ASTM E1325-02(2008)
(Terminology)Standard Terminology Relating to Design of Experiments
Standard Terminology Relating to Design of Experiments
SIGNIFICANCE AND USE
This standard is a subsidiary to Terminology E 456.
It provides definitions, descriptions, discussion, and comparison of terms.
SCOPE
1.1 This standard includes those statistical items related to the area of design of experiments for which standard definitions appears desirable.
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Designation: E1325 − 02(Reapproved 2008) An American National Standard
Standard Terminology Relating to
Design of Experiments
This standard is issued under the fixed designation E1325; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision.Anumber in parentheses indicates the year of last reapproval.A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope k of different versions from the t versions of a single
principal factor arranged so that every pair of versions
1.1 This standard includes those statistical items related to
occurs together in the same number, λ, of blocks from the b
the area of design of experiments for which standard defini-
blocks.
tions appears desirable.
DISCUSSION—The design implies that every version of the principal
factor appears the same number of times r in the experiment and that
2. Referenced Documents
the following relations hold true: bk = tr and r (k−1)=λ(t − 1).
2.1 ASTM Standards:
For randomization, arrange the blocks and versions within each
E456Terminology Relating to Quality and Statistics blockindependentlyatrandom.Sinceeachletterintheaboveequations
represents an integer, it is clear that only a restricted set of combina-
3. Significance and Use tions(t, k, b, r,λ)ispossibleforconstructingbalancedincompleteblock
designs. For example, t =7, k =4, b =7, λ=2. Versions of the
3.1 This standard is a subsidiary to Terminology E456.
principal factor:
3.2 It provides definitions, descriptions, discussion, and
Block1 1236
comparison of terms. 2 2347
3 3451
4 4562
4. Terminology
5 5673
6 6714
aliases, n—in a fractional factorial design,twoormoreeffects
7 7125
which are estimated by the same contrast and which,
therefore, cannot be estimated separately.
n completely randomized design, n—a design in which the
DISCUSSION—(1) The determination of which effects in a 2 factorial
treatments are assigned at random to the full set of experi-
are aliasedcanbemadeoncethe defining contrast(inthecaseofahalf
mental units.
replicate) or defining contrasts (for a fraction smaller than ⁄2) are
stated. The defining contrast is that effect (or effects), usually thought
DISCUSSION—No block factors are involved in a completely random-
tobeofnoconsequence,aboutwhichallinformationmaybesacrificed
ized design.
fortheexperiment.Anidentity, I,isequatedtothe defining contrast(or
2 2 2
defining contrasts) and, using the conversion that A = B = C = I, the
completely randomized factorial design, n—a factorial ex-
multiplication of the letters on both sides of the equation shows the
periment (including all replications) run in a completely
aliases.Intheexampleunderfractionalfactorialdesign, I =ABCD.So
randomized design.
2 2 2
that: A = A BCD = BCD, and AB = A B CD=CD.
(2) With a large number of factors (and factorial treatment combi-
composite design, n—a design developed specifically for
1 1
nations) the size of the experiment can be reduced to ⁄4, ⁄8,orin
fitting second order response surfaces to study curvature,
k n-k
general to ⁄2 to form a 2 fractional factorial.
constructed by adding further selected treatments to those
(3) There exist generalizations of the above to factorials having
n
obtained from a 2 factorial (or its fraction).
more than 2 levels.
DISCUSSION—If the coded levels of each factor are−1 and+1 in the
n
balanced incomplete block design (BIB), n—an incomplete
2 factorial(seenotation2underdiscussionfor factorial experiment),
block design in which each block contains the same number the(2n +1)additionalcombinationsfora central composite designare
(0, 0, ., 0), (6a, 0, 0, ., 0) 0, 6a, 0, ., 0) ., (0, 0, ., 6 a). The
n
minimum total number of treatments to be tested is (2 +2n +1) for
1 n
ThisterminologyisunderthejurisdictionofASTMCommitteeE11onQuality
a2 factorial. Frequently more than one center point will be run. For n
and Statistics and is the direct responsibility of Subcommittee E11.10 on Sampling
=2, 3 and 4 the experiment requires, 9, 15, and 25 units respectively,
/ Statistics. The definitions in this standard were extracted from E456–89c.
although additional replicate runs of the center point are usual, as
Current edition approved April 1, 2008. Published May 2008. Originally
n
compared with 9, 27, and 81 in the 3 factorial. The reduction in
approved in 1990. Last previous edition approved in 2002 as E1325–02. DOI:
experiment size results in confounding, and thereby sacrificing, all
10.1520/E1325-02R08.
information about curvature interactions.The value of a can be chosen
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
to make the coefficients in the quadratic polynomials as orthogonal as
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
possibletooneanotherortominimizethebiasthatiscreatedifthetrue
Standards volume information, refer to the standard’s Document Summary page on
the ASTM website. form of response surface is not quadratic.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E1325 − 02 (2008)
first question it might be logical to ask is whether there is an overall
confounded factorial design, n—a factorial experiment in
linear trend.This could be done by comparing A and A , the extremes
1 3
which only a fraction of the treatment combinations are run
of A in the experiment. A second question might be whether there is
in each block and where the selection of the treatment
evidencethattheresponsepatternshowscurvatureratherthanasimple
combinations assigned to each block is arranged so that one
linear trend. Here the average of A and A could be compared to A .
1 3 2
or more prescribed effects is(are) confounded with the block
(If there is no curvature, A should fall on the line connecting A and
2 1
effect(s), while the other effects remain free from confound-
A or, in other words, be equal to the average.) The following example
ing. illustrates a regression type study of equally spaced continuous
variables. It is frequently more convenient to use integers rather than
NOTE 1—All factor level combinations are included in the experiment.
fractions for contrast coefficients. In such a case, the coefficients for
Contrast 2 would appear as (−1,+2,−1).
DISCUSSION—Example: Ina2 factorial with only room for 4
treatments per block, the ABC interaction
Response A A A
1 2 3
(ABC: − (1) + a+b−ab+c−ac−bc+abc ) can be sacrificed Contrast coefficients for question 1 −1 0 +1
Contrast 1 −A . + A
through confounding with blocks without loss of any other effect if the 1 3
1 1
Contrast coefficients for question 2 − ⁄2 +1 − ⁄2
blocks include the following.
1 1
Contrast 2 − ⁄2 A + A − ⁄2A
1 2 3
Block 1 Block 2
Treatment (1) a
Example 2: Another example dealing with discrete versions of a
Combination ab b
factor might lead to a different pair of questions. Suppose there are
(Code identification shown in discus- ac c
three sources of supply, one of which, A , uses a new manufacturing
sion under factorial experiment) bc abc
technique while the other two, A and A use the customary one. First,
2 3
does vendor A with the new technique seem to differ from A and A ?
1 2 3
The treatments to be assigned to each block can be determined
Second, do the two suppliers using the customary technique differ?
once the effect(s) to be confounded is(are) defined. Where only one
Contrast A and A . The pattern of contrast coefficients is similar to that
2 3
term is to be confounded with blocks, as in this example, those with
for the previous problem, though the interpretation of the results will
a positive sign are assigned to one block and those with a negative
differ.
sign to the other. There are generalized rules for more complex situ-
Response A A A
1 2 3
ations. A check on all of the other effects (A, B, AB, etc.) will show
Contrast coefficients for question 1 −2 +1 +1
the balance of the plus and minus signs in each block, thus eliminat-
Contrast 1 −2A +A +A
1 2 3
ing any confounding with blocks for them. Contrast coefficients for question 2 0 −1 +1
Contrast 2 . − A + A
2 3
confounding, n—combining indistinguishably the main effect
The coefficients for a contrast may be selected arbitrarily provided
of a factor or a differential effect between factors (interac-
the ∑a =0 condition is met. Questions of logical interest from an
i
tions) with the effect of other factor(s), block factor(s) or
experiment may be expressed as contrasts with carefully selected
interaction(s).
coefficients. See the examples given in this discussion. As indicated
in the examples, the response to each treatment combination will
NOTE 2—Confounding is a useful technique that permits the effective
have a set of coefficients associated with it. The number of linearly
use of specified blocks in some experiment designs.This is accomplished
independent contrasts in an experiment is equal to one less than the
by deliberately preselecting certain effects or differential effects as being
number of treatments. Sometimes the term contrast is used only to
oflittleinterest,andarrangingthedesignsothattheyareconfoundedwith
refer to the pattern of the coefficients, but the usual meaning of this
block effects or other preselected principal factor or differential effects,
term is the algebraic sum of the responses multiplied by the appro-
while keeping the other more important effects free from such complica-
priate coefficients.
tions.Sometimes,however,confoundingresultsfrominadvertentchanges
to a design during the running of an experiment or from incomplete
contrast analysis, n—a technique for estimating the param-
planningofthedesign,anditservestodiminish,oreventoinvalidate,the
effectiveness of an experiment. eters of a model and making hypothesis tests on preselected
linear combinations of the treatments (contrasts). See Table
contrast, n—alinearfunctionoftheobservationsforwhichthe
1 and Table 2.
sum of the coefficients is zero.
NOTE 4—Contrast analysis involves a systematic tabulation and analy-
NOTE 3—With observations Y , Y , ., Y , the linear function
1 2 n
sis format usable for both simple and complex designs. When any set of
a Y + a Y +.+ a Y is a contrast if, and only if ∑a =0, where the a
1 2 2 1 n i i
orthogonal contrasts is used, the procedure, as in the example, is
values are called the contrast coefficients.
straightforward. When terms are not orthogonal, the orthogonalization
DISCUSSION—Example 1: A factor is applied at three levels and the
process to adjust for the common element in nonorthogonal contrast is
resultsarerepresentedby A ,A , A .Ifthelevelsareequallyspaced,the also systematic and can be programmed.
1 2 3
TABLE 1 Contrast Coefficient
Source Treatments (1) ab ac bc ad bd cd abcd
Centre X +1 +1 +1 +1 +1 +1 +1 +1 See Note 1
A(+BCD): pH (8.0; 9.0) X −1 +1 +1 −1 +1 −1 −1 +1
3 3
B(+ ACD): SO (10 cm ;16cm ) X −1 +1 −1 +1 −1 +1 −1 +1
4 2
C(+ ABD): Temperature (120°C; 150°C) X −1 −1 +1 +1 −1 −1 +1 +1
D(+ABC): Factory (P; Q) X −1 −1 −1 −1 +1 +1 +1 +1
AB + CD X X =X +1 +1 −1 −1 −1 −1 +1 +1
1 2 12
AC+BD X X =X +1 −1 +1 −1 −1 +1 −1 +1 See Note 2
1 3 13
AD+BC X X =X +1 −1 −1 +1 +1 −1 −1 +1
1 4 14
NOTE 1—The center is not a constant (∑X ≠ 0) but is convenient in the contrast analysis calculations to treat it as one.
i
NOTE2—Oncethecontrastcoefficientsofthemaineffects(X ,X ,X and X )arefilledin,thecoefficientsforallinteractionandothersecondorhigher
1 2 3 4
order effects can be derived as products (X = X X) of the appropriate terms.
ij i i
E1325 − 02 (2008)
TABLE 2 Contrast Analysis
Regression coefficient
Contrast Divisor Student’s t ratio
B =( X Y)/ X
Source o o
j ij I ij
2 2
i i
X Y X ( X Y)/s X
o o o o
ij i ij ij i ij
i i i i
œ
2 2 2
X : Centre ^ X Y
0 0 o X (^ X Y)/s o X B 5so X Yd/o X
0 0 œ 0 0 0 0
2 2 2
X : A + BCD ^ X Y
X (^X Y)/s X B 5 X Y / X
1 1 o o so d o
1 1 œ 1 1 1 1
2 2 2
X : B + ACD ^ X Y
2 2 X (^X Y)/s X B 5s X Yd/ X
o 2 o o o
2 œ 2 2 2 2
2 2 2
X : C + ABD ^ X Y
X (^ X Y)/s X B 5 X Y / X
3 3 o o so d o
3 3 œ 3 3 3 3
2 2 2
X : D + ABC ^X Y
4 4 X (^X Y)/s X B 5s X Yd/ X
o 4 o o o
4 œ 4 4 4 4
2 2 2
X :AB+CD ^ X Y
X (^X Y)/s X B 5 X Y / X
12 12 o o so d o
12 12 œ 12 12 12 12
2 2 2
X :AC+BD ^ X Y
13 13 X (^X Y)/s X B 5s X Yd/ X
o 13 o o o
13 œ 13 13 13 13
2 2 2
X :AD+BC ^ X Y
14 14 o X (^X Y)/s o X B 5so X Yd/o X
14 14 œ 14 14 14 14
NOTE 1—The notation for contrast analysis usually uses Y to indicate the response variable and X the predictor variables.
NOTE 2—The measure of experimental error, s, can be obtained in various ways. If the experiment is replicated, s is the square root of the pooled
variancesofthepairsforeachtreatmentcombination.(Eachrowof Xvalueswouldbeexpandedtoaccountfortheadditionalobservationsinthecontrast
analysiscomputations).Ifsomeeffectswerefelttobepseudo-replicates(example,nointeractionswerelogical)multiplyingthecontrastbytheregression
coefficient of these terms forms a sum of squares (as in analysis of variance) and these would be summed and divided by the number of terms involved
togive s .Also,inmanyexperiments,pastexperiencemayalreadyprovideanestimateofthiserror.Assumedmodel:Y=B +B X +B X +B X +e
0 1 1i 2 3i 4 4i
). In a simple 2-level experiment such as this, the regression coefficient measures the half-effect of shifting a factor, say pH, between its low and high
level, or the effect of shifting from a center level to the high level. In general, substitution of the appropriate contrast coefficients for the X terms in the
model will permit any desired comparisons. The difference between quantitative and qualitative factors lies in the interpretation. Since a unit of X
represents a pH shift of 0.5, there is a meaningful translation into physical units. On the other hand, the units of the qualitative variable (factories) have
no significance other than for identification and in the substitution process to obtain estimates of the average response values.
DISCUSSION—Example: Half-replicate of a 2 factorial experiment
evolutionary operation (EVOP), n— a sequential form of
with factors A, B and C (X , X and X being quantitative, and factor D
1 2 3 experimentation conducted in production facilities during
(X ) qualitative. Defining contrast I= +ABCD=X X X X (see
4 1 2 3 4
regular production.
fractional factorial design and orthogonal design for derivation of
the contrast coeffıcients ). NOTE 6—The principal theses of EVOPare that knowledge to improve
the process should be obtained along with a product, and that designed
dependent variable, n—see response variable.
experiments using relatively small shifts in factor levels (within produc-
tion tolerances) can yield this
...
This document is not anASTM standard and is intended only to provide the user of anASTM standard an indication of what changes have been made to the previous version. Because
it may not be technically possible to adequately depict all changes accurately, ASTM recommends that users consult prior editions as appropriate. In all cases only the current version
of the standard as published by ASTM is to be considered the official document.
An American National Standard
Designation:E1325–91(Reapproved 1997) Designation: E 1325 – 02 (Reapproved 2008)
Standard Terminology Relating to
Design of Experiments
This standard is issued under the fixed designation E1325; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision.Anumber in parentheses indicates the year of last reapproval.A
superscript epsilon (e) indicates an editorial change since the last revision or reapproval.
1. Scope
1.1 This standard includes those statistical items related to the area of design of experiments for which standard definitions
appears desirable.
2. Referenced Documents
2.1 ASTM Standards:
E456 Terminology relatedRelating to Quality and Statistics
3. Significance and Use
3.1 This standard is a subsidiary to Terminology E456.
3.2 It provides definitions, descriptions, discussion, and comparison of terms.
4. Terminology
aliases, n—in a fractional factorial design, two or more effects which are estimated by the same contrast and which, therefore,
cannot be estimated separately.
n
DISCUSSION—(1)Thedeterminationofwhicheffectsina2 factorialarealiasedcanbemadeoncethedefiningcontrast(inthecaseofahalfreplicate)
or defining contrasts(forafractionsmallerthan ⁄2)arestated.The defining contrastisthateffect(oreffects),usuallythoughttobeofnoconsequence,
about which all information may be sacrificed for the experiment.An identity, I, is equated to the defining contrast (or defining contrasts) and, using
2 2 2
the conversion that A = B = C = I, the multiplication of the letters on both sides of the equation shows the aliases. In the example under fractional
2 2 2
factorial design, I =ABCD. So that: A = A BCD = BCD, and AB = A B CD=CD.
k
1 1 1
( 2) With a large number of factors (and factorial treatment combinations) the size of the experiment can be reduced to ⁄4 , ⁄8 , or in general to ⁄2
n-k
to form a 2 fractional factorial.
(3) There exist generalizations of the above to factorials having more than 2 levels.
balanced incomplete block design (BIB), n—an incomplete block design in which each block contains the same number k of
different versions from the t versions of a single principal factor arranged so that every pair of versions occurs together in the
same number, l, of blocks from the b blocks.
DISCUSSION—The design implies that every version of the principal factor appears the same number of times r in the experiment and that the
following relations hold true: bk = tr and r (k−1)= l(t − 1).
For randomization, arrange the blocks and versions within each block independently at random. Since each letter in the above equations represents
aninteger,itisclearthatonlyarestrictedsetofcombinations(t,k,b,r, l)ispossibleforconstructingbalancedincompleteblockdesigns.Forexample,
t =7, k =4, b =7, l=2. Versions of the principal factor:
Block 1 1236
2 2347
3 3451
4 4562
5 5673
6 6714
7 7125
This terminology is under the jurisdiction of ASTM Committee E11 on Quality and Statistics and is the direct responsibility of Subcommittee E11.70E11.10 on
Editorial/Terminology.Sampling. The definitions in this standard were extracted from E456–89c.
Current edition approved Feb. 22,1991. April 1, 2008. Published April 1991.May 2008. Originally published as E1325–90.approved in 1990. Last previous edition
E1325–90a.approved in 2002 as E1325–02.
ForreferencedASTMstandards,visittheASTMwebsite,www.astm.org,orcontactASTMCustomerServiceatservice@astm.org.For Annual Book ofASTM Standards
, Vol 14.02.volume information, refer to the standard’s Document Summary page on the ASTM website.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.
E 1325 – 02 (2008)
completely randomized design, n—a design in which the treatments are assigned at random to the full set of experimental units.
DISCUSSION—No block factors are involved in a completely randomized design.
completely randomized factorial design, n—a factorial experiment (including all replications) run in a completely randomized
design.
composite design, n—a design developed specifically for fitting second order response surfaces to study curvature, constructed
n
by adding further selected treatments to those obtained from a 2 factorial (or its fraction).
n
DISCUSSION—If the coded levels of each factor are−1 and+1 in the 2 factorial (see notation 2 under discussion for factorial experiment ), the
(2n +1) additional combinations for a central composite design are (0, 0, ., 0), (6a, 0, 0, ., 0) 0, 6a, 0, ., 0) ., (0, 0, ., 6 a). The minimum
n n
total number of treatments to be tested is (2 +2n +1)fora2 factorial. Frequently more than one center point will be run. For n =2, 3 and 4 the
experiment requires, 9, 15, and 25 units respectively, although additional replicate runs of the center point are usual, as compared with 9, 27, and 81
n
in the 3 factorial. The reduction in experiment size results in confounding, and thereby sacrificing, all information about curvature interactions. The
value of a can be chosen to make the coefficients in the quadratic polynomials as orthogonal as possible to one another or to minimize the bias that
is created if the true form of response surface is not quadratic.
confounded factorial design, n—a factorial experiment in which only a fraction of the treatment combinations are run in each
block and where the selection of the treatment combinations assigned to each block is arranged so that one or more prescribed
effects is(are) confounded with the block effect(s), while the other effects remain free from confounding.
NOTE 1—All factor level combinations are included in the experiment.
DISCUSSION—Example: Ina2 factorial with only room for 4 treatments per block, the ABC interaction
(ABC:−(1)+a+b−ab+c−ac−bc+abc)canbesacrificedthroughconfoundingwithblockswithoutlossofanyothereffectiftheblocksinclude
the following.
Block 1 Block 2
Treatment (1) a
Combination ab b
(Code identification shown in discus- ac c
sion under factorial experiment) bc abc
The treatments to be assigned to each block can be determined once the effect(s) to be confounded is(are) defined. Where only one term is to be
confounded with blocks, as in this example, those with a positive sign are assigned to one block and those with a negative sign to the other. There
aregeneralizedrulesformorecomplexsituations.Acheckonalloftheothereffects(A, B,AB,etc.)willshowthebalanceoftheplusandminussigns
in each block, thus eliminating any confounding with blocks for them.
confounding,n—combiningindistinguishablythemaineffectofafactororadifferentialeffectbetweenfactors(interactions)with
the effect of other factor(s), block factor(s) or interaction(s).
NOTE 2—Confounding is a useful technique that permits the effective use of specified blocks in some experiment designs. This is accomplished by
deliberately preselecting certain effects or differential effects as being of little interest, and arranging the design so that they are confounded with block
effects or other preselected principal factor or differential effects, while keeping the other more important effects free from such complications.
Sometimes, however, confounding results from inadvertent changes to a design during the running of an experiment or from incomplete planning of the
design, and it serves to diminish, or even to invalidate, the effectiveness of an experiment.
contrast, n—a linear function of the observations for which the sum of the coefficients is zero.
NOTE 3—WithobservationsY ,Y ,.,Y ,thelinearfunctiona Y + a Y +.+ a Y isacontrastif,andonlyif (a =0,wherethea valuesarecalled
1 2 n 1 1 2 2 1 n i i
the contrast coefficients.
DISCUSSION—Example 1: Afactor is applied at three levels and the results are represented by A ,A , A . If the levels are equally spaced, the first
1 2 3
question it might be logical to ask is whether there is an overall linear trend. This could be done by comparing A and A , the extremes of A in the
1 3
experiment. A second question might be whether there is evidence that the response pattern shows curvature rather than a simple linear trend. Here
the average of A and A could be compared to A . (If there is no curvature, A should fall on the line connecting A and A or, in other words, be
1 3 2 2 1 3
equaltotheaverage.)Thefollowingexampleillustratesaregressiontypestudyofequallyspacedcontinuousvariables.Itisfrequentlymoreconvenient
to use integers rather than fractions for contrast coefficients. In such a case, the coefficients for Contrast 2 would appear as (−1,+2,−1).
Response A A A
1 2 3
Contrast coefficients for question 1 −1 0 +1
Contrast 1 −A . +A
1 3
1 1
Contrast coefficients for question 2 − ⁄2 +1 − ⁄2
1 1
Contrast 2 − ⁄2 A +A − ⁄2 A
1 2 3
Example 2: Another example dealing with discrete versions of a factor might lead to a different pair of questions. Suppose there are three sources of supply, one of
which, A , uses a new manufacturing technique while the other two, A and A use the customary one. First, does vendor A with the new technique seem to differ
1 2 3 1
from A and A ? Second, do the two suppliers using the customary technique differ? Contrast A and A . The pattern of contrast coefficients is similar to that for the
2 3 2 3
previous problem, though the interpretation of the results will differ.
Response A A A
1 2 3
Contrast coefficients for question 1 −2 +1 +1
Contrast 1 −2A +A +A
1 2 3
Contrast coefficients for question 2 0 −1 +1
Contrast 2 . −A +A
2 3
The coefficients for a contrast may be selected arbitrarily provided the (a =0 condition is met. Questions of logical interest from an experiment
i
maybeexpressedascontrastswithcarefullyselectedcoefficients.Seetheexamplesgiveninthisdiscussion.Asindicatedintheexamples,theresponse
E 1325 – 02 (2008)
toeachtreatmentcombinationwillhaveasetofcoefficientsassociatedwithit.Thenumberoflinearlyindependentcontrastsinanexperimentisequal
to one less than the number of treatments. Sometimes the term contrast is used only to refer to the pattern of the coefficients, but the usual meaning
of this term is the algebraic sum of the responses multiplied by the appropriate coefficients.
contrast analysis, n—a technique for estimating the parameters of a model and making hypothesis tests on preselected linear
combinations of the treatments (contrasts). See Table 1 and Table 2.
NOTE 4—Contrast analysis involves a systematic tabulation and analysis format usable for both simple and complex designs. When any set of
orthogonal contrasts is used, the procedure, as in the example, is straightforward.When terms are not orthogonal, the orthogonalization process to adjust
for the common element in nonorthogonal contrast is also systematic and can be programmed.
DISCUSSION—Example: Half-replicate of a 2 factorial experiment with factors A, B and C (X , X and X being quantitative, and factor D ( X )
1 2 3 4
qualitative. Defining contrast I = +ABCD=X X X X (see fractional factorial design and orthogonal design for derivation of the contrast
1 2 3 4
coeffıcients).
dependent variable, n—see response variable.
design of experiments, n—the arrangement in which an experimental program is to be conducted, and the selection of the levels
(versions)ofoneormorefactorsorfactorcombinationstobeincludedintheexperiment.Synonymsincludeexperimentdesign
and experimental design.
DISCUSSION—The purpose of designing an experiment is to provide the most efficient and economical methods of reaching valid and relevant
conclusions from the experiment. The selection of an appropriate design for any experiment is a function of many considerations such as the type of
questionstobeanswered,thedegreeofgeneralitytobeattachedtotheconclusions,themagnitudeoftheeffectforwhichahighprobabilityofdetection
(power) is desired, the homogeneity of the experimental units and the cost of performing the experiment.Aproperly designed experiment will permit
relatively simple statistical interpretation of the results, which may not be possible otherwise.The arrangement includes the randomization procedure
for allocating treatments to experimental units.
experimental design, n—see design of experiments.
experimental unit, n—a portion of the experiment space to which a treatment is applied or assigned in the experiment.
NOTE 5—The unit may be a patient in a hospital, a group of animals, a production batch, a section of a compartmented tray, etc.
experiment space, n—the materials, equipment, environmental conditions and so forth that are available for conducting an
experiment.
DISCUSSION—That portion of the experiment space restricted to the range of levels (versions) of the factors to be studied in the experiment is
sometimes called the factor space. Some elements of the experiment space may be identified with blocks and be considered as block factors.
evolutionary operation (EVOP), n— a sequential form of experimentation conducted in production facilities during regular
production.
NOTE 6—The principal theses of EVOP are that knowledge to improve the process should be obtained along with a product, and that designed
experimentsusingrelativelysmallshiftsinfactorlevels(withinproductiontolerances)canyieldthisknowledgeatminimumcost.Therangeofvariation
of the factors for any one EVOP experiment is usually quite small in order to avoid making out-of-tolerance products, which may require considerable
replication, in order to be able to clearly detect the effect of small changes.
n
2 factorial experiment, n—a factorial experiment in which n factors are studied, each of them in two levels (versions).
n
DISCUSSION—The 2 factorial is a special case of the general factorial. (See factorial experiment (general).)Apopular code is to indicate a small
letter when a factor is at its high level, and omit the letter when it is at its low level. When factors are at their low level the code is ( 1).
Example (illustrating the discussion )—A 2 factorial with factors A, B, and C:
Level
Factor A Low High Low High Low High Low High
Factor B Low Low High High Low Low High High
Factor C Low Low Low Low High High High High
Code (1) a b ab c ac bc abc
TABLE 1 Contrast Coefficient
Source Treatments (1) ab ac bc ad bd cd abcd
Centre X +1 +1 +1 +1 +1 +1 +1 +1 See Note 1
A(+BCD): pH (8.0; 9.0) X −1 +1 +1 −1 +1 −1 −1 +1
3 3
B(+ACD): SO (10 cm ;16cm ) X −1 +1 −1 +1 −1 +1 −1 +1
4 2
C(+ABD): Temperature (120°C; 150°C) X −1 −1 +1 +1 −1 −1 +1
...
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