1
1. TAGUCHI ANALYSIS
1.1 Background
The principles of Dr. Genichi Taguchi's Quality Engineering, or Taguchi Meth-
o ds, as they are often called in the United States, have had a dramatic impact on
U.S. industry, particularly since the early 1980's, as more and more companies have
started various programs and training metho ds with the hop e of achieving some of
the successes that various Japanese companies have exp erienced over the last thirty
or more years. Case studies detailing successes at the Ina Seito Tile Co., the Electri-
cal Communication Lab oratory Japan , the Kawasaki Oil Re nery,aswell as seven
other examples are given in Taguchi [130 ]. Many consultants travel throughout the
U.S. presenting seminars and training workshops concerned with the learning and
implementation of Taguchi metho ds, to ols and applications. Taguchi metho ds have
b een used in such situations as VLSI circuit design Welch et al [142], non-destructive
testing McEwan, Belavendram, and Ab ou-Ali [93], rob ot pro cess capability Black
and Jiang [14], thickness of aluminum dep osits on silicon wafers Aceves, Hernandez,
and Murphy [2], injection molded plastic comp onents Hanrahan and Baltus [63],
as well as cruise control vacuum valves, shrinkage of sp eedometer cable, computer
resp onse time, p owers supply circuits, noise reduction in hydraulic rollers, and in re-
ducing the rework rate of diesel injectors Wu [143 ]. More examples may b e found in
Dehned [37]. The American Supplier Institute holds an annual symp osium devoted
solely to Taguchi Metho ds, with the rst held in 1983.
In most quality or quality control situations, the goal is to pro duce output as
uniformly near a target value as p ossible. The reduction of variation is now regarded
2
as a fundamental concept, b oth for those concerned with manufacturing and pro duc-
tion and also for those fo cused on service or task pro cessing. The Taguchi metho d is
aimed at the manufacturing situation.
To understand the workings of Taguchi metho ds one must consider three p ossible
situations concerning a qualitycharacteristic of interest. These three are: larger val-
ues are b etter, smaller values are b etter, target value is b est. A qualitycharacteristic
is an imp ortant dimension, prop erty or attribute of a manufactured output. The
three cases will b e considered separately.
For the case where a target value is b est, the aim of Taguchi metho ds is twofold:
center some measured qualitycharacteristic around a target value, and minimize the
variation around this target value. The famous Sony-USA compared with Sony-Japan
example illustrates that b eing centered at the target value is not enough, that b eing
correctly centered at the target and b eing within sp eci cation limits is b etter, but the
b est situation is one of b eing centered at the target and having minimal variation
around that target. See Sullivan, [124 ] .
The realization that both the average output and the variability of the output
are crucial is a tremendous improvementover the traditional engineering practice
of only considering whether or not a part falls within sp eci cation limits. With
regard to the implementation of not the concept of this idea, Taguchi has exerted
a broad in uence and deserves much praise. Any deviation from the optimal value is
considered bad, the severity of the deviation is usually measured by a quadratic loss
function. The loss to the manufacturer or so ciety is considered to b e prop ortional to
the square of the distance from the optimal value. Mathematically, this loss can b e
2
written: l Y = k Y T , where Y is the observed value and T is the optimal or
target value. The constant, k , can b e determined based on the loss at a xed value of
Y T . This xed value may represent the edge of the sp eci cation limits, or mightbe
apoint at which it is felt that, say, 50 of the customers would complain or require
warrantywork. The squared error loss is motivated bya Taylor series expansion. See
Taguchi, [129] and [130 ] . Minimizing this loss or exp ected loss is consistent with
3
the two goals of the Taguchi Metho d. The squared error loss function is a continuous
function, and is strictly increasing in jY T j, as opp osed to the discrete pass/fail
nature of sp eci cation limits. This loss function is also e ectiveinconvincing p eople
to worry ab out variation even when the pro cess is on target and mayeven b e within
the desired sp eci cation limits. In particular, if the loss is expressed in dollar units,
those at the managementlevel will now b e more concerned ab out reducing variation.
Consider next the situation where smaller values are b etter, but the resp onse or
qualitycharacteristic of interest is non-negative. In this situation we are concerned
2
with the same loss, k Y T , except nowwe set the target value to zero and hop e
2
to minimize E Y , which represents the mean squared error, or mse, around zero.
The nal situation is that in which the resp onse is again non-negative, but here
2
larger values are considered to b e b etter. The goal is to maximize E Y . Here the
2
Taguchi practitioner considers E 1=Y and again tries to minimize this quantity.
The quantities to b e minimized stem from the desire to measure and reduce the
variation relative to the target value. However, the Taguchi metho d do es not base its
analysis directly on the ab ove quantities; the exact quantity to b e optimized for each
of these three situations will b e given later. In using the recipro cal squared as the
quantityofinterest, the larger is b etter situation now b ecomes that of smaller values
b eing b etter, and we pro ceed as in the smaller is b etter situation.
The rst step in implementing the Taguchi Metho d is that of parameter design.
Parameters are various inputs susp ected or known to have an e ect on the quality
characteristic, and might include such things as rate of ow, temp erature, humidity
or reaction time. Parameter design is an investigation to determine whichvariables
or parameters have an e ect on the qualitycharacteristic. This attempts to improve
on quality while still at the design stage, as opp osed to intensive sampling plans that
insp ect their way to high qualityby `catching' p o or output. Designing high quality
and lowvariabilityinto the pro duct will not only b e muchcheap er than extensiveor
exhaustive insp ection, but also will allow a pro duct to go from blueprint to usable
pro duct in much less time. Variables susp ected to have some e ect on the quality
4
characteristic of interest are known as factors. At this p ointany exp ert knowledge of
the sub ject matter should b e applied in the selection of variables to include as factors.
Those variables susp ected to have the largest e ects on the qualitycharacteristic are
generally considered rst but it is very p ossible to unknowingly omit variables that
should have b een considered and also to include others that have no e ect on the
qualitycharacteristic. Once chosen, these variables are then classi ed into one of
two categories: control variables or noise variables. Control variables represent those
parameters or factors which can b e controlled by the designer or manufacturer. In
contrast, noise variables are those parameters whichmayhave an e ect on the quality
characteristic but generally cannot b e controlled by the designer. Variables that are
very exp ensiveorvery dicult to control may also b e considered noise variables.
In order to screen these variables for their e ect on the qualitycharacteristic, a
classical Taguchi designed exp erimenttakes place. Various settings are selected for
eachofthecontrol variables and various settings are temporarily xed for eachof
the noise variables. The controllable factors are placed into what is known as the
inner array. The inner array is somewhat similar to the classical `design matrix'.
Eachrow of this inner array represents one particular combination of settings for the
control factors, or one `design p oint'. Thus an inner array with eightrows represents
eight p ossible con gurations of the control factors. The noise factors are placed into
what is termed the outer array. The settings for the outer array should representa
range of p ossible noise conditions that will b e encountered in everyday pro duction,
although in practice it is very common to consider only two or three levels for each
of the control factors. Again, these settings are xed solely for the purp ose of the
exp eriment, in general they will b e random. If the noise is assumed to have a linear
e ect, the two settings would generally b e , when the noise factor is
noise noise
susp ected or known to have a quadratic e ect on the qualitycharacteristic, the
p p
3=2; ; + 3=2 .\Thesechoices three settings would generally b e
noise noise noise
are apparently based on the assumption that the noise factors have approximately
symmetric distributions" Kackar, [78] . The twoarrays are then crossed, which
5
means that every exp erimental condition called for by the inner array of control factors
takes place at every condition called for by the outer array of noise factors. In this way
all of the p oints in the inner array are exp osed to the same values of the noise factors
in the outer array; it is hop ed that the temp orarily xed settings of the noise factors at
the exp erimentation level will approximate the random conditions that will o ccur at
the pro duction level. Data measurements on the qualitycharacteristic of interest are
then collected. In some cases the qualitycharacteristic will b e obvious: nal weightof
an exp ensive pro duct sold byweight, diameter of a cylinder, length of a shaft, shipping
weightofabox of cereal, numb er of defectivewelds on an assembly, time to complete
a crucial task, purity of a nal pro duct, numb er of blemishes on a car do or, number of
cycles until failure of a key mechanical part, or the amount of stress b efore failure of a
cable or attaching mechanism. The question of what to measure is b est answered by
a p erson with exp ert knowledge in the area, not by an exp ert in Taguchi metho ds or
in statistical analysis. Particularly in the engineering sciences, anyknowledge of the
underlying physical structure should b e used as an aid in cho osing whichvariables to
include in the design, and also in determining what qualitycharacteristic should b e
measured. This sub ject matter knowledge may also servetopoint out what ranges
of these variables should b e considered, ranges that mayormay not coincide with
the ranges of the `knobs' that can b e turned. Moreover, sub ject matter knowledge
mayserve to p oint out that the control variables or `knobs' available representsome
transformation of the underlying variables that drive the pro cess or reaction. An
example of this would b e where the qualitycharacteristic dep ends critically on the
density of a mixture, but the `knob' available to the exp erimenter is blending sp eed.
In general, for an inner arraywithm settings of the control factors and n con g-
urations of the noise variables, the Taguchi metho d requires mn observations, even
more if there are replications. The two arrays are most often balanced, which means
that for any column of either array, an equal numb er of observations are taken for
each setting that is given in the array.Furthermore, for any pair of columns from the
same array , anycombinations of factors that do o ccur will o ccur the same number of
6
times. Thus if the setting where factor A is at level one and factor B is at level three
o ccurs four times in the inner array, then any combination of factors A and B that
do es o ccur must o ccur exactly four times in the inner array. The arrays are generally
0
is zero if i 6= j ,wherex x also orthogonal, which means that the vector pro duct x
i j
i
~ ~ ~
and x represent columns from the same array, suitably scaled, centered at zero. This
j
~
balance in the arrays allows the exp erimenter to determine the e ects of individual
factors byaveraging over the levels of the other factors. The data collected consist
of measurements of some qualitycharacteristic, collected at each setting dictated by
the two crossed arrays. This characteristic might b e, for example, p ercent purity,
yield, or length of some critical dimension. Controllable factors to b e placed into the
inner arraymight include engine sp eed, amount of a certain chemical, thickness of
a metal part, amount of adhesive applied, voltage through a resistor, gas ow, mix-
ing sp eed, spring tension, initial load etc. These representvariables that can easily
and e ectively b e controlled by the exp erimenterormanufacturer. Noise factors in
the outer array could include ambienthumidity in a large factory, a particular lot
or batchofa key ingredient, cleanliness of a part, the supplier of a key input, or
amount of time since last lubrication. The noise variables and control variables to b e
included in any exp eriment tend to b e particular to the situation b eing considered;
those most familiar with the output of interest will often have a general idea of what
forces, controllable or otherwise, have an in uence on the output of interest.
The inner array is sometimes referred to as the design space, while the outer
array is sometimes referred to as the environment space. The settings in eachof
the arrays generally represent a factorial design or a fractional factorial design. A
factorial design is one where every value or level of each factor o ccurs with all levels
of every other factor. Consider an inner array with three factors. If one factor has
twolevels, one has three levels and one has four, a factorial design for this array
would require 2 3 4 = 24 observations, more if replications are desired. This inner
arraywould then b e crossed with the outer array, requiring even more observations. A fractional factorial design is a design that considers only some carefully selected
7
Table 1.1 Taguchi Crossed Arrays, 32 observations
z =1 z =1 z =1 z =2 z =2 z =1 z =2 z =2
1 2 1 2 1 2 1 2
x 11112222 1 1 11 22 2 2 11112222 11112222
1
x 11221122 1 1 22 11 2 2 11221122 11221122
2
x 12121212 1 2 12 12 1 2 12121212 12121212
3
1; 2;::: 9; 10;::: 17; 18;::: 25; 26;:::
fraction or p ortion of a factorial design. Typically two or three settings are considered
for eachofthevariables, and a fractional factorial design would reduce the number
of observations required bypowers of 1/2 or p owers of 1/3, by running, say, only
one half or one ninth of the p ossible design p oints required for the factorial design.
The design p oints actually chosen for the fractional factorial design usually, under
certain conditions, maintain the orthogonality and balance of the full design. Such
designs can often provide sucient information while saving time and money.For
example, imagine ve design variables, eachwithtwo levels. These could b e placed
1
5 5 2
fraction of a 2 exp eriment. The into a 2 fractional factorial design, whichisa
4
full factorial design calls for 2 2 2 2 2 = 32 observations, while this quarter
3
fraction calls for only eight. The outer array mightthenbea2 design, whichisan
exp erimenthaving three noise factors, eachattwolevels. The two arrays, inner
and outer, would then still b e crossed, requiring 88=64 runs. If constraints dictated
that only 32 runs were p ossible, the exp erimenter mightchose to run only one half
3 3 1
of the 2 outer array, 2 , giving a 50 savings in time and money.To illustrate
the crossing of the two arrays, see table 1.1, where the following layout is presented.
3 2
The inner arrayofthecontrol factors, x ;x;x,isa2 design, crossed with a 2
1 2 3
outer array of noise factors, z and z , and would require 32 = 84 observations. In
1 2
3 2
the Taguchi notation, this would b e describ ed as a L 2 crossed with a L 2 . This
8 4
mayormay not b e a layout recommended byTaguchi, it is included only to illustrate
the crossing of the inner and outer arrays.
8
The Taguchi metho d includes the noise factors in the exp eriment for the purp ose
of identifying control factors settings which are robust against noise, i.e. those settings
of the design factors which pro duce the smallest variation in the resp onse across the
di erentlevels of the noise factors. Some combinations of control factor settings may
yield output that is a ected by the noise factors, thus causing the resp onse to vary
around its mean, while for others combinations the output is insensitivetothechanges
in the noise factors. Phadke [111 ]. Similarly, the qualitycharacteristic might exhibit
more variability at, say, the low setting of a certain control factor than it do es at the
high setting. For other control factors, the variation in the qualitycharacteristic might
b e nearly constant across the levels of those control factors, while the average quality
characteristic mightormight not change across the levels of these control factors. For
any control factor, there are four p ossible situations { e ect or no e ect on the mean,
coupled with e ect or no e ect on the variation. If the exp erimenter is fortunate,
some factors may arise that impact the mean but not the variation. Factors with this
typ e of e ect have b een termed \p erformance measures indep endent of adjustment"
or PerMIA - L eon, Sho emaker, and Kackar [88]. Such `adjustment' factors maybe
used to move a pro cess onto or towards its target value, without a ecting the variation
around the target. In a two-step adjustment pro cess, the variation is minimized by
the appropriate settings of the factors that a ect the variation, and then the output
is centered at the target value by the appropriate settings of the factors that only
in uence the mean. Factors which app ear to have little or no impact on either the
mean or the variation are typically set to the level representing the lowest cost.
In order to achieve its goals, the Taguchi metho d analyses not the measured
resp onse but rather some transformation of that resp onse, dep ending on the situation.
As mentioned previously,theTaguchi metho d aims to minimize the exp ected loss,
2
E Y T , but it do es not base the analysis on this quantity directly. A signal to
noise ratio is calculated, the choice of the particular signal to noise ratio dep ends
on the desired outcome of the resp onse: target or nominal is b est, smaller is b etter,
larger is b etter. The last two are considered equivalentby taking recipro cals. For
9
all three situations, the analysis falls under Taguchi's idea of Signal to Noise ratio,
although only in the target is b est case are b oth signal and noise considered. Taguchi
is said to have coined the term `signal to noise' in order that it app eal to the electrical
engineers with whichhewas working during the 1950s - Barker [10] .
For the target is b est situation, deviations from target in either direction must b e
2
considered. In order to minimize the E Y T the exp erimentermust b e concerned
with b oth variance and bias. In this situation Taguchi suggests that the following
2 2
y =s , which is to b e maximized. This signal to noise ratio b e considered: 10 log
10
ratio is calculated for eachlevel of eachcontrol factor, averaging over the various
noise levels, and over the other control factors. This ratio is also calculated for each
`design p oint', averaging over the settings of the noise factors in the inner array.It
y=s,which should b e noted that maximizing this ratio is equivalent to maximizing
is equivalent to minimizing the sample co ecientofvariation, s= y . In calculating
this ratio, the Taguchi practitioner is simply p erforming a data transformation. For
P
1
2
the larger is b etter situation the signal to noise ratio is 10 log 1=y , for the
i
10
n
P
1
2
smaller is b etter situation the signal to noise ratio is 10 log y : In all cases,
10
i
n
the goal is to maximize the signal to noise ratio.
By analyzing the signal to noise ratio that is recommended for a particular sit-
uation, the Taguchi practitioner hop es to nd which levels of design variables were
insensitivetothevariation in the noise variables, i.e. which setting in the inner array
gave the smallest variation in resp onse across the outer array. By constructing the
same signal to noise ratio for each level of eachcontrol factor, across all other factors,
the researcher may also discover that some factors a ect the mean while others have
an e ect on the variation. Such factors are known as lo cation e ects and disp ersion
e ects, resp ectively Box and Meyer [19].
As stated previously,theTaguchi metho d calls for this signal to noise ratio to b e
calculated for the data coming from each exp erimental setup of control factors. Thus
this ratio is calculated for eachrow of the inner array. Based on the calculated signal
to noise ratios, with higher b eing b etter, the `b est' levels of each of the control factors
10
are assigned. For `lo cation e ects', the signal to noise ratio allows the researcher to see
the magnitude of the e ect on the qualitycharacteristic, measured across the levels of
the noise factors. If the e ect is small the exp erimentermaycho ose the control factor
setting that represents the lowest cost. Mean plots may also b e calculated for each
of the levels of the control factors; the extenttowhich these agree with the signal
to noise ratio plots will dep end on the nature of the dep endence b etween and .
This graphical technique often accompanies the more classical technique of Analysis
of Variance ANOVA . Since the Taguchi design crosses the twoarrays, it is p ossible
to conduct an ANOVA on b oth the mean and the signal to noise ratio. Under the
Taguchi system it is quite typical to verify the `optimal' results with exp erimental
follow-up.
Another approach whichmay b e imp ossible in some situations is to rst mini-
mize the variation bypicking the b est levels of factors that a ect only the variation,
and the bring the pro cess `on target' by selecting the b est levels of other factors
that in uence only the mean. The hop e here is that the appropriate signal to noise
ratio can b e minimized in a two stage fashion. If the pro cess target changes, the
exp erimenter then only has to worry ab out one set of factors, since the other factors
have b een selected to minimize variation indep endently of the mean. In many ap-
plications of Taguchi metho ds, this partitioning of variables into disjoint categories
based on whether they a ect the mean or the variance is an assumption made b efore
the data are collected. For a multiplicative error mo del, whichsays that the error
increases in prop ortion to the mean, the maximization of the signal to noise ratio
2
and the minimization of E Y T are equivalent. The Taguchi idea is to use a
two-stage minimization pro cedure, rst maximizing the signal to noise ratio over the
factors that e ect the variation, then adjusting the pro cess to target by selecting the
appropriate levels of those factors that e ect the mean but not the variation. Under
2 2
the assumption of a multiplicative error, wehave E Y = ; V ar Y = ,thus
2 2 2 2
the mse is T + , and is minimized over at = T= 1 + . The signal
2 2
to noise ratio is 10 log E Y =V ar Y , which can b e written 10 log 1= ,and
10 10
11
is to b e maximized. Thus the variance could rst b e minimized by maximizing the
signal to noise ratio , and would haveaminimum value, say . The other factors
min
2
could b e used to set the resp onse at = T= 1 + . Therefore, maximizing the
min
min
signal to noise ratio will b e equivalent to minimizing the mse in a two step fashion.
More information on the details of the two-stage minimization, the assumptions re-
quired and a comparison of signal to noise ratio against mse for various mo dels are
given in L eon, Sho emaker, and Kackar [88 ], as well as Box [21 ]. Note that in an
2 2 2
additivemodel,E Y = + ,andthus the signal to noise ratio for the smaller is
P
1
2
y , clearly mixes together the e ect on the mean and b etter situation, 10 log
10
i
n
the e ect on the variance.
A third approach Vining and Myers [139] is to set the mean at the appropri-
ate level while at the same time minimizing the variance within the framework of
a Resp onse Surface mo del. Details concerning resp onse surface metho ds rsm and
mo dels will b e given in the following chapters. Within this context, the exp erimenter
is concerned with b oth lo cation and spread, and the mathematical mo deling that
takes places amounts to constrained optimization within the framework of a general
linear mo del. For example, in the target is b est situation this metho d calls for the
2
exp erimenter to minimize sub ject to the constraint = T: For the other two cases
more extreme is b etter the exp erimenter places the constraints on and optimizes
sub ject to the various restrictions. Lagrange multipliers are then used to determine
the optimal levels of the input factors. Quite often, however, an improvementinmean
resp onse will b e accompanied by increased variation, and thus the nal choice of de-
sign settings will often represent a tradeo b etween minimum variance and minimum
bias. Contour graphs for b oth the mean and the standard deviation are very useful
in determining what settings represent the b est choice. The approachgiven by these
authors amounts to mo deling b oth the mean and the variance but do es not suggest
how noise variables could b e incorp orated into the analysis.
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1.2 Potential Flaws
Within the statistical community the acceptance and implementation of so called
`Taguchi-Metho ds' has not matched the optimism of industry.Many statisticians see
a lot of go o d things within the Taguchi framework but p oint out many shortcomings
as well. For a discussion involving b oth scho ols of thought, see Barker et al [9], as
well as Pignatiello and Ramb erg [112 ].
Although fractional factorial designs can provide a great deal of information with
small to mo derate sample sizes, the crossing of the twoarrays in the Taguchi design
often requires that a large numb er of observations b e taken, with more observations
almost always meaning more time and greater cost. This may b e particularly bad if
the exp eriment is of the `screening' typ e, where a large numb er of factors, b oth noise
and control, are b eing considered. The large numb er of runs called for by the crossed
inner and outer arrays is one of the frequent criticisms of the Taguchi metho d.
Furthermore, there are those in the statistical community who view this crossing
of the arrays as inecient in terms of information loss of information . Because the
two arrays are crossed, the overall resolution of the design may not b e as high as it
could b e with a di erent design. In many cases, this crossing results in the inabilityto
handle or estimate signi cantinteractions. Some suggest combining the twotyp es of
factors into a single combined array, i.e. consider a single fractional factorial design
with sucient resolution to estimate the main e ects as well as the interactions of
interest, including control-control interactions as well as control-noise interactions. In
considering a single array, it will often b e p ossible to nd a higher resolution design
that requires the same numb er of observations as the Taguchi crossed arrays, or to
achieve the same design resolution with less exp erimental runs. If certain interac-
tions are susp ected to exist, this can b e considered at the design stage and a design
with an appropriate aliasing structure that allows for the estimation of the desired
interactions can b e found. The fact that Taguchi designs do not handle interactions
easily is another ma jor shortcoming. Taguchi designs do allowforinteractions if it
13
is known whichinteractions exist before the data are collected; for unknown inter-
actions b etween control factors the Taguchi metho dology falls short. The Taguchi
metho d do es advo cate that follow-up exp erimentation take place, whichseemstobe
a go o d pro cedure but it may somewhat awed not only due to the inadequacy of the
signal to noise ratio but also to due to the p o or prop erties of the original design. See
Lorenzen in Nair, [103], pages 138-139 where the follow-up design had the same
numb er of runs but higher resolution. In this case the `b est p oint' was con rmed
but an improvementof30was also present . Most factorial or fractional factorial
designs can b e augmented by adding center p oints and/or axial p oints. Such designs
are commonly used within the context of resp onse surface mo deling and exp erimenta-
tion. These and similar designs of this form are the Central Comp osite Design CCD ,
the Box-Behnken three level designs, as well as Plackett-Burnam designs. Boxand
Drap er [20], and Myers, Khuri, and Carter [97] . Incorp orating these ideas within
the context of resp onse surface metho dology allows for sequential investigation, a fea-
ture that, as is often noted, is not generally available with the Taguchi metho dology,
except in the sense of follow-up con rmation.
2
It must b e rememb ered that the goal is to nd settings that minimize k Y T
while the tactic is to pick settings that minimize the signal to noise ratio. That
the two are equivalent relies directly on the multiplicative error assumption and the
further assumption that the parameters can b e partitioned into distinct and disjoint
subsets, one containing `adjustment' factors which are used to adjust to the optimal
value, one containing `disp ersion' factors whichhave an e ect on the the variation
but not the mean, and one containing factors that in uence neither the mean nor the
variation. However,inanadditive error mo del, where the variance is unrelated to
the mean, the minimizations of the signal to noise ratio and the mse lead to di erent
`b est' p oints. That the set of parameters can b e partitioned as required is based on
the a priori assumption that the signal to noise ratio is a function that dep ends only
on some subset of the parameters. The question of when and whether the two are
equivalentiscovered in L eon, Sho emaker, and Kackar [88]. Furthermore, many
14
di erent data p oints representing drastically di erent means and variances can give
rise to the same signal to noise ratio see Box [21] .
Within the context of a statistical analysis, data transformations are usually p er-
formed in order to stabilize the variance of the resp onse. Therefore, in any particular
situation, the actual transformation should b e stem from a physical or data based un-
derstanding of the relation b etween the mean and the variance, see Box and Jones
[22 ]. The recipro cal transformation or recipro cal squared may b e highly non-linear
and is taken indep endent of the actual data values. The e ectiveness of any trans-
formation clearly is related to the actual data; the e ect of the recipro cal squared
transformation for y between .3 and 2 is very di erent than the same transformation
for y between 20 and 25. The e ect of a transformation is well known to b e related
to the magnitude of the sample quantity, y =y . Another criticism of the signal
max min
to noise ratio and the issue of data transformation is that b oth the presence and
magnitude of e ects are related to the metric chosen by the exp erimenter. For this
reason many p eople feel that if a transformation is to b e done, it should b e one sug-
gested by the data, not one chosen b eforehand. Lamb da plots Box [21 ] are helpful
in determining an appropriate transformation, the Taguchi signal to noise ratio is
just one transformation. It should b e noted that this transformation is known to b e
helpful when is prop ortional to Bartlett [12].
To summarize, the ma jor complaints against the Taguchi Metho d are large and
exp ensive designs, the p o or abilitytoallowforinteractions, the lack of sequential
investigation, the a priori assumption of a multiplicative error mo del, and the inade-
quacy of the signal to noise ratio. New ideas that fall mostly within the framework of
RSM but could apply some of the Taguchi metho ds include multiple resp onse analy-
sis, sequential design/analysis, the realization that system variabilitymay itself vary
within the design region, and non-parametric design Myers, Khuri, and Carter [97].
15
2. GENERAL ALTERNATIVES TO TAGUCHI
Many of those who most strongly voice criticismsoftheTaguchi metho d favor
what is known as Resp onse Surface Metho dology, or RSM. Resp onse surface metho ds
and metho dologies b egan in 1951, with the work of Box and Wilson Myers, Khuri,
and Carter [97 ]. In this review pap er, the authors prop ose one de nition for RSM as
\acol lection of tools in design or data analysis that enhance the exploration of a region
of design variables in one or moreresponses ". The interest in exploring the resp onse
throughout a region is very much di erent than the Taguchi metho d of `picking the
b est' from what was observed. Applications of RSM in biological sciences, fo o d
science, so cial sciences, physical sciences, engineering and other general industrial
situations are also o ered in this pap er. The underlying premise of RSM is that
some resp onse, typically denoted , is a function of some set of design variables, say
x ;x ; :::; x . Although the exact functional relationship b etween and x is generally
1 2 k
~
unknown, the assumption is made that this exact relation can b e approximated, at
0
least lo cally,by a p olynomial function of the x s. Actual data collected, typically
denoted y , is expressed as y = + ", Box and Drap er [20 ]. In most cases the
approximating function will b e a rst or second degree p olynomial. The two most
common mo dels are thus:
= + x + ::: + x 2.1
0 1 1 k k
and
k k
X X X
2
= + x + x + x x 2.2
0 i i ii i ij i j
i=1 i=1 i 0 0 0 B x, + x and = + x In vector notation these can b e written as = + x 0 0 ~ ~ ~ ~ ~ ~ ~ resp ectively. Here, x = x ;x ; :::; x isak -vector of input variables, while , , 1 2 k 0 ~ ~ 16 and B represent a scalar parameter, a k -dimensional parameter vector, and a k by k ~ symmetric matrix of parameters, resp ectively. Some assumptions are required on the 2 error asso ciated with the resp onse, assuming that this error followaN 0; distri- 2 bution for some allows the researcher to employmany of the to ols that accompany classical least squares theory. Inherent to the RSM framework is the idea of sequential investigation. This concept is not a part of the Taguchi metho dology other than in follow-up or con rmation of the `b est' design p oint. Sequential investigation lo oks for a p ossible improvement in the qualitycharacteristic. This improvementmay require settings of factors that were not part of the original exp eriment. By tting a mo del, it b ecomes p ossible to estimate the resp onse and p ossibly the variation throughout the entire design region, suggesting areas where the resp onse may b e closer to target, or other areas where the variation is much less, or is more stable. In contrast to sequential investigation, the Taguchi metho d lo oks at optimization, generally consid- ering only design p oints at which data were actually collected. Remember that the Taguchi inner array is often highly fractionated, which means that many `p otential' design p oints were not included in the inner array. In using resp onse surface metho ds the researcher conducts a designed exp eriment, estimates the necessary parameters to t a mo del, checks the adequacy of the mo del lack of t test , determines or guesses where improvement will or mighttake place, and then conducts another exp eriment with design p oints concentrated around the area of susp ected improvement. This pro- cess could b e rep eated as often as needed but careful planning and exp erimentation will keep the numb er of exp eriments and observations to a minimum. This parallels the idea of trying to understand the pro cess, a ma jor tenet of those in quality circles. In most resp onse surface situations, all of the `x'variables in the mo del can gener- ally b e placed at arbitrary levels and are not considered random. To compare directly with the Taguchi metho dology, the resp onse surface mo del needs to b e expanded to consider noise variables as well as p otential interactions b etween noise variables and control variables. In many cases it will b e relatively easy to observe and record the noise variables. In such situations it may also b e p ossible to place a non-sub jective 17 prior distribution on the noise variables. Based on an initial exp eriment, the qual- itycharacteristic could b e mo deled as a resp onse of both the noise variables and the control variables, typically a p olynomial of rst or second degree. Sp eci cally,ifwe let Y denote some qualitycharacteristic of interest, we can t a general mo del of the form: 0 0 0 0 0 Y = Y = + x + x B x + z + z C z + z D x + " 2.3 xz 0 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Box and Jones [22 ]. In this mo del, x represents a k -vector of design variables, z ~ ~ represents a p-dimensional random vector of noise or environmental variables, and " 2 represents error, usually one assumes " N 0; . The parameters , , B , , C , 0 ~ ~ ~ ~ 2 D , and are generally unknown. ~ Within the context of sequential investigation, the researcher nowwould liketo know which design p oints should b e considered for the next exp eriment, making use of the current tted mo del, realizing that the tted mo del is based on parameter estimates, not parameters, and after integrating over the p-dimensional prior on z , ~ the vector of noise variables. Many criterion are available for deciding what makes a new design p oint or a new design matrix optimal. These include D-optimality, A-optimality, minimum bias, minimum variance of prediction, etc. A very common goal is to minimize exp ected mean square relative to a desired or sp eci ed target value. When the concern is for what will happ en in the future, the mean square er- ror of prediction b ecomes a ma jor concern Allen [4], Dwivedi and Srivastava [41]. Noise factors may b e regarded as random regressors, thus changing the interpretation of the regression mo del slightly, see Narula [101 ], Reilman, Gunst, and Lakshmi- narayanan [115 ], Sha er [122], Walls and Weeks [140 ], and Kackar and Harville [77 ] for various viewp oints concerning random regressors. In some instances, the re- sp onse surface may dep end on b oth quantitative and qualitative factors, see Drap er and John [40]. Various approaches to the dual-resp onse problem are considered in Castillo and Montgomery [29 ], Vining and Myers [139 ] and Gan [51]. 18 3. RESPONSE SURFACE ALTERNATIVES Avery common statistical measure of p erformance is squared error loss, or ex- p ected squared error loss. This concept has b een used by statisticians for manyyears but the increased use of Taguchi metho ds has made many p eople, particularly en- gineers and others in industrial situations, realize that any deviation from target is detrimental, and that considering only whether or not a part is within sp eci cation limits can lead to a false sense of security ab out a pro cess. Insp ection plans concerned only ab out sp eci cation limits without considering variation around a target value can also lead to distributions that are almost uniformly spread across the sp eci ca- tion limits, as opp osed to b eing concentrated at the target value. The previously mentioned example of Sony-USA compared to Sony-Japan is a classic example of this phenomenon. Sullivan, [124 ] . Within the framework of a Taguchi design the quan- 2 titytominimize would b e E Y T , where Y is the characteristic of interest and T represents the target value. Inner and outer arrays would b e crossed and signal to noise ratios would b e calculated to determine which settings of the design variables would minimize squared error loss. Recall that although the goal is to minimize the mse,theTaguchi criterion is actually some signal to noise ratio. Here a di erent formulation will b e considered in an e ort to improve on standard industry practice. Let Y represent the qualitycharacteristic of interest and recall the general resp onse surface mo del 0 0 0 0 0 D x + " 3.1 C z + z + z B x + z + x Y = Y = + x xz 0 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Box and Jones [22]. In this mo del, x represents a k -dimensional vector of design ~ variables, z represents a p-dimensional random vector of noise variables, and " repre- ~ 2 sents error, usually it is assumed that " N 0; . The parameters , , B , , C , 0 ~ ~ ~ ~ 19 2 2 D , and are generally unknown. This mo del allows for pure quadratic e ects x or i ~ 2 z ,aswell as second order terms suchas x x ;xz ; and z z . i j i j i j j If we consider the noise variables as b eing part of an environment space, wecould rst integrate over this environment space and then attempt to minimize the mean squared error. Sp eci cally, the squared error loss would b e: Z Z Z 2 Y T f z f y jz dy jz dz 3.2 z Y jZ Y jZ Z Z ~ ~ ~ ~ p 1 ~ ~ ~ 2 ;A and, indep endentof z ;" N 0; , while all Here it assumed that z N e p ~ ~ ~ ~ ,is other terms are constants, p ossibly unknown. It is assumed that the mean, e ~ known and that the covariance matrix, A,isknown and is p ositive de nite. ~ In the most general case, namely 0 0 0 0 0 Y = + x + x B x + z + z C z + z D x + " xz 0 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0 0 0 B x +z + z C z + z D x + " ~ ~ ~ ~ ~ ~ ~ ~ ~ wehave 0 0 0 0 C + D x + + tr AC 3.3 E Y = + x + x B x + e 0 e e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0 0 0 0 2 C z AD x + Var z D A + x Var Y = + ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0 0 0 0 0 0 C z ] 3.4 ;z C z +cov z D x;z D x +cov z ;z + 2[cov z ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ where 0 0 C AC C z =2tr C AC A +4 Var z e e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0 0 0 0 0 0 0 0 AC D C z =2x D x;z A ; cov z D D x =x ;z cov z e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0 0 0 cov z ;z C z =2 AC e ~ ~ ~ ~ ~ ~ ~ ~ ~ thus 2 0 0 0 0 0 + + Var z onl y 3.5 Var Y =x D AD x +2x D A +4x D AC e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 2 See Seb er, p4112, p40 4, [121 ] where is the variance in the resp onse and Var z onl y represents variation due to the noise variables but indep endentofthe ~ 20 value of x. These pieces may b e ignored when determining the values of x that ~ ~ minimize the mse: Therefore, ignoring some terms indep endentofx ~ 2 0 0 0 0 0 E Y T = x D AD x +2x D A +4x D AC e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 2 0 0 0 0 0 0 + + x + x B x + x D C + + + tr AC T 0 e e e e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 2 the rst term is Var Y , the second is the squared bias, E Y T . Considered as function of x,thismay b e written as ~ 0 0 0 0 0 0 0 AD + + D + D +2B + + C + tr AC T ]x x [D e e 0 e e e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0 0 0 0 0 + tr AC T ] C + + + + D +2x [D A +2C e e 0 e e e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0 2 0 0 0 0 + x B x +2x B x x + D e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ This is fourth degree function of x, but all of the third and fourth order terms will ~ vanish if the mo del contains no second order terms in x, i.e. when the B x term is ~ ~ 0 : strictly linear in x,when B x = + x 0 ~ ~ ~ ~ 3.1 Case 1 In the simplest case the mo del is given by: 0 0 0 + " B x + z + x y = + x xz 0 ~ ~ ~ ~ ~ ~ ~ This mo del is quadratic in x and linear or rst order in z ,withnointeraction b etween ~ ~ noise variables and design variables. For this mo del, it can b e shown that, after integrating over z , Y follows a normal distribution, with mean and variance ~ 2 0 0 0 0 ; + A + x + x B x + 0 e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ In this situation the mean squared error reduces to the following quadratic function of B x : ~ 2 2 0 0 2 2 0 T T + + A + E Y T = B x +2B x e e ~ ~ ~ ~ ~ ~ ~ ~ ~ 0 2 2 0 = B x + T + + A e ~ ~ ~ ~ ~ ~ 21 0 This quadratic is minimized at B x =T ; which is the same as E Y =T . e ~ ~ ~ Note that the previous derivations explicitly assume that all necessary parameters are known. In practice this will rarely if ever o ccur. The implications of using parameter 0 estimates are discussed in the next chapter. The squared error loss at B x =T e ~ ~ 2 0 is + A . Notice that this is simply the overall variance of Y; which is equivalent ~ ~ ~ to the mean squared error when the design p oints are such that E Y =T , i.e. when an unbiasedness condition is satis ed. For this mo del, the variance of Y do es not involve x,andthus the minimum mse o ccurs when new design p oints satisfy E Y ~ = T . Sp eci cally, this means that new design p oints should satisfy either 0 0 a + x = T 0 e ~ ~ ~ ~ 0 0 0 or b + x + x B x = T 0 e ~ ~ ~ ~ ~ ~ ~ Knowing the marginal distribution of y in this situation allows for con dence intervals to b e pro duced; prediction intervals are more complicated since any future value of Y will also involve a random value z . ~ 3.2 Case 2 In this case the mo del is given by: 0 0 0 0 C z + "; + z B x + z + x y = + x xz 0 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ This allows for pure quadratic terms and interactions within b oth the design region and the noise region but do es not allow for interactions across the two regions. For this mo del the mse is 0 2 0 + tr AC T ] C + B x +2B x [ e e e ~ ~ ~ ~ ~ ~ ~ ~ ~ 2 0 0 2 0 2 0 0 2 + + A + T + E [T z C z ] +2E [z z C z ] T z z e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ As in case 1, Var Y doesnotinvolve x, so the mse is again minimizedby requiring ~ new design p oints to b e unbiased. The mse is again a quadratic function of B x , ~ and is minimized at 0 0 tr AC ; C B x =T e e e ~ ~ ~ ~ ~ ~ ~ ~ 22 which means wemust lo ok for design p oints satisfying 0 0 0 tr AC = T a + x C e 0 e e ~ ~ ~ ~ ~ ~ ~ ~ ~ 0 0 0 0 or b + x + x B x = T C tr AC 0 e e e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ For this mo del, it is much more complicated to integrate over the densityof z ,since ~ the conditional densityofY nowcontains terms involving up to the fourth p ower of the z . Itiseasytoshow, however, that, in this case, using 3.4 and 3.5 , i 0 0 0 0 E Y = + x + x + tr AC ; C + B x + 0 e e e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 2 0 0 0 Var Y = + A +2tr C AC A +4 C AC +4 AC e e e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 3.3 Case 3 As the mo dels presented haveallowed for random e ects to in uence the resp onse, it seems reasonable, or even probable, that this in uence may dep end on the levels of certain control variables. The Taguchi metho d implicitly assumes that the variance of the qualitycharacteristic is not equal at all control settings given in the inner array. This essentially implies that the design space and the environment space interact. The simplest mo del to handle this situation is given by: 0 0 0 0 D x + "; + z B x + z + x y = + x xz 0 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0 0 = B x +z + z D x + " ~ ~ ~ ~ ~ ~ 0 + D x +" = B x +z ~ ~ ~ ~ ~ This represents a mo del that is linear in z , quadratic in x and allows for interaction ~ ~ between design and noise variables. As far as the random variable z is concerned, ~ this is equivalent to mo del one, with the substitution b ecoming + D x. Since this ~ ~ ~ ~ substitution involves x, the choice of optimal new design p oints will not, however, ~ b e the same as in mo del one. Furthermore, Var Y nowdoesdependonx and thus ~ b eing unbiased maynotgive the minimum mse. This mo del has mse 0 2 0 + D x T ]+ + D x A + D x B x +2B x [ e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 2 2 0 + D x T + + e ~ ~ ~ ~ 23 Since the di erence b etween this mo del and mo del one involves only x,we can again ~ calculate the densityofY,afterintegrating over z , as b eing normal, with ~ 0 0 2 0 0 E Y = + x + x B x + + D x ; Var Y = + + D x A + D x 0 e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Notice that the mse is no longer a quadratic function of B x asitnowalsoinvolves ~ quadratic terms in b oth B x andD x,aswell as the higher order terms stemming ~ ~ ~ from the pro duct of B x and D x.For this mo del, the variance of Y is now a quadratic ~ ~ ~ 0 1 0 function of x and is minimized at x = D AD D A . Satisfying the minimum ~ ~ ~ ~ ~ ~ ~ ~ variance condition may place the mean resp onse well away from the target value or require values of x out of the range of the design space. ~ 3.3.1 Case 3, Second Order Mo del If the mo del do es allow for or require second order terms in x; i.e. B x = ~ ~ 0 0 + x + x B x, then the mean square will contain third and fourth order terms due 0 ~ ~ ~ ~ ~ 2 the B x D x term and the B x term. Sp eci cally,wehave ~ ~ ~ ~ 0 0 0 0 0 0 0 2 T ]x D +2B + D + D +2 AD + [D E Y T = x 0 e e e e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0 0 0 0 T ] + A + + D [D + 2x e 0 e ~ ~ ~ ~ ~ ~ ~ ~ ~ 0 2 0 0 0 0 + x B x +2x B x x + x D e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Obviously the higher order terms involving x make it more dicult to minimize the ~ mse and stay within the design or feasibilityregion. 3.3.2 Case 3, First Order Mo del 0 If the term B x isgiven by B x = + x , then the mse is again a quadratic 0 ~ ~ ~ ~ function of x = x ;x ;;:::;x , although not the same function as in case 1 or case 1 2 k ~ 2. Under the assumption that B =0, the mean squared error is ~ ~ 0 0 0 0 0 0 D ]x D + D +2 AD + [D x e e e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0 0 0 0 T ] + +2x [D A + + D e 0 e ~ ~ ~ ~ ~ ~ ~ ~ ~ 2 2 0 T + + + 0 e ~ 24 Notice the ab ove function is convex in x,thus a unique minimum o ccurs but sucha ~ pointmay not b e in the design region. 3.4 Case 4 Dep ending on the situation, it may b e necessary to include second order terms in the noise variables in order to more accurately mo del the resp onse. Generally a mo del that includes these second order terms in the noise variables would also contain terms up to second degree in the control variables, as well as p ossible interactions b etween the noise and control variables. A mo del that allows for all of this is given by: 0 0 0 0 0 y = + x + x B x + z + z C z + z D x + " xz 0 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ The mse is nowgiven by: 2 0 2 C + tr AC T ]+ [ + D x T ] B x +2B x [ + D x + e e e e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0 0 0 2 0 2 2 +[ + D x A + D x ] + 2E [z + D x z C z ] + + E [T z C z ] T ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ This can b e simpli ed to again, ignoring terms indep endentof x ~ 0 0 0 0 2 C + tr AC T + B x B x +2x x e e e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0 0 0 0 D + x B x x +2x e ~ ~ ~ ~ ~ ~ ~ ~ 0 0 0 0 0 0 0 0 D x AD x +2x D x + x D x + x D + x e e e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0 0 0 0 0 0 0 + tr AC T C + + tr AC T +2x D C + +2x e e e e e e e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0 0 0 0 +2x D A +4x D AC e ~ ~ ~ ~ ~ ~ ~ ~ ~ All of the ab ovemay b e factored and group ed as 0 0 0 0 0 0 0 0 x [D AD + D + D +2 + tr AC T ]x C + D +2B + e 0 e e e e e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0 0 0 0 0 +2x [D A +2C + + D + + C + tr AC T ] e e 0 e e e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0 0 0 0 0 2 D + x B x x B x +2x + x e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 25 3.4.1 Case 4, Second Order Mo del When the resp onse is quadratic in x,themse is fourth degree in x.Itmayprove ~ ~ dicult to minimizethe mse, due to the higher order terms in x. A slightly di erent ~ approachwould b e to require that our new design p oints are unbiased, and then minimize the variance of Y that dep ends on x,namelywewant ~ 0 0 0 0 0 minfxg x D AD x +2x D A +4x D AC e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ sub ject to: 0 0 0 0 0 0 tr AC = T C + x + x B x + x D e e 0 e e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ We also have the adder restriction that the the p ointmust b e in the design region. In general, minimizing the variance while requiring unbiasedness means wemust minimize a second order function of x sub ject to a second order condition on x. ~ ~ Dep ending on the circumstance, it may happ en that the bias relative to target is more imp ortantthanthevariance. One metho d to handle the relative imp ortance between the variance and the squared bias is to minimize the function 2 g x = Var Y + 1 bias ; 0 <<1 ~ for various values of ; with = .5 b eing equivalent to minimizing mse.For more details on this and howtocho ose , see Box and Jones [22]. 3.4.2 Case 4, First Order Mo del 0 If the mo del is linear in x, meaning that the B x term is + x , then the two 0 ~ ~ ~ ~ conditions are for the constrained optimization problem of minimizing the variance sub ject to the unbiasedness condition ignoring the constraint of b eing within the design space are equivalentto 0 0 0 c such that x d = k minfxg x H x + x ~ ~ ~ ~ ~ ~ ~ ~ 0 0 0 Here, H = D AD and c =2D A +4D .Thus the minimum variance unbiased AC e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ condition takes the form of a somewhat standard quadratic programming problem, 26 and can b e solved fairly easily see [118 ] if the matrix H , is p ositive semide nite, ~ as it generally will b e; however the programming problem is solved under the added condition that x > 0 ,which presents a second condition whichmay b e imp ossible to ~ ~ meet. A further simpli cation under the assumption that B =0 is that the entire ~ ~ mse simpli es to ignoring terms indep endentof x ~ 0 0 0 0 0 0 0 + tr AC T ]x C + +2 + + D x [D AD + + D e e 0 e e e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0 0 0 0 0 + 2x [D + + D A +2C + tr AC T ] C + + e e e 0 e e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Notice that this is now only second order in x and may b e minimized muchmore ~ easily, as compared to the previous mo del , namely 0 1 0 0 0 0 0 + tr AC T ] C + +2B + + D AD + + D = [D x e e 0 e e e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0 0 0 0 + tr AC T ] C + + + + D A +2C [D e e 0 e e e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Here, x is the p oint that minimizes the mse: As the mo del gets more complicated, ~ the corresp onding mse also b ecomes more complicated, and will involve higher p owers of x, higher moments of z and the corresp onding cross-pro ducts. It can b e shown, ~ ~ however, that, although the marginal density is not normal, one still has, using 3.4 and 3.5 , 0 0 + tr AC C + D x + E Y =B x + e e e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 2 0 0 0 Var Y = + + D x A + D x +2tr C AC A +4 C AC +4 + D x AC e e e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 27 4. NEW DESIGN POINTS - EFFECT OF PARAMETER ESTIMATION For eachcasegiven in chapter three, the solution determined by E Y =T is 0 actually a surface, either of rst or second degree in x = x ;x ;:::x ; minimizing 1 2 k ~ the mse is equivalent to nding the level curves of a function that maybeupfourth order in x.However, the development in the previous chapter is based on the assump- ~ tion that all of the necessary parameters are known, an assumption that will rarely, if ever, b e met in practice. Since the goal is to select new design p oints, the mse of prediction is a critical issue. The variance in prediction stems from three sources: the variance in the resp onse, the variance asso ciated with the noise variables, and the variance induced by making predictions based on parameter estimates. While we hop e to minimize the mse of some future observation, we can only do this condition- ally, conditional on the parameter estimates wehave at the current time. Thus while wewould like to minimize the true mse, dropping some terms indep endentof x ~ 0 0 0 0 0 0 2 AC D A +4x D AD x +2x D E [ Y T ] = x e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0 2 0 0 0 0 0 C + + + x + x B x + x D + + tr AC T ; 0 e e e e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ wewillhave to consider an estimate of the true mse,which, again ignoring some terms indep endentofx isgiven by: ~ 0 0 0 d 0 0 0 ^ ^ ^ ^ ^ 4.1 C A D A ^ +4x D D x +2x A D MSE = x e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0 0 0 2 0 0 0 ^ ^ ^ ^ ^ ^ + x + + x + x D C T 4.2 + tr A C ^ + B x + 0 e e e e ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ The rst term represents the variance asso ciated with the noise-control factor interac- tions, the second term is the squared bias relative to the target value. The estimates could b e based on classical least squares theory, but this is not required.