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Craig (1962) described an application of order statistics from a binomial distribution and gave an approximate expression for the mean value of the minimum of two binomial variables having the same probability of success and the same number of trials. In this note we suggest another approximation for the mean and also for the standard deviation of the minimum, using normal order statistics tabulated by Teichroew (1956).

SUMMARY This paper considers a wide class of latent structure models. These models can serve as possible explanations of the observed relationships among a set of m manifest polytomous variables. The class of models considered here includes both models in which the parameters are identifiable and also models in which the parameters are not. For each of the models considered here, a relatively simple method is presented for calculating the maximum likeli- hood estimate of the frequencies in the m-way contingency table expected under the model, and for determining whether the parameters in the estimated model are identifiable. In addition, methods are presented for testing whether the model fits the observed data, and for replacing unidentifiable models that fit by identifiable models that fit. Some illus- trative applications to data are also included. This paper deals with the relationships among m polytomous variables, i.e. with the analysis of an m-way contingency table. These m variables are manifest variables in that, for each observed individual in a sample, his class with respect to each of the m variables is observed. We also consider here polytomous variables that are latent in that an individ- ual's class with respect to these variables is not observed. The classes of a latent variable will be called latent classes. Consider first a 4-way contingency table which cross-classifies a sample of n individuals with respect to four manifest polytomous variables A, B, C and D. If there is, say, some latent dichotomous variable X, so that each of the n individuals is in one of the two latent classes with respect to this variable, and within the tth latent class the manifest variables (A, B, C, D) are mutually independent, then this two-class latent structure would serve as a simple explanation of the observed relationships among the variables in the 4-way con- tingency table for the n individuals. There is a direct generalization when the latent variable has T classes. We shall present some relatively simple methods for determining whether the observed relationships among the variables in the m-way contingency table can be explained by a T-class structure, or by various modifications and extensions of this latent structure. To illustrate the methods we analyze Table 1, a 24 contingency table presented earlier by Stouffer & Toby (1951, 1962, 1963), which cross-classifies 216 respondents with respect to whether they tend towards universalistic values ( + ) or particularistic values (-) when confronted by each of four different situations of role conflict. The letters A, B, C and D in

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

Now that satisfactory and probably final solutions have been obtained for a wide variety of statistical problems concerned with a single normally distributed variable, more and more attention has recently been given to the solution of multivariate problems. The multiple correlation methods of the old large sample theory have been replaced in many instances by others for which " studentized " test criteria are available, often having sampling distributions that are already familiar in univariate problems. In a recent paper on " The statistical utilization of multiple measurements", R. A. Fisher (1938a) has shown the connexion between certain of these methods: the D2-statistic work of Mahalanobis, the discriminant function methods of the Galton Laboratory and the generalized " Student's " ratio of Hotelling. A similar very general problem was dealt with some time ago by S. S. Wilks (1932), while mention may also be made of two papers by D. G. Lawley (1938 a, b) and a paper by P. L. Hsu (1938). The purpose of the methods put forward is to obtain information regarding the mean values of a number, say q, of correlated variables in one or more, say k, populations from which random samples have been drawn. If we denote by x8 a value of the sth variable (s = 1, 2, ..., q), then in all this work it has been assumed not only that x8 is normally distributed, but that it has the same variance o2 in every population sampled. Further, it is assumed that if x. is a second variable the correlation coefficients psu between x8 and x. is the same in all populations. The estimates of variance and covariance required in order to "studentize" the function of the sample means are therefore obtained by pooling together the sums of squares and sums of products from all samples. While it is true that even if oa, and psu are not the same in all populations the error involved may not be very large, it is however important to have available some means of testing the basic hypothesis which assumes homogeneity throughout the populations. Such a test has been derived by S. S. Wilks (1932) by an extension of Neyman & Pearson's likelihood ratio method of approach. Hitherto the somewhat lengthy computations required to obtain the moments of the sampling distribution of the test criterion have probably discouraged its use. The objects of the present paper are as follows: (a) In the simple but commonly met case, where the k samples are of the same

The main intent of this paper is to introduce a new statistical procedure for testing a complete sample for normality. The test statistic is obtained by dividing the square of an appropriate linear combination of the sample order statistics by the usual symmetric estimate of variance. This ratio is both scale and origin invariant and hence the statistic is appropriate for a test of the composite hypothesis of normality. Testing for distributional assumptions in general and for normality in particular has been a major area of continuing statistical research-both theoretically and practically. A possible cause of such sustained interest is that many statistical procedures have been derived based on particular distributional assumptions-especially that of normality. Although in many cases the techniques are more robust than the assumptions underlying them, still a knowledge that the underlying assumption is incorrect may temper the use and application of the methods. Moreover, the study of a body of data with the stimulus of a distributional test may encourage consideration of, for example, normalizing transformations and the use of alternate methods such as distribution-free techniques, as well as detection of gross peculiarities such as outliers or errors. The test procedure developed in this paper is defined and some of its analytical properties described in ? 2. Operational information and tables useful in employing the test are detailed in ? 3 (which may be read independently of the rest of the paper). Some examples are given in ? 4. Section 5 consists of an extract from an empirical sampling study of the comparison of the effectiveness of various alternative tests. Discussion and concluding remarks are given in ?6. 2. THE W TEST FOR NORMALITY (COMPLETE SAMPLES) 2 1. Motivation and early work This study was initiated, in part, in an attempt to summarize formally certain indications of probability plots. In particular, could one condense departures from statistical linearity of probability plots into one or a few 'degrees of freedom' in the manner of the application of analysis of variance in regression analysis? In a probability plot, one can consider the regression of the ordered observations on the expected values of the order statistics from a standardized version of the hypothesized distribution-the plot tending to be linear if the hypothesis is true. Hence a possible method of testing the distributional assumptionis by means of an analysis of variance type procedure. Using generalized least squares (the ordered variates are correlated) linear and higher-order

When estimating the size of a finite population, it is possible to consider, as an alternative to the capture-recapture method, a sequential scheme. Suppose an urn contains an unknown number, N, balls, initially all white. A single ball is drawn at random and if it is white it is painted black and returned to the urn, while if it is black, indicating that it has already been sampled at least once before, it is returned unchanged. Thus initially nearly all drawings will be of white balls, but after a long time mostly black balls will appear. Somewhere in between we wish to stop sampling and produce an estimate of N. This scheme was first proposed by Goodman (1949, 1953). The most recent treatment of this problem is by Samuel (1968, 1969), who considered asymptotic distributions of sample sizes and maximum likelihood estimates for each of four stopping rules. The crucial question of the choice between these rules remained unanswered, however, since the balance between cost of sampling and risk due to inaccurate estimation was not considered. It is precisely this which is the aim of the present paper. A bibliography of previous, non-Bayesian, work is given by Samuel (1968). Since preparing the first draft of this paper, the author has seen an unpublished report by D. G. Hoel and W. E. Lever in which this problem is formulated in a Bayesian framework, and a small numerical solution given for a quadratic loss function, and with an upper limit on the value of N.

Professor William Hovanitz called my attention to the following problem: An observer catches butterflies, marks them, and immediately releases them. It is assumed that a butterfly, no matter how many times it has been caught before, has the same susceptibility to capture as any other butterfly in the population which is supposed stable while the captures are being made. Records are kept of f, the frequency of cases in which the same butterfly is caught x times, x = 1, 2, ..., until a total of s captures of r different butterflies have been made. (Efx = r; Zxfx = s.) The number, fo, of butterflies which escape is not observed; the problem is to estimate from the values offx the total population n of butterflies on the area assumed well defined. The estimation of biological populations by means of capture-recapture data is by no means a new problem, though papers dealing with it from the mathematical-statistical point of view are largely quite recent. (In particular, see the papers by Leslie & Chitty (1951), Bailey (1951), Moran (1951, 1952), and the bibliographies quoted by them.) However, the experimental conditions and the mathematical models for the present study arpear to differ in essential ways from those previously considered. The important point of departure is that each butterfly on being netted is immediately marked (with a spot of nail polish) and released. The butterflies (Colias eurytheme) were caught in one of two isolated alfalfa fields, which they inhabit, in southern California. Each catch was made during the same day at times when the butterflies were freely flying. Thus, it seemed reasonable to assume that the population was stable during a catch. The experimenter, Prof. Hovanitz, endeavoured to give each butterfly an equal chance of capture, walking in straight lines across the field and deviating in direction before reaching a boundary only when he noticed that a butterfly just caught tended to fly down his path. One check of the suitability of a mathematical model is to test the agreement of the experimental results with respect to the number of butterflies caught once, twice, etc., with those predicted from the model. I will return to this point at the end of the paper. Two mathematical models seem appropriate to serve as a basis for discussion of this estimation problem. It is of some interest to see that both lead to approximately the same estimates with little difference in their precision for large samples. It may be of more interest that for both models in which the population size is regarded as a parameter, though maximum likelihood estimates exist and agree substantially with moment estimates in all sixteen of the actual field experiments for which I have data, nevertheless with increasing sample size meaningful solutions of the likelihood equation do not exist.

SUMMARY This paper discusses from a Bayesian viewpoint some aspects of the estimation of latent roots and vectors of the covariance matrix of the bivariate normal distribution. The joint distribution of (i) the angle of the canonical transformation, and (ii) the ratio of the larger root to the total variance is considered in detail and illustrated by an example. Also discussed is the problem of making inferences about the larger roots, and several simple approximations to the distribution are considered. Finally, a generalization is given of one of the approximation methods to latent roots of higher dimensional covariance matrices. In this paper some aspects of the estimation of latent roots and vectors of the covariance matrix of the bivariate normal distribution are considered from a Bayesian viewpoint. In ? 2 the joint posterior distribution is obtained for (i) the angle 0 associated with the canonical transformation, and (ii) the ratio V of the larger root to the total variance, i.e. the sum of the two roots. Some properties of the joint distribution and the associated marginal distributions are discussed and a numerical example is given. Attention is then turned, in ? 3, to the problem of making inferences about the larger root and several simple approximations to its distribution are considered. In particular, it is demonstrated that the distribution can be closely approximated by a scaled inverted x2 distribution upon equating moments.

In a recent note Bartlett (1954) has summarized various approximate x2 tests (see also other papers referred to therein). Our object in this paper is to extend certain of these, listed as III a, b, c, which may be regarded as tests involving the latent roots of sample covariance and correlation matrices. We shall be interested in those cases where the effects of the k largest latent roots have been removed and where a hypothesis of equality of the remaining roots is made. Such hypotheses are made in principal components and factor analyses.

The purpose of this paper is to show the geometrical significance of the moments of order statistics derived from normal populations. It appears that these, as well as the moment-generating function of the square of any order statistic, are intimately related to the contents of the members of a class of hyperspherical simplices. Further geometrical interpretations, together with the extension of the present results to bivariate moments and other properties, will be provided in subsequent publications. The problem of order statistics in normal populations has been extensively considered in the literature. For example,t Tippett (1925) gives the second, third and fourth moments of the extreme order statistics for a few sample sizes. Hojo (1931) examines the sampling variation of the median, quartiles and interquartile distance in samples from normal populations and computes a large number of integrals for this purpose. Cole (1951) produces a very simple recurrence relationship between the 'normalized' moments which enables the normalized moments for all samples of size no greater than n to be obtained, by successive differencing, from the 'normalized' moments of the extreme order statistics in samples of size m < n.4 Hastings, Mosteller, Tukey & Winsor (1947) give, among other results, the means, variances, covariances and correlations of order statistics in samples of ten or less from a normal population, some of the results being given to only two decimal places because of the extreme labour of the computation. Jones (1948) and Godwin (1949a,b) have obtained exact values for some of the lower moments. In particular, Godwin (1949a, p. 283) proceeds in a more systematic manner, and approaches most closely the essential idea upon which this paper is based. In general, it may be said that many statisticians have attacked the problem in a rather disjointed and fragmentary manner, usually from the small sample end, but have failed to develop a systematic attack which shall at the same time throw light on the interconnexion between the moments and enable the computation of the moments to become an economic proposition. It is believed that these conditions are met by this paper.

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)

SUMMARY The point of change in mean in a sequence of normal random variables can be estimated from a cumulative sum test scheme. The asymptotic distribution of this estimate and associated test statistics are derived and numerical results given. The relation to likelihood inference is emphasized. Asymptotic results are compared with empirical sequential results, and some practical implications are discussed. The cumulative sum scheme for detecting distributional change in a sequence of random variables is a well-known technique in quality control, dating from the paper of Page (1954) to the recent expository account by van Dobben de Bruyn (1968). Throughout the literature on cumulative sum schemes the emphasis is placed on tests of departure from initial conditions. The purpose of this paper is to examine a secondary aspect: estimation of the index T in a sequence {xt}, where the departure from initial conditions has taken place. The work is closely related to an earlier paper by Hinkley (1970), in which maximum likelihood estimation and inference were discussed. We consider specifically sequences of normal random variables x1, ..., xT, say, where initially the mean 00 and the variance o2 are known. A cumulative sum, cusum, scheme is used to detect possible change in mean from 00, and for simplicity suppose that it is a one-sided scheme for detecting decrease in mean. Then the procedure is to compute the cumulative sums t

SUMMARY This paper is concerned with the application of several closed sequential procedures to the testing of an exponential parameter. In particular, the procedures of Armitage (1957), Schneiderman & Armitage (1962a,b) and Anderson (1960) are considered. In certain of these cases either the properties or bounds on the properties of the sequential test can be obtained without approximations. There are certain situations in which the risk of a long sequential experiment is unacceptable. Some common examples are to be found in the area of medical trials and in various types of industrial experimentation. Such situations have brought about the study of alternatives to the sequential probability ratio test (SPRT) in which a procedure's boundaries are closed. One class of such tests is the 'restricted' procedures of Armitage (1957) and the 'wedge' plans of Schneiderman & Armitage (1962a, b). Another test which also belongs to this category is a modification of the SPRT as described by Anderson (1960) and by Donnelly in his unpublished Ph.D. thesis. These tests, which are concerned with normal means, require an approximation of the random walk by a diffusion process for an evaluation of their properties. In this paper we shall apply the above type regions to the testing of the parameter from an exponential distribution and show that in certain cases the properties of the test can be obtained without approximations.

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

On account of the unique property of samples from a normal population that the ratio n+1 (x-Fb)/V(n + 1)/s (where jt is the population mean, x = x1/(n + 1) and ns2 =Z(Xi= 1 is the ratio of a normal deviate to a stochastically independent estimate of its variance, Student's t-test is a suitable test of significance for the mean of a normal population. However, in a variety of cases, it is necessary to test for the mean of a population which does not follow the Gaussian law. Efforts have, therefore, been made to see how far Student's distribution may be used for the purpose in non-normal populations. Due, mainly, to the analytical difficulties of the problem, no extensive theoretical discussion has yet been given. Thus, Pearson & Adyanthaya (1929), Rietz (1939) and Nair (1941) have given experimental treatments, while the theoretical discussions of some others (Rider, 1929; Perlo, 1933; Laderman, 1939) have dealt only with trivially small sample sizes. The papers by Bartlett (1935) and Geary (1936, 1947) give results true for any sample size, though they are based on certain assumptions and approximations. The present paper deals with the population considered by Geary in his 1936 paper, subject to the same approximations. The second contribution by Geary (in which is derived the t-distribution in samples from a population which departs more from normality than that considered in the 1936 paper) came to my notice too late to be made use of in the present work; but it is proposed to consider it later on. Geary (1936) has obtained the distribution of the ratio (x -It) (n + 1)/s in the case of an asymmetrical population, whose fourth and higher cumulants are zero, by neglecting squares and higher powers of the third cumulant. We know from this how far the probability of an error of the first kind (i.e. the probability of rejecting the null hypothesis when it is true) in such a population differs from that for a normal distribution, provided we may neglect the square of the standardized third cumulant yl. Here again, on account of analytical difficulties, it is not possible, except for very small sample sizes, to consider the effect of terms containing higher powers of yl. However, we can assume the result derived by Geary to be correct for very small values of yl, as also for large sample sizes-but in such cases the deviation from values of the normal theory is practically negligible. Even then, it is of interest to know whether, in using the usual tables of the t-test (based on the normal distribution), we are committing the greater error in the probability of an error of the first kind or in that of an error of the second kind. In the present paper are derived the values of the probability of an error of the second kind (and hence, of the power of the test) when the usual t-tables are used to define the critical region. It may be mentioned here that this problem is only a special case of a general investigation, on which the writer is engaged, into the effect, on statistical tests, of differences between the actual and the assumed distribution laws of the universe sampled. The solution of these problems is hampered by analytical difficulties in the derivation of the probability laws (and particularly of power functions), and the present case is one of the few in which a mathematical, though only approximate, solution has been found possible.

SUMMARY The probability integral for the distribution of range of a sample from a population whose distribution can be represented by the first few terms of an Edgeworth series has been obtained in this paper. The numerical values of the corrective functions arising due to nonnormality are tabulated. The new theoretical results are compared with the earlier results, where available. The distribution of range of samples from nonnormal populations was first studied empirically by Pearson & Adyanthdya (1928) and later, among others, by Pearson (1950), Cox (1954) and David (1954). These studies have been limited mainly to the mean range and to the probability integral in some simple nonnormal cases, and from these likely effects of nonnormality on the distribution of range have been conjectured. Singh (1967) obtained some theoretical results regarding the expectation and the variance of range of samples from a population whose distribution can be represented by the first few terms of an Edgeworth series. These results provided some additional information regarding the effects of parental excess and skewness on the mean and variance of the range. In the present paper the probability integral for the distribution of range of samples from the same type of population has been obtained and evaluated for small samples to examine the effects of parental excess and skewness.

(1) A PAPER* by Eleanor Pairmian anld Karl Pearson, F.R.S., was published in 1918 on "Corrections for the Moment-Coefficients of Limited Range Frequency Distributions when there are Finite or Iinfinite Ordinates and any Slopes at the Terminals of the Range." In the second part of that paper, dealing with cases of asymptotic frequency, the auxiliarv curve selected to give the first five frequencies was

In a recent paper, 'Further contributions to the theory of paired comparisons', M. G. Kendall (1955) considers paired comparison designs, in which each pair of judges have certain comparisons in common. Such designs should prove useful for testing concordance between judges. He notes that designs of an optimum kind which balance by numbers of comparisons, objects compared, numbers of observers on given comparisons and so forth are rather rare. It is the object of this paper to obtain some paired comparison designs which have a high degree of symmetry. These designs have been defined in ?2, and certain inequalities between the parameters are obtained in ? 3. In ?? 4 and 5, two special classes of these designs have been investigated, and explicit designs for small values of n (the number of objects to be compared) have been given in Tables 1 and 2. The method of analysis would, to a certain extent, depend on what use the experimenter wants to make of the design. This question will be considered in a future communication. My thanks are due to Professor Kendall for suggesting the problem, and for helpful discussion during the preparation of the paper.

Student's t-statistic provides a suitable test of significance for the mean when the sample comes from a normal population. The power of the normal theory test has been studied by Neyman (1935); Neyman & Tokarska (1936) and Johnson & Welch (1939). As in many cases the samples appear to belong to populations other than the Gaussian, it is necessary to see how far the normal theory test can be assumed to be valid in controlling the Type I and Type II errors of inference on non-normal samples. The effect of non-normality on the Type I error of Student's t-test was studied experimentally by Pearson & Adyanthaya (1929) and theoretically by Bartlett (1935), Geary (1936, 1947) and Gayen (1949). Assuming the parent population to be specified by the first two terms of the Edgeworth series, Geary (1936) obtained the approximate distribution of t for any sample size, and later this work was extended by Gayen (1949) by including in the distribution the effects of parental kurtosis A4 = fl2-3 and A2 = IA. Apart from the pioneer empirical study by Pearson & Adyanthaya (1929, pp. 276-80), the effect of non-normality on the Type II error (and hence on the power) of the t-test was first studied by Ghurye (1949). He has, however, considered only the effect of skewness of the population, for he started with the joint distribution of the mean and the variance for populations specified by the first two terms of the Edgeworth series (Geary, 1936). In this paper, it has been possible to study the effects of kurtosis and skewness of the parent population which may be assumed to cover a larger range of nonnormality. Gayen's (1949) formulae for the joint distribution of the sample mean and variance for the first four terms of the Edgeworth population have been utilized for the derivation of the corrective terms of the power function. The mnethod followed by Ghurye for the evaluation of the corrective term of the power function due to A3 appears to be satisfactory for derivation of the effects due to higher odd-order cumulants. But for those of the even-order cumulants his method does not appear to be useful, as Ghurye himself encountered some 'analytic difficulties'. In this paper, by a different approach it has been possible to evaluate integrals involved in the power function due to A4 and A3. The non-normal population considered here is supposed to be characterized by non-zero values of the standardized third and fourth cumulants. Since the effects of the higher-order terms depending on A5,6, A3A4, A2, ... are assumed to be negligible, the population covered is only moderately non-normal. Too high values of A3 and A4 can also not be permitted as they will make f(x) negative at one or both tails, and will give rise to subsidiary modes. To ensure a positive definite, unimodal frequency function, A4 should lie roughly between 0 and 2-4 and A3 02 (Barton & Dennis, 1952).t Also it is found possible in this paper to calculate in the non-normal case the critical region

A great deal of use has undoubtedly been made of least squares regression methods in circumstances in which they are known to be inapplicable. In particular, they have often been employed for the analysis of time series and similar data in which successive observations are serially correlated. The resulting complications are well known and have recently been studied from the standpoint of the econometrician by Cochrane & Orcutt (1949). A basic assumption underlying the application of the least squares method is that the error terms in the regression model are independent. When this assumption—among others—is satisfied the procedure is valid whether or not the observations themselves are serially correlated. The problem of testing the errors for independence forms the subject of this paper and its successor. The present paper deals mainly with the theory on which the test is based, while the second paper describes the test procedures in detail and gives tables of bounds to the significance points of the test criterion adopted. We shall not be concerned in either paper with the question of what should be done if the test gives an unfavourable result.

SUMMARY This paper develops formulae for the first two moments of estimates of covariance for the general multivariate 'one-way' and 'two-way' models. The results are used to obtain the large sample dispersion matrix for estimated coefficients of two types of genetic selection indexes. These dispersion matrices provide the necessary extension of known results in balanced models to the unbalanced case. The problems of estimation associated with variance and covariance component analysis with unbalanced data have been of major concern to the statistical geneticist. This is primarily because most of the selection procedures applied to livestock require a knowledge of genetic variances and covariances which usually have to be estimated from the analyses of hierarchical models. Invariably, there is a marked lack of balance in the data thus rendering standard formulae for the variances of the estimates inapplicable. Serious consideration to these problems has been given by Henderson (1953), Searle (1956) and Hartley & Rao (1967). Searle gave particular attention to the one-way analysis of variance and covariance and used matrix methods to calculate the moments of the various estimators. Other work in this area concerns analysis of variance models of varying complexity; see Searle (1958, 1961), Mahamunulu (1963) and Blischke (1966). It is the purpose of the present paper to extend and complement existing results. With the advent of high speed computers, matrix operations can be handled with great speed and hence formulae for expectations and covariances of sums of squares and products can be left in a general computable form. Thus, explicit algebraic evaluation of each case is, in most cases, not only time-consuming but unnecessary. We consider here the general one-way and two-way analysis of covariance model with fixed and random effects. The number of variables included in the analysis is assumed to be arbitrary and this seems to lead to somewhat involved notation and algebra. However, general results are required in order to solve a number of practical problems. We give two examples from statistical genetics. In the theory of animal breeding interest centres around certain phenotypic and genetic parameters. Suppose that k characters of a particular breed of animal are relevant from the point of view of a selection programme. Then we let P and G be the phenotypic and additive genetic covariance matrices for the k characters and we consider two types of selection index which are based on these matrices.

Experiments involving paired comparisons have mainly concerned the situation where each judge compares all possible pairs of treatments or objects. In certain types of experiments, this may require an excessive number of comparisons to be made by any observer. To overcome this handicap, Bose (1956) and Kendall (1955) constructed certain designs, symmetrical with respect to objects and judges, which do not require each judge to compare all possible pairs of objects. However, neither Bose nor Kendall proposed any procedure of analysis in their respective papers. The purpose of this paper is to consider the problem of analysis of the Bose-Kendall paired comparison designs. This analysis is carried out in the tradition of the fundamental Bradley-Terry (1952) paper concerning the situation where all possible pairs are compared by each judge. Using the Bradley-Terry mathematical model, likelihood ratio tests are constructed in detail for certain classes of hypotheses, and are stated for some additional situations of interest. To exemplify the test procedures, the proposed analysis is applied to an experiment involving pairwise comparison of handwriting specimens.

In this paper we consider the following problem. Given a multivariate normal population with known variance matrix, what test is appropriate to determine whether the means are slipped to the right? In the case when the population is univariate normal, this problem can be solved by the ordinary one-sided test using either the normal or the t-distribution functions. It is the purpose of this paper to develop what may be termed a multivariate analogue of the one-sided test of significance.

SUMMARY This paper follows the development of an aspect of the Theory of Errors known as the Hypothesis of Elementary Errors. This development produced a scientific tradition in statistical inference theory concerned with the causes and effects of nonnormality of statistical data, a tradition sometimes called the Scandinavian school. Its founder was the Swedish astronomer C. V. L. Charlier. Being concerned mainly with the period of Charlier and his followers, this paper first outlines briefly some of the nineteenth-century thinking that formed the essential background for the Scandinavian school. The principal contributions of Charlier and his followers are discussed. Their ideas are compared with simultaneous developments in England, starting from the time of Karl Pearson. 1. SOME NINETEENTH-CENTURY DEVELOPMENTS ON THE CONTINENT The contribution by Gauss that attached his name to the normal law of error was his famous 'first proof' (1809) of the principle of least squares. Thus the normal law of error was called the Gaussian law of error. Being concerned with astronomical measurements, Gauss's problem was that of the combination and adjustment of observations. The essence of his 'first proof' was the derivation of the normal law of errors through the Principle of the Arithmetic Mean, from which the Principle of Least Squares immediately followed. On the other hand, Laplace's (1810, 1812) slightly later derivation of the normal law of errors, quite different from Gauss's, essentially consisted in the first demonstration of the Central Limit Theorem. The analytical technique of derivation of the normal law of error used by Laplace, which included extensive use of characteristic functions, became an important tool in Charlier's work on nonnormality. Hence Laplace's work on the normal law, and its subsequent clarification by Poisson (1837), constitutes the point of departure for the trend of thought to be followed in this paper. While the analytical techniques stemmed from Laplace, the leading theoretical concept in Charlier's work consisted in proliferations of Hagen's (1837) original formulation of the Hypothesis of Elementary Error. Hagen's simple form of the hypothesis stated that the observational error is the algebraic sum of an infinite number of elementary errors, all having the same absolute value and the same probability of being positive as negative. The normal law of error resulted from considering the general term in the expansion of

Since Tippett (1925) tabulated the mean range of random samples from a normal population, the range has been extensively used for the rapid estimation of dispersion. Moreover, much work has been done recently on quick significance tests in which the root-mean-square estimate of dispersion in the t-test, analysis of variance, etc., is replaced by an estimate derived from the range. All these uses of the range rest on the assumption of normality,, and so it is of interest to examine the distribution of range from non-normal populations. This was first done by E. S. Pearson & Adyanthaya (1929), and their work, and later work by Shone (1949), has been summarized, discussed and extended by Pearson (1950). The conclusion was that in small samples the ratio of mean range to populIation standard deviation is not much affected by the form of the population, but that the coefficient of variation of range depends fairly critically on the population. In large samnples the distribution of range is, of course, determined bv the tails of the population and so is very sensitive to nonnorlnality. The main object of the present paper is to predict the mean and coefficient of variation of range in small random samples of n (n < 5) fromn a population of given skewness and kurtosis and then to show how these results can be uised to assess the effect of non-normality on the comiimon applications of the range. Such a prediction can only be approximate because there is no functional relation between the distribution of range and population skewness and kurtosis. There are four ways of proceeding: (i) by evaltuating numerically the single and double integrals for the mean and mean square of range for a representative selection of non-normal populations; (ii) by sampling experiments; (iii) by a derivation of upper and lower limnits for the miiean andl coefficient of variation of range of populations with given properties. This was first done by Plackett (1947), who obtained an upper bound to the ratio of mean range to population standard deviation in samples of given size from an arbitrary population. An imiportant extension of Plackett's work (Hartley & David, 1954) appeared as the present paper was being completed

Garwood (1940) considered the time for which a single minor road vehicle had to wait to cross a major road at a junction controlled by a vehicle-actuated signal. The problem was essentially one of waiting for a gap of at least T in the major road flow, with gaps assumed to be exponentially distributed. Tanner (1951) used the same model for pedestrians crossing the road. With the arrival of pedestrians forming a Poisson process, he found the distribution of the number waiting at any instant. Mayne (1954) generalized this result for an arbitrary distribution of gaps. Weiss & Maradudin (1962) introduced a gap acceptance function az(t) which is the probability that the minor road vehicle will accept a gap of duration t. Using the general gap distribution they derived the distribution of delay to each of two vehicles arriving simultaneously. Oliver & Bisbee (1962) also assumed a general gap distribution and that a vehicle would not cross the main road if there was an arrival due on it within a time T. They further imposed the restriction that no two vehicles could cross in the same gap. Thus minor road vehicles, arriving at random, were forced to queue. By considering a Markov chain embedded at the times of crossing they found the distribution of queue size at these epochs. Gaver (1963) has used a model which differs from the others in not requiring the protection of a gap which is distinctly larger than the time it takes the minor road vehicle to cross the junction. This latter time is supposed to be a random service time in the queueing of minor road vehicles. On reaching the head of the queue a vehicle must wait for a gap which is greater than its service time. Gaps are distributed exponentially and alternate with blocks which have an arbitrary distribution, and during which no vehicle may cross. This is equivalent to pre-emptive-repeat-identical queueing with active interruptions which he studied in his 1962 paper. He derives the distribution of waiting time. When service time is constant this model is equivalent to that of ? 5 below with a = T. Tanner (1962) also considers this form of main road flow, and in particular the blocks are considered as being formed by the passage of bunches of vehicles which have been formed by queueing in a system with constant service time, and taking account of the usual constant T. Minor road vehicles arrive at random and cross the road with constant headway during a gap. He derives the mean waiting time. This is very similar to Gaver's model with constant service time except that at least one vehicle crosses in every gap, provided that a vehicle is waiting. In this paper we assume the same system of gaps and blocks, or greens and reds. We also assume that minor road vehicles arrive in bunches, but during a gap they cross one at a time with an interval of at least a between them. The equilibrium distribution of waiting time is obtained. It is shown how the system of greens and reds may be generated by a more general form of main road flow than Tanner's.

SUMMARY Somerville (1954) proposed a one-stage and a two-stage procedure for the selection of the population with the largest mean from a set of normal populations with unknown means and a common, known variance. The two-stage procedure eliminates one population after the first stage. For these procedures he assumed that a certain loss was incurred when an incorrect selection was made and also that a cost was due to sampling. The sample sizes which would minimize the maximum expected loss, the maximum being taken over all possible configurations of the true population means, were derived. For the special case of three populations he showed that the two-stage procedure, using appropriate allocations of observations between stages, has a smaller maximum expected loss than does the onestage procedure. This paper generalizes Somerville's formulation of the two-stage procedure to an arbitrary finite number of populations and presents results for the special case of four populations, where two two-stage procedures are considered. For the problem of selecting the population with the largest mean from a set of normal populations with unknown means and a common, known variance, Somerville (1954) proposed a one-stage and a two-stage procedure, which eliminates one population after the first stage. He assumed that a certain loss was incurred when an incorrect selection was made and also assumed a cost due to sampling. For these procedures, he derived the sample sizes which minimize the maximum expected loss, the maximum being taken over all possible configurations of the true population means. He showed, for the special case of three populations that the two-stage procedure, using appropriate allocations of observations between stages, has a smaller maximum expected loss than does the one-stage procedure. Other approaches to this problem have been made by Bechhofer, Sobel and Gupta (see Bechhofer, 1954, and Gupta, 1965). In this paper the formulation of Somerville's two-stage procedure is extended to an arbitrary, finite number of populations. For the case of four populations, the expected losses are analysed in detail and numerical results obtained. Two possible two-stage procedures are considered: in the first procedure, one population is discarded and in the second, two populations are discarded after the first stage. It is found that while both procedures, using appropriate allocations, have smaller maximum expected losses than does the corresponding one-stage procedure, which one of the two-stage procedures is the better depends on the allocation. These numerical results required the evaluation of certain multivariate normal integrals with arbitrary correlation matrices and arbitrary limits of integration. This was accomplished for 4- and 5-variate integrals using a method suggested

In an earlier paper* we have endeavoured to emphasise the importance of placing in a logical sequence the stages of reasoning adopted in the solution of certain statistical problems, which may be termed problems of inference. In testing whether a given sample, Z, is likely to have been drawn from a population H, we have started from the simple principle that appears to be used in the judgments of ordinary life-that the degree of confidence placed in an hypothesis depends upon the relative probability or improbability of alternative hypotheses. From this point of view any criterion which is to assist in scaling the degree of confidence with which we accept or reject the hypothesis that X has been randomly drawn from HI should be one which decreases as the probability (defined in some definite manner) of alternative hypotheses becomes relatively greater. Now it is of course impossible in practice to scale the confidence with which we form a judgment with any single numerical criterion, partly because there will nearly always be present certain a priori conditions and limitations which cannot be expressed in exact terms. Yet though it may be irmpossible to bring the ideal situation into agreement with the real, some form of numerical measure is essential as a guide and control. In our previous paper we have made use of the criterion of likelihood. That there may be other forms of criteria or that this one can be interpreted in a different manner is very possible, but our object has been to find a single principle connecting the various sampling tests already in use, and one which could be extended to new problems.

SUMMARY An epidemic model is discussed in which there is an initial introduction of a carrier or carriers into a population and further carriers may be created from the susceptibles. The probability distribution for the number of survivors is derived and approximations and numerical illustrations are given. In a recent paper Weiss (1965) has considered an epidemic in a closed population which is spread not by the infected individuals but by carriers. He supposed that one or more carriers were introduced into the population and that these carriers infected susceptible individuals until detected, the time before detection having an exponential distribution. At no time during the epidemic were any additional carriers introduced or created and the epidemic terminated as soon as all the carriers were detected or all susceptibles had been infected. Weiss's model has the mathematical advantage that it is completely soluble in terms of elementary functions; see Dietz (1966) and Downton (1967b). It is, however, unrealistic in its assumption that no new carriers can be created. The present paper discusses a model in which it is supposed that after the initial introduction of a carrier or carriers, no new carriers are introduced from outside the population, but that new carriers may be created from the susceptibles in that population. The model is appropriate for the situation where a proportion 7T of those infected contract the disease in such a mild form that their symptoms are unnoticeable even though they are capable of passing on the disease. Such subelinically infected persons would then act as carriers until detected. For such a case we would expect oT to be small; oT = O corresponds to the Weiss model. The main aim of this paper is to obtain approximate expressions for the probability distribution and moments of the number of susceptibles surviving the epidemic in that case. The ultimate behaviour of the epidemic does not, however, depend upon oT in a simple way so that while valid approximations have been obtained for small it, even if this results in a large epidemic, it turns out that these approximations may be used in certain cases even when or is not small. In particular, they may be employed even if 7t = 1, when the mathematics becomes identical with that of the Kermack-McKendrick (1927) model for the general stochastic epidemic provided attention is confined to small subcritical epidemics. The stochastic model describing the process is defined in ? 2, where an explicit expression for the probability distribution of the number of survivors is given, together with an indication of the ways in which it may be obtained. In ? 3 the deterministic analogue of the stochastic model is discussed, providing the deterministic approximation to the mean number of survivors. This approximation is valid for small epidemics in large populations. Returning to the stochastic model ? 4 shows that the survivor probabilities may be expressed in terms of

SUMMARY A generalization of the Kruskal-Wallis test, which extends Gehan's generalization of Wilcoxon's test, is proposed for testing the equality of K continuous distribution functions when observations are subject to arbitrary right censorship. The distribution of the censoring variables is allowed to differ for different populations. An alternative statistic is proposed for use when the censoring distributions may be assumed equal. These statistics have asymptotic chi-squared distributions under their respective null hypotheses, whether the censoring variables are regarded as random or as fixed numbers. Asymptotic power and efficiency calculations are made and numerical examples provided. A generalization of Wilcoxon's statistic for comparing two populations has been proposed by Gehan (1965a) for use when the observations are subject to arbitrary right censorship. Mantel (1967), as well as Gehan (1965b), has considered a further generalization to the case of arbitrarily restricted observation, or left and right censorship. Both of these authors base their calculations on the permutation distribution of the statistic, conditional on the observed censoring pattern for the combined sample. However, this model is inapplicable when there are differences in the distribution of the censoring variables for the two populations. For instance, in medical follow-up studies, where Gehan's procedure has so far found its widest application, this would happen if the two populations had been under study for different lengths of time. This paper extends Gehan's procedure for right censored observations to the comparison of K populations. The probability distributions of the relevant statistics are here considered in a large sample framework under two models: Model I, corresponding to random or unconditional censorship; and Model II, which considers the observed censoring times as fixed numbers. Since the distributions of the censoring variables are allowed to vary with the population, Gehan's procedure is also extended to the case of unequal censorship. For Model I these distributions are theoretical distributions; for Model II they are empirical. Besides providing chi-squared statistics for use in testing the hypothesis of equality of the K populations against general alternatives, the paper shows how single degrees of freedom may be partitioned for use in discriminating specific alternative hypotheses. Several investigators (Efron, 1967) have pointed out that Gehan's test is not the most efficient against certain parametric alternatives and have proposed modifications to increase its power. Asymptotic power and efficiency calculations made below demonstrate that their criticisms would apply equally well to the test proposed here. Hopefully some of the modifications they suggest can likewise eventually be generalized to the case of K

The present investigation was suggested by the problem of planning the storage capacity of a reservoir, where one would like to know the distribution of the water level over a given number n of years. The level after r years may be regarded as the sum of r annual increments, and we may approximate this real problem by the ideal one where the annual increments are independent variates with a common distribution. Applications to other storage problems are obvious. In the case of very large n the asymptotic range of the consequent random walk has been discussed by Feller (1951). In the present paper we are concerned with an exact formulation for the case of finite n, taking the increments to be standard normal variates, and for this case we obtain a simple expression for the mean range. The paper falls into two parts. In the first (??2-5), tlle problem is reduced to that of evaluating the integral

It is clear that a p-variate body of data arranged in a q-way classification will formally look llke a (p+q)-dimensional contingency table, but a distinction can be made between a 'variate' and a ‘way of classification’ in that along the direction of a 'variate' the marginal frequencies are supposed to be stochastic variates while along a 'way of classification' the marginal frequencies are supposed to be fixed. When, along certain directions, the marginals are fixed, an approach based on a conditional probability argument has been used. In the present paper (i) the conditional probability approach is abandoned and we start either from a single multinomial distribution or a product of an appropriate number of different multinomial distributions according as, with multi-way frequency data, all ways are 'variates' or some are 'variates' and some are 'ways of classification'. (ii) Also the hypotheses that are posed are of different kinds altogether according as we have a 'multivariate analysis' situation or an 'analysis of variance' situation. The hypotheses that are meaningful for one situation would not be too meaningful for the other and vice versa. Since the conditional probability approach is altogether abandoned, the mathematical theorems to which appeal is made are the two theorems as stated and proved by Cramer (1946, chapter 30) and a number of other such theorems which have been proved the same way and which, between them, take care of all the hypotheses discussed in this paper. When all ways are 'variates' the hypotheses are analogous to those in the usual multivariate analysis, and when some ways are 'variates' and some are 'ways of classification' the hypotheses are analogous to those in the analysis of variance. The general methods discussed in this paper arose out of an attempt to analyse a large mass of categorical data. The analysis has been carried out, and a few typical cases (together with the numerical analysis) illustrating different parts of the theoretical development will be presented in a subsequent paper. In this paper only the large-sample tests are considered. How 'large' the sample size has to be for the validity of the use of these asymptotic techniques, or in other words, some results on the nature of the approximation involved will also be discussed in a later paper.