1、Designation: E1325 15E1325 16 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 revision, the year of
2、 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 which standard defini
3、tionsappear 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. Terminologyaliases, nin a
4、 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 of a half replicate) or de
5、finingcontrasts (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 defining contrasts) and, usi
6、ng 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 factorial treatment combination
7、s) 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 each block contains the s
8、ame 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 same number of times r in th
9、e 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 set of combinations (t,
10、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 Subcommittee E11.10 on Sa
11、mpling /Statistics.Current edition approved Oct. 1, 2015April 1, 2016. Published October 2015April 2016. Originally approved in 1990. Last previous edition approved in 20082015 asE1325 02 (2008).E1325 15. DOI: 10.1520/E1325-15.10.1520/E1325-16.2 For referencedASTM standards, visit theASTM website, w
12、ww.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 standard and is intended only to provide the user of an ASTM standard an indication of w
13、hat 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 cases only the current versionof the standard as published by ASTM is to be considered t
14、he official document.Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States1Block 1 1 2 3 62 2 3 4 73 3 4 5 14 4 5 6 25 5 6 7 36 6 7 1 47 7 1 2 5block factor, na factor that indexes division of experimental units into disjoint subsets.DISCUS
15、SIONBlocks are sets of similar experimental units intended to make variability within blocks as small as possible, so that treatment effects will be moreprecisely estimated. The effect of a block factor is usually not of primary interest in the experiment. Components of variance attributable to bloc
16、ksmay be of interest. The origin of the term “block” is in agricultural experiments, where a block is a contiguous portion of a field divided intoexperimental units, “plots,” that are each subjected to a pletely randomized design, na design in which the treatments are assigned at random to the full
17、set of experimental units.DISCUSSIONNo block factors are involved in a completely randomized pletely randomized factorial design, na factorial experiment (including all replications) run in a completely posite design, na design developed specifically for fitting second order response surfaces to stu
18、dy 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 the 2n factorial (see notation 2 under discussion for factorial experiment), the (2n + 1) additionalcombinations for a
19、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. Frequently more than one center point will be run. For n = 2, 3 and 4 the experiment requires, 9, 15,and 25 units res
20、pectively, 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, and thereby sacrificing, all information about curvature interactions. The value of a can be chosento make the coef
21、ficients 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 factorial design, na factorial experiment in which only a fraction of the treatment combinations are run in eachblock
22、 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 other effects remain free from confounding.NOTE 1All factor level combinations are included in the experiment.DISCUSS
23、IONExample: 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 without loss of any other effect if the blocks include the following:Block 1 Block 2Treatment (1) aCombination ab b(Code
24、 identification shown in discus-sion under factorial experiment)acbccabcThe treatments to be assigned to each block can be determined once 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 t
25、o one block and thosewith a negative sign to the 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 and minus signs in each block, thus eliminating any confounding with blocks forthem.confounding, ncom
26、bining indistinguishably the main effect of a factor or a differential effect between factors (interactions) withthe effect of other factor(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. Thi
27、s is accomplished bydeliberately preselecting certain effects or differential effects as being of little interest, and arranging the design so that they are confounded with blockE1325 162effects or other preselected principal factor or differential effects, while keeping the other more important eff
28、ects 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 thedesign, and it serves to diminish, or even to invalidate, the effectiveness of an experiment.contrast, na linear functi
29、on of the observations for which the sum of the coefficients is zero.NOTE 3With observations Y1, Y2, ., Yn, the linear function a1Y1 + a2Y2 + . + 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 level
30、s 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 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 evidenc
31、e that the response pattern shows curvature rather than a simple linear trend. Here the average of A1 andA3 could be compared to A2. (If there 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
32、type study of equally spaced continuous variables. It is frequently more convenient to use integers ratherthan fractions for contrast coefficients. 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 .
33、+ A3Contrast coefficients for question 2 12 +1 12Contrast 2 12 A1 + A2 12A3Example 2:Another example dealing with discrete versions of a 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
34、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 suppliers using the customary technique differ? Contrast A2 and A3.The patternof contrast coefficients is similar to that for the previous problem, though the interpret
35、ation of the results will differ.Example 2: Another example dealing with discrete versions of a factor might lead to a different pair of questions. Suppose there are three sources ofsupply, one of which, A1, uses a new manufacturing technique while the other two, A2 and A3 use the customary one. Fir
36、st, does vendor A1 with the newtechnique seem to differ fromA2 andA3? Second, do the two suppliers using the customary technique differ? ContrastA2 andA3. The pattern of contrastcoefficients is similar to that for the previous problem, though the interpretation of the results will differ.Response A1
37、 A2 A3Contrast 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 provided the ai = 0 condition is met. Questions of logical interest from an experimentmay be expressed as
38、 contrasts with carefully selected coefficients. See the examples given in this discussion. As indicated in the examples, theresponse 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 th
39、an 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.The coefficients for a contrast may be selected arbitrarily provided t
40、he ai = 0 condition is met. Questions of logical interest from an experimentmay be expressed as contrasts with carefully selected coefficients. See the examples given in this discussion.As indicated in the examples, the responseto each treatment combination will have a set of coefficients associated
41、 with it. The number of linearly independent contrasts in an experiment is equalto 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 meaningof this term is the algebraic sum of the responses multiplied by the a
42、ppropriate coefficients.contrast analysis, na technique for estimating the parameters of a model and making hypothesis tests on preselected linearcombinations of the treatments (contrasts). See Table 1 and Table 2.NOTE 4Contrast analysis involves a systematic tabulation and analysis format usable fo
43、r 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 orthogonalization process to adjustfor the common element in nonorthogonal contrast is also systematic and can be programmed.DISCUS
44、SIONExample: 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 X4 (see fractional factorial design and orthogonal contrasts for derivation of the contrast coeffcients).TABLE 1 Con
45、trast 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(+ 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
46、 +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 163design of experiments, nthe arrangement in which an experimental program is to be conducted, and the selection of the levels(versions) of one or more factors or
47、 factor combinations to be included in the experiment. Synonyms include experiment designand experimental design.DISCUSSIONThe purpose 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
48、 appropriate design for any experiment is a function of many considerations such as the type of questions to beanswered, the degree of 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 experim
49、ental units and the cost of performing the experiment. A properly designed experiment will permit relativelysimple statistical interpretation 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 assigne
