ASTM E1325-15
(Terminology)Standard Terminology Relating to Design of Experiments
Standard Terminology Relating to Design of Experiments
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Designation: E1325 − 15 AnAmerican National Standard
Standard Terminology Relating to
1
Design of Experiments
This standard is issued under the fixed designation E1325; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision.Anumber in parentheses indicates the year of last reapproval.A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope k of different versions from the t versions of a single
principal factor arranged so that every pair of versions
1.1 This standard includes those statistical items related to
occurs together in the same number, λ, of blocks from the b
the area of design of experiments for which standard defini-
blocks.
tions appear desirable.
DISCUSSION—The design implies that every version of the principal
factor appears the same number of times r in the experiment and that
2. Referenced Documents
the following relations hold true: bk = tr and r (k−1)=λ(t − 1).
2
2.1 ASTM Standards:
For randomization, arrange the blocks and versions within each
E456Terminology Relating to Quality and Statistics blockindependentlyatrandom.Sinceeachletterintheaboveequations
represents an integer, it is clear that only a restricted set of combina-
3. Significance and Use tions(t, k, b, r,λ)ispossibleforconstructingbalancedincompleteblock
designs. For example, t =7, k =4, b =7, λ=2. Versions of the
3.1 This standard is a subsidiary to Terminology E456.
principal factor:
3.2 It provides definitions, descriptions, discussion, and
Block1 1236
comparison of terms. 2 2347
3 3451
4 4562
4. Terminology
5 5673
6 6714
aliases, n—in a fractional factorial design,twoormoreeffects
7 7125
which are estimated by the same contrast and which,
therefore, cannot be estimated separately.
n completely randomized design, n—a design in which the
DISCUSSION—(1) The determination of which effects in a 2 factorial
treatments are assigned at random to the full set of experi-
are aliasedcanbemadeoncethe defining contrast(inthecaseofahalf
1
mental units.
replicate) or defining contrasts (for a fraction smaller than ⁄2) are
stated. The defining contrast is that effect (or effects), usually thought
DISCUSSION—No block factors are involved in a completely random-
tobeofnoconsequence,aboutwhichallinformationmaybesacrificed
ized design.
fortheexperiment.Anidentity, I,isequatedtothe defining contrast(or
2 2 2
defining contrasts) and, using the conversion that A = B = C = I, the
completely randomized factorial design, n—a factorial ex-
multiplication of the letters on both sides of the equation shows the
periment (including all replications) run in a completely
aliases.Intheexampleunderfractionalfactorialdesign, I =ABCD.So
randomized design.
2 2 2
that: A = A BCD = BCD, and AB = A B CD=CD.
(2) With a large number of factors (and factorial treatment combi-
composite design, n—a design developed specifically for
1 1
nations) the size of the experiment can be reduced to ⁄4, ⁄8,orin
fitting second order response surfaces to study curvature,
k n-k
1
general to ⁄2 to form a 2 fractional factorial.
constructed by adding further selected treatments to those
(3) There exist generalizations of the above to factorials having
n
obtained from a 2 factorial (or its fraction).
more than 2 levels.
DISCUSSION—If the coded levels of each factor are−1 and+1 in the
n
balanced incomplete block design (BIB), n—an incomplete
2 factorial(seenotation2underdiscussionfor factorial experiment),
block design in which each block contains the same number the(2n +1)additionalcombinationsfora central composite designare
(0, 0, ., 0), (6a, 0, 0, ., 0) 0, 6a, 0, ., 0) ., (0, 0, ., 6 a). The
n
minimum total number of treatments to be tested is (2 +2n +1) for
1 n
ThisterminologyisunderthejurisdictionofASTMCommitteeE11onQuality
a2 factorial. Frequently more than one center point will be run. For n
and Statistics and is the direct responsibility of Subcommittee E11.10 on Sampling
=2, 3 and 4 the experiment requires, 9, 15, and 25 units respectively,
/ Statistics.
although additional replicate runs of the center point are usual, as
Current edition approved Oct. 1, 2015. Published October 2015. Originally
n
compared with 9, 27, and 81 in the 3 factorial. The reduction in
approved in 1990. Last previous edition approved in 2008 as E1325–02 (2008).
experiment size results in confounding, and thereby sacrificing, all
DOI: 10.1520/E1325-15.
information about curvature interactions.The value of a can be chosen
2
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
to make the coefficients in the quadratic polynomials as orthogonal as
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
possibletooneanotherortominimizethebiasthatiscreatedifthetrue
Standards
...
This document is not an ASTM 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. Because
it 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 version
of the standard as published by ASTM is to be considered the official document.
Designation: E1325 − 02 (Reapproved 2008) E1325 − 15 An American National Standard
Standard Terminology Relating to
1
Design of Experiments
This standard is issued under the fixed designation E1325; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope
1.1 This standard includes those statistical items related to the area of design of experiments for which standard definitions
appearsappear desirable.
2. Referenced Documents
2
2.1 ASTM Standards:
E456 Terminology Relating to Quality and Statistics
3. Significance and Use
3.1 This standard is a subsidiary to Terminology E456.
3.2 It provides definitions, descriptions, discussion, and comparison of terms.
4. Terminology
aliases, n—in a fractional factorial design, two or more effects which are estimated by the same contrast and which, therefore,
cannot be estimated separately.
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 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 as
E1325 – 02.E1325 – 02 (2008). DOI: 10.1520/E1325-02R08.10.1520/E1325-15.
2
For referenced ASTM standards, visit the ASTM website, www.astm.org, or contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM Standards
volume information, refer to the standard’s Document Summary page on the ASTM website.
DISCUSSION—
n
(1) The determination of which effects in a 2 factorial are aliased can be made once the defining contrast (in the case of a half replicate) or defining
1
contrasts (for a fraction smaller than ⁄2) are stated. The defining contrast is that effect (or effects), usually thought to be of no consequence, about which
all information may be sacrificed for the experiment. An identity, I, is equated to the defining contrast (or defining contrasts) and, using the conversion
2 2 2
that A = B = C = I, the multiplication of the letters on both sides of the equation shows the aliases. In the example under fractional factorial design,
2 2 2
I = ABCD. So that: A = A BCD = BCD, and AB = A B CD = CD.
k
1 1 1
(2) With a large number of factors (and factorial treatment combinations) the size of the experiment can be reduced to ⁄4, ⁄8, or in general to ⁄2
n-k
to form a 2 fractional factorial.
(3) There exist generalizations of the above to factorials having more than 2 levels.
balanced incomplete block design (BIB), n—an incomplete block design in which each block contains the same number k of
different versions from the t versions of a single principal factor arranged so that every pair of versions occurs together in the
same number, λ, of blocks from the b blocks.
DISCUSSION—
The design implies that every version of the principal factor appears the same number of times r in the experiment and that the following relations
hold true: bk = tr and r (k − 1) = λ(t − 1).
For randomization, arrange the blocks and versions within each block independently at random. Since each letter in the above equations represents
an integer, it is clear that only a restricted set of combinations (t, k, b, r, λ) is possible for constructing balanced incomplete block designs. For example,
t = 7, k = 4, b = 7, λ = 2. Versions of the principal factor:
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E1325 − 15
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completely randomized design, n—a design in which the treatments are assigned at random to the full set of experimental units.
DISCUSSION—
No block factors are involved in a completely randomized design.
completely randomized factorial design, n—a factorial experiment (including all replications) run in a completely randomized
design.
composite design, n—a design developed specifically for fitting second order response surfaces to study curvature, constructed by
n
adding further selected treatments to those obtained from a 2 factorial (or its fraction).
DISCUSSION—
n
If the coded levels of each factor are −
...
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