ASTM E1325-91(1997)
(Terminology)Standard Terminology Relating to Design of Experiments
Standard Terminology Relating to Design of Experiments
SCOPE
1.1 This standard includes those statistical items related to the area of design of experiments for which standard definitions appears desirable.
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Standards Content (Sample)
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Designation: E 1325 – 91 (Reapproved 1997) An American National Standard
Standard Terminology Relating to
Design of Experiments
This standard is issued under the fixed designation E 1325; 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 (e) indicates an editorial change since the last revision or reapproval.
factor appears the same number of times r in the experiment and that
1. Scope
the following relations hold true: bk 5 tr and r (k −1)5l(t − 1).
1.1 This standard includes those statistical items related to
For randomization, arrange the blocks and versions within each
the area of design of experiments for which standard defini-
block independently at random. Since each letter in the above equations
tions appears desirable.
represents an integer, it is clear that only a restricted set of combina-
tions (t, k, b, r, l) is possible for constructing balanced incomplete
2. Referenced Documents
block designs. For example, t 5 7, k 5 4, b 5 7, l5 2. Versions of
the principal factor:
2.1 ASTM Standards:
Block 1 1236
E 456 Terminology related to Quality and Statistics
2 2347
3 3451
3. Significance and Use
4 4562
5 5673
3.1 This standard is a subsidiary to Terminology E 456.
6 6714
3.2 It provides definitions, descriptions, discussion, and
7 7125
comparison of terms.
4. Terminology completely randomized design, n—a design in which the
treatments are assigned at random to the full set of experi-
aliases, n—in a fractional factorial design, two or more effects
mental units.
which are estimated by the same contrast and which,
therefore, cannot be estimated separately. DISCUSSION—No block factors are involved in a completely random-
ized design.
n
DISCUSSION—(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
completely randomized factorial design, n—a factorial ex-
replicate) or defining contrasts (for a fraction smaller than ⁄2) are
periment (including all replications) run in a completely
stated. The defining contrast is that effect (or effects), usually thought
randomized design.
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 composite design, n—a design developed specifically for
2 2 2
defining contrasts) and, using the conversion that A 5 B 5 C 5 I,
fitting second order response surfaces to study curvature,
the multiplication of the letters on both sides of the equation shows the
constructed by adding further selected treatments to those
aliases. In the example under fractional factorial design, I 5 ABCD.
n
obtained from a 2 factorial (or its fraction).
2 2 2
So that: A 5 A BCD 5 BCD, and AB 5 A B CD 5 CD.
( 2) With a large number of factors (and factorial treatment
DISCUSSION—If the coded levels of each factor are − 1 and + 1 in the
1 1
combinations) the size of the experiment can be reduced to ⁄4, ⁄8,orin n
2 factorial (see notation 2 under discussion for factorial experiment),
k n-k
general to ⁄2 to form a 2 fractional factorial.
the (2n + 1) additional combinations for a central composite design are
(3) There exist generalizations of the above to factorials having
(0, 0, ., 0), (6a, 0, 0, ., 0) 0, 6a, 0, ., 0) ., (0, 0, ., 6 a). The
more than 2 levels. 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 n
balanced incomplete block design (BIB), n—an incomplete
5 2, 3 and 4 the experiment requires, 9, 15, and 25 units respectively,
block design in which each block contains the same number
although additional replicate runs of the center point are usual, as
k of different versions from the t versions of a single
n
compared with 9, 27, and 81 in the 3 factorial. The reduction in
principal factor arranged so that every pair of versions
experiment size results in confounding, and thereby sacrificing, all
occurs together in the same number, l, of blocks from the b
information about curvature interactions. The value of a can be chosen
blocks. 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
DISCUSSION—The design implies that every version of the principal
form of response surface is not quadratic.
confounded factorial design, n—a factorial experiment in
This terminology is under the jurisdiction of ASTM Committee E-11 on Quality
and Statistics and is the direct responsibility of Subcommittee E11.60 on Terminol-
which only a fraction of the treatment combinations are run
ogy. The definitions in this standard were extracted from E 456 – 89c.
in each block and where the selection of the treatment
Current edition approved Feb. 22,1991. Published April 1991. Originally
combinations assigned to each block is arranged so that one
published as E 1325 – 90. Last previous edition E 1325 – 90a.
Annual Book of ASTM Standards, Vol 14.02. or more prescribed effects is(are) confounded with the block
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NOTICE: This standard has either been superseded and replaced by a new version or discontinued.
Contact ASTM International (www.astm.org) for the latest information.
E 1325
evidence that the response pattern shows curvature rather than a simple
effect(s), while the other effects remain free from confound-
linear trend. Here the average of A and A could be compared to A .
1 3 2
ing.
(If there is no curvature, A should fall on the line connecting A and
2 1
NOTE 1—All factor level combinations are included in the experiment.
A or, in other words, be equal to the average.) The following example
DISCUSSION—Example: Ina2 factorial with only room for 4
illustrates a regression type study of equally spaced continuous
treatments per block, the ABC interaction
variables. It is frequently more convenient to use integers rather than
(ABC: − (1) + a+b−ab+c−ac−bc+abc) can be sacrificed
fractions for contrast coefficients. In such a case, the coefficients for
through confounding with blocks without loss of any other effect if the
Contrast 2 would appear as (−1, + 2, − 1).
blocks include the following.
Response A A A
1 2 3
Block 1 Block 2 Contrast coefficients for question 1 −1 0 +1
Contrast 1 −A . +A
Treatment (1) a
1 3
1 1
Combination ab b Contrast coefficients for question 2 − ⁄2 +1 − ⁄2
1 1
(Code identification shown in discus- ac c Contrast 2 − ⁄2A +A − ⁄2A
1 2 3
sion under factorial experiment) bc abc
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
The treatments to be assigned to each block can be deter-
which, A , uses a new manufacturing technique while the other two, A and A use the
1 2 3
mined once the effect(s) to be confounded is(are) defined.
customary one. First, does vendor A with the new technique seem to differ from A
1 2
and A ? Second, do the two suppliers using the customary technique differ? Contrast A
Where only one term is to be confounded with blocks, as in this
2 and A . The pattern of contrast coefficients is similar to that for the previous prob-
example, those with a positive sign are assigned to one block
lem, though the interpretation of the results will differ.
and those with a negative sign to the other. There are
Response A A A
1 2 3
generalized rules for more complex situations. A check on all
Contrast coefficients for question 1 −2 +1 +1
Contrast 1 −2A +A +A
of the other effects (A, B, AB, etc.) will show the balance of the 1 2 3
Contrast coefficients for question 2 0 −1 +1
plus and minus signs in each block, thus eliminating any
Contrast 2 . −A +A
2 3
confounding with blocks for them.
The coefficients for a contrast may be selected arbitrarily
confounding, n—combining indistinguishably the main effect
provided the (a 5 0 condition is met. Questions of logical
i
of a factor or a differential effect between factors (interac-
interest from an experiment may be expressed as contrasts with
tions) with the effect of other factor(s), block factor(s) or
carefully selected coefficients. See the examples given in this
interaction(s).
discussion. As indicated in the examples, the response to each
treatment combination will have a set of coefficients associated
NOTE 2—Confounding is a useful technique that permits the effective
with it. The number of linearly independent contrasts in an
use of specified blocks in some experiment designs. This is accomplished
by deliberately preselecting certain effects or differential effects as being
experiment is equal to one less than the number of treatments.
of little interest, and arranging the design so that they are confounded with
Sometimes the term contrast is used only to refer to the pattern
block effects or other preselected principal factor or differential effects,
of the coefficients, but the usual meaning of this term is the
while keeping the other more important effects free from such complica-
algebraic sum of the responses multiplied by the appropriate
tions. Sometimes, however, confounding results from inadvertent changes
coefficients.
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
contrast analysis, n—a technique for estimating the param-
effectiveness of an experiment.
eters of a model and making hypothesis tests on preselected
contrast, n—a linear function of the observations for which linear combinations of the treatments (contrasts). See Table
the sum of the coefficients is zero. 1 and Table 2.
NOTE 3—With observations Y , Y , ., Y , the linear function a Y + a NOTE 4—Contrast analysis involves a systematic tabulation and analy-
1 2 n 1 1
2Y + . + a Y is a contrast if, and only if (a 5 0, where the a values are sis format usable for both simple and complex designs. When any set of
2 1 n i i
called the contrast coefficients. orthogonal contrasts is used, the procedure, as in the example, is
DISCUSSION—Example 1: A factor is applied at three levels and the straightforward. When terms are not orthogonal, the orthogonalization
results are represented by A ,A , A . If the levels are equally spaced, process to adjust for the common element in nonorthogonal contrast is
1 2 3
the first question it might be logical to ask is whether there is an overall also systematic and can be programmed.
linear trend. This could be done by comparing A and A , the extremes DISCUSSION—Example: Half-replicate of a 2 factorial experiment
1 3
of A in the experiment. A second question might be whether there is with factors A, B and C (X , X and X being quantitative, and factor D
1 2 3
TABLE 1 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 5 X +1 +1 −1 −1 −1 −1 +1 +1
1 2 12
AC+BD X X 5 X +1 −1 +1 −1 −1 +1 −1 +1 See Note 2
1 3 13
AD+BC X X 5 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 5 X X ) of the appropriate terms.
ij i i
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E 1325
TABLE 2 Contrast Analysis
Student’s t ratio
Contrast Divisor Regression coefficient
1 2 2
X Y X B 5 ( X Y )/ X
Source ( ( ( X Y )/s X ( (
ij i ij ( ij i ( ij j ij i ij
=
i i i i i i
2 2
X : Centre ( X Y ( X B 5 (( X Y)/( X
0 0 ((X Y)/s =( X 0 0
0 0
0 0
2 2
X : A + BCD ( X Y ( X B 5 (( X Y)/( X
1 1 1 ((X Y)/s =( X 1 1 1
2 2
X : B + ACD ( X Y ( X B 5 (( X Y)/( X
2 2 2 (( X Y)/s =( X 2 2 2
2 2
2 2
X : C + ABD ( X Y ( X B 5 (( X Y)/( X
((X Y)/s ( X
3 3 3 = 3 3 3
3 3
2 2
X : D + ABC (X Y ( X B 5 (( X Y)/( X
4 4 ((X Y)/s ( X 4 4
4 = 4
4 4
2 2
X : AB+CD ( X Y ( X B 5 (( X Y)/( X
12 12 12 ((X Y)/s =( X 12 12 12
2 2
X : AC+BD ( X Y
( X B 5 (( X Y)/( X
13 13 13 ((X Y)/s =( X 13 13 13
13 13
2 2
X : AD+BC ( X Y ( X B 5 (( X Y)/( X
((X Y)/s ( X
14 14 14 = 14 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). 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 5 B +B X +B
0 1 1i
2X +B X +e ). In a simple 2-level experiment such as this, the regression coefficient measures the half-effect of shifting a factor, say pH, between
3i 4 4i
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.
( X ) qualitative. Defining contrast I 5 + ABCD 5 X X X X (see NOTE 6—The principal theses of EVOP are that knowledge to improve
4 1 2 3 4
fractional factorial design and orthogonal design for derivation of the process should be obtained along with a product, and that de
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