Understanding the Structure of the Experimental Design
Bate and Chatfield (2016a) describe a
five-stage process to identify the structure of the experimental design.
These stages are summarised in Table 1.
Table 1: Overview of the proposed five-stage process to identify the
Layout Structure (Bate and Chatfield, 2016a).
|
Stage
|
Outcome
|
Purpose of stage
|
Properties of the outcomes from the stage
|
What the stage outcomes are derived from
|
|
1
|
A list of factors
|
Researcher should aim to identify the factors that are a) ‘of interest’
and/or b) inherent to the experimental material or c) additional
potential sources of variability.
|
The list must contain all factors the researcher wishes to include in
the experimental design.
|
Hypotheses to be tested and background knowledge of the experimental
material.
|
|
2
|
Factor levels
|
To define the factor levels – this is required so that the structure of
the experimental design can be identified algorithmically.
|
For each factor, the factor levels should be specified uniquely.
|
Background knowledge of the practicalities of the experiment and how the
experiment will be managed.
|
|
3
|
Experiment Scheme
|
The researcher defines the properties the experimental design should
have, identify the class of design they wish to use and potentially the
non-randomised experimental design itself.
|
A list of the factors and factor levels that define the design, a list
of the properties the design should have which should respect the
relationships between inherent factors, the choice of design class and,
if possible, the non-randomised design.
|
The factors, factor levels and background knowledge of the experimental
material.
|
|
4
|
Layout Structure
|
To summarise the relationships between the factors in the experimental
design to help identify the structure of the experimental design.
|
A list of factors and generalised factors and the relationships between
the factors.
|
Factors, factor levels, nested/crossed/equivalent relationships between
the factors as defined in the Experiment Scheme and the experimental
design plan (generated using the Experimental Factors table and Layout
Structure table).
|
|
|
Hasse diagram of the Layout Structure
|
To visualise the structure of the experimental design.
|
A diagram illustrating the crossed/nested/equivalent relationships
between the factors/generalised factors in the design.
|
Layout Structure.
|
|
5
|
Degrees of freedom
|
To identify potential weaknesses in the design (i.e. lack of
replication) and degrees of freedom in a statistical analysis using a
model based on the factors/generalised factors in the Layout Structure.
|
For each factor/generalised factor the number of levels of the factor,
the number of levels present in the design and the associated degrees of
freedom.
|
Factor levels, experimental design and Hasse diagram.
|
Stage 1. Identifying the experimental factors
The process begins by identifying all of the effects (and hence
corresponding factors) that may influence the outcome of the response
variable or are needed to define the structure of the experimental
material. The scientist should consider the following:
S1(a) All ‘factors of interest’ which include, but
are not limited to, ‘treatment’ factors.
S1(b) All factors that either define the physical
structure of the experimental material or are inherent to the conduct of
the proposed experiment, (the inherent factors) not already considered
in S1(a).
S1(c) Factors that identify likely experimental
units, if not already defined in S1(a) and S1(b). At this stage it is
useful for the scientist to consider which factors they plan to assign
to experimental units (‘treatment’ factors) and what experimental units
they are to be assigned to. Factors should be included that uniquely
index any likely experimental units.
S1(d) The factor that identifies the observational
units, if not already defined in S1(a-c).
S1(e) Any additional effects that are likely sources
of variability in the experiment and, unless choosing to control them at
a specific level, ensure that there are factors included in the process
to account for them (for example, additional blocking factors or effects
of time that have not been considered inherent in S1(b)).
Once the set of factors has been generated, it may be decided to
ignore some of the inherent factors when constructing the design as they
do not correspond to potential sources of variability (the ignored
inherent factors). Care should be taken when doing this however,
inherent factors should be retained in the process if the scientist
cannot rule them out as potential sources of variability. Even if it is
intended to ignore an inherent factor in the design, that factor should
be retained in the process at this stage as a prompt to the researcher
that the design will be ‘non-conforming’ in stage 3.
Stage 2. Specifying the factor levels
At this stage the researcher should make an initial assessment of the
factor level labels using the following process:
S2(a) The factor level labels must be uniquely
specified and have a practical meaning within the experiment.
S2(b) Following an assessment of the design it may
be the case that the factor levels change, for example if the
experimental design is amended to more efficiently account for nuisance
sources of variability.
So, for example, if there are twenty subjects in an experiment, ten
per treatment arm (say treatment or control), then the subjects should
be labelled 1 to 20 and not 1 to 10 for the treatment group, 1 to 10 for
the control group.
Stage 3. Choosing the Experiment Scheme
Once the factors and levels have been identified in stages 1 and 2,
the Experiment Scheme describes the planning stages of the
experiment and the decisions taken before randomising the experimental
material. It consists of:
S3(a) A list of the factors and factor levels, as
described in stages 1 and 2.
S3(b) A list of all the properties that the
experimental design should have (for example, a decision to equally
replicate the treatments or a decision that all the treatments should be
administered within each block).
S3(c) The type of design required (for example,
block, split-plot, cross-over, etc.).
S3(d) The non-randomised design itself, which
specifies the non-randomised allocation of treatments to the
experimental units, i.e. a list of the combinations of the levels of the
factors present in the design for each observational unit. If the
non-randomised design is unavailable, then the randomised allocation can
be used instead.
Stage 4. Identifying the structure of the experimental
design and constructing the Layout Structure
Once the factors have been identified, the factor levels chosen and
the Experiment Scheme devised, the Layout Structure can be
constructed. The Layout Structure consists of:
S4(a) A list of ‘structural’ objects consisting of
the factors that define the experimental design (identified in stage
1).
S4(b) The generalised factors (that correspond to
combinations of the factors) that are implied by the structure of the
experimental design.
S4(c) A description of the nested, crossed and
equivalent relationships between the factors, as exhibited in the
experimental design.
An algorithmic approach that can be used to generate the Layout
Structure is described in Bate and Chatfield (2016a). The Layout
Structure can be visualised using a generalisation of the Hasse Diagram
and the hassediagrams package can be used to generate the
diagram in R, see below.
Stage 5. Assigning the levels and degrees of
freedom
The levels and degrees of freedom can now be determined for the
factors and generalised factors present in the Layout Structure.
S5(a) For each generalised factor present in the
Layout Structure we begin by calculating the total number of
combinations of the levels of the factors that are present in the
design.
S5(b) We also calculate the total number of
potential levels. This is the theoretical number of levels rather than
the number of observed levels. If a factor is nested within another
factor, then the number of potential levels for this factor is taken as
the number of levels that the nested factor has (and not the number of
potential levels of the nested factor crossed with the factor which
nests it, as usually this will be practically meaningless).
S5(c) For equivalent factors/generalised factors
with different numbers of potential levels, it is the highest number
that is used as the number of potential levels.
S5(d) To calculate the degrees of freedom for the
factor/generalised factor \(G_{u}\),
all degrees of freedom from the factors and generalised factors that
nest \(G_{u}\) in the Layout Structure
are subtracted from the number of levels of \(G_{u}\) present in the design. This is
similar to the approach outlined by Tjur
(1984) and is defined as the subtraction approach.
Example: Split-plot split-block experiment
Federer (1975), Example 6.1, describes
a split-plot split-block experiment consisting of two treatment
factors, TA (levels: \(a_0, a_1\) and
\(a_2\)) and TB (levels: \(b_0, b_1, b_2\) and \(b_3\)). The experiment was broken down into
three blocks, where each block consists of a \(4 \times 3\) grid of plots. The rows and
columns that define the individual plots within blocks are labelled
across blocks. This implies the researcher believes there are systematic
effects across the rows and columns of the plots within the blocks. So
the Row factor is defined as a factor with four levels and the Column
factor with three levels. In the Experiment Scheme it is decided to use
a Latin square to allocate the levels of treatment TA to the columns of
the design and a Youden rectangle to allocate the levels of treatment TB
to the rows of the design. The experiment scheme is given in Table 2,
and the Hasse diagram of the Layout Structure for this design is given
in Figure 1. Note there are some warning messages indicating shared
degrees of freedom not included.
Table 2: The experiment scheme for the split-plot split-block experiment
including TA, TB, Block, Row and Column levels.
|
Block I
|
|
|
|
|
|
\(c_1\)
|
\(c_2\)
|
\(c_3\)
|
|
\(r_1\)
|
\(a_0, b_0\)
|
\(a_1, b_0\)
|
\(a_2, b_0\)
|
|
\(r_2\)
|
\(a_0, b_1\)
|
\(a_1, b_1\)
|
\(a_2, b_1\)
|
|
\(r_3\)
|
\(a_0, b_2\)
|
\(a_1, b_2\)
|
\(a_2, b_2\)
|
|
\(r_4\)
|
\(a_0, b_3\)
|
\(a_1, b_3\)
|
\(a_2, b_3\)
|
|
|
|
|
|
|
Block II
|
|
|
|
|
|
\(c_1\)
|
\(c_2\)
|
\(c_3\)
|
|
\(r_1\)
|
\(a_1, b_3\)
|
\(a_2, b_3\)
|
\(a_0, b_3\)
|
|
\(r_2\)
|
\(a_1, b_0\)
|
\(a_2, b_0\)
|
\(a_0, b_0\)
|
|
\(r_3\)
|
\(a_1, b_1\)
|
\(a_2, b_1\)
|
\(a_0, b_1\)
|
|
\(r_4\)
|
\(a_1, b_2\)
|
\(a_2, b_2\)
|
\(a_0, b_2\)
|
|
|
|
|
|
|
Block III
|
|
|
|
|
|
\(c_1\)
|
\(c_2\)
|
\(c_3\)
|
|
\(r_1\)
|
\(a_2, b_2\)
|
\(a_0, b_2\)
|
\(a_1, b_2\)
|
|
\(r_2\)
|
\(a_2, b_3\)
|
\(a_0, b_3\)
|
\(a_1, b_3\)
|
|
\(r_3\)
|
\(a_2, b_0\)
|
\(a_0, b_0\)
|
\(a_1, b_0\)
|
|
\(r_4\)
|
\(a_2, b_1\)
|
\(a_0, b_1\)
|
\(a_1, b_1\)
|
Using the structure of the experimental design and the randomisation
to construct a mixed model
Once the first 5 stages have been performed, and the Layout Structure
generated and the Hasse Diagram of the Layout Structure produced, the
researcher can then move onto the randomisation and identify a suggested
mixed model. These final three stages are summarised in Table 3 (see
Bate and Chatfield, 2016b).
Table 3: Overview of the proposed three-stage process to identify the
Restricted Layout Structure.
|
Stage
|
Components produced
|
Purpose of component
|
Properties of the component
|
What the component is derived from
|
|
6
|
Randomisation
|
Randomise the experimental material.
|
A randomised plan of the experimental design.
|
The randomisation performed and the un-randomised experimental design.
|
|
|
Description of the randomisation
|
Defining the randomisation in terms of randomisation objects to aid in
model selection.
|
A list of randomisation objects that define the randomisation performed
with randomisation arrows connecting them (randomisation objects
‘involved in the randomisation’).
|
Knowledge of the randomisation performed.
|
|
7
|
Categorisation of factors as fixed or random
|
To aid in the analysis and also model selection.
|
A list of random factors and a list of fixed factors.
|
Knowledge of the properties of the factors, the hypotheses being tested
and the nesting of factors in the design.
|
|
|
Restricted Layout Structure
|
To construct a list of the randomisation objects (those involved in the
randomisation and the randomisation objects that nest them) to aid in
model development.
|
A list of randomisation objects.
|
The Layout Structure and the randomisation object pairs that are
connected by randomisation arrows.
|
|
|
Hasse diagram of the Restricted Layout Structure
|
To visualise the structure of the experiment as implied by the
randomisation performed.
|
A diagram illustrating the crossed and nested relationship between the
randomisation objects.
|
Layout Structure and the Restricted Layout Structure.
|
|
8
|
The mixed model formation
|
Identify the terms to include in the mixed model.
|
A list of model terms.
|
The objects in the Restricted Layout Structure.
|
Stage 6b. Describing the randomisation
In order to construct a randomisation-based statistical model, the
randomisation is defined using an arrow notation and associated
randomisation objects. This approach is an adaptation of the method
described in Bate and Chatfield (2016b). The terminology
randomisation objects is used to differentiate them from the
structural objects (the factors/generalised factors) that
describe the structural relationships between the factors as defined by
the experimental design.
Consider a randomisation that involves two factors/generalised
factors \(F_{u}\) and \(F_{v}\), where the levels of \(F_{u}\) are randomly assigned to the levels
of \(F_{v}\) (\(F_{u}\) is randomised to \(F_{v}\) for short):
If the levels of \(F_u\) are
randomised to the levels of \(F_v\),
and no other factor/generalised factor is involved in the randomisation,
then write:
\[
F_u \rightarrow F_v. \tag{1}
\]
If the levels of \(F_u\) are
randomised to the levels of \(F_v\)
separately within each level of a third factor/generalised factor \(F_w\), then write:
\[
F_u \rightarrow F_v[F_w]. \tag{2}
\]
If the levels of \(F_u\) are
randomised to the levels of the generalised factor \(F_v \wedge F_w\) by independently
randomising the levels of \(F_v\) and
\(F_w\), then write:
\[
F_u \rightarrow F_v \otimes F_w, \tag{3}
\] where the \(\otimes\) symbol
denotes that the levels of factors \(F_v\) and \(F_w\) have been permuted
separately.
If the levels of the generalised factor \(F_u \wedge F_v\) are randomised to the
levels of the factor \(F_w\) by
independently randomising the levels of \(F_u
\wedge F_v\), then write:
\[
F_u \wedge F_v \rightarrow F_w. \tag{4}
\]
If the levels of the generalised factor \(F_u \wedge F_v\) are randomised to the
levels of the factor \(F_w\) by
randomising the levels of \(F_u\)
independently to the randomisation of the levels of \(F_v\), then write:
\[
F_u \otimes F_v \rightarrow F_w. \tag{5}
\]
Randomisation objects at the beginning and end of the randomisation
arrows are said to be involved in the randomisation. Each
randomisation object (factor or generalised factor) has a corresponding
structural object; these are formally defined in Table 4. The levels of
the randomisation objects are equivalent to the levels of the
corresponding factors/generalised factors. Note the relationship between
the structural object \(F_v \wedge
F_w\) (in the Layout Structure) and the randomisation objects
\(F_v \wedge F_w\) and \(F_v \otimes F_w\) (at either end of a
randomisation arrow). The levels of both objects correspond to the
combinations of \(F_v\) and \(F_w\).
Table 4: Randomisation objects and corresponding structural objects that
share the same levels.
|
Randomisation object
|
Corresponding structural object
|
|
\(F_v\)
|
\(F_v\)
|
|
\(F_v \wedge F_w\)
|
\(F_v \wedge F_w\)
|
|
\(F_v [F_w]\)
|
\(F_v (F_w)\)
|
|
\(F_v \otimes F_w\)
|
\(F_v \wedge F_w\)
|
|
\(\{ F_v \otimes F_w \} [F_x]\)
|
\(F_v \wedge F_w (F_x)\)
|
|
\(F_v \otimes \{ F_w [F_x] \}\)
|
\(F_v \wedge F_w (F_x)\)
|
|
\(F_v [F_w \otimes F_x]\)
|
\(F_v (F_w \wedge F_x)\)
|
|
\(F_x \otimes F_v \otimes F_w\)
|
\(F_x \wedge F_v \wedge F_w\)
|
As commented by Brien and Bailey
(2006), randomisation arrows can be aligned in either direction.
In the proposed approach, if the two randomisation objects at either end
of an arrow have different numbers of levels, then the arrow begins at
the randomisation object that corresponds to the structural object with
fewer levels. If the randomisation objects at either end of an arrow
correspond to equivalent structural objects, then the arrow can point in
either direction.
As with structural objects, it is possible to define a nesting
hierarchy for randomisation objects. This nesting hierarchy will be used
to identify randomisation objects to be included in the Restricted
Layout Structure. The nesting of randomisation objects, defined as
randomisation-nesting, is given in Table 5. Note that any
object which is a randomisation-nest object is also defined as a
randomisation object. These randomisation-nesting relationships share
many similarities with the structural-nesting of objects in the Layout
Structure. For example, consider the randomisation object \(F_v [F_w]\). As the levels of \(F_v\) are randomised separately within each
level of \(F_w\), then this implies the
researcher believes that \(F_v\) nests
\(F_w\) in the experimental design. For
\(F_v \otimes F_w\), the levels of
\(F_v\) and \(F_w\) are independently randomised, and
this implies the researcher believes that \(F_v\) is crossed with \(F_w\) in the experimental design. The
randomisation-nesting reflects the implications that the choice of
randomisation has on the perceived structure of the experimental
design.
Table 5: Randomisation objects and the objects that randomisation-nest
them.
|
Randomisation object
|
Randomisation-nest objects
|
|
\(F_v\)
|
Mean
|
|
\(F_v \wedge F_w\)
|
Mean, \(F_w\)
|
|
\(F_v [F_w]\)
|
Mean, \(F_w\)
|
|
\(F_v \otimes F_w\)
|
Mean, \(F_v, F_w\)
|
|
\(\{ F_v \otimes F_w \} [F_x]\)
|
Mean, \(F_v [F_x], F_w [F_x],
F_x\)
|
|
\(F_v \otimes \{ F_w [F_x] \}\)
|
Mean, \(F_v \otimes F_x, F_w [F_x],
F_v, F_x\)
|
|
\(F_v [F_w \otimes F_x]\)
|
Mean, \(F_v \otimes F_x, F_v,
F_w\)
|
|
\(F_x \otimes F_v \otimes F_w\)
|
Mean, \(F_x, F_v, F_w, F_x \otimes
F_v, F_x \otimes F_w, F_v \otimes F_w\)
|
Stage 7a. Defining the experimental factors as fixed or
random
Once the randomisation has been defined and performed, so that the
final allocation of the factor levels to the experimental units is
confirmed, the researcher can then consider the model for the
statistical analysis. The Restricted Layout Structure, which consists of
a list of randomisation objects, is generated by considering the Layout
Structure, the randomisation performed and also whether the experimental
factors are fixed or random.
In this paper it is assumed that the experimental factors can be
defined as either fixed or random, see Armitage,
Berry, and Matthews (2008). For many factors this is
straightforward, but some factors could be considered as either fixed or
random, depending on the specific question the analyst wishes to answer
(page 27 from Brien (1992)) or the
physical practicalities Preece (2001). If
one or more of the factors present in a generalised factor are random,
then it is assumed that all equivalent structural objects (in the Layout
Structure) and any corresponding randomisation objects are also defined
as random.
A generalised factor will usually be defined as fixed if all the
factors that define the generalised factor are themselves fixed.
However, if a generalised factor is equivalent to a factor (or another
generalised factor) that is random, then declaring all equivalent
factors/generalised factors as random for the purpose of developing the
Restricted Layout Structure is likely to be the most sensible
choice.
Stage 7b. Constructing the Restricted Layout
Structure
The Restricted Layout Structure is constructed by first considering
the set of randomisation objects that are required to define the
randomisation arrows for the experiment. Additional randomisation
objects are then added to this set using the procedural instructions
described below, which select objects for which it is considered that
statistical inference will be supported by the randomisation. These
instructions are based on randomisation-nesting rather than the
structural-nesting defined by the experimental design. If a
non-conforming randomisation is employed so that the
randomisation-nesting relationships contradict the structural-nesting
relationships, then the randomisation-nesting overrides the
structural-nesting when following the instructions.
To generate the Restricted Layout Structure, the procedural
instructions S7b(i)-S7b(iv) should be followed:
S7b(i) To begin with, the randomisation objects that
are involved in the randomisation are included in the Restricted Layout
Structure.
S7b(ii) All randomisation objects that
randomisation-nest the randomisation objects in S7b(i) are
included in the Restricted Layout Structure.
S7b(iii) Also included is the randomisation object
that corresponds to the Mean and the randomisation object that
corresponds to the generalised factor that defines the observational
unit, regardless of whether the latter is involved in the randomisation
or not.
S7b(iv) All randomisation objects that correspond to
a fixed generalised factor of the form \(F_1
\wedge F_2 \wedge \dots \wedge F_x\) are included in the
Restricted Layout Structure if: (a) the corresponding generalised
factors exist in the Layout Structure, and (b) the fixed factors that
define them (i.e., \(F_1, F_2, \dots,
F_x\)) are themselves randomised to other factors/generalised
factors.
For example, if the fixed factors \(F_x,
F_y\) and \(F_z\) are involved
in the randomisation, then by S7b(iv) the fixed randomisation
objects \(F_x \wedge F_y, F_x \wedge F_z, F_y
\wedge F_z\) and \(F_x \wedge F_y
\wedge F_z\) are included in the Restricted Layout Structure if
they exist in the Layout Structure.
The purpose of S7b(i)-S7b(iv) is to create a Restricted
Layout Structure that can be used to construct a statistical model that
contains terms that are aligned to the choice of experimental design and
randomisation. These choices, it is assumed, were made to allow the
researcher to assess the factors ‘of interest’ and also the random
effects corresponding to potential sources of variability.
S7b(i) and S7b(ii) identify those terms for inclusion
as dictated by the randomisation selected, S7b(iii) implies the
supremum and infimum of the set of all the experimental factors will be
present in the final model, while S7b(iv) attempts to identify
randomisation objects corresponding to interactions that are ‘of
interest’.
Stage 7c. Illustrating the Restricted Layout Structure using
a Hasse Diagram
Once the Restricted Layout Structure has been constructed, it can be
visualised using a Hasse diagram. These diagrams may only involve
randomisation objects and hence illustrate the randomisation-nesting, or
where desirable for understanding, include a combination of
randomisation and structural objects. They share many similarities with
the Hasse diagrams of the Layout Structure but have some additional
features, for example, the inclusion of the randomisation arrows.
The Hasse diagram of the Restricted Layout Structure is constructed
by amending the Layout Structure Hasse diagram. Firstly, the labels of
the structural objects (factors and generalised factors) are replaced
with the corresponding randomisation objects. Structural objects in the
Layout Structure that do not have corresponding entries in the
Restricted Layout Structure should be removed, and any new lines
included that are required to indicate the direct nesting of the
randomisation objects.
Sometimes it may be useful to retain the structural labels in the
Hasse diagram of the Restricted Layout Structure alongside the
randomisation objects. Alternatively, for a complementary perspective,
two versions could be generated, one with the structural objects and one
with the randomisation objects. This can help the researcher to better
understand the structure of the experimental design. When both types of
objects are included in a label, the structural and randomisation
objects should be separated by a ‘|’ symbol rather than an ‘=’ symbol as
the latter symbol is used to indicate structural equivalence. Enclosing
both labels in a box to highlight that they correspond to each other is
also useful for clarity.
Once the entries in the diagram have been re-labelled, information
about the randomisation applied can also be included by adding the
randomisation arrows to the diagram. Where applicable, these arrows will
replace existing lines on the diagram that indicate the nesting of the
randomisation objects. Finally, the randomisation objects that
correspond to random factors/generalised factors are underlined to
differentiate them from those that correspond to fixed
factors/generalised factors.
Consider the randomisation of \(F_u\) to the levels of \(F_v\), i.e., \[
F_u \rightarrow F_v. \tag{6}
\] where at least one level of \(F_u\) is randomly assigned to two or more
levels of \(F_v\). Hence, \(F_u\) will structurally nest \(F_v\), and hence \(F_v\) is not included in the Layout
Structure as it is equivalent to \(F_u \wedge
F_v\). When creating the Hasse diagram of the Restricted Layout
Structure, the structural object \(F_u \wedge
F_v\) is re-labelled as the randomisation object \(F_v\), and the randomisation arrow points
to this entry.
Assume that factor/generalised factor \(F_u\) is fixed and \(F_w\) is random. A dotted line between them
indicating partial crossing provides a visual reminder that if a mixed
model approach is used to analyse the data, then it may be the case that
information about \(F_u\) can be
recovered both within \(F_w\) levels as
well as between \(F_w\) levels.
In extreme circumstances, these terms may have zero degrees of
freedom, given the other terms included in the model. An example of such
a design is given in section 7 of Payne
(1998), where the random Block factor is partially crossed with
\(T_1 \wedge T_2\). Partial crossing
can occur intentionally in certain types of design, e.g., fractional
factorials, but can also occur, perhaps unintentionally, when
non-conforming randomisations are performed.
Stage 8. Selecting the mixed model
The choice of method of analysis, and the final maximal model to be
fitted in the analysis, is in some respect down to the analyst. In most
cases, a maximal model that consists of the terms corresponding to the
factors and generalised factors present in the Restricted Layout
Structure provides a satisfactory model that forms the basis of the
statistical analysis when the data becomes available. This maximal model
is based on the underlying structure of the experimental design and the
randomisation performed and hence the terms in the model reflect the
decisions taken when identifying the factors of interest and the
potential sources of variability.
When using the Restricted Layout Structure to define the terms to be
included in the mixed model, any entries in the Restricted Layout
Structure containing \(\wedge\) or
\(\otimes\) are replaced with the
interaction symbol \(*\) and the \([ \dots ]\) brackets are replaced with the
nested notation \(( \dots )\). So, for
example, entries in the Restricted Layout Structure such as \(F_v \wedge F_w\) and \(F_v \otimes F_w\) would correspond to the
\(F_v * F_w\) interaction term in the
mixed model, and \(F_v [F_w]\) would
correspond to the nested term \(F_v
(F_w)\). Table 6 defines the randomisation objects and
corresponding mixed model terms.
Table 6: Corresponding randomisation objects and mixed model terms.
|
Randomisation object
|
Mixed model term
|
|
\(F_v\)
|
\(F_v\)
|
|
\(F_v \wedge F_w\)
|
\(F_v * F_w\)
|
|
\(F_v [F_w]\)
|
\(F_v (F_w)\)
|
|
\(F_v \otimes F_w\)
|
\(F_v * F_w\)
|
|
\(\{ F_v \otimes F_w \} [F_x]\)
|
\(F_v * F_w (F_x)\)
|
|
\(F_v \otimes \{ F_w [F_x] \}\)
|
\(F_v * F_w (F_x)\)
|
|
\(F_v [F_w \otimes F_x]\)
|
\(F_v (F_w * F_x)\)
|
Example: Split-plot split-block experiment
(continued)
Returning to the example, the levels of treatment TA are randomised
to the columns of the design using a Latin Square. This square is
defined by the combinations of Block and Column (i.e., the levels of the
Block\(\wedge\)Column generalised
factor), hence we write:
\[
\text{TA} \rightarrow \text{Block} \otimes \text{Column}, \tag{7}
\]
and similarly for treatment TB,
\[
\text{TB} \rightarrow \text{Block} \otimes \text{Row}. \tag{8}
\]
Both of these randomisations are conforming randomisations, given the
experiment scheme.
The Hasse diagram of the Restricted Layout Structure is given in
Figure 2.
The mixed model corresponding to this Restricted Layout Structure
includes the terms TA, TB, TA\(*\)TB,
Block, Column, Row, Block\(*\)Column
and Block\(*\)Row, where TA, TB and
TA\(*\)TB are fixed and Block, Column,
Row, Block\(*\)Column and Block\(*\)Row are random.
Figure 2: Hasse diagram of the Restricted Layout Structure for the
split-plot split-block experiment.
Disclaimer
The content of this article are reflective of the authors own
personal opinions and not necessarily those of the affiliated
institutions.
