ASTM_E_1325_-_16.pdf

Designation:E132516An 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,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 halfreplicate)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)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 to14,18,or ingeneral 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 different versions from the t versions of a singleprincipal factor arranged so that every pair of versionsoccurs together in the same number,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 following relations hold true:bk=tr and r(k 1)=(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,)is possible for constructing balanced incomplete blockdesigns.For example,t=7,k=4,b=7,=2.Versions of theprincipal factor:Block11236223473345144562556736671477125block factor,na factor that indexes division of experimentalunits into disjoint subsets.DISCUSSIONBlocks 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 pletely randomized design,na design in which thetreatments are assigned at random to the full set of experi-mental units.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,constructed 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 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 serviceastm.org.For Annual Book of ASTMStandards volume information,refer to the standards 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)fora 2nfactorial.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 addi