There is another set of values,
the `canonical_efficiency_factors`, that
are used to evaluate a design but which
has not yet been discussed. Let be the number of units
receiving treatment (this is the general diagonal element of ) and let be the diagonal matrix with
the along the diagonal. The *canonical efficiency
factors*

for design are the largest eigenvalues of . The remaining eigenvalue of is .

In the incomplete block design, the variance of the estimator of
is equal to
, while the
variance in a completely randomized design with the same replication
is
, where the two values of are
the variances per plot in the incomplete block design and the
completely randomized design respectively. Therefore the relative
efficiency is

The first part of this, which depends on the design but not on the values of the plot variances, is called the

which is equal to if is an eigenvector of with eigenvalue .

Since is symmetric, it can orthogonally diagonalized. The
contrast is called a *basic contrast* if for
an eigenvector of which is not a multiple of , where
is the all- vector. The basic contrasts span the space of
all treatment contrasts; moreover, if is orthogonal to
then the estimators of and are
uncorrelated (and independent if the errors are normally distributed).

Each efficiency factor lies between 0 and 1; at the extremes are contrasts that cannot be estimated (efficiency factor ) and contrasts that are estimated just as well as in an unblocked design with the same (efficiency factor ). Thus is the proportion of information lost to blocking when estimating a corresponding basic contrast (or any contrast in its eigenspace); is the proportion of information retained. Design is disconnected if and only if .

The comparison to a completely randomized design
*with the same replication
numbers* is the key concept here.
Efficiency factors evaluate design over the
universe of all designs with
the same replications
as ,
constraining the earlier discussed reference universe of competitors
with the given and block size distribution.
This constrained universe of comparison is typically justified as
follows: the replication numbers have been purposefully chosen (and thus fixed)
to reflect relative interest in the treatments, or the replication
numbers are forced by the availablity of the material (for example,
scarce amounts of seed of new varieties but plenty of the control
varieties),
so the task is to determine a best (in whatever sense) design within
those constraints. The idealized best (in every sense) is
the completely randomized design (no blocking) *so long as this
does not increase the variance per plot*. Though experimental
material at hand has forced blocking, the unobtainable CRD can still
be used as a fixed basis for comparison.

Variances of contrasts estimated with a CRD exactly mirror the
selected sample sizes. If the replication numbers are intended
to reflect relative interest in treatments, then a reasonable design
goal is to find for which variances of all contrast estimators
enjoy the same relative magnitudes as in the CRD. This is exactly
the property of *efficiency balance*: design is
*efficiency balanced* if its canonical efficiency factors are all equal:
.

For equal block sizes (), the only equireplicate, binary, efficiency balanced designs are the BIBDs. Unfortunately, an unequally replicated design cannot be efficiency balanced if the block sizes are constant and it is binary. Thus in many instances the best hope is to approximate the relative interest intended by the choice of sample sizes. Approximating efficiency balance (seeking small dispersion in the efficiency factors) will then be a design goal, typically in conjunction with seeking a high overall efficiency factor as measured through one or more summary functions of the canonical efficiency factors. The harmonic mean of the canonical efficiency factors (see below) is often called ``the'' efficiency factor of a design; if the value is 0.87, for instance, then use of blocks has resulted in an overall 13% loss of information.

For an equireplicate design (all are equal--to say)
the canonical efficiency factors are just 1/ times the inverses
of the canonical variances; some statisticians consider them a more
interpretable alternative to the canonical variances in this case.
If all the efficiency factors are 1, the design is
*fully efficient*, a property achieved
in the equiblocksize case (with ) only by complete block
designs. Consequently, efficiency factors for equireplicate designs
can also be interpreted as
summarizing the loss of information when using incomplete blocks
(block sizes smaller than ) rather than complete blocks.

The external representation contains the following commonly
used
`summaries_of_efficiency_factors`. In terms of these
measures, an optimal design is one which *maximizes* the value.
Each summary measure induces a design ordering which is identical to
that for one of the `optimality_criteria` above, based on the
canonical variances, *provided* the set
of competing designs is restricted to be equireplicate. More generally, these
measures should only be used to compare designs with the
same replication numbers.

`harmonic_mean`

This is the harmonic mean of the efficiency factors. Equivalent to (produces the same design ordering as) in the equireplicate case.`geometric_mean`

This is the geometric mean of the efficiency factors. Equivalent to (produces the same design ordering as) in the equireplicate case.`minimum`

The smallest efficiency factor (). Equivalent to in the equireplicate case.

The Introduction gives an example of a block design which is called the
Fano plane. It is a BIBD for treatments in blocks of size .
As with any BIBD, it is `pairwise_balanced`,
`variance_balanced`, and `efficiency_balanced`, and
it is optimal with respect to all of the `optimality_criteria`
over its entire reference universe. Here are all of the
`statistical_properties`, that have been discussed so far, for
this example:

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