Alternate Dispersion Measures in Replicated Factorial Experiments

Transcription

1 Alternate Diperion Meaure in Replicated Factorial Experiment Neal A. Mackertich The Raytheon Company, Sudbury MA Jame C. Benneyan Northeatern Univerity, Boton MA Peter D. Krau The Raytheon Company, Tewkbury MA Abtract Any of everal tatitic traditionally are ued to detect diperion effect in the analyi of replicated factorial experiment, including the within-run tandard deviation, the natural logarithm, variou ignal-to-noie ratio, and other. Thi tudy examine the relative performance of each approach uing recent experimental deign, with typically producing the bet reult. An alternate approach, baed on the abolute deviation from the within-run mean, alo i hown to increae detection but at the expene of uncontrolled fale alarm. Introduction While deigned experiment hitorically have focued primarily on optimizing the mean of one or more repone, they increaingly are being ued to identify factor that affect repone diperion. Example include increaed interet in variance reduction a a primary objective, increaed emphai on proce and product robutne, and work on deign for etimating variance function (e.g., Box (1988), Box and Jone (1992), Byrne and Taguchi (1986), Davidian and Carroll (1987), Phadke (1989), and Vining and Schaub (1996)). In order to detect difference in repone variance due to factor effect, any of everal tatitic traditionally are ued in the analyi of replicated factorial experiment. Thee include the within-run tandard deviation, the natural logarithm, the within-run variance 2, variou ignal-to-noie ratio, and other. Thi heightened emphai on repone variance increae the importance of undertanding the relative performance of alternate poible meaure for detecting true variance effect in typical experimental cenario. The current tudy alo i motivated by the uggetion that the bet meaure for detecting difference in ome value of interet might not necearily be one of the mot familiar tatitic for etimating that parameter. We examined the relative performance of conventional and alternate type of diperion meaure in recent experimental cenario, a well a another type of diperion meaure motivated during the coure of thi tudy by a deire to increae power by preerving aociated degree of freedom. 1

2 Alternate Diperion Meaure in DOE Reult ummarized below indicate that tend to be the bet traditional meaure for detecting variance effect, while larger-the-better and maller-the-better ignal-to-noie ratio tend to be quite poor. Two alternate tatitic uggeted during the coure of thi tudy, the abolute deviation from the within-run mean, y i,j - y i, and an approximately normalized tranform of thi tatitic, y i,j - y i.42 increae power in everal example over,, and popular ignal-to-noie ratio, although at the expene of uncontrolled type I error rate. Traditional Diperion Meaure in DOE The mot common meaure ued in replicated deign to identify factor and interaction affecting repone diperion are well-known in the literature (for example, ee Box (1988), Box, Hunter, and Hunter (1978), Montgomery (1984), Myer and Montgomery (1995), Schmidt and Launby (1995), Taguchi (1986)) and include: the within-run ample tandard deviation,, of all repone replicate at each et of experimental condition, the within-run ample variance, 2, of all repone replicate at each et of experimental condition, the logarithm or natural logarithm of or (+1), and Taguchi' three mot common "ignal-to-noie" ratio (i.e., "nominal-the-bet", "maller-the-better", and "larger-the better"). The within-run tandard deviation and variance 2 are direct etimate of their theoretical counterpart σ and σ 2 and need no further motivation. The uual rationale for taking logarithm of the ample tandard deviation i to approximately normalize in order to obtain more accurate reult via tandard tatitical tet for ignificance, with ome mall contant (by convention 1) often firt added to avoid the potential problem of taking a logarithm of zero (uch a due to rounding in data collection). The three mot tandard ignal-to-noie ratio typically are referred to a "nominal-the-bet", "maller-the-better", and "larger-the-better" and have been propoed for the three common ituation for which the experimental objective either are to: achieve repone value a cloe a poible to a deired target value (and with a minimum variability about that target a poible), uch a for a manufacturing dimenion or output voltage, minimize all repone value a much a poible (and with a minimum variability a poible), uch a for percent hrinkage or deviation from round, or maximize all repone value a much a poible (and with a minimum variability a poible), uch a for bond trength or time until failure. 2

3 Alternate Diperion Meaure in DOE Thee three ignal-to-noie ratio uually are etimated, repectively, by: "Nominal-the-bet": S/N ^ N = 10 log y 2 2, "Smaller-the-better": S/N ^ "Larger-the-better": S/N ^ S = -10 log 1 n L = -10 log 1 n n 2 y i i=1 n i=1, and 1. 2 y i A dicued by other, note that an important ditinction of ignal-to-noie ratio i that rather than decoupling the mean and variance into eparate analye, they attempt to form a ingle combined metric of both central tendency and variability (e.g., Gunter (1988) and Hunter (1987)). Alo note that although Taguchi (1986) propoed many other ignal-to-noie ratio (for example, a le common nominal-the-bet ignal-to-noie ratio ha the form S/N N2 = 10 log 2 ), the above three almot excluively tend to be ued in practice. For each ratio and their aociated lo function, the general rationale i to penalize in ome way for deviation from the deired value in term of both location and diperion in a ingle meaure. For further dicuion, ee Box (1988), Hunter (1985, 1987), Phadke (1989), Pignatello and Ramberg (1991), Taguchi (1986), and other. Regardle of which meaure i ued, ignificant factor or interaction typically are identified via the analyi of variance, F tet, t tet, probability plot, dot plot, and the like (ee Box, Hunter, and Hunter (1978), Daniel (1976), and Montgomery (1984)). Unlike the cae of teting for mean effect, however thee tet now are conducted on the 2 k within-run ummary tatitic (in the cae of two-level factorial deign), rather than the n individual outcome within each of the 2 k run, reulting in the lo of 2 k (n - 1) degree of freedom and the need for either at leat one empty column or variance pooling to identify diperion effect. In addition to the above traditional meaure, alo of interet here i whether ome alternative could reult in better variance effect detection. A uggeted earlier, the bet tet criterion for detecting difference in a proce parameter (in thi cae repone variance σ 2 ) may not necearily be baed on the mot familiar direct etimate of that parameter (e.g., the ample variance 2 ). Thi notion i imilar to a tatement by Box (1988) that "the two deiderata - the bet choice of performance meaure and the bet way to employ the data to etimate it - are ditinct and frequently attainable, and they ought not be confued." Following thi reaoning, one general type of alternate meaure might be baed on replacing each of the n oberved repone within a given run i with ome function of that value in uch a manner that each new value now individually provide ome type of meaure of diperion. The primary motivation for thi approach i that each tranformed value now itelf become a type of individual diperion meaure, rather than all within-run i replicate being aggregated into a ingle collective diperion meaure (uch a i, ln( i +1), S/N N, i, et cetera). Statitical analyi then might be conducted on thee individual value, rather than on the traditional ummary meaure, much a one would conduct analyi for mean effect uing 3

4 Alternate Diperion Meaure in DOE each of the n within-run obervation a individual point etimate of the run repone mean µ i. Such an approach would thereby increae the number of aociated degree of freedom, which in turn may reult in tronger detection power, uch a due to a tighter null reference ditribution (although ee below caution). The general tructure of a 2 2 or L 4 replicated deign i hown in Figure 1a, with all of the n repone within each run i replaced in Figure 1b by the correponding abolute deviation of each obervation from the within-run mean, y i,j - y i, where the ubcript i denote a given et of experimental condition (i.e., a run) and the ubcript j denote a given replication of the experiment under thee condition. The abolute deviation value then are analyzed a if they were individual repone uing traditional analyi of variance or other method, analogou to the approach employed in the tudy of central tendency, with the total original number of degree of freedom preerved. Thi may be epecially advantageou for highly Traditional Meaure Experimental Factor Setting Replicate j Reult, y i,j Mean Variance Run (i) A B C (AB) n y i i ln( i +1) S/N N,i y 1,1 y 1,2... y 1,n y 1 1 ln( 1 +1) S/N N, y 2,1 y 2,2... y 2,n y 2 2 ln( 2 +1) S/N N, y 3,1 y 3,2... y 3,n y 3 3 ln( 3 +1) S/N N, y 4,1 y 4,2... y 4,n y 4 4 ln( 4 +1) S/N N,4 Figure 1a: Traditional Structure of Replicated 2 2 Deign Uing Either,, or S/N N a Diperion Meaure Alternate Meaure Experimental Factor Setting Some Function of Replicate Reult Mean Variance Run (i) A B C (AB) n y i y i y i y 1,1 - y 1 y 1,2 - y 1... y 1,n - y 1 y 1 y 1 y y 2,1 - y 2 y 2,2 - y 2... y 2,n - y 2 y 2 y 2 y y 3,1 - y 3 y 3,2 - y 3... y 3,n - y 3 y 3 y 3 y y 4,1 - y 4 y 4,2 - y 4... y 4,n - y 4 y 4 y 4 y 4 Figure 1b: Alternate Structure of Replicated 2 2 Deign Uing y i,j - y i a Diperion Meaure 4

5 Alternate Diperion Meaure in DOE replicated experiment, fairly aturated deign, or mall deign, but alo could produce unpredictable type I error. (Taking abolute value prevent the n within-run (y i,j - y i ) deviation term from mathematically canceling each other out to um to 0.) Comparion of Meaure Approach In order to compare the relative performance of each meaure, everal factor and interaction were included in tandard 2 k-p factorial experimental deign, and the ability to detect different magnitude variance effect of a factor wa examined. Repone value for each run were generated from normal ditribution with parameter pecified by tandard firt-order additive model and µ = β 0 + β r X r r + β r,m X r r m r m X m σ (γ ) = γ 0 + γ r X r + γ X + γ r,m X r X m, r r where X r = Coded etting of factor r, r m r m β r = Mean coefficient (half-effect) of factor r, β r,m = Mean coefficient (half-effect) of interaction X r X m, r m, γ r = Standard deviation coefficient (half-effect) of factor r, r, and γ r,m = Standard deviation coefficient (half-effect) of interaction X r X m, r m, and γ = Standard deviation coefficient (half-effect) of factor S (varied in analyi). Each of the factor etting are determined from the experimental layout and coded a +1 or -1 in the uual manner, with the half-effect ize γ of factor S varied a decribed below. Note that γ repreent the degree to which factor S affect the repone tandard deviation, with γ = 0 indicating that no true effect exit and larger value repreenting larger effect. By iterating acro a range of value for γ, the operating characteritic for each diperion meaure were etimated by filling the experimental array with imluated repone value uing the µ and σ model equation and coded factor etting, calculating each diperion meaure, and conducting analye of variance on thee reult in the uual manner. Thi proce wa repeated 100,000 time at each increment of γ to achieve reaonably accurate reult, with the probability of each meaure ignaling a variance effect for each value of γ etimated a Pr(Signal γ ) = Number of Time F tet wa Significant Total Number of Simulation Replication. 5

6 Alternate Diperion Meaure in DOE A 2 V 5 1 Example A an illutration, in the following analyi four replicate were generated for each of the 16 experimental run of a 2 V 5 1 factorial experiment, with ix hypotheized factor and interaction aigned to column, uing mean and tandard deviation model equation µ = X 1-5X 2 + 7X 3-4X 2 X 3 + 5X 1 X 4 σ = 10 + X X 2 - X 3 + γ 4 X X X 2 X X 1 X 4, uch that the overall repone mean and variance with all factor et at their midpoint are µ Y = 100 and σ Y = 10, repectively (i.e., when all coded term are zero uch that no half-effect are added or ubtracted). The half-effect γ 4 for factor 4 then wa incremented iteratively from γ 4 = 0 to γ 4 = 4 by 0.05 (correponding to half-effect ranging from 0% to 40% of total repone tandard deviation at center point), at each increment of γ 4 repeating the imulation and ubequent analyi of variance (at an α =.05 ignificance level) 100,000 time. Thee reult are hown in Figure 2, with value along the ordinate repreenting the ize of the half-effect of Factor 4 relative to the repone tandard deviation if all X i are et equal to zero, γ 4 /γ 0. Thi caling can be thought of a half the relative reduction in σ poible by changing between X 4 = -1 and X 4 = +1 (auming no active interaction). Alo included in thi analyi i the abolute deviation term raied to the 0.42 power, y i,j - y i.42, a an attempt ^.42 Etimated Probability of Detecting Effect var ^.42 Var Relative Contribution to Standard Deviation at Center Point Figure 2: Relative Performance of Alternate v Traditional Meaure, 2 V 5 1 Example (4 replicate) 6

7 Alternate Diperion Meaure in DOE Probability of Detection Value of γ 4 and γ 4 /γ 0 (Relative Contribution of Factor 4 to Variability) Diperion γ 4 = 0 γ 4 = 1 γ 4 = 2 γ 4 = 3 γ 4 = 4 Meaure (γ 4 /γ 0 = 0) (γ 4 /γ 0 =.1) (γ 4 /γ 0 =.2) (γ 4 /γ 0 =.3) (γ 4 /γ 0 =.4) y j - y y j - y S/N N S/N N S/N S S/N L Table 1: Comparion of Diperion Meaure Performance, 2 V 5 1 Example to normalize the abolute deviation term before forming the tet tatitic a explained below in the Dicuion ection. The ordering from top to bottom of the legend in Figure 2 reflect the relative ordering of each meaure power from bet to wort. Fale alarm probabilitie for each meaure and their power to detect everal effect ize alo are tabulated in Table 1 for further comparion. A thee reult illutrate, the choice of diperion meaure can make a ignificant difference in both the probability of detecting true diperion effect (i.e., power) and the probability of erroneouly ignaling when no effect truly exit (i.e., ignificance). In thi example,,, S/N N, and S/N N2 have comparable operating characteritic and fale alarm probabilitie cloe to the intended α = 0.05 ( α ˆ ), α ˆ , α ˆ S/N(N) , and α ˆ S/N(N2) ). Note that and exhibit lightly higher power for all effect ize, which agree with tudie reported elewhere (Schmidt and Launby (1995), Box (1988), Gunter (1988)). The within-run variance, 2, ha ignificantly lower power, with S/N S and S/N L both eentially being uele with α ˆ S/N(S) and α ˆ S/N(L) and with negligible detection power acro all effect ize. A pointed out by Hunter (1987) and Montgomery (1996), thi i not urpriing given that thee meaure ineparably confound location and diperion effect. Awarene of thee tradeoff can be important information to practitioner in electing a repone meaure and interpreting analyi reult. Interetingly, the abolute deviation meaure exhibit conitently higher power than the other, although at the expene of higher fale alarm probabilitie ( α ˆ = and α ˆ = , repectively), poibly due to the effect of taking abolute value on term lightly le than the average. 7

8 Alternate Diperion Meaure in DOE Adjutment for Equal Fale Alarm Probabilitie Becaue the difference in fale alarm rate noted above make direct comparion difficult, a an exercie the (etimated) ignificance level for each meaure were adjuted empirically to all equal α ˆ = With γ 4 = 0, all 100,000 F tet value for each meaure were orted to locate the empirical 5 th percentile, which then wa ued a an empirical critical value for that meaure, denoted herein a F'. Thee empirical ' Fˆ α =.05 value then were ued in place of the uual critical value a the half-effect γ 4 increaed iteratively a previouly. The reultant α-adjuted operating characteritic for the ame 2 V 5 1 example a above are compared in Figure 3 and Table 2. A hown, the ame performance comparion and ordering a previouly till hold after adjuting all meaure for equal pecificity, but with maller difference in relative power. The conventional meaure,, S/N N, and S/N N2 again are grouped together, with the ample variance 2 exhibiting ignificantly le power, and S/N S and S/N L again being eentially uele for detecting diperion effect. Similar to previouly, both abolute deviation tatitic have greater power than traditional meaure Etimated Probability of Detecting Effect ^.42 var ^.42 Var Relative Contribution to Standard Deviation at Center Point Figure 3: Relative Performance of Alternate v Traditional Meaure, 2 V 5 1 Example (4 replicate) (After empirical adjutment for equal fale alarm probabilitie) Of coure, determining the 5 th empirical percentile of the correponding F value or otherwie adjuting for a deired α probability will not be poible in practice, but the above example illutrate that relative improvement are poible even with fale alarm probabilitie omehow et equal. A illutrated by the following example, imilar benefit alo are 8

9 Alternate Diperion Meaure in DOE Probability of Detection Value of γ 4 and γ 4 /γ 0 (Relative Contribution of Factor 4 to Variability) Diperion γ 4 = 0 γ 4 = 1 γ 4 = 2 γ 4 = 3 γ 4 = 4 Meaure (γ 4 /γ 0 = 0) (γ 4 /γ 0 =.1) (γ 4 /γ 0 =.2) (γ 4 /γ 0 =.3) (γ 4 /γ 0 =.4) y j - y y j - y S/N N S/N N S/N S S/N L Table 2: Comparion of Diperion Meaure Performance, 2 V 5 1 Example (After empirical adjutment for equal fale alarm probabilitie) obtained under other experimental condition (deign ize, number of replicate, degree of aturation). A Second Example (2 3 ) A fairly pare deign wa ued in the above example, reulting in a good number of degree of freedom aociated with the SSE term to etimate between treatment variance (8 for traditional meaure veru 56 for the abolute deviation meaure). Alternatively, if a more aturated 2 3 deign with 5 replicate were ued to tet which of five factor or interaction appear ignificant, the error term would have only 2 degree of freedom uing traditional meaure veru 34 for the abolute deviation meaure. Ue of maller or more aturated deign therefore may impact traditional meaure more adverely than the abolute deviation meaure. In order to examine the relative performance of each meaure in uch cae, 5 replicate for each run of a full factorial 2 3 experimental deign were generated uing the mean and tandard deviation model equation µ = X X 2-1.5X 3-3.5X 1 X X 2 X 3 σ = γ 1 X X X X 1 X X 2 X 3 with the tandard deviation half-effect of factor 1, γ 1, incremented a above and with each analyi of variance again uing α = 0.05 ignificance level. Thee reult are hown in Fig- 9

10 Alternate Diperion Meaure in DOE ure 4a, with Figure 4b again baed on empirically adjuted ˆ α = 0.05 ignificance level. To facilitate further comparion, Table 3a and 3b alo ummarize thee reult ^.42 Etimated Probability of Detecting Effect var ^.42 Var Relative Contribution to Standard Deviation at Center Point Figure 4a: Relative Performance of Alternate v. Traditional Diperion Meaure, 2 3 Example (5 replicate) Etimated Probability of Detecting Effect ^.42 var ^.42 Var Relative Contribution to Standard Deviation at Center Point Figure 4b: Relative Performance of Alternate v. Traditional Diperion Meaure, 2 3 Example (5 replicate) (After empirical adjutment for equal fale alarm probabilitie) 10

11 Alternate Diperion Meaure in DOE Probability of Detection Value of γ 1 and γ 1 /γ 0 (Relative Contribution of Factor 1 to Variability) Diperion γ 1 = 0 γ 1 =.75 γ 1 = 1.5 γ 1 = 2.25 γ 1 = 3 Meaure (γ 1 /γ 0 = 0) (γ 1 /γ 0 =.1) (γ 1 /γ 0 =.2) (γ 1 /γ 0 =.3) (γ 1 /γ 0 =.4 y j - y y j - y S/N N S/N N S/N S S/N L Table 3a: Comparion of Diperion Meaure Performance, 2 3 Example Probability of Detection Value of γ 1 and γ 1 /γ 0 (Relative Contribution of Factor 1 to Variability) Diperion γ 1 = 0 γ 1 =.75 γ 1 = 1.5 γ 1 = 2.25 γ 1 = 3 Meaure (γ 1 /γ 0 = 0) (γ 1 /γ 0 =.1) (γ 1 /γ 0 =.2) (γ 1 /γ 0 =.3) (γ 1 /γ 0 =.4 y j - y y j - y S/N N S/N N S/N S S/N L Table 3b: Comparion of Diperion Meaure Performance, 2 3 Example (After empirical adjutment for equal fale alarm probabilitie) A previouly, note that, S/N N, and S/N N2 exhibit higher power and roughly the ame fale alarm rate a, S/N L and S/N S again are fairly uele, and y i,j - y i and y i,j - y i.42 exhibit higher power acro all magnitude of variance effect. In contrat with the firt example, note that each meaure exhibit lower power a γ 1 increae than previouly, which may 11

12 Alternate Diperion Meaure in DOE be attributable to the difference in deign aturation and degree of freedom aociated with the error term. Ue of the abolute deviation alo may be appealing for another reaon, namely that if the above deign had been fully aturated, tandard ANOVA or other analyi of thee term offer an alternative to the problematic practice of pooling up or down umof-quare that are etimated to have negligible effect. A dicued by Montgomery (1997), "the pooling of mean quare (variance) i a procedure that ha long been known to produce coniderable bia in tet reult." Number of Replicate In order to examine the effect of the number of replication per run on the relative performance of each meaure, Figure 5 and 6 illutrate the effect of uing only n = 2 replicate per run and increaing to 8 replicate per run, repectively, for the ame 2 3 example a above (where previouly n = 5 replicate). Figure 5a and 6a are for the unadjuted cae and Figure 5b and 6b are for the cae with all fale alarm probabilitie empirically adjuted to α ˆ = 0.05 in the manner decribed above. Table 4 alo compare the performance of, y i,j - y i, and y i,j - y i.42 acro a range of other number of replicate. A thee reult illutrate, note that the difference in performance can vary ignificantly for different number of replicate, with the term conitently exhibiting among the bet detection power than other traditional meaure acro all effect and replicate ize. The abolute deviation term exhibit better power but again at the expene of uncontrolled type I error rate. The y i,j - y i fale detection rate for the n = 2 replicate cae i very inflated above the deired α = 0.05 well beyond any ueful point, perhap due to the lack of central limit effect, wherea it appear to decreae to 0 a n increae. A a general rule, therefore, ue of thi meaure for a mall number of replication hould be avoided; mot cae uing four or more replication examined to-date reulted in fale detection rate of roughly 0.10 or lower (with α =.05).. Alo note that none of the meaure are effective in thi example for n = 2. Table 4ummarize the benefit of larger number of replicate for on both power and convergence to the deired α. Dicuion In all examined cae, and both nominal ignal-to-noie ratio conitently produced equal or better power than other traditional meaure, followed by and 2, while the largerthe-better and maller-the-better ratio were relatively uele. The increaed power of y i,j - y i and y i,j - y i.42 to detect true diperion effect i intereting, although at the expene of higher the fale alarm rate. In one 2 V 5 1 example, power to detect diperion half-effect of roughly 25% of the average total repone variance (when all other factor are at their center point) wa increaed to approximately 0.64 from approximately 0.51 for the conventional,, S/N N1, and S/N N2 meaure. Other example ugget imilar benefit for different experiment ize, aturation level, and number of replicate. 12

13 Alternate Diperion Meaure in DOE Etimated Probability of Detecting Effect ^.42 ^.42 var Var Relative Contribution to Standard Deviation at Center Point Figure 5a: 2 Replicate per Run, 2 3 Example 0.3 Etimated Probability of Detecting Effect ^.42 var ^.42 Var Relative Contribution to Standard Deviation at Center Point Figure 5b: 2 Replicate per Run, 2 3 Example (After empirical adjutment for equal fale alarm probabilitie) 13

14 Alternate Diperion Meaure in DOE 1 Etimated Probability of Detecting Effect ^.42 var ^.42 Var Relative Contribution to Standard Deviation at Center Point Figure 6a: 8 Replicate per Run, 2 3 Example 1 ^.42 Etimated Probability of Detecting Effect ^.42 var Var Relative Contribution to Standard Deviation at Center Point Figure 6b: 8 Replicate per Run, 2 3 Example (After empirical adjutment for equal fale alarm probabilitie) 14

15 Alternate Diperion Meaure in DOE Probability of Detection Value of γ 1 and γ 1 /γ 0 (Relative Contribution of Factor 1 to σ) Number of Diperion γ 1 = 0 γ 1 =.75 γ 1 = 1.5 γ 1 = 2.25 γ 1 = 3 Replicate Meaure (γ 1 /γ 0 = 0) (γ 1 /γ 0 =.1) (γ 1 /γ 0 =.2) (γ 1 /γ 0 =.3) (γ 1 /γ 0 = y j - y y j - y y j - y y j - y y j - y y j - y y j - y y j - y y j - y y j - y y j - y y j - y y j - y y j - y y j - y y j - y Table 4: Effect of Number of Replicate (unadjuted 2 3 cae, deired α =.05) Although the abolute deviation meaure are relatively eay to implement by hand or in a preadheet and preent little additional work for the analyt, their fale alarm rate can vary dramatically when uing traditional analyi of variance method. Ideally a more exact mathematical tet baed on the underlying random variable or reference ditribution therefore could be developed. For example, auming Y ~ normal, the abolute deviation from the mean ha a folded or half normal ditribution related to a central chi denity with one degree of freedom (Johnon, Kotz, and Balakrihnan (1994)), uggeting ome type of tet of 15

16 Alternate Diperion Meaure in DOE the upper chi tail. Alternatively, the abolute deviation can be tranformed to approximate normality by raiing it to a power omewhere between 0.35 and 0.55 dependent upon the pecific criteria, with Y - Y.4168 being baed on the Kullback-Leibler information. Reference Box, G. E. P. (1988), Signal-to-Noie Ratio, Performance Criteria, and Tranformation (with dicuion), Technometric, 30, Box, G. E. P., Hunter, W. G., and Hunter, J. S. (1978), Statitic for Experimenter, New York: John Wiley & Son. Byrne, D. M., and Taguchi, S. (1987), The Taguchi Approach to Parameter Deign, Quality Progre, 20(12), Dec 1987, pp Daniel, C. (1976), Application of Statitic to Indutrial Experimentation, New York: John Wiley & Son. Davidian, M. and Carroll R. J. (1987), Variance Function Etimation, Journal of the American Statitical Aociation, 82, Gunter, B. (1988), Dicuion: Signal-to-Noie Ratio, Performance Criteria, and Tranformation, Technometric, 30, Hunter, J. S. (1985), Statitical Deign Applied to Product Deign, Journal of Quality Technology, 17(4), Hunter, J. S. (1987), Signal to Noie Ratio Debated, Quality Progre 20(5), May 1987, pp Johnon, N. L., Kotz, S., and Balakrihnan, N. (1994), Continuou Univariate Ditribution, New York: John Wiley & Son. Montgomery, D. C. (1997), Deign and Analyi of Experiment, 4th ed., New York: John Wiley & Son. Montgomery, D. C. (1996), Introduction to Statitical Quality Control, 3rd ed., New York: John Wiley & Son. Myer, R. H., and Montgomery, D. C. (1995), Repone Surface Methodology, New York: John Wiley & Son. Phadke, M. S. (1989), Quality Engineering Uing Robut Deign, Princeton NJ: Prentice-Hall. Pignatiello, J. J. and Ramberg, J. S. (1991), "The Ten Triumph and Tragedie of Genichi Taguchi, Quality Engineering 4(2), Schmidt, S. R., and Launby, R.G. (1995), Undertanding Indutrial Deigned Experiment, 4th ed., Colorado Spring, CO: Air Academy Pre. Taguchi, G. (1986). Introduction to Quality Engineering, Dearborn, MI: American Supplier Intitute. Vining, G. G. and Schaub, D. (1996), Experimental Deign for Etimating Both Mean and Variance Function, Journal of Quality Technology, 28(2),