1、Designation: E1325 02 (Reapproved 2008)E1325 15 An American National StandardStandard Terminology Relating toDesign of Experiments1This standard is issued under the fixed designation E1325; the number immediately following the designation indicates the year oforiginal adoption or, in the case of rev
2、ision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon () indicates an editorial change since the last revision or reapproval.1. Scope1.1 This standard includes those statistical items related to the area of design of experiments for whi
3、ch standard definitionsappearsappear desirable.2. Referenced Documents2.1 ASTM Standards:2E456 Terminology Relating to Quality and Statistics3. Significance and Use3.1 This standard is a subsidiary to Terminology E456.3.2 It provides definitions, descriptions, discussion, and comparison of terms.4.
4、Terminologyaliases, nin a fractional factorial design, two or more effects which are estimated by the same contrast and which, therefore,cannot be estimated separately.DISCUSSION(1) The determination of which effects in a 2n factorial are aliased can be made once the defining contrast (in the case o
5、f a half replicate) or definingcontrasts (for a fraction smaller than 12) are stated.The defining contrast is that effect (or effects), usually thought to be of no consequence, about whichall information may be sacrificed for the experiment.An identity, I, is equated to the defining contrast (or def
6、ining contrasts) and, using the conversionthat A2 = B2 = C2 = I, the multiplication of the letters on both sides of the equation shows the aliases. In the example under fractional factorial design,I = ABCD. So that: A = A2BCD = BCD, and AB = A2B2CD = CD.(2) With a large number of factors (and factor
7、ial treatment combinations) the size of the experiment can be reduced to 14, 18, or in general to 12kto form a 2 n-k fractional factorial.(3) There exist generalizations of the above to factorials having more than 2 levels.balanced incomplete block design (BIB), nan incomplete block design in which
8、each block contains the same number k ofdifferent versions from the t versions of a single principal factor arranged so that every pair of versions occurs together in thesame number, , of blocks from the b blocks.DISCUSSIONThe design implies that every version of the principal factor appears the sam
9、e number of times r in the experiment and that the following relationshold true: bk = tr and r (k 1) = (t 1).For randomization, arrange the blocks and versions within each block independently at random. Since each letter in the above equations representsan integer, it is clear that only a restricted
10、 set of combinations (t, k, b, r, ) is possible for constructing balanced incomplete block designs. For example,t = 7, k = 4, b = 7, = 2. Versions of the principal factor:1 This terminology is under the jurisdiction of ASTM Committee E11 on Quality and Statistics and is the direct responsibility of
11、Subcommittee E11.10 on Sampling /Statistics. The definitions in this standard were extracted from E456 89c.Current edition approved April 1, 2008Oct. 1, 2015. Published May 2008October 2015. Originally approved in 1990. Last previous edition approved in 20022008 asE1325 02.E1325 02 (2008). DOI: 10.1
12、520/E1325-02R08.10.1520/E1325-15.2 For referencedASTM standards, visit theASTM website, www.astm.org, or contactASTM Customer Service at serviceastm.org. For Annual Book of ASTM Standardsvolume information, refer to the standards Document Summary page on the ASTM website.This document is not an ASTM
13、 standard and is intended only to provide the user of an ASTM standard an indication of what changes have been made to the previous version. Becauseit may not be technically possible to adequately depict all changes accurately, ASTM recommends that users consult prior editions as appropriate. In all
14、 cases only the current versionof the standard as published by ASTM is to be considered the official document.Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States1colwidth=“19.00*“/COLSPECBlock 1 1 2 3 62 2 3 4 73 3 4 5 14 4 5 6 25 5 6 7 3
15、6 6 7 1 47 7 1 2 5completely randomized design, na design in which the treatments are assigned at random to the full set of experimental units.DISCUSSIONNo block factors are involved in a completely randomized pletely randomized factorial design, na factorial experiment (including all replications)
16、run in a completely posite design, na design developed specifically for fitting second order response surfaces to study curvature, constructed byadding further selected treatments to those obtained from a 2n factorial (or its fraction).DISCUSSIONIf the coded levels of each factor are 1 and + 1 in th
17、e 2n factorial (see notation 2 under discussion for factorial experiment), the (2n + 1) additionalcombinations for a central composite design are (0, 0, ., 0), (6a, 0, 0, ., 0) 0, 6a, 0, ., 0) ., (0, 0, ., 6 a). The minimum total number of treatmentsto be tested is (2n + 2n + 1) for a 2n factorial.
18、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 in the 3n factorial. Thereduction in experiment size results in confounding
19、, and thereby sacrificing, all information about curvature interactions. The value of a can be chosento 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 formof response surface is not quadratic.confounded
20、factorial design, na factorial experiment in which only a fraction of the treatment combinations are run in eachblock and where the selection of the treatment combinations assigned to each block is arranged so that one or more prescribedeffects is(are) confounded with the block effect(s), while the
21、other effects remain free from confounding.NOTE 1All factor level combinations are included in the experiment.DISCUSSIONExample: In a 23 factorial with only room for 4 treatments per block, the ABC interaction (ABC: (1) + a + b ab + c ac bc + abc) can besacrificed through confounding with blocks wit
22、hout loss of any other effect if the blocks include the following.following:Block 1 Block 2Treatment (1) aCombination ab b(Code identification shown in discus-sion under factorial experiment)acbccabcBlock 1 Block 2Treatment (1) aCombination ab b(Code identification shown in discus-sion under factori
23、al experiment)acbccabcThe 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 tobe 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
24、other.There are generalized rules for more complex situations. A check on all of the other effects (A, B, AB, etc.) will show the balance of the plus andminus signs in each block, thus eliminating any confounding with blocks for them.The treatments to be assigned to each block can be determined once
25、 the effect(s) to be confounded is(are) defined. Whereonly one term is to be confounded with blocks, as in this example, those with a positive sign are assigned to one block and thosewith a negative sign to the other. There are generalized rules for more complex situations. A check on all of the oth
26、er effects (A,B, AB, etc.) will show the balance of the plus and minus signs in each block, thus eliminating any confounding with blocks forthem.confounding, ncombining indistinguishably the main effect of a factor or a differential effect between factors (interactions) withthe effect of other facto
27、r(s), block factor(s) or interaction(s).NOTE 2Confounding is a useful technique that permits the effective use of specified blocks in some experiment designs. This is accomplished bydeliberately preselecting certain effects or differential effects as being of little interest, and arranging the desig
28、n so that they are confounded with blockeffects or other preselected principal factor or differential effects, while keeping the other more important effects free from such complications.E1325 152Sometimes, however, confounding results from inadvertent changes to a design during the running of an ex
29、periment or from incomplete planning of thedesign, and it serves to diminish, or even to invalidate, the effectiveness of an experiment.contrast, na linear function of the observations for which the sum of the coefficients is zero.NOTE 3With observations Y1, Y2, ., Yn, the linear function a1Y1 + a2Y
30、2 + . + a1Yn is a contrast if, and only if ai = 0, where the ai values are calledthe contrast coefficients.DISCUSSIONExample 1: A factor is applied at three levels and the results are represented by A1,A2, A3. If the levels are equally spaced, the first question it mightbe logical to ask is whether
31、there is an overall linear trend. This could be done by comparing A1 and A3, the extremes of A in the experiment.Asecondquestion might be whether there is evidence that the response pattern shows curvature rather than a simple linear trend. Here the average of A1 andA3 could be compared to A2. (If t
32、here is no curvature, A2 should fall on the line connecting A1 and A3 or, in other words, be equal to the average.)The following example illustrates a regression type study of equally spaced continuous variables. It is frequently more convenient to use integers ratherthan fractions for contrast coef
33、ficients. In such a case, the coefficients for Contrast 2 would appear as (1, + 2, 1).Response A1 A2 A3Contrast coefficients for question 1 1 0 +1Contrast 1 A1 . + A3Contrast coefficients for question 2 12 +1 12Contrast 2 12 A1 + A2 12A3Example 2: Another example dealing with discrete versions of a
34、factor might lead to a different pair of questions. Suppose there are three sourcesof supply, one of which, A1, uses a new manufacturing technique while the other two, A2 and A3 use the customary one. First, does vendor A1 withthe new technique seem to differ from A2 and A3? Second, do the two suppl
35、iers using the customary technique differ? Contrast A2 and A3. Thepattern of contrast coefficients is similar to that for the previous problem, though the interpretation of the results will differ.Response A1 A2 A3Contrast coefficients for question 1 2 +1 +1Contrast 1 2A1 +A2 +A3Contrast coefficient
36、s for question 2 0 1 +1Contrast 2 . A2 + A3The coefficients for a contrast may be selected arbitrarily provided the ai = 0 condition is met. Questions of logical interest from an experi-ment may be expressed as contrasts with carefully selected coefficients. See the examples given in this discussion
37、. As indicated in the examples,the response to each treatment combination will have a set of coefficients associated with it. The number of linearly independent contrasts in anexperiment is equal to one less than the number of treatments. Sometimes the term contrast is used only to refer to the patt
38、ern of the coefficients,but the usual meaning of this term is the algebraic sum of the responses multiplied by the appropriate coefficients.contrast analysis, na technique for estimating the parameters of a model and making hypothesis tests on preselected linearcombinations of the treatments (contra
39、sts). See Table 1 and Table 2.NOTE 4Contrast analysis involves a systematic tabulation and analysis format usable for both simple and complex designs. When any set oforthogonal contrasts is used, the procedure, as in the example, is straightforward. When terms are not orthogonal, the orthogonalizati
40、on process to adjustfor the common element in nonorthogonal contrast is also systematic and can be programmed.DISCUSSIONExample: Half-replicate of a 24 factorial experiment with factors A, B and C (X1, X2 and X3 being quantitative, and factor D (X4) qualitative. Definingcontrast I = + ABCD = X1X2X3
41、X4 (see fractional factorial design and orthogonal designcontrasts for derivation of the contrast coeffcients).dependent variable, nsee response variable.design of experiments, nthe arrangement in which an experimental program is to be conducted, and the selection of the levels(versions) of one or m
42、ore factors or factor combinations to be included in the experiment. Synonyms include experiment designand experimental design.DISCUSSIONTABLE 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) X1 1 +1 +1 1 +1 1 1 +1B(
43、+ ACD): SO4 (10 cm3; 16 cm3) X2 1 +1 1 +1 1 +1 1 +1C(+ ABD): Temperature (120C; 150C) X3 1 1 +1 +1 1 1 +1 +1D(+ABC): Factory (P; Q) X4 1 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 +1E1325 153The purpose
44、of designing an experiment is to provide the most efficient and economical methods of reaching valid and relevant conclusions from theexperiment. The selection of an appropriate design for any experiment is a function of many considerations such as the type of questions to beanswered, the degree of
45、generality to be attached to the conclusions, the magnitude of the effect for which a high probability of detection (power) isdesired, the homogeneity of the experimental units and the cost of performing the experiment. A properly designed experiment will permit relativelysimple statistical interpre
46、tation of the results, which may not be possible otherwise. The arrangement includes the randomization procedure forallocating treatments to experimental units.experimental design,nsee design of experiments.experimental unit, na portion of the experiment space to which a treatment is applied or assi
47、gned in the experiment.NOTE 5The unit may be a patient in a hospital, a group of animals, a production batch, a section of a compartmented tray, etc.experiment space, nthe materials, equipment, environmental conditions and so forth that are available for conducting anexperiment.DISCUSSIONThat portio
48、n of the experiment space restricted to the range of levels (versions) of the factors to be studied in the experiment is sometimes called thefactor space. Some elements of the experiment space may be identified with blocks and be considered as block factors.evolutionary operation (EVOP), n a sequent
49、ial form of experimentation conducted in production facilities during regularproduction.NOTE 6The principal theses of EVOP are that knowledge to improve the process should be obtained along with a product, and that designedexperiments using relatively small shifts in factor levels (within production tolerances) can yield this knowledge at minimum cost. The range of variationof the factors for any one EVOP experiment is usually quite small in o
