Efficiency factors

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 $r_{i}$ be the number of units receiving treatment $i$ (this is the general diagonal element of $A'_d A_d$) and let $R$ be the diagonal matrix with the $\sqrt{r_{i}}$ along the diagonal. The canonical efficiency factors

$$e_{d1} \leq e_{d2} \leq \cdots \leq e_{d,v-1}$$

for design $d$ are the $v-1$ largest eigenvalues of $F_d=R^{-1} C_d R^{-1}$. The remaining eigenvalue of $F_d$ is $0$.

In the incomplete block design, the variance of the estimator of $x'\tau$ is equal to $x'C_d^{-}x\sigma^2_{\rm {IBD}}$, while the variance in a completely randomized design with the same replication is $x'R^{-2}x\sigma^2_{\rm {CRD}}$, where the two values of $\sigma^2$ are the variances per plot in the incomplete block design and the completely randomized design respectively. Therefore the relative efficiency is

$$\frac{x'R^{-2}x}{x'C_d^-x}\times \frac{\sigma^2_{\rm CRD}}{\sigma^2_{\rm IBD}}$$

The first part of this, which depends on the design but not on the values of the plot variances, is called the efficiency factor for the contrast $x'\tau$. Put $R^{-1}x = u$. Then the efficiency factor for $x'\tau$ is

$$\frac{u'u}{u'F_d^-u},$$

which is equal to $\varepsilon$ if $u$ is an eigenvector of $F$ with eigenvalue $\varepsilon$.

Since $F_d$ is symmetric, it can orthogonally diagonalized. The contrast $x'\tau$ is called a basic contrast if $x=Ru$ for an eigenvector $u$ of $F_d$ which is not a multiple of $Ru_0$, where $u_0$ is the all-$1$ vector. The basic contrasts span the space of all treatment contrasts; moreover, if $u_1$ is orthogonal to $u_2$ then the estimators of $(Ru_1)'\tau$ and $(Ru_2)'\tau$ 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 $= 0$) and contrasts that are estimated just as well as in an unblocked design with the same $\sigma^2$ (efficiency factor $= 1$). Thus $1-e_{di}$ is the proportion of information lost to blocking when estimating a corresponding basic contrast (or any contrast in its eigenspace); $e_{di}$ is the proportion of information retained. Design $d$ is disconnected if and only if $e_{d1}=0$.

The comparison to a completely randomized design with the same replication numbers is the key concept here. Efficiency factors evaluate design $d$ over the universe of all designs with the same replications $r_{1},\ldots,r_{v}$ as $d$, constraining the earlier discussed reference universe of competitors with the given $v$ 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 $d$ 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 $d$ is efficiency balanced if its canonical efficiency factors are all equal: $e_{d1} = e_{d2} = \ldots = e_{d,v-1}$.

For equal block sizes $k$ ($<v$), 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 $r_{i}$ are equal--to $r$ say) the canonical efficiency factors are just 1/$r$ 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 $k \leq v$) 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 $v$) 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

$(v-1) / \sum (1/e_{di})$
This is the harmonic mean of the efficiency factors. Equivalent to (produces the same design ordering as) $\Phi_1$ in the equireplicate case.

geometric_mean

$exp(\sum {log(e_{di})/(v-1)} )$
This is the geometric mean of the efficiency factors. Equivalent to (produces the same design ordering as) $\Phi_0$ in the equireplicate case.

minimum

The smallest efficiency factor ($e_{d1}$). Equivalent to $E_1$ in the equireplicate case.

The Introduction gives an example of a block design which is called the Fano plane. It is a BIBD for $7$ treatments in $7$ blocks of size $3$. 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:

<statistical_properties precision="9">
    <canonical_variances no_distinct="1" ordered="true">
        <value multiplicity="6"><d>0.428571429</d></value>
    </canonical_variances>
    <pairwise_variances>
        <function_on_ksubsets_of_indices domain_base="points" k="2" n="7"
         ordered="true">
            <map>
                <preimage>
                    <entire_domain>
                    </entire_domain>
                </preimage>
                <image><d>0.857142857</d></image>
            </map>
        </function_on_ksubsets_of_indices>
    </pairwise_variances>
    <optimality_criteria>
        <phi_0>
            <value><d>-5.08378716</d></value>
            <absolute_efficiency><z>1</z></absolute_efficiency>
            <calculated_efficiency><z>1</z></calculated_efficiency>
        </phi_0>
        <phi_1>
            <value><d>0.428571429</d></value>
            <absolute_efficiency><z>1</z></absolute_efficiency>
            <calculated_efficiency><z>1</z></calculated_efficiency>
        </phi_1>
        <phi_2>
            <value><d>0.183673469</d></value>
            <absolute_efficiency><z>1</z></absolute_efficiency>
            <calculated_efficiency><z>1</z></calculated_efficiency>
        </phi_2>
        <maximum_pairwise_variances>
            <value><d>0.857142857</d></value>
            <absolute_efficiency><z>1</z></absolute_efficiency>
            <calculated_efficiency><z>1</z></calculated_efficiency>
        </maximum_pairwise_variances>
        <E_criteria>
            <E_value index="1">
                <value><d>0.428571429</d></value>
                <absolute_efficiency><z>1</z></absolute_efficiency>
                <calculated_efficiency><z>1</z></calculated_efficiency>
            </E_value>
            <E_value index="2">
                <value><d>0.857142857</d></value>
                <absolute_efficiency><z>1</z></absolute_efficiency>
                <calculated_efficiency><z>1</z></calculated_efficiency>
            </E_value>
            <E_value index="3">
                <value><d>1.28571429</d></value>
                <absolute_efficiency><z>1</z></absolute_efficiency>
                <calculated_efficiency><z>1</z></calculated_efficiency>
            </E_value>
            <E_value index="4">
                <value><d>1.71428571</d></value>
                <absolute_efficiency><z>1</z></absolute_efficiency>
                <calculated_efficiency><z>1</z></calculated_efficiency>
            </E_value>
            <E_value index="5">
                <value><d>2.14285714</d></value>
                <absolute_efficiency><z>1</z></absolute_efficiency>
                <calculated_efficiency><z>1</z></calculated_efficiency>
            </E_value>
            <E_value index="6">
                <value><d>2.57142857</d></value>
                <absolute_efficiency><z>1</z></absolute_efficiency>
                <calculated_efficiency><z>1</z></calculated_efficiency>
            </E_value>
        </E_criteria>
    </optimality_criteria>
    <other_ordering_criteria>
        <trace_of_square_of_C>
            <value><d>32.6666667</d></value>
            <absolute_comparison><z>1</z></absolute_comparison>
            <calculated_comparison><z>1</z></calculated_comparison>
        </trace_of_square_of_C>
        <max_min_ratio_canonical_variances>
            <value><d>1.0</d></value>
            <absolute_comparison><z>1</z></absolute_comparison>
            <calculated_comparison><z>1</z></calculated_comparison>
        </max_min_ratio_canonical_variances>
        <max_min_ratio_pairwise_variances>
            <value><d>1.0</d></value>
            <absolute_comparison><z>1</z></absolute_comparison>
            <calculated_comparison><z>1</z></calculated_comparison>
        </max_min_ratio_pairwise_variances>
        <no_distinct_canonical_variances>
            <value><z>1</z></value>
            <absolute_comparison><z>1</z></absolute_comparison>
            <calculated_comparison><z>1</z></calculated_comparison>
        </no_distinct_canonical_variances>
        <no_distinct_pairwise_variances>
            <value><z>1</z></value>
            <absolute_comparison><z>1</z></absolute_comparison>
            <calculated_comparison><z>1</z></calculated_comparison>
        </no_distinct_pairwise_variances>
    </other_ordering_criteria>
    <canonical_efficiency_factors no_distinct="1" ordered="true">
        <value multiplicity="6"><d>0.777777778</d></value>
    </canonical_efficiency_factors>
    <functions_of_efficiency_factors>
        <harmonic_mean alias="A">
            <value><d>0.777777778</d></value>
        </harmonic_mean>
        <geometric_mean alias="D">
            <value><d>0.777777778</d></value>
        </geometric_mean>
        <minimum alias="E">
            <value><d>0.777777778</d></value>
        </minimum>
    </functions_of_efficiency_factors>
</statistical_properties>