ASTM E1325-21
(Terminology)Standard Terminology Relating to Design of Experiments
Standard Terminology Relating to Design of Experiments
ABSTRACT
This standard includes those statistical items related to the area of design of experiments for which standard definitions appear desirable. It provides definitions, descriptions, discussion, and comparison of terms.
SIGNIFICANCE AND USE
3.1 This standard is a subsidiary to Terminology E456.
3.2 It provides definitions, descriptions, discussion, and comparison of terms.
SCOPE
1.1 This standard includes those statistical items related to the area of design of experiments for which standard definitions appear desirable.
1.2 The values stated in SI units are to be regarded as standard. No other units of measurement are included in this standard.
1.3 This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
General Information
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This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the
Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
Designation: E1325 − 21 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.Anumber in parentheses indicates the year of last reapproval.A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
stated. The defining contrast is that effect (or effects), usually thought
1. Scope
tobeofnoconsequence,aboutwhichallinformationmaybesacrificed
1.1 This standard includes those statistical items related to
fortheexperiment.Anidentity, I,isequatedtothe defining contrast(or
2 2 2
the area of design of experiments for which standard defini-
defining contrasts) and, using the conversion that A = B = C = I, the
tions appear desirable.
multiplication of the letters on both sides of the equation shows the
aliases. In the example under fractional factorial design, I= ABCD.So
1.2 The values stated in SI units are to be regarded as
2 2 2
that: A= A BCD= BCD, and AB= A B CD=CD.
standard. No other units of measurement are included in this
(2) With a large number of factors (and factorial treatment combi-
standard. 1 1
nations) the size of the experiment can be reduced to ⁄4, ⁄8,orin
k n–k
1
general to ⁄2 to form a 2 fractional factorial.
1.3 This international standard was developed in accor-
(3) There exist generalizations of the above to factorials having
dance with internationally recognized principles on standard-
more than 2 levels.
ization established in the Decision on Principles for the
analysis of variance (ANOVA), n—statistical models and
Development of International Standards, Guides and Recom-
mendations issued by the World Trade Organization Technical associated procedures, in which the observed variance is
partitioned into components due to different explanatory
Barriers to Trade (TBT) Committee.
variables.
2. Referenced Documents
analysis of variance table, n—a tabular summary of results
2
2.1 ASTM Standards:
from a regression model or an experimental design for the
E456Terminology Relating to Quality and Statistics
purpose of evaluating effects of factors.
E1488GuideforStatisticalProcedurestoUseinDeveloping
DISCUSSION—Theanalysisofvarianceforadesignedexperimentlists
and Applying Test Methods
factors and, for each factor, the degrees of freedom, sum of squares,
mean square (sum of squares divided by degrees of freedom), and may
3. Significance and Use
list test statistics (F-ratio) or expected values of mean squares as
3.1 This standard is a subsidiary to Terminology E456.
functions of components of variance.
Example:Analysis of variance of a randomized block design with k
3.2 It provides definitions, descriptions, discussion, and
blocksand ttreatments.Theresponsefortreatment iinblock jis x ,the
ij
comparison of terms.
blockaverageis x¯ ,thetreatmentaverageis x¯ ,andtheoverallaverage
·j i·
is x¯ . σ is the component of variance due to blocks. τ is the treatment
·· b i
4. Terminology
effectwith τ 50.Asignificancetestoftreatmentsisthemeansquarefor
(
i
treatments, divided by the mean square for error. See Table 1.
aliases, n—in a fractional factorial design,twoormoreeffects
which are estimated by the same contrast and which,
balanced incomplete block design (BIB), n—an incomplete
therefore, cannot be estimated separately.
block design in which each block contains the same number
n
DISCUSSION—(1) The determination of which effects in a 2 factorial
k of different versions from the t versions of a single
are aliasedcanbemadeoncethe defining contrast(inthecaseofahalf
principal factor arranged so that every pair of versions
1
replicate) or defining contrasts (for a fraction smaller than ⁄2) are
occurs together in the same number, λ, of blocks from the b
blocks.
1
ThisterminologyisunderthejurisdictionofASTMCommitteeE11onQuality
DISCUSSION—The design implies that every version of the principal
and Statistics and is the direct responsibility of Subcommittee E11.10 on Sampling
factor appears the same number of times r in the experiment and that
/ Statistics.
the following relations hold true: bk= tr and r (k−1)= λ(t − 1).
CurrenteditionapprovedJune1,2021.PublishedJuly2021.Originallyapproved
For randomization, arrange the blocks and versions within each
in 1990. Last previous edition approved in 2016 as E1325–16. DOI: 10.1520/
blockindependentlyatrandom.Sinceeachletterintheaboveequations
E1325-21.
2 represents an integer, it is clear that only a restricted set of combina-
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
tions(t, k
...
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 − 16 E1325 − 21 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 appear
desirable.
1.2 The values stated in SI units are to be regarded as standard. No other units of measurement are included in this standard.
1.3 This international standard was developed in accordance with internationally recognized principles on standardization
established in the Decision on Principles for the Development of International Standards, Guides and Recommendations issued
by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
2. Referenced Documents
2
2.1 ASTM Standards:
E456 Terminology Relating to Quality and Statistics
E1488 Guide for Statistical Procedures to Use in Developing and Applying Test Methods
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.
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. = 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
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.
Current edition approved April 1, 2016June 1, 2021. Published April 2016July 2021. Originally approved in 1990. Last previous edition approved in 20152016 as
E1325 – 15.E1325 – 16. DOI: 10.1520/E1325-16.10.1520/E1325-21.
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’sstandard’s Document Summary page on the ASTM website.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
1
---------------------- Page: 1 ----------------------
E1325 − 21
n-k n – k
to form a 2 fractional factorial.
(3) There exist generalizations of the above to factorials having more than 2 levels.
analysis of variance (ANOVA), n—statistical models and associated procedures, in which the observed variance is partitioned
into components due to different explanatory variables.
analysis of variance table, n—a tabular summary of results from a regression model or an experimental design for the purpose
of evaluating effects of factors.
DISCUSSION—
The analysis of variance for a designed experiment lists factors and, for each factor, the degrees of freedom, sum of squares, mean square (sum of
squares divided by degrees of freedom), and may list test statistics (F-ratio) or expected values of mean squares as functions of components of variance.
Example: Analysis of variance of a randomized block design with k blocks and t treatments. The response for treatment i in block j is x , the block
ij
average is x¯ , the treatment average is x¯ , and the overall average is x¯ . σ is the component of variance due to blocks. τ is the treatment effect with
·j i· ·· b i
τ 50. A significance test of treatments is the mean square for treatments, divided by the mean square for error. See Table 1.
(
i
balanced incomplete blo
...
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