ASTM E1325-21

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

• Status: Published
• Publication Date: 31-May-2021
• ICS: 01.040.19 - Testing (Vocabularies)
• ICS: 19.020 - Test conditions and procedures in general
• Technical Committee: E11 - Quality and Statistics
• Drafting Committee: E11.10 - Sampling / Statistics

Relations

• Refers: ASTM E1488-23 - Standard Guide for Statistical Procedures to Use in Developing and Applying Test Methods
• Effective Date: 01-Nov-2023
• Refers: ASTM E456-13a(2022)e1 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 01-Apr-2022
• Refers: ASTM E456-13A(2017)e3 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 01-Oct-2017
• Refers: ASTM E456-13A(2017)e1 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 01-Oct-2017
• Refers: ASTM E456-13ae1 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 15-Nov-2013
• Refers: ASTM E456-13ae2 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 15-Nov-2013
• Refers: ASTM E456-13a - Standard Terminology Relating to Quality and Statistics
• Effective Date: 15-Nov-2013
• Refers: ASTM E456-13ae3 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 15-Nov-2013
• Refers: ASTM E456-13 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 15-Aug-2013
• Refers: ASTM E1488-12e1 - Standard Guide for Statistical Procedures to Use in Developing and Applying Test Methods
• Effective Date: 01-Aug-2012
• Refers: ASTM E1488-12 - Standard Guide for Statistical Procedures to Use in Developing and Applying Test Methods
• Effective Date: 01-Aug-2012
• Refers: ASTM E456-12e1 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 01-May-2012
• Refers: ASTM E456-12 - Standard Terminology Relating to Quality and Statistics
• Effective Date: 01-May-2012
• Refers: ASTM E1488-09 - Standard Guide for Statistical Procedures to Use in Developing and Applying Test Methods
• Effective Date: 15-May-2009
• Refers: ASTM E1488-08a - Standard Guide for Statistical Procedures to Use in Developing and Applying Test Methods
• Effective Date: 01-Oct-2008

Overview

ASTM E1325-21, published by ASTM International, is the "Standard Terminology Relating to Design of Experiments." This standard provides authoritative definitions, descriptions, and comparisons of statistical terms essential in the field of experimental design. By standardizing terminology, ASTM E1325-21 supports clear communication, reproducibility, and best practices in experimental planning, execution, and analysis across industries.

This terminology standard complements and is subsidiary to ASTM E456, focusing specifically on concepts and terms in the design of experiments (DOE). The definitions reflect internationally recognized principles, supporting global harmonization and the comparability of scientific results.

Key Topics

ASTM E1325-21 covers fundamental terms used in the design and statistical analysis of experiments, including but not limited to:

• Design of Experiments (DOE): The planning and structuring of experiments, selection of factor levels, and arrangement for efficient, unbiased results.
• Factor and Level: An independent variable and its specific values or settings within an experiment.
• Experimental Unit: The smallest entity to which a treatment is independently applied.
• Completely Randomized Design: Assignment of treatments entirely at random to all experimental units.
• Block Factor / Blocking: Grouping similar units to reduce variance and isolate treatment effects.
• Analysis of Variance (ANOVA): Statistical method partitioning observed variance to evaluate factor effects.
• Contrast and Contrast Analysis: Linear combinations of treatment means used for hypothesis testing.
• Interaction: When the effect of one factor varies according to the levels of another factor.
• Fractional Factorial and Confounded Designs: Reducing the number of experimental runs by confounding or aliasing factors or interactions.
• Balanced Incomplete Block Design (BIB): Experimental arrangement to ensure evenness or balance when not all treatments can be tested together.

The standard also explains the rationale and meaning of concepts such as main effect, component of variance, nested and hierarchical experiments, evolutionary operation (EVOP), method of least squares, and more.

Applications

ASTM E1325-21 is essential wherever statistical design of experiments is practiced to ensure effective communication and consistent application of DOE principles. Practical uses include:

• Industrial Quality and Process Optimization: Manufacturers and engineers designing experiments to optimize product quality, reliability, or process efficiency.
• Scientific Research: Laboratory and field research across disciplines (chemistry, biology, materials science, pharmaceuticals) where complex experiments need reproducibility.
• Product Development: Teams evaluating multiple variables affecting product performance using factorial, fractional factorial, or response surface designs.
• Agricultural Studies: Field trials employing randomized block or Latin square designs to study treatments across different environments.
• Regulatory and Compliance Settings: Ensuring statistical terminology aligns in documents, protocols, and compliance reports, especially in international environments.

By using ASTM E1325-21, organizations can standardize experiment planning, compare results meaningfully, improve statistical robustness, and reduce ambiguity in technical communication.

Related Standards

• ASTM E456: Terminology Relating to Quality and Statistics; parent terminology standard for statistics and quality assurance.
• ASTM E1488: Guide for Statistical Procedures to Use in Developing and Applying Test Methods; offers more practical guidance on statistical analysis within test methods.
• ISO/IEC Standards: International norms on statistics, measurement, and laboratory practices.
• Industry-specific guidelines: Regulatory or sector-specific documents referencing DOE terminology and requirements.

Incorporating ASTM E1325-21 helps align experimental practices with global best practices and fosters consistency across industries and borders, supporting better data-driven decisions and innovations.

Standards Content (Sample)

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
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
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
2.1 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
replicate) or defining contrasts (for a fraction smaller than ⁄2) are
occurs together in the same number, λ, of blocks from the b
blocks.
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, b, r,λ)ispossibleforconstructingbalancedincompleteblock
contact ASTM Customer Service at [e-mail]. For Annual Book of ASTM
designs.Forexample, t=7, k=4, b=7,λ=2.Versionsoftheprincipal
Standards volume information, refer to the standard’s Document Summary page on
the ASTM website. factor:
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E1325 − 21
TABLE 1 Example Analysis of Variance Table
Source Degrees of Freedom Sum of Squares (SS) Mean Square (MS) Expected Mean Square
k
SS
Blocks
Blocks 2 2
k21 σ 1tσ
t sx¯ 2 x¯ d b
o
·j ··
k21
j51
t
SS k
Treatments
Treatments
2 2 2
t21
k sx¯ 2 x¯ d σ 1 oτ
o i· ·· i
i51 t21 t21
t k
SS
Error 2
Error
sk 2 1dst 2 1d σ
x 2 x¯ 2 x¯ 1 x¯
o o s d
ij i· ·j ··
k 2 1 t 2 1
i51j51 s ds d
t k
Total
x 2 x¯
s d
o o ij ··
i51j51
information about curvature interactions.The value of a can be chosen
Block1 1236
to make the coefficients in the quadratic polynomials as orthogonal as
2 2347
3 3451 possibletooneanotherortominimizethebiasthatiscreatedifthetrue
4 4562
form of response surface is not quadratic.
5 5673
6 6714
confounded factorial design, n—a factorial experiment in
7 7125
which only a fraction of the treatment combinations are run
in each block and where the selection of the treatment
block factor, n—a factor that indexes division of experimental
combinations assigned to each block is arranged so that one
units into disjoint subsets.
or more prescribed effects is(are) confounded with the block
effect(s), while the other effects remain free from confound-
DISCUSSION—Blocks are sets of similar experimental units intended
tomakevariabilitywithinblocksassmallaspossible,sothattreatment
ing.
effects will be more precisely estimated. The effect of a block factor is
NOTE 1—All factor level combinations are included in the experiment.
usually not of primary interest in the experiment. Components of
varianceattributabletoblocksmaybeofinterest.Theoriginoftheterm 3
DISCUSSION—Example: Ina2 factorial with only room for 4
“block” is in agricultural experiments, where a block is a contiguous
treatments per block, the ABC interaction
portion of a field divided into experimental units, “plots,” that are each
(ABC: − (1) + a+b−ab+c−ac−bc+abc) can be sacrificed
subjected to a treatment.
through confounding with blocks without loss of any other effect if the
blocks include the following:
completely randomized design, n—a design in which the
Block 1 Block 2
treatments are assigned at random to the full set of experi-
Treatment (1) a
mental units.
Combination ab b
DISCUSSION—No block factors are involved in a completely random-
(Code identification shown in discus- ac c
ized design.
sion under factorial experiment) bc abc
completely randomized factorial design, n—a factorial ex-
The treatments to be assigned to each block can be determined once the
effect(s) to be confounded is(are) defined. Where only one term is to be
periment (including all replications) run in a completely
confounded with blocks, as in this example, those with a positive sign are
randomized design.
assigned to one block and those with a negative sign to the other. There are
generalized rules for more complex situations. A check on all of the other
component of variance, n—apartofatotalvarianceidentified
effects (A, B, AB, etc.) will show the balance of the plus and minus signs in
with a specified source of variability. E1488 each block, thus eliminating any confounding with blocks for them.
confounding, n—combining indistinguishably the main effect
DISCUSSION—Components of variance are used here in the context of
experimental designs with random effects. If a response variable X is
of a factor or a differential effect between factors (interac-
the sum of two statistically independent variables Y and Z, then the
tions) with the effect of other factor(s), block factor(s) or
Variance (X) = Variance (Y) + Variance (Z). Variance (Y) and Variance
interaction(s).
(Z) are then components ofVariance (X), or variance components of X.
NOTE 2—Confounding is a useful technique that permits the effective
composite design, n—a design developed specifically for
use of specified blocks in some experiment designs.This is accomplished
fitting second order response surfaces to study curvature,
by deliberately preselecting certain effects or differential effects as being
constructed by adding further selected treatments to those oflittleinterest,andarrangingthedesignsothattheyareconfoundedwith
n
block effects or other preselected principal factor or differential effects,
obtained from a 2 factorial (or its fraction).
while keeping the other more important effects free from such complica-
DISCUSSION—If the coded levels of each factor are−1 and+1 in the
n tions.Sometimes,however,confoundingresultsfrominadvertentchanges
2 factorial(seenotation2underdiscussionfor factorial experiment),
to a design during the running of an experiment or from incomplete
the (2n + 1) additional combinations for a central composite design are
planningofthedesign,anditservestodiminish,oreventoinvalidate,the
(0, 0, ., 0), (6a, 0, 0, ., 0) 0, 6a, 0, ., 0) ., (0, 0, ., 6 a). The
effectiveness of an experiment.
n
minimum total number of treatments to be tested is (2 +2n+1) for a
n
2 factorial. Frequently more than one center point will be run. For
contrast, n—alinearfunctionoftheobservationsforwhichthe
n=2, 3 and 4 the experiment requires, 9, 15, and 25 units respectively,
sum of the coefficients is zero.
although additional replicate runs of the center point are usual, as
n
compared with 9, 27, and 81 in the 3 factorial. The reduction in
NOTE 3—With observations Y , Y , ., Y , the linear function
1 2 n
experiment size results in confounding, and thereby sacrificing, all a Y + a Y +.+ a Y is a contrast if, and only if ∑a =0, where the a
1 1 2 2 1 n i i
E1325 − 21
values are called the contrast coefficients. orthogonal contrasts is used, the procedure, as in the example, is
straightforward. When terms are not orthogonal, the orthogonalization
DISCUSSION—Example 1: A factor is applied at three levels and the
process to adjust for the common element in nonorthogonal contrast is
results are represented by A , A , A . If the levels are equally spaced,
1 2 3
also systematic and can be programmed.
thefirstquestionitmightbelogicaltoaskiswhetherthereisanoverall
DISCUSSION—Example: Half-replicate of a 2 factorial experiment
linear trend.This could be done by comparing A and A , the extremes
1 3
with factors A, B and C (X , X , and X being quantitative, and factor
of A in the experiment. A second question might be whether there is
1 2 3
D (X ) qualitative. Defining contrast I= +ABCD=X X X X (see
evidencethattheresponsepatternshowscurvatureratherthanasimple
4 1 2 3 4
fractional factorial designand orthogonal contrastsforderivationof
linear trend. Here the average of A and A could be compared to A .
1 3 2
the contrast coeffıcients).
(If there is no curvature, A should fall on the line connecting A and
2 1
A or, in other words, be equal to the average.) The following example
design of experiments, n—the arrangement in which an
illustrates a regression type study of equally spaced continuous
experimental program is to be conducted, and the selection
variables. It is frequently more convenient to use integers rather than
of the levels (versions) of one or more factors or factor
fractions for contrast coefficients. In such a case, the coefficients for
Contrast 2 would appear as (−1,+2,−1). combinations to be included in the experiment. Synonyms
include experiment design and experimental design.
Response A A A
1 2 3
Contrast coefficients for question 1 −1 0 +1
DISCUSSION—The purpose of designing an experiment is to provide
Contrast 1 −A . + A
1 3
the most efficient and economical methods of reaching valid and
1 1
Contrast coefficients for question 2 − ⁄2 +1 − ⁄2
relevant conclusions from the experiment. The selection of an appro-
1 1
Contrast 2 − ⁄2 A + A − ⁄2 A
1 2 3
priate design for any experiment is a function of many considerations
Example 2: Another example dealing with discrete versions of a
such as the type of questions to be answered, the degree of generality
factor might lead to a different pair of questions. Suppose there are
to be attached to the conclusions, the magnitude of the effect for which
three sources of supply, one of which, A , uses a new manufacturing
a high probability of detection (power) is desired, the homogeneity of
technique while the other two, A and A use the customary one.
2 3
the experimental units and the cost of performing the experiment. A
First, does vendor A with the new technique seem to differ from A
1 2
properly designed experiment will permit relatively simple statistical
and A ? Second, do the two suppliers using the customary technique
interpretation of the results, which may not be possible otherwise. The
differ? Contrast A and A . The pattern of contrast coefficients is
2 3
arrangement includes the randomization procedure for allocating
similar to that for the previous problem, though the interpretation of
treatments to experimental units.
the results will differ.
experimental design, n—see design of experiments.
Response A A A
1 2 3
Contrast coefficients for question 1 −2 +1 +1
experimental unit, n—a portion of the experiment space to
Contrast 1 −2A +A +A
1 2 3
which a treatment is applied or assigned in the experiment.
Contrast coefficients for question 2 0 −1 +1
Contrast 2 . − A + A
2 3
NOTE 5—The unit may be a patient in a hospital, a group of animals, a
The coefficients for a contrast may be selected arbitrarily provided
production batch, a section of a compartmented tray, etc.
the ∑a =0 condition is met. Questions of logical interest from an
i
experiment space, n—the materials, equipment, environmen-
experiment may be expressed as contrasts with carefully selected
tal conditions and so forth that are available for conducting
coefficients. See the examples given in this discussion. As indicated
in the examples, the response to each treatment combination will
an experiment.
have a set of coefficients associated with it. The number of linearly
DISCUSSION—That portion of the experiment space restricted to the
independent contrasts in an experiment is equal to one less than the
range of levels (versions) of the factors to be studied in the experiment
number of treatments. Sometimes the term contrast is used only to
is sometimes called the factor space. Some elements of the experiment
refer to the pattern of the coefficients, but the usual meaning of this
spacemaybeidentifiedwithblocksandbeconsideredasblockfactors.
term is the algebraic sum of the responses multiplied by the appro-
evolutionary operation (EVOP), n—a sequential form of
priate coefficients.
experimentation conducted in production facilities during
contrast analysis, n—a technique for estimating the param-
regular production.
eters of a model and making hypothesis tests on preselected
NOTE 6—The principal theses of EVOPare that knowledge to improve
linear combinations of the treatments (contrasts). See Table
the process should be obtained along with a product, and that designed
2 and Table 3.
experiments using relatively small shifts in factor levels (within produc-
NOTE 4—Contrast analysis involves a systematic tabulation and analy- tion tolerances) can yield this knowledge at minimum cost. The range of
sis format usable for both simple and complex designs. When any set of variation of the factors for any one EVOP experiment is usually quite
TABLE 2 Contrast Coefficient
Source Treatments (1) ab ac bc ad bd cd abcd
Centre X +1 +1 +1 +1 +1 +1 +1 +1 See Note 1
A(+ BCD): pH (8.0; 9.0) X −1 +1 +1 −1 +1 −1 −1 +1
3 3
B(+ ACD): SO (10 cm ;16cm ) X −1 +1 −1 +1 −1 +1 −1 +1
4 2
C(+ ABD): Temperature (120°C; 150°C) X −1 −1 +1 +1 −1 −1 +1 +1
D(+ ABC): Factory (P; Q) X −1 −1 −1 −1 +1 +1 +1 +1
AB + CD X X = X +1 +1 −1 −1 −1 −1 +1 +1
1 2 12
AC + BD X X = X +1 −1 +1 −1 −1 +1 −1 +1 See Note 2
1 3 13
AD + BC X X = X +1 −1 −1 +1 +1 −1 −1 +1
1 4 14
NOTE 1—The center is not a constant (∑X ≠ 0) but is convenient in the contrast analysis calculations to treat it as one.
i
NOTE2—Oncethecontrastcoefficientsofthemaineffects(X ,X ,X ,and X )arefilledin,thecoefficientsforallinteractionandothersecondorhigher
1 2 3 4
order effects can be derived as products (X = X X) of the appropriate terms.
ij i i
E1325 − 21
TABLE 3 Contrast Analysis
Contrast Divisor Student’s t ratio Regression coefficient
Source
1 2 2 2
X Y X X Y ⁄s X B5 X Y ⁄ X
o o o o o o
ij i ij ij i ij j ij i ij
s d s d
i i i i i i
œ
2 2 2
X :Centre X Y X X Y ⁄s X B 5 X Y / X
o o so d o so d o
0 0 0 0 œ 0 0 0 0
2 2 2
X :A1BCD oX Y o X so X Yd⁄s o X B 5so X Yd/o X
1 1 1 1 œ 1 1 1 1
2 2 2
X :B1ACD oX Y o X so X Yd⁄s o X B 5so X Yd/o X
2 2 2 2 2 2 2 2
œ
2 2 2
X :C1ABD X Y X s X Yd⁄s X B 5s X Yd/ X
o o o o o o
3 3 3 3 œ 3 3 3 3
2 2 2
X :D1ABC X Y X s X Yd⁄s X B 5s X Yd/ X
o o o o o o
4 4 4 4 œ 4 4 4 4
2 2 2
X :AB1CD X Y X s X Yd⁄s X B 5s X Yd/ X
o o o o o o
12 12 12 12 œ 12 12 12 12
2 2 2
X :AC1BD X Y X s X Yd⁄s X B 5s X Yd/ X
o o o o o o
13 13 13 13 œ 13 13 13 13
2 2 2
X :AD1BC X Y X X Y ⁄s X B 5 X Y / X
o o so d o so d o
14 14 14 14 œ 14 14 14 14
NOTE 1—The notation for contrast analysis usually uses Y to indicate the response variable and X the predictor variables.
NOTE 2—The measure of experimental error, s, can be obtained in various ways. If the experiment is replicated, s is the square root of the pooled
variancesofthepairsforeachtreatmentcombination.(Eachrowof Xvalueswouldbeexpandedtoaccountfortheadditionalobservationsinthecontrast
analysiscomputations.)Ifsomeeffectswerefelttobepseudo-replicates(example,nointeractionswerelogical)multiplyingthecontrastbytheregression
coefficient of these terms forms a sum of squares (as in analysis of variance) and these would be summed and divided by the number of terms involved
to give s . Also, in many experiments, past experience may already provide an estimate of this error. Assumed model:
Y=B +B X +B X +B X +e. In a simple 2-level experiment such as this, the regression coefficient measures the half-effect of shifting a factor,
0 1 1i 2 3i 4 4i
say pH, between its low and high level, or the effect of shifting from a center level to the high level. In general, substitution of the appropriate contrast
coefficients for the X terms in the model will permit any desired comparisons. The difference between quantitative and qualitative factors lies in the
interpretation. Since a unit of X represents a pH shift of 0.5, there is a meaningful translation into physical units. On the other hand, the units of the
qualitativevariable(factories)havenosignificanceotherthanforidentificationandinthesubstitutionprocesstoobtainestimatesoftheaverageresponse
values.
small in order to avoid making out-of-tolerance products, which may
require considerable replication, in order to be able to clearly detect the
Example (illustrating contrast)—Two-factor, two-level factorial 2 with factors A
effect of small changes. and B: A =[a−(1)]+[ab−b]. This is the contrast of A at the low level of B plus
the contrast of A at the high level of B. B =[b−(1)]+[ab−a].This is the contrast
factor, n—independent variable in an experimental design.
of B at the low level of A plus the contrast of B at the high level of A: AB
=[ab−b]−[a−(1)]=[ab−a]−[b−(1)]. This is the contrast of the contrasts of
DISCUSSION—Factors can include controllable factors that are of
A at the high level of B and the low level of B or the contrast of the contrasts of B
at the high level of A and at the low level of A.
interest for the experiment, block factors that are created to enhance
Eachcontrastcanbederivedfromthedevelopmentofa symbolic productoftwo
precision of the factors of interest, and uncontrolled factors that might
factors, these factors being of the form (a ± 1), (b ± 1), using − 1 when the capital
be measured in the experiment. Design of an experiment consists of
letter (A, B) is included in the contrast and + 1 when it is not.
allocatinglevelsofeachcontrollableexperimentalfactortoexperimen-
Example:
tal units.
A:(a−1)(b+1)
B:(a+1)(b−1)
n
2 factorial experiment, n—a factorial experiment in which n
AB:(a−1)(b−1)
factors are studied, each of them in two levels (versions).
n
These expressions are usually written in a standard order, in this case:
DISCUSSION—The2 factorialisaspecialcaseofthegeneralfactorial.
(See factorial experiment (general).) A popular code is to indicate a
A:−(1)+ a − b + ab
small letter when a factor is at its high level, and omit the letter when
B:−(1)− a + b + ab
it is at its low level. When factors are at their low level the code is (1).
AB:(1)− a − b + ab
Example (illustrating the discussion)—A2 factorial with factors A,
Note that the coefficient of each treatment combination inAB ( + 1 or − 1) is the
B, and C:
n
product of the corresponding coefficients inAand B.This property is general in 2
Level n
factorial experiments. After grouping, the A term 2 represents the effect of A
Factor A Low High Low High Low High Low High
averagedoverthetwolevelsofB,thatis,amaineffectoraverageeffect.Similarly,
Factor B Low Low High High Low Low High High
B represents the average effect of B over both levels ofA. TheAB term contrasts
Factor C Low Low Low Low High High High High
the effect ofAat the high and the low levels of B (or the effect of B at the high and
Code (1) a b ab c ac bc abc
low levels of A), that is an interaction or differential effect.
This example is, of course, the simplest case, but it illustrates the basic
This type of identification has advantages for defining blocks, confounding and
principles. The contrasts may appear more complex as additional factors are
aliasing. See confounded factorial design and fractional factorial design.
introduced.
Factorial experiments regardless of the form of analysis used, essentially
involve contrasting the various levels (versions) of the factors.
E1325 − 21
contrasts, but this entails a loss of information.
factorial experiment (general), n—in general, an experiment
Fractional factorial designs are often used very effectively in
in which all possible treatments formed from two or more
screening tests to determine which factor or factors are large contribu-
fa
...

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
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
2.1 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
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
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.
For referenced ASTM standards, visit the ASTM website, www.astm.org, or contact ASTM Customer Service at [e-mail]. 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
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 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(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, = 7, k = 4, = 4, b = 7, λ = 2. = 7, λ = 2. Versions of the principal factor:
Block 1 1 2 3 6
2 2 3 4 7
3 3 4 5 1
4 4 5 6 2
5 5 6 7 3
6 6 7 1 4
7 7 1 2 5
block factor, n—a factor that indexes division of experimental units into disjoint subsets.
DISCUSSION—
Blocks are sets of similar experimental units intended to make variability within blocks as small as possible, so that treatment effects will be more
precisely estimated. The effect of a block factor is usually not of primary interest in the experiment. Components of variance attributable to blocks
may be of interest. The origin of the term “block” is in agricultural experiments, where a block is a contiguous portion of a field divided into
experimental units, “plots,” that are each subjected to a treatment.
completely randomized design, n—a design in which the treatments are assigned at random to the full set of experimental units.
TABLE 1 Example Analysis of Variance Table
Source Degrees of Freedom Sum of Squares (SS) Mean Square (MS) Expected Mean Square
k
SS
Blocks 2 2
Blocks
k21 σ 1tσ
t x¯ 2 x¯ b
o s d
·j ··
j51 k21
t
SS k
Treatments Treatments
t21 2 2 2
k x¯ 2 x¯ σ 1 τ
o s d o
i· ·· i
t21 t21
i51
t k
SS
Error 2
Error
sk 2 1dst 2 1d σ
sx 2 x¯ 2 x¯ 1 x¯ d
oo
ij i· ·j ··
i51j51 sk 2 1dst 2 1d
t k
Total
sx 2 x¯ d
oo
ij ··
i51j51
E1325 − 21
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.
component of variance, n—a part of a total variance identified with a specified source of variability. E1488
DISCUSSION—
Components of variance are used here in the context of experimental designs with random effects. If a response variable X is the sum of two statistically
independent variables Y and Z, then the Variance (X) = Variance (Y) + Variance (Z). Variance (Y) and Variance (Z) are then components of Variance
(X), or variance components of X.
composite design, n—a design developed specifically for fitting second order response surfaces to study curvature, constructed
n
by adding further selected treatments to those obtained from a 2 factorial (or its fraction).
DISCUSSION—
n
If the coded levels of each factor are − 1 and + 1are −1 and +1 in the 2 factorial (see notation 2 under discussion for factorial experiment), the (2n
+ 1) + 1) additional combinations for a central composite design are (0, 0, ., 0), (6a, 0, 0, ., 0) 0, 6a, 0, ., 0) ., (0, 0, ., 6 a). The minimum
n n
total number of treatments to be tested is (2 + 2+ 2n + 1) for a 2 factorial. 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, as compared with 9, 27, and
n
81 in the 3 factorial. The reduction in experiment size results in confounding, and thereby sacrificing, all information about curvature interactions.
The value of a can be chosen to make the coefficients in the quadratic polynomials as orthogonal as possible to one another or to minimize the bias
that is created if the true form of response surface is not quadratic.
confounded factorial design, n—a factorial experiment in which only a fraction of the treatment combinations are run in each
block and where the selection of the treatment combinations assigned to each block is arranged so that one or more prescribed
effects is(are) confounded with the block effect(s), while the other effects remain free from confounding.
NOTE 1—All factor level combinations are included in the experiment.
DISCUSSION—
Example: In a 2 factorial with only room for 4 treatments per block, the ABC interaction (ABC: − (1) + a + b − ab + c − ac − bc + abc) can be
sacrificed through confounding with blocks without loss of any other effect if the blocks include the following:
Block 1 Block 2
Treatment (1) a
Combination ab b
(Code identification shown in discus- ac c
sion under factorial experiment) bc abc
Block 1 Block 2
Treatment (1) a
Combination ab b
(Code identification shown in discus- ac c
sion under factorial experiment) bc abc
The treatments to be assigned to each block can be determined once the effect(s) to be confounded is(are) defined. Where only one term is to be confounded with
blocks, as in this example, those with a positive sign are assigned to one block and those with 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 for them.
The treatments to be assigned to each block can be determined once the effect(s) to be confounded is(are) defined. Where
only one term is to be confounded with blocks, as in this example, those with a positive sign are assigned to one block and those
with 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 for
them.
confounding, n—combining indistinguishably the main effect of a factor or a differential effect between factors (interactions)
with the effect of other factor(s), block factor(s) or interaction(s).
NOTE 2—Confounding is a useful technique that permits the effective use of specified blocks in some experiment designs. This is accomplished by
E1325 − 21
deliberately preselecting certain effects or differential effects as being of little interest, and arranging the design so that they are confounded with block
effects or other preselected principal factor or differential effects, while keeping the other more important effects 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 the
design, and it serves to diminish, or even to invalidate, the effectiveness of an experiment.
contrast, n—a linear function of the observations for which the sum of the coefficients is zero.
NOTE 3—With observations Y , Y , ., Y , the linear function a Y + a Y + . + a Y is a contrast if, and only if ∑a = 0, where the a values are called
1 2 n 1 1 2 2 1 n i i
the contrast coefficients.
DISCUSSION—
Example 1: A factor is applied at three levels and the results are represented by A ,A , A . If the levels are equally spaced, the first question it might
1 2 3
be logical to ask is whether there is an overall linear trend. This could be done by comparing A and A , the extremes of A in the experiment. A second
1 3
question might be whether there is evidence that the response pattern shows curvature rather than a simple linear trend. Here the average of A and
A could be compared to A . (If there is no curvature, A should fall on the line connecting A and A or, in other words, be equal to the average.)
3 2 2 1 3
The following example illustrates a regression type study of equally spaced continuous variables. It is frequently more convenient to use integers rather
than fractions for contrast coefficients. In such a case, the coefficients for Contrast 2 would appear as (−1, + 2, − 1).(−1, +2, −1).
Response A A A
1 2 3
Contrast coefficients for question 1 −1 0 +1
Contrast 1 −A . + A
1 3
1 1
Contrast coefficients for question 2 − ⁄2 +1 − ⁄2
1 1
Contrast 2 − ⁄2 A + A − ⁄2A
1 2 3
Example 2: Another example dealing with discrete versions of a factor might lead to a different pair of questions. Suppose there are three sources
of supply, one of which, A , uses a new manufacturing technique while the other two, A and A use the customary one. First, does vendor A with
1 2 3 1
the new technique seem to differ from A and A ? Second, do the two suppliers using the customary technique differ? Contrast A and A . The pattern
2 3 2 3
of contrast coefficients is similar to that for the previous problem, though the interpretation of the results will differ.
Response A A A
1 2 3
Contrast coefficients for question 1 −1 0 +1
Contrast 1 −A . + A
1 3
1 1
Contrast coefficients for question 2 − ⁄2 +1 − ⁄2
1 1
Contrast 2 − ⁄2 A + A − ⁄2A
1 2 3
Example 2: Another example dealing with discrete versions of a factor might lead to a different pair of questions. Suppose there are three sources
of supply, one of which, A , uses a new manufacturing technique while the other two, A and A use the customary one. First, does vendor A
1 2 3 1
with the new technique seem to differ from A and A ? Second, do the two suppliers using the customary technique differ? Contrast A and A .
2 3 2 3
The pattern of contrast coefficients is similar to that for the previous problem, though the interpretation of the results will differ.
Response A A A
1 2 3
Contrast coefficients for question 1 −2 +1 +1
Contrast 1 −2A +A +A
1 2 3
Contrast coefficients for question 2 0 −1 +1
Contrast 2 . − A + A
2 3
Response A A A
1 2 3
Contrast coefficients for question 1 −2 +1 +1
Contrast 1 −2A +A +A
1 2 3
Contrast coefficients for question 2 0 −1 +1
Contrast 2 . − A + A
2 3
The coefficients for a contrast may be selected arbitrarily provided the ^a = 0 condition is met. Questions of logical interest from an experiment
i
may be expressed as contrasts with carefully selected coefficients. See the examples given in this discussion. As indicated in the examples, the
response to each treatment combination will have a set of coefficients associated with it. The number of linearly independent contrasts in an
experiment is equal to 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 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 the ∑a = 0 condition is met. Questions of logical interest from an experiment may
i
be expressed as contrasts with carefully selected coefficients. See the examples given in this discussion. As indicated in the examples, the response to
each treatment combination will have a set of coefficients associated with it. The number of linearly independent contrasts in an experiment is equal to
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 meaning of this
term is the algebraic sum of the responses multiplied by the appropriate coefficients.
contrast analysis, n—a technique for estimating the parameters of a model and making hypothesis tests on preselected linear
combinations of the treatments (contrasts). See Table 12 and Table 23.
E1325 − 21
TABLE 12 Contrast Coefficient
Source Treatments (1) ab ac bc ad bd cd abcd
Centre X +1 +1 +1 +1 +1 +1 +1 +1 See Note 1
A(+BCD): pH (8.0; 9.0) X −1 +1 +1 −1 +1 −1 −1 +1
A( + BCD): pH (8.0; 9.0) X −1 +1 +1 −1 +1 −1 −1 +1
3 3
B(+ ACD): SO (10 cm ; 16 cm ) X −1 +1 −1 +1 −1 +1 −1 +1
4 2
3 3
B( + ACD): SO (10 cm ; 16 cm ) X −1 +1 −1 +1 −1 +1 −1 +1
4 2
C(+ ABD): Temperature (120°C; 150°C) X −1 −1 +1 +1 −1 −1 +1 +1
C( + ABD): Temperature (120°C; 150°C) X −1 −1 +1 +1 −1 −1 +1 +1
D(+ABC): Factory (P; Q) X −1 −1 −1 −1 +1 +1 +1 +1
D( + ABC): Factory (P; Q) X −1 −1 −1 −1 +1 +1 +1 +1
AB + CD X X = X +1 +1 −1 −1 −1 −1 +1 +1
1 2 12
AB + CD X X = X +1 +1 −1 −1 −1 −1 +1 +1
1 2 12
AC + BD X X = X +1 −1 +1 −1 −1 +1 −1 +1 See Note 2
1 3 13
AC + BD X X = X +1 −1 +1 −1 −1 +1 −1 +1 See Note 2
1 3 13
AD + BC X X = X +1 −1 −1 +1 +1 −1 −1 +1
1 4 14
AD + BC X X = X +1 −1 −1 +1 +1 −1 −1 +1
1 4 14
NOTE 1—The center is not a constant (∑X ≠ 0) but is convenient in the contrast analysis calculations to treat it as one.
i
NOTE 2—Once the contrast coefficients of the main effects (X , X , X , and X ) are filled in, the coefficients for all interaction and other second or higher
1 2 3 4
order effects can be derived as products (X = X X ) of the appropriate terms.
ij i i
NOTE 4—Contrast analysis involves a systematic tabulation and analysis format usable for both simple and complex designs. When any set of orthogonal
contrasts is used, the procedure, as in the example, is straightforward. When terms are not orthogonal, the orthogonalization process to adjust for the
common element in nonorthogonal contrast is also systematic and can be programmed.
DISCUSSION—
Example: Half-replicate of a 2 factorial experiment with factors A, B and C (X , X , and X being quantitative, and factor D (X ) qualitative. Defining
1 2 3 4
contrast I = + ABCD = X X X X (see fractional factorial design and orthogonal contrasts for derivation of the contrast coeffıcients).
1 2 3 4
design of experiments, n—the arrangement in which an experimental program is to be conducted, and the selection of the levels
(versions) of one or more factors or factor combinations to be included in the experiment. Synonyms include experiment design
and experimental design.
DISCUSSION—
The purpose of designing an experiment is to provide the most efficient and economical methods of reaching valid and relevant conclusions from the
experiment. The selection of an appropriate design for any experiment is a function of many considerations such as the type of questions to be
answered, the degree of generality to be attached to the conclusions, the magnitude of the effect for which a high probability of detection (power) is
desired, the homogeneity of the experimental units and the cost of performing the experiment. A properly designed experiment will permit relatively
simple statistical interpretation of the results, which may not be possible otherwise. The arrangement includes the randomization procedure for
allocating treatments to experimental units.
experimental design, n—see design of experiments.
experimental unit, n—a portion of the experiment space to which a treatment is applied or assigned in the experiment.
NOTE 5—The unit may be a patient in a hospital, a group of animals, a production batch, a section of a compartmented tray, etc.
experiment space, n—the materials, equipment, environmental conditions and so forth that are available for conducting an
experiment.
DISCUSSION—
That portion of the experiment space restricted to the range of levels (versions) of the factors to be studied in the experiment is sometimes called the
factor space. Some elements of the experiment space may be identified with blocks and be considered as block factors.
evolutionary operation (EVOP), n—a sequential form of experimentation conducted in production facilities during regular
production.
NOTE 6—The principal theses of EVOP are that knowledge to improve the process should be obtained along with a product, and that designed experiments
E1325 − 21
TABLE 2 Contrast Analysis
Regression coefficient
Contrast Divisor Student’s t ratio
B = ( X Y ) / X
o o
Source j ij I ij
2 2
i i
X Y X ( X Y )/s X
o o o o
ij i ij ij i ij œ
i i i i
œ
2 2 2
X : Centre ^ X Y
0 0 X (^ X Y)/s X B 5s X Yd/ X
o 0 o o o
0 œ 0 0 0 0
2 2 2
X : A + BCD ^ X Y
X (^X Y)/s X B 5 X Y / X
1 1 o o so d o
1 1 œ 1 1 1 1
2 2 2
X : B + ACD ^ X Y
2 2 X (^X Y)/s X B 5s X Yd/ X
o 2 o o o
2 œ 2 2 2 2
2 2 2
X : C + ABD ^ X Y
X (^ X Y)/s X B 5 X Y / X
3 3 o o so d o
3 3 œ 3 3 3 3
2 2 2
X : D + ABC ^X Y
4 4 X (^X Y)/s X B 5s X Yd/ X
o 4 o o o
4 œ 4 4 4 4
2 2 2
X : AB + CD ^ X Y
12 12 o X (^X Y)/s o X B 5so X Yd/o X
12 12 œ 12 12 12 12
2 2 2
X : AC + BD ^ X Y
X (^X Y)/s X B 5s X Yd/ X
13 13 o o o o
13 13 œ 13 13 13 13
2 2 2
X : AD + BC ^ X Y
14 14 o X (^X Y)/s o X B 5so X Yd/o X
14 14 14 14 14 14
œ
TABLE 3 Contrast Analysis
Contrast Divisor Student’s t ratio Regression coefficient
Source
1 2 2 2
X Y X X Y ⁄s X B 5 X Y ⁄ X
oo o o o o o o
ij i ij œ ij i ij j ij i ij
s d s d
i i i i i i i i
œ
2 2 2
X :Centre X Y X X Y ⁄s X B 5 X Y / X
o o so d o so d o
0 0 0 0 œ 0 0 0 0
2 2 2
X :A1BCD oX Y o X so X Yd⁄s o X B 5so X Yd/o X
1 1 1 1 œ 1 1 1 1
2 2 2
X :B1ACD oX Y o X so X Yd⁄s o X B 5so X Yd/o X
2 2 2 2 œ 2 2 2 2
2 2 2
X :C1ABD X Y X s X Yd⁄s X B 5s X Yd/ X
o o o o o o
3 3 3 3 œ 3 3 3 3
2 2 2
X :D1ABC X Y X s X Yd⁄s X B 5s X Yd/ X
o o o o o o
4 4 4 4 œ 4 4 4 4
2 2 2
X :AB1CD X Y X s X Yd⁄s X B 5s X Yd/ X
o o o o o o
12 12 12 12 œ 12 12 12 12
2 2 2
X :AC1BD X Y X X Y ⁄s X B 5 X Y / X
o o so d o so d o
13 13 13 13 œ 13 13 13 13
2 2 2
X :AD1BC X Y X X Y ⁄s X B 5 X Y / X
o o so d o so d o
14 14 14 14 œ 14 14 14 14
NOTE 1—The notation for contrast analysis usually uses Y to indicate the response variable and X the predictor variables.
NOTE 2—The measure of experimental error, s, can be obtained in various ways. If the experiment is replicated, s is the square root of the pooled variances
of the pairs for each treatment combination. (Each row of X values would be expanded to account for the additional observations in the contrast analysis
computations).computations.) If some effects were felt to be pseudo-replicates (example, no interactions were logical) multiplying the contrast by the
regression coefficient of these terms forms a sum of squares (as in analysis of variance) and these would be summed and divided by the number of terms
involved to give s . Also, in many experiments, past experience may already provide an estimate of this error. Assumed model:
Y = B + B X + B X + B X + e + e.). In a simple 2-level experiment such as this, the regression coefficient measures the half-effect of shifting a
0 1 1i 2 3i 4 4i
factor, say pH, between its low and high level, or the effect of shifting from a center level to the high level. In general, substitution of the appropriate
contrast coefficients for the X terms in the model will permit any desired comparisons. The difference between quantitative and qualitative factors lies
in the interpretation. Since a unit of X represents a pH shift of 0.5, there is a meaningful translation into physical units. On the other hand, the units of
the qualitative variable (factories) have no significance other than for identification and in the substitution process to obtain estimates of the average
response values.
using relatively small shifts in factor levels (within production tolerances) can yield this knowledge at minimum cost. The range of variation of the factors
for any one EVOP experiment is usually quite small in order to avoid making out-of-tolerance products, which may require considerable replication, in
order to be able to clearly detect the effect of small changes.
factor, n—independent variable in an experimental design.
DISCUSSION—
Factors can include controllable factors that are of interest for the experiment, block factors that are created to enhance precision of the factors of
interest, and uncontrolled factors that might be measured in the experiment. Design of an experiment consists of allocating levels of each controllable
experimental factor to experimental units.
n
2 factorial experiment,n—a factorial experiment in which n factors are studied, each of them in two levels (versions).
DISCUSSION—
n
The 2 factorial is a special case of the general factorial. (See factorial experiment (general).) A popular code is to indicate a small letter when a
factor is at its high level, and omit the letter when it is at its low level. When factors are at their low level the code is (1).
E1325 − 21
Example (illustrating the discussion)—)—AA2 2 factorial with factors A, B, and C:
Level
Factor A Low High Low High Low High Low High
Factor B Low Low High High Low Low High High
Factor C Low Low Low Low High High High High
Code (1) a b ab c ac bc abc
This type of identification has advantages for defining blocks, confounding and aliasing. See confounded factorial design and fractional factorial design.
Factorial experiments regardless of the form of analysis used, essentially involve contrasting the various levels (versions) of the factors.
Example (illustrating contrast)—Two-factor, two-level factorial 2 with factors A and B: A = [a − (1)] + [ab − b]. This is the contrast of A at the low level of B plus the
contrast of A at the high level of B. B = [b − (1)] + [ab − a]. This is the contrast of B at the low level of A plus the contrast of B at the high level of A: AB = [ab − b] − [a
− (1)] = [ab − a ] − [b − (1)]. This is the contrast of the contrasts of A at the high level of B and the low level of B or the contrast of the contrasts of B at the high level of
A and at the low level of A.
Each contrast can be derived from the development of a symbolic product of two factors, these factors being of the form (a ± 1), (b ± 1), using − 1 when the capital letter
(A, B) is included in the contrast and + 1 when it is not.
Example:
A: (a − 1)(b + 1)
B: (a + 1)(b − 1)
AB: (a − 1)(b − 1)
These expressions are usually written in a standard order, in this case:
A: −(1) + a − b + ab
B: −(1) − a + b + ab
AB: (1) − a − b + ab
n
Note that the coefficient of each treatment combination in AB ( + 1 or − 1) is the product of the corresponding coefficients in A and B. This property is general in 2 factorial
n
experiments. After grouping, the A term 2 represents the effect of A averaged over the two levels of B, that is, a main effect or average effect. Similarly, B represents the
average effect of B over both levels of A. The AB ter
...