## Glossary

**analysis formulae** The formulae displaying the relationships between the factors within a tier and incorporating intertier interactions. Their derivation is based on the tiers.

**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 randomization employed in the experiment. There is a a major column for each analysis formula.

**coincident multiple randomizations** Involves three tiers of factors. Tier 2 and 3 factors are randomized to the same tier 1 factors in two separate randomizations. Diagram

**composed multiple randomizations** Involves three tiers of factors. Tier 3 factors are randomized to tier 2 factors in one randomization and tier 2 factors are randomized to tier 1 factors in the second randomization. Each randomization involves different units. Like randomized-inclusive randomizations, the aim of composed randomizations is to randomize tier 3 factors to tier 2 factors and then to to randomize these factors to tier 1 factors. The difference is that only for composed randomizations can tier 3 factors be ignored in the second randomization, where tier 2 factors are randomized to tier 1 factors. Diagram

**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. 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 randomization employed in the experiment. There is a a major column for each analysis formula, consisting of the sources, degress of freedom and, for nonorthogonal experiments, the efficiency factors.

**double multiple randomizations** Involves three tiers of factors. Tier 3 factors are randomized to both tier 1 and 2 factors in two randomizations. Diagram

**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.

**experimental unit** For each generalized factor that is allocated, it is the generalized factor with the most factors to which it is allocated.

(a) **factor allocation** Involves the allocation of one or more sets of objects to another set of objects. This allocation may involve both randomization and systematic ellements.

**the factor-allocation description** 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.

**factor-allocation diagram** A diagram that displays the factor allocation in an experiment. It is an extension of the randomization diagram introduced by Brien and Bailey (2006) to describe, not only randomized, but also nonrandomized allocation of factors. The conventions used in them are described here.

**factor-allocation mixed models** An extension of randomization-based mixed models that includes systematically assigned terms. The terms in the model are derived from the factor-allocation diagram.

**generalized factor** A factor derived from the observed levels combinations of two or more of the original factors in the experiment. For example, suppose A, B and C are 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 original factors. A generalized pseudofactor involves pseudofactors and, perhaps, original 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).

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

**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 randomizations** Involves three tiers of factors. Tier 2 and 3 factors are randomized to different sets of tier 1 factors in two separate randomizations. 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.

**intertier interactions** Interactions that involve factors from different tiers. 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 within a tier.

**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 randomization. It may be a standard design or a design with multiple randomizations.

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

**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 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.

**multiple randomizations** Occurs when more than a single randomization is performed in an experiment. There are six types of multiple randomizations: composed, randomized-inclusive, unrandomized-inclusive, coincident, double and independent. Experiments with multiple randomizations are multitiered.

**multistage experiments** Experiments conducted in several distinct time intervals with randomization 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 randomization, where we are specific about how a randomization is defined.
As it involves multiple - two or more - randomizations it has more than two tiers, and hence it is multitiered. Multitiered experiments include two-phase, some superimposed, many animal and some plant experiments. However, the majority of textbook designs, including split-plots designs, involve a single randomization and so are two-tiered.

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

**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 or has something allocated to it, for example treatments are allocated to units in a standard design.

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

**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 randomization is involved and the study is an observational study. If a randomization is involved then there maybe a single randomization or multiple randomizations. Multiple randomizations cannot be composed or randomized-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.

**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 subspaces of 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. The pseudofactors here represent groups of treatments to be applied to rows or columns in a row-column design. The 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).

**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 randomizations in a multitiered experiment. The panels for the tiers are enclosed in a dashed oval in a factor-allocation diagram.

(a) **randomization** It is one form of factor allocation. A single randomization involves two sets of objects, one of which is randomized to the other. We shall refer to the two sets as the unrandomized and randomized objects for this randomization. Each set of objects is indexed by a set of factors that is referred to as a tier. The function of a randomization is to confound sources from the randomized tier with certain sources from the unrandomized tier. This is achieved by assigning, usually at random, the levels combinations of the factors from the randomized tier to those of the factors from the unrandomzied tier using the following procedure:

- for the unrandomized 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 unrandomized 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 randomized tier, randomly choose a permutation of the unrandomized objects from the set all possible permuations and apply it to the levels combinations of factors from the unrandomized 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 unrandomized factors. The confounding resulting from a randomization can be displayed using a decomposition table

**randomization diagram** A diagram introduced by Brien and Bailey (2006) for displayng the randomization in an experiment. Factor-allocation diagrams are an extension of them. 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.

**randomization-based mixed model** A mixed model for an experiment in which there has only been randomization and the model is based on the randomization employed. In our case we take this to mean that the tiers have been used to formulate the terms in the model as described in the procedure for deriving the decomposition table. Factor-allocation models are an extension of them to include systematically assigned terms,

**randomized-inclusive multiple randomizations** Involves three tiers of factors. Tier 3 factors are randomized to tier 2 factors in the first randomization. Then the combined tier 3 and 2 factors are randomized to the tier 1 factors. Each randomization involves different units. Like composed randomizations, the aim of randomized-inclusive randomizations is to randomize tier 3 factors to tier 2 factors and then to to randomize these factors to tier 1 factors. The difference is that only for composed randomizations can tier 3 factors be ignored in the second randomization, where tier 2 factors are randomized to tier 1 factors. Diagram

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

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

**single-stage experiments** Experiments consisting of only one time interval during which there is randomization (single or multiple) followed by the conduct of the experiment. Single-stage experiments involving multiple randomizations 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. 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 nesting 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.

**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 balance** A nonorthogonal experiment that is balanced with respect to specific structures. That is, the sources in each structure are orthogonal and any relevant relationships between sources from different structures are balanced in that there is a single angle between them so that they have a single efficiency factor. For further details see Brien and Bailey (2009).

**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, the terms are represented by generalized factors. 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 a randomization. All of the factors in a tier have the same status in the randomization. The factors in different tiers must have been associated using a randomization; those in the same tier have not. In standard textbook designs, where there is only one randomization,
the two tiers are often referred to as block factors and treatment factors, the block factors being the unrandomized factors and the treatment factors being the randomized factors. However, just two names, like 'block' and 'treatment', are inadequate for experiments involving more than two tiers. Instead a general term for set of factors grouped according to the randomization is used.

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

**two-phase experiments** Experiments consisting of 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 multitiered, with multiple randomizations, as at least one randomization is perfomed in each phase. 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. Another source of two-phase experiments are experiments that involve a storage phase for the second phase.

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

More generally, experiments can be multiphase.

**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 to which a one or more other sets of objects is assigned. For a standard design only one set of objects, called the treatments, are assigned. 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 randomized.

**unrandomized-inclusive multiple randomizations** Involves three tiers of factors. Tier 2 factors are randomized to tier 1 factors in the first randomization. Then tier 3 factors are randomized to the combined tier 1 and 2 factors. Diagram