1、Designation: E 1325 02 (Reapproved 2008)An American National StandardStandard Terminology Relating toDesign of Experiments1This standard is issued under the fixed designation E 1325; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision,
2、the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon (e) indicates an editorial change since the last revision or reapproval.1. Scope1.1 This standard includes those statistical items related tothe area of design of experiments for which stan
3、dard defini-tions appears desirable.2. Referenced Documents2.1 ASTM Standards:2E 456 Terminology Relating to Quality and Statistics3. Significance and Use3.1 This standard is a subsidiary to Terminology E 456.3.2 It provides definitions, descriptions, discussion, andcomparison of terms.4. Terminolog
4、yaliases, nin a fractional factorial design, two or more effectswhich are estimated by the same contrast and which,therefore, cannot be estimated separately.DISCUSSION(1) The determination of which effects in a 2nfactorialare aliased can be made once the defining contrast (in the case of a halfrepli
5、cate) or defining contrasts (for a fraction smaller than12) arestated. The defining contrast is that effect (or effects), usually thoughtto be of no consequence, about which all information may be sacrificedfor the experiment.An identity, I, is equated to the defining contrast (ordefining contrasts)
6、 and, using the conversion that A2= B2= C2= I, themultiplication of the letters on both sides of the equation shows thealiases. In the example under fractional factorial design, I = ABCD. Sothat: A = A2BCD = BCD, and AB = A2B2CD=CD.( 2) With a large number of factors (and factorial treatmentcombinat
7、ions) the size of the experiment can be reduced to14 ,18 ,orin general to12kto form a 2n-kfractional factorial.(3) There exist generalizations of the above to factorials havingmore than 2 levels.balanced incomplete block design (BIB), nan incompleteblock design in which each block contains the same
8、numberk of different versions from the t versions of a singleprincipal factor arranged so that every pair of versionsoccurs together in the same number, l, of blocks from the bblocks.DISCUSSIONThe design implies that every version of the principalfactor appears the same number of times r in the expe
9、riment and thatthe following relations hold true: bk = tr and r (k 1)=l(t 1).For randomization, arrange the blocks and versions within eachblock independently at random. Since each letter in the above equationsrepresents an integer, it is clear that only a restricted set of combina-tions (t, k, b, r
10、, l) is possible for constructing balanced incompleteblock designs. For example, t =7,k =4,b =7,l = 2. Versions of theprincipal factor:Block11236223473345144562556736671477125completely randomized design, na design in which thetreatments are assigned at random to the full set of experi-mental units.
11、DISCUSSIONNo block factors are involved in a completely random-ized pletely randomized factorial design, na factorial ex-periment (including all replications) run in a completelyrandomized posite design, na design developed specifically forfitting second order response surfaces to study curvature,co
12、nstructed by adding further selected treatments to thoseobtained from a 2nfactorial (or its fraction).DISCUSSIONIf the coded levels of each factor are 1 and + 1 in the2nfactorial (see notation 2 under discussion for factorial experiment),the (2n + 1) additional combinations for a central composite d
13、esign are(0, 0, ., 0), (6a, 0, 0, ., 0) 0, 6a, 0, ., 0) ., (0, 0, ., 6 a). Theminimum total number of treatments to be tested is (2n+2n + 1) for a2nfactorial. 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 addit
14、ional replicate runs of the center point are usual, ascompared with 9, 27, and 81 in the 3nfactorial. The reduction inexperiment size results in confounding, and thereby sacrificing, allinformation about curvature interactions. The value of a can be chosento make the coefficients in the quadratic po
15、lynomials as orthogonal aspossible to one another or to minimize the bias that is created if the true1This terminology is under the jurisdiction ofASTM Committee E11 on Qualityand Statistics and is the direct responsibility of Subcommittee E11.10 on Sampling.The definitions in this standard were ext
16、racted from E 456 89c.Current edition approved April 1, 2008. Published May 2008. Originallyapproved in 1990. Last previous edition approved in 2002 as E 1325 02.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at serviceastm.org. For Annual Book
17、of ASTMStandards volume information, refer to the standards Document Summary page onthe ASTM website.1Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.form of response surface is not quadratic.confounded factorial design, na factorial
18、experiment inwhich only a fraction of the treatment combinations are runin each block and where the selection of the treatmentcombinations assigned to each block is arranged so that oneor more prescribed effects is(are) confounded with the blockeffect(s), while the other effects remain free from con
19、found-ing.NOTE 1All factor level combinations are included in the experiment.DISCUSSIONExample: Ina23factorial with only room for 4treatments per block, the ABC interaction(ABC: (1) + a+bab+cacbc+abc) can be sacrificedthrough confounding with blocks without loss of any other effect if theblocks incl
20、ude the following.Block 1 Block 2Treatment (1) aCombination ab b(Code identification shown in discus-sion under factorial experiment)acbccabcThe treatments to be assigned to each block can be determined oncethe effect(s) to be confounded is(are) defined. Where only one term isto be confounded with b
21、locks, as in this example, those with a positivesign are assigned to one block and those with a negative sign to theother. There are generalized rules for more complex situations.Acheckon all of the other effects (A, B, AB, etc.) will show the balance of theplus and minus signs in each block, thus e
22、liminating any confoundingwith blocks for them.confounding, ncombining indistinguishably the main effectof a factor or a differential effect between factors (interac-tions) with the effect of other factor(s), block factor(s) orinteraction(s).NOTE 2Confounding is a useful technique that permits the e
23、ffectiveuse of specified blocks in some experiment designs. This is accomplishedby deliberately preselecting certain effects or differential effects as beingof little interest, and arranging the design so that they are confounded withblock effects or other preselected principal factor or differentia
24、l effects,while keeping the other more important effects free from such complica-tions. Sometimes, however, confounding results from inadvertent changesto a design during the running of an experiment or from incompleteplanning of the design, and it serves to diminish, or even to invalidate, theeffec
25、tiveness of an experiment.contrast, na linear function of the observations for whichthe sum of the coefficients is zero.NOTE 3With observations Y1, Y2, ., Yn, the linear function a1Y1+ a2Y2+ . + a1Ynis a contrast if, and only if (ai= 0, where the aivalues arecalled the contrast coefficients.DISCUSSI
26、ONExample 1: A factor is applied at three levels and theresults are represented by A1,A2, A3. If the levels are equally spaced,the first question it might be logical to ask is whether there is an overalllinear trend. This could be done by comparing A1and A3, the extremesof A in the experiment. A sec
27、ond question might be whether there isevidence that the response pattern shows curvature rather than a simplelinear trend. Here the average of A1and A3could be compared to A2.(If there is no curvature, A2should fall on the line connecting A1andA3or, in other words, be equal to the average.) The foll
28、owing exampleillustrates a regression type study of equally spaced continuousvariables. It is frequently more convenient to use integers rather thanfractions for contrast coefficients. In such a case, the coefficients forContrast 2 would appear as (1, + 2, 1).Response A1A2A3Contrast coefficients for
29、 question 1 1 0 +1Contrast 1 A1. +A3Contrast coefficients for question 2 12 +1 12Contrast 2 12 A1+A212 A3Example 2: Another example dealing with discrete versions of a factor mightlead to a different pair of questions. Suppose there are three sources of supply,one of which, A1, uses a new manufactur
30、ing technique while the other two, A2and A3use the customary one. First, does vendor A1with the new techniqueseem to differ from A2and A3? Second, do the two suppliers using the custom-ary technique differ? Contrast A2and A3. The pattern of contrast coefficients issimilar to that for the previous pr
31、oblem, though the interpretation of the resultswill differ.Response A1A2A3Contrast coefficients for question 1 2 +1 +1Contrast 1 2A1+A2+A3Contrast coefficients for question 2 0 1 +1Contrast 2 . A2+A3The coefficients for a contrast may be selected arbitrarily providedthe (ai= 0 condition is met. Ques
32、tions of logical interest from anexperiment may be expressed as contrasts with carefully selectedcoefficients. See the examples given in this discussion. As indicated inthe examples, the response to each treatment combination will have aset of coefficients associated with it. The number of linearly
33、indepen-dent contrasts in an experiment is equal to one less than the number oftreatments. Sometimes the term contrast is used only to refer to thepattern of the coefficients, but the usual meaning of this term is thealgebraic sum of the responses multiplied by the appropriate coeffi-cients.contrast
34、 analysis, na technique for estimating the param-eters of a model and making hypothesis tests on preselectedlinear combinations of the treatments (contrasts). See Table1 and Table 2.NOTE 4Contrast analysis involves a systematic tabulation and analy-sis format usable for both simple and complex desig
35、ns. When any set oforthogonal contrasts is used, the procedure, as in the example, isstraightforward. When terms are not orthogonal, the orthogonalizationTABLE 1 Contrast CoefficientSource Treatments (1) ab ac bc ad bd cd abcdCentre X0+1 +1 +1 +1 +1 +1 +1 +1 See Note 1A(+BCD): pH (8.0; 9.0) X11 +1 +
36、1 1 +1 1 1 +1B(+ACD): SO4(10 cm3;16cm3) X21 +1 1 +1 1 +1 1 +1C(+ABD): Temperature (120C; 150C) X31 1 +1 +1 1 1 +1 +1D(+ABC): Factory (P; Q) X41 1 1 1 +1 +1 +1 +1AB + CD X1X2=X12+1 +1 1 1 1 1 +1 +1AC+BD X1X3=X13+1 1 +1 1 1 +1 1 +1 See Note 2AD+BC X1X4=X14+1 1 1 +1 +1 1 1 +1NOTE 1The center is not a c
37、onstant (Xifi 0) but is convenient in the contrast analysis calculations to treat it as one.NOTE 2Once the contrast coefficients of the main effects ( X1,X2,X3and X4) are filled in, the coefficients for all interaction and other second or higherorder effects can be derived as products (Xij= XiXi) of
38、 the appropriate terms.E 1325 02 (2008)2process to adjust for the common element in nonorthogonal contrast isalso systematic and can be programmed.DISCUSSIONExample: Half-replicate of a 24factorial experimentwith factors A, B and C (X1, X2and X3being quantitative, and factor D( X4) qualitative. Defi
39、ning contrast I = +ABCD=X1X2X3X4(seefractional factorial design and orthogonal design for derivation ofthe contrast coeffcients).dependent variable, nsee response variable.design of experiments, nthe arrangement in which anexperimental program is to be conducted, and the selectionof the levels (vers
40、ions) of one or more factors or factorcombinations to be included in the experiment. Synonymsinclude experiment design and experimental design.DISCUSSIONThe purpose of designing an experiment is to providethe most efficient and economical methods of reaching valid andrelevant conclusions from the ex
41、periment. The selection of an appro-priate design for any experiment is a function of many considerationssuch as the type of questions to be answered, the degree of generalityto be attached to the conclusions, the magnitude of the effect for whicha high probability of detection (power) is desired, t
42、he homogeneity ofthe experimental units and the cost of performing the experiment. Aproperly designed experiment will permit relatively simple statisticalinterpretation of the results, which may not be possible otherwise. Thearrangement includes the randomization procedure for allocatingtreatments t
43、o experimental units.experimental design, nsee design of experiments.experimental unit, na portion of the experiment space towhich a treatment is applied or assigned in the experiment.NOTE 5The unit may be a patient in a hospital, a group of animals, aproduction batch, a section of a compartmented t
44、ray, etc.experiment space, nthe materials, equipment, environmen-tal conditions and so forth that are available for conductingan experiment.DISCUSSIONThat portion of the experiment space restricted to therange of levels (versions) of the factors to be studied in the experimentis sometimes called the
45、 factor space. Some elements of the experimentspace may be identified with blocks and be considered as block factors.evolutionary operation (EVOP), n a sequential form ofexperimentation conducted in production facilities duringregular production.NOTE 6The principal theses of EVOP are that knowledge
46、to improvethe process should be obtained along with a product, and that designedexperiments using relatively small shifts in factor levels (within produc-tion tolerances) can yield this knowledge at minimum cost. The range ofvariation of the factors for any one EVOP experiment is usually quitesmall
47、in order to avoid making out-of-tolerance products, which mayrequire considerable replication, in order to be able to clearly detect theeffect of small changes.2nfactorial experiment, na factorial experiment in which nfactors are studied, each of them in two levels (versions).DISCUSSIONThe 2nfactori
48、al is a special case of the general factorial.(See factorial experiment (general).) A popular code is to indicate asmall letter when a factor is at its high level, and omit the letter whenit is at its low level. When factors are at their low level the code is ( 1).Example (illustrating the discussio
49、n)A 23factorial with factors A,B, and C:LevelFactor A Low High Low High Low High Low HighFactor B Low Low High High Low Low High HighFactor C Low Low Low Low High High High HighCode (1) a b ab c ac bc abcThis type of identification has advantages for defining blocks,confounding and aliasing. See confounded factorial design andfractional factorial design.Factorial experiments regardless of the form of analysis used,essentially involve contrasting the various levels (
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