ISO 3534-3:2013 統計—用語, 語彙と記号—パート3：実験計画 | ページ 6

3.1.1

experiment

purposive investigation of a system through selective adjustment of controllable conditions and allocation of resources

Note 1 to entry: A system is an interacting combination of elements, viewed in relation to function. Deliberate alterations or adjustments are made to a system in order to improve or to understand it. In other words, an experiment is a systematic and objective means of getting unambiguous and valid answers to the questions that the experimenter has in mind by varying controllable factors in a predetermined manner.

Note 2 to entry: A critical aspect to an experiment is control — the investigator has the capability to vary settings, input materials, assignment of procedures to individuals and so forth with the intention of obtaining an understanding of the system efficiently. By proper design and conduct of the experiment, it is possible to attribute causation to the impact of the settings.

Note 3 to entry: Experiments are different from observational studies where the investigators may determine which units are to be studied and the observational process to be observed, but the assignment of experimental treatments (3.1.13) is outside their control.

3.1.2

model

formalized representation of outcomes of an experiment (3.1.1)

Note 1 to entry: The model consists of three parts. The first part is the response variable (3.1.3) that is being modelled. The second part is the deterministic or the systematic part of the model that includes predictor variable(s) (3.1.4) . Finally, the third part is the residual error (3.1.6) that can involve pure random error (3.1.9) and misspecification error (3.1.10) . The model applies for the experiment as a whole and for separate outcomes denoted with subscripts. The model is a mathematical description that relates the response variable to predictor variables and includes associated assumptions. Outcomes refer to recorded or measured observations of the response variable.

Note 2 to entry: The model is a simplified representation of the actual system where only key or fundamental features are considered.

Note 3 to entry: The above description of a model not only applies to the classical linear models with additive error but also to generalized linear models ここで, the error can be described by a variety of distributions including the binomial, Poisson, exponential, gamma and normal distributions. Linearity occurs in Example 4 with a logarithmic transformation applied to the deterministic part of the function. Although the examples given in this terminological entry are linear in the parameters, this is not intended to suggest that such a case will apply in all experimental design situations.

3.1.3

response variable

output variable

variable representing the outcome of an experiment (3.1.1)

Note 1 to entry: The term “dependent variable” is not recommended as a synonym due to potential confusion with “independence” (see ISO 3534-1:2006, 2.4).

Note 2 to entry: It may be that the response variable is vector-valued because several responses are recorded from each experimental unit (3.1.24) .

Note 3 to entry: The response variable is likely influenced by one or more predictor variables (3.1.4) , the nature of which can be useful in controlling or optimizing the response variable.

3.1.4

predictor variable

variable that can contribute to the explanation of the outcome of an experiment (3.1.1)

Note 1 to entry: A predictor variable can be used to model the impact of a categorical factor, e.g. at two levels. For multiple levels of a factor, two or more predictor variables can be devised to represent the distinct categorical levels.

Note 2 to entry: A predictor variable can include a random element in it or it can, for example, be from a set of qualitative classes which can be observed or assigned without random error.

Note 3 to entry: The term “predictor variable” is typically used in the development of a mathematical relationship among the response variable (3.1.3) and the available predictor variable(s) or functions of predictor variables. The term “factor” tends to be used operationally as a means to assess the response variable as particular factors vary.

Note 4 to entry: “Independent variable” is not recommended as a synonym due to potential confusion with “independence” (see ISO 3534-1:2006, 2.4). Other terms sometimes substituted for predictor variable include “input variable”, “descriptor variable” and “explanatory variable”.

3.1.5

factor

feature under examination as a potential cause of variation

Note 1 to entry: The extent to which a given factor can be controlled dictates its potential role in a designed experiment. Factors can be controllable (fixed), modifiable (controllable only for short duration or at considerable expense) or uncontrollable (random).

Note 2 to entry: A factor could be associated with the creation of blocks (3.1.25) .

3.1.6

residual error

error term

random variable representing the difference between the response variable (3.1.3) and its prediction based on an assumed model (3.1.2)

Note 1 to entry: The predicted value of the response variable is based upon an assumed model, the parameters of which are estimated from the data. The residual error is that part of the response variable that is unexplained by those predictor variables, which have been included in the model, and may be due to both systematic and chance causes.

Note 2 to entry: For the purpose of this definition, the term “predicted response value” is understood to be the estimated response for that experimental treatment (3.1.13) determined from the empirical model derived from the data of the experiment (3.1.1) using the assumed model.

Note 3 to entry: Residual error includes pure random error (3.1.9) and misspecification error (3.1.10) . The expectation of the residual error is assumed to be zero.

Note 4 to entry: The variance of the residual error is usually estimated in an experiment by subtracting the pooled sum of squares for terms included in the assumed model from the total sum of squares and dividing by the corresponding difference in degrees of freedom (3.1.32) .

Note 5 to entry: The term “residual error” is used in practice in two different ways. For this part of ISO 3534, the term is used as a random variable associated with the difference between the response variable which is a random variable and the prediction of the response variable which is based on an assumed model.

Note 6 to entry: In cases in which the residual error is estimated from data, the terms sample residual error or empirical residual error are used.

3.1.7

residual

observed value of the residual error (3.1.6)

3.1.8

variance component

part of the total variance of a response variable (3.1.3)

Note 1 to entry: A variance component can be either an individual variance component that is part of the overall variability of the response variable or it could be due to a random variable when modelling the response variable as a sum of independent error terms.

Note 2 to entry: Other models can be envisaged that include nested (see 3.2.21) or crossed factors (see 3.2.1).

Note 3 to entry: In the simplest case, it is conceivable to imagine a model in which the variance of the residual error is the sole variance component (e.g. an experiment where no factors are varied and the experiment consists of repeated measurements on a single unit).

3.1.9

pure random error

pure error

part of the residual error (3.1.6) associated with replicated observations

Note 1 to entry: It is a common characteristic of experiments (3.1.1) that, when repeated, results vary from trial to trial, although the experimental materials, environmental conditions and the experimental operations have been carefully controlled. Thus, pure random error is a common occurrence in spite of the best efforts of the experimenter. This variation introduces a degree of uncertainty into conclusions drawn from the results, and consequently, should be considered when reaching decisions.

Note 2 to entry: If only the centre point (3.1.39) in an experimental design (3.1.28) were replicated, then the sample variance of responses at the centre point provides an estimate of the variance of the pure error. If replicates (3.1.36) took place at multiple treatment combinations, then an overall estimate of the variance of the pure error can be achieved by pooling the estimates at these experimental treatments (3.1.13) .

EXAMPLE:

Returning to Example 3 in 3.1.2, an estimate of the variance of pure error for fixed (i, j) is
where i = 1,..., I; j = 1,..., J; k = 1,..., n_ij ; .

Note 3 to entry: The term “pure error” is used in practice in two different ways. For this part of ISO 3534, the term is used as a random variable and refers to a population variance (σ²) in association with a mathematical model. From the mathematical perspective, the pure error can be construed as ε_ij in Example 2, as ε_ijk in Example 3, and ε_i in Example 4 all from 3.1.2.

Note 4 to entry: In cases in which the pure error can be estimated from data (i.e. there are replicates), pure error actually refers to the “sample” or “empirical” pure error which in conjunction with the estimated residual error (3.1.6) provides the basis for a lack of fit test of the model. If the estimated residual error based on a model is reasonably close to the estimated pure error, then the model does not exhibit substantial lack of fit. Since the residual error includes all sources of variation for the difference between response variable and the predictive model, the residual error includes the contribution of pure error. Of the examples illustrating models in 3.1.2, only Example 3 having replicates would facilitate direct estimation of the pure error.

3.1.10

misspecification error

part of the residual error (3.1.6) not accounted for by pure random error (3.1.9)

Note 1 to entry: Misspecification error can be attributed to predictor variables (3.1.4) or a function of the predictor variables that are erroneously omitted from the model of the response variable (3.1.3) .

Note 2 to entry: There could be non-attributable factors including fixed or random factors that may not have been incorporated in the model. This occurs, for example, if the true model is quadratic but the fitted model is linear. As being unknown to the experimenter, these factors are effectively included in the variation from one trial to the next. Inherent factors that actually impact the response variable but were omitted in the model could lead to systematic errors in the experimental results. It may be possible to mitigate this problem through careful selection of the model and by randomization (3.1.30) .

Note 3 to entry: Of related interest to residual error (3.1.6) , pure random error (3.1.9) and misspecification error are the terms repeatability standard deviation (ISO 3534-2:2006, 3.3.7) and reproducibility standard deviation (ISO 3534-2:2006, 3.3.12) which apply in the experimental design context directly if the actual design of the experiment is in accordance with repeatability conditions (ISO 3534-2:2006, 3.3.6) or reproducibility conditions (ISO 3534-2:2006, 3.3.11), respectively.

3.1.11

design region

design space

set of allowable values for the predictor variables (3.1.4)

Note 1 to entry: In some situations, the design region is determined by the range of the individual predictor variables and is rectangular (or hyper-rectangular in higher dimensions). However, if the range of one predictor variable could influence the reasonable values of another predictor variable, then the region need not be rectangular. For example, in the simple case of an experiment (3.1.1) in baking a cake, the duration in the oven logically depends on the temperature setting.

3.1.12

factor level

setting, value or assignment of a factor (3.1.5)

Note 1 to entry: The factor levels can be represented through the values of the predictor variables (3.1.4) in the model.

Note 2 to entry: Responses observed at the various levels of a factor provide information for determining the effect of the factor within the range of levels of the experiment (3.1.1) . Extrapolation beyond the range of these levels is usually inappropriate without a firm basis for assuming model (3.1.2) relationships. Interpolation within the range may depend on the number of levels and the spacing of these levels. It is usually reasonable to interpolate, although it is possible to have discontinuous or multi-modal relationships that cause abrupt changes within the range of the experiment. The levels may be limited to certain selected fixed values (whether these values are or are not known) or they may represent purely random selection over the range to be studied.

3.1.13

run

experimental treatment

specific settings of every factor (3.1.5) used on a particular experimental unit (3.1.24)

Note 1 to entry: Ultimately, the impact of the factors will be captured through their representation in the predictor variables (3.1.4) and the extent to which the model matches the outcome of the experiment (3.1.1) .

3.1.14

factor effect

factor (3.1.5) that influences the response variable (3.1.3)

Note 1 to entry: Factor effects can include main effects (3.1.15) , dispersion effects (3.1.16) and confounded effects (3.1.18) .

3.1.15

main effect

factor effect (3.1.14) applicable in the context of linearly structured models with respect to expectation

Note 1 to entry: Linear structured models include additive, linear models which in turn include the class of models related to factorial experiments (3.2.1) and fractional factorial experiments (3.2.3) . Main effects are mostly readily understood in a model with zero interactions.

Note 2 to entry: The main effect can be estimated by averaging the response variable over all other runs provided the experiment is fully balanced.

3.1.16

dispersion effect

factor effect (3.1.14) in the context of linearly structured models with respect to variation

Note 1 to entry: It is important to recognize that a dispersion effect (3.1.16) could be significant, whereas the main effect corresponding to the same factor could have little impact. This situation affords the opportunity to achieve low variability or consistency in the responses by using a factor that does not necessarily impact the overall level of the response.

3.1.17

interaction

influence of one factor (3.1.5) on one or more other factors’ impact on the response variable (3.1.3)

Note 1 to entry: An interaction is present if the apparent influence of one factor on the response variable (3.1.1) depends upon one or more other factors. In such a situation, these two (or more) factors are said to interact. Interactions can be incorporated into the model (3.1.2) by defining a new predictor variable that is a function of two or more factors. An interaction reflects the dependence of the level of one factor on the level(s) of another or other factors by providing the differential comparison of the responses for each level of the factor on each of the levels of the other factor(s).

Note 2 to entry: Interaction indicates an inconsistency of the main effect (3.1.15) of a factor on the response variable depending on the level of another factor. Figure 1 indicates these phenomena ranging from very strong interaction, to limited interaction, to no interaction. The presence of an interaction ought to be assessed in relation its estimated uncertainty via an appropriate test of significance.

Note 3 to entry: Interactions are considered initially involving only two factors and are referred to as either two-way interactions or first order interactions. Of course, it is possible that three factors, for example A, B, and C, interact in the sense that the first order interaction of ab depends on the level of factor C. In this case, there is a second order interaction. Similarly, third, fourth, and higher order interactions can be conceived. First order interactions are relatively easy to explain graphically and in words compared to higher-order interactions.

Note 4 to entry: Example 3 in 3.1.2 provides a formal model representation of an experiment (3.1.1) with two factors and the two-way or first order interaction τ_ij between them.

Figure 1 — Interaction plots

Key

Y	Response
X1	No interaction
X2	Limited interaction
X3	Very strong interaction

3.1.18

confounded effect

factor effect (3.1.14) that is indistinguishable from another factor effect

Note 1 to entry: A confounded effect is sometimes created at the design stage in order to accommodate blocks (3.1.25) or to introduce another factor without increasing the number of experimental units (3.1.24) under consideration. A confounded effect could be a high order interaction (3.1.17) .

3.1.19

confounding

equating two or more factor effects (3.1.14) so as to be indistinguishable from each other

Note 1 to entry: At the design stage, confounding is an important technique which permits, for example, the effective use of specified blocks (3.1.25) in some designed experiments (3.1.27) . This is accomplished by deliberately pre-selecting certain factor effects [typically high-order interactions (3.1.17) ] as being of little interest, and arranging the design so that it confounds them with block effects, while keeping the other more important main effects (3.1.15) and key interactions free from such complications. Confounding may be deliberately used to diminish the number of trials of the experimental plan (3.1.29) . Sometimes, however, confounding results from inadvertent changes to a design during the running of an experiment (3.1.1) or from incomplete planning of the design, and it serves to diminish, or even to invalidate, the effectiveness of an experiment.

Note 2 to entry: At the analysis stage, confounding is a device that sacrifices information about contrasts (3.1.22) of insignificant effects (typically corresponding to high-order interactions) in order to improve the precision with which more relevant contrasts can be estimated.

3.1.20

alias

factor effect (3.1.14) that is equal to another factor effect or a function of other factor effects

Note 1 to entry: A main effect (3.1.15) that is deliberately confounded in an experiment (3.1.1) with another factor effect (such as a high order interaction (3.1.17) is an alias of that factor effect; the effects are aliases of each other.

3.1.21

curvature

departure from a straight line relationship between the response variable (3.1.3) and a predictor variable (3.1.4)

Note 1 to entry: Curvature has meaning with quantitative predictor variables, but not with categorical (nominal) or non-quantitative (ordinal) predictor variables. Detection of curvature requires more than two levels of the factors. In some instances, replicated centre points (the factor set midway between the high and low settings of the factors) can allow the detection and assessment of curvature. Alternatively, an expanded range of the factor levels (3.1.12) can be necessary to observe curvature.

3.1.22

contrast

linear function of the values of the response variables (3.1.3) for which the sum of the coefficients is zero when not all coefficients equal to zero

Note 1 to entry: With observations y₁, y₂,…, y_n , the linear function with not all a_i ’s equal to zero. The role of contrasts in the design of experiments is to compare and to investigate effects as given in the subsequent examples.

3.1.23

orthogonal contrast

set of contrasts (3.1.22) whose coefficients satisfy the condition that, if multiplied in corresponding pairs, the sum of the products equals zero

Note 1 to entry: The purpose of orthogonal contrasts is to facilitate independent tests of hypotheses of interest in the experimental situation. In Example 1, the two tests can be conducted independently (one test has no bearing on the other and provided further that the residual error follows a normal distribution). In Example 2, on the other hand, the tests are dependent, indicating for instance, that a rejection in one test could suggest a more likely rejection of the other.

3.1.24

experimental unit

basic unit of the experimental material

Note 1 to entry: The experimental unit could be the entire manufacturing process in which case a run corresponds to the set up of the process and the response variable (3.1.3) (possibly multivariate) corresponds to characteristics of the product. In another situation, the experimental unit could be individual subjects in a study (laboratory animals or human participants) each of which receives a particular experimental treatment.

Note 2 to entry: In agricultural settings, the basic unit of the experimental material is a plot of land. Although the agricultural setting suggests plot as a natural description of a experimental unit, the concept is more general and can be applied to other contexts. Plot is then replaced by the generic term experimental unit.

3.1.25

block

collection of experimental units (3.1.24)

Note 1 to entry: To be effective, the blocks should represent sets of experimental units that are homogeneous in some sense. The term “block” originated in agricultural experiments (3.1.1) in which a field was subdivided into sections having common conditions, such as exposure to the wind, proximity to underground water or thickness of the arable layer. In other situations, blocks are based on batches of raw material, operators, the number of units studied in a day, and so forth. More generally, blocks can consist of regions of a country, groups of factories, time frames (e.g. shifts in a manufacturing facility) and so forth.

Note 2 to entry: Generally, recognition of the existence of blocks may affect how the experimental treatments (3.1.13) of interest are assigned to experimental units. Operationally, additional “artificial” treatments are created within the model (3.1.2) that designate the assignment to blocks. The treatments thus consist of the factor effects (3.1.14) of direct interest in the study and the treatments that relate to blocking assignments. The idea is to enhance the possibility of recognizing important factor effects which would otherwise be obscured had the blocks not been established.

3.1.26

blocking

arrangement of experimental units into blocks (3.1.25)

Note 1 to entry: Within each block, the residual error (3.1.6) can be expected to be smaller than would be expected should a similar number of units be randomly assigned to the experimental treatment (3.1.13) without regard to blocks.

Note 2 to entry: Blocks are usually selected to allow for the effects of assignable causes, in addition to those introduced as factors to be studied (factor of primary interests), which it may be difficult, or even impossible to keep constant for all of the experimental units in the complete experiment. The effect of these assignable causes may be minimized within blocks, thus a more homogeneous experiment sub-space is obtained. The analysis of the experimental results must account for the effect of blocking the experiment.

3.1.27

designed experiment

experiment (3.1.1) with an explicit objective and structure of implementation

Note 1 to entry: The purpose of a properly designed experiment is to provide the most efficient and economical method of reaching valid and relevant conclusions from the experiment.

Note 2 to entry: Associated with a designed experiment is an experimental design (3.1.28) that includes the response variable (3.1.3) or variables and the experimental treatments (3.1.13) with prescribed factor levels (3.1.12) . A class of models that relates the response variable to the predictor variables could also be envisaged.

3.1.28

experimental design

assignment of experimental treatments (3.1.13) to each experimental unit (3.1.24)

Note 1 to entry: The assignment of experimental treatments could also include the time order or randomized order in which the treatments are applied.

Note 2 to entry: An experimental design can be considered as a scheme assigning experimental treatments (levels of a factor or a combination of such levels) involved in an experiment to the experimental units. The design matrix (3.2.25) includes the specified levels of each factor in the experiment (as well as other columns which give values of the predictor variables used at the analysis stage).

Note 3 to entry: An experimental design provides the determination as to how the observations/measurements should be taken to answer a research question in a valid, efficient and economical way.

Note 4 to entry: This definition does not preclude the possibility of designs that are known to be relatively inefficient (e.g. “one-factor-at-a-time” designs are experimental designs according to the definition but that does not suggest that they are to be recommended.

3.1.29

experimental plan

specification of the intended procedure for the implementation of an experiment (3.1.1)

Note 1 to entry: The experimental plan should ideally offer the possibility of providing the most efficient and economical method of reaching valid and relevant conclusions from a designed experiment (3.1.27) . The selection of an appropriate design for an experiment will depend on considerations such as the type of questions to be addressed, the degree of generality to be attached to the conclusions, the magnitude of the effect from which a high probability of detection (power) is desired, the homogeneity of the experimental units (3.1.24) and the cost and the method of performing the experiment. The experimental plan establishes a protocol for the conduct of the experiment.

Note 2 to entry: A properly designed experiment will frequently lead to effective results from relatively simple statistical analysis and interpretation of the results. However, a poorly designed experiment may not meet the experimental objectives in spite of sophisticated analyses of the results.

Note 3 to entry: The development of an experimental plan can be quite arduous. A detailed description of the process is given in Annex D.

3.1.30

randomization

strategy in which each experimental unit (3.1.24) has an equal chance of being assigned a particular experimental treatment (3.1.13)

Note 1 to entry: Randomization attempts to protect against biases due to causes not taken into account explicitly in the model (3.1.2) . Randomization may further neutralize potential temporal or spatial effects. The equal chance of assignment could be within a subset of the collection of experimental units.

Note 2 to entry: From a practical standpoint, sampling without replacement may govern the allocation of experimental units to treatments so that the final experimental unit drawn seemingly is not “randomly” chosen at this stage. However, at the onset of the allocation, each experimental unit had an equal chance of being chosen so that ultimately, a random allocation has occurred.

3.1.31

orthogonal array

set of experimental treatment (3.1.13) combinations, in which for every pair of factors, each treatment combination occurs the same number of times across the possible factor levels (3.1.12)

Note 1 to entry: The concept of strength in relation to orthogonal arrays arises with screening designs (3.2.8) , which is one possible use of orthogonal arrays. A design of strength d is a complete factorial design in any d factors. Strength 1 implies that the levels of each factor occur the same number of times (which is sometimes called a balanced factor). An orthogonal array has strength 2. The subset size d is known as the strength.

3.1.32

degrees of freedom

v

number of linearly independent effects that can be estimated

Note 1 to entry: Informally, the degrees of freedom are the number of quantities that are free to vary without restriction.

Note 2 to entry: Degrees of freedom is commonly associated with the denominator of a variance calculation. The value of the degrees of freedom is the sample size minus the number of constraints associated with the quantity being computed. Estimating the population mean by the sample mean in a variance calculation reduces the degrees of freedom by 1, yielding degrees of freedom of n−1, with n as the sample size. One degree of freedom is eliminated since knowing the sample mean and n−1 values from the sample establishes the remaining data value.

Note 3 to entry: Degrees of freedom are the parameters of certain theoretical distributions that occur as sample distributions in statistical estimation and testing; for example, the chi-square distribution, the F-distribution, and the t-distribution.

3.1.33

one-factor experiment

designed experiment (3.1.27) in which a single factor (3.1.5) is investigated as to its effect (if any) on the response variable (3.1.3)

3.1.34

two-factor experiment

designed experiment (3.1.27) in which two distinct factors (3.1.5) are simultaneously investigated for possible effects on the response variable (3.1.3)

Note 1 to entry: If the two factors operate without interacting with each other, the term main effect (3.1.15) necessarily still applies. Namely, for each factor the main effect is its contribution to the mean of the response variable. See Example 2 of 3.1.2.

3.1.35

k-factor experiment

multi-factor experiment

designed experiment (3.1.27) in which k distinct factors (3.1.5) are simultaneously investigated for possible effects on the response variable (3.1.3)

3.1.36

replication

multiple occurrences of a given treatment combination or setting of predictor variables (3.1.4)

Note 1 to entry: Experimental constraints may dictate that replications take place sequentially rather than in a randomized order. Informally, such a situation would correspond to repetition, but universal concurrence on this term does not exist. Hence, for purposes of this part of ISO 3534, replication will be the term involving the attainment of different response values for a fixed set of levels of predictor variables.

Note 2 to entry: A run is repeated a number of times in order to obtain a more reliable estimate than is possible from a single observation. The function of replication is two-fold: (a) it provides an estimate of the pure error, and (b) it adds to the confidence in the experimental results.

Note 3 to entry: Replication as used in this part of ISO 3534 should be distinguished from the concepts of repeatability and reproducibility given in ISO 3534-2:2006) which relate particularly to measurement systems analysis. In experimental situations relevant to this part of ISO 3534, replication includes contributions from the process itself in addition to measurement uncertainty.

3.1.37

cube point

vector of factor level (3.1.12) settings of the form (a₁, a₂, …, a_k ) where each a_i equals +1 or −1 as a notation for the coded levels of the factors (3.1.5)

Note 1 to entry: These points are precisely the type of points found in a two-level full factorial experiment (3.2.2) or fractional factorial experiment (3.2.3) with k factors. As many as 2 ^k cube points can be used in the context of a central composite design [see Example 1 in (3.2.19)].

3.1.38

star point

vector of factor level (3.1.12) settings of the form (a₁, a₂, …, a_k ) ここで, one a_i equals αor−α and the other a_i ’s equal 0, as notation for the coded levels of the factors (3.1.5)

Note 1 to entry: All star points have a single non-zero component equal to +αor−α. In k-factor central composite designs, typically a total of 2 ^k star points are employed.

3.1.39

centre point

vector of factor level (3.1.12) settings of the form (a₁, a₂, …, a_k ) ここで, all a_i equal 0, as notation for the coded levels of the factors (3.1.5)

Note 1 to entry: All entries of centre points are zero, so the vectors are of the form (0, 0, ... , 0) corresponding to the centre of the experimental design (3.1.28) in the coded variables. The number of these points, for example n₀, is chosen to achieve various objectives in response surface designs (3.2.19) . Centre points are sometimes replicated (3.1.36) to obtain an estimate of the pure error (3.1.9) of the process under investigation. Graphical depictions of two designs with star (five pointed star), cube (cubes on the corners) and centre points (⊕) are given in Figures 2 and 3.

Figure 2 — Two-dimensional illustration of star, cube and centre points

Figure 3 — Three-dimensional illustration of star, cube and centre points (one cube point hidden)

3.1.40

rotatability

characteristic of a designed experiment (3.1.27) for which the response variable (3.1.3) that is predicted from a fitted model (3.1.2) has the same variance at all equal distances from the centre of the design

Note 1 to entry: A design is rotatable if the variance of the predicted response at any point x depends only on the distance of x from the centre point (3.1.39) . A design with this property can be rotated around its centre point without changing the prediction variance at x .

Note 2 to entry: Rotatability is a desirable property for response surface designs (3.2.19) .

3.2.1

factorial experiment

designed experiment (3.1.27) with one or more factors (3.1.5) and with at least two levels applied for one of the factors

Note 1 to entry: The term “factorial experiment” is more general than full factorial experiment (3.2.2).

Note 2 to entry: Crossed factors: two factors are crossed if every level of one occurs with every level of the other in the experiment (3.1.1). Nested factors: factor A is nested within another factor B if the levels or values of A are different for every level or value of B. Nested factors or effects have a hierarchical relationship. (See 3.2.21).

3.2.2

full factorial experiment

factorial experiment (3.2.1) consisting of all possible combinations of the levels of the factors (3.1.5)

Note 1 to entry: All interactions (3.1.17) and main effects (3.1.15) can be estimated from a full factorial experiment.

Note 2 to entry: A full factorial experiment is usually described symbolically as the product of the number of levels of each factor. For example, an experiment based on 3 levels of factor A, 2 levels of factor B and 4 levels of factor C would be referred to as a 3 × 2 × 4 factorial. The product of these numbers (24 in this case) indicates the total number of distinct runs.

Note 3 to entry: When a full factorial experiment includes factors all having the same number of levels, the description is usually given in terms of the number of levels raised to a power equal to the number of factors, k. Thus, an experiment with two factors each at three levels would be referred to as a 3² full factorial (k being equal to 2) and requires 9 experimental units which are given different experimental treatments.

3.2.3

fractional factorial experiment

factorial experiment (3.2.1) consisting of a subset of the full factorial experiment (3.2.2)

Note 1 to entry: Typically, the fraction is a simple proportion of the full set of possible experimental treatment combinations. For example, half-fractions, quarter-fractions, and so forth are common.

Note 2 to entry: All interactions (3.1.17) and main effects (3.1.15) cannot be estimated from a fractional factorial experiment.

Note 3 to entry: Fractional factorial designs are experimental designs (3.1.28) consisting of a carefully chosen subset (fraction) of the experimental runs of a full factorial design. The subset may be chosen so as to exploit the main effects and low-order interactions to expose information about the most important features of the problem studied, while using a fraction of the effort of a full factorial design in terms of experimental runs and resources and thereby yielding screening designs (3.2.8) . In other situations, the subset may be chosen to account for inhomogeneity among the experimental units, thereby yielding, for example, Latin square (3.2.11) or Graeco-Latin square designs (3.2.12) .

3.2.4

two-level experiment

full factorial experiment (3.2.2) in which all factors (3.1.5) assume at most two factor levels (3.1.12)

3.2.5

2 ^k factorial experiment

full factorial experiment (3.2.2) with k factors (3.1.5) , each at two factor levels (3.1.12)

Note 1 to entry: Two level full factorial experiments are full factorial experiments in which each of the p available factors is investigated at only two levels. The early stages of experimentation usually involve the investigation of a large number of potential factors to discover the “vital few” factors. Two level factorial experiments are used during these stages to quickly filter out unwanted effects so that attention can then be focused on the important ones.

3.2.6

2 ^{k −p} fractional factorial experiment

fractional factorial experiment (3.2.3) the size of which is a 2 ^−p fraction of the size of the 2 ^k factorial experiment (3.2.5)

Note 1 to entry: For a large number of factors (3.1.5) , 2 ^k may require more runs than are feasible. Through careful selection, nearly the same amount of information can be obtained from the fractional factorial experiment as the full factorial experiment (3.1.2) . In particular, the selection is typically made so that main effects (3.1.15) and interactions (3.1.17) that are expected to be of practical importance are confounded (3.1.19) only with interactions expected to be negligible.

Note 2 to entry: For p equal to 1, the resulting fractional factorial experiment is a half-fraction; for p equal to 2, the resulting fractional factorial experiment is a quarter-fraction; and so forth.

Note 3 to entry: A 2 ^k−p fractional factorial experiment is constructed by considering the k factors to be in two groups, a primary one with k−p factors and a secondary one with p factors. The k−p factors in the primary group are allocated to a full factorial with 2 ^k−p experimental units which are the number of experimental units of the design. The levels of each of the factors of the secondary group for each experimental unit are defined in terms of levels of factors of the primary group. The set of p equations that define the factors of the secondary group in terms of the factors of the primary group is called the generating relation, because it generates the design. The p equations of the generating relation can be used to calculate the 2 ^p − 1 equations of the defining relation that can be used to determine properties of the design. In the previous example, k=6, p=2; the primary factors were A, B, C and D and the secondary group were E and F; the generating relations were E=ABC and F=BCD; the defining relation was I = ABCE = BCDF = ADEF.

3.2.7

design resolution

length of the shortest word in the defining relation associated with a 2 ^k-p fractional factorial experiment (3.2.6)

Note 1 to entry: Design resolution indicates the extent of aliasing (3.1.20) among main effects (3.1.15) and two-way and higher order interactions (3.1.17) .

Note 3 to entry: The higher the resolution, the more effects (main or interactions) can be estimated unambiguously provided the higher order interactions are negligible. Given a choice of two potential designs involving the same number of factors and experimental units, the design with the higher resolution should be selected. Fortunately, for most cases of k and p of practical interest, the most appropriate defining relations are recorded (see, for example, Reference [2], p. 410; or Reference [3], p. 272).

Note 4 to entry: Full factorial designs (3.2.1) have no confounding. For most practical purposes, a resolution V design is excellent and a resolution IV design may be adequate. Resolution III designs are useful as economical screening designs (3.2.8) .

Note 5 to entry: Factor names are presumed to be expressed by single letters for purposes of the definition.

3.2.8

screening design

designed experiment (3.1.27) that identifies a subset of factors (3.1.5) for further study

Note 1 to entry: Such experimental designs (3.1.28) generally focus on the investigation of main effects (3.1.15) , with the presence of interactions (3.1.17) leading to complications in the analysis, and possibly the need for additional experimental treatments (3.1.13) to resolve ambiguities.

Note 2 to entry: The primary purpose of an experiment with a screening design is to select or to screen out the few important main effects from the many less important ones. Screening designs are also sometimes termed main effects designs.

Note 4 to entry: Aguchi’s orthogonal array (3.1.31) L12 is equivalent to the Plackett-Burman 12-run design given above. L20 is equivalent to the Plackett- Burman 20-run design. As a word of caution, the L array convention typically gives the entries of the design matrix in a different order than that provided by the Hadamard construction.

Note 5 to entry: Plackett-Burman designs can be adapted for use in a supersaturated setting (i.e. cases with more factors than experimental runs). For further details on supersaturated designs, see References [5] and [6].

3.2.9

block design

designed experiment (3.1.27) that uses blocks (3.1.25)

Note 1 to entry: A block design takes advantage of the homogeneity of subsets of the experimental units (3.1.24) . Inhomogeneity among experimental units, if ignored in the experimental design, can reduce the amount of information obtained from the experiment (3.1.1) by increasing the observed variation. Accounting for this situation in the design can increase the experiment's capability of meeting the design objective. Block designs are intended to counteract the impact of known nuisance effects or effects that contribute to the response but are not inherently of interest relative to the objective of the experiment. If the presumption of homogeneity is incorrect, then blocking incurs a cost in reducing the degrees of freedom.

3.2.10

randomized block design

block design (3.2.9) with an arrangement of ν experimental treatments (3.1.13) in b blocks (3.1.25) such that each treatment appears exactly once in each block

Note 1 to entry: These designs are useful for one-way elimination of heterogeneity setting, i.e. when the heterogeneity in the experimental units is caused by a single factor and levels of this factors causing heterogeneity is same as the number of treatments.

Note 2 to entry: Since each of the treatments appear in each block, therefore, it is a complete block design and a better term for this is “randomized complete block design”. The number of replications of treatments are same as number of blocks. The treatments are randomly allocated to each block.

Note 3 to entry: This is the simplest design using all the three principles of design as given by Fisher (replication, randomization and local control).

3.2.11

Latin square design

fractional factorial design (3.2.3) with an arrangement of v² runs (3.1.13) in v² positions arranged in v rows and v columns, such that every symbol occurs precisely once in each row and precisely once in each column

Note 1 to entry: Latin square designs can investigate at most three factors (rows, columns and letters). The symbol v represents the order of the Latin square.

Note 4 to entry: A basic assumption is that these block factors do not interact (3.1.17) with the factor of most interest under study, or among themselves. If this assumption is not valid, the measure of residual error (3.1.6) is increased and the effect of the factor confounded with such interactions (3.1.17) . Sometimes other factors of interest are used in the block positions so that there may be three factors without any designation of being block factors. This is equivalent to a fractional factorial experiment (3.2.3) with the assumption of no interactions. Some designs of fractional factorial experiments form Latin squares and it may be more desirable to approach the problem from the fractional factorial experiment viewpoint in order to more fully understand the assumptions being made concerning interactions.

3.2.12

Graeco-Latin square design

fractional factorial design (3.2.3) involving 4 factors (3.1.5) , each at h levels (3.1.12) , in which the combination of the levels of any one of the factors with the levels of the other three factors appears once and only once in an experiment of size h²

Note 1 to entry: A Graeco-Latin square design involves four factors and there are h² (h ≥ 3, a positive integer) experimental units (3.1.24) classified according to three block (3.1.25) factors (for example, row factor, column factor and Greek letter) each factor having h levels. There are h levels of the factor of primary interest which are allocated to the h² experimental units at random in such a manner that each experimental treatment (3.1.13) appears in every row and column precisely once and also appears with a Greek letter precisely once.

Note 2 to entry: Two Latin squares are said to be orthogonal, if each letter in one square coincides exactly once with each letter of the other square. Pairs of Latin square designs that are orthogonal can be combined to produce Graeco-Latin square designs.

Note 3 to entry: Graeco-Latin square designs allow the incorporation of three blocking variables, all of which have the same number of levels as the number of levels of the factor of primary interest.

3.2.13

incomplete block design

block design (3.2.9) with at least one block (3.1.25) containing less than all of the experimental treatments (3.1.13)

Note 1 to entry: In effect, a randomized block design (3.2.10) can be construed as a “complete” block design emphasizing that a sufficient number of experimental units are available within each block to accommodate the required number of treatments.

Note 2 to entry: If material is divided into blocks and it is desired to allocate certain treatments to the block, the treatments may be too numerous for them to appear in each block. When a block contains less than a complete replication of the treatments, it is called incomplete.

3.2.14

balanced incomplete block design

BIBD

incomplete block design (3.2.13) in which each block (3.1.25) contains the same number k of different levels (3.1.12) from the l levels of the factor of primary interest (3.1.5) arranged so that every pair of levels occurs in the same number λ of blocks from the b blocks

Note 1 to entry: The term “balanced” refers to consistent number of pairings, “incomplete” refers to the inability to examine every level of the factor of primary interest in each block, and “block” refers to the strategy of conducting the experiments (3.1.1) on homogeneous sets of experimental units (3.1.24) .

Note 3 to entry: For randomization (3.1.30) , arrange the blocks and levels within each block independently at random.

3.2.15

partially balanced incomplete block design

PBIBD

incomplete block design (3.2.13) in which each block (3.1.25) contains the same number k of different levels (3.1.12) from the l levels of the principal factor (3.1.5) , arranged so that not all pairs of levels occur together in the same number of the b blocks