## Glossary

**Note:** An exposition of the use of this terminology in the context of an example is available in Brien (2017c, Section 2).

**aliasing** Relates to the relationships between sources involving factors from the same tier; that is, two sources whose factors were both allocated in the same allocation or both recipients in the same allocation. Aliasing between two such sources can arise in two circumstances: (i) considering the two sources in isolation, or (ii) when the two sources are merged or combined with a source involving their recipient factors. There are are three possibilities for the two sources: (i) they are orthogonal so that their associated effects can be estimated independently, (ii) one is aliased with the other in that there is no information about one of the sources after the other has been included in the analysis, or (iii) they are partially aliased because there is some information about both sources available. Technicaly, this can be viewed as reflecting the relationships between the subspaces of the data space for the sources. For instance, if a factorial experiment involves two treatment factors and the numbers of observations for the combinations are disproprtionate then the main effects of the two factors will be partially aliased. (cf. confounding)

**allocated factors/objects** The set of factors that index the set of objects that are allocated in an allocation to a set of recipient objects and factors. The set of allocated factors will occupy a panel in the factor-allocation diagram, so that they form a tier, and the panel is named for the objects.

**allocated-inclusive multiple allocations** (formerly randomized-inclusive) Involves three tiers of factors. Tier 3 factors are allocated to tier 2 factors in the first allocation. Then the combined tier 3 and 2 factors are allocated to the tier 1 factors. Each allocation involves different units. Like composed allocations, the aim of allocated-inclusive allocations is to allocate tier 3 factors to tier 2 factors and then to to allocate these factors to tier 1 factors. The difference is that only for composed allocations can tier 3 factors be ignored in the second allocation, where tier 2 factors are allocated to tier 1 factors. Allocated-inclusive multiple allocations form a chain of allocations. Allocated-inclusive multiple randomizations result when both allocations involve only randomization. Diagram

(an) **allocation** Involves the allocation of one or more sets of objects to another set of objects. In a single allocation one set of objects is termed the allocated set of objects and the other set is termed the recipient set of objects. Each set of objects is indexed by a set of factors that is referred to as a tier. An allocation may involve both randomization and systematic elements; the systematic elements can be viewed as placing further restrictions on the permutations to be used in carrying out the randomization element of the allocation. An outcome of an allocation is that allocated sources are confounded with certain recipient sources and this can be displayed using a decomposition table

**allocation-based mixed models** Are models based on the allocations employed in the design for an experiment, and so are obtained after the design is obtained (cf. anticipated model). Brien (2017c, Figure 2) shows the derivation of such a model from the factor-allocation diagram using a three-stage process that is analogous to that of Brien and Demétrio (2009). At each stage of this process a particular allocation-based model is obtained:

- initial allocation model: The model that reflects the allocations in the experiment. It can be obtained directly from the factor-allocation diagram and specified using intratier formulae.
- homogeneous allocation model: The model derived from the initial allocation model by swapping terms from being fixed to random and vice versa and by adding terms.
- prior allocation model: The model obtained from the homogeneous allocation model by reparameterizing fixed and random terms, if required, and removing terms to ensure that the variance matrix for the model is nonsingular. The prior allocation model is derived prior to the data being available for analysis; it represents what is considered at that point in time to be the maximal model.

The models are expressed in the form of symbolic mixed models, being a list of fixed terms and a list of random terms. Brien and Demétrio (2009) include extra considerations in the model formulation process for longitudinal experiments.

Randomization-based mixed models are a restricted class of allocation-based mixed models.

**analysis formulae** The formulae, one for each tier, displaying the relationships between the factors within a tier and incorporating any intertier interactions. They are used in Deriving the decomposition table for an experiment.

**analysis-of-variance table** A table that includes Sources, degrees of freedom, sums of squares, mean squares, F statistics and perhaps p-values. Optionally, efficiency factors and expected mean squares may be added. The sources should be genuine sources based on only genuine factors. That is, they are derived from skeleton analysis-of-variance tables by adding statistics calculated from observed data, which in turn are derived from decomposition tables. As for decomposition tables, we derive them from the tiers so that they display the confounding arising from the allocation employed in the experiment. There is a a *major column* for each analysis formula.

**anatomy** (of a design) The aliasing and confounding between the sources derived from the terms in a mixed model for an experiment. It is displayed in a decomposition table and is useful for checking the properties of a proposed experimental design.

**anova-applicable experiment** An experiment that is structure balanced and for which the estimates of the fixed effects for each source are to be obtained from a single source: the combination of information from different sources is not required. Consequently, analyzing the data from the experiment using an analysis of variance is feasible.

**anticipated model** A mixed model with terms for the effects that are anticpated to occur in an experiment and are formulated before a design is obtained (cf. allocation models). It would be formed before a design is chosen and used to inform the selection of a design. Deciding on the terms to include often requires consultation with the researcher. Generally, the objective in looking for a design is to find one that is optimal for this model. However, the anticipated model is often not explicitly stated, but has a subliminal presence, and so the search for an optimal design can be an informal process.

**canonical (covariance) component** A variance component that, except for the component for an identity term, is allowed to be negative (Bailey and Brien, 2016). Components for identity terms must be positive definite to ensure that the variance matrix is positive definite. The component for a term is denoted by ‘φ’ with a subscript that consists of the initial letters of the tier factors in the generalized factor for the term.

**canonical efficiency factors** Reflect the amount of information shared by two sources. They are the nonzero eigenvalues of a certain product of the projection matrices for the two sources and take values between zero and one (Brien, C.J., and Bailey, 2009). For an allocated source confounded with a recipient source, it measures the amount of confounding; a value of one for all canonical efficiency factors means that all the informated about the allocated source is confounded with the recipient source, while values of less than one for some or all canonical efficiency factors indicates that they are partially confounded. Simularly, for two aliased sources. The number of nonzero eigenvalues is equal to the degrees of freedom for the combined source that results from combining the two sources. Summary statistics are calculated from the canonical efficiency factors to provide efficiency criteria.

**chain of allocations** Allocations in which the recipient factors/objects for one allocation are allocated to the receipient factors/objects for another allocation. This contrasts with two-one and one-two allocations. Composed and allocation-inclusive multiple allocations produce a chain of allocations. Diagram

**coincident multiple allocations** Involves three tiers of factors. Tier 2 and 3 factors are allocated to the same tier 1 factors in two separate allocations. Coincident multiple randomizations result when both allocations involve only randomization. Diagram

**composed multiple allocations** Involves three tiers of factors. Tier 3 factors are allocated to tier 2 factors in one allocations and tier 2 factors are allocated to tier 1 factors in the second allocations. Each allocations involves different units. Like allocated-inclusive allocations, the aim of composed allocations is to allocate tier 3 factors to tier 2 factors and then to allocate these factors to tier 1 factors. The difference is that only for composed allocated can tier 3 factors be ignored in the second allocations, where tier 2 factors are allocated to tier 1 factors. Composed multiple allocations form a chain of allocations. Composed multiple randomizations result when both allocations involve only randomization. Diagram

**confounding** Relates to the relatonship between a source involving allocated factors, with some recipient factors if it is an intertier interaction, and a source involving recipient factors that have received the allocated factors either directly or indirectly. There are three possibilities for the allocated source: (i) it is orthogonal to the recipient source, (ii) it is confounded with the recipient sources in that all the information about it contained in the recipient source, or (iii) it is partially confounded because only some of the information about it is obtainable from the recipient sources. Technically, the subspace for the allocated source is: (i) for orthogonality, orthogonal to the subspace the recipient source, (ii) for confounded, a subspace of the subspace for the recipient source, or (iii) for partially confounded, nonorthogonal to the subspace for the recipient source. It differs from exhaustive confounding in that it is primarily a property of an allocated source. (cf. alaising)

**consonant randomization** Occurs when the nesting of the factors being randomized matches the nesting of the factors to which they are randomized. When this is not the case, the randomization is not consonant. For example, a nested factor is to be randomized to a non-nested factor results in a nonconsant randomization. A sign that a randomization may not be consonant is that the number of levels combinations of the factors being randomized is greater than those of the factors to which they are randomized.

**decomposition table** A table with Sources, degrees of freedom and, optionally, efficiency factors (Brien, C.J., and Bailey, 2009). It displays the anatomy for an experiment. The sources might include pseudosources. They have not had genuine factors substituted for pseudofactors in the sources and they do not include the expected mean squares. That is, it is a precursor to both a skeleton analysis-of-variance table and an analysis-of-variance table. They are useful in assessing the properties of a proposed design, irrespective of whether an analysis of variance is to be used to analyse the data. Here we derive them from the tiers so that they display the confounding arising from the allocation employed in the experiment. There is a a *major column* for each analysis formula, consisting of the sources, degrees of freedom and, for nonorthogonal experiments, the efficiency factors.

**double multiple allocations** Involves three tiers of factors. Tier 3 factors are allocated to both tier 1 and 2 factors in two allocations. Double multiple randomizations result when both allocations involve only randomization. Diagram

**efficiency criteria** Are calculated from the canonical efficiency factors to summarize aspects of the amount of information shared by two sources that are either confounded or aliased: the *A-efficiency criterion* is the harmonic mean of the canonical efficiency factors; the *M-efficiency criterion* is the arithmetic mean of the canonical efficiency factors; the *E-efficiency criterion* is the minimum of the canonical efficiency factors; the *S-efficiency criterion* is the variance of the canonical efficiency factors; the *X-efficiency criterion* is the maximum of the canonical efficiency factors; the *order* is the number of canonical efficiency factors; *df-orthogonal* is the number of canonical efficiency factors equal to one.

**exhaustive confounding** Is a particular type of confounding that occurs when all the degrees of freedom for a recipient source are confounded by one or more recipient sources. The recipient source is said to be exhausted. That is, the subspace for the recipient source is completely occupied by subspaces associated with allocated sources. There will be no Residual degrees of freedom for the recipient source. It differs from (partial) confounding in that it is a property of a recipient source.

**experimental design** A prescription for the allocation of factors from one or more sets of objects onto another set of objects. The latter objects are the units for the design. Generally, the objective in looking for a design is to find one that is optimal for the anticipated model for the experiment.

**experimental unit** For each generalized factor that is allocated, it is the generalized factor with the most factors to which it is allocated. This differs from the definition that states that experimental unit for an allocation is the smallest entity to which the allocated objects are assigned.

(the) **factor-allocation description** (Brien, Harch, Correll and Bailey, 2011) Describes an experiment in terms of the allocation of multiple sets of objects, along with their associated tiers and the nesting and crossing relations among the factors within each tier. This information can be exhibited in a factor-allocation diagram. (Further discussion is here.)

**factor-allocation diagram** A diagram that displays the allocation(s) in an experiment. It is based on a set of panels that contains the set of factors indexing a set of objects. It is an extension by Brien, Harch, Correll and Bailey (2011) of the randomization diagram introduced by Brien and Bailey (2006) to describe, not only randomized, but also nonrandomized allocation of factors. An example and the conventions used in them are described here.

**first-order balanced design** A design that is structure balanced except that not all subspaces for allocated sources, when combined with a recipient source, are orthogonal; however, the nonzero canonical efficiency factors between any pair of allocated sources combined with the same recipient source are all equal. That is, the design does not exhibit adjusted orthogonality (Eccleston and Russel, 1992).

**first-phase design** The design for the first phase, which may be a standard design, a design with multiple allocations or an observational-study design.

**generalized factor** A factor derived one or more of the tier factors in the experiment. When there is more than one tier factor in the generalized factor, it has been called a joint factor and the levels of such a generalized factor are the observed levels combinations of the constituent tier factors. For example, suppose A, B and C are tier factors with levels *a*, *b* and *c* levels, respectively, and all combinations of the three factors are observed. Then A∧B∧C is the generalized factor with *abc* levels formed from all combinations of the levels of the three tier factors. A generalized pseudofactor involves pseudofactors and, perhaps, tier factors.

**grazing trials** Grazing trials are experiments that involve animals grazing pasture and/or crops. These experiments are often inherently multitiered as they involve the multiple randomizations of treatments to plots and the randomization of plots to animals. Brien, C.J., and Demétrio (1998a) discuss the design and analysis of grazing trials from a multitiered perspective. In particular, this clarifies the identification of the experimental units for such experiments. A more extensive discussion of the analysis of animal experiments is downloadable from Brien, C.J. and Demétrio, C.G.B. (1998b).

**homogeneous allocation model** The second of three allocation-based models to produce in formulating the mixed model for a designed experiment. Convert the initial allocation model to the homogeneous allocation model by (i) adding intertier interactions that are of interest to the researcher (e.g. interactions of factors such as Sites, Sex, Judges and so on with Treatments), (ii) augmenting the model with potentially important terms not taken into account in constructing the experimental design (e.g. sources of variation identified during or after the running of the experiment or that could not be incorporated into the design), and (iii) deciding if any terms need to be swapped from fixed to random and vice versa. The model is homogeneous in the sense that all the fixed terms have a parameter for each level of its genearlized factor and the random effects for each term are assumed independently and identically distributed.

**identity term** A term that uniquely indexes a set of objects with as many elements as there are observational units..

**incoherent unrandomized-inclusive multiple randomizations** A factor that nests factors in the first randomization of unrandomized-inclusive randomizations becomes nested with these, or equivalent, factors in the second randomization.

**independent multiple allocations** Involves three tiers of factors. Tier 2 and 3 factors are allocated to different sets of tier 1 factors in two separate allocations. Independent multiple randomizations result when both allocations involve only randomization. Diagram

**indicator variable** An indicator variable is a varaiable for the level of a (generalized) factor that takes the values 0 or 1, 0 if the observation does not have the level and 1 if it does.

**initial allocation model** Is the first of three allocation-based models to produce in formulating the mixed model for a designed experiment. Obtained directly from the factor-allocation diagram by forming terms from all posible combinations of the factors within a panel, subject to the restriction that terms involving a nested factor must include all its nesting factors. The fixed terms are those that come from the factors in a panel that was only ever allocated; the remaining terms are the random terms. The models can be compressed by using an intratier formula for the fixed terms and the sum of one or more intratier formulae for the random terms. (See Brien and Demétrio, 2009, Table 1 for notation.)

When the set of objects, and their factors, are allocated using only randomization that respects the nesting relationships between the factors as exhibited in the factor-allocation diagram then the initial allocation model is designated as the randomization model. When at least one allocated factor has not been randomized, but the factors to which they are allocated have been taken into account in the design construction, the initial allocation model is termed a quasi-randomization model. On the other hand, if not all the factors and their nesting relationships, shown in a factor-allocation diagram, were taken into account in the design construction, the initial allocation model is called an assumed model.

**intertier interactions** Interactions that involve factors from different tiers. The intertier interaction ‘belongs’ to the tier of the factor(s) in it that were allocated to the tiers of the remaining factors. It is a more general term for block-treatment interactions. See Brien and Bailey (2006, section 7.1) for a fuller discusion of them.

**intratier formulae** The formulae derived from the relationships between the factors tier. In a factor-allocation diagram, these are the factors within a panel.

**laboratory allocation** Occurs when a later phase is a laboratory phase. It involves the allocation of units from the previous phase to laboratory units, and perhaps of laboratory treatments to either earlier-phase units or laboratory units.

**laboratory phase** The phase of an experiment in which material produced from the previous phase is processed in a laboratory. Processing in the laboratory is to be interpreted broadly as further processing, measurement, testing and so on, even if, strictly speaking, a laboratory is not involved.

**laboratory-phase design** An experimental design for achieving the laboratory allocation. It may be a standard design or a design with multiple allocations.

**laboratory replication** is the repeated processing of the same product from the previous phase(s), which may require a separate portion for each processing.

**laboratory treatments** Treatments that are introduced during a laboratory phase.

**locations** is the generic name given to the laboratory units, which are often the times at which an analysis is performed, or positions in a machine each time a set of specimens are processed together.

**multiphase design** An experimental design for a multiphase experiment. It will be the combination of several standard designs.

**multiphase experiments** (also **multi-phase experiments**.) An experiment involving several phases, the most commmon being two-phase experiments. Each phase is based on a different set of units. (For more detail and examples of applications see Brien, Harch, Correll and Bailey, 2011; Brien, 2017c).

**multiple allocations/randomizations** Occurs when more than a single allocation is performed in an experiment. There are six types of multiple allocations: composed, allocated-inclusive, recipient-inclusive, coincident, double and independent. Multiple randomizations result when all of the allocations involve only randomization. Experiments with multiple allocations are multitiered.

**multistage experiments** Experiments conducted in several distinct time intervals with the allocation of factors
for each interval. They include two-phase, superimposed and changeover experiments (but not longitudinal studies), as well as multistage reprocessing or same-unit experiments, in which the response is measured only after several stages of processing the same units (examples are available)

**multitiered experiment** An experiment that involves more than one allocation, where we are specific about how it is possible to achieve the randomization involved.
As it involves multiple - two or more - allocations it has more than two tiers, and hence it is multitiered. Multitiered experiments include multiphase, some superimposed, many animal and some plant experiments. However, the majority of textbook designs, including split-plots designs, involve a single allocation and so are two-tiered.

**nonconsonant randomization** A single randomization that is not consonant.

**nonorthogonal design** An experimental design for which one or more nonzero canonical efficiency factors are not equal to one for at least one pair of sources, one of which is allocated and the other of which is a recipient source.

**normal two-phase experiments** A two-phase experiment of the type first described by McIntyre (1955) in that they have a single randomization in each phase.

**object** Something that either is allocated, the allocated objects, or has something allocated to it, the recipient objects. In a standard design, treatments are the allocated objects and units are the recipient objects.

**observational unit** The smallest physical entity that yields a single value of a response variable and that is the unit that is never allocated.

**one-two allocations** Allocations in which there are two sets of recipient objects to which the one set of allocated objects are allocated. This contrasts with a chain of allocations and two-one allocations. Double multiple allocations are the only known type of allocation that is a one-two allocation. Diagram

**orthogonal design** An experimental design for which (i) all the nonzero canonical efficiency factors between any allocated and any recipient source must equal one and the number of nonzero canonical efficiency factors equals the degrees of freedom of the allocated source, and (ii) a set of orthogonal subspaces must result from combining the subspace for a recipient source with the subspace for each allocated source confounded with it; this second condition is a form of adjusted orthogonality (Eccleston and Russel, 1992).

**panel** A box or oval in a factor-alocation diagram that lists the factors in a tier, along with their numbers of levels and nesting relations. A factor that is nested is followed by `in' and a list of the first letters of the names of the factors within which it is nested.

**phase** The period of time during which a set of units are engaged in producing their outcome. The outcome can be material for processing in the next phase, or values for response variables, or both. Only the final phase need have a response variable. Also, one phase might overlap another phase.

If the units are a random sample, then no allocation is involved and the study is an observational study. If a allocation is involved then there maybe a single allocation or multiple allocations. Multiple allocations cannot be composed or allocated-inclusive because the units cannot change during a phase. Clearly, the use of phase here is to be distinguished from its use in clinical trials.

**prior allocation model** The third of three allocation-based models to produce in formulating the mixed model for a designed experiment. Derive it from the homogeneous allocation model by (i) changing the parameterization of those random terms for which it is to be assumed that their effects display unequal variance or are correlated, (ii) consider removing random terms because of potential singularities in the variance matrix, and (iii) reparameterize those fixed for which linear or curved trends are to be fitted.

**pseudofactor** A factor that groups levels of another factor, but does not represent a new source of differences in the experiment. It is usually used as to identify a set of subspaces for a term that will produce a balanced analysis. They occur more frequently in multitiered than in two-tiered experiments. They were introduced by Yates in connection with balanced lattice square designs. Yates' pseudofactors represent groups of treatments to be applied to rows or columns in a row-column design. His pseudofactors still represent Treatment contrasts -- it is just that recognition of these Treatment groups aids the analysis in that it is balanced with respect to the pseudofactors. For a fuller discussion see Monod and Bailey (1992) and for their application in multiphase experiments see Brien, Harch, Correll and Bailey, 2011.

**pseudosource** A source that includes at least one pseudofactor.

**pseudotier** The combination of two or more tiers that are to be treated as a single set of factors for one of the allocations in a multitiered experiment. The panels for the tiers are enclosed in a dashed, rounded rectangle in a factor-allocation diagram.

(a) **randomization** An allocation that involves only randomization. Thus, a single randomization involves two sets of objects, the allocated objects that are randomized to the recipient objects. We also refer to the recipient objects as the unrandomized objects and the allocated objects as the randomized objects for this randomization. A randomization is achieved by allocating, at random, the levels combinations of the allocated tier of factors to the recipient tier of factors using the following procedure:

- for the recipient objects, write down an unrandomized list of the levels combinations of factors from both tiers according to the design being used;
- identify the set of all permutations of the levels combinations for the factors from the recipient tier for the design being used --- this set involves restrictions imposed by the design being used;
- to randomize the levels combinations of factors from the allocated tier, randomly choose a permutation of the recipient objects from the set all possible permuations and apply it to the levels combinations of factors from the recipient tier.

You now have the list, for all factors, of levels combinations that will occur in this experiment. It is likely to be desirable to arrange the list in standard order for the recipient factors. The confounding resulting from a randomization can be displayed using a decomposition table

**randomization-based mixed model** An allocation-based mixed model for an experiment in which there has only been randomization.

**randomization diagram** A diagram introduced by Brien and Bailey (2006) for displayng the randomization in an experiment. Factor-allocation diagrams are an extension of randomization diagrams and were introduced by Brien, Harch, Correll and Bailey (2011). An example and the conventions used in them are described here.

**randomization model** A model whose properties are solely derived from the randomization (for more information see Bailey (1981)). The corresponding mixed model can be obtained by employing the procedure for deriving the decomposition table, but omitting any intertier interactions.

**randomized-inclusive multiple randomizations** (deprecated) Allocated-inclusive multiple allocations that only involve randomization. Now referring to them as allocated-inclusive multiple randomizations would be more appropriate. Diagram

**recipient factors/objects** The set of factors that index the objects to which another set of objects are allocated in an allocation. The set of factors will occupy a panel in the factor-allocation diagram, so that they form a tier, and the panel is named for the objects.

**recipient-inclusive multiple allocations** (formerly unrandomized-inclusive) Involve three tiers of factors. Tier 2 factors are allocated to tier 1 factors in the first allocation. Then tier 3 factors are allocated to the combined tier 1 and 2 factors. Recipient-inclusive multiple randomizations result when both allocations involve only randomization. Diagram

**second-phase design** An experimental design for the second phase. It may be a standard design or a design with multiple allocations. If the second phase is a laboratory phase, it is a laboratory-phase design.

**set of generalized factors** for a tier/panel Consists of the generalized factors formed from all subsets of the factors within that tier/panel, except that nested factors never occur without the factors that nest them.

(the) **single-set description** The smallest set of factors, including the factors of interest to the researcher, that is sufficient to uniquely index the units in the experiment. This may necessitate the incorporation of a single synthetic factor that is nested within the other factors and whose levels differ for the repeats of the combinations of the other factors (Brien, Harch, Correll and Bailey, 2011, Section 3.2). It differs from the factor-allocation description in that the latter identifies multiple sets of objects, along with their associated factors. (See an example of its use.)

**single-stage experiments** Experiments consisting of only one time interval during which there is an allocation (single or multiple) followed by the conduct of the experiment. Single-stage experiments involving multiple allocations include some grazing and some plant experiments. Examples are available.

**skeleton analysis-of-variance table** A table that is the same as a decomposition table, except that any pseudosource is replaced with the corresponding genuine source and, optionally, expected mean squares may be added manually using rules for doing this. So a decomposition table is a precursor to a skeleton analysis-of-variance table, which in turn is a precursor for an analysis-of-variance table. The analysis-of-variance table includes statistics computed from observed data.

**source** A subspace of the data space that is orthogonal to all term subspaces for terms derived from the factors in the same tier. The dimension of a source subspace is called its degrees of freedom. Sources occur in the decomposition table and are of two forms a) A#B#C meaning the interaction of A, B and C; b) A#B[C∧D] meaning the interaction of A and B nested within the observed combinations of C and D. (cf term) There is a simple rule for deriving a source from a term. Some sources may be genuine sources and others pseudosources. A genuine source is comprised only of genuine factors whereas a pseudosource includes at least one pseudofactor. (cf. term)

**standard design** Defined to be the result obtained from allocating one set of objects to a second set of objects, the latter to be called the units.

**storage experiments** Experiments in which the effect of storage on products is investigated. They typically involve two, or more, phases the first phase being the production phase and the subsequent phases the storage phases. Examples are available.

(a) **structure** The set of subspaces or sources associated with a set of objects. The set of subspaces consists of a subspace for each term and has a corresponding source in the decomposition table. In many cases a structure will be derived the crossing and nesting relations between a set of factors indexing a set of objects. It may include sources derived from pseudofactors.

**structure-balanced design** A design that is balanced with respect to specific structures. To be structure balanced a design must meet all the conditions for an orthogonal design, except that, while all the nonzero canonical efficiency factors between an allocated and a recipient source must be equal, they do not have to equal one. If they are all equal, but not to one, for one or more pairs of an allocated and a recipient source then the design is nonorthogonal with structure balance. A design that is not structure balanced is said to be unbalanced. For further details see Brien and Bailey (2009) or Brien (2017c).

**symbolic mixed model** Contain a list of the fixed terms followed by a list of the random terms, the lists being separated by a vertical line (|). Identity terms are underlined. Functions that indicate that trend are to be fitted or that correlation between random is to be included. The notation is outlined by Brien and Demétrio (2009, Table 1).

**term** A potential term in a mixed model corresponding to a generalized (pseudo)factor. Such terms are parameterized in terms of indicator variables with the number of associated parameters equal to the number of levels of the generalized factor. In symbolic models, each terms is represented by a generalized factor. For each term, there is a term subpace of the data space that is of dimension equal to the number of its parameters. The terms can be derived from analysis formulae by expanding these formulae. (cf source)

**tier** A set of factors indexing one of the sets of objects involved in an allocation. All of the factors in a tier have the same status in the allocation and will occupy a single panel in the factor-allocation diagram for an experiment. The factors in different tiers must have been associated using an allocation; those in the same tier have not. In standard textbook designs, where there is only one allocation,
the factors in the two tiers are often referred to as the unit (or block) factors and the treatment factors, the unit factors being the recipient factors and the treatment factors being the allocated factors. However, just two names, like 'unit' and 'treatment', are inadequate for experiments involving more than two tiers. Instead a general term for set of factors grouped according to the allocations is used.

**tier factor** A factor that is a member of a tier. That is, not a generalized factor that is is comprised of several factors.

**treatment** A, perhaps conceptual, object that is allocated to one or more units or recipient objects.

**two-one allocations** Allocations in which two sets of allocated objects are allocated to the one set of recipient objects. This contrasts with a chain of allocations and one-two allocations. Recipient-inclusive, coincident and independent multiple allocations lead to two-one allocations. Diagram

**two-phase experiments** Multiphase experiments that consist of just two phases. First described by McIntyre (1955) and discussed by Cox (1958, Section 5.5), although Cox used the term ‘stage’ rather than ‘phase’.

That is, they are experiments in which a) two separate time periods are involved and b) the units on which the experiment is conducted in the second phase are completely different to those in the first phase. They are usually multitiered because they generally employ multiple allocations, at least one in each phase; two-phase experiments that employ an observational-study design in the first phase may not be multitiered. A very common type of two-phase experiment is one that involves a field phase and a laboratory phase. In the field phase, a field trial is carried out on field units and then the produce from that experiment is taken in to the laboratory where the experiment involves laboratory units. Other sources of two-phase experiments include (i) experiments that involve a storage phase for the second phase (ii) greenhouse experiments that involve two greenouse phases, the plants being moved from one room/greenhouse in the first phase to another room/greenouse in the next phase, and (iii) experiments that involve multiple phases in the laboratory, such as some gene-expression and plant pathology experiments. (For a review of where multiphase experiments have been applied see Brien, 2017c.)

A subset of these are the normal two-phase experiments.

More generally, experiments can be multiphase.

**unbalanced designs** Those that do not meet the conditions for structure balance.

**unique indexing** The indexing of a set of objects by a set of factors in which each object has a unique combination of the levels of the factors.

**units** (for an experimental design) Form a set of objects, the recipient objects, to which one or more other sets of objects is allocated. For a standard design only one set of objects, called the treatments, are allocated to the one set of recipient objects, called the units. For other designs, one or more sets of treatments or of units may be assigned. The units for an observational study design are the selected objects. Observational units are the units that are never allocated.

**unrandomized-inclusive multiple randomizations** (deprecated) Recipient-inclusive multiple allocations that only involve randomization. Now referring to them as recipient-inclusive multiple randomizations would be more appropriate. Diagram.