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IN this paper Haldane points out (p. 142) a difference between his results for the mean and variance of x2 in a 2 x n-fold contingency table when the expectation p is fixed and the results obtained by me in my paper (Annals of Eugenics, Vol. VII, part III, p. 211). The difference is that I have (n 1) throughout where Haldane has n. Haldane writes, "my own results would appear to be slightly more accurate than Cochran's", which might, I think, give the impression that both Haldane's results and mine are only approximations. In fact, both results are mathematically exact, the difference between them being one of definition of x2. My paper is almost entirely concerned with the distribution of x2 when the expectation p is not known. In the results which I gave for the distribution of x2 when p is known, I retained the term S (x -x-)2 in the numerator of x2 instead of S (x np)2, to facilitate comparison between this and my other results. Thus my x2 has (n 1) degrees of freedom, whereas Haldane's x2 has n degrees of freedom. Unfortunately I did not emphasize this point in the passage concerned, and as it may have appeared misleading to others besides Haldane, I welcome this opportunity of drawing attention to it. Haldane's x2 is, of course, the one which is normally appropriate in testing the departure from independence when the expectation is known.

In an earlier paper Kume & Wood (2005) showed how the normalizing constant of the Fisher– Bingham distribution on a sphere can be approximated with high accuracy using a univariate saddlepoint density approximation. In this sequel, we extend the approach to a more general setting and derive saddlepoint approximations for the normalizing constants of multicomponent Fisher– Bingham distributions on Cartesian products of spheres, and Fisher–Bingham distributions on Stiefel manifolds. In each case, the approximation for the normalizing constant is essentially a multivariate saddlepoint density approximation for the joint distribution of a set of quadratic forms in normal variables. Both first-order and second-order saddlepoint approximations are considered. Computational algorithms, numerical results and theoretical properties of the approximations are presented. In the challenging high-dimensional settings considered in this paper the saddlepoint approximations perform very well in all examples considered. Some key words: Directional data; Fisher matrix distribution; Kent distribution; Orientation statistics.

SUMMARY The asymptotic distribution of an eigenprojection for a sample correlation matrix is obtained. In particular, it is shown that the rank of the asymptotic covariance matrix depends on distributional parameters in a somewhat complicated manner. The results obtained in this paper can be used to determine this rank. Some applications of the asymptotic distribution of these eigenprojections to inferential problems involving principal components subspaces are given. Many statistical applications involve the analysis of a sample covariance matrix or sample correlation matrix. As a result of the complexity of the exact joint distribution of the elements of these matrices, most of these applications utilise the corresponding asymptotic distributions. Since the asymptotic distribution for the sample covariance matrix is somewhat simpler than that of the sample correlation matrix, covariance-matrix based analyses have received more attention in the literature. Unfortunately, in practice, analyses based on correlation matrices are more prevalent than those based on covariance matrices. As a result, in those situations in which correlation-matrix based procedures have not yet been developed, practitioners will often use procedures that have been developed for a corresponding analysis based on a sample covariance matrix. This results in a procedure which, at best, can only be described as a crude approximation. In this paper, we review some of the asymptotic properties of the sample correlation matrix. In particular, we give the asymptotic distribution of eigenprojections computed from the sample correlation matrix. The associated asymptotic covariance matrix of this distribution is obtained and its rank is determined. Throughout this paper, we focus on the sample correlation matrix computed from the usual sample covariance matrix. However, it should be noted that the asymptotic results obtained can be easily generalised to the correlation matrix computed from any asymptotically normal estimate of the covariance matrix. We also consider some applications which utilise eigenprojections of correlation matrices; these are some inference problems in a principal components analysis of a correlation matrix. In some of these situations, straightforward generalisations of the covariance-matrix based test statistics are not possible because of some complications associated with the rank of the asymptotic covariance matrix of the eigenprojections. In these cases

SUMMARY In this paper we consider the problem of sequentially applying v treatments to n experimental units over time or space where there may be an unknown trend effect which can be expressed as a polynomial function of the order in which the observations are taken. A method is given for constructing ordered applications of treatments to experimental units which produce easily computed and optimal estimates for treatment differences that are orthogonal to the unknown trend effect. In this paper we consider experimental situations where v treatments are to be applied to n experimental units sequentially over space or time and where there may be an unknown time or spatial trend. Any sequential or ordered application of treatments to experimental units shall be called a run order or design and be denoted by d. We assume that the treatments are to be applied to experimental units at equally spaced intervals and that the trend effect can be represented as a polynomial function of the positions in

SUMMARY Williams & Bjerknes (1972) suggested a model for the growth of abnormal cells in a plane. Abnormal cells are assumed to divide k times faster than normal cells. In this paper an unbiased minimum variance estimator of k is obtained and the estimates obtained from simulated growths are compared with the actual values. A method of estimating k from the ' crinkliness' of a growth, as suggested by Mollison (1972), is also considered. 1. THE MODEL In the human body ceLls are all the time dividing and reproducing. In some circumstances it is reasonable to assume that: (i) cells are of two types, normal and abnormal; (ii) if a cell in the basal layer of the epithelium divides, then the daughter cell is of the same type as the mother cell, and both of them remain in the epithelium displacing a neighbouring cell; (iii) cells can divide in the basal layer and nowhere else. Using these assumptions as a basis, Williams & Bjerknes (1972) constructed a stochastic model which they hoped represented cancer growths; of course, they fully understood the tentative nature of the study. In this paper we consider the same model. The basal layer is assumed to be a two-dimensional plane, in which cells are located at every vertex of a regular lattice. At the start there is one abnormal cell, and all the other cells are normal. A daughter cell is equally likely to displace any one of its nearest neighbours. The cellular division times are assumed to be exponentially distributed with the abnormal cells dividing k times faster than the normal cells; k can be considered to be the carcinogenic advantage. Williams & Bjerknes (1972), Mollison (1972), and Downham & Morgan (1973 a, b) investigate such properties as the times taken for the growth to reach a given size, the number of cells in the growth at a given time, and the probability that the growth regresses to zero, i.e. the probability that the abnormal cells are forced from the basal layer by the normal cells. Richardson (1973) obtains some general results for a similar model. In this paper inferential arguments are used to obtain minimum variance estimators of k. For this model it is necessary to consider only the cell divisions around the periphery of the growth of abnormal cells, since any other division does not alter the number of abnormal cells in the basal layer; that is, we need study only divisions for those cells which have at least

SUMMARY Under the common principal component model the covariance matrices Ti of k populations are assumed to have identical eigenvectors, that is, the same orthogonal matrix diagonalizes all Ti simultaneously. This paper modifies the common principal component model by assuming that only q out of p eigenvectors are common to all Ti,, while the remaining p - q eigenvectors are specific in each group. This is called a partial common principal component model. A related modification assumes that q eigenvectors of each matrix span the same subspace, a problem that was first considered by Krzanowski (1979). For both modifications this paper derives the normal theory maximum likelihood estimators. It is shown that approximate maximum likelihood estimates can easily be computed if estimates of the ordinary common principal component model are available. The methods are illustrated by numerical examples.

SUMMARY Sequential x2 and F tests are proposed for comparing more than two treatments. Extending earlier results of the author, one obtains approximations to the significance level and power of the tests, and approximate confidence intervals for the noncentrality parameter O., For each fixed 0 a confidence region is given for the angle c which the treatment effect vector in canonical coordinates makes with a fixed direction, and hence an overall confidence region is obtained for the pair (0, co). For comparing two treatments in clinical trials, Armitage (1975) has suggested a class of sequential tests which he calls repeated significance tests. These tests have been studied numerically by McPherson & Armitage (1971) and theoretically by Siegmund (1977, 1978), who also described a method for obtaining confidence intervals related to these tests and suggested a modification of the tests to increase the precision of the estimates; see also Siegmund & Gregory (1980). The goal of the present paper is to study analogous procedures for comparing more than two treatments. Simple examples involving three treatments might arise in comparing a drug at maximal dose, a drug at minimal dose and placebo, or drug A, drug B and drug A together with B. For theoretical analysis it is convenient to assume that observations are made sequentially on vectors Xn = (X1n, ..., Xrn)' (n = 1, 2, ...), where r denotes the number of treatments and Xij is the response of the jth patient assigned to treatment i. The results of this paper should be applicable to randomized schemes of allocating treatments which approximate this situation. The Xij are assumed to be independently and normally distributed with mean uij and common variance U2. The following discussion is restricted to the special case ,Usj = ,ui for all j. The theory can easily be modified to accommodate stratification on some concomitant variable, provided that the allocation of treatments is approximately balanced within strata. The one-way analysis of variance model with uij = ui for all j is customarily reparameterized in terms of ,u = r-1 2jp and ao = jug-,u, so that E(Xij) = Ju+oi with Z ox = 0. The log likelihood ratio statistic for.testing Ho: 1 = 0 against H1: Z a2 > 0 based on n observations is

Let (Y1J, Y2J) have a bivariate normal distribution with expectations P, and [12, variances a2 and U2 and correlation coefficient p. Furthermore, let the variance U3 be the sum of two components a2 and y7. The problem treated in this paper is then to use these data to test the hypothesis H: y2 = y2. Under H, with Ty2 = 22 = y2 the relation between the correlation coefficient and the variance components is taken to be p = r2/(o2 +?72). Bhoj (1979) has given a test for the case with additional observations on both responses and Ekbohm (1981) suggested a test applicable also when K1 or K2 equals zero. The purpose of this paper is to suggest a few more approaches and to compare tests. The following notation is used: Y Y2 are the means for the first J individuals and Y(Ki) Y(K2) for the final sets. Further i ~~~~Y2 J J ai= ,(Y..J))2 (i= 1, 2), a12= Z (Y1j-Y4J))(Y2J(4)), j=1 j 1 J+Ki J+K2 b= Z (y 1.j._/K,))2, b2 = Z (Y2j-(K2))2, r = al2/y/(al a2). j=J+ 1j=i+

SUMMARY For the analysis of square contingency tables having ordered categories, this paper introduces the diagonals-parameter symmetry model and other related multiplicative models. When these models are applied to the 4 x 4 table on unaided vision first analysed by Stuart (1953, 1955), a possible explanation is obtained for (i) the 'strange residual pattern' noted by McCullagh (1978), both in his recent analysis of the data using the palindromic symmetry model and in the analysis of the data using the quasisymmetry model as fitted by Plackett (1974, p. 61) and others; (ii) the observed asymmetry in the table; (iii) the inhomogeneity observed between the row marginal, i.e. right eye vision, and the column marginal, left eye vision; and (iv) the observed association between right eye and left eye vision. The models introduced in this paper are included within the general class of multiplicative models considered by Goodman (1972), but they are different from the specific kinds of models considered in that article.

Watson (1955) gave an inequality for the minimum efficiency of least squares as measured by the ratio of the generalized variances of the efficient and the inefficient estimators of the vector of regression coefficients. The inequality had been conjectured by J. Durbin. The proof given by Watson was faulty and was acknowledged as such by Watson (1967). This paper gives a proof for the inequality, which up to the present has remained a conjecture. The paper owes much to conversations over several months with J. Durbin and K. G. Binmore. Bloomfield & Watson (1975) have independently obtained a proof of the same inequality. The inequality asserts that

Papers that have been written on the use of the rank order statistic for the two-sample problem under the Lehmann alternatives have focused on the use of rank tests to determine whether there is a difference in the two populations. In particular, the test proposed by Savage (1956) is the rank test that is usually suggested to be used in this situation. In this paper a Bayesian analysis of the problem based on the rank order statistic is presented. Suppose that a random sample of m observations is obtained from distribution 1 and an independent random sample of n observations is obtained from distribution 2, and N = m+n. The rank order statistic will be denoted by z = (z1, ...,ZN), where zi is 0 if the ith smallest observation is from distribution 1 and zi is 1 otherwise (i -1, ..., N). Let F(x) and G(x) be the distribution functions of distributions 1 and 2, respectively, both assumed to be continuous. If G(x) is the Lehmann alternative 1{1 F(x)}k(k > 0), which holds, for example, if the population distributions are both exponential or both Weibull, then the rank order probability is

Let Xl, X2, X3 and X4 obey a quadrivariate niormal distribution with zero means, aind let P4 be the value of the quadrivariate normal integral, i.e. the probability that Xl, X2, X3 and X4 are simultaneously positive. The generalized tetrachoric series for P4, as given by Aitken (unpublished), Kendall (1941, 1945) (see also Kendall & Stuart, 1958, pp. 350-4), and Moran (1948), are not well suited for computation. For the case in which all six correlation coefficients are equal, the series has been sumlmed approximately by McFadden (1956). For special numerical values of the correlation matrix, exact results for P4 have been given by Schlafli (1858, 1860), Anis & Lloyd (1953), and Plackett (1954). Methods for numerical integration in more general cases have been given by Plackett (1954), Ihm (1959), and John (1959). Numerical methods for integration when all the correlation coefficients are equal have been provided by Ruben (1954) and Moran (1956). In this paper we present two series expansions for P4 which are well suited for computation. The first case occurs when X1, X2, X3 and X4 are successive measurements from a stationary Gaussian Markov process (with zero mean), or, equivalently, when the inverse of the correlation matrix has zero elements except on the main diagonal and immediately adjacent to it. The second case occurs when the correlation matrix itself has zero elements except on the main diagonal and adjacent to it. We shall then show that these two cases are related by a simple transformation. In a previous paper, McFadden (1955, eq. (39)) has expressed P4 in terms of the various product moments of the set of random variables Yi which assume the sign of the normal variables Xi. Let Y.=1 when X. O' ---1 when Xi< Oj (i= 1,2,3,4). (1)

IN genetical practice we are constantly presented with large numbers of small samples from populations consisting of several well-defined classes. For example in the mouse we can readily obtain hundreds of litters containing anything from one up to about twelve members. Their totals may agree satisfactorily with expectation on a Mendelian basis, for example J coloured, i white, or 9 grey, H black, i white. But we desire to know whether the individual litters can be regarded as random samples from such a population. In addition the problem of homogeneity may arise. That is to say the population as a whole may not conform to any particular expectation. But we may desire to know whether the litters can be regarded as random samples of the population given by the totals. It has long been known that when the numbers expected in any observation are small, the distribution of x2 departs from that given by Pearson (1900). The mean appears sometimes, but not always, to be equal to the number of degrees of freedom. But the variance is no longer exactly equal to twice that number. Exact expressions for it in certain cases have been given by Pearson (1932) and Cochran (1936). These are based on an ingenious application of the theory of multiple contingency by Pearson. It will be shown in this paper that the first few moments can often be calculated by entirely elementary methods involving nothing more advanced than the multinomial theorem. In an accompanying paper (Griineberg and Haldane, 1937) they will be applied to actual data on mice. We first study the distribution of x2 in a n-fold table with n 1 degrees of freedom, then in a (m x n)-fold table with m (n-i) degrees of freedom. For genetical work we are particularly interested in the (n x 2)-fold table with n degrees of freedom. As a limiting case of the 2-fold table with 1 degree of freedom we derive the moments of the variance of samples from a Poisson series, and thence the distribution of x2 in a n-fold table with n degrees of freedom. The important case of the (m x n)-fold table with (m 1) (n 1) degrees of freedom remains to be investigated. Consider a sample of 8 individuals falling into n classes. Let the expected and observed numbers in these classes be

SUMMARY In this paper the locally most powerful rank test for testing the symmetry axis of a symmetric circular distribution is derived. Then an efficiency expression for a class of linear rank tests is obtained. The results are applied to the class of vonMises distributions and efficiency results for the sign test and the Wilcoxon test are calculated. 1. INTRODIUCTION Circular distributions are encountered in many areas of scientific investigation. Perhaps the most important example is the distribution of phases of periodic phenomena, for example, in biology and physics. Another area of application is the analysis of directions, for example, in earth sciences, migration, etc. Several examples are given by Batschelet (1965). Assume that Xl, ..., Xn are independent observations from a random variable, which takes its values on the unit circle. In this paper we are concerned with the problem of finding nonparametric tests for the hypothesis that the underlying distribution is symmetric with respect to the horizontal axis against the alternative of a displaced or rotated centre of symmetry. We proceed as follows: First we obtain a locally most powerful rank test against rotation alternatives, ? 2. After studying the distribution of a class of related test statistics under the hypothesis as well as under alternatives, ? 3, we derive the efficiency of such test sequences with respect to parametric competitors and we show that a fully efficient nonparametric test exists, ? 4. Finally, we apply some of the results to the class of von Mises distributions, ? 5.

Many problems of statistical inference, notably the ones in multiple decisions and life testing involve the use of ordered Xi's. In an earlier paper Gupta, Pillai & Steck (1964) considered the distribution of linear functions of X(j)'s and gave closed form results for the case when the random variables are equally correlated. Also in the same paper closed form expressions for the distribution of the range W = X(l) - X(1> were obtained for n = 3 and 4 for the general case. In this paper we study the characteristic functions of individual order statistics and also the linear functions for the bivariate and trivariate cases for the general correlation matrix. Formulae for the expected values of X(f), X&2i) and X(j) X(j) and the first and second moments of a linear function of the X(j)'s are obtained. The joint distribution of the range and the mid-range is given in a closed form in the trivariate case, from which the distributions of the mid-range and the mid-range/range ratio are derived, again in closed forms. Best linear unbiased estimators of the common mean of three correlated normal variables have been obtained and tabulation of the coefficients made for different sets of values of pij's. Applications in the fields of life testing and time series analysis are discussed.

are known to admit a sufficient estimator of 0 in realizations of the chain starting with a fixed state and consisting of a fixed number of transitions. This paper considers whether transition probabilities of the same form will admit a sufficient estimator of 0 in other finite regular, but not positively regular, Markov chains. For chains with an irreducible subset of two or more states, in which a realization starts from a fixed state and consists of a fixed number of transitions, these probabilities are found to admit a maximum-likelihood estimator of the function g(6) = -A'(O)/A'(0), which is sufficient and unbiased. There is some difference in chains with an absorbing state, in which realizations start from a fixed state but continue until the absorbing state is reached; in this sequential case, the maximum-likelihood estimator, with the number of transitions in the realization, together provide a sufficient estimator of the function g(O), which in general is no longer unbiased. We restrict ourselves to the particular case where a certain linear relation is satisfied. This gives rise to some simple stochastic matrices admitting a sufficient estimator of 0, which consist of probabilities of the forms 0 and 1 -0; in some of these cases, unbiased sufficient estimators of 0 reduce to known results of Girshick, Mosteller & Savage (1946). Some non-regular finite and infinite chains with absorbing states, associated with random walks, whose matrices consist of various patterns of probabilities 0 and 1 -0, are also found to admit a sufficient estimator of 0. The paper ends with the examination of such an example, arising in sequential estimation.

Full Bayesian computational inference for model determination in undirected graphical models is currently restricted to decomposable graphs, except for problems of very small scale. In this paper we develop new, more efficient methodology for such inference, by making two contributions to the computational geometry of decomposable graphs. The first of these provides sufficient conditions under which it is possible to completely connect two disconnected complete subsets of vertices, or perform the reverse procedure, yet maintain decomposability of the graph. The second is a new Markov chain Monte Carlo sampler for arbitrary positive distributions on decomposable graphs, taking a junction tree representing the graph as its state variable. The resulting methodology is illustrated with numerical experiments on three specific models., 22 pages, 7 figures, 1 table. V2 as V1 except that Fig 1 was corrected. V3 has significant edits, dropping some figures and including additional examples and a discussion of the non-decomposable case. V4 is further edited following review, and includes additional references

This paper introduces, constructs and studies a new class of arrays, called strong orthogonal arrays, as suitable designs for computer experiments. A strong orthogonal array of strength t enjoys better space-filling properties than a comparable orthogonal array in all dimensions lower than t while retaining the space-filling properties of the latter in t dimensions. Latin hypercubes based on strong orthogonal arrays of strength t are more space-filling than comparable orthogonal array-based Latin hypercubes in all g dimensions for any 2 ≤ g ≤ t - 1. Copyright 2013, Oxford University Press.

This paper investigates the definition and the estimation of the Fréchet mean of a random rigid body motion in ℝ-super-p. The sample space SE(p) contains objects M&equals;(R,t) where R is a p×p rotation matrix and t is a p×1 translation vector. This work is motivated by applications in biomechanics where the posture of a joint at a given time is expressed as M&isinv;SE(3), the rigid body displacement needed to map a system of axes on one segment of the joint to a similar system on the other segment. This posture can also be reported as M-super- - 1&equals;(R-super-T, - R-super-Tt) by interchanging the role of the two segments. Several definitions of a Fréchet mean for a random motion are proposed using weighted least squares distances. A special emphasis is given to a Fréchet mean that is equivariant with respect to the inverse transform; this means that if P is the Fréchet mean for M then P-super- - 1 is the Fréchet mean for M-super- - 1, where M is a random SE(p) object. The sampling properties of moment estimators of the Fréchet means are studied in a large concentration setting, where the scatter of the random Ms around their mean value P is small, and as the sample size goes to ∞. Some simple exponential family models for SE(p) data that generalize Downs' (1972) Fisher--von Mises matrix distribution for rotation matrices are introduced and the least squares mean values for these distributions are calculated. Asymptotic comparisons between the estimators presented in this work are carried out for a particular model when p&equals;2. A numerical example involving the motion of the ankle is presented to illustrate the methodology. Copyright 2012, Oxford University Press.

Generalised minimum aberration is a recently-established design criterion for the whole class of orthogonal arrays and fractional factorial designs. The criterion is, as its name suggests, a generalisation of minimum aberration for regular designs and of minimum G 2 -aberration for two-level designs. The aim of the criterion is to find designs which minimise in a certain sense the aliasing between main effects and interactions. In this paper, theoretical results are developed for finding symmetrical orthogonal arrays with generalised minimum aberration for more than two factor levels.

The current literature on fractional factorial plans in block designs centres around orthogonal blocking which may not, however, always be attainable because of practical restrictions on the block size. For general factorials, including asymmetric ones, sufficient conditions are indicated in this paper for a main effect plan to be universally optimal under possibly non-orthogonal blocking. A construction procedure is given using generalised Youden designs in conjunction with orthogonal arrays. We also illustrate how the procedure can be applied to obtain optimal main effect plans in the practically important situation where each factor has two or three levels and the block size is small.

SUMMARY Grenander & Miller (1994) describe a model for representing amorphous twodimensional objects with no obvious landmark. Each object is represented by n vertices around its perimeter, and is described by deforming an n-sided regular polygon using edge transformations. A multivariate normal distribution with a block circulant covariance matrix is used to model these edge transformations. The purpose of this paper is to describe in detail the statistical properties of this multivariate model and the eigenstructure of the covariance matrix. Various special cases of the model are considered, including articulated models and conditional Markov random field models. We consider maximum likelihood based inference and the model is applied to some datasets to explore shape variability.

SUMMARY We consider the principal components analysis of g groups of m variables for those situations in which, for each group, the first k principal components account for most of the total variability of observations in that group. If the set of these gk principal component vectors spans a space of dimension r, where r is less than m, then it will be possible simultaneously to reduce the dimensionality, for all groups, from m to r while retaining most of the within-group variability. Methods are already available for determining if r = k, in which case the g groups have a common principal component subspace. In this paper, we develop a general procedure for determining if r = s for arbitrary s. This can then be used repeatedly to determine r when r > k.

This paper considers nonparametric estimation in a varying coefficient model with repeated measurements (Y ij , X ij , t ij ), for i = 1 n and j = 1 n i , where X ij = (X ij .,,X ijk ) T and (Y ij , X ij , t ij ) denote the jth outcome, covariate and time design points, respectively, of the ith subject. The model considered here is Y ij = X ij T β(t ij )+ e i (t ij ), where β(t)=(β 0 (t)., β k (t)) T , for k≥ 0, are smooth nonparametric functions of interest and e i (t) is a zero-mean stochastic process. The measurements are assumed to be independent for different subjects but can be correlated at different time points within each subject. Two nonparametric estimators of β(t), namely a smoothing spline and a locally weighted polynomial, are derived for such repeatedly measured data. A crossvalidation criterion is proposed for the selection of the corresponding smoothing parameters. Asymptotic properties, such as consistency, rates of convergence and asymptotic mean squared errors, are established for kernel estimators, a special case of the local polynomials. These asymptotic results give useful insights into the reliability of our general estimation methods. An example of predicting the growth of children born to HIV infected mothers based on gender, HIV status and maternal vitamin A levels shows that this model and the corresponding nonparametric estimators are useful in epidemiological studies.

SUMMARY This paper discusses the application of generalised linear methods to bilinear models by criss-cross regression. It proposes an extension to segmented bilinear models in which the expectation matrix is linked to a sum in which each segment has specified row and column covariance matrices as well as a coefficient parameter matrix that is specified only by its rank. This extension includes a variety of biadditive models including the generalised Tukey degree of freedom for non-additivity model that consists of two bilinear segments, one of which is constant. The extension also covers a variety of other models for which least squares fits had not hitherto been available, such as higher-way layouts combined into the rows and columns of a matrix, and a harmonic model which can be reparameterised so a lower rank fit is equivalent to a constant phase parameter. A number of practical applications are provided, including displaying fits by biplots and using them to diagnose models.

SUMMARY Serially balanced sequences of types 1 and 2 are used for designing experiments where one experimental unit receives several treatments over successive periods. Both the direct effect of a treatment applied in a certain period, as well as the residual effect of the treatment applied in the previous period, are of interest. Optimality properties of such sequences have been studied in the literature. However, these sequences have block sizes equal to the number of treatments v and so they pose a problem to the practitioner when v is large. In this paper, a class of sequences with incomplete blocks is proposed and their construction and analysis given. These sequences can be used to estimate direct and residual effects of treatments with high efficiencies. The sequences may be replicated over any number of units when the experiment is run with more than one unit. In many situations, experiments are to be designed where each experimental unit receives some or all of the treatments, one at a time, over successive periods of time. In such designs, each treatment has a 'direct effect' in the period in which it is applied and also a 'residual effect' in the following period. These experiments are called cross-over experiments or, following Hedayat & Afsarinejad (1975), repeated measurements experiments. The corresponding designs are called cross-over or repeated measurements designs. Such experiments are widely used in clinical trials, agricultural experiments, pharmaceutical experiments and in many other fields. One form of such experiments is one in which the number of experimental units must be very small, sometimes only one, but a far greater number of periods may be used. For such experiments, Finney & Outhwaite (1956) introduced serially balanced sequences of types 1 and 2, primarily for use in bioassay, where all the trials are made on one subject. A typical situation for using these sequences is one where many tests can be made in fairly rapid succession, so that one or more treatments may be assigned to the one subject. One of the earliest such examples is in Finney (1955, p. 132), where in the assay of histamine, the experimental unit is an isolated strip of guinea- pig's gut and a sequence of doses is applied to it. Serially balanced sequences are particularly useful here since, with repeated use of one strip of gut, trends in responsiveness may occur and sets of successive doses can be grouped into blocks so as to permit the elimination of the trend. For details of use of such sequences and for discussion and construction of these sequences we refer to Sampford (1957), Finney (1955), Finney & Outhwaite (1955), Finney (1960) and Street & Street (1987). Moreover, if more than one subject is to be used, then these sequences may be easily replicated over the available units. Optimality properties of repeated measurements designs have been investigated by, among others, Hedayat & Afsarinejad (1975, 1978), Cheng & Wu (1980), Magda (1980), Kunert (1983, 1984, 1985), Dey, Gupta & Singh (1983) and Bose (1993). Optimality properties of type 1 and type 2

Canonical correlation analysis measures the linear relationship between two random vectors X 1 and X 2 as the maximum correlation between linear combinations of X 1 and linear combinations of X 2 . Several generalisations of canonical correlation analysis to k > 2 random vectors X 1 ,...,X k have been proposed in the literature (Kettenring, 1971, 1985), based on the principle of maximising some generalised measure of correlation. In this paper we propose an alternative generalisation, called common canonical variates, based on the assumption that the canonical variates have the same coefficients in all k sets of variables. This generalisation is applicable in situations where all X i have the same dimension. We present normal theory maximum likelihood estimation of common canonical variates, and illustrate their use on a morphometric data set.

SUMMARY The paper aims to derive optimal two-treatment cross-over designs for experiments with many periods, when the observations are correlated and residual effects are present. The case of small numbers of periods was treated by Laska & Meisner (1985) and Matthews (1987). Their results are substantially affected by end effects arising because the first and last periods behave differently. We look at designs which maximize an upper bound of the information matrix and show that these designs are highly efficient, especially when the number of periods is large. Assume we have b experimental units, each of which is exposed to a series of treatments. The number of treatments is t and each unit receives k different or identical treatments in successive periods. The design d determines which treatment is applied to which unit at which period. We denote the set of all such cross-over designs by ft,b,k. The measurement on the jth period of the uth unit is denoted by Yd,u,j if design d E itb,k is applied. Williams (1949) introduced a model for this situation which assumes that

SUMMARY Suppose that a homogeneous group of n subjects is observed on T occasions. Each subject has one of K levels of response at each of the T occasions. The cross-classification of the recorded responses over time forms a K T contingency table. This paper discusses the design matrix for the marginal homogeneity model, which assumes that the first-order marginal probabilities at each time are equal. The marginal homogeneity model is a linear model for the vector of cell probabilities. We obtain the design matrix by linearly transforming the parameter space of the cell probabilities; our method will work for any T and any K.

The estimation of parameters in an absorbing Markov chain has been discussed by Gani (1956), Bhat & Gani (1960), Bhat (1961) and possibly by other authors. Asymptotic theory for the distribution of maximum-likelihood estimators applies if a large number of independent replicates of the chain is available. These replicates, however, could be considered as occurring sequentially in time, so that the chain has implicitly been altered in such a way that the absorbing state has been replaced by an 'instantaneous return' state; once it is reached, a new replicate is commenced starting from the original initial state. In this paper, we concern ourselves with the asymptotic theory of maximum-likelihood estimators from a single realization of truly absorbing chains. It is clear that if the number of states is kept fixed there can be no asymptotic theory, since with probability one, an absorbing state will be reached after a finite number of transitions, and no further informa. tion can be obtained by continuing observation. In ? 2, we show by means of two simple examples that asymptotic theory may, at least in some cases, be available if the number of non-absorbing states is large. In the remaining sections, forming the main part of the paper, we discuss a more complex example, a population genetic model of Moran (1958a). We do not prove that the conjectured asymptotic theory holds for the estimation of the parameter, but produce numerical evidence from simulation studies to support this conjecture. Further research is needed to clarify the general problem of inference in absorbing Markov chains. The problem can be made to depend on the theory of positively regular chains but the latter is itself incomplete for this purpose.

Recent papers by Hotelling & Pabst (1936), Pitman (1937), Kendall (1938) and Kendall, Kendall & Babington-Smith (1939) discuss the distribution of the correlation coefficient when members of the sample corresponding to the two variates are permuted randomly relative to one another. In the case of rank correlation, the characteristics of the population sampled are generally unknown, and a significance test has to be based on the distribution obtained from the sample in this way. Hotelling & Pabst prove that as the sample size is increased, Spearman's p tends to follow a normal distribution law. Kendall's measure of rank correlation, r, in which all possible corresponding pairs in two given rankings are assigned marks + 1 according to whether they agree or differ in order, follows a specially simple distribution law which tends rapidly to the normal form and becomes highly correlated with Spearman's p for samples of moderate size. The present paper discusses the properties of the class of correlation coefficients I obtained on replacing Kendall's marks + 1 by a more general system of scores. By an empirical argument Kendall et al. showed it to be likely that the correlation between r and p is

are possible. We set out in this paper five different methods of randomization in a circle and two possible criteria which may be used to test for randomness. A method of obtaining the moments of these criteria is given and an approximation to the P.D.F. is suggested. Certain practical difficulties have been encountered in applying the theory in that there is uncertainty in the identification of the chromosomes, the shadow of the wall of the nucleus is not always visible and when visible is only approximately circular and so on. We give here theory only and reserve the practical applications for another paper 2. NOTATIONS The plane area is assumed to be a circle of unit radius and an arbitrary axis of reference is supposed. n random chords are supposed drawn on this circle. If we consider a random point (R, 0) (O < R < 1, 0 < 0 < 27T), we write its P.D.F. asf(R, 0). If we consider a random chord it is sometimes convenient to consider the perpendicular, p, to it from the centre, O < p < 1, and the angle a the perpendicular makes with the reference line. The P.D.F. of the centre of the chord we will denote by g(p, a). It is sometimes convenient to work withf (R, 0)

SUMMARY Various expectations concerning the convex hull of N independently and identically distributed random points in the plane or in space are evaluated. Integral expressions are given for the expected area, expected perimeter, expected probability content and expected number of sides. These integrals are shown to be particularly simple when the underlying distribution is normal or uniform over a disk or sphere. In two recent papers Renyi & Sulanki (1963, 1964) have given expressions for the expected area, perimeter, and number of vertices of the convex hull of N independently and identically selected random points in the plane. In these papers, limit theorems for asymptotically large N receive the greatest attention. Here the emphasis will be on the development of convenient formulae for fixed values of N. Some new results for random convex hulls in the plane are derived, such as the expected probability content, and also various expectations concerned with random convex hulls in three and higher dimensions. Special attention is given to the case of normally distributed points and also to the case of points drawn uniformly from an ellipse or an ellipsoid. Historically, calculating the expected probability content of three random points in two dimensions is known as 'Sylvester's problem', and has been solved explicitly for many different distributions by Deltheil (1926, p. 42). The corresponding problem of four points in three dimensions is connected with the name of Hostinsky (1925). A discussion of Sylvester's problem is given in Kendall & Moran's monograph, Geometrical Probability (1963). 2. THE EXPECTED NUMBER OF VERTICES, FACES, AND EDGES

SUMMARY Let Y(x) be the toxicity associated with some dosage level x. Let the conditional distribution of Y be normal with mean linear in x. The optimal dosage is defined to be the largest value of x such that the probability that Y(x) does not exceed -1 is at least y, where -j is some threshold toxicity level and y is some specified tolerance probability. Recently, Eichhorn and Zacks have introduced the problem of sequentially and consistently estimating the optimal dosage at the same time controlling the probability that a dose does exceed the optimal dose. In this paper sequential search procedures are defined which control the probability that the toxicity exceeds -q, the threshold value. Two models are considered corresponding to two different variance functions.

Let X1,..., Xn represent a sample of points on a line of length L whose left endpoint is 0, and let X(1) < ... < X(n) denote the order statistics of the X's. The problem of testing the hypothesis that the points placed along the line arise as a random sample from a uniform distribution has been of considerable interest for some time. Many of the tests are based on the spacings between the ordered points, that is on, D1 = X(1), Di = X(i)X(i1) (i = 2, ...,n) and Dn+1 = L-X(n) (Pyke, 1965). The present paper considers a slight modification of a test of randomness of points on a line which was introduced by Pinder & Witherick (1973) and which is also based on spacings. The test statistic, called the linear nearest neighbour statistic, is the onedimensional analogue of the areal two-dimensional nearest neighbour statistic proposed by Clark & Evans (1954), which has found considerable use in the biological sciences for assessing the randomness of points in two dimensions (Pielou, 1969, Chapter 10). The linear nearest neighbour statistic was proposed to study the clustering, dispersion, i.e. equal spacing, or randomness of points; for example, stores, towns, housing sites, along natural or manmade 'lines' such as streets and waterways. Various uses of the statistic in geography are given by Pinder & Witherick (1973), and an archaeological application is discussed by Stark & Young (1981). Thus far, exact and approximate distributions of the statistic are unknown. To define the linear nearest neighbour statistic let Mi be the distance between X(i) and the point nearest it, that is Mi = min (Di, Di+ ) (i = 1, ..., n). When L is known, the linear nearest neighbour statistic is n M= ZMi/L.

SUMMARY An expansion of the score statistic as a polynomial in the random variable can be used to provide a nondecreasing sequence of lower bounds for the Fisher information when the moments are known as functions of the parameter. If the random variable is asymptotically normal, the first three bounds provide the first three terms in the asymptotic expansion for the Fisher information. This leads to a definition of efficiency which may be used to advantage with small sample sizes, providing lower bounds for the efficiency of any statistic whose moments are known but whose exact distribution is intractable. Fisher (1925) suggested that the efficiency of a statistic be measured by its squared correlation with the maximum likelihood estimate. This idea was extended by Rao (1962) who defined first- and second-order efficiency by considering how well a linear or quadratic function of the statistic correlated with the derivative of the log likelihood. In ? 2 of the present paper, we develop this further by looking at polynomial expansions of the score statistic to obtain a nondecreasing sequence of lower bounds for the Fisher information 4y(0) of a statistic Y = y(X) which is some function of the data X. These bounds may be expressed in terms of the moments of the statistic and their first derivatives with respect to the unknown parameter 0. If we return to Fisher's (1925) definition of efficiency as the ratio of fy(0)/Ox(0), whre Ox(0) is the Fisher information in the whole sample, we can use this to obtain an efficiency for Y which is invariant under one-one transformations and which does not need to appeal to asymptotic arguments. The results are particularly useful for random variables whose moments are known but whose exact sampling distribution is either unknown or intractable. In ? 3, we consider other implications, such as the possibility of using different sequences of functions and the question of convergence. A recent application of these results by Brockwell & Brown (1981) is given as an example. Rao (1962) pointed out that the limit of the ratio fy(0)/fx(O) in large samples was the asymptotic squared correlation between the statistic Y and the derivative of the log likelihood function, lx(O) = alx(0)/aO, for the full data set. When this ratio tends to one, both Fisher and Rao suggested using lim {fx(0) -fy(0)} as n -+ oo as a means of discriminating between alternative statistics. This asymptotic concept of second-order

SUMMARY This paper generalizes the procedure proposed by Girshick (1946) for sequential comparison of two sequences of independent and identically distributed random variables to the sequential comparison of two positive regular Markov chains, both defined on the same finite state space.

SUMMARY The extremal quotient, Q, is defined for nonnegative observations as the ratio of the largest observation in a sample to the smallest observation. This paper gives the exact distribution of this quotient based on a sample drawn from a gamma distribution. The statistic is scalefree, and its distribution depends only on the sample size n and on the shape parameter y of the underlying gamma distribution. Applications include certain areas of industrial quality control and Hartley's maximum F ratio, Fmax, for testing heterogeneity of variances in a multisample, equal degree of freedom situation. The kth moment of Q exists only if k < ; the first and second moments are evaluated for various values of n and . Comparisons are made with David's tables of the percentage points of F.max

We first show a simplification and a refinement of the result obtained by Shibata ('1976). It is an evaluation of the mean squared error of estimates of regression parameters when the number of regression variables is selected by a certain procedure. The evaluation yields a good approximation to the efficiency of a selection procedure when the number of regression variables is small. The approximate efficiency of FPE (Akaike, 1970), AIC (Akaike, 1973) or Cp (Mallows, 1973) turns out to be not satisfactorily high, although they are all asymptotically efficient in the sense of Shibata (1981). Based on such an approximation, a procedure for choosing a out of the family FPE, (Bhansali & Downham, 1977) is suggested. A further generalization is also developed. Consider a regression model y = XJI(k)l+ e, where y' = (yl yj) is a vector of observations, X is an n x K design matrix and f,(k)' = (fl1, ..., fk, 0, ..., 0) is a vector of regression parameters. For simplicity we assume e' = (El, ..., EJ) is a vector of independent normally distributed random variables with mean zero and variance a'. We call the above model 'model k'. In this paper, we are concerned only with the problem of selecting the number k from a given range 1 < k < K. Under the model k, we have a least squares estimate of the regression parameters, 43(k)' = (/h, ...,/f1k), which is a solution of X(k)'X(k) /(k) = X(k)'y. Here X(k) is an n x k submatrix consisting of the first k column vectors of the design matrix X. Hereafter, f,(k) is occasionally considered as the corresponding K-dimensional vector with the undefined entries being zero. Suppose that observations yl,..., yn come from a model ko (1 < ko < K) with the regression parameter ,B such that f3k0 * 0. Our main concern is the mean squared error

This paper introduces the notion of affine (μ1,…,μ1)-resolvability and explores the interrelations between: (a) affine (μ1,…,μ1)-resolvability, (b) variance-balance, and (c) the relation b = v+t-1, where b is the number of blocks. It is seen that, while (a) and (b) imply (c), and (b) and (c) imply (a), the relation (a) and (c) imply (b) is not in general true. A necessary and sufficient condition under which (a) and (c) imply (b) has been derived and certain nonexistence results follow. The last section states an open problem in this connexion and indicates the link with a problem in factorial designs.

Suppose that Y is an n x 1 observable random vector, normally distributed, with mean XfB and dispersion D(F1, ..., Fk), where X is an n x q matrix of known constants, ,B is a q x 1 vector of unknown parameters, Fj = ((ojtu)), say, is an unknown p x p symmetric positive-definite matrix for j = 1, ..., k, and D(F1, ..., Fk) is an n x n symmetric positivedefinite matrix which is a known linear function of the vector of -kp(p + 1) parameters ojtu (j = 1, ..., k; t ) u). In addition, suppose that there are observable p x p random matrices Si (j = 1, ..., k), with Y, S1, ..., Sk mutually independent, such that nj Sj Wp(nj, F), where Wp(n, F) denotes the Wishart distribution of size p with n degrees of freedom and associated dispersion F. Suppose that we are interested in testing the hypothesis Ho: H,B = h versus H1: H/I * h where H is an r x q matrix of known constants with rank r such that H/I is estimable and h is an r x 1 vector of known constants in the column space of H. The problem is to test Ho, the probability of type I error to be approximately oc for all values of the unknown parameters. In the literature, two methods of obtaining a test in particular examples and in general have been tractable. One is an expansion method following Welch (1947) for the Behrens-Fisher problem, James (1951) for the one-way classification, James (1956) for the common mean problem and James (1954) for the problem in general. The other is an approximate degrees of freedom method following Welch (1947) for the Behrens-Fisher problem, Welch (1951) for the one-way classification, Meier (1953) for the common mean problem and Johansen (1980) for the problem in general. Welch (1947) showed that the approximate degrees of freedom test and the expansion test in the Behrens-Fisher problem are equivalent to terms of order nj-'. Welch (1951) showed that the approximate degrees of freedom test of Welch (1951) and the expansion test of James (1951) for the one-way classification are equivalent to terms of order nj-l. Johansen (1980) stated that it is difficult to compare his approximate degrees of freedom test with the expansion test of James (1954). It is the purpose of this paper to show that equivalence to terms of order n-' holds between the general solutions of Johansen (1980) and James (1954) by deriving the approximate degrees of freedom test of Johansen using the methods of James and by deriving a simpler version of the expansion test.

SUMMARY One important practical application of principal component analysis is to reduce a large number of variables, say p, to a smaller number, m, by making use of the first m principal components. This technique can easily be extended to two or more groups if the subspaces spanned by the first m principal components are the same for all groups. In this paper we develop an approximate procedure for testing such a hypothesis of common subspaces when two groups are involved. The adequacy of the approximation is investigated by a simulation and the method is illustrated by a numerical example.

Let S = {s} be the set of subsets of {1, ..., N} such that each s eS contains n elements. We consider in this paper only designs p(s) with fixed sample size, that is P(S) > 0 only if s eS and ESp(s) = 1, where Es denotes summation over s eS. Let p(s) be independent of the Yk, where Yk is the value of the variable of interest for the population unit labelled k (k = 1, ..., N). We consider a superpopulation model: Yl, ..., YN is a realization of Y1, .,YN with joint distribution 6. If d(. ) denotes expectationwith respect to 6, let, for k, I = 1, ..., N, 6 be further specified by

Upper bounds for efficiency factors can be useful in helping us to construct incomplete block designs that have good statistical properties. A bound gives us an idea of how high we might expect the efficiency factor to be, and therefore gives us a standard against which to judge any design we might construct (Pearce, 1968; Jarrett, 1977; Williams & Patterson, 1977). In this paper we derive a bound for the minimum canonical efficiency factor (Kiefer, 1959; Shah, 1960).

SUMMARY A test for the maximum of a large number of values of Kendall's ' is obtained by an application of a Poisson limit theorem for symmetric statistics proved by Silverman & Brown (1978). The test depends on the number of comparisons rather than the sample size being large. It is shown that the asymptotics are always practically useful. A table of critical values for the test statistic is given, together with an illustrative application. Given the measurements of k characteristics for each of n individuals, suppose that one wishes to decide whether any of the characteristics are associated and, if they are, to determine which ones. There are numerous measures of association between two sets of observations which could be calculated for each of the 1 k(k -1) distinct comparisons, the product-moment coefficient of correlation, Spearman's rank correlation coefficient and Kendall's T, being the most widely used. The ?k(k -1) different coefficients are strongly dependent and this dependence should be taken into account in any suggested testing procedure. If all the observations are normally distributed, one could calculate the productmoment coefficients of correlation and study them either by a graphical technique (Hills, 1969) or by using a formal testing procedure (Moran, 1980). In the latter paper, bounds are derived on the distribution of the maximum of a set of dependent correlation coefficients under the hypothesis of the independence of the k characteristics. These bounds are very accurate in the tail of the distribution; of course it is the tail of the distribution which is used in any test. Recently, Eagleson (1983) has derived an asymptotic formula for the tail of the distribution of the maximum of a set of Spearman's rank correlation coefficients and has shown that Moran's bounds continue to hold in this case. The asymptotic results are based on an application of a limit result for symmetric statistics and refer to large values of k, the number of characteristics, rather than to n, the number of observations. A comparison of Moran's bounds and Eagleson's approximation shows that the approximation is remarkably good, even for k as small as 5. In this note we point out that arguments analogous to those of Eagleson (1983) provide a similar result for the distribution of the maximum of a set of Kendall's -'s. This leads to a test procedure for the hypothesis that all the k characteristics are independent

SUMMARY The presence of serial correlation affects the design strategy of a statistical experiment. It is the object of this paper to construct robust and efficient designs under such conditions. Special consideration is given to the two-level factorial designs and weighing designs.

In comparing two treatments in a clinical trial, the randomized play-the-winner rule tends to assign more study subjects to the better treatment. It is applicable when patients have delayed responses to treatments. It is not deterministic and is less vulnerable to experimental bias than other adaptive designs. In this paper, an efficient algorithm is provided to construct exact permutation tests for testing the equality of the two treatments with the randomized play-the-winner rule. The test procedure is illustrated with a real-life example. The example shows that the design used in the trial should not be ignored in the analysis.

SUMMARY Rank tests are proposed and studied for the hypothesis of equal treatment effects in randomized block experiments. The procedures are applicable to samples containing arbitrarily right-censored observations as well as uncensored observations. In the case of logistic and double exponential rank scores, the tests are shown to be censored data extensions of the well-known Friedman (1937) and Brown & Mood (1951) procedures respectively. The application of the techniques to problems in survival analysis is also discussed. variate is subjected to arbitrary right censorship. In the context of biological assays, Sampford & Taylor (1959) have largely dealt with parametric normal theory approaches to both the estimation and testing of treatment effects. Patel (1975) has discussed a censored data modification of Friedman's (1937) statistic for testing the global hypothesis of equal treatment effects, while L. J. Wei has in unpublished work proposed a class of test procedures based on Gehan's (1965) scoring scheme. Downton (1976) and Morton (1978) have also discussed nonparametric tests for block designs. The latter develops a class of rank statistics arising by forming a linear combination of score components from Cox's (1975) partial likelihood. The present approach is to use the marginal probability of a generalized rank vector rather than a partial likelihood in the formation of a class of tests. The resulting rank tests differ from Morton's in several respects. For example, multiple distinct censored values between two consecutive uncensored observations would receive identical scores with the present scheme, distinct scores with Morton's. The specific objective of this paper is to propose a class of linear rank tests, derived under optimal local power considerations, for the global hypothesis of equal treatment effects. To develop this class of tests we exploit the concept of the accumulated rank vector probability of Prentice (1978). The principal theoretical results concerning the derivation of the test statistic and its null hypothesis permutation distribution follow in ? 2. The scores for the test statistic are given in explicit form for selected densities in ? 3. In ? 4 the test statistic is shown, when dealing with only uncensored data, to reduce to Friedman's (1937) statistic with logistic scores and to Brown & Mood's (1951) median test with double exponential scores. A discussion of the application of the techniques to both the log linear and proportional hazards models in survival analysis is the subject of ? 5.

SUMMARY Under the usual additive model, a design of n factors at levels mi(i = 1, ..., n) is defined to be connected if the rank of its information matrix is equal to 2mi - n + 1. The ith factor is said to be connected if the coefficient matrix of the reduced normal equations obtained by eliminating all other factors has rank mi-1. This paper obtains simple necessary and sufficient conditions for connectedness . This is achieved by restricting the class of designs under consideration through the concept of orthogonality between pairs of factors. The degree of restriction imposed on the class of designs is mild, and some interesting and practical results are obtained.

SUMMARY A method is developed for identifying effects and confounding patterns in factorial designs generated by Patterson's (1976) DSIGN algorithm. The method is an extension of that com- monly used for symmetrical designs with prime number of levels. Das (1964) described an equivalent method in which some of the treatment factors are designated as basic factors and the others as added factors. Levels of added factors are derived by combination of the levels of the basic factors over GF(t). White & Hultquist (1965) extended the field method to designs with numbers of levels of treatment factors still prime or prime power but possibly differing from one factor to another. John & Dean (1975) described the construction of a particular class of single replicate block designs, which they call generalized cyclic designs. The levels of treatment factors are now integers 0, 1, ..., t - 1. The essential feature of the method is that the m-tuples giving the treatments of a particular block constitute an Abelian group, the intrablock subgroup. The method gives the same single-replicate designs as the field method when t is prime. Treatment levels are, however, differently represented when t is a power of a prime and so, in general, different designs are obtained. Unlike the field method, the generalized cyclic method is also available when t is neither prime nor prime power. Dean & John (1975) extended their method to asymmetrical designs. Patterson (1976) described a general computer algorithm, called DSIGN, in which levels of treatment factors are derived by linear combinations of levels of plot and block factors. The method provides finite-field, generalized cyclic and other designs. It is not restricted to block designs but is available for any of the simple block structures defined by Nelder (1965) with fractional, single or multiple replication of treatments. In the present paper we are concerned with identification of effects and confounding in the DSIGN method. We tackle the problem by inspecting linear combinations of levels of treatment factors. This approach is, of course, familiar for finite-field designs: Kempthorne (1947) has