# ASTM E1325-16

Designation: E1325 − 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 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 tothe area of design of experiments for which standard defini-tions appear 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, andcomparison of terms.4. Terminologyaliases, n—in 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 halfreplicate) or defining contrasts (for a fraction smaller than1⁄2) 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 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 treatment combi-nations) the size of the experiment can be reduced to1⁄4,1⁄8,oringeneral to1⁄2kto form a 2n-kfractional factorial.(3) There exist generalizations of the above to factorials havingmore than 2 levels.balanced incomplete block design (BIB), n—an incompleteblock design in which each block contains the same numberk of different versions from the t versions of a singleprincipal factor arranged so that every pair of versionsoccurs together in the same number, λ, of blocks from the bblocks.DISCUSSION—The design implies that every version of the principalfactor appears the same number of times r in the experiment and thatthe following relations hold true: bk = tr and r (k −1)=λ(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, λ) is possible for constructing balanced incomplete blockdesigns. For example, t =7, k =4, b =7, λ = 2. Versions of theprincipal factor:Block11236223473345144562556736671477125block factor, n—a factor that indexes division of experimentalunits into disjoint subsets.DISCUSSION—Blocks are sets of similar experimental units intendedto make variability within blocks as small as possible, so that treatmenteffects will be more precisely estimated. The effect of a block factor isusually not of primary interest in the experiment. Components ofvariance attributable to blocks may be of interest.The origin of the term“block” is in agricultural experiments, where a block is a contiguousportion of a field divided into experimental units, “plots,” that are eachsubjected to a treatment.completely randomized design, n—a design in which thetreatments are assigned at random to the full set of experi-mental units.DISCUSSION—No block factors are involved in a completely random-ized design.completely randomized factorial design, n—a factorial ex-periment (including all replications) run in a completelyrandomized design.composite design, n—a design developed specifically forfitting second order response surfaces to study curvature,constructed by adding further selected treatments to thoseobtained from a 2nfactorial (or its fraction).DISCUSSION—If 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 are1This terminology is under the jurisdiction ofASTM Committee E11 on Qualityand Statistics and is the direct responsibility of Subcommittee E11.10 on Sampling/ Statistics.Current edition approved April 1, 2016. Published April 2016. Originallyapproved in 1990. Last previous edition approved in 2015 as E1325 – 15. DOI:10.1520/E1325-16.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at service@astm.org. For Annual Book of ASTMStandards volume information, refer to the standard’s Document Summary page onthe ASTM website.Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States1(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) fora2nfactorial. 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, 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 orthogonal aspossible to one another or to minimize the bias that is created if the trueform of response surface is not quadratic.confounded factorial design, n—a factorial 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 confound-ing.NOTE 1—All factor level combinations are included in the experiment.DISCUSSION—Example: Ina23factorial with only room for 4treatments per block, the ABC interaction(ABC: − (1) + a+b−ab+c−ac−bc+abc) can be sacrificedthrough 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 bedetermined once the effect(s) to be confounded is(are) defined.Where only one term is to be confounded with blocks, as in thisexample, those with a positive sign are assigned to one blockand those with a negative sign to the other. There aregeneralized rules for more complex situations. A check on allof the other effects (A, B, AB, etc.) will show the balance of theplus and minus signs in each block, thus eliminating anyconfounding with blocks for them.confounding, n—combining 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 2—Confounding 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 confounded 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 incompleteplanning of the design, and it serves to diminish, or even to invalidate, theeffectiveness of an experiment.contrast, n—a linear function of the observations for which thesum of the coefficients is zero.NOTE 3—With observations Y1, Y2, ., Yn, the linear function a1Y1+ a2Y2+ . + a1Ynis a contrast if, and only if ∑ai= 0, where the aivalues are called the contrast coefficients.DISCUSSION—Example 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 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 the 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 forContrast 2 would appear as (−1, + 2, − 1).Response A1A2A3Contrast coefficients for question 1 −1 0 +1Contrast 1 −A1. + A3Contrast coefficients for question 2 −1⁄2 +1 −1⁄2Contrast 2 −1⁄2 A1+ A2−1⁄2 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 manufacturing technique while the other two, A2andA3use the customary one. First, does vendor A1with the new technique seem todiffer from A2and A3? Second, do the two suppliers using the customary techniquediffer? Contrast A2and A3. The pattern of contrast coefficients is similar to that forthe previous problem, though the interpretation of the results will 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 provided the ^ai=0condition is met. Questions of logical interest from an experiment may beexpressed as contrasts with carefully selected coefficients. See the examplesgiven in this discussion. As indicated in the examples, the response to eachtreatment combination will have a set of coefficients associated with it.The numberof linearly independent contrasts in an experiment is equal to one less than thenumber of treatments. Sometimes the term contrast is used only to refer to thepattern of the coefficients, but the usual meaning of this term is the algebraic sumof the responses multiplied by the appropriate coefficients.contrast analysis, n—a 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 4—Contrast analysis involves a systematic 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 isalso systematic and can be programmed.DISCUSSION—Example: Half-replicate of a 24factorial experimentwith factors A, B and C (X1, X2and X3being quantitative, and factor D(X4) qualitative. Defining contrast I = +ABCD=X1X2X3X4(seefractional factorial design and orthogonal contrasts for derivation ofthe contrast coeffıcients).design of experiments, n—the arrangement in which anexperimental 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.DISCUSSION—The purpose of designing an experiment is to providethe most efficient and economical 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 the effect for whicha high probability of detection (power) is desired, the homogeneity ofthe experimental units and the cost of performing the experiment. AE1325 − 162properly designed experiment will permit relatively simple statisticalinterpretation of the results, which may not be possible otherwise. Thearrangement includes the randomization procedure for allocatingtreatments to experimental units.experimental design, n—see design of experiments.experimental unit, n—a portion of the experiment space towhich a treatment is applied or assigned in the experiment.NOTE 5—The unit may be a patient in a hospital, a group of animals, aproduction batch, a section of a compartmented tray, etc.experiment space, n—the materials, equipment, environmen-tal conditions and so forth that are available for conductingan experiment.DISCUSSION—That portion of the experiment space restricted to therange of levels (versions) 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 duringregular production.NOTE 6—The 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 ofvariation 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.factor, n—independent variable in an experimental design.DISCUSSION—Factors can include controllable factors that are ofinterest for the experiment, block factors that are created to enhanceprecision of the factors of interest, and uncontrolled factors that mightbe measured in the experiment. Design of an experiment consists ofallocating levels of each controllable experimental factor to experimen-tal units.2nfactorial experiment, n—a factorial experiment in which nfactors are studied, each of them in two levels (versions).DISCUSSION—The 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 at their low level the code is (1).TABLE 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(+ ACD): SO4(10 cm3;16cm3) X2−1 +1 −1 +1 −1 +1 −1 +1C(+ ABD): Temperature (120°C; 150°C) 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