common variance σ 2. X C Now, partitioning X matrix into two subgroups as X =, where the X X matrix is of order (n k), X C are the observations in X matrix corresponding to the observed response variable and X * in X matrix denotes to the censored response variable. Thus, the data matrix is obtained as follows: Y Y com cen X C εc = β + X ε. (13) Where Y com is (m 1), and X C is an (m k) matrix with rank k. However, equation (13) is decomposed into two parts. The first part produces a sub-model for the completely observed data, which is given by, Ycom = XCβC + ε C (14) T 1 T with the residual sum of squares (RSS) of Ycom ( I XC ( XC XC) XC) Ycom and the corresponding normal equation gives the least square estimate (LSE) of β C, $ T 1 T β c = ( X C XC) XCY com. (15) As a classical predictor of the (n - m) vector Y cen, the estimate $ β c from the complete case, is added to the predicted value for censored observation based on Pakistan Institute of Quality Control 7

8 the corresponding factor levels from censored response. The elements of ε c are assumed to be independently distributed with mean 0 and variance σ 2. The residual sum of square (RSS) is 2 2 S(β, Y cen ) = Ycom Xcom + Ycen Xcen = S 1 (β) + S 2 (Y cen, β), (16) which must now be minimized with respect to both β (parameter) and Y cen (the vector of unknown replacing the censored values). Yate s (1933) method is based on minimization of S (β, Y cen ) first with respect to β and then with respect to Y cen. Although in some experiments the censored data affect the distribution of the remaining observations, it is implicitly assumed throughout the analysis that the distribution of the observed random variables of the model is the same with or without censored data Dodge (1985). It is assumed that the lifetime before and after censoring follows the same distribution. Therefore, the failure time after censored time depends on the β coefficient of the distribution of failure time before censoring. Hence, we estimate predicted failure times (P ft ) using the obtained β coefficient. The predicted failure times are estimated by considering the corresponding factor level combination. The following imputation formula is used for censored response data: $ Y cen = T cen + P ft, (17) Where T cen (censoring time) is the actual experimental ending time and P ft is the predicted failure time beyond censoring time. The P ft is estimated as X i β c, where i represent the corresponding runs for factor level combination. NUMERICAL EXAMPLE In this section, we adopt a numerical example from clutch spring experiment provided by Lakey and Rigdon (1993) for RDE with six control factors. Lakey and Rigdon (1993) analyzed the data without considering noise factors in the experimental plan and treated censoring times as the actual failure time. This experiment involves 3 6 full factorial design and the six design factors are: (A) shapes (3 levels), (B) hole ratio (2 levels), (C) coining (2 levels), (D) stress (σ t = 40, 65, 90), (E) stress (σ c = 140, 170, 200), (F) shot-peening (3 levels), and (G) outside perimeter planing (3 levels), and the response variable is the lifetime of clutch spring. A full experiment involving six factors with three levels (L: low, M: medium and H: high) require 3 6 = 729 runs to run a full factorial experiment. Instead, a 1/27 th fraction of the experiment as a fraction of 729 runs are taken in the control array and 3 clutch springs are made according to the specifications at the 27 different design factors varied with three noise factors. Three springs are made at each treatment and the experimental setup, and the failure data are shown in table 2. Each of the 81 springs is subjected to sequences of 10,000 compressions until either (i) the spring fails, in which case the failure time is the number of 10,000 compression stages that the unit went through, or (ii) the spring goes through 11 sessions of 10,000 compressions. Since test terminates after 11 cycles at 10,000 compressions (prior to failure of all units), then the test results in censored data. A star (*) mark expresses censored response data in the table 2 and the expression for censoring are shown by the figure 4. Table 1: Clutch spring experiment data Run Factor Response No D E A BC F G Y 1 Y 2 Y * * Pakistan Institute of Quality Control 8

9 * 11* * 11* 11* * 11* * 11* 11* * 11* 11* * 11* 11* * 11* 11* * 11* 11* * 11* 11* * 11* * 11* 11* Censored data in the response variables are expressed by a star (*) mark in table 1 and their corresponding imputed values are shown in table 2 with double star (**) mark. The joint effect of B and C factor is treated as one of the design factors in the design and the three levels of the BC factor are actually the (B L C L : low, low), (B H C L : high, low) and (B L C H : low, high). Since, factors B and C, both have two levels, the combination of B and C results in a design factor with three levels. However, all factors except A and BC are quantitative variables. Factors A (shape) and BC (combinations of B and C) are nominal or class variables, (not a quantitative variable). In the experimental design, we treat the stress variables as design factors, although they seem to be noise variables and we consider three noise factors: temperature (H), humidity (I), and usage (J). Then, for each row in the control array, the noise factors are varied according to the noise array, resulting in the experimental data shown in last three columns of table 1. It has been noted that Taguchi s accumulating analysis is incorrect, inefficient and complicated to handle censored data (Box and Jones (1986), Nair (1986), and Box, Bisgaard and Fung (1987)). Now, using the proposed method for imputing censored data in a RDE by treating the mean and SNR B as response variables, since the objective is to maximize clutch spring lifetime. Using the equation 9, we obtain the SNR B response (biggest is best quality characteristics) shown in the last column of table 2, for instance, log = SNR B2 = Table 2: Mean and SNR response when bigger response is better Run Response Mean SNR No Y 1 Y 2 Y ** ** ** 14** Pakistan Institute of Quality Control 9

10 ** 14** 13** ** 12** ** 14** 14** ** 13** 14** ** 14** 13** ** 15** 13** ** 15** 15** ** 14** 15** ** 14** ** 14** 15** Recall that for all cases censoring time was 11 and using this censoring time we estimated all imputed data for censored response. In estimating the imputed data for the first response we obtained the following regression coefficients: β 0 (2.918), β 1 (0.891), β 2 (-0.427), β 3 (0.680), β 4 (-.147), β 5 (0.867), and β 6 (-0.100). Using these coefficients we obtained the estimated P ft for x 11 as Hence, the imputed value of x 11 would be $ Y cen (14.198) which was expressed as 14 with double star (**) mark in column 2 of table 2. The computation of imputed censored data for the first response provides: x 11 (14.198), x 13 (13.138), x 18 (14.418), x 20 (15.383), x 21 (16.683), x 22 (13.882), x 23 (15.923), and x 27 (15.162). Similarly, in estimating the imputed data for the second response we obtained the following regression coefficients: β 0 (2.562), β 1 (0.537), β 2 (-0.349), β 3 ( ), β 4 (0.25), β 5 (0.447), and β 6 (0.159). The computation of imputed censored data for the first response provides: x 5 (13.313), x 11 (14.161), x 12 (14.188), x 13 (13.799), x 18 (12.726), x 20 (14.198), x 21 (14.976), x 22 (14.581), x 23 (14.137), x 24 (13.574) and x 27 (13.513). These estimates are shown in column 3 of table 2. Furthermore, we use the following regression coefficients for the third response: β 0 (2.56), β 1 (0.724), β 2 (0.921), β 3 (-0.530), β 4 (0.145), β 5 (0.26), and β 6 (-0.425). The computation procedure provides the following estimates for the imputed data of the third response: x 2 (11.915), x 3 (11.365), x 5 (13.521), x 11 (12.784), x 12 (11.799), x 13 (13.665), x 18 (14.230), x 20 (13.218), x 21 (12.668), x 22 (14.534), x 23 (14.824), x 24 (13.494), and x 27 (15.100). These estimates are shown in column 4 of table 2. ANALYSIS AND FINDINGS The estimates of the average SNR for all factor levels are calculated using table 1 and 2 and are plotted in figure 5. The average SNR of the first level of factor D in table 1 is calculated as the average of the nine run results conducted with level 1 of the factor D. Similarly, the average of the second level of factor D is calculated as the average of the run 10 through 18 results conducted with level 2 of the factor D and the average of the third level of factor D is calculated as the average of the run 19 through 27 results conducted with level 3 of the factor D. Figure 5 shows that the factor D and F causes large variations in the SNR, whereas factor G causes a small change in the SNR. The effects of other factors are within this range. Pakistan Institute of Quality Control 10

11 D E D D E E A A A B C B C B C F F F G G G A verag es of SN R D E A B C F G Facto rs Figure 5: Average SNR for clutch spring experiment In addition, figure 5 interprets the linearity of the factors related to SNR response. For three levels of a particular factor, when the difference between levels 1 & 2 and levels 2 & 3 is equal and these levels appear in the chronological (proper) order (either 1, 2, 3 or 3, 2, 1), then the effect of that factor is linear. Nonetheless, for a particular factor if difference between levels is unequal or if they are not in the proper order, then the effect of that factor is not linear. For example, factor D has approximately a linear response while the factor F has nonlinear response. The average SNR for each level of the six factors are given in table 3. The table reveals that factors D and F are more significant than other factors. Clearly, level 2 of factor F appears to be the best choice because it corresponds to the highest SNR. Table 3: Average SNR for clutch spring experiment data Factor A BC D E F G Average SNR Level 1 Level 2 Level (B L C L ) (B H C L ) (B L C H ) Factor A, BC, E and G appear to have little significance with respect to SNR. For factor BC it is seen that level 2 is better than level 1 and 3. As for factor A, it is seen that level 3 is better than level 1 and 2. If there are very little differences among the mean SNR of the three levels of a factors, then either level could be chosen. In this case, decision should be based on the analysis of mean response. To determine the relative importance of the six factorial effects, standard ANOVA is performed to identify which factors have a significant effect on the SNR. These factors are identified as control factors and implying that they control the variability. Whenever the MSS for a particular factor is smaller than mean sum of square residual (MSE), then a pooled ANOVA is formed by pooling the sum of squares of the factors with RSS. The ANOVA is performed assuming the response SNR follows a normal distribution with constant variance. The ANOVA Pakistan Institute of Quality Control 11

12 result shows that the effects of factor D and F are the main contributors followed by factor A, BC, and E and factor G is small contributor. The ANVOA results show that the factor D (significant at 0.003) and factor F (significant at 0.000) for SNR response. Hence both D and F are the most important factors for SNR response. The ANOVA result also shows that the mean sum squares (MSS) for factor G (6.6) is smaller than the MSE (11.33). Hence, we perform a pooled ANOVA by pooling the MSS of G with MSE. The estimates of the average means for all factor levels are calculated using table 1 and 2. The average of the first, second and third level of factor D are calculated as the average of the first nine run results with level 1, the run results with level 2, and the run results with level 3 of the factor D, respectively. The average means for the six factors with three levels are shown in figure 4. It can be seen that the factor D and F causes large variations in the mean response. Nevertheless, factor G causes a small change in the means. Figure 6 shows that for factor D the difference between levels 1 & 2 and 2 & 3 are equal and these levels appear in the proper order (i.e. they are in order 1, 2, 3 or 3, 2, 1), thus the effect of factor D is linear. Nonetheless, for factor E and F, neither the differences between levels are equal nor the orders are proper, thus the effect of factor E and F are non-linear. 12 Averag es of M eans D D D E E E A A A BC BC BC F F F G G G D E A B C F G Factors Figure 6: Average of means for clutch spring experiment The averages of mean response in table 4 reveals that factors D and F are more significant than other factors and level 2 of factor F appears to be the best choice since it corresponds to the highest average of the mean response. Factor A, BC, E and G appear to be insignificant with respect to their corresponding means. For factor BC that level 2 is little better than level 1 and 3. Also, table 4 indicates level 3 of factor A is preferable over level 2, whereas level 2 of factor E is preferable over level 1. The ANOVA result shows that the effects of factor D and F are larger than the effects of factor A, BC, and E. Factor G has a small contribution. The ANOVA results show that the factor D (significant at 0.003) and factor F (significant 0.000) for mean response. Hence both D and F are the most important factors for mean response. The results also show that the MSS for factor G (3.5) is smaller than the MSE (6.394). Then, a pooled ANOVA is performed by pooling the MSS of G with MSE. Table 4: Average of means for clutch spring experiment Pakistan Institute of Quality Control 12

13 Factor A BC D E F G Average of Means Level 1 Level 2 Level (B L C L ) 8.56 (B H C L ) 5.48 (B L C H ) The percentage contributions (PC) are equal to the percentage of the total sum of squares (TSS) explained by the factors after an appropriate estimates of the RSS are removed. The larger PC indicates the greater sensitivity of the response variable with respect to the changing the level of that factor. The formula for computing the PC for a factor A is given below: SS MSS d. f TSS A R A 100 (18) where SS A is the sum of squares of factor A, MSS R is the mean sum of squares of residual, the TSS is the total sum of squares, and d.f A is the degrees of freedom of factor A. Here, the degrees of freedom associated with all factors A, BC, D, E, and F was 2 and the residual degrees of freedom 16 results in total degrees of freedom 26. Using equation 18, we estimate percentage contributions for both mean and SNR response shown on the last two columns of the table 5. Table 5: Percentage contributions for the mean and SNR response Source Sum of Squares Sum of Squares Percentage contribution for mean response A BC D E F Residua l Total Percentage contribution for SNR response Clearly, table 5 depicts that both factor D and F contributes highest percentage for mean and SNR response. On the other hand, other factors such as A, BC, and E have the lowest percentage contribution for mean and SNR response. Recall that the MSS for factor G is smaller than MSE, and therefore we pooled it by pooling the sum of squares of the factor G with RSS. Furthermore, to separate the joint effect of BC factor into two individual effects (such as factor effect of B and C) in the pooled ANOVA, the Phadke, et. al. (1983) approach could be used. CONCLUSIONS In this study, we use mean and signal-to-noise ratio as the response variable since Pakistan Institute of Quality Control 13

14 AND DISCUSSIONS the objective is to maximize the product lifetime. The analysis of variance was used to determine which factor influence mean and which influence SNR of clutch spring experimental data. Graphical display shows both factor D and F causes large variation in the mean and SNR response. It was found that effect of factor D is linear while factor F is not linear for SNR. Nevertheless, effect of factor D is linear while factor E and F is not linear for mean response. Therefore, the factor D causes large variation, as well as its effect is linear for both mean and SNR. The marginal means for each factor in both the mean and SNR graphs show that factors D and F are more significant than other factors. Clearly, level 2 of factor F appears to be the best choice as it corresponds to the highest mean and SNR. The ANOVA results show that the both D and F factors are significant for mean and SNR response. Hence both D and F are the most important factors for mean and SNR response. However, mean, SNR and ANOVA results reveal that factor G has a small contribution. Findings show that the optimal parameter design is A H B H C L D H E L F M for SNR B and A H B H C L D H E M F M for mean lifetime. Hence, this factor level combination reduces the variation in the lifetime as well as maximizing the mean lifetime of clutch spring. The RDE provides a systematic and efficient approach for finding optimum combination of design parameters so that the product is functional, exhibits a high level of performance, and is robust to noise factors. This study shows that RDE method is applicable for analyzing right-censored data after a suitable imputation of censored data. Principal benefits include considerable time and resource savings; determination of important factors affecting operation, performance; quantitative measures of the robustness (sensitivity) of the near optimum results; and high quality solutions. Overall results suggest that RDE is a powerful tool, which offers simultaneous improvements in quality, reliability and productivity. REFERENCES 1. Box, G. E. P., Hunter, W. G. and Hunter, J. S Statistics for Experimenters: An Introduction to Design, Data Analysis, and Model Building. New York: John Wiley. 2. Box, G. E. P., Bisgaard, S. and Fung, C An Explanation and Critique of Taguchi s Contributions to Quality Engineering: Journal of Quality and Reliability Engineering International. 4: Box, G. E. P. and Jones, S An Investigation of the Method of Accumulation Analysis. R - 9, Center for Quality & Productivity Improvement, University of Wisconsin. 4. Dogde, Y Analysis of Experiments with Missing Data. New York: John Wiley. 5. Freeny, A. E. and Nair, V. N Robust Parameter Design with Uncontrolled Noise Variables: Statistica Sinica. 2: Hamada, M Reliability Improvement via Taguchi s Robust Design: Journal of Quality and Reliability Engineering International. 9: Kackar, Raghu Off-Line Quality Control, Parameter Design, and the Taguchi Method: Journal of Quality Technology, 17(4): Lakey, M. J. and Rigdon, S. E Reliability Improvement using Experimental Design: 1993 ASQC Quality Congress Transactions Leon, R. V., Shoemaker, A. C. and Kacker, R. N Performance Measures Independent of Adjustment- An Explanation and Extension of Taguchi s Signal-to-Noise Ratios: Technometrics, 29(3): Logothetis N. and Salmon J. P Tolerance Design and Analysis of Audio- Pakistan Institute of Quality Control 14

15 Circuits. Taguchi Methods: Proceedings of the 1988 European Conference, Elsevier Applied Science, UK, Michaels, S. E The Usefulness of Experimental Designs (with discussion), Applied Statistics. Journal of Royal Statistical Society, C, 13(3), Miller. A. and Wu, C. F. J Parameter Design for Signal Response Systems, University of Michigan, Technical Report No Nair, V. J Testing in Industrial Experiments with Ordered Categorical Data: Technometrics, 28(4): Nair, V. J. (ed) Taguchi's Parameter Design- A Panel Discussion: Technometrics, 34(2): Phadke. M. S Quality Engineering using Robust Design. New Jersey: Prentice Hall. 16. Phadke. M. S., Kacker, R.N., Speeney, D.V. and Greico, M.J Off-Line Quality Control for Integrated Circuit Fabrication using Experimental Design: Bell System Technical Journal, 62: Quinlan, J. and et.al Product Improvement by Application of Taguchi Methods. Taguchi Methods: Applications in World Industry, IFS Pub Taguchi, G Taguchi Methods, vol.-1, R&D. Michigan: ASI Press. 19. Taguchi, G Experimental Designs. 3 rd Edition, 1, Maruzen Publishing Company, Tokyo, Japan. 20. Taguchi, G Experimental Designs. 3 rd Edition, 2, Maruzen Publishing Company, Tokyo, Japan. 21. Taguchi, G Introduction to Quality Engineering. Asian Productivity Organization, Tokyo, Japan. 22. Taguchi, G System of Experimental Design. Unipub/ Kraus International Publications, New York, USA. 23. Taguchi, G. and Wu, Y. I Introduction to Off-line Quality Control. Central Japan Quality Control Association, Nagaya, Japan. 24. Yates, F The Analysis of Replicated Experiments when the Field Results are Incomplete. Emp. Journal of Experiments in Agriculture. 1: Pakistan Institute of Quality Control 15