A history of the design of experiments as seen through the papers in
100 years of the journal Biometrika
Annotated Partial List of papers on the Design of
Experiments that have appeared in Biometrika
This list purports to contain the details of all papers on the design
of experiments which have appeared in Biometrika. In addition,
there are papers from Biometrika which, in the opinion of
R. A. Bailey or A. C. Atkinson, are relevant to the development of
design. Comments are the opinions of RAB or ACA, and should probably
carry a government health warning.

``Student'':
Tables for estimating the probability that
the mean of a unique sample of observations
lies between \infty and any given distance
of the mean of the population from which the
sample is drawn.
Biometrika,
11,
1917,
pp. 414417.
Suggests using days/tests as variance
components, and doing replicates on wellseparated days.

K. Smith:
On the standard deviations of adjusted and interpolated values of an
observed polynomial function and its constants and the guidance they
give towards a proper choice of the distribution of observations.
Biometrika,
12,
1918,
pp. 185.

Egon S. Pearson:
On the variation in personal equation and the
correlation of successive judgments.
Biometrika,
14,
1922,
pp. 23102.
When one person makes
a series of similar measurements over time,
the correlations between measurements depend
on time difference.

``Student'':
On testing varieties of cereals.
Biometrika,
15,
1923,
pp. 271293.
Amendment and correction 16, 1924, p. 411.
If plots are too small
there is competition between them. Recommends
discarding the outer rows of all plots at harvest.
Systematic layouts ABC ... ABC ... lead to bias in
estimates of treatment differences. He recommends
the halfdrill strip ABBAABBA ...

``Student'':
Errors of routine analysis.
Biometrika,
19,
1927,
pp. 151164.
Expands on the concerns
in his 1917 paper.

E. S. Pearson:
The analysis of variance in cases of nonnormal variation.
Biometrika,
23,
1931,
pp. 114133.

``Student'':
The Lanarkshire milk experiment.
Biometrika,
23,
1931,
pp. 398406.
A badly designed experiment.
Lack of objective randomization led to too many
undernourished children being allocated to
milk instead of control; and the difference
between raw and pasteurized milk was confounded
with schools so there was insufficient power
to detect an important difference.

B. L. Welch:
On the ztest in randomized blocks and Latin squares.
Biometrika,
29,
1937,
pp. 2152.
Assumes arbitrary
fixed plot effects additive with treatment effects.
For `randomized blocks', expectations of differences
and of mean squares agree with those of normal
theory. Because the plot effects are fixed,
the usual mean squares are not independent.
The tail probabilities of the variance
ratio may not be those given by normal theory, but several sets
of uniformity data show good agreement. Actually he uses as statistic
the treatment sum of squares, calculates formulae
for its mean and variance, and applies a normal
approximation to that. For Latin squares he
randomizes by choosing at random from among
all Latin squares of the same size. The variance of
the treatments sum of squares depends on the
number of intercalates in the squares. Now the
normal approximation to this agrees slightly
less well with normal theory.

E. S. Pearson:
Some aspects of the problem of randomization.
Biometrika,
29,
1937,
pp. 5364.
Following some of Fisher's
examples, assumes that the purpose of randomization
is to do a randomization test. He points out
that this test depends on the statistic used.

E. J. G. Pitman:
Significance tests which may be applied to samples from any
populations. III. The analysis of variance test.
Biometrika,
29,
1938,
pp. 322335.
Similar conclusions to Welch.

``Student'':
Comparison between balanced and random arrangements of field plots.
Biometrika,
29,
1938,
pp. 363379.
A very important paper.
Until this point he and R. A. Fisher had corresponded
and agreed. Now he needed to publicly disagree.
He clearly shows that `balance' for suspected trend
leads to smaller bias in estimates of treatment
effects but tends to overestimate error (if trend
is not fitted). This agrees with much of Fisher
(except apparently in the paper
which Student is attacking), but Student concludes that
you should balance, Fisher that you should block
and randomize. Student uses uniformity data
quite differently from Welch. He calculates
the variance ratio for a particular layout
and some particular values of the variance
of treatment effects (assuming an additive model).
So he is assessing power, whereas Welch and Pitman
had been doing randomization tests on uniformity
data so were assessing significance.

J. Neyman and E. S. Pearson:
Note on some points in ``Student's'' paper on
``Comparison between balanced and random arrangements of field plots''.
Biometrika,
29,
1938,
pp. 380388.
Amplification of Student's
paper; their attempt to write what they thought
he was writing when he died.

E. S. Pearson:
Some aspects of the problem of randomization. II. An illustration of
``Student's'' inquiry into the effect of ``balancing'' in agricultural
arrangements.
Biometrika,
30,
1938,
pp. 159179.
Mathematics underpinning
Student's last paper. Balanced arrangements
have less power than randomized when treatment
differences are small compared to error, but
the situation reverses as the relative size of
treatment differences increase.

F. Yates:
The comparative advantages of systematic and randomized arrangements
in the design of agricultural and biological experiments.
Biometrika,
30,
1939,
pp. 440466.
He thinks that the purposes
of randomization are to avoid accidental bias,
to make the results credible, and to obtain
unbiased estimators both of treatment differences
and of their variances. So he is interested
in estimation rather than testing. He supports
Gosset's criticisms of Fisher, but goes on
to consider the possibility of randomizing
designs which have complicated ways of
allowing for trend, such as the Latin square
with balanced corners and the Latin square
with split plots.

Harold Jeffreys:
Random and systematic arrangements.
Biometrika,
31,
1939,
pp. 18.
He agrees with Fisher
that you should `balance or eliminate the
larger systematic effects as accurately as
possible and randomize the rest'.

R. L. Plackett and J. P. Burman:
The design of optimum multifactorial experiments.
Biometrika,
33,
1946,
pp. 305325.
Main effects plans for
symmetrical factorial experiments with all treatment factors
equireplicate. Introduced what are now called
orthogonal arrays of strength two. For factors
at 2 levels they use what are now called Hadamard
matrices, constructed by Payley's method.
For p levels with p prime, use
affine geometries via sets of mutually orthogonal
Latin squares. Note the equivalence of orthogonal
arrays with affine resolvable designs.

R. L. Plackett:
Some generalizations in the multifactorial design.
Biometrika,
33,
1946,
pp. 328332.
Orthogonal arrays of higher
strength. Also collapse of levels of any
factor in an OA (each new level corresponding
to the same number of old ones).

O. Kempthorne:
A simple approach to confounding and fractional replication in
factorial experiments.
Biometrika,
34,
1947,
pp. 255272.
Uses affine geometry over GF(p) to
reexpress the factorial designs previously given by Fisher and Finney
in terms of elementary Abelian groups.

N. L. Johnson:
Alternative systems in the analysis of variance.
Biometrika,
35,
1948,
pp. 8087.
Some discussion of
how the randomization procedure can justify the
assumption of a simple model, in a simple case.

K. A. Brownlee, B. K. Kelly and P. K. Loraine:
Fractional replication arrangements for factorial experiments with
factors at two levels.
Biometrika,
35,
1948,
pp. 268276.
Fractional factorials
with factors at two and/or four levels. Classification
according to the numbers of words of each length
in the defining contrasts subgroup.

K. A. Brownlee and P. K. Loraine:
The relationship between finite groups and
completely orthogonal squares, cubes and hypercubes.
Biometrika,
35,
1948,
pp. 277282.
Construct mutually orthogonal squares
of prime order by using elementary Abelian
groups and confounding (of course, this is the
same as the finite field method, which they don't
say: Stevens 1939 is in the references but not
cited). Extension to orders 4 and 8 by using
pseudofactors. In a cube, each plane section
should be MOLS, so p2 treatment factors
can be used. And so on. These are saturated
orthogonal arrays of strength n in p^{n} plots.

W. L. Stevens:
Statistical analysis of a nonorthogonal trifactorial experiment.
Biometrika,
35,
1948,
pp. 346367.
Introduced `sweeping' for the convergent
iterative fit of nonorthogonal terms (but did not call it
`sweeping').

K. S. Banerjee:
How balanced incomplete block designs may be made to furnish
orthogonal estimates in weighing designs.
Biometrika,
37,
1950,
pp. 5058.
Uses the blocks of an
BIBD as the weighings in a spring balance design,
plus a certain number of empty weighings to
determine the bias.

E. S. Pearson and H. O. Hartley:
Charts of the power function for analysis
of variance tests, derived from the noncentral
Fdistribution.
Biometrika,
38,
1951,
pp. 112130.

K. S. Banerjee:
Some observations on the practical aspects
of weighing designs.
Biometrika,
38,
1951,
pp. 248251.
More examples illustrating Banerjee 1950.

D. R. Cox:
Some systematic experimental designs.
Biometrika,
38,
1951,
pp. 312323.
Plots in a line, with a
loworder polynomial trend. We want treatment
contrasts to be orthogonal to this trend as
far as possible. If the design is symmetric
about its midpoint then treatment contrasts
are orthogonal to oddorder orthogonal polynomials,
so trialanderror solutions orthogonal to
the quadratic orthogonal polynomial are orthogonal to cubic
trend.

H. D. Patterson:
The construction of balanced designs for experiments involving
sequences of treatments.
Biometrika,
39,
1952,
pp. 3248.
Changeover designs for firstorder residual
effects. Each treatment occurs equally often in each period. Subjects
form a balanced (complete or incomplete) block design. Each treatment
follows each other treatment equally often. Similar condition for
(any period, last period) within subjects. Various constructions. Abelian
group construction from one or more initial blocks with some
conditions. Rowcomplete Latin squares.
Suitably labelled sets of mutually orthogonal Latin squares.

G. E. P. Box:
Multifactor designs of first order.
Biometrika,
39,
1952,
pp. 4957.

R. M. Williams:
Experimental designs for serially correlated observations.
Biometrika,
39,
1952,
pp. 151167.
Plots in a long line. Plot effects are
correlated in an AR1 or AR2. He considers designs in which there
are successive complete blocks, possibly with borders. This is
to eliminate trend, although he doesn't seem to fit trend in the
analysis. For AR1 he imposes some neighbour conditions just to simplify
the analysis: each treatment is equally often neighbour either to
every other treatment and never to itself, or to every
treatment. These are versions of undirectional neighbour balance at
distance 1. For AR2 he also requires undirectional neighbour balance
at distance 2.

K. D. Tocher:
A note on the design problem.
Biometrika,
39,
1952,
p. 189.
Short proof that orthogonal
designs are optimal.

Ralph Allan Bradley and Milton E. Terry:
Rank analysis of incomplete block designs.
I. The method of paired comparisons.
Biometrika,
39,
1952,
pp. 324345.
Each block is a subset of treatments, which are
ranked within that block by a judge. No implications for design.

G. E. P. Box and J. S. Hunter:
A confidence region for the solution of
a set of simultaneous equations with an
application to experimental design.
Biometrika,
41,
1954,
pp. 190199.

D. R. Cox:
The design of an experiment in which certain
treatment arrangements are inadmissible.
Biometrika,
41,
1954,
pp. 287295.
Block structure is sets * periods.
Treatment structure may be factorial. Within each set, treatment
levels can only increase. With 2 treatments and 4 periods the best design has
one set each of 0000 and 1111, and two each of 0001, 0011 and 0111.
Similar solutions for two treatments in 2 or 3 periods (one of constant, two
of rest; and 2 of constant, 1 of rest, respectively). A clever
trialanderror combination for 3 treatment factors at 2 levels in
8 sets and 4 periods. Also a small design for two factors at 3 levels.

Ralph Allen Bradley:
Rank analysis of incomplete block designs.
II. Additional tables for the method of paired comparisons.
Biometrika,
41,
1954,
pp. 502537.
With correction 51, 1964, p. 288.
Continuation of Bradley and Terry.

J. A. Nelder:
The interpretation of negative components of variance.
Biometrika,
41,
1954,
pp. 544548.
The randomization model, or the normal model
with correlated errors, can both lead to negative components of
variance. So it is not good practice to routinely replace any negative
estimate by zero.

M. B. Wilk:
The randomization analysis of a generalized randomized block design.
Biometrika,
42,
1955,
pp. 7079.
Each treatment occurs equally
often in all blocks. Randomization argument shows
that there is no test for block effects (of course!).
He does not assume plottreatment additivity so his
results are not concordant with other randomization arguments.

Fred C. Andrews and Herman Chernoff:
A largesample bioassay design with random
doses and uncertain concentration.
Biometrika,
42,
1955,
pp. 307315.
Given a fixed quantity
of fluid treatment, how much should be set aside
to estimate its concentration and what fractions
of the rest should be given to the experimental
animals?

Ralph Allan Bradley:
Rank analysis of incomplete block designs.
III. Some largesample results on estimation
and power for a method of paired comparisons.
Biometrika,
42,
1955,
pp. 450470.
More continuation of
Bradley and Terry.

R. C. Bose:
Paired comparison designs for testing concordance
between judges.
Biometrika,
42,
1955,
pp. 113121.
v judges assess r pairs from
n objects to choose which they prefer. Judges should form a
balanced incomplete block design with respect to pairs. In addition,
each object is assessed a times by each judge. These are called
linked paired comparison designs. Construction when
a=2: take a symmetric BIBD with concurrence 2 and block
size n; delete one block; for every pair in that block, find
the other block containing them; the remaining labels in that block are
the judges to assess that pair. Construction for even n: make a
resolved BIBD on all pairs from n; then allocate judges to replicates
according to another BIBD. For odd n, start with a 2resolved design.

J. C. Butcher:
Treatment variances for experimental designs
with serially correlated observations.
Biometrika,
43,
1956,
pp. 208212.
Extension of R. M. Williams 1952. For an
AR(m) process he requires designs neighbourbalanced at all distances
up to m, where neighbourbalance means that the number of
ordered pairs of plots at that distance receiving a pair of treatments
depends only on whether the treatments are equal.

W. D. Ray:
Sequential analysis applied to certain experimental designs in the
analysis of variance.
Biometrika,
43,
1956,
pp. 388403.
No implications for design.

P. Armitage:
Restricted sequential procedures.
Biometrika,
44,
1957,
pp. 926.
Two treatments. Patients
enter sequentially, their response is known before
the next one arrives. Fixed maximum number of
patients but also a stopping rule to stop
the trial when a given statistic reaches a
certain boundary.

John W. Wilkinson:
An analysis of paired comparison designs with
incomplete repetitions.
Biometrika,
44,
1957,
pp. 97113.
Applies the BradleyTerry
analysis to incomplete designs like Bose 1956.

S. C. Pearce:
Experimenting with organisms as blocks.
Biometrika,
44,
1957,
pp. 141149.
Assume that each treatment has a direct
effect on its plot, and the same remote effect on every other plot in
the same block. Then the difference between the two can be estimated
withinblocks, while (direct + (block size1)*remote) can be estimated
between blocks. So we want incomplete block designs with good
efficiency both between and within blocks.

D. R. Cox:
The use of a concomitant variable in selecting an experimental design.
Biometrika,
44,
1957,
pp. 150158.
With amendment 44, 1957, p. 534.
Given a reading x on each plot which is
thought to be highly correlated with the eventual response on that
plot under treatment, how should treatments be allocated? If
correlation is high, try to make treatment contrasts orthogonal to
x and fit x; otherwise discrete blocking may
be used to approximate x.

G. P. Sillitto:
An extension property of a class of balanced
incomplete block designs.
Biometrika,
44,
1957,
pp. 278279.
Write down the +1, 1 incidence matrix. If its
rows are orthogonal then the design is in the class. The tensor product of
any two such matrices gives another design in the class.

D. R. Cox:
The interpretation of the effects of nonadditivity in the Latin
square.
Biometrika,
45,
1958,
pp. 6973.
Follows the WilkKempthorne argument.
Randomization involves an element of random sampling. Discusses what
can be done if treatments and plots are not assumed to be additive.

Rita J. Maurice:
Selection of the population with the largest mean when comparisons can
be made only in pairs.
Biometrika,
45,
1958,
pp. 581586.
Teams meet in pairs and win/lose.
For 4 teams, comparison of BIBD with knockout.

G. E. P. Box and H. L. Lucas:
Design of experiments in nonlinear situations.
Biometrika,
46,
1959,
pp. 7790.

J. D. Biggers:
The estimation of missing and mixedup observations
in several experimental designs.
Biometrika,
46,
1959,
pp. 91105.
In simple designs, insert missing values so
as to minimize the residual sum of squares.

H. A. David:
Tournaments and paired comparisons.
Biometrika,
46,
1959,
pp. 139149.
Compares knockout tournaments with round robins.

R. G. Mitton and F. R. Morgan:
The design of factorial experiments: a survey
of some schemes requiring not more than 256
treatment combinations.
Biometrika,
46,
1959,
pp. 251259.
Table of factorial designs
for factors with 2, 4 or 8 levels. May be in blocks
and/or fractions. Constructed by the group method.
All main effects and twofactor interactions
are estimable.

W. A. Glenn:
A comparison of the effectiveness of tournaments.
Biometrika,
47,
1960,
pp. 253262.
Six types of tournament. Each game has two
players and results in win/lose. If the criterion is the expected
number of games to produce an overall winner then the simple knockout
is best. If the criterion is the probability that the best player
wins (assuming a total order on ability) then the best is the knockout
with each game replaced by the best of three. For a mixture of these
two criteria, one or other of these two tournaments is usually best.

S. C. Pearce:
Supplemented balance.
Biometrika,
47,
1960,
pp. 263271.
An incompleteblock design with all blocks of
the same size and with one control treatment is defined to have
supplemented balance if all noncontrol treatments have the same
replication and have the same concurrence with the control, and if all
pairs of noncontrol treatments have the same concurrence.
Example for 4 noncontrol treatments in 4 blocks of size 7: all blocks
like AABCDOO, where O is the control. Example for 6
noncontrol treatments in 9 blocks of size 6: all blocks like
ABCDEO and three blocks of ABCDEF.

John Leroy Folks and Oscar Kempthorne:
The efficiency of blocking in incomplete block designs.
Biometrika,
47,
1960,
pp. 273283
Randomization argument. They define a design to
be a master plan plus randomization instructions.
For the structure groups/blocks/plots they assume that each treatment
occurs equally often in each group, and give a formula for the
estimated variance in an unblocked design as a linear combination of
two mean squares: that for error (eliminating both treatments and
blocks) and that for blocks eliminating both treatments and
groups. Hence the efficiency factor of the block design can be used to
calculate the efficiency of the blocked design relative to the unblocked.

M. Atiqullah:
On a property of balanced designs.
Biometrika,
48,
1961,
pp. 215218.
Incomplete block designs, not necessarily with
equal replication or equal block sizes. (a) Theorem 1: variance
balance if and only if the information matrix for treatments is
completely symmetric. (b) If the design is binary and
variancebalanced in n plots then the (harmonic mean)
efficiency factor is (nb)/[n(11/v)] with
b blocks and v treatments. (c) Theorem 2: If the design
is binary, connected, equireplicate and variancebalanced then
b >= v.

M. J. R. Healy and J. C. Gower:
Aliasing in partially confounded factorial experiments.
Biometrika,
48,
1961,
pp. 218220.
Discusses the possible aliasing among
orthogonal polynomial contrasts when block designs are constructed by
using confounding and Abelian groups. All examples have factors with 2
or 3 levels. There is no aliasing when the design has factorial
balance. There is an incomplete definition of partial aliasing.

M. A. Schneiderman and P. Armitage:
A family of closed sequential procedures.
Biometrika,
49,
1962,
pp. 4156.
With correction 56, 1969, p. 457.
Sequential experiments to assess the effect of
a single treatment. Each result known before next experimental unit is
used. Stopping rules for one and twosided null hypotheses.

Damaraju Raghavarao:
On balanced unequal block designs.
Biometrika,
49,
1962,
pp. 561562.
Short proof of Fisher's inequality, and of its
extension for resolved designs, for variancebalanced equireplicate
block designs (no assumption of equal block sizes).

A. Zinger:
A note on optimal allocation for a oneway layout.
Biometrika,
49,
1962,
pp.563564.
Variance components for the structure
batches/objects. Given a fixed number of objects to sample, the
optimal distribution among batches depends on the null hypothesis
relating the variance components.

Theodore Colton:
Optimal drug screening plans.
Biometrika,
50,
1963,
pp. 3145.
Testing a drug for efficacity. Two or more
stages, several experimental units per stage. A hypothesis test, with
the possibility of stopping the trial, after each stage. Optimal
numbers of units at each stage.

B. Kurkjian and M. Zelen:
Applications of the calculus of factorial arrangements.
I: Block and direct product designs.
Biometrika,
50,
1963,
pp. 6373.
Defined a block design for factorial treatments
to have `Property A' if its information matrix is a linear
combination of the relationship (R) matrices for the factorial
association scheme. So this is the same as Yates' `factorial
balance'. Proved that property A implies various things.

K. Hinkelmann and O. Kempthorne:
Two classes of group divisible partial diallel crosses.
Biometrika,
50,
1963,
pp. 281291.
In a partial diallel cross the observed
treatments are a subset of all unordered pairs from the set of
parents; no pair is observed more than once; each parent occurs
equally often. This is equivalent to an equireplicate incomplete
block design with block size 2.
They define an IBD to be Generalized Group Divisible (m)
(GGD/m) if it is partially balanced with respect to a
hierarchical (i.e. multiply nested) association scheme
N_{1} / N_{2} / ... /
N_{m}; and to be Extended Group Divisible
(m) (EGD/m) if it is PB wrt a factorial association
scheme N_{1} × N_{2} × ... ×
N_{s}, where m=2^{s} 1.
With block size two, any subset of the associate classes gives a PB
design. They give some efficient designs PB for those two association
schemes.

G. E. P. Box and N. R. Draper:
The choice of second order rotatable design.
Biometrika,
50,
1963,
pp. 335352.
With correction 52, 1965, p.305.
Factorial designs to fit unknown function eta
over region R of interest, using an approximating `graduating'
function g. If R is spherical then bias is minimized by
rotatable designs. We want to minimize J, which is the weighted
integral of the expectation of the square of the difference between
eta and the fitted value; J is the sum of V, the average
weighted variance, and B, the average squared bias. Specific
results are given for the case that eta is cubic and g is quadratic.

G. R. Hext:
The estimation of secondorder tensors,
with related tests and designs.
Biometrika,
50,
1963,
pp. 353375.

R. O. Collier and F. B. Baker:
The randomization distribution of Fratios for the splitplot
designan empirical investigation.
Biometrika,
50,
1963,
pp. 431438.
Splitplot structure 4/6/4. Analysis using
different error mean squares for the splitplot treatment factor and
the interaction. Randomization simulation. Conclusion that Ftests are
OK at upper tails 0.1, 0.05 and 0.025.

P. E. King:
Optimal replication in sequential drug screening.
Biometrika,
51,
1964,
pp. 110.
Using a few stages to successively
choose smaller subsets of a large number
of promising new drugs: cf. plant breeding.

N. R. Draper and W. E. Lawrence:
Designs which minimize model inadequacies: cuboidal regions of interest.
Biometrika,
52,
1965,
pp. 111118.

E.R. Muller:
A method of constructing balanced incomplete
block designs.
Biometrika,
52,
1965,
pp. 285288.
Uses a complete set of
mutually orthogonal squares of side n
to produce a BIBD for n
treatments in n(n1) blocks of
any size up to n.

J. A. John:
A note on the analysis of incomplete block
experiments.
Biometrika,
52,
1965,
pp. 633636
The analysis involves
inverting a certain matrix. For partially
balanced designs with 1 or 2 associate classes
the inverse has the same pattern as the
original matrix, so inversion is easy.
Special case of BoseMesner algebra (1959).

Raymond O. Collier, Jr. and Frank B. Baker:
Some Monte Carlo results on the power
of the Ftest under permutation in
the simple randomized block design.
Biometrika,
53,
1966,
pp. 199203.
For eight or fifteen
complete blocks of three treatments, the
randomization distribution of the variance
ratio is close to that of F, at least
in the upper tail.

Max H. Myers, Marvin A. Schneiderman
and Peter Armitage:
Boundaries for closed (wedge) sequential
t test plans.
Biometrika,
53,
1966,
pp.431437.
Development of Schniederman
and Armitage 1962.

R. Bohrer:
On Bayes sequential design with two random variables.
Biometrika,
53,
1966,
pp. 469475.
Sequential allocation
of two treatments when the response is known
after each individual.

D. A. Preece:
Some balanced incomplete block designs for two sets of treatments.
Biometrika,
53,
1966,
pp. 497506.
With correction 56, 1969, p. 691.
Two sets of v treatments.
Each experimental unit receives one treatment
from each set. The experimental units are
grouped into b blocks with b > v.
Each set of treatments is balanced with
respect to blocks and to the other set of treatments,
and is overall totally balanced with respect to
blocks and the other set of treatments taken
together.

E.R. Muller:
Balanced confounding of factorial experiments.
Biometrika,
53,
1966,
pp. 507524.
Many constructions for asymmetrical factorial
designs in incomplete blocks. The designs have factorial balance
and are constructed from symmetrical factorials
and BIBDs. Example 1 for n × m with m < n:
start with a balanced lattice for n^{2} treatments regarded
as all combinations of two factors with n levels. Partition
levels of second factor into m sets. If these are not of
equal size, apply a 2transitive set of permutations to these m.
Example 2 for m=2: start with a BIBD for n treatments.
Turn each block into one of size n by combining level 1 of
second factor with the previous block and level 2 with
its complement; then repeat the other way round.

N. R. Draper and W. G. Hunter:
Design of experiments for parameter estimation in multiresponse
situations.
Biometrika,
53,
1966,
pp. 525533.

N. R. Draper and W. E. Lawrence:
The use of second order `spherical' and `cuboidal' designs in the
wrong regions.
Biometrika,
53,
1966,
pp. 596599.

H. O. Hartley and J. N. K. Rao:
Maximum likelihood estimation for the mixed analysis of variance model.
Biometrika,
54,
1967,
pp. 93103.
Precursor to Patterson and Thompson 1971.

M. G. Mostafa:
Designs for the simultaneous estimation of
functions of variance components from twoway
crossed classifications.
Biometrika,
54,
1967,
pp. 127131.
Structure is (rows * columns) /plots
with variance components. Sample r rows
and r columns, then a single plot in
each cell except for two plots in either one
or two transversals.

N. R. Draper and W. G. Hunter:
The use of prior distributions in the design of experiments for
parameter estimation in nonlinear situations.
Biometrika,
54,
1967,
pp. 147153.

D. W. Alling:
Tests of relatedness.
Biometrika,
54,
1967,
pp. 459469
Treatments are ordered
pairs from a set of biological or chemical
substances.

C. McGilchrist:
Analysis of plant competition experiments
for different ratios of species.
Biometrika,
64,
1967,
pp. 471477.
Some plots contain
monocultures, other a mixture of two
species in unequal proportions. The response
on each species in each plot is measured.

D. A. Preece:
Nested balanced incomplete block designs.
Biometrika,
54,
1967,
pp. 479486.
Small blocks nested
in large blocks. If either size of block is
ignored, treatments form a balanced incompleteblock
design with respect to the other blocks.

N. R. Draper and W. G. Hunter:
The use of prior distributions in the design of experiments for
parameter estimation in nonlinear situations: multiresponse case.
Biometrika,
54,
1967,
pp. 662665.

V. D. Fedorov, V. N. Maximov and V. G.Bogorov:
Experimental development of nutritive media
for microorganisms.
Biometrika,
55,
1968,
pp. 4351.
Ten treatments factors.
An initial screening experiment in 16 runs, followed
by steepest ascent on the three most important
factors. The first stage used random balance,
which is less efficient than an orthogonal array.

P. J. Laycock and S. D. Silvey:
Optimal designs in regression problems with a general convex loss function.
Biometrika,
55,
1968,
pp. 5366.

H. D. Patterson:
Serial factorial design.
Biometrika,
55,
1968,
pp. 6781.
Factorial design for several years. Factors at
2 levels, 2^{N} plots. Every set of consecutive
N years has a complete replicate. Denote effect of A in
year i by A_{i}. Serial means that if, say,
A_{1} A_{2} A_{4}
is a defining contrast then so is
A_{2} A_{3} A_{5}.
If A_{i} is confounded with A_{j}
then we must assume that residual effects are zero j  i
years after application.

S. C. Pearce:
The mean efficiency of equireplicate designs.
Biometrika,
55,
1968,
pp. 251253.
Harmonic mean efficiency factor. Those for a
design (E) and its dual (F) are related by
(v1)F + (bv)EF = (b1)E.

I. I. Berenblut:
Changeover designs balanced for the linear component of first residual
effects.
Biometrika,
55,
1968,
pp. 297303.

D. G. Hoel:
Closed sequential tests of an exponential
parameter.
Biometrika,
55,
1968,
pp. 387391.
Like Armitage 1957 and
Schneiderman and Armitage 1962 but the
response is exponential instead of normal.

E. J. Snell and J. BryanJones:
A design balanced for trend.
Biometrika,
55,
1968,
pp. 535539.
Linear time trend.
Nonnormal responses. Design so that treatment
effects are nearly orthogonal to linear time.

A. C. Atkinson:
The use of residuals as a concomitant variable.
Biometrika,
56,
1969,
pp. 141.
Given onedimensional
neighbour correlation between responses, a more
efficient analysis than randomized blocks. Then
designs with exact neighbour balance are much
better than randomized blocks, which in turn
are much better than systematic designs.

P. Davies:
The choice of variables in the design of experiments for linear
regression.
Biometrika,
56,
1969,
pp. 5563.

O. Kempthorne and T. E. Doerfler:
The behaviour of some significance tests under experimental
randomization.
Biometrika,
56,
1969,
pp. 231248.
Clear description of how to randomize a design
for a fixed number of plots. Recommendation
of randomization test rather than tests based
on distributional assumptions. For two
treatments, the randomization test applied
to the difference between means is better than
the sign test or Wilcoxon.

A. W. Davis and W. B. Hall:
Cyclic changeover designs.
Biometrika,
56,
1969,
pp. 283293.
Row * columns structure. Design preserved by a
regular cyclic group. Residual effects in the model.

N. E. Day:
A comparison of some sequential designs.
Biometrika,
56,
1969,
pp. 301311.
Sequential in the sense of knowing the results
to date.

D. H. Rees:
The analysis of variance of some nonorthogonal
designs with splitplots.
Biometrika,
56,
1969,
pp. 4354.
Given two incomplete block designs for
disjoint sets of treatments, form the direct product of the sets of
treatments in each pair of blocks, one from each design.
The treatments from the first design are allocated
to whole plots in the new design, those from the
second to subplots. If the two initial designs are
generally or partially balanced with respect
to any treatment structures then the new design
is generally or partially balanced with respect
to the cross of those structures.

A. Hedayat and W. T. Federer:
An application of group theory to the existence and nonexistence of
orthogonal Latin squares.
Biometrika,
56,
1969,
pp. 547551.
Let G be cyclic of order n and
let q be the smallest prime dividing n. Then

there are q1 MOLS based on G
(this follows directly from Mann's 1942 automorphism method!)
 if q=2 then no Latin square based on G has a
transversal
(proved earlier by Singer, 1960, who is cited, and by Euler, 1779, as
cited in Denes and Keedwell's book).

S. M. Stigler:
The use of random allocation for the control of selection bias.
Biometrika,
56,
1969,
pp. 553560.
Patients arrive sequentially but total number
is fixed. Two treatments. Best design is
complete randomization of the equireplicate
assignment.

M. Stone:
The role of experimental randomization in
Bayesian statistics: finite sampling and two
Bayesians.
Biometrika,
56,
1969,
pp. 681683.
If one Bayesian collects
the data and another Bayesian analyses them,
randomization will make their results more
credible to other scientists.

G. N. Wilkinson:
A general recursive procedure for analysis of variance.
Biometrika,
57,
1970,
pp. 1946.

A. C. Atkinson:
The design of experiments to estimate the slope of a response surface.
Biometrika,
57,
1970,
pp. 319328.

N. G. Becker:
Mixture designs for a model linear in the proportions.
Biometrika,
57,
1970,
pp. 329338.

S. C. Pearce:
The efficiency of block designs in general.
Biometrika,
57,
1970,
pp. 339346.
Extension of Pearce 1968 to block designs
whose block sizes and treatment replications
are not necessarily severally equal.

J. Robinson:
Blocking in incomplete split plot designs.
Biometrika,
57,
1970,
pp. 347350.
Block structure is
blocks/wholeplots/subplots. Treatment structure is A × B. A is applied
as a BIBD on wholeplots in blocks, B is applied as a BIBD on subplots
in wholeplots. Hence the design is partially balanced with respect to
the rectangular association scheme A × B.

A. Hedayat, E. T. Parker and W. T. Federer:
The existence and construction of two families of designs for two
successive experiments.
Biometrika,
57,
1970,
pp. 351355.
Amendment 58, 1971, p. 687.
They use a common transversal of a pair of n
× n MOLS to construct n × (n +1) double Youden
rectangles. They claim that such a transversal exists for all n
other than 3, 6.

M. Sobel and G. H. Weiss:
Playthewinner sampling for selecting the better of two binomial
populations.
Biometrika,
57,
1970,
pp. 357365.
Each treatment results
in success or failure before the next patient is
allocated. If success, keep the same treatment
for the next patient; otherwise, switch.

M. A. Kastenbaum, D. G. Hoel and K.O. Bowman:
Sample size requirements: oneway analysis
of variance.
Biometrika,
57,
1970,
pp. 421430.
Tables relating the following, so that any
one can be calculated from the others with
no need for iteration: significance level,
power, number of treatments (up to 6),
number of samples per treatment, and standardized
range, which is defined to be the maximum
absolute difference between treatment means
divided by the error standard deviation.

H. D. Patterson:
Nonadditivity in changeover designs for a
quantitative factor at four levels.
Biometrika,
57,
1970,
pp. 537549.
Improvement on Berenblut's
designs, partly by using the ideas of serial
factorial design.

P. A. K. CoveyCrump and S. D. Silvey:
Optimal regression designs with previous observations.
Biometrika,
57,
1970,
pp. 551566.
Design uses Hadamard matrices.

M. A. Kastenbaum, D. G. Hoel and K.O. Bowman:
Sample size requirements: randomized block
designs.
Biometrika,
57,
1970,
pp. 573577.
Similar to K, H and B above but for block designs
with up to 5 blocks, each of which contains
all treatment equally often. They assume a
blockbytreatment interaction.

M. J. Box:
An experimental design criterion for precise
estimation of a subset of the parameters in a nonlinear model.
Biometrika,
58,
1971,
pp. 149153.
Doptimality for estimating interesting
parameters in the presence of nuisance parameters, cf. treatments in
the presence of blocks.

D. R. Cox:
A note on polynomial response functions for mixtures.
Biometrika,
58,
1971,
pp. 155159.

S. C. Pearce:
Precision in block experiments.
Biometrika,
58,
1971,
pp. 161167.
If the contrast between treatments i and
j is orthogonal to blocks then those two treatments may be
merged without affecting the precision of other contrasts, otherwise
precision generally increases.

John S. deCani:
On the number of replications of a paired comparison.
Biometrika,
58,
1971,
pp. 169175.
Two teams play, outcome is win/lose, winner is
the first to win k games. Minimize k given the relative
costs of another game and a wrong decision, and the probability that
team A wins each game.

Milton Sobel and Yung Liang Tong:
Optimal allocation of observations for partitioning a set of normal
populations in comparison with a control.
Biometrika,
58,
1971,
pp. 177181.
Given k equireplicated treatments and
one control, what should the relative replication of the control be
if we want to correctly detect those treatments which are a specified
amount worse/better than control? The square root of kl, where
l is the ratio of control variance to treatment variance.

A. T. James and G. N. Wilkinson:
Factorization of the residual operator and canonical decomposition of
nonorthogonal factors in the analysis of variance.
Biometrika,
58,
1971,
pp. 279294.
The critical angles between the blocks subspace
and the treatments subspace of the data space give the canonical
efficiency factors.

B. Efron:
Forcing a sequential experiment to be balanced.
Biometrika,
58,
1971,
pp. 403417.
Randomizing treatment labels then doing
ABAB... or even ABBAABBA... can lead to selection bias if entry to the
trial is not blind; or accidental bias if there are unknown nuisance
effects. Randomization gives a basis for inference. He proposes,
within each block, allocate to the underrepresented treatment (of
two) with probability p, or one half if both treatments have
been used equally. Here p is specified and greater than one
half: he recommends 2/3. He compares this design with random permuted
blocks in terms of correct guesses of next treatment (RPB is better
for block length greater than 18) and with ``complete randomization''
for accidental bias (CR is worse if there are trends with small
frequencies). Using a randomization test should give similar results
under all three methods of randomization. Nothing about estimating
variances.

B. J. Flehinger and T. A. Louis:
Sequential treatment allocation in clinical trials.
Biometrika,
58,
1971,
pp. 419426.
Two treatments, response is survival time,
censored results to date are known. Allocate next patient to the
treatment which is better so far by a criterion which combines number
of deaths and death rate. Terminate as soon as data summary gets into
a certain region. These rules appear to decrease the number of
patients given the inferior treatment.

D. F. Andrews:
Sequentially designed experiments for screening
out bad models with F tests.
Biometrika,
58,
1971,
pp. 427432.

H. D. Patterson and R. Thompson:
Recovery of interblock information when block sizes are unequal.
Biometrika,
58,
1971,
pp. 545554.
Estimate components of variance by maximizing
the likelihood of the data orthogonal to the treatments space. Assume
normality. Does not assume orthogonal block structure or any kind of balance.

I. Guttman:
A remark on the optimal regression designs with previous observations
of CoveyCrump and Silvey.
Biometrika,
58,
1971,
pp. 683685.

K. O. Bowman:
Tables of sample size requirement.
Biometrika,
59,
1972,
p. 234.
Announcement of extension of the two sets of
tables given by Kastenbaum, Hoel and Bowman, 1970.

A. C. Atkinson:
Planning experiments to detect inadequate regression models.
Biometrika,
59,
1972,
pp. 275293.

B. Afonja:
Minimal sufficient statistics for variance
components for a general class of designs.
Biometrika,
59,
1972,
pp. 295302.
If the concurrence matrix has at most three
eigenvalues then a generalized inverse of the information matrix is a
linear combination of I, J and the concurrence
matrix. (Special case of James and Wilkinson 1971; also includes PBIBD(2).)

David G. Hoel, Milton Sobel and George H. Weiss:
A twostage procedure for choosing the better of two binomial populations.
Biometrika,
59,
1972,
pp. 317322.
Use first stage to estimate the probabilities
of each winning, then use these estimates to decide the strategy for
the second stage.

W. C. Guenther:
On the number of replications of a paired comparison:
an easy solution with standard tables.
Biometrika,
59,
1972,
pp. 481483.
Improvement on deCani, 1971.

Agnes M. Herzberg and D. R. Cox:
Some optimal designs for interpolation and extrapolation.
Biometrika,
59,
1972,
pp. 551561.

R. J. Brooks:
A decision theory approach to optimal regression design.
Biometrika,
59,
1972,
pp. 563571.

J. A. Cornell:
A note on the equality of least squares estimates
using secondorder equiradial rotatable designs.
Biometrika,
59,
1972,
pp. 686687.
These designs have the good property that
weighted and unweighted least squares give the same result if variance
is proportional to distance from the origin (special case of:
treatment projector commutes with the variance matrix).

A. C. Atkinson:
Multifactor second order designs for cuboidal regions.
Biometrika,
60,
1973,
pp. 1519.

S. D. Silvey and D. M. Titterington:
A geometric approach to optimal design theory.
Biometrika,
60,
1973,
pp. 2132.

H. D. Patterson:
Quenouille's changeover designs.
Biometrika,
60,
1973,
pp. 3345.
Designs for t treatments
in 2t periods and t^{2} subjects
with treatments orthogonal to periods, to subjects
and to firstorder residual effects. Construction
uses Eulerian trails in digraphs.

W. B. Hall and E. R. Williams:
Cyclic superimposed designs.
Biometrika,
60,
1973,
pp. 4753.
Rowcolumn designs for
two sets of t treatments, constructed
cyclically.

J. A. John:
Generalized cyclic designs in factorial experiments.
Biometrika,
60,
1973,
pp. 5563.
Introduced Abelian group designs specially for
factorial experiments. Proved that Abelian group designs have
orthogonal factorial structure.

P. J. Brown:
Aspects of design for binary key models.
Biometrika,
60,
1973,
pp. 309318.
Binary responses on 2^{k}
factorials. Which models can be specified by
fewer factors? Good designs are linked to good
linear codes.

Lynda V. White:
An extension of the General Equivalence Theorem to nonlinear models.
Biometrika,
60,
1973,
pp. 345348.

D. A. Preece, S. C. Pearce and J. R. Kerr:
Orthogonal designs for threedimensional experiments.
Biometrika,
60,
1973,
pp. 349358.
Latin cubes of first and second order.
Block structures X*Y*Z and
X/(Y*Z) and
(X/Y)*Z have valid randomizations,
but X+Y+Z does not.

Roger R. Davidson and Daniel L. Soloman:
A Bayesian approach to paired comparison experimentation.
Biometrika,
60,
1973,
pp. 477487.

D. F. Andrews and Agnes M. Herzberg:
A simple method for constructing exact tests for sequentially designed
experiments.
Biometrika,
60,
1973,
pp. 489497.

G. H. Freeman:
Experimental designs with unequal concurrences
for estimating direct and remote effects of treatments.
Biometrika,
60,
1973,
pp. 559563.
Designs with supplemented balance
for experiments with control treatments when there are direct
and remote effects.

Lawrence L. Kupper and Edward F. Meydrech:
A new approach to mean squared error estimation of response surfaces.
Biometrika,
60,
1973,
pp. 573579.

S. Geisser:
A predictive approach to the random effect model.
Biometrika,
61,
1974,
pp. 101107.

S. D. Silvey and D. M. Titterington:
A Lagrangian approach to optimal design.
Biometrika,
61,
1974,
pp. 299302.

R. J. Brooks:
On the choice of an experiment for prediction in linear regression.
Biometrika,
61,
1974,
pp. 303311.

I. I. Berenblut and G. I. Webb:
Experimental design in the presence of autocorrelated errors.
Biometrika,
61,
1974,
pp. 427437.
In changeover designs,
if the problem is autocorrelated errors within subjects
rather than residual effects, then numerical
investigation shows that the designs of E. J. Williams
appear to perform well in terms of the Dcriterion.

S. C. Pearce, T. Calinski and T. F. de C. Marshall:
The basic contrasts of an experimental design with special reference
to the analysis of data.
Biometrika,
61,
1974,
pp. 449460.
In the context of block designs, basic
contrasts are defined to be the eigenvectors of the information matrix.

H. Taheri and D. Young:
A comparison of sequential sampling procedures for selecting the
better of two binomial populations.
Biometrika,
61,
1974,
pp. 585592.
Compares Sobel and Weiss 1970
with other methods.

D. Conniffe and Joan Stone:
The efficiency factor of a class of incomplete block designs.
Biometrika,
61,
1974,
pp. 633636.
Upper and lower bounds for the harmonicmean
efficiency factor for given sizes of concurrences. Heuristic: make the
concurrences as equal as possible.

A. C. Atkinson and V. V. Fedorov:
The design of experiments for discriminating between two rival models.
Biometrika,
62,
1975,
pp. 5770.

D. R. Jensen, L. S. Mayer and R. H. Myers:
Optimal designs and largesample tests for linear hypotheses.
Biometrika,
62,
1975,
pp. 7178.

B. A. Worthington:
General iterative method for analysis of
variance when block structure is orthogonal.
Biometrika,
62,
1975,
pp. 113120.
An extension of Stevens 1948
sweeping to many block systems which together
form an orthogonal block structure. No necessity
for general balance. Far fewer sweeps than Wilkinson 1970.

J. Kiefer:
Optimal design: variation in structure and
performance under change of criterion.
Biometrika,
62,
1975,
pp. 277288.

A. C. Atkinson and V. V. Fedorov:
Optimal design: experiments for discriminating between several models.
Biometrika,
62,
1975,
pp. 289303.

P. J. Laycock:
Optimal design: regression models for directions.
Biometrika,
62,
1975,
pp. 305311.

D. M. Titterington:
Optimal design: some geometrical aspects of Doptimality.
Biometrika,
62,
1975,
pp. 313320.

G. Berry:
Design of carcinogenesis experiments using the Weibull distribution.
Biometrika,
62,
1975,
pp. 321328.
When should the remaining
animals be killed?

Gerald R. Chase and David G. Hoel:
Serial dilutions: error effects and optimal
designs.
Biometrika,
62,
1975,
pp.329334.
Several dilutions necessary
before plating out.

L. L. Pesotchinsky:
Doptimum and quasiDoptimum
second order designs on a cube.
Biometrika,
62,
1975,
pp. 335340.
With correction, 63, 1976, p.412.

J. Eccleston and K. Russell:
Connectedness and orthogonality in multifactor designs.
Biometrika,
62,
1975,
pp. 341345.
Given factors A, B
and C. Adjust A and B for
C. If the results are orthogonal to each other
then A and B are defined to have
adjusted orthogonality with respect to C.

George E. P. Box and Norman R. Draper:
Robust designs.
Biometrika,
62,
1975,
pp. 347352.

T. A. Louis:
Optimal allocation in sequential tests comparing
the means of two Gaussian populations.
Biometrika,
62,
1975,
pp. 359369.
With correction 63, 1976, p. 218.

D. Conniffe and J. Stone:
Some incomplete block designs of maximum efficiency.
Biometrika,
62,
1975,
pp. 685686.
With comment, 63, 1976, p. 686.
Group divisible designs with two groups and
betweengroups concurrence equal to one more than the betweengroups
concurrence are Aoptimal (result often attributed to Cheng 1978).

E. R. Williams:
Efficiencybalanced designs.
Biometrika,
62,
1975,
pp. 686689.
A block design is defined
to be efficiencybalanced if all treatment
contrasts have the same efficiency relative to
an unblocked design with all the same replications.

Mark C. K. Yang:
A design problem for determining the population direction of movement.
Biometrika,
63,
1976,
pp. 7782.
Where to place traps on the seabed to
estimate the direction in which lobsters travel through the area.

H. D. Patterson and E. R. Williams:
A new class of resolvable incomplete block designs.
Biometrika,
63,
1976,
pp. 8392.
Introduction of \alphadesigns.

David G. Herr:
A geometric characterization of connectedness
in a twoway design.
Biometrika,
63,
1976,
pp. 93100.
Reproduces the geometric
results of James and Wilkinson but for factors
called rows and columns rather than blocks
and treatments.

F. Downton:
Nonparametric tests for block experiments.
Biometrika,
63,
1976,
pp. 137141.
Responses have Lehmann distribution.
Rank tests.

Ralph A. Bradley and Abdalla T. ElHelbawy:
Treatment contrasts in paired comparisons:
basic procedures with application to factorials.
Biometrika,
63,
1976,
pp. 255262.
Treatments are factorial.
Each pair of treatments are compared several times:
each response is a simple preference. Analysis
to identify factorial effects.

G. H. Freeman:
A cyclic method of constructing regular group
divisible incomplete block designs.
Biometrika,
63,
1976,
pp. 555558.
Some Abelian group designs which are groupdivisible.
No mention of difference sets!

KimLian Kok and H. D. Patterson:
Algebraic results in the theory of serial
factorial design.
Biometrika,
63,
1976,
pp. 559565.
Changeover designs for
t treatments in 2t periods and t^{2}
subjects, such that treatments are orthogonal to subjects and every
ordered pair of treatments occurs once in each consecutive
pair of periods. Direct effects are orthogonal to residual effects
and to the interaction, but the latter two are orthogonal only under extra
conditions. Uses Nelder's ideas.

S. M. Lewis and J. A. John:
Testing main effects in fractions of asymmetrical
factorial experiments.
Biometrika,
63,
1976,
pp. 678680.
Discussion of difficulty in testing hypotheses
in irregular fractions.

Camille Duby, Xavier Guyon and Bernard Prum:
The precision of different experimental designs for a random field.
Biometrika,
64,
1977,
pp. 5966.
Covariance decays exponentially with distance.
Numerical comparison of various designs. Plots should be long and thin
but blocks square. No one type of design is uniformly best.

Richard G. Jarrett:
Bounds for the efficiency factor of block designs.
Biometrika,
64,
1977,
pp. 6772.
Bounds based on the
variance of the concurrences or on the dual design.

Ramon C. Littell and James M. Boyett:
Designs for R×C factorial paired
comparison experiments.
Biometrika,
64,
1977,
pp. 7377.
Comparison of the following
two designs for a twofactor factorial: (i)
compare all pairs of treatments (ii) compare
only those pairs that agree on one factor.

James M. Lucas:
Design efficiencies for varying numbers of centre points.
Biometrika,
64,
1977,
pp. 145147.

I. N. Vuchkov:
A ridgetype procedure for design of experiments.
Biometrika,
64,
1977,
pp. 147150.

Stuart J. Pocock:
Group sequential methods in the design and analysis of clinical trials.
Biometrika,
64,
1977,
pp. 191199.
Two treatments. Random permuted blocks of
fixed even size. Interim assessment and possible stopping after
each block.

A. P. Dawid:
Invariant distributions and analysis of variance models.
Biometrika,
64,
1977, pp. 291297.

R. J. Brooks:
Optimal regression design for control in linear regression.
Biometrika,
64,
1977,
pp. 319325.

J. A. Eccleston and K. G. Russell:
Adjusted orthogonality in nonorthogonal designs.
Biometrika,
64,
1977,
pp. 339345.
Two factors have adjusted orthogonality with
respect to a set of other factors if the following two subspaces are
orthogonal to each other: for each factor, form the sum of the space
for that factor and the space U for the set of factors, then take
the orthogonal complement of U in that.

R. A. Bailey, F. H. L. Gilchrist and H. D. Patterson:
Identification of effects and confounding patterns
in factorial designs.
Biometrika,
64,
1977,
pp. 347354.
Possibly asymmetric
factorial designs constructed by the design
key. How to identify confounded effects.

R. A. Bailey:
Patterns of confounding in factorial designs.
Biometrika,
64,
1977,
pp. 597603.
MR 59#18945
General factorial
designs constructed by the Abelian group
method. How to find the confounded effects, and
how to construct a design with given confounding.

D. R. Bellhouse:
Some optimal designs for sampling in two dimensions.
Biometrika,
64,
1977,
pp. 605611.
With correction 66, 1979, p. 402.

A. Dey:
Construction of regular group divisible designs.
Biometrika,
64,
1977,
pp. 647649.
Given two symmetric block designs for
v treatments and an involution between them with a special
condition, construct a group divisible design for v groups
of size 2.

J. A. Robinson:
Sequential choice of an optimal dose: a prediction intervals approach.
Biometrika,
65,
1978,
pp. 7578.
Trying to estimate the optimal dose without
exceeding it.

L. J. Wei:
On the random allocation design for the control of selection bias in
sequential experiments.
Biometrika,
65,
1978,
pp. 7984.
Two treatments with prescribed
replications. Comparison of a single random permuted block with
independent binomial allocation until one treatment is used up. The
criterion is the experimenter's ability to bias the experiment.

K. D. Glazebrook:
On the optimal allocation of two or more treatments
in a controlled clinical trial.
Biometrika,
65,
1978,
pp. 335340.
Several treatments. One patient at a time, with
known multinomial outcome before the next patient is
allocated. Outcomes have costs, which we want to minimize.

R. G. Jarrett and W. B. Hall:
Generalized cyclic incomplete block designs.
Biometrika,
65,
1978,
pp. 397401.
Extended the meaning of `generalized cyclic' to
semiregular action of an Abelian group.

E. R. Jones and T. J. Mitchell:
Design criteria for detecting model inadequacy.
Biometrika,
65,
1978,
pp. 541551.

S. D. Silvey:
Optimal design measures with singular information matrices.
Biometrika,
65,
1978,
pp. 553559.

K. Sinha:
A resolvable triangular partially balance incomplete block design.
Biometrika,
65,
1978,
p. 665.
Doubling the classical selfdual design for 15
treatments in blocks of size 3 is not resolvable. Here is a design for
the same size and the same association scheme which is resolvable.

Peter D. H. Hill:
A note on the equivalence of Doptimal
design measures for three rival linear models.
Biometrika,
65,
1978,
pp. 666667.

M. Shafiq and W. T. Federer:
Generalized Nary balanced block designs.
Biometrika,
66,
1979,
pp. 115123.
The elements of the incidence matrix are
N successive terms in an arithmetic progression.

P. G. Hoel and R. I. Jennrich:
Optimal designs for dose response experiments in cancer research.
Biometrika,
66,
1979,
pp. 307316.

Sarah C. Cotter:
A screening design for factorial experiments with interactions.
Biometrika,
66,
1979,
pp. 317320.
All factors at two levels. Treatments are: all
0; one 1 and rest 0; one 0 and rest 1; all 1. Hence modelindependent
tests of main effects.

M. Singh and A. Dey:
Block designs with nested rows and columns.
Biometrika,
66,
1979,
pp. 321326.
Block structure is blocks /(rows *
columns). Balance in the bottom stratum only.

K. G. Russell:
Balancing carryover effects in round robin tournaments.
Biometrika,
67,
1980,
pp. 127131.
Each round consists of several draws, each of
which consists of matches between two teams. In a round, all pairs
play each other exactly once. If A plays B in one
draw and C in the next, then C receives a carryover
from B.
``Balance'' means that each team receives carryover from all but one
of the other teams in each round, with carryover from the other team
at the round borders. Construction of balanced draws when the number
of teams is a power of 2.

I. Ford and S. D. Silvey:
A sequentially constructed design for estimating a nonlinear
parametric function.
Biometrika,
67,
1980,
pp. 381388.
Sequential design for estimating maximum of
quadratic function ax+bx^{2} on [1,1]. All
observations are taken at the two ends.

I. N. Vuchkov and E. B. Solakov:
The influence of experimental design on robustness to nonnormality of
the F test in regression analysis.
Biometrika,
67,
1980,
pp. 489492.
Robustness mostly affected by the distribution
of the replications among the support points.

J. M. Steele:
Efron's conjecture on vulnerability to bias in a method for balancing
sequential trials.
Biometrika,
67,
1980,
pp. 503504.
Proved a conjecture about Efron's biased coin
design.

A. F. M. Smith and I. Verdinelli:
Bayes designs for inference using a hierarchical linear model.
Biometrika,
67,
1980,
pp. 613619.

M. Jacroux:
On the determination and construction of Eoptimal block
designs with unequal numbers of replicates.
Biometrika,
67,
1980,
pp. 661667.
Given b blocks of size k and
v treatments. If replications are specified then a design is
Eoptimal if all its concurrences are at least
r_{min}(k1)(v1); when k=2 this
can happen only if v1 divides
r_{min}. Otherwise, similar results with
r_{min} replaced by the integer part of
bv/k.

E. R. Williams and D. Ratcliff:
A note on the analysis of lattice designs with repeats.
Biometrika,
67,
1980,
pp. 706708.
Simplify some formulae for combining
information from bottom two strata of square and rectangular lattice
designs.

H. Chernoff and A. J. Petkau:
Sequential medical trials involving paired data.
Biometrika,
68,
1981,
pp. 119132.
Two treatments allocated sequentially. Do the
first so many as randomized blocks of two, then allocate all the
remainder to the better so far.

C.S. Cheng and C. F. Wu:
Nearly balanced incomplete block designs.
Biometrika,
68,
1981,
pp. 493500.
Replications and also concurrences as equal as
possible.

W. B. Hall and R. G. Jarrett:
Nonresolvable incomplete block designs with few replicates.
Biometrika,
68,
1981,
pp. 617627.
Tables of designs using the debased meaning of
`generalized cyclic'.

A. C. Atkinson:
Optimum biased coin designs for sequential clinical trials with
prognostic factors.
Biometrika,
69,
1982,
pp. 6117.

H. L. Agrawal and J. Prasad:
Some methods of construction of balanced incomplete block designs with
nested rows and columns.
Biometrika,
69,
1982,
pp. 481483.
The structure is blocks/(rows *
columns). Their definition requires balance only in the bottom stratum
but many of the constructions give balance in every stratum.

R. J. Martin:
Some aspects of experimental design and analysis when errors are
correlated.
Biometrika,
69,
1982,
pp. 597612.
Twodimensional designs with correlated
errors. Defines a design to be treatmentbalanced if the information
matrix for treatments is completely symmetric: this depends on the
correlation structure as well as the design.

R. A. Bailey:
Restricted randomization.
Biometrika,
70,
1983,
pp. 183198.
MR 85j:62075
Restricted randomization using twotransitive
groups to retain validity while avoiding bad patterns. Tables given
for various numbers of plots and numbers of factor levels.

K. Afsarinejad:
Balanced repeated measurements designs.
Biometrika,
70,
1983,
pp. 199214.
A crossover trial is defined to be balanced if
all treatments occur equally often in each period and each treatment
is preceded equally often by every other treatment but never by
itself. It is called extrabalanced if the final phrase is replaced by
`every treatment including itself'. There is no requirement on the
sets of treatments allocated to subjects. Cyclic constructions are
given for both types of design in the case that the number of subjects
is equal to the number of treatments, which is greater than the number
of periods.

W. J. Welch:
A mean squared error criterion for the design of experiments.
Biometrika,
70,
1983,
pp. 205213.
Uses a linear combination of variance and
bias to choose design points in a continuous region.

L. Paterson:
Circuits and efficiency in incomplete block designs.
Biometrika,
70,
1983,
pp. 215225.
He conjectures that optimal designs have smaller
number of short circuits.

S. W. Bergman and B. W. Turnbull:
Efficient sequential designs for destructive life testing with
application to animal serial sacrifice experiments.
Biometrika,
70,
1983,
pp. 305314.
Choosing times to sacrifice. The variable of
interest is the time to onset of something whose presence can be
measured only by sacrifice.

S. C. Gupta and B. Jones:
Equireplicate balanced block designs with unequal block sizes.
Biometrika,
70,
1983,
pp. 433440
Variancebalanced block designs with two or
three block sizes, obtained by putting together two or three group
divisible designs with the same association scheme.

L. Paterson:
An upper bound for the minimal canonical efficiency factor of
incomplete block designs.
Biometrika,
70,
1983,
pp. 441446.
Uses induced subgraphs of the treatmentsblocks
graph of an incomplete block design to find this upper bound.

D. Robinson:
A comparison of sequential treatment allocation rules.
Biometrika,
70,
1983,
pp. 492495.
Sequential designs for two treatments with
instant information. Compares methods. Want to maximize the use of the
better treatment and minimize the risk of choosing the wrong treatment
at the end of the experiment.

R. T. Smythe and L. J. Wei:
Significance tests with restricted randomization design.
Biometrika,
70,
1983,
pp. 496500.
Two treatments assigned sequentially to
subjects. ``Complete randomization'' means that each subject is
independently assigned each treatment with equal probability. Urn
design means start with a balls of each of two colours; draw
one and replace, together with b balls of the other colour; and
so on. Asymptotic null distribution of the test
statistic when an urn design is used.

A. Giovagnoli and I. Verdinelli:
Bayes Doptimal and Eoptimal block designs.
Biometrika,
70,
1983,
pp. 695706.
Designs for one control and other test
treatments. With prior information the optimal designs are sometimes,
but not always, the same as the classical ones.

D. Bellhouse:
Optimal randomization for experiments in which autocorrelation is
present.
Biometrika,
71,
1984,
pp. 155160.
For certain patterns of autocorrelation the
optimal design is systematic, the only randomization being the
randomization of treatment labels.

R. Mukerjee and S. Kageyama:
On resolvable and affine resolvable variancebalanced designs.
Biometrika,
72,
1985,
pp. 165172.
In an incomplete block design, links between
(a) variance balance (b) affine (generalized) resolvability (c)
extension of Fisher's equality. Also between these and
proportional arrays and factorial designs.

R. Mukerjee and S. Huda:
Minimax second and thirdorder designs to
estimate the slope of a response surface.
Biometrika,
72,
1985,
pp. 173178.
Rotatable designs for 8 or fewer factors, which
minimize the maximum variance of the estimated slope averaged over all
directions.

N. R. Draper and D. Faraggi:
Role of the Papadakis estimator in one and
twodimensional field trials.
Biometrika,
72,
1985,
pp. 223226.
The Papadakis estimator is appropriate for one
pattern of neighbour correlation but not for another. In both cases,
appropriately neighbourbalanced designs lead to simpler estimators.

J. Kunert:
Optimal repeated measurements designs for correlated observations and
analysis by weighted least squares.
Biometrika,
72,
1985,
pp. 375389.
Crossover designs with no residual effects but
withinpatient correlation. Special kinds of Latin square are optimal.

C.Y. Suen and I. M. Chakravarti:
Balanced factorial designs with twoway
elimination of heterogeneity.
Biometrika,
72,
1985,
pp. 391402.
Rowcolumn designs with factorial balance.

R. A. Ipinyomi and J. A. John:
Nested generalized cyclic rowcolumn designs.
Biometrika,
72,
1985,
pp. 403409.
Structure is blocks/(rows * columns). Design
is preserved by a semiregular cyclic group.

D. Steinberg:
Model robust response surface designs: scaling twolevel factorials.
Biometrika,
72,
1985,
pp. 513526.
Choice of scale for continuous variables with
only two levels of each to be used.

P. J. Green:
Linear models for field trials, smoothing and crossvalidation.
Biometrika,
72,
1985,
pp. 527537.
Strictly speaking not about design, but one of
a clutch of papers that appeared in the mid 1980s suggesting that the
classical method of analysing field trials is inadequate. The new
methods have implications for design.

P. S. Gill and G. K. Shukla:
Efficiency of nearest neighbour balanced
designs for correlated observations.
Biometrika,
72,
1985,
pp. 539544.
For complete blocks with neighbour correlation
within blocks, designs with nondirectional neighbour balance are efficient.

I. Ford, D. M. Titterington and C. F. J. Wu:
Inference and sequential design.
Biometrika,
72,
1985,
pp. 545551.
Difficulties in nonasymptotic inference can
affect choice of sequential design.

C. F. J. Wu:
Asymptotic inference from sequential design in a nonlinear situation.
Biometrika,
72,
1985,
pp. 553558.

V. V. Fedorov and V. Khabarov:
Duality of optimal designs for model discrimination and parameter estimation.
Biometrika,
73,
1986,
pp. 183190.

J. D. Spurrier and D. Edwards:
An asymptotically optimal subclass of balanced
incomplete block designs for comparison with
a control.
Biometrika,
73,
1986,
pp. 191199.
A block design with one control treatment is
called a balanced control incomplete block design if it consists of a
BIBD on the noncontrol treatments with, say, c copies of the
control adjoined to each block. Among designs with supplemented
balance, the following are asymptotically optimal: unions of at most
two BCIBDs with appropriate values of c.

D. Bellhouse:
Randomization in the analysis of covariance
Biometrika,
73,
1986,
pp. 207211.
Restrict randomization so the the treatment
space is almost orthogonal (this concept is defined by an arbitrary
constant) to the covariate space. Empirical study to see if the usual
distributional assumptions still seem OK.

R. J. Martin:
On the design of experiments under spatial correlation.
Biometrika,
73,
1986,
pp. 247277.
Correction 75, 1988, p. 396.
Spatial correlation instead of rows and
columns. Some sort of neighbour balance is desirable, but so is valid
restricted randomization.

E. R. Williams:
A neighbour model for field experiments.
Biometrika,
73,
1986,
pp. 279287.
Fixed effects of `replicates' (superblocks);
withinblock covariance is an affine function of distance between
plots. Suggested limited method of randomization.

L. J. Paterson and P. Wild:
Triangles and efficiency factors.
Biometrika,
73,
1986, pp. 289299.
The number of triangles in the
varietyconcurrence graph gives an upper bound on the (harmonicmean)
efficiency factor. It is easy to calculate when the design has a large
automorphism group.

J. A. John and J. A. Eccleston:
Rowcolumn \alphadesigns.
Biometrika,
73,
1986,
pp. 301306.
Given a cyclic group G and a subgroup H.
Also an array A such that each column is a transversal to the
cosets of H and each row is a partial transversal. Extend this to
a rowcolumn design by replacing each column a by all columns
ah for h in H. Then withincosetsofH
contrasts are orthogonal to rows, while betweencosetsofH
contrasts are orthogonal to columns. Hence adjusted
orthogonality. Hence general balance.

C.S. Cheng:
A method for constructing balanced incomplete
block designs with nested rows and columns.
Biometrika,
73,
1986,
pp. 695700.
Structure is blocks/(rows *
columns). Construction uses a BIBD to create a larger such design from
a smaller. The construction preserves these two properties, if the
smaller design has either: (a) general balance (b) balance in the
bottom stratum.

C.M. Yeh:
Conditions for universal optimality of block designs.
Biometrika,
73,
1986,
pp. 701706.
If an IBD has information matrix L
with maximal trace and, for each other design in the class with
information matrix C, L is a positive linear combination of
the treatmentpermuted forms of C, then the first design is
universally optimal.

R. A. Bailey:
Oneway blocks in twoway layouts.
Biometrika,
74,
1987,
pp. 2732.
MR 88e:62187
Valid restricted randomization for rectangular
layouts where rows are complete blocks; bad patterns in columns are avoided.

R. J. Brooks:
Optimal allocation for Bayesian inference
about an odds ratio.
Biometrika,
74,
1987,
pp. 196199.
Two populations (treatments) with different
proportions. How should a fixed number of samples (experimental units)
be distributed?

R. Morton:
A generalized linear model with nested strata of extraPoisson
variation.
Biometrika,
74,
1987,
pp. 247257.
Block structure is a/b/c. The model is
multiplicative, conditional on higherstratum terms.
Not explicitly design, but could have implications for the design of
experiments for nonnormal data.

J. N. S. Matthews:
Optimal crossover designs for the comparison of two treatments in the
presence of carryover effects and autocorrelated errors.
Biometrika,
74,
1987,
pp. 311320.
With correction 75, 1988, p. 396.
Crossover designs for two treatments. Such a
design is defined to be dual balanced if each sequence and its dual
occur equally often, where the dual is obtained by interchanging the
treatments. Efficient designs for three or four periods tabulated by
autoregression parameter.

J. G. Pigeon and D. Raghavarao:
Crossover designs for comparing treatments with a control.
Biometrika,
74,
1987,
pp. 321328.
Crossover designs with one control
treatment. Such a design is defined to be a control balanced residual
effects design if it is the obvious generalization of a balanced
control incomplete block design. Constructions from balanced residual
effects designs; from pairwise balanced designs; from an Abelian group
on the noncontrol treatments.

C.S. Cheng and K.C. Li:
Optimality criteria in survey sampling.
Biometrika,
74,
1987,
pp. 337345.

H. A. David:
Ranking from unbalanced pairedcomparison data.
Biometrika,
74,
1987,
pp. 432436.
Tournament in which each pair plays at most
once. Suggested method of ranking the players. This has implications
for the design of such a tournament.

K. Sinha:
A method of construction of regular group divisible designs.
Biometrika,
74,
1987,
pp. 443444.
If there are m groups and
k=m1 and the withingroup concurrence is zero then each
block contains one element from each of m1 groups. Adjoin to
it the whole of the remaining group.

W. T. Federer and D. Raghavarao:
Response models and minimal designs for mixtures
of n of m items useful for
intercropping and other investigations.
Biometrika,
74,
1987,
pp. 571577.
Each experimental unit receives a subset of the
set of items. Response may be either total for that unit or for each
item in the unit. Effects like main effects and interactions are
defined, generalizing those for the diallel cross (n=2).

D. J. Fletcher:
A new class of changeover designs for factorial experiments.
Biometrika,
74,
1987,
pp. 649654.
Changeover designs made from Abelian groups,
hence easy calculations of efficiency factors for direct and residual
factorial effects.

J. Kunert:
Neighbour balanced block designs for correlated
errors.
Biometrika,
74,
1987,
pp. 717724.
Improves the results of Gill and Shukla,
1985, and extends to incomplete blocks. Semibalanced arrays can be
used to give designs with the required neighbour balance.

A. Azzalini and A. Giovagnoli:
Some optimal designs for repeated measurements
with autoregressive errors.
Biometrika,
74,
1987,
pp. 725734.
Also extends the results of Gill and Shukla.

V. K. Gupta and A. K. Nigam:
Mixed orthogonal arrays for variance estimation
with unequal numbers of primary selections per stratum.
Biometrika,
74,
1987,
pp. 735742.
Not strictly speaking design, but typical of
papers that show how the same idea (here orthogonal arrays) can be useful
in both design and sampling.

P. R. Wild and E. R. Williams:
The construction of neighbour designs.
Biometrika,
74,
1987,
pp. 871876.
Modification of the alphadesign construction
to produce incomplete block designs with desirable neighbour
properties. Generalized cyclic in the second sense.

R. Mukerjee and S. Huda:
Optimal design for the estimation of variance components.
Biometrika,
75,
1988,
pp. 7580.
Variance components for several completely
crossed factors.

S. D. Oman and E. Seiden:
Switchback designs.
Biometrika,
75,
1988,
pp. 8189.
Crossover designs for three periods when each
subject has its own linear change of effect over time. A switchback
design uses the same number of the sequences
(i,j,i) and (j,i,j). The
design is balanced if each ordered pair is used in this way equally
often. Extra conditions if subjects are blocked. Constructions from
resolved BIBDs with block size 2.

D. R. Cox:
A note on design when response has an exponential family distribution.
Biometrika,
75,
1988,
pp. 161164.
The potential loss from unnecessary blocking or
from ignoring blocks at the design stage.

P. F. Thall, R. Simon and S. S. Ellenberg:
Twostage selection and testing designs for
comparative clinical trials.
Biometrika,
75,
1988,
pp. 303310.
At the first stage, randomly allocate all new
treatments and control to equal numbers of subjects. Each subject
either fails or succeeds. Then either accept the hypothesis of no
difference between any treatments, or choose best new treatment and
proceed to a comparison of that against control on equal numbers of
new subjects.

L. J. Wei:
Exact twosample tests based on the randomized playthewinner rule.
Biometrika,
75,
1988,
pp. 603606.
Proposes that a randomization test should be
used that takes note of the sequential randomization procedure
actually used. Application to very dodgy design with treatments
allocated to 11 and 1 patients respectively.

V. K. Gupta and R. Singh:
On Eoptimal block designs.
Biometrika,
76,
1989,
pp. 184188.
Extension of Jacroux 1980 to blocks of unequal size.

G. M. Constantine:
Robust designs for serially correlated observations.
Biometrika,
76,
1989,
pp. 245251.
Correlations (possibly unequal) between nearest
neighbours only. For a maineffects factorial at two levels, change as
many factors as possible each time that the correlation is negative
and as few as possible (cf. Gray codes) each time that it is positive.

D. L. Zimmerman and D. A. Harville:
On the unbiasedness of the Papadakis estimator and other nonlinear
estimators of treatment contrasts in fieldplot experiments.
Biometrika,
76,
1989,
pp. 253259.
Possibly different spatial models for the true
data and the method of analysis. Conditions which imply that treatment
estimators are unbiased.

P. R. Sreenath:
Construction of some balanced incomplete
block designs with nested rows and columns.
Biometrika,
76,
1989,
pp. 399402.
Structure is blocks/(rows *
columns). Variancebalance in the bottom stratum. Constructions from
finite fields.

A. C. Atkinson and A. N. Donev:
The construction of exact Doptimum
experimental designs with application to blocking
response surface designs.
Biometrika,
76,
1989,
pp. 515526.

S. Gupta:
Efficient designs for comparing test treatments with a control.
Biometrika,
76,
1989,
pp. 783787.
Relates Pearce's supplemented balance to more
recent literature on this topic.

M. Jacroux and R. S. Ray:
On the construction of trendfree run orders of treatments.
Biometrika,
77,
1990,
pp. 187191.
A generalization of the foldover method to
increase the degree of polynomial trend orthogonal to treatments.

Nizam Uddin and John P. Morgan:
Some constructions for balanced incomplete
block designs with nested rows and columns.
Biometrika,
77,
1990,
pp. 193202.
Balance in bottom stratum only, in definition,
but most examples are balanced throughout. Abelian group construction.

C. B. Begg:
On inference from Wei's biased coin design
for clinical trials (with discussion).
Biometrika,
77,
1990,
pp. 467484.
Arguments about Wei 1988. Gems from the
discussion include Pocock's `sophisticated analysis should not be used
to rescue problems in design' and Kempthorne's asking how you can have
a sequential design without a stopping rule.

D. J. Schaid, S. Wieand and T. M. Therneau:
Optimal twostage screening designs for survival comparisons.
Biometrika,
77,
1990,
pp. 507513.
A twostage design intended to minimize the
number of patients assigned to treatments with no survival
improvement over the standard. In stage 1 assign standard and all new
treatments to the same number of patients. Then if all new vs standard
statistics are below a low boundary indicating no improvement,
stop. If any are above a high boundary indicating definite
improvement, accept all of those and stop. Otherwise go on to Stage 2
with just those new treatments whose statistics fall between the boundaries.

H. Chernoff and Y. Haitovsky:
Locally optimal design for comparing two
probabilities from binomial data subject to misclassification.
Biometrika,
77,
1990,
pp. 797805.

N. Uddin:
Some series constructions for minimal size equineighboured balanced
incomplete block designs with nested rows and columns.
Biometrika,
77,
1990,
pp. 829833.
Neighbour balance at all distances within rows
and within columns of the structure blocks / (rows * columns). Also
balance in the bottom stratum. Constructions from finite fields.

YouGan Wang:
Gittins indices and constrained allocation in clinical trials.
Biometrika,
78,
1991,
pp. 101111.
Two treatments, each succeeds or fails,
sequential allocation knowing previous results. Want to maximize
number of successes, or the probability that the better treatment has
the larger proportion of successes at the end of the trial.

D. S. Coad:
Sequential tests for an unstable response variable.
Biometrika,
78,
1991,
pp. 113121.
Similar problem but with normal responses and
linear time trend. Discussion of both tests and allocation rules.

C.F. J. Wu:
Balanced repeated replications based on mixed
orthogonal arrays.
Biometrika,
78,
1991,
pp. 181188.

K. G. Russell:
The construction of good changeover designs
when there are fewer units than treatments.
Biometrika,
78,
1991,
pp. 305313.
Firstorder residual effects. The same number of
periods as treatments, but fewer subjects.

Joachim Kunert:
Crossover designs for two treatments and correlated errors.
Biometrika,
78,
1991,
pp. 315324.
Also residual effects. Many periods.

ChingShui Cheng and David M. Steinberg:
Trend robust twolevel factorial designs.
Biometrika,
78,
1991,
pp. 325336.
Autoregressive or other timeseries
errors. Designs orthogonal to loworder polynomial trend may not be
efficient unless there are many changes of level.

Frances P. Stewart and Ralph A. Bradley:
Some universally optimal rowcolumn designs with empty nodes.
Biometrika,
78,
1991,
pp. 337348.
Designs for three factors such that each pair
are either orthogonal or a BIBD and treatments (one of the three) have
total balance with respect to the other two. Very much in the
PearcePreece OTT school.

S. M. Lewis and A. M. Dean:
On general balance in rowcolumn designs.
Biometrika,
78,
1991,
pp. 595600.
Review literature on rowcolumn designs to see
which are generally balanced. Relationship with adjusted orthogonality.

A. C. Ponce de Leon and A. C. Atkinson:
Optimum experimental design for discriminating between two rival
models in the presence of prior information.
Biometrika,
78,
1991,
pp. 601608.

Ashish Das and Sanpei Kageyama:
A class of Eoptimal proper efficiencybalanced designs.
Biometrika,
78,
1991,
pp. 693696.
The designs are not equireplicate. They are
obtained from BIBDs by merging some pairs of treatments.

M. Zelen and Y. Haitovsky:
Testing hypotheses with binary data subject to misclassification
errors: analysis and experimental design.
Biometrika,
78,
1991,
pp. 857865.
Really sampling. Want to maximize power and minimize cost.

LingYau Chan:
Optimal design for estimation of variance in nonparametric regression
using first order differences.
Biometrika,
78,
1991,
pp. 926929.

Gregory M. Constantine:
On the information and precision matrices of varietal contrasts.
Biometrika,
79,
1992,
pp. 214216.
Relationship between the information matrix for
treatments and the covariances of estimators of simple differences.

Patrick deFeo and Raymond H. Myers:
A new look at experimental design robustness.
Biometrika,
79,
1992,
pp. 375380.

R. J. Martin and J. A. Eccleston:
Recursive formulae for constructing block designs with dependent
errors.
Biometrika,
79,
1992,
pp. 426430.
How to update the information matrix when new
plots are added to a block, or treatments transposed.

A. Gerami and S. M. Lewis:
Comparing dual with single treatments in block designs.
Biometrika,
79,
1992,
pp. 603610.
Two quantitative treatment factors, but double
placebo is unethical.

WengKee Wong:
A unified approach to the construction of minimax designs.
Biometrika,
79,
1992,
pp. 611619.

R. J. Boys and K. D. Glazebrook:
A robust design of a screen for a binary response.
Biometrika,
79,
1992,
pp. 643650.

J. A. John and Deborah J. Street:
Bounds for the efficiency factor of rowcolumn designs.
Biometrika,
79,
1992,
pp. 658661.
With correction 80, 1993, pp. 712713.
Structure is either rows * columns or blocks
/(rows * columns). In the second case, each block is a single
replicate. Efficiency factor in the bottom stratum only.
One upper bound given for generally balanced designs,
another in terms of various concurrences.

Friedrich Pukelsheim and Sabine Rieder:
Efficient rounding of approximate designs.
Biometrika,
79,
1992,
pp. 763770.

Timothy E. O'Brien:
A note on quadratic designs for nonlinear regression models.
Biometrika,
79,
1992,
pp. 847849.

C. F. J. Wu and Runchu Zhang:
Minimum aberration designs with twolevel and fourlevel factors.
Biometrika,
80,
1993,
pp. 203209.
Designs made from elementary Abelian 2groups
by using pseudofactors for the 4level factors. Obvious definition of
wordlength, hence of minimum aberration.

R. R. Sitter:
Balanced repeated replications based on orthogonal multiarrays.
Biometrika,
80,
1993,
pp. 211221.
Sampling by using a generalization of
orthogonal multiarrays (which are related to semiLatin squares).

H. Monod and R. A. Bailey:
Twofactor designs balanced for the neighbour effect of one factor.
Biometrika,
80,
1993,
pp. 643659.
Factorial designs in space. One factor has
neighbour effects, so its levels should occur equally often next to
all treatments.

C. F. J. Wu:
Construction of supersaturated designs through partially aliased
interactions.
Biometrika,
80,
1993,
pp. 661669.
Squeezing a quart out of a pint Hadamard matrix.

Kuemhee Chough Carrière and Gregory C. Reinsel:
Optimal twoperiod repeated measurement designs
with two or more treatments.
Biometrika,
80,
1993,
pp.924929.
Residual effects. If subject effects are random
then a design with all ordered pairs of treatments equally often as
treatment sequences is optimal for estimation of direct effects. A
different conclusion if subject effects are fixed.

Jim Burridge and Paola Sebastiani:
Doptimal designs for generalised linear models with
variance proportional to the square of the mean.
Biometrika,
81,
1994,
pp. 295304.

Sudhir Gupta and Sanpei Kageyama:
Optimal complete diallel crosses.
Biometrika,
81,
1994,
pp. 420424.
To estimate general combining ability in a
block design, use all pairs in a nested BIBD on the parents.

Holger Dette and Linda M. Haines:
Eoptimal designs for linear and nonlinear
models with two parameters.
Biometrika,
81,
1994,
pp. 739754.

Sharon L. Lohr:
Optimal Bayesian design of experiments for the
oneway random effects model.
Biometrika,
82,
1995,
pp. 175186.

R. A. Bailey, H. Monod and J. P. Morgan:
Construction and optimality of affineresolvable designs.
Biometrika,
82,
1995,
pp. 187200.
Such designs are optimal among resolved block
designs. They are combinatorially equivalent to orthogonal arrays of
strength 2.

Clemens Elster and Arnold Neumaier:
Screening by conference designs.
Biometrika,
82,
1995,
pp. 589602.
Twolevel fractional factorials for screening
for a few effective factors among many as a preliminary to a more
detailed experiment on those few. Of course, all classical fractions
are based on the assumption that certain interactions are zero. They
propose having assumptionindependent tests of each main effect by
having at least one setting of all the other factors where this factor
takes both its values. Hence recommend designs based on conference
matrices. But if, say, the problem is twofactor interactions, then
classical resolution 4 fractions are better.

R. A. Bailey, J.M. Azaïs and H. Monod:
Are neighbour methods preferable to analysis of variance for
completely systematic designs? `Silly designs are silly!'.
Biometrika,
82,
1995,
pp. 655659.
The usual analysis of a completeblock design
remains valid under a range of assumptions, but if the treatments have
the same order in each block then any analysis which estimates a
smaller variance than randomized blocks is suspect.

Morris L. Eaton, Alessandra Giovagnoli and Paola Sebastiani:
A predictive approach to the Bayesian design
problem with application to normal regression models.
Biometrika,
83,
1996,
pp. 111125.

S. Altan and D. Raghavaro:
Nested Youden square designs.
Biometrika,
83,
1996,
pp. 242245..
Structure is 2 blocks /(3*4) and there are
seven treatments. They recommend two Youden squares with one treatment
in common. They do not acknowledge that these are a (very!) special
case of the rowregular designs introduced by Bagchi, Mukhopadhyay and
Sinha, by Chang and Notz, and by Morgan and Uddin.

R. J. Boys, K. D. Glazebrook and C. M. McCrone:
A Bayesian model for the optimal ordering of a collection of screens.
Biometrika,
83,
1996,
pp. 472476.

Aloke Dey and Chand K. Midha:
Optimal block designs for diallel crosses.
Biometrika,
83,
1996,
pp. 484489.
Since triangular designs (in blocks) have
their treatments indexed by the unordered pairs from a set, it is
natural to use them as block designs for diallel crosses. They are
variance balanced for general combining effects.

Giovanni Pistone and Henry P. Wynn:
Generalised confounding with Gröbner bases.
Biometrika,
83,
1996,
pp. 653666.
Continous factors. Choose a set of combinations
so that certain loworder polynomial models are all distinct on that
set. But they may not be closed under intersection without violating
marginality.

Holger Dette and Weng Kee Wong:
Robust optimal extrapolation designs.
Biometrika,
83,
1996,
pp. 667680.

E. R. Williams and J. A. John:
A note on optimality in lattice square designs.
Biometrika,
83,
1996,
pp. 709713.
Block structure is squares/ (rows *
columns). Consider resolvable designs only. Among those, optimality
for rows or columns separately means that they should both be square
lattice designs. Among lattice squares, a certain simple condition
gives optimality.

George Box and John Tyssedal:
Projective properties of certain orthogonal arrays.
Biometrika,
83,
1996,
pp. 950955.
An orthogonal array has projectivity r
if every set of r constraints has all combinations of levels at
least once. Useful for screening designs.

Mausumi Bose:
Some efficient incomplete block sequences.
Biometrika,
83,
1996,
pp. 956961.
Circular blocks of size one fewer than the
number of treatments, and directional neighbour balance at distance one.
For this application, circles are opened into lines with a single
border plot, and direction is time. Cf same designs used for treatment
interference in space, or dancing girls.

D. K. Ghosh and J. Divecha:
Two associate class partially balanced incomplete block designs and
partial diallel crosses.
Biometrika,
84,
1997,
pp. 245248.
Take a PBIB and replace each block by the set
of unordered pairs from that block. Assume general combining effects
only. Efficiencies are related to those of the original PBIB, but
there are probably more efficient ways of blocking that set of pairs.

France Mentré, Alain Mallet and Dohar Baccar:
Optimal design in randomeffects regression models.
Biometrika,
84,
1997,
pp. 429442.

Nizam Uddin and John P. Morgan:
Efficient block designs for settings with spatially correlated errors.
Biometrika,
84,
1997,
pp. 443454.
With correction 86 1999, p. 233.
Structure is blocks /(rows * 2 columns). Within
blocks there are spatial correlations. Good designs when the block
size is equal to, or one less than, the number of treatments.

H. B. Kushner:
Optimality and efficiency of twotreatment
repeated measurements designs.
Biometrika,
84,
1997,
pp. 455468.
With correction 86, 1999, p. 234.
Residual effects and correlated errors within
subjects. Improves many earlier results.

Tomas Philipson and Jeffrey Desimone:
Experiments and subject sampling.
Biometrika,
84,
1997,
pp. 619630.
Repeated measures for one new treatment plus
control. Subjects can elect to drop out of the trial (i.e. revert to
the control) as they see their own results. Only under very strong
conditions do randomization and blinding have the properties usually
claimed.

P. Hu and M. Zelen:
Planning clinical trials to evaluate early detection programmes.
Biometrika,
84,
1997,
pp. 817829.
Longterm evaluation of two treatments. How to
choose (a) numbers of subjects (b) number of medical examinations (c)
time interval between examinations (d) length of followup, to
maximize power.

Rahul Mukerjee:
Optimal partial diallel crosses.
Biometrika,
84,
1997,
pp. 939948.
General combining ability only. For an
unblocked fraction, the set of withingroup pairs (groups of equal
size) is optimal. Also some results on block designs.

ChingShui Cheng:
Some hidden projection properties of orthogonal arrays with strength
three.
Biometrika,
85,
1998,
pp. 491495.
Given an orthogonal array of strength three
with all factors at two levels. If the number of units is not a
multiple of sixteen then the projection onto any five factors allows
the estimation of all main effects and twofactor interactions if all
higherorder interactions are negligible.

Inchi Hu:
On sequential designs in nonlinear problems.
Biometrika,
85,
1998,
pp. 496503.

Holger Dette and Weng Kee Wong:
Bayesian Doptimal designs on a fixed number of design points
for heteroscedastic models.
Biometrika,
85,
1998,
pp. 869882.

R. J. Martin and J. A. Eccleston:
Variancebalanced changeover designs for dependent observations.
Biometrika,
85,
1998,
pp. 883892.
Residual effects and arbitrary pattern of
correlation within subjects. Designs constructed from orthogonal
arrays of strength two are variancebalanced.

Holger Dette and Timothy E. O'Brien:
Optimality criteria for regression models
based on predicted variance.
Biometrika,
86,
1999,
pp. 93106.

FengShui Chai and Rahul Mukerjee:
Optimal designs for diallel crosses
with specific combining abilities.
Biometrika,
86,
1999,
pp. 453458.
The eigenspaces of the triangular association
scheme are precisely the general combining ability and specific
combining ability, so any triangular design (in blocks), when used for
a halfdiallel, gives the diallel equivalent of factorial balance.

KaiTai Fang and Rahul Mukerjee:
A connection between uniformity and aberration in regular fractions of
twolevel factorials.
Biometrika,
87,
2000,
pp. 193198.

Isabella Verdinelli:
A note on Bayesian design for the normal linear model with unknown
error variance.
Biometrika,
87,
2000,
pp. 222227.

PiWen Tsai, Steven G. Gilmour and Roger Mead:
Projective threelevel main effects designs robust to model uncertainty.
Biometrika,
87,
2000,
pp. 467475.

E. R. Williams and J. A. John:
Updating the average efficiency factor in $\alpha$designs.
Biometrika,
87,
2000,
pp. 695699.
R. A. Bailey
Page maintained by
R. A. Bailey.
Page modified 7/8/01
H. Jeffreys:
Distance apart in regression. Systematic and ignored polynomial patterns.
Biometrika,
31,
1939,
pp. 18.
Wrong date! What is this paper?