Statistical Properties

For a statistician, a block design is a plan for an experiment. The
points of the block design are usually called *treatments,* a
general terminology encompassing any set of *v* distinct
experimental conditions of interest. The purpose of the experiment is to
compare the treatments in terms of the magnitudes of change they induce
in a response variable, call it . These magnitudes are called
*treatment effects.*

In a typical experiment (there are *many* variations on this, but
we stick to the basics to start), each treatment is employed for the
same number of experimental runs. Each *run* is the application
of the treatment to an individual *experimental unit*
(also called *plot*) followed by
the observation of the response . An experiment to compare
treatments using runs (or ``replicates'') requires a total of
experimental units.

If the experimental units are *homogeneous* (for the purposes
of the experiment, essentially undifferentiable) then the assignment of
the treatments, each to units, is made completely at random.
Upon completion of the experiment, differences in treatment effects are
assessed via differences in the means of the observed values for
the treatments (each mean is the average of observations). This
simplest of experiments is said to follow a *completely randomized
design* (it is *not* a block design).

The concept of a *blocked* experiment comes into play when the
experimental units are *not* homogeneous. A *block* is just a
subset of the experimental units which are essentially undifferentiable,
just as described in the previous paragraph. If we can partition our
heterogeneous units into sets (blocks) of homogeneous units
each, then after completion of the experiment, when the statistical
analysis of results is performed, we are able to isolate the variability
in response due to this systematic unit heterogeneity.

To make clear the essential issue here, consider a simple example. We have fertilizer cocktails (the treatments) and will compare them in a preliminary greenhouse experiment employing potted tobacco plants (the experimental units). If the pots are identically prepared with a common soil source and each receiving a single plant from the same seed set and of similar size and age, then we deem the units homogeneous. Simply randomly choose two pots for the application of each cocktail. This is a completely randomized design. At the end of the experimental period (two months, say) we measure = the total biomass per pot.

Now suppose three of the plants are clearly larger than the remaining
three. The statistically ``good'' design is also the intuitively
appealing one: make separate random assignments of the three cocktails
to the three larger plants, and to the three smaller plants, so that
each cocktail is used once with a plant of each size. We have blocked
(by size) the 6 units into two homogeneous sets of 3 units each, then
randomly assigned treatments within blocks. Notice that there are
3!3!=36 possible assignments here; above there were 6!=720 possible
assignments. Because this is called a *complete* block
design.

The statistical use of the term ``block design'' should now be clear: a block design is a plan for an experiment in which the experimental units have been partitioned into homogeneous sets, telling us which treatment each experimental unit receives. The external representation is a bit less specific: each block of a block design in external representation format tells us a set of treatments to use on a homogeneous set (block) of experimental units but without specifying the exact treatment-to-unit map within the block. The latter is usually left to random assignment, and moreover, does not affect the standard measures of ``goodness'' of a design (does not affect the information matrix; see below), so will not be mentioned again.

There are solid mathematical justifications for why the complete block design in the example above is deemed ``good,'' which we develop next. This development does not require that , nor that the block sizes are all the same, nor that each treatment is assigned to the same number of units. However, it does assume that the block sizes are known, fixed constants, as determined by the collection (of fixed size) of experimental units at hand. Given the division of units into blocks, we seek an assignment of treatments to units, i.e. a block design, that optimizes the precision of our estimates for treatment effects. From this perspective, two different designs are comparable if and only if they have the same , , and block sizes (more precisely, block size distribution).

Statistical estimation takes place in the context of a model for the observations . Let denote the observation on unit in block . Of course we must decide what treatment is to be placed on that unit - this is the design decision. Denote the assigned treatment by . Then the standard statistical model for the block design (there are many variations, but here this fundamental, widely applicable block design model is the only one considered) is

where is the treatment effect mentioned earlier, is the effect of the block (reflecting how this homogeneous set of units differs from other sets), is an average response (the treatment and block effects may be thought of as deviations from this average), and is a random error term reflecting variability among homogeneous units, measurement error, and indeed whatever forces that play a role in making no experimental run perfectly repeatable. In this model the 's have independent probability distributions with common mean 0 and common (unknown) variance .

With the total number of experimental units in a block design, the design map (note: symbol is used both for the map and the block design itself) from plots to treatments can be represented as an incidence matrix, denoted . Also let be the treatment/block incidence matrix, let be the diagonal matrix of block sizes ( for equisized blocks), and write

which is called the *information matrix* for design (note:
denotes the transpose of a matrix ). Why this name? Estimation
focuses on comparing the treatment effects: every *treatment
contrast*
with is of possible
interest. All contrasts are *estimable* (can be linearly and
unbiasedly estimated) if and only if the block design is connected. For
disconnected designs, all contrasts within the connected treatment
subsets span the space of all estimable contrasts. For a given design
, we employ the *best* (minimum variance) linear unbiased
estimators for contrasts. The variances of these estimators, and their
covariances, though best for given , are a function of . In fact,
if is the vector of contrast coefficients then the variance
of contrast
is

where is the Moore-Penrose inverse of (if is the spectral decomposition of , then ). The information carried by is the precision of our estimators: large information corresponds to small variances as determined by .

We wish to make variances small through choice of . That is, we
choose so that is (in some sense) small. *Design
optimality criteria* are real-valued functions of
that it is desirable
to minimize. Obviously a design criterion may also be thought of as a
function of itself, which we do when convenient.

With this background, let us turn now to what has been implemented for
the external representation of `statistical_properties`:

statistical_properties = element statistical_properties { attribute precision { xsd:positiveInteger } , canonical_variances ? , pairwise_variances ? , optimality_criteria ? , other_ordering_criteria ? , canonical_efficiency_factors ? , functions_of_efficiency_factors ? , robustness_properties ? }

The elements of `statistical_properties` are quantities which
can be calculated starting from the information matrix .

- Canonical variances
- Pairwise variances
- Optimality criteria
- Other ordering criteria
- Efficiency factors
- Robustness properties
- Computational details
- Design orderings based on the information matrix