1、Designation: E 1325 02An 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, the year of last r
2、evision. 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 standard defini-tions
3、appears desirable.2. Referenced Documents2.1 ASTM Standards:E 456 Terminology related to Quality and Statistics23. Significance and Use3.1 This standard is a subsidiary to Terminology E 456.3.2 It provides definitions, descriptions, discussion, andcomparison of terms.4. Terminologyaliases, nin a fra
4、ctional 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 halfreplicate) or defining c
5、ontrasts (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) and, using the con
6、version 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 treatmentcombinations) the size of t
7、he 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 numberk of differen
8、t 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 experiment and thatthe
9、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, l) is possible fo
10、r 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.DISCUSSIONNo block
11、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,constructed by adding
12、 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 design are(0, 0, .,
13、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 additional replicate run
14、s 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 polynomials as orthog
15、onal aspossible to one another or to minimize the bias that is created if the trueform of response surface is not quadratic.confounded factorial design, na factorial experiment inwhich only a fraction of the treatment combinations are run1This terminology is under the jurisdiction ofASTM Committee E
16、11 on Qualityand Statistics and is the direct responsibility of Subcommittee E11.10 on Sampling.The definitions in this standard were extracted from E 456 89c.Current edition approved Oct. 10, 2002. Published December 2002. Originallypublished as E 1325 90. Last previous edition E 1325 91(1997)2Annu
17、al Book of ASTM Standards, Vol 14.02.1Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.in each block and where the selection of the treatmentcombinations assigned to each block is arranged so that oneor more prescribed effects is(are)
18、confounded with the blockeffect(s), while the other effects remain free from confound-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 sacrificedth
19、rough confounding with blocks without loss of any other effect if theblocks include 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)
20、 to be confounded is(are) defined. Where only one term isto be confounded with blocks, 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,
21、AB, etc.) will show the balance of theplus and minus signs in each block, thus eliminating 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 facto
22、r(s) orinteraction(s).NOTE 2Confounding is a useful technique that permits the effectiveuse 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 c
23、onfounded withblock effects or other preselected principal factor or differential 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 incompletepl
24、anning of the design, and it serves to diminish, or even to invalidate, theeffectiveness 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
25、 only if (ai= 0, where the aivalues arecalled the contrast coefficients.DISCUSSIONExample 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. Thi
26、s could be done by comparing A1and A3, the extremesof A in the experiment. A second 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 t
27、he line connecting A1andA3or, in other words, be equal to the average.) The following 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 fo
28、rContrast 2 would appear as (1, + 2, 1).Response A1A2A3Contrast coefficients for 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. S
29、uppose there are three sources of supply,one of which, A1, uses a new manufacturing 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 A2an
30、d A3. The pattern of contrast coefficients issimilar to that for the previous problem, 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
31、a contrast may be selected arbitrarily providedthe (ai= 0 condition is met. Questions 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 combin
32、ation will have aset of coefficients associated with it. The number of linearly 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 thealg
33、ebraic sum of the responses multiplied by the appropriate coeffi-cients.contrast 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 sys
34、tematic tabulation and analy-sis format usable for both simple and complex designs. When any set oforthogonal contrasts is used, the procedure, as in the example, isstraightforward. When terms are not orthogonal, the orthogonalizationprocess to adjust for the common element in nonorthogonal contrast
35、 isalso systematic and can be programmed.DISCUSSIONExample: Half-replicate of a 24factorial experimentwith factors A, B and C (X1, X2and X3being quantitative, and factor DTABLE 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
36、.0; 9.0) X11 +1 +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
37、center is not a constant (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 produc
38、ts (Xij= XiXi) of the appropriate terms.E1325022( X4) qualitative. Defining 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 anexperi
39、mental program is to be conducted, and the selectionof the levels (versions) 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 e
40、conomical methods of reaching valid andrelevant conclusions from the experiment. 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 th
41、e effect for whicha high probability of detection (power) is desired, the 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. Thearra
42、ngement includes the randomization procedure for allocatingtreatments to 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
43、, a group of animals, aproduction batch, a section of a compartmented tray, 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
44、) of the factors to be studied in the experimentis sometimes called the 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 duringre
45、gular production.NOTE 6The principal theses of EVOP are that knowledge 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 ofvari
46、ation of the factors for any one EVOP experiment is usually quitesmall 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
47、 studied, each of them in two levels (versions).DISCUSSIONThe 2nfactorial 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
48、at their low level the code is ( 1).Example (illustrating the discussion)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 identificati
49、on 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 (versions) of thefactors.Example (illustrating contrast)Two-factor, two-level factorial 22with factors A and B: A =a(1)+abb. This is the contrast of Aat the low level of B plus the contrast of A at the high level of B. B =b
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