Biometrika (1970), 57, 2, p. 339
Printed in Cheat Britain
339
The efficiency of block designs in general
B Y S. C. PEARCE
East Mailing Research Station
SXJMMABY
1. INTRODUCTION
In experimentation, restrictions of block size may well preclude the use of well tried
designs like randomized blocks. Many alternatives have been suggested but even so the
experimenter may have to devise a novel approach in order to obtain the highest precision
possible with the experimental material available. Consequently information about the
precision and efficiency of block designs in general is desirable; a number of special cases is
not enough.
A general approach to the problem was given by Tocher (1952). He started with the
incidence matrix n, the elements of which give the number of times each treatment occurs
in each block, there being a row for each treatment and a column for each block. Hence
nl = r, the vector of treatment replications, and n'l = k, the vector of block sizes, where
1 is a vector of ones. It will be seen that N, the total number of plots, equals both r ' l and
k'l. He adopted a notation whereby xa* represents a diagonal matrix with elements,
x\, x%,..., where xv x2>..., are the elements of a vector x.
A different notation was used by Jones (1959), who worked with two design matrices,
D for blocks and A for treatments. Each has a column for each plot and a row for each block
or treatment. An element equals zero unless the plot is in the block or receives the treatment,
in which case it equals one. Then n = AD', k* = DD', r* = AA', Dl = k, Al = r and
D'l = 1 = A'l. If the data are written as a vector y, the block totals B equal Dy and the
treatment totals T equal Ay. The grand total 0 equals both B'l and T ' l . The general
mean, GjN, will be written as m.
Tocher drew attention to the important matrix SI such that
SI-1 = r»- nk-*n' + rr'IN.
(1)
Thu8 if
'
O = T-nk-*B,
(2)
then the vector of treatment parameters y is estimated by SIQ. If the general mean m is
added to each element, the result is the vector of adjusted treatment means a, i.e.
a = ml + J2Q.
(3)
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BIM 57
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By the efficiency of a design is meant its precision relative to that of an orthogonal design
with the same treatment replications. The general case of orthogonal block designs is first
studied, because it provides the basis of comparison. An expression is then given for the
efficiency of a general block design with respect to the estimation of treatment means. It
leads to a relationship between the mean efficiency of a design and its dual. However,
efficiency depends upon the equations of constraint that are adopted. For the estimation of
differences of treatment means no design is ever more efficient than for the estimation
of the means themselves. Also, for this purpose, the efficiency depends upon a set of
quantities, which are here named 'quasi-replications', rather than upon the actual
replications.
340
S. C. PEAKCE
I t has the covariance matrix
ry «
,^.
where a2 is the error mean square in the analysis of variance. The sum of squares attributable
to treatments in the same analysis is
,_.
n , o n
All these results were obtained using the equations of constraint
k'p = 0 = r'y,
(6)
where p is the vector of block parameters.
Note that Sl~H = r. Therefore,
o
\ t)
2. ORTHOGONAL DESIGNS
Since orthogonal designs have a special role in the calculations of efficiency they will be
examined first. Orthogonality may be defined in several ways but it will appear that all
definitions are equivalent.
It is usually defined by the independence of the adjusted treatment means a, i.e. by all
the off-diagonal elements of Si, and hence of S2"1 also, being zero. If this is so, Sl~x = r* and
nk~*n' — rr'/N = 0. To generalize an argument first given by Tocher (1952), the ith
diagonal element of nk~*n' — rr'/N is
(ytyy
{{aMpl,}
= 0.
Therefore, for all i and,?', n^ = rikj/Nt i.e.
n = rk'/tf.
(8)
That is to say, all blocks are made up proportionately in the same way with respect to
treatments and all treatments are distributed in proportionately the same way between
the blocks. Conversely, substituting (8) into (1) we find that
SI = r-«,
(9)
i.e. the adjusted treatment means are distributed independently.
Another definition makes use of the property that the actual treatment means r~*T
equal the adjusted ones, ml + SIQ, whatever the data may be. If
(10)
then
r-*Ay = (11 'IN + £2A - S2nk~*D) y.
Since, relative to the design, y is an arbitrary vector, the matrices by which it is premultiplied
on the two sides of the equation must be equal. If each is premultiplied by r* and postmultiplied by D', the result leads to (8). Conversely, if (8) is substituted in (2), it appears
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This paper is concerned with the precision and efficiency of block designs in the general
case. Where a design is required to estimate a quantity and does so with variance 6<r2, it
will be said to have a precision of 1/0 for that estimate. If an orthogonal design with the
same treatment replications would have given a variance 0<r2, the efficiency of the first
design is denned as <pjd. If there are several quantities to be estimated, a mean value for
efficiency will be expressed as the harmonic mean of the individual values.
Efficiency of block designs in general
341
that 0 = T —mr. Since (8) implies (9) also, the right hand side of (10) reduces to the left
hand side by virtue of (7).
Again, orthogonality may be defined by the property of the adjusted treatment sum of
squares Q'SIQ being always equal to the unadjusted T'r~*T — G*jN, whatever the data.
If that is true,
After omitting the arbitrary vector y, the two matrices left must be equal. If each is premultiplied by A and postmultiplied by D', it follows that
(r*-nk-*n')ft(n-n) =
n-rk'/N.
Q'SIQ = ( T ' - r a r ' ) r - ' ( T - m r )
= T'r- J T-G 2 /iV.
Since all the commonly recognized properties of orthogonal designs are necessary and
sufficient conditions of (8) and hence of one another, it will be convenient to write
n^rk'/N + A,
(11)
where A may be termed the nonorthogonality matrix, having all elements equal to zero
in the orthogonal case. It may be noted that
Al = n l - r k ' l / i V = 0
and similarly
A 1 = Q
(12)
Consequently, if A does have a nonzero element, it must have at least four to make all
rows and columns sum to zero.
3. EFFICIENCY WITH RESPECT TO MEANS
Substituting (11) into (1), we see that
A'.
(13)
From this point it will be convenient to adopt a subsidiary notation and write the square
roots of the block sizes as K and the square roots of the treatment replications as p, i.e.
2
K2* = k* and p * = r*. Hence, writing % = P~*AK~* and H = %%', we obtain
Because H can be obtained as the product of a matrix and the transpose of that matrix, it
cannot have any negative eigenvalues.
From (14) it follows that
,T „ ,
Writing
V.O +
we find that
V=
W'
=
=
SI
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Since the left hand side equals 0, it follows that (8) must hold. Conversely, substituting (8)
into (2) and making use of (9), we see that
342
S. C. PEAKCE
-1
because of (7). Since (I — H ) may be written as (p*V)(p'V)', like H it has no negative
eigenvalues.
Let the eigenvalues of H be AA, each corresponding to an eigenvector uft, then
where AA > 0 for all h. Further, the eigenvalues of (I — H) are (1 — Aft) and those of (I — H)" 1
are 1/(1 — Ah). Since none of these last is negative,
0 ^ Ah < 1.
(16)
x
£2 = p-*(I + H + H 2 +...)p-' s ,
ClV
%u;)}pi
(17)
(18)
h
where
.. ,.,
, .
Consider the efficiency of the design for estimating adjusted treatment means, dt = £lu
and <f>t = l/r t . Writing v for the number of treatments and & for the mean efficiency, i.e. the
harmonic mean of the values ^JO^ we find that
h
say, and hence
= v + E,
g = v\{v + E)
(19)
(20)
4. RELATIONSHIP BETWEEN A DESIGN AND ITS DVAJ.
In experimentation it sometimes happens that the blocks correspond to the residual
effect of former treatments or in some other way are of interest on their own account. It
may then be important to dualize the design and to study the block differences ehminating
those due to treatments instead of vice versa. For the dual design, A' replaces A, %' replaces
%, while H ' is replaced by
R3 . = „, _ ,
Since the trace of a product of two matrices does not depend on the order in which they are
multiplied, and since H^ can be made into HJ' by transferring the initial %' to the end, it
follows that
Replacing v, the number of treatments, by b, the number of blocks, we see that (19) becomes
?* =b + E.
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If any AA equals 1, Si~ will be singular, which it is in certain special cases, for example,
when the design is totally confounded, but these will not be considered here.
From these results it is justifiable to expand (I — H)" 1 and to write
Efficiency of block designs in general
From (20),
343
E = v{\-S)\S.
Hence>
*„ = b§\{v + (b - v) g).
(21)
This result is already known for equi-replicate designs (Pearce, 1968) but is here seen to
apply generally. Note that if b > v, then $* > S. Hence, of the two designs that which on
average has fewer plots for the groups, blocks or treatments, under study has the higher
mean efficiency, i.e. the mean precisions are less disparate than would be the case had the
design been orthogonal.
5. EFFECT OF EQUATIONS OF CONSTRAINT
6. EFFICIENCY WITH RESPECT TO DIFFERENCES OF MEANS
Considering now the efficiency of a general design with respect to the estimation of the
difference between two adjusted treatment means, of and ajt we shall see that
6ij = (Qii + Qjj-2Qij)
,
,
,
0tf =(*/»••+V»V)-
whereas
The efficiency is 0#/0#.
Before proceeding to the evaluation of the mean efficiency, it will be helpful to prove a
LEMMA.
Let uhi be the element of uh in (15) that corresponds to treatment i, then
i
Let
.
z
Mj = PiUhi + PiUhj>
Una = Piuu~ Piuu,
Then
*
I
(22)
S{
^
.
(23)
The upper bound for Xh is clearly 2(v — 1); the lower bound depends upon the maximum
values
\
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These results were obtained using the conventional equations of constraint set out in (6).
They are not altered by the substitution of some other linear function for ry = 0; such a
change may lead to all estimates of yi being increased by a constant quantity but since the
same quantity will be subtracted from the estimate of a, the values of a = a l + y will remain
the same. A change in the relationship, k'(S = 0, will, however, alter a with no compensating
change in y and will, therefore, affect the estimate of a.
This will not affect differences of adjusted treatment means and therefore the efficiency
of their estimation will remain unchanged. It will, however, affect the adjusted treatment
means themselves and therefore the results obtained in the last two sections will no longer
apply. In general the choice of equations of constraint can be arbitrary, but sometimes it is
determined by the problem under study.
344
S. C. P B A R C E
Since S^Ai = 1, «£< + «« < 1. If u = \uhi\, then \uhj\ ^ (l-v?)l.
I zMj I < Piu+PAl
~ u<i)h = VMP
sa
Therefore,
y•
Differentiating yhii by u to find a maximum, we find that when u2 = p\\{p\ +pj), VMJ n a s i*8
greatest value, which is (pf +pj)$- Returning to (23), we see that the lower bound of XAis
(v- ), i.e.
v-1 ^Xh^2(v-1).
(24)
On the basis of (24), the derivation of $', the mean efficiency of the design with respect
to differences between treatment means, can proceed. We have
l
On account of (18),
s
»(»-i)iSJ
o
Since, from (19), J^/ih = E, and making use of (24), we obtain
h
l+Ejv < \\S' < 1 + 2E/V,
/)<^</.
(25)
It appears that no design ever has a higher mean efficiency for estimating the differences
between treatment means than it has for estimating the means themselves. On the other
hand, the difference need not be large. The greatest range in the possible values of &' comes
when $ = 0-586, the bounds being then 0-414 and S. For designs with usefully high efficiency
(«? ^ 0-9, say) £' will never fall much below S.
For designs in which all blocks are the same size and all treatments are equally replicated,
$' is completely determined by $ (Pearce, 1968), the relationship being
S' = (v-\)£\{v-8).
(26)
7. QTJASI-BEPLICATION
Although with differences of means, as with means, it is usual to determine efficiency with
reference to what might have been achieved using an orthogonal design with the same
replications, r, the practice obscures the role of replication when only contrasts are of
interest.
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1
g'
Efficiency of block designs in general
345
It is in fact possible to determine the precision of estimation of a difference of means in
another way. Take the matrix nk~*n' and set all its diagonal elements equal to zero, the
resulting matrix being written M. Let
Ml = q.
(27)
The elements ofq will be termed the 'quasi-rephcations' of the treatments. Letr —q = d,
then nk-*n' = M + d*. From (1), it follows that
ft-i = (d» + q*) - (M + d*) + IT'/ N
q'l = U,
t o 1 = q ' - M + qq'/tf.
Since co""1! = q, it is permissible to write wq = 1, analogous to (7). Clearly,
to- 1 = n - i + qq'/tf-rr'/iV.
Premultiplying both sides by to and post multiplying by SI, we see that
Si = <o + lq'ft/ V - tori '/JV.
For calculating the precision of estimation of treatment contrasts, SI and to lead to the same
results, for if c'a is a linear function of adjusted treatment means such that c'l = 0, then
c'£2c = c'toc. In fact, it has proved possible to express precision without reference to the
actual replications r by using the quasi-replications q instead.
This result becomes explicable when q is examined further. If a treatment i is studied to
find how far its mean differs from that of all the other treatments, the precision of estimation of the difference is qt. To show this result, consider the experiment that arises when all
treatments other than i are merged. Since for all designs, SI-1! = r, and a diagonal element
„
V
„,„
i-1i-riljy
,
r
The determinant of S2-1 is Nqv Inverting the matrix, we obtain £2u + £222 — 2Q12 = l/g1,
thus justifying the assertion. In fact, q measures the precision with which each treatment is
compared with the others. For purposes of studying contrasts it is more relevant than r, but
the two cannot be divorced entirely because the more plots there are of a treatment,
provided they are dispersed over the blocks, the better the comparisons are between that
treatment and the others. On the other hand, the importance of q provides a warning
against using blocks that are too small. In the early days of statistics as applied to experimentation, most work was being done in an agricultural context in which large blocks led to
undesirably large values of the error variance. In such circumstances it was reasonable to
explore the uses of small blocks, but that end is not to be pursued at the expense of quasireplications that are too small.
Note that
_ , „ ,, „
S(/*)
(29)
If blocks are kept small, i.e. k} is small, a design may result that is poor at estimating contrasts. On the other hand, large blocks facilitate comparisons; see Pearce (1964). In practice
a balance has to be preserved between high precision and a reasonable level of error variance.
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T 6ot
346
S. C. PEABCE
REFERENCES
JONES, R. M. (1959). On a property of incomplete blocks. J. R. Statist. Soc. B 21, 172-9.
PEABCE, S. C. (1964). Experimenting with blocks of natural size. Biometrics 20, 699-706.
PEAECE, S. C. (1968). The mean efficiency of equi-replicate designs. Biometrika 55, 251-3.
TOCHEB, K. D. (1952). The design and analysis of block experiments (with discussion). J. R. Statist.
Soc. B 14, 45-100.
[Received August 1969. Revised December 1969]
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