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. 414-417.
Suggests using days/tests as variance components, and doing replicates on well-separated 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. 1-85.
Egon S. Pearson: On the variation in personal equation and the correlation of successive judgments. Biometrika, 14, 1922, pp. 23-102.
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. 271-293. 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 half-drill strip ABBAABBA ...
``Student'': Errors of routine analysis. Biometrika, 19, 1927, pp. 151-164.
Expands on the concerns in his 1917 paper.
E. S. Pearson: The analysis of variance in cases of non-normal variation. Biometrika, 23, 1931, pp. 114-133.
``Student'': The Lanarkshire milk experiment. Biometrika, 23, 1931, pp. 398-406.
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 z-test in randomized blocks and Latin squares. Biometrika, 29, 1937, pp. 21-52.
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. 53-64.
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. 322-335.
Similar conclusions to Welch.
``Student'': Comparison between balanced and random arrangements of field plots. Biometrika, 29, 1938, pp. 363-379.
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. 380-388.
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. 159-179.
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. 440-466.
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. 1-8.
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. 305-325.
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. 328-332.
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. 255-272.
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. 80-87.
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. 268-276.
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. 277-282.
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 p-2 treatment factors can be used. And so on. These are saturated orthogonal arrays of strength n in pⁿ plots.
W. L. Stevens: Statistical analysis of a non-orthogonal tri-factorial experiment. Biometrika, 35, 1948, pp. 346-367.
Introduced `sweeping' for the convergent iterative fit of non-orthogonal 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. 50-58.
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 non-central F-distribution. Biometrika, 38, 1951, pp. 112-130.
K. S. Banerjee: Some observations on the practical aspects of weighing designs. Biometrika, 38, 1951, pp. 248-251.
More examples illustrating Banerjee 1950.
D. R. Cox: Some systematic experimental designs. Biometrika, 38, 1951, pp. 312-323.
Plots in a line, with a low-order 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 odd-order orthogonal polynomials, so trial-and-error 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. 32-48.
Change-over designs for first-order 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. Row-complete Latin squares. Suitably labelled sets of mutually orthogonal Latin squares.
G. E. P. Box: Multi-factor designs of first order. Biometrika, 39, 1952, pp. 49-57.
R. M. Williams: Experimental designs for serially correlated observations. Biometrika, 39, 1952, pp. 151-167.
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. 324-345.
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. 190-199.
D. R. Cox: The design of an experiment in which certain treatment arrangements are inadmissible. Biometrika, 41, 1954, pp. 287-295.
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 trial-and-error 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. 502-537. 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. 544-548.
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. 70-79.
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 plot-treatment additivity so his results are not concordant with other randomization arguments.
Fred C. Andrews and Herman Chernoff: A large-sample bioassay design with random doses and uncertain concentration. Biometrika, 42, 1955, pp. 307-315.
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 large-sample results on estimation and power for a method of paired comparisons. Biometrika, 42, 1955, pp. 450-470.
More continuation of Bradley and Terry.
R. C. Bose: Paired comparison designs for testing concordance between judges. Biometrika, 42, 1955, pp. 113-121.
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 2-resolved design.
J. C. Butcher: Treatment variances for experimental designs with serially correlated observations. Biometrika, 43, 1956, pp. 208-212.
Extension of R. M. Williams 1952. For an AR(m) process he requires designs neighbour-balanced at all distances up to m, where neighbour-balance 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. 388-403.
No implications for design.
P. Armitage: Restricted sequential procedures. Biometrika, 44, 1957, pp. 9-26.
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. 97-113.
Applies the Bradley-Terry analysis to incomplete designs like Bose 1956.
S. C. Pearce: Experimenting with organisms as blocks. Biometrika, 44, 1957, pp. 141-149.
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 within-blocks, while (direct + (block size-1)*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. 150-158. 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. 278-279.
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 non-additivity in the Latin square. Biometrika, 45, 1958, pp. 69-73.
Follows the Wilk-Kempthorne 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. 581-586.
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 non-linear situations. Biometrika, 46, 1959, pp. 77-90.
J. D. Biggers: The estimation of missing and mixed-up observations in several experimental designs. Biometrika, 46, 1959, pp. 91-105.
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. 139-149.
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. 251-259.
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 two-factor interactions are estimable.
W. A. Glenn: A comparison of the effectiveness of tournaments. Biometrika, 47, 1960, pp. 253-262.
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. 263-271.
An incomplete-block design with all blocks of the same size and with one control treatment is defined to have supplemented balance if all non-control treatments have the same replication and have the same concurrence with the control, and if all pairs of non-control treatments have the same concurrence.
Example for 4 non-control treatments in 4 blocks of size 7: all blocks like AABCDOO, where O is the control. Example for 6 non-control 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. 273-283
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. 215-218.
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 variance-balanced in n plots then the (harmonic mean) efficiency factor is (n-b)/[n(1-1/v)] with b blocks and v treatments. (c) Theorem 2: If the design is binary, connected, equi-replicate and variance-balanced then b >= v.
M. J. R. Healy and J. C. Gower: Aliasing in partially confounded factorial experiments. Biometrika, 48, 1961, pp. 218-220.
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. 41-56. 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 two-sided null hypotheses.
Damaraju Raghavarao: On balanced unequal block designs. Biometrika, 49, 1962, pp. 561-562.
Short proof of Fisher's inequality, and of its extension for resolved designs, for variance-balanced equi-replicate block designs (no assumption of equal block sizes).
A. Zinger: A note on optimal allocation for a one-way layout. Biometrika, 49, 1962, pp.563-564.
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. 31-45.
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. 63-73.
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. 281-291.
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 equi-replicate 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 N₁ / N₂ / ... / N_m; and to be Extended Group Divisible (m) (EGD/m) if it is PB wrt a factorial association scheme N₁ × N₂ × ... × 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. 335-352. 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 second-order tensors, with related tests and designs. Biometrika, 50, 1963, pp. 353-375.
R. O. Collier and F. B. Baker: The randomization distribution of F-ratios for the split-plot design---an empirical investigation. Biometrika, 50, 1963, pp. 431-438.
Split-plot structure 4/6/4. Analysis using different error mean squares for the split-plot treatment factor and the interaction. Randomization simulation. Conclusion that F-tests 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. 1-10.
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. 111-118.
E.-R. Muller: A method of constructing balanced incomplete block designs. Biometrika, 52, 1965, pp. 285-288.
Uses a complete set of mutually orthogonal squares of side n to produce a BIBD for n treatments in n(n-1) blocks of any size up to n.
J. A. John: A note on the analysis of incomplete block experiments. Biometrika, 52, 1965, pp. 633-636
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 Bose-Mesner algebra (1959).
Raymond O. Collier, Jr. and Frank B. Baker: Some Monte Carlo results on the power of the F-test under permutation in the simple randomized block design. Biometrika, 53, 1966, pp. 199-203.
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.431-437.
Development of Schniederman and Armitage 1962.
R. Bohrer: On Bayes sequential design with two random variables. Biometrika, 53, 1966, pp. 469-475.
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. 497-506. 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. 507-524.
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 n² 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 2-transitive 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. 525-533.
N. R. Draper and W. E. Lawrence: The use of second order `spherical' and `cuboidal' designs in the wrong regions. Biometrika, 53, 1966, pp. 596-599.
H. O. Hartley and J. N. K. Rao: Maximum likelihood estimation for the mixed analysis of variance model. Biometrika, 54, 1967, pp. 93-103.
Precursor to Patterson and Thompson 1971.
M. G. Mostafa: Designs for the simultaneous estimation of functions of variance components from two-way crossed classifications. Biometrika, 54, 1967, pp. 127-131.
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 non-linear situations. Biometrika, 54, 1967, pp. 147-153.
D. W. Alling: Tests of relatedness. Biometrika, 54, 1967, pp. 459-469
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. 471-477.
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. 479-486.
Small blocks nested in large blocks. If either size of block is ignored, treatments form a balanced incomplete-block 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 non-linear situations: multiresponse case. Biometrika, 54, 1967, pp. 662-665.
V. D. Fedorov, V. N. Maximov and V. G.Bogorov: Experimental development of nutritive media for micro-organisms. Biometrika, 55, 1968, pp. 43-51.
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. 53-66.
H. D. Patterson: Serial factorial design. Biometrika, 55, 1968, pp. 67-81.
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, A₁ A₂ A₄ is a defining contrast then so is A₂ A₃ A₅. 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 equi-replicate designs. Biometrika, 55, 1968, pp. 251-253.
Harmonic mean efficiency factor. Those for a design (E) and its dual (F) are related by (v-1)F + (b-v)EF = (b-1)E.
I. I. Berenblut: Changeover designs balanced for the linear component of first residual effects. Biometrika, 55, 1968, pp. 297-303.
D. G. Hoel: Closed sequential tests of an exponential parameter. Biometrika, 55, 1968, pp. 387-391.
Like Armitage 1957 and Schneiderman and Armitage 1962 but the response is exponential instead of normal.
E. J. Snell and J. Bryan-Jones: A design balanced for trend. Biometrika, 55, 1968, pp. 535-539.
Linear time trend. Non-normal 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. 1-41.
Given one-dimensional 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. 55-63.
O. Kempthorne and T. E. Doerfler: The behaviour of some significance tests under experimental randomization. Biometrika, 56, 1969, pp. 231-248.
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 change-over designs. Biometrika, 56, 1969, pp. 283-293.
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. 301-311.
Sequential in the sense of knowing the results to date.
D. H. Rees: The analysis of variance of some non-orthogonal designs with split-plots. Biometrika, 56, 1969, pp. 43-54.
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 non-existence of orthogonal Latin squares. Biometrika, 56, 1969, pp. 547-551.
Let G be cyclic of order n and let q be the smallest prime dividing n. Then
1. there are q-1 MOLS based on G (this follows directly from Mann's 1942 automorphism method!)
2. 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. 553-560.
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. 681-683.
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. 19-46.
A. C. Atkinson: The design of experiments to estimate the slope of a response surface. Biometrika, 57, 1970, pp. 319-328.
N. G. Becker: Mixture designs for a model linear in the proportions. Biometrika, 57, 1970, pp. 329-338.
S. C. Pearce: The efficiency of block designs in general. Biometrika, 57, 1970, pp. 339-346.
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. 347-350.
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. 351-355. 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: Play-the-winner sampling for selecting the better of two binomial populations. Biometrika, 57, 1970, pp. 357-365.
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: one-way analysis of variance. Biometrika, 57, 1970, pp. 421-430.
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 change-over designs for a quantitative factor at four levels. Biometrika, 57, 1970, pp. 537-549.
Improvement on Berenblut's designs, partly by using the ideas of serial factorial design.
P. A. K. Covey-Crump and S. D. Silvey: Optimal regression designs with previous observations. Biometrika, 57, 1970, pp. 551-566.
Design uses Hadamard matrices.
M. A. Kastenbaum, D. G. Hoel and K.O. Bowman: Sample size requirements: randomized block designs. Biometrika, 57, 1970, pp. 573-577.
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 block-by-treatment 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. 149-153.
D-optimality 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. 155-159.
S. C. Pearce: Precision in block experiments. Biometrika, 58, 1971, pp. 161-167.
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. 169-175.
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. 177-181.
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. 279-294.
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. 403-417.
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. 419-426.
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. 427-432.
H. D. Patterson and R. Thompson: Recovery of inter-block information when block sizes are unequal. Biometrika, 58, 1971, pp. 545-554.
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 Covey-Crump and Silvey. Biometrika, 58, 1971, pp. 683-685.
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. 275-293.
B. Afonja: Minimal sufficient statistics for variance components for a general class of designs. Biometrika, 59, 1972, pp. 295-302.
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 two-stage procedure for choosing the better of two binomial populations. Biometrika, 59, 1972, pp. 317-322.
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. 481-483.
Improvement on deCani, 1971.
Agnes M. Herzberg and D. R. Cox: Some optimal designs for interpolation and extrapolation. Biometrika, 59, 1972, pp. 551-561.
R. J. Brooks: A decision theory approach to optimal regression design. Biometrika, 59, 1972, pp. 563-571.
J. A. Cornell: A note on the equality of least squares estimates using second-order equiradial rotatable designs. Biometrika, 59, 1972, pp. 686-687.
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. 15-19.
S. D. Silvey and D. M. Titterington: A geometric approach to optimal design theory. Biometrika, 60, 1973, pp. 21-32.
H. D. Patterson: Quenouille's changeover designs. Biometrika, 60, 1973, pp. 33-45.
Designs for t treatments in 2t periods and t² subjects with treatments orthogonal to periods, to subjects and to first-order residual effects. Construction uses Eulerian trails in digraphs.
W. B. Hall and E. R. Williams: Cyclic superimposed designs. Biometrika, 60, 1973, pp. 47-53.
Row-column designs for two sets of t treatments, constructed cyclically.
J. A. John: Generalized cyclic designs in factorial experiments. Biometrika, 60, 1973, pp. 55-63.
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. 309-318.
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. 345-348.
D. A. Preece, S. C. Pearce and J. R. Kerr: Orthogonal designs for three-dimensional experiments. Biometrika, 60, 1973, pp. 349-358.
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. 477-487.
D. F. Andrews and Agnes M. Herzberg: A simple method for constructing exact tests for sequentially designed experiments. Biometrika, 60, 1973, pp. 489-497.
G. H. Freeman: Experimental designs with unequal concurrences for estimating direct and remote effects of treatments. Biometrika, 60, 1973, pp. 559-563.
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. 573-579.
S. Geisser: A predictive approach to the random effect model. Biometrika, 61, 1974, pp. 101-107.
S. D. Silvey and D. M. Titterington: A Lagrangian approach to optimal design. Biometrika, 61, 1974, pp. 299-302.
R. J. Brooks: On the choice of an experiment for prediction in linear regression. Biometrika, 61, 1974, pp. 303-311.
I. I. Berenblut and G. I. Webb: Experimental design in the presence of autocorrelated errors. Biometrika, 61, 1974, pp. 427-437.
In change-over 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 D-criterion.
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. 449-460.
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. 585-592.
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. 633-636.
Upper and lower bounds for the harmonic-mean 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. 57-70.
D. R. Jensen, L. S. Mayer and R. H. Myers: Optimal designs and large-sample tests for linear hypotheses. Biometrika, 62, 1975, pp. 71-78.
B. A. Worthington: General iterative method for analysis of variance when block structure is orthogonal. Biometrika, 62, 1975, pp. 113-120.
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. 277-288.
A. C. Atkinson and V. V. Fedorov: Optimal design: experiments for discriminating between several models. Biometrika, 62, 1975, pp. 289-303.
P. J. Laycock: Optimal design: regression models for directions. Biometrika, 62, 1975, pp. 305-311.
D. M. Titterington: Optimal design: some geometrical aspects of D-optimality. Biometrika, 62, 1975, pp. 313-320.
G. Berry: Design of carcinogenesis experiments using the Weibull distribution. Biometrika, 62, 1975, pp. 321-328.
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.329-334.
Several dilutions necessary before plating out.
L. L. Pesotchinsky: D-optimum and quasi-D-optimum second order designs on a cube. Biometrika, 62, 1975, pp. 335-340. With correction, 63, 1976, p.412.
J. Eccleston and K. Russell: Connectedness and orthogonality in multi-factor designs. Biometrika, 62, 1975, pp. 341-345.
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. 347-352.
T. A. Louis: Optimal allocation in sequential tests comparing the means of two Gaussian populations. Biometrika, 62, 1975, pp. 359-369. With correction 63, 1976, p. 218.
D. Conniffe and J. Stone: Some incomplete block designs of maximum efficiency. Biometrika, 62, 1975, pp. 685-686. With comment, 63, 1976, p. 686.
Group divisible designs with two groups and between-groups concurrence equal to one more than the between-groups concurrence are A-optimal (result often attributed to Cheng 1978).
E. R. Williams: Efficiency-balanced designs. Biometrika, 62, 1975, pp. 686-689.
A block design is defined to be efficiency-balanced 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. 77-82.
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. 83-92.
Introduction of \alpha-designs.
David G. Herr: A geometric characterization of connectedness in a two-way design. Biometrika, 63, 1976, pp. 93-100.
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. 137-141.
Responses have Lehmann distribution. Rank tests.
Ralph A. Bradley and Abdalla T. El-Helbawy: Treatment contrasts in paired comparisons: basic procedures with application to factorials. Biometrika, 63, 1976, pp. 255-262.
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. 555-558.
Some Abelian group designs which are group-divisible. No mention of difference sets!
Kim-Lian Kok and H. D. Patterson: Algebraic results in the theory of serial factorial design. Biometrika, 63, 1976, pp. 559-565.
Change-over designs for t treatments in 2t periods and t² 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. 678-680.
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. 59-66.
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. 67-72.
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. 73-77.
Comparison of the following two designs for a two-factor 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. 145-147.
I. N. Vuchkov: A ridge-type procedure for design of experiments. Biometrika, 64, 1977, pp. 147-150.
Stuart J. Pocock: Group sequential methods in the design and analysis of clinical trials. Biometrika, 64, 1977, pp. 191-199.
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. 291-297.
R. J. Brooks: Optimal regression design for control in linear regression. Biometrika, 64, 1977, pp. 319-325.
J. A. Eccleston and K. G. Russell: Adjusted orthogonality in nonorthogonal designs. Biometrika, 64, 1977, pp. 339-345.
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. 347-354.
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. 597-603. 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. 605-611. With correction 66, 1979, p. 402.
A. Dey: Construction of regular group divisible designs. Biometrika, 64, 1977, pp. 647-649.
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. 75-78.
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. 79-84.
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. 335-340.
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. 397-401.
Extended the meaning of `generalized cyclic' to semi-regular action of an Abelian group.
E. R. Jones and T. J. Mitchell: Design criteria for detecting model inadequacy. Biometrika, 65, 1978, pp. 541-551.
S. D. Silvey: Optimal design measures with singular information matrices. Biometrika, 65, 1978, pp. 553-559.
K. Sinha: A resolvable triangular partially balance incomplete block design. Biometrika, 65, 1978, p. 665.
Doubling the classical self-dual 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 D-optimal design measures for three rival linear models. Biometrika, 65, 1978, pp. 666-667.
M. Shafiq and W. T. Federer: Generalized N-ary balanced block designs. Biometrika, 66, 1979, pp. 115-123.
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. 307-316.
Sarah C. Cotter: A screening design for factorial experiments with interactions. Biometrika, 66, 1979, pp. 317-320.
All factors at two levels. Treatments are: all 0; one 1 and rest 0; one 0 and rest 1; all 1. Hence model-independent tests of main effects.
M. Singh and A. Dey: Block designs with nested rows and columns. Biometrika, 66, 1979, pp. 321-326.
Block structure is blocks /(rows * columns). Balance in the bottom stratum only.
K. G. Russell: Balancing carry-over effects in round robin tournaments. Biometrika, 67, 1980, pp. 127-131.
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 carry-over from B. ``Balance'' means that each team receives carry-over from all but one of the other teams in each round, with carry-over 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. 381-388.
Sequential design for estimating maximum of quadratic function ax+bx² 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. 489-492.
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. 503-504.
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. 613-619.
M. Jacroux: On the determination and construction of E-optimal block designs with unequal numbers of replicates. Biometrika, 67, 1980, pp. 661-667.
Given b blocks of size k and v treatments. If replications are specified then a design is E-optimal if all its concurrences are at least r_min(k-1)(v-1); when k=2 this can happen only if v-1 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. 706-708.
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. 119-132.
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. 493-500.
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. 617-627.
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. 61-17.
H. L. Agrawal and J. Prasad: Some methods of construction of balanced incomplete block designs with nested rows and columns. Biometrika, 69, 1982, pp. 481-483.
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. 597-612.
Two-dimensional designs with correlated errors. Defines a design to be treatment-balanced 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. 183-198. MR 85j:62075
Restricted randomization using two-transitive 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. 199-214.
A cross-over 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 extra-balanced 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. 205-213.
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. 215-225.
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. 305-314.
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. 433-440
Variance-balanced 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. 441-446.
Uses induced subgraphs of the treatments-blocks graph of an incomplete block design to find this upper bound.
D. Robinson: A comparison of sequential treatment allocation rules. Biometrika, 70, 1983, pp. 492-495.
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. 496-500.
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 D-optimal and E-optimal block designs. Biometrika, 70, 1983, pp. 695-706.
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. 155-160.
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 variance-balanced designs. Biometrika, 72, 1985, pp. 165-172.
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 third-order designs to estimate the slope of a response surface. Biometrika, 72, 1985, pp. 173-178.
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 two-dimensional field trials. Biometrika, 72, 1985, pp. 223-226.
The Papadakis estimator is appropriate for one pattern of neighbour correlation but not for another. In both cases, appropriately neighbour-balanced designs lead to simpler estimators.
J. Kunert: Optimal repeated measurements designs for correlated observations and analysis by weighted least squares. Biometrika, 72, 1985, pp. 375-389.
Cross-over designs with no residual effects but within-patient correlation. Special kinds of Latin square are optimal.
C.-Y. Suen and I. M. Chakravarti: Balanced factorial designs with two-way elimination of heterogeneity. Biometrika, 72, 1985, pp. 391-402.
Row-column designs with factorial balance.
R. A. Ipinyomi and J. A. John: Nested generalized cyclic row-column designs. Biometrika, 72, 1985, pp. 403-409.
Structure is blocks/(rows * columns). Design is preserved by a semi-regular cyclic group.
D. Steinberg: Model robust response surface designs: scaling two-level factorials. Biometrika, 72, 1985, pp. 513-526.
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 cross-validation. Biometrika, 72, 1985, pp. 527-537.
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. 539-544.
For complete blocks with neighbour correlation within blocks, designs with non-directional neighbour balance are efficient.
I. Ford, D. M. Titterington and C. F. J. Wu: Inference and sequential design. Biometrika, 72, 1985, pp. 545-551.
Difficulties in non-asymptotic inference can affect choice of sequential design.
C. F. J. Wu: Asymptotic inference from sequential design in a nonlinear situation. Biometrika, 72, 1985, pp. 553-558.
V. V. Fedorov and V. Khabarov: Duality of optimal designs for model discrimination and parameter estimation. Biometrika, 73, 1986, pp. 183-190.
J. D. Spurrier and D. Edwards: An asymptotically optimal subclass of balanced incomplete block designs for comparison with a control. Biometrika, 73, 1986, pp. 191-199.
A block design with one control treatment is called a balanced control incomplete block design if it consists of a BIBD on the non-control 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. 207-211.
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. 247-277. 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. 279-287.
Fixed effects of `replicates' (superblocks); within-block 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. 289-299.
The number of triangles in the variety-concurrence graph gives an upper bound on the (harmonic-mean) efficiency factor. It is easy to calculate when the design has a large automorphism group.
J. A. John and J. A. Eccleston: Row-column \alpha-designs. Biometrika, 73, 1986, pp. 301-306.
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 row-column design by replacing each column a by all columns ah for h in H. Then within-cosets-of-H contrasts are orthogonal to rows, while between-cosets-of-H 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. 695-700.
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. 701-706.
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 treatment-permuted forms of C, then the first design is universally optimal.
R. A. Bailey: One-way blocks in two-way layouts. Biometrika, 74, 1987, pp. 27-32. 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. 196-199.
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 extra-Poisson variation. Biometrika, 74, 1987, pp. 247-257.
Block structure is a/b/c. The model is multiplicative, conditional on higher-stratum terms. Not explicitly design, but could have implications for the design of experiments for non-normal 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. 311-320. With correction 75, 1988, p. 396.
Cross-over 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. 321-328.
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 non-control treatments.
C.-S. Cheng and K.-C. Li: Optimality criteria in survey sampling. Biometrika, 74, 1987, pp. 337-345.
H. A. David: Ranking from unbalanced paired-comparison data. Biometrika, 74, 1987, pp. 432-436.
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. 443-444.
If there are m groups and k=m-1 and the within-group concurrence is zero then each block contains one element from each of m-1 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. 571-577.
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 change-over designs for factorial experiments. Biometrika, 74, 1987, pp. 649-654.
Change-over 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. 717-724.
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. 725-734.
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. 735-742.
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. 871-876.
Modification of the alpha-design 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. 75-80.
Variance components for several completely crossed factors.
S. D. Oman and E. Seiden: Switch-back designs. Biometrika, 75, 1988, pp. 81-89.
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. 161-164.
The potential loss from unnecessary blocking or from ignoring blocks at the design stage.
P. F. Thall, R. Simon and S. S. Ellenberg: Two-stage selection and testing designs for comparative clinical trials. Biometrika, 75, 1988, pp. 303-310.
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 two-sample tests based on the randomized play-the-winner rule. Biometrika, 75, 1988, pp. 603-606.
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 E-optimal block designs. Biometrika, 76, 1989, pp. 184-188.
Extension of Jacroux 1980 to blocks of unequal size.
G. M. Constantine: Robust designs for serially correlated observations. Biometrika, 76, 1989, pp. 245-251.
Correlations (possibly unequal) between nearest neighbours only. For a main-effects 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 field-plot experiments. Biometrika, 76, 1989, pp. 253-259.
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. 399-402.
Structure is blocks/(rows * columns). Variance-balance in the bottom stratum. Constructions from finite fields.
A. C. Atkinson and A. N. Donev: The construction of exact D-optimum experimental designs with application to blocking response surface designs. Biometrika, 76, 1989, pp. 515-526.
S. Gupta: Efficient designs for comparing test treatments with a control. Biometrika, 76, 1989, pp. 783-787.
Relates Pearce's supplemented balance to more recent literature on this topic.
M. Jacroux and R. S. Ray: On the construction of trend-free run orders of treatments. Biometrika, 77, 1990, pp. 187-191.
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. 193-202.
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. 467-484.
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 two-stage screening designs for survival comparisons. Biometrika, 77, 1990, pp. 507-513.
A two-stage 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. 797-805.
N. Uddin: Some series constructions for minimal size equineighboured balanced incomplete block designs with nested rows and columns. Biometrika, 77, 1990, pp. 829-833.
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.
You-Gan Wang: Gittins indices and constrained allocation in clinical trials. Biometrika, 78, 1991, pp. 101-111.
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. 113-121.
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. 181-188.
K. G. Russell: The construction of good change-over designs when there are fewer units than treatments. Biometrika, 78, 1991, pp. 305-313.
First-order residual effects. The same number of periods as treatments, but fewer subjects.
Joachim Kunert: Cross-over designs for two treatments and correlated errors. Biometrika, 78, 1991, pp. 315-324.
Also residual effects. Many periods.
Ching-Shui Cheng and David M. Steinberg: Trend robust two-level factorial designs. Biometrika, 78, 1991, pp. 325-336.
Autoregressive or other time-series errors. Designs orthogonal to low-order polynomial trend may not be efficient unless there are many changes of level.
Frances P. Stewart and Ralph A. Bradley: Some universally optimal row-column designs with empty nodes. Biometrika, 78, 1991, pp. 337-348.
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 Pearce-Preece OTT school.
S. M. Lewis and A. M. Dean: On general balance in row-column designs. Biometrika, 78, 1991, pp. 595-600.
Review literature on row-column 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. 601-608.
Ashish Das and Sanpei Kageyama: A class of E-optimal proper efficiency-balanced designs. Biometrika, 78, 1991, pp. 693-696.
The designs are not equi-replicate. 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. 857-865.
Really sampling. Want to maximize power and minimize cost.
Ling-Yau Chan: Optimal design for estimation of variance in nonparametric regression using first order differences. Biometrika, 78, 1991, pp. 926-929.
Gregory M. Constantine: On the information and precision matrices of varietal contrasts. Biometrika, 79, 1992, pp. 214-216.
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. 375-380.
R. J. Martin and J. A. Eccleston: Recursive formulae for constructing block designs with dependent errors. Biometrika, 79, 1992, pp. 426-430.
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. 603-610.
Two quantitative treatment factors, but double placebo is unethical.
Weng-Kee Wong: A unified approach to the construction of minimax designs. Biometrika, 79, 1992, pp. 611-619.
R. J. Boys and K. D. Glazebrook: A robust design of a screen for a binary response. Biometrika, 79, 1992, pp. 643-650.
J. A. John and Deborah J. Street: Bounds for the efficiency factor of row-column designs. Biometrika, 79, 1992, pp. 658-661. With correction 80, 1993, pp. 712-713.
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. 763-770.
Timothy E. O'Brien: A note on quadratic designs for nonlinear regression models. Biometrika, 79, 1992, pp. 847-849.
C. F. J. Wu and Runchu Zhang: Minimum aberration designs with two-level and four-level factors. Biometrika, 80, 1993, pp. 203-209.
Designs made from elementary Abelian 2-groups by using pseudofactors for the 4-level factors. Obvious definition of word-length, hence of minimum aberration.
R. R. Sitter: Balanced repeated replications based on orthogonal multi-arrays. Biometrika, 80, 1993, pp. 211-221.
Sampling by using a generalization of orthogonal multi-arrays (which are related to semi-Latin squares).
H. Monod and R. A. Bailey: Two-factor designs balanced for the neighbour effect of one factor. Biometrika, 80, 1993, pp. 643-659.
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. 661-669.
Squeezing a quart out of a pint Hadamard matrix.
Kuemhee Chough Carrière and Gregory C. Reinsel: Optimal two-period repeated measurement designs with two or more treatments. Biometrika, 80, 1993, pp.924-929.
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: D-optimal designs for generalised linear models with variance proportional to the square of the mean. Biometrika, 81, 1994, pp. 295-304.
Sudhir Gupta and Sanpei Kageyama: Optimal complete diallel crosses. Biometrika, 81, 1994, pp. 420-424.
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: E-optimal designs for linear and non-linear models with two parameters. Biometrika, 81, 1994, pp. 739-754.
Sharon L. Lohr: Optimal Bayesian design of experiments for the one-way random effects model. Biometrika, 82, 1995, pp. 175-186.
R. A. Bailey, H. Monod and J. P. Morgan: Construction and optimality of affine-resolvable designs. Biometrika, 82, 1995, pp. 187-200.
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. 589-602.
Two-level 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 assumption-independent 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 two-factor 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. 655-659.
The usual analysis of a complete-block 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. 111-125.
S. Altan and D. Raghavaro: Nested Youden square designs. Biometrika, 83, 1996, pp. 242-245..
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 row-regular 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. 472-476.
Aloke Dey and Chand K. Midha: Optimal block designs for diallel crosses. Biometrika, 83, 1996, pp. 484-489.
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. 653-666.
Continous factors. Choose a set of combinations so that certain low-order 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. 667-680.
E. R. Williams and J. A. John: A note on optimality in lattice square designs. Biometrika, 83, 1996, pp. 709-713.
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. 950-955.
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. 956-961.
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. 245-248.
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 random-effects regression models. Biometrika, 84, 1997, pp. 429-442.
Nizam Uddin and John P. Morgan: Efficient block designs for settings with spatially correlated errors. Biometrika, 84, 1997, pp. 443-454. 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 two-treatment repeated measurements designs. Biometrika, 84, 1997, pp. 455-468. 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. 619-630.
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. 817-829.
Long-term evaluation of two treatments. How to choose (a) numbers of subjects (b) number of medical examinations (c) time interval between examinations (d) length of follow-up, to maximize power.
Rahul Mukerjee: Optimal partial diallel crosses. Biometrika, 84, 1997, pp. 939-948.
General combining ability only. For an unblocked fraction, the set of within-group pairs (groups of equal size) is optimal. Also some results on block designs.
Ching-Shui Cheng: Some hidden projection properties of orthogonal arrays with strength three. Biometrika, 85, 1998, pp. 491-495.
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 two-factor interactions if all higher-order interactions are negligible.
Inchi Hu: On sequential designs in nonlinear problems. Biometrika, 85, 1998, pp. 496-503.
Holger Dette and Weng Kee Wong: Bayesian D-optimal designs on a fixed number of design points for heteroscedastic models. Biometrika, 85, 1998, pp. 869-882.
R. J. Martin and J. A. Eccleston: Variance-balanced change-over designs for dependent observations. Biometrika, 85, 1998, pp. 883-892.
Residual effects and arbitrary pattern of correlation within subjects. Designs constructed from orthogonal arrays of strength two are variance-balanced.
Holger Dette and Timothy E. O'Brien: Optimality criteria for regression models based on predicted variance. Biometrika, 86, 1999, pp. 93-106.
Feng-Shui Chai and Rahul Mukerjee: Optimal designs for diallel crosses with specific combining abilities. Biometrika, 86, 1999, pp. 453-458.
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 half-diallel, gives the diallel equivalent of factorial balance.
Kai-Tai Fang and Rahul Mukerjee: A connection between uniformity and aberration in regular fractions of two-level factorials. Biometrika, 87, 2000, pp. 193-198.
Isabella Verdinelli: A note on Bayesian design for the normal linear model with unknown error variance. Biometrika, 87, 2000, pp. 222-227.
Pi-Wen Tsai, Steven G. Gilmour and Roger Mead: Projective three-level main effects designs robust to model uncertainty. Biometrika, 87, 2000, pp. 467-475.
E. R. Williams and J. A. John: Updating the average efficiency factor in $\alpha$-designs. Biometrika, 87, 2000, pp. 695-699.

R. A. Bailey

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H. Jeffreys: Distance apart in regression. Systematic and ignored polynomial patterns. Biometrika, 31, 1939, pp. 1-8.
Wrong date! What is this paper?