dr abdul hannan chowdhury design of experiments in life data analysis with censored data social...

8/2/2019 Dr Abdul Hannan Chowdhury Design of Experiments in Life Data Analysis With Censored Data Social Quality Case S

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Abdul H. ChowdhurySchool of BusinessNort h South Universit y, Dhaka, Bangladeshe-mail: [email protected]

Nasser FardSchool of EngineeringNortheastern University, Boston, USAe-mail: n. [email protected]

iqc .com.pk


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Pakistans 10th

International Convention on Quality Improvement ICQI`2006

Pakistan Institute of Quality Control 2

DESIGN OF EXPERIMENTS IN

LIFE-DATA ANALYSIS THECENSORED DATA

AUTHORS Abdul H. ChowdhurySchool of BusinessNort h South University, Dhaka, Bangladeshe-mail: [email protected]

Nasser FardSchool of EngineeringNortheastern University, Boston, USAe-mail: [email protected]

ABSTRACT Traditionally, design of experiments (DOE) has widely been used for productquality improvement. Analysts are often challenged in analyzing life-testingexperiments with censored data because it encounters many unforeseensituations and the test could be terminated before the predetermined testingtime. In life-testing experiment a typical analysis of variance (ANOVA) assumesthat the observed responses (life-times) are normally distributed and they are not

subject to censoring. This paper presents a robust design experimental t echniqueto analyze right censored data. The procedure also introduced an imputationtechnique for censored data in DOE problem for quality improvement. This paperprovides description of DOE terminologies and different classes of experimentalplans for qualit y improvement for a mult i-factor/ mult i-level experiment . Todemonstrate the application of the proposed method numerical example hasbeen given. Optimal parameter designs for reducing variation in the lifetime andmaximizing the mean lifetime are identified

The options that support a cultural change include leadership activities, usingstories and sett ing agendas, involving employees, qualit y ini t iat ives and processredesign. For any TQM implementation program to succeed in an organization,

the understanding, measuring and changing of organizational culture is a pre-requisite. TQM implementation programs should be accompanied wit h theinfusion and reinforcement of a cultural change.

Keywords: Total Quality Management, Organizational Culture, Behavior,Socialabil it y, Solidarit y, Myths, Planning, Implementat ion, Inst it ut ionalizat ion,Quali ty Initi at ives

INTRODUCTION

In life testing experimentation, reduction of variation in a manufactured productis an integral part of quality improvement. Recent development in qualityimprovement methods have led to considerable interest in the design of

experiment (DOE). In the tradit ional DOE, the expectation of the response is aprimary concern, rather than the variability, and the variability is onlyconsidered with respect to inference on the expectation (Box, Hunter andHunter, 1978). On the other hand, in a robust design of experiment (RDE), thevariabil it y of t he response is of as much interest as the mean. Ongoing


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researchers have emphases on applying RDE to reduce variation, improveproduct/ process qualit y and to achieve robust reliabil it y. 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 uncontrol lable variati ons, by exploi t ing therelationships 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 froma designed experiment using RDE technique. In life testing experiment a typicalANOVA assumes that the observed responses are normal ly dist ributed and theyare not subject to censoring. The paper provides descript ion of DOEterminologies and different classes of experimental plans for qualityimprovement in studying mult i factor/ mult i levels experiment wit h censoredresponse data. This paper also introduced an imputation technique for censoreddata in DOE problem for quality improvement. The paper also describes anapplication of RDE method to improve and to achieve robust reliability. Anindust rial experiment is revisit ed and reanalyzed to obtain t he optimal parameterdesign through imputati on of r ight-censored data.

DOE VERSUSRDE

Michaels (1964) worked on robustness and robust design and discussed howrobustness is improved by exploiting interactions between design andenvironmental variables. Taguchi (1987, 1991) developed RDE methods tooptimize the industrial process experiments using the DOE philosophy. Hisphilosophy provided a new dimension in the quality improvement field thatdiffers from traditional DOE practices. Taguchi considered three stages in aproduct/ process improvement: system, parameter and tolerance design anddesigned experiments by applying orthogonal array (OA). The traditional DOEuses OA to study parameter space t hat contains a large number of design factors,whereas the RDE uses OA, t o study the parameter space that contains a largenumber of decision variables, with a small number of experiments. However, OA

reduces the number of experimental configurations significantly, since Taguchisimpl if ied t heir use by providing tabulated sets of standard OA and correspondingli near graphs to fi t a specif ic project. Using OA, t he robust design approachimproved the efficiency of generating the necessary information to designsystems which are robust to variations in manufacturing processes and operatingconditions. As a result, development time and research and development costs isreduced considerably.

The RDE is a systematic method for producing high quality product by reducingthe cost with a low operating cost. Taguchi (1987) defined two types ofapplication for RDE which he referred to as static and dynamic characteristics.The applications of static characteristics involve situations where obtaining thevalue of a quality characteristic (y), close to a specified target value is the mainobject ive. Nevert heless, the applicat ions of dynamic characterist ics involvesituation where the performance of the system is determined by relationshipbetween a signal factor, and an observe response. For more detail about staticcharacteristics, consult Nair (1992) and Box (1988) and for dynamiccharacteristics, consult Phadke (1989) and Miller & Wu (1995). Freeny and Nair(1992) proposed a method to analyze a situation where noise variables areuncontrolled but observable. Their method involved treating the noise variable ascovariates and modeling both the location parameters and the regressioncoefficients (equivalent to signal-to-noise ratio) as functions of the designfactors. Bryne and Taguchi (1986), Phadke (1989), Taguchi (1986) and Logothet isand Salmon (1988) presented several applications of RDE to optimize qualitycharacteristics.

RDETERMINOLOGYS

Several t erminologies are used in RDE method, f or instance, loss functi on, controlfactors, noise factors and signal to noise ratio (SNR). Loss function is a measureof cost associated with the deviation of product characteristic from its target


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value. Taguchi recommends using the quadratic loss function in equation 1 toapproximate the loss (Leon, Shoemaker, and Kacker, 1987; Phadke, 1989). Theequation for the loss function is of the second order in terms of deviation ofquality characteristic,

L y t k y t( , ) ( )= 2 , (1)

where ybe the quality characteristic of a product, t be the target value for yandk is a constant (quality loss co-efficient) that depends on the cost structure of amanufact uring process. Constant k is determined by using a simple economicargument (Taguchi and Wu, 1980 and Phadke, 1989). The term (y - t) representsthe deviation of quality characteristic y from the target value t. By takingexpectation of equation 1 with respect to y, we obtain t he expected loss

E L y t kE y t[ ( , )] [( ) ]= 2 . (2)

If yis nonnegative and t = 0, then the loss funct ion reduces to

E L y t kE y[ ( , )] [ ]= =0 2 . (3)

Note t hat it is possible to est imate the expected loss if a random sample f rom thedistribution of yis obtained.

In RDE two types qualit y systems are of great import ance, they are off-l ine andon-line quality system. The off-line quality system is all activities, which takeplace before production begins (activities in product and process design). Theseactivities usually include market research, product and process development. Theoff-line quality control system reduces performance variation and hence theproduct s li fetime cost . The on-line quali ty system is all act ivit ies, which takeplace 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 variat ion t hataffects product output. Noise factors are those parameters that areuncontrollable or too expensive to control. There are three types of noise: Outer(environmental), Inner (deteriorative), and Product-to-product (manufacturingimperf ection) noise. Taguchi s RDE uses a statistical measure of performancecall ed signal-t o-noise rat io (SNR). The SNR is a measure of how sensit ive a systemis to noise (or variance). An insensitive (robust) system will have a high SNR. TheSNR is a performance measure t o choose control levels that best cope wit h noiseand 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). TheSNR depends on the crit erion for t he qualit y characterist ic t o be opt imized.

Although there are many dif ferent possible SNR, three of them are consideredstandard and are generally applicable in situations given below:

Smallest -is-best quali ty character istic:

The measure of performance characteristic y has a nonnegativedist ributi on, the target t is set to zero, and the loss L(y) increases as yincreasesfrom zero. In this case, the expected loss in equation 3 is proportional to

MSE x E y( ) [ ]= 2 (4)

and Taguchi (1976, 1977) recommends using the perf ormance measure

SNR MSE x= 10 10log ( ) . (5)

The larger the performances measure, the smaller is the mean squared error


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(MSE). The performance statistic can then be estimated as

SNRn

yS i= 101

10

2log [ ] . (6)

This is employed when the object ive is to make the response as small as possible.

Biggest -is-best qualit y character isti c:

The measure of performance characteristic yhas a nonnegative distribution, thetarget 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, where1

y

is the performance characteristic. The target t of1

yis set to zero. In this case,

the expected loss in equation 3 is proportional to

MSE x E y( ) [( / ) ]= 1 2 (7)

and the performance measure is

SNR MSE x= 10 10log ( ) . (8)

The larger the performances measure, the smaller is the mean squared error(MSE). The performance statistic reduces to

SNRn y

B

i

= 101 1

10 2log [ ] . (9)

This is employed when the object ive is to make the response as large as possibl e.

Nominal-is-best quality characteristic:

The performance characteristics y has a specific target t = t0 and the lossfunction L(y) increases as ydeviates from t0 in either direction. In this case, theexpected loss in equation 2 is proportional to

MSE x E y t( ) [( ) ]= 2 . (10)

The performance statistic can then be estimated as

SNR ysT =

10 10

2

2log [ ] . (11)

This is employed when the obj ective is closeness to target .

Regardless of the type of quality characteristic, the transformations are such thatthe SNR is always int erpreted the same, the larger the SNR rat io t he bet ter. Ahigh value of the SNR represents a small variance and t herefore a small quali tyloss.

DOE AND RDE

PLAN

In practice, three factors such as signal (m), noise (n), and control (x) factorsaffect the qualit y characterist ics of a product/ processes. The signal factors areselected by the experimenters based on the experience and are set by the usersor operator to express the intended value for the response of the product. Noisefactors are those factors whose settings (levels) are uncontrollable or expensiveto control by the designer. On the other hand, control factors are those factorsspecif ied by the designer and it is the designer s duty to determine the best


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values for t hese factors. Figure 1 shows a typical block diagram of aproduct/ process, which is used to represent a process or a system, usuallyreferred to as p-diagram(Phadke, 1989).

Noise factors (n)

Signal f actors (m)Product / Process

Response (y)/ Quali t y characteri st ics

Control f act ors (x)

Figure 1: Block diagram of a product/ process in RDE

Suppose that in a sample of (Xi, Yi), where independent variable (Xi) arecompletely observed and in the response variable (Yi) some values are censored.This structure is expressed by a monotone pattern of censored data expressed infigure 2. This patt ern helps to classif y the censored data st rategy according towhether the probability of the response: depends on response variable Yand Xvariable as well; depends on Xvariable but not on Yvariable; and is independentof both Xand Yvariable.

X Y

x11 xk1 y1x12 xk2 y2

: ycom: ym-1

: : ym: : ym+1: : ym+2

x1(n-1) xk(n-1) : ycenx1n xkn yn-1

yn

Figure 2: Patt ern f or censored data in t he DOE

In RDE planning, it is important to know which control factor changemanufacturing cost and which do not. To solve the RDE problem, Taguchisuggests a two-part experimenting strategy, and formulates the RDE problem tochoose the levels of the control factors (x) to minimize the expected loss causedby the noise factors (n) illustrated in figure 3. In RDE, three quality

characterist ics such as the mean (y ), the variance (s2) and t he SNR are used as

the response variables. Estimates of y and s2 are obtained from the repeated

product of each setting and the estimates of SNR for different cases are given inequat ion 6, 9 and 11. Whenever an experiment involves repeated observations ateach of the trial conditions, the SNR provides a practical way to measure and

control combined influence of deviation of the population mean from the targetand the variation around the mean.


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Control Factor (x) Noise Factor (n)Response (Y)Run D E A BC F G HI J Y1 Y2 Y3

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

2 -1 -1 0 0 0 0 -10 1 y1,1 ycen y1,3

3 - 1 -1 1 1 1 10 1 -1

: : : :1 - 1 0

: : : : : : :

: 10 0 -1 -1 0 -1-1 : : :

: 11 0 -1 0 10 0 : :: : 12 0 -1 1-1 1 1 : :

: : : : : : : : : :

: : : : : : : : : :

251 1 -1 1 1 0 -1 01 ycen y27,2 y27,326 1 1 0 -1 -1 1 01 -1

27 1 1 1 0 0 -11 - 1 1

Figure 3: Pattern for Censored data in RDE

METHODOLOGY

The analysis of censored response data in a designed experiment requi res somemodifications in the usual procedures since the actual failure time would begreater than the censored time.

Suppose that the response Y is written as (y1, y2,,ym, ym+1,,yn) and theresponse column is divided into two sub groups, such as Ycom = (y1, y2,,ym) forthe completely observed response and Ycen = (ym+1,,yn) for the censoredresponse variables. Let , the response variable Y be arranged according to

YY

Yicom

cen

=

, and the design matrix (consists only control factor) be expressed by

X, then confine the structure of the data set as follows:Y

YX

com

cen

= + . (12)

Where, Y is an n-vector of possible observations, Xis an (n k) design matrix (n

> k), is a k-factor of unknown parameters, while is a vector of random errors.It is assumed that errors are independently normally distributed with means zero

and a common variance 2.

Now, partitioning X matrix into two subgroups as XX

X

C=

, where the X

matrix is of order (n k), XCare the observations in Xmatrix corresponding to

the observed response variable andX* in

Xmatrix denotes to the censoredresponse variable. Thus, the data matrix is obtained as follows:

Y

Y

X

X

com

cen

C C

=

+

. (13)

Where Ycom is (m 1), and XCis an (mk) matrix with rank k. However, equation(13) is decomposed into two parts. The f irst part produces a sub-model f or thecompletely observed data, which is given by,

Y Xcom C C C = + (14)

with the residual sum of squares (RSS) of ( )

Y I X X X X Y com C C T

C C

T

com( )1

and

the corresponding normal equation gives the least square estimate (LSE) of C,

( )$

c CT C CT comX X X Y=1

. (15)

As a classical predictor of the (n - m) vector Ycen, the estimate$c from the

complete case, is added to the predicted value for censored observation based on


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the corresponding factor levels from censored response. The elements of c are

assumed to be independently dist ributed wi th mean 0 and variance 2. Theresidual sum of square (RSS) is

S(, Ycen) = Y X Y X com com cen cen + 2 2

= S1() + S2(Ycen, ), (16)

which must now be minimized with respect to both (parameter) and Ycen (thevector of unknown replacing the censored values). Yate s (1933)method is based

on minimization of S (, Ycen) first with respect to and then with respect toYcen.

Although in some experiments the censored data affect the distribution of theremaining observations, it is implicitly assumed throughout the analysis that thedistribution of the observed random variables of the model is the same with orwithout censored data Dodge (1985). It is assumed that t he lifet ime before andafter censoring follows the same distribution. Therefore, the failure time after

censored time depends on the coefficient of the distribution of failure timebefore censoring. Hence, we estimate predicted failure times (Pft) using the

obtained coefficient. The predicted failure times are estimated by consideringthe corresponding factor level combination. The following imputation formula isused for censored response data:

$Ycen = Tcen+ Pf t, (17)

Where Tcen (censoring time) is the actual experimental ending time and Pft is the

predicted failure time beyond censoring time. The Pf t is est imated asXi c ,

where i represent the corresponding runs for factor level combination.

NUMERICAL

EXAMPLE

In this section, we adopt a numerical example from clutch spring experimentprovided by Lakey and Rigdon (1993) for RDE with six control factors. Lakey and

Rigdon (1993) analyzed the data without considering noise factors in theexperimental plan and treated censoring times as the actual failure time. Thisexperiment involves 36 full factorial design and the six design factors are: (A)

shapes (3 levels), (B) hole ratio (2 levels), (C) coining (2 levels), (D) stress (t =40, 65, 90), (E) stress (c = 140, 170, 200), (F) shot-peening (3 levels), and (G)outside perimeter planing (3 levels), and the response variable is the lifetime ofclutch spring. A full experiment involving six factors with three levels (L: low, M:medium and H: high) require 36 = 729 runs to run a full factorial experiment.Instead, a 1/ 27th fraction of the experiment as a fraction of 729 runs are taken inthe control array and 3 clutch springs are made according to the specifications atthe 27 diff erent 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 tosequences of 10,000 compressions until either (i) the spring fails, in which casethe failure time is the number of 10,000 compression stages that the unit wentthrough, or (i i) the spring goes through 11 sessions of 10,000 compressions. Sincetest terminates after 11 cycles at 10,000 compressions (prior to failure of allunit s), then the test result s in censored data. A star (*) mark expresses censoredresponse data in the table 2 and the expression f or censoring are shown by t hefigure 4.

Table 1: Clut ch spring experiment data

Run Factor Response

No D E A BC F G Y1 Y2 Y3

1 -1 -1 -1 -1 -1 -1 1 1 12 -1 -1 0 0 0 0 4 5 11*3 -1 -1 1 1 1 1 2 2 11*


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4 -1 0 -1 -1 0 1 2 3 35 -1 0 0 0 1 -1 5 11* 11*6 -1 0 1 1 -1 0 1 1 17 -1 1 -1 -1 1 0 1 1 38 -1 1 0 0 -1 1 1 1 29 -1 1 1 1 0 -1 3 3 4

10 0 -1 -1 0 -1 -1 1 1 211 0 -1 0 1 0 0 11* 11* 11*12 0 -1 1 -1 1 1 6 11* 11*13 0 0 -1 0 0 1 11* 11* 11*14 0 0 0 1 1 -1 2 2 215 0 0 1 -1 -1 0 1 2 216 0 1 -1 0 1 0 2 3 417 0 1 0 1 -1 1 2 2 218 0 1 1 -1 0 -1 11* 11* 11*19 1 -1 -1 1 -1 -1 3 4 420 1 -1 0 -1 0 0 11* 11* 11*21 1 -1 1 0 1 1 11* 11* 11*22 1 0 -1 1 0 1 11* 11* 11*

23 1 0 0 -1 1 -1 11* 11* 11*24 1 0 1 0 -1 0 5 11* 11*25 1 1 -1 1 1 0 4 4 626 1 1 0 -1 -1 1 2 2 327 1 1 1 0 0 -1 11* 11* 11*

Censored data in the response variables are expressed by a star (*) mark in table1 and their corresponding imputed values are shown in table 2 with double star(**) mark. The joint eff ect of B and C factor is treated as one of t he designfactors in the design and the three levels of the BC factor are actually the (BLCL:low, low), (BHCL: high, low) and (BLCH: low, high). Since, factors B and C, bothhave two levels, the combination of B and C results in a design factor with threelevels. However, all factors except A and BC are quant it ative variables. Factors A(shape) and BC (combinations of B and C) are nominal or class variables, (not aquantitative variable). In the experimental design, we treat the stress variablesas design factors, alt hough they seem to be noise variables and we consider threenoise factors: t emperature (H), humidity (I), and usage (J). Then, f or each row inthe 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 hasbeen noted t hat Taguchi s accumulati ng analysis is incorrect, ineff icient andcomplicated to handle censored data (Box and Jones (1986), Nair (1986), andBox, Bisgaard and Fung (1987)). Now, using the proposed method for imputingcensored data in a RDE by t reat ing t he mean and SNRB as response variables,since the objective is to maximize clutch spring lifetime. Using the equation 9,we obtain t he SNRB response (biggest is best quality characteristics) shown in the

last column of table 2, for instance,

SNRB2 2 2 2101

3

1

4

1

5

1

121438= + +

=log . .

Table 2: Mean and SNR response when bigger response is better

Run Response Mean SNR

No Y1 Y2 Y3

1 1 1 1 1.00 0.002 4 5 12** 7.00 14.383 2 2 11** 5.00 7.714 2 3 3 2.67 8.03

5 5 13** 14** 10.67 17.696 1 1 1 1.00 0.007 1 1 3 1.67 1.538 1 1 2 1.33 1.25


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9 3 3 4 3.33 10.2310 1 1 2 1.33 1.2511 14** 14** 13** 13.00 22.7012 6 14** 12** 10.67 18.7713 13** 14** 14** 13.67 22.7014 2 2 2 2.00 6.02

15 1 2 2 1.67 3.0116 2 3 4 3.00 8.5017 2 2 2 2.00 6.0218 14** 13** 14** 13.67 22.7019 3 4 4 3.66 11.0420 15** 14** 13** 14.00 22.8821 17** 15** 13** 15.00 23.3722 14** 15** 15** 14.67 23.3123 16** 14** 15** 15.00 23.4824 5 14** 14** 10.67 17.7625 4 4 6 4.67 12.9326 2 2 3 2.33 6.9127 15** 14** 15** 14.33 23.31

Recall that for all cases censoring time was 11 and using this censoring time weestimated all imputed data for censored response. In estimating the imputed

data for the first response we obtained the following regression coefficients: 0(2.918), 1 (0.891), 2 (-0.427), 3 (0.680), 4 (-.147), 5 (0.867), and 6 (-0.100).Using these coefficients we obtained the estimated Pf t for x11 as 3.198. Hence,

the imputed value of x11 would be$Ycen (14.198) which was expressed as 14 wit h

double star (**) mark in column 2 of table 2. The computation of imputedcensored data for the first response provides: x11 (14.198), x13 (13.138), x18(14.418), x20 (15.383), x21 (16.683), x22 (13.882), x23 (15.923), and x27 (15.162).Simi larl y, i n est imat ing the imputed data for t he second response we obtained

the following regression coefficients: 0 (2.562), 1 (0.537), 2 (-0.349), 3 (-0.0782), 4 (0.25), 5 (0.447), and 6 (0.159). The computation of imputedcensored data for the first response provides: x5 (13.313), x11 (14.161), x12(14.188), x13 (13.799), x18 (12.726), x20 (14.198), x21 (14.976), x22 (14.581), x23(14.137), x24 (13.574) and x27 (13.513). These est imat es are shown in column 3 oftable 2. Furthermore, we use the following regression coefficients for the third

response: 0 (2.56), 1 (0.724), 2 (0.921), 3 (-0.530), 4 (0.145), 5 (0.26), and6 (-0.425). The computation procedure provides the following estimates for theimputed data of the third response: x2 (11.915), x3 (11.365), x5(13.521), x11(12.784), x12 (11.799), x13 (13.665), x18 (14.230), x20 (13.218), x21 (12.668), x22(14.534), x23 (14.824), x24 (13.494), and x27 (15.100). These est imates are shownin column 4 of t able 2.

ANALYSIS ANDFINDINGS

The est imates of t he average SNR for all factor levels are calculated using table 1and 2 and are plotted in f igure 5. The average SNR of t he fi rst l evel of factor D intable 1 is calculated as the average of the nine run results conducted with level 1of t he factor D. Similarly, the average of t he second level of f actor D iscalculated as the average of the run 10 through 18 results conducted with level 2of the factor D and the average of the third level of factor D is calculated as theaverage 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 variat ions in t he SNR,whereas factor G causes a small change in the SNR. The effects of other factorsare wit hin this range.


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D L 1

D L 2

D L 3

E L 2

F L 1

F L 2

F L 3

G L 2

G L 3G L 1

B C L 2

B C L 3B C L 1

A L 1

A L 2

A L 3

E L 3

E L 1

0

2

4

6

81 0

1 2

1 4

1 6

1 8

2 0

L

1

L

2

L

3

L

1

L

2

L

3

L

1

L

2

L

3

L

1

L

2

L

3

L

1

L

2

L

3

L

1

L

2

L

3

D E A B C F G

F a c t o r s

A

verag

es

o

f

S

N

R

Figure 5: Average SNR for clutch spring experiment

In addit ion, f igure 5 interpret s the li neari ty of t he factors related to SNRresponse. For three levels of a particular factor, when the difference betweenlevels 1 & 2 and levels 2 & 3 is equal and these levels appear i n 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 ifthey are not in the proper order, then the effect of that factor is not linear. Forexample, factor D has approximately a linear response while the factor F hasnonlinear response. The average SNR for each level of the six factors are given intable 3. The table reveals that factors D and F are more significant than otherfactors. Clearly, level 2 of factor F appears to be the best choice because itcorresponds to the highest SNR.

Table 3: Average SNR for clutch spring experiment data

Factor Average SNR

Level 1 Level 2 Level 3

A

BC

D

E

F

G

9.92

11.92 (BLCL)

6.76

13.56

5.25

12.86

13.48

14.47 (BHCL)

12.41

13.55

18.91

11.52

14.10

11.11 (BLCH)

18.33

10.38

13.33

13.12

Factor A, BC, E and G appear to have little significance with respect to SNR. Forfactor BC it is seen that level 2 is better than level 1 and 3. As for factor A, it isseen that level 3 is better than level 1 and 2. If there are very little differencesamong the mean SNR of t he three levels of a factors, then eit her level could bechosen. In this case, decision should be based on the analysis of mean response.To determine the relative importance of the six factorial effects, standardANOVA is performed to identify which factors have a significant effect on theSNR. These factors are identif ied as control factors and implying that t hey

cont rol the variabil it y. Whenever the MSS for a part icular factor is smaller t hanmean sum of square residual (MSE), then a pooled ANOVA is formed by poolingthe sum of squares of t he factors with RSS. The ANOVA is perf ormed assuming theresponse SNR follows a normal dist ribut ion wi th constant variance. The ANOVA


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result shows that the effects of factor D and F are the main contributors followedby factor A, BC, and E and factor G is small contributor. The ANVOA results showthat the factor D (signifi cant at 0.003) and factor F (signifi cant at 0.000) for SNRresponse. Hence both D and F are t he most impor tant factors for SNR response.The ANOVA result also shows that the mean sum squares (MSS) for factor G (6.6)is smaller t han 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 usingtable 1 and 2. The average of the first, second and third level of factor D arecalculated as the average of t he fi rst nine run result s wit h level 1, the run 10 - 18results with level 2, and the run 19 - 27 results with level 3 of the factor D,respectively. The average means for the six factors with three levels are shown infigure 4. It can be seen that the factor D and F causes large variat ions in themean response. Nevert heless, factor G causes a small change in t he means.Figure 6 shows that for factor D the difference between levels 1 & 2 and 2 & 3are 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, thusthe effect of factor E and F are non-linear.

E L 2A L 3 B C L 2

G L 3

F L 2

F L 1

F L 3

G L 1

G L 2B C L 3

B C L 1

A L 2

A L 1

D L 3

D L 2

D L 1E L 3

E L 1

0

2

4

6

8

1 0

1 2

L

1

L

2

L

3

L

1

L

2

L

3

L

1

L

2

L

3

L

1

L

2

L

3

L

1

L

2

L

3

L

1

L

2

L

3

D E A B C F G

F a c t o r s

A

verag

es

o

f

M

ean

s

Figure 6: Average of means for clut ch spring experiment

The averages of mean response in t able 4 reveals that factors D and F are moresignificant than other factors and level 2 of factor F appears to be the bestchoice since it corresponds to the highest average of the mean response. FactorA, BC, E and G appear to be insignificant with respect to their correspondingmeans. For factor BC that level 2 is little better than level 1 and 3. Also, table 4indicates level 3 of factor A is preferable over level 2, whereas level 2 of factor Eis preferable over level 1. The ANOVA result shows that the effects of factor Dand F are larger than the effects of factor A, BC, and E. Factor G has a smallcontribution. The ANOVA results show that the factor D (significant at 0.003) andfactor F (signif icant 0.000) f or mean response. Hence both D and F are t he mostimport ant factors for mean response. The result s also show that the MSS forfactor G (3.5) is smaller t han the MSE (6.394). Then, a pooled ANOVA isperformed by pool ing the MSS of G wi th MSE.

Table 4: Average of means for clut ch spring experiment


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Factor Average of Means

Level 1 Level 2 Level 3

A

BCD

E

F

G

5.15

6.96 (BLCL)3.74

7.85

2.77

7.22

7.48

8.56 (BHCL)6.78

8.00

10.70

6.29

8.37

5.48 (BLCH)10.48

5.15

7.52

7.48

The percentage contributions (PC) are equal t o the percentage of t he tot al sumof squares (TSS) explained by the factors after an appropriate est imates of theRSS are removed. The larger PC indicates the greater sensitivity of the responsevariable 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 SSA is the sum of squares of factor A, MSSR is the mean sum of squares ofresidual, the TSSis the total sum of squares, and d.fA is the degrees of freedomof 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 offreedom 26. Using equat ion 18, we est imate percentage contr ibut ions for bothmean and SNR response shown on the last two columns of the table 5.

Table 5: Percent age contribut ions for the mean and SNR response

Source Sum ofSquares

Sum ofSquares

Percent agecontributionfor meanresponse

Percent agecontributionfor SNRresponse

A

BC

D

E

FResidual

49.8

42.6

205.1

46.4

286.5158.3

91.4

55.3

603.1

60.9

849.7287

3.81

2.89

23.50

3.37

33.8132.62

2.85

1.00

29.13

1.29

41.7923.94

Total 788.7 1947.4 100 100

Clearly, table 5 depicts that both factor D and F contributes highest percentagefor mean and SNR response. On the other hand, other f actors such as A, BC, andE have the lowest percent age contribut ion for mean and SNR response. Recallthat t he MSS for factor G is smaller t han MSE, and therefore we pooled it bypooling the sum of squares of t he factor G wit h RSS. Furt hermore, to separatethe joint effect of BC factor into two individual effects (such as factor ef fect of B

and C) in the pooled ANOVA, t he Phadke, et . al. (1983) approach could be used.

CONCLUSIONS In this study, we use mean and signal-t o-noise rat io as the response variable since


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ANDDISCUSSIONS

the objective is to maximize the product lifetime. The analysis of variance wasused to determine which factor inf luence mean and which inf luence SNR ofclutch spring experimental data. Graphical display shows both factor D and Fcauses large variat ion in the mean and SNR response. It was found that effect offactor D is l inear whil e factor F is not l inear for SNR. Nevertheless, eff ect offactor 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 meanand SNR. The marginal means for each f actor in both the mean and SNR graphsshow that factors D and F are more significant than other factors. Clearly, level 2of factor F appears to be the best choice as it corresponds to the highest meanand SNR. The ANOVA result s show that the both D and F factors are signif icant formean and SNR response. Hence both D and F are the most impor tant factors formean and SNR response. However, mean, SNR and ANOVA results reveal thatfactor G has a small contribution. Findings show that the optimal parameterdesign is AHBHCLDHELFM for SNRB and AHBHCLDHEMFM for mean lifetime. Hence, thisfactor level combination reduces the variation in the lifetime as well asmaximizing the mean li fetime of clutch spring.

The RDE provides a systematic and efficient approach for finding optimumcombination of design parameters so that the product is functional, exhibits ahigh level of performance, and is robust to noise factors. This study shows thatRDE method is applicable f or analyzing right -censored data after a suit ableimputation of censored data. Principal benefits include considerable time andresource savings; determination of important factors affecting operation,performance; quantitative measures of the robustness (sensitivity) of the nearoptimum results; and high quality solutions. Overall results suggest that RDE is apowerful tool, which offers simultaneous improvements in quality, reliability andproductivity.

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2. Box, G. E. P., Bisgaard, S. and Fung, C. 1987. An Explanation and Cri t ique ofTaguchi s Contr ibut ions to Qualit y Engineering: Journal of Quality andReliability Engineering International. 4: 123 131.

3. Box, G. E. P. and Jones, S. 1986. An Invest igat ion of t he Method ofAccumulation Analysis. R - 9, Center for Quality & Productivity Improvement,University of Wisconsin.

4. Dogde, Y. 1985. Analysis of Experiment s with Missing Data. New York: JohnWiley.

5. Freeny, A. E. and Nair, V. N. 1992. Robust Parameter Design withUncontrolled Noise Variables: Statistica Sinica. 2: 313 - 334.

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10.Logothet is N. and Salmon J. P. 1988. Tolerance Design and Analysis of Audio-


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Circui ts. Taguchi Methods: Proceedings of the 1988 European Conference,Elsevier Applied Science, UK, 161 175.

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12.Miller. A. and Wu, C. F. J. 1995. Parameter Design for Signal ResponseSystems, Universit y of Michigan, Technical Report No. 251.

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dr abdul hannan chowdhury design of experiments in life data analysis with censored data social...

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