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Neyman & Pearson's (1933) conception of the power of a test of a statistical hypothesis, Ho, was developed, in the first instance as a means of guiding the choice between alternative tests. This, it was shown, could be done by comparing the effectiveness of the tests in discriminating between Ho and a set of admissible alternative hypotheses regarded as most relevant to the question under test. Where there is no doubt about the most appropriate test and no sequential scheme of sampling is possible, the power function may play a useful part in indicating, before the data are collected, how large the samples should be to avoid an inconclusive result. If this procedure is to be easily applied, a ready means must be available of calculating the power of the test for a given significance level and sample size. The tables of the power function of the t-test (Neyman & Tokarska, 1936) and Tang's Tables (1938) applicable in the Analysis of Variance, are examples of such aids. The present paper aims at providing in simple, if approximate, form a means of determining the power function of the test for the difference between two proportions. The test may be briefly outlined as follows. In two 'infinite' populations the proportions of individuals possessing a character A are pl(A) and p2(A) respectively. Random samples of m and n are drawn from the two populations and the result is represented in a 2 x 2 table, thus

sider the problem of estimating the set of parameters (ca,, ..., op, /,1 ... ,/&), given consecutive observations (xl, x2, ..., xn) from the series, and show that for large n, the method of estimation for a moving-average model (obtained by putting coi = 0, 1 < i < p) which is described in a previous paper (Walker, 1961 b), referred to subsequently as I, can readily be adapted to the above model. As in I we shall assume that the representation of the moving-average residual process is 'regular', i.e. that the moduli of the roots of the equation

This paper presents procedures for multivariate analysis of variance which simultaneously test for significance all variables together as well as all subsets of variables, or the differences between all samples as well as between the samples of any subgroup. These procedures ensure a predetermined probability of making no type I error on any of the tests and are coherent in preserving the implication relations which exist between hypotheses on different groups of samples. The procedure using Roy's maximum characteristic root statistics is found to be preferable to all other procedures of this kind.

SUMMARY Laguerre series expansions are developed of the frequency distributions of the 'nonnormal' variance-ratios used for testing (i) the homogeneity of a set of means in case of oneway classification for analysis of variance with non-identical group to group error distributions; and (ii) the compatibility of two population variances. The effect of non-normality on the distributions is illustrated numerically. The effect of non-normality on the frequency distributions of the variance-ratios used for testing (a) the equality of a set of means in one-way classification for analysis of variance; and (b) the compatibility of two population variances, was first studied by E. S. Pearson (1931) by way of sampling experiments. Theoretical investigations were carried out by Bartlett (1935), Geary (1936, 1947) and Gayen (1950). Their investigations were essentially of the same type, basing their derivations on the first few terms of the Edgeworth's series expansion of the probability density function of the population. They did not consider the case when, in analysis of variance, the error distribution from group to group is not identical. It has been possible here to tackle this general problem in an entirely different way. The distribution of 'between' and 'within' sum of the squares is formally expanded in terms of Laguerre polynomials and Gamma density functions. The first few coefficients are worked out in terms of population cumulants. The 'non-normal' sampling distribution of two independent variances is also given. The results obtained are important not only on their own account, but in giving an elegant and simple method of judging the degree of approximation of the expressions for large n, n being the sample size. Similar expansions of Student's t distribution and power-function of F-test have been obtained elsewhere (Tiku, 1962, 1963). Since this investigation was completed my attention has been called to a paper by Khamis (1960) advocating the use of these polynomials.

1. In psychological work the problem of comparing two different rankings of the same set of individuals may be divided into two types. In the first type the individuals have a given order A which is objectively defined with reference to some quality, and a characteristic question is: if an observer ranks the individuals in an order B, does a comparison of B with A suggest that he possesses a reliable judgment of the quality, or, alternatively, is it probable that B could have arisen by chance? In the second type no objective order is given. Two observers consider the individuals and rank them in orders A and B. The question now is, are these orders sufficiently alike to indicate similarity of taste in the observers, or, on the other hand, are A and B incompatible within assigned limits of probability? An example of the first type occurs in the familiar experiments wherein an observer has to arrange a known set of weights in ascending order of weight; the second type would arise if two observers had to rank a set of musical compositions in order of preference. The measure of rank correlation proposed in this paper is capable of being applied to both problems, which are, in fact, formally very much the same. For purposes of simplicity in the exposition it has, however, been thought convenient to preserve a distinction between theni.

It is often possible, in certain types of mass production, to use a large sample of articles for simple routine inspection and to find with ease the articles with more extreme values of the characteristic measured. Examples of this are articles which undergo a routine check on their length or weight, in which case extreme values can be sorted out either by sight, or by use of GO-NO GO checks on a balance set at two suitable weights. In these cases, a great deal of labour can be saved, if the dispersion is estimated from these extreme values, which may comprise only about 5 % of the total. Such an estimate of dispersion may be used in controlling variability by specifying limits for this estimate. One miiethod of specifying the variability, which avoids the complication of subdividing the sample, is to lay down limits for the difference between the sum of the r highest and r lowest values observed in the sample. In this paper it will be shown how the mean, variance, and also higher moments of this difference can be found. Approximate formulae, which are reasonably easy to calculate, are given for the mean and variance of the difference. These should be satisfactory for most p)ractical purposes. In Table 1 are given exact values of the mean and variance of the difference in the case when the parent population is Gaussian (normal) for selected sample sizes and values of r. The mean and variance with other parent populations may be calculated by applying equations (22) and (25) to Tables 3 and 4.

Consider anl experimernt in which the experimental units are arranrged in sets of k units, a set corresponding, for example, to a single production run of an indlustrial process. Suppose also that the ki units in each set are arranged in order corresponding to the first, second, etc., period of the set. If there are both systemiatic differences between periods and an appreciable component of variance between sets, an experiment to compare a number of treatments, using these experi;nental units, would ordinarily be arranged irn a Latin or Youden square, chosen to eliminate differences between periods and sets. Suppose, however, that for practical reasons there is a restriction on the order of treatments within each set, such as that the level of the treatment must not decrease from one period to the next in a set. Then Latin and Youden squares cannot be used. This paper is concerned with designs for such a situation; the method of construction is described and designs are listed for a few special cases. I have not hesitated to suggest designs that are nearly optimum whenever the determination of the strictly optimum design would be a very lengthy process. The particular application for which this work was done concerned the properties of alloys cast from high-purity metals. Because of the high cost of the raw materials it was especially important to use an efficient experimental design. From a 20 lb. melt it was possible to cast four 5 lb. ingots, the smallest size convenient for further processing before testing. (In the language of the first paragraph the melt is a set, and the four ingots are the periods within a set, so that k = 4.) After an ingot had been cast, the composition of the remaining molten metal could be changed only by the addition of one of the alloying metals, no technique being available for selectively reducing the concentration of an alloying element. Therefore we have just the situation described above, in which the level of treatment cannot decrease from one period to the next in a set. A further similar application is to certain large-scale experiments in steel-works, in which it is possible to tap several ladles from one cast of an open-hearth furnace. The effect of the addition of several elements could, with the present designs, be investigated by varying the chemical composition in successive ladles, rather thani by the more costly and cumbersome procedure of varying the composition from cast to cast.

SUMMARY The comparison of sensitivities is considered in detail for a random effects model in which the different methods of measurement are applied to the same experimental units. An exact distribution is obtained and tabulated and approximations to it considered. Other applications of the results are outlined. The problem of comparing two or more methods of measuring the effects of experimental treatments arises in various fields of research. Cochran (1943) discussed this problem in considerable detail under the assumption that analysis of variance techniques were applicable. He suggested that a comparison of the sensitivities of two scales of measurement, or two experimental techniques, should depend both on the magnitudes of treatment effects and the experimental errors associated with the measurement scales, or experimental techniques. A method of measurement for which the differences between treatment effects show up well, and which, at the same time, has a relatively small error variance, will be considered a sensitive method. In this paper we distinguish between three types of situation in experimentation: (i) Type 1. The different methods of measurement are applied to the same experimental units. This can be done whenever not more than one method affects the experimental units. The results of the methods will be correlated. (ii) Type 11. The different methods of measurement are applied to separate subsamples of the same experimental units. Here only the 'treatment effects' for the methods will be correlated. (iii) Type 111. The different methods of measurement are applied to independent experimental units. Schumann & Bradley (1957, 1959) considered the type III situation. Following Cochran's line of thought, they effected a comparison of the sensitivities of two experiments through a comparison of two variance ratios. Their work was restricted to independent experiments and they were mainly concerned with similar experiments in which the variance ratios compared had identical degrees of freedom. They calculated critical values of the ratios of two independent F's to be used in hypothesis tests to compare sensitivities. Schumann & Bradley assumed that analysis of variance techniques were applicable and considered both model I (fixed effects model) and model II (random effects model) of the analysis of variance. Dar (1962) derived a normal approximation to the Schumann & Bradley density function for the ratio of two F's. He showed that the logarithm of the ratio was approximately normally distributed for large degrees of freedom. He also extended the theory to obtain an approximate chi-square test for the comparison of the sensitivities of several

In works on classical renewal theory (for example, Cox, 1962), the basic ingredient is the distribution of life-times, or failure-times. This function, representing the stochastic duration of the article subject to renewal, can be conveniently measured in the specific applications which led to the development of the theory. From the abstract point of view, however, the importance of renewal processes goes far beyond the problems which contributed the name 'renewal'. In the family of point processes, a renewal process represents the first large step away from a Poisson process; being defined by adding together independent equi-distributed random variables, it is a process of natural simplicity. When we formulate probability models for special point processes, it is reasonable, there- fore, to consider various renewal processes, regardless of whether or not the renewal model has any relevance to the problem at hand. The present study is motivated by two such examples in which the distribution of life-times is theoretically or practically unknowable. The first example is accident distribution: accidents are given in statistical sources as so many per hour, or per day, or per year. If one wanted to measure the quantity analogous to life-time, one would have to record the exact instant of every accident, and most of the data from the past, or otherwise beyond the control of the statistician, would be useless. The other example is the instant in time, or position in space, of vehicles. The quantity corresponding to life-time is the time, or distance, between consecutive vehicles—a quantity difficult to measure without sophisticated instrumentation. As in the first example, the more convenient statistic (and the one which is usually given by traffic authorities) is the number of vehicles in a fixed time period or fixed length of roadway. We shall call the probability distributions formed by the number of events in a fixed interval counting distributions. The main purpose of this paper is to isolate those discrete distributions which can possibly serve as counting distributions for renewal processes. In developing the results, we will use terminology and notation which reflect a point of view somewhat more general than that of the original 'renewal-motivated' renewal theory.

Fisher (1928) expressed the distribution function of the non-central %2 in terms of Bessel functions with imaginary argument. Wishart (1932) and Tang (1938) evaluated the probability integrals of the non-central %2 and F distributions; they involve a heavy amount of labour. In this paper Laguerre series expansions of these distributions are derived. In ? 2 series expansions of the distributionp (X) of X = ,y'2 (denoting a non-central chi-squared by x'2) is developed in terms of Laguerre polynomials, namely

SUMMARY This paper provides exact tests in connexion with the distribution introduced by Fisher to describe observations on a sphere. The tests are for the dispersion parameter of the population, and for the modal direction when the dispersion is known. Approximate tests are also examined. The tests may be adapted to provide confidence intervals.

Three different tests have been proposed for the Wilks-Lawley hypothesis of the homogeneity of group means in multivariate analysis (i.e. of the equality of np-variate normal means). These tests are: the likelihood ratio test (Wilks, 1932); the analysis of variance analogue (Lawley, 1938) and the generalized Mahalanobis distance statistic (Pillai, 1954)*. We will call these test statistics W, L and D, respectively; their expressions as functions of the observations are given in ? 1-2. below. Under normal theory these tests are asymptotically equivalent. That is, suitably scaled, they have the same asymptotic distribution both when the null and the alternative hypotheses are true. This extends to non-normal theory, on central limit theorem considerations, provided the homoscedasticity and independence assumptions are retained. They are of course completely equivalent for all sample sizes in the univariate case, and also in the two-sample case. There is little to choose between them in respect of ease of computation and whilst intuitively it may be felt that W would be more seriously affected by departures from normality in small samples, relatively little is known of the robustness of the tests. (Barton & David, 1962 have studied the randomization theory of D and their results are available for the investigation of the effects of nonnormality on the first kind of error entailed by its use.) The null hypothesis percentage points of W were tabulated by Pearson & Wilks (1933) and Pillai (1954), gives those of D to a high degree of approximation. Both tabulations are for the usual percentage points, taken over a small range of values of N and p in the case of Pearson & Wilks, and a wide range of N andp in the case of Pillai; here N is the total sample size. The upshot of these considerations is that the distinction between the tests rests almost entirely on the comparison of their powers in small samples. Such comparison is made in the present paper for selected values of the several parameters concerned. In particular, the calculations have been restricted to the bivariate case p = 2, although the methods have been designed with the aim of application when p has any value. The overall aim has been to develop simple approximate methods of evaluating powers, checking the accuracy of approximation by comparison with exact values at points where these are obtainable by means of the theorems of Kshirsagar (1961) or the Fisher-Hsu-Roy theorem (cf. Hsu, 1940

SUMMARY A study has been made to facilitate the design of block experiments in circumstances where considerable balance can be introduced but not so much as to permit the adoption of a standard design. First, to provide a theoretical basis, the effect of merging treatments is considered. It is then shown that many standard designs owe their distinctive characteristics to features that can often be introduced in nonstandard cases. Finally, it is assumed that one treatment has been allocated provisionally to plots. Limits are found for the precision of estimation of its mean however the other treatments are allocated. Similar expressions are given for the case in which two treatments have been allocated and it is desired to estimate the difference betweeni their means. In the past many specific experimental designs have been studied and expressions published for their precision of estimation, both of treatment means and of differences betweeii them. At the other extreme, methods have been developed that facilitate the analysis of data from experiments that have hardly any restrictions on design at all; for example, Tocher (1952), Kuiper (1952), as developed by Corsten (1958), Nelder (1965a,b) and Wilkinson (1970). Sucli methods imply ways of determining precision in particular cases and the lemma of Freeman (1957) also is useful for that purpose. Sometimes, however, the designer of experiments is at neither extreme. He cannot secure the complete regularity of the publislhed designs, yet he can build in enoug,h pattern to obtain precise estimation of certain important effects. Classes of design useful in such circumstances lhave been reviewed previously (Pearce, 1963). The various formulae for standard errors have muclh in common, which suggests the existence of expressions more general than any now known. In this paper attention will be confined to block designs. The conclusions are in three parts. First in ? 3 some results will be presented relating to the effect of merging two treatments to form one. The results are of interest on their own account because treatments are sometimes specified conditionally; for example, 'trees will be irrigated whenever the soil moisture deficit exceeds 8 cm', and in unexpected circumstances may become merged unintentionally. The results are needed here, however, as a basis for what follows. Secondly, a number of relationships between treatments will be presented in ? 4, together with the consequent effects on precision. Thirdly, it will be supposed that one or two of the treatments have already been provisionally allocated to plots. In ? 5 limits of precision will be given that apply regardless of the allocation of the others.

SUMMARY The distribution of extreme values in sets of independent random variables has been very thoroughly treated (Gumbel, 1958) but few methods have been suggested for dealing with correlated variables. In this paper an approximate formula is developed for the distribution of extreme values in a normal correlated population, when the correlation matrix has dominant elements adjacent to the leading diagonal. The formula is then used to investigate extremes in a random assembly. We consider a one-dimensional interval covered by a number n of subintervals placed at random. The number, N, of subintervals covering any point is then a random variable correlated from point to point and a method is suggested whereby the extreme values of N can be estimated. Some numerical results are given, together with an asymptotic formula for large n.

SUMMARY Consider a finite set of points, located on the circumference of a circle. Several tests have been proposed of the hypothesis that the points constitute a random sample from a uniform distribution. In this paper we study a test statistic defined as the maximal number of points that can be covered by some semicircle. Exact and asymptotic distributions under the null hypothesis, and under a certain alternative hypothesis, are given together with some tables. A related test statistic is studied briefly. An expression is obtained concerning most powerful invariant tests of the hypothesis of a uniform circular distribution. In 1965, Dr G. Borenius described an unpublished experiment with a bubble chamber, where points representing events were observed through a circular window. A natural hypothesis was that the events occurred at random with a constant probability density within the circle. In one case it was observed that 67 out of 100 events fell within a suitably chosen semicircle. The question then arose whether this asymmetry should be judged inconsistent with the hypothesis of a uniform distribution. More generally, suppose that n points are observed and that each point is moved radially to the circumferences of the circle. We then have a sample of n points on the circumference and want to test the hypothesis that the underlying probability distribution is uniform over the circumference. The test statistic suggested by the foregoing paragraph is the maximal number of points in the sample that can be covered by a suitably chosen semicircle. In the following this test statistic will be denoted by N. We reject the hypothesis if N is too large. Many other tests of the same hypothesis have been proposed. For example, the classical Kolmogorov-Smirnov test has been adapted to circular distributions by Kuiper; see Kuiper (1960) and Stephens (1965). Watson (1961) did the same thing for the Crame6r-von Mises test. A detailed study of the null distribution of Watson's test statistic has been made by Stephens (1963, 1964). For a general review of statistical methods in connexion with circular distributions, see Batschelet (1965). The problem of determining the distribution of N under the hypothesis is purely combinatorial. It was solved by Borenius for sample sizes n up to n = 15 and the general solution was inductively conjectured by him. He thus found that the above-mentioned observation, N = 67 for a sample of size n = 100, corresponds to a level of significance P = 1U6 0/. Dr S. Johansen, University of Copenhagen, has told the author that he, too, has found the null-distribution of N. This was done in connexion with an application to the study of

SUMMARY An asymptotic approximation is given to the expectation of the size of a selected subset from a set of t objects, where the design is a balanced paired comparison experiment of n replicates. The paper also contains an excerpt (Table 4) of a table of expected sizes of the selected subset, expressed as proportions of t. The complete set of tables, which may be obtained by writing to the author, includes entries for combinations of t = 3(1) 20, n = 1(1) 20(5) 50(10) 100 and preference probabilities p = 0-60(0410) 1 00 at each level of selection P* = 0 75, 0 90, 0 95, 0 975, 0 99. Examples of applications are directed towards an optimization of designs for selection.

SUMMARY The contagious disease model proposed by Bailey (1956a,b) predicts the frequency distribution of time intervals between the appearance of subsequent cases. Using data from households of two or three susceptibles into which a disease, measles, had been introduced, Bailey used a maximum likelihood procedure to determine various disease parameters, for example, length of the latent and infectious periods. In the original calculations, a desk calculator scoring procedure was used to maximize the likelihood. The present paper describes the application of a computer maximization technique to this problem. The computation is of course more rapid and the use of a computer permits the automatic determination of starting points for the computation, the calculation of goodness of fit y2 and facilitates variations of the model, such as the addition of an extra parameter to allow for the misclassification of chains. The methods recommended are illustrated both by the previous measles data and by new data on infectious hepatitis. The mathematical theory of contagious disease has two distinct, though interrelated aspects. The first deals with general theory, which attempts to provide broad insights and to suggest further research. The second is concerned with studying the finer structure of the relevant processes, and entails, in particular, the estimation of parameters and the statistical fitting of models to actual data. Although in the latter case interesting or elegant mathematics may be involved, this is largely irrelevant to most applications. The important thing is to be able to compute estimates of parameters, together with their standard errors, and

SUMMARY This paper deals with the application to various problems in time series analysis of the maximum-likelihood ratio procedure proposed by Cox for situations in which there are rival hypotheses Ho and H1, such that probability distributions of data under Ho and H1 constitute 'separate' parametric families. The large-sample theory developed by Cox, outlined in ? 2, is extended to two main types of problem. In the first, the time series is generated by an autoregression of specified order p under one hypothesis and by a movingaverage of specified order q under the other; the particular case p = q = 1 is treated in ? 3 and the general case in ? 4. In the second, the series is generated by a simple harmonic oscillation with superimposed random error under one hypothesis and by an autoregression under the other; this is treated in ? 5. More general types of problem are considered in ? 6, but for these the computations required by the procedure may be impracticable.

The method of maximum likelihood was proposed as a general method of estimation by Fisher (1912). However, rigorous proofs of the asymptotic properties of the maximum likelihood estimators (hereafter called ML estimators) were not developed until the work of Dugue (1946), Wald (1943, 1949), Cramer (1946), Huzurbazar (1948) and Chanda (1954), all of whom published within the last 25 years. Each of these authors assumed observations from a single population. Neyman & Scott (1948) and Kraft & LeCam (1956) have pointed out situations wherein the ML estimator is inefficient and not consistent. Their examples involved sampling from a number of subpopulations with different distributions but with only a finite number of observations from each of the subpopulations. Some situations arise where observations do not come from a single population but from distinct but related populations, related in the sense that some populations reasonably may be assumed to have some parameters in common. We shall call such populations associated. We consider two cases of associated populations. In the first, a finite number of populations are considered and limiting results regarding ML estimators are obtained as numbers of observations from the distinct populations become large in constant ratios. In the second, one observation is obtained from each population and the number of populations sampled becomes large. The second case may include the first as some populations sampled are identical but for applications to investigations of specific ML estimators it seems better to consider the two cases separately. In this paper we extend the basic theory on ML estimation to associated populations. Under regularity conditions developed, the ML estimators for parameters in associated populations are shown to be consistent and asymptotically normal with a variance-covariance matrix as derived. The asymptotic X2-distribution of -2 ln A, A being the likelihood ratio, holds and, for the non-null distributions, the parameter of non-centrality is set forth. Our objective is to set forth results as simply as possible and in usable forms; more generalized theorems could be obtained. We limit consideration to ML estimators that are roots of the normal equations for the maximization process. The consistency of the more generally defined estimator can be proved for associated populations following the approach of Wald (1943). Our demonstrations follow those of Chanda closely; most of the results for associated populations are easy generalizations of results for a single population and we state results without complete proofs when this is so. Two simple examples involving associated populations are included. One of these applies to the work of Dykstra (1960) which is dependent upon results given here.

SUMMARY In recent years methods of data analysis for point processes have received some attention, for example, by Cox & Lewis (1966) and Lewis (1964). In particular Bartlett (1963a,b) has introduced methods of analysis based on the point spectrum. Theoretical models are relatively sparse. In this paper the theoretical properties of a class of processes with particular reference to the point spectrum or corresponding covariance density functions are discussed. A particular result is a self-exciting process with the same second-order properties as a certain doubly stochastic process. These are not distinguishable by methods of data analysis based on these properties.

SUMMARY In an earlier paper Sobel & Weiss (1970) examined two sampling schemes for choosing the binomial population with the larger probability of success. The two sequential allocation procedures were discussed in the light of their ability to reach a decision while using as small a number as possible of tests with the poorer population. It was shown that neither sampling scheme was uniformly better than the other with respect to that criterion. However, one sampling procedure, the Play-the-Winner rule (Robbins, 1956; Zelen, 1969), was superior, provided that the probability of success of the better population was greater than a number roughly equal to A. Here we investigate a two-stage procedure in which the first stage is used to estimate the probability of success of the better population. It is shown that a fairly small estimation stage, approximately five tests with each population, allows one to benefit considerably from the choice between the two sampling methods available in the second stage.

In two previous papers, a model for multivariate paired comparisons was proposed and the associated methodology and large sample properties were developed. The present study considers the problem of relating the response pattern on overall quality to that on a specified set of attributes. A regression equation is derived for a joint distribution of responses to dichotomous items and is applied to the multivariate paired comparison model. A test of the significance of the responses to specified attributes in estimating the responses to overall quality is presented. (Author)

SUMMARY A Bayesian procedure is obtained for the simultaneous estimation of the parameters of m binomial distributions. The method uses logistic transformations for the parameters and an exchangeable prior distribution. Information is combined between the binomial distributions to obtain estimates which, under certain circumstances, will be superior to the usual proportions. This paper is intended as a forerunner to a more general theory for the analysis of nonlinear models.

This model seems particularly appropriate if columns are time-related variables. For example, an industrial plant has five different machines which are supposedly identical. The number of unacceptable parts each machine produces in a day is recorded. There is only a short number of days for which comparable data are available on all machines simultaneously. It is known that the number of bad parts varies daily, due to humidity changes, etc. It is desired to evaluate results relative to the hypothesis that the machines do not differ in their rate of unacceptable parts. The data are naturally categorized as a two-way table with, say, machines as rows and days as columns. However, the days within a machine are not randomly selected. Therefore it is doubtful that the errors within rows would be independent, but there seems no reason to assume that the error terms on separate machines are related. A model such as one described by (1.1) and (1.2) would therefore seem to be appropriate for such a situation. If the matrix V = {vi>} is known, inferences on the row or column effects pose no new problems; see Aitken (1935). Note that in the usual analysis of variance situation, V is assumed equal to I, and hence o2 is the onlyunknown covariance parameter. If V is unknown, no simple general solution exists in practice. This paper is concerned with inferences on the

For the past 40 years, there has been confusion on the differences between tests of significance and tests of hypotheses to the point where data interpretation is presented as an accept-reject process. This paper discusses the case of inference for a simple experiment, the paired design. It gives a rationale of the significance tester, and proceeds to a comparison of three tests of significance which can be viewed as competitors in the case of this design. These are the sign test, S, the Wilcoxon test, W, and the Fisher randomization test, R. The conclusion is that there is indeed an ordering of the tests, which is R preferred to W preferred to S. The S test should never be used if the others are possible. An Appendix gives a proof of monotonicity of the R test with respect to a shift alternative.

The technique of regression analysis has been in general use for many years. Its wide acceptance has led people in many fields to adjust the technique in various ways to answer questions which are outside its field of applicability. Thus, an engineer might feel that the tensile strength of cement depends on the ratio of cement to aggregate, amount of water used, temperature and humidity of surrounding air during hardening, pressure applied, coarseness of the aggregate used, etc. To estimate the relationship of strength to the other variables mentioned he might very well assume a linear regression model, perform an experiment or series of experiments, and use the usual least squares technique to estimate the unknown coefficients in the model. At some point in this procedure he may decide that all the independent variables or predictors he has included in his model are not really necessary to predict tensile strength 'accurately'. To decide the point objectively he tests the hypothesis (or hypotheses) that the coefficients of the doubtful predictors are equal to zero. If he accepts the hypothesis, he simply deletes the corresponding predictors from his model and, if he rejects the hypothesis, he retains the doubtful variables in his model. Similarly, a chemist might use the same type of procedure in deciding which of a group of predictors is necessary to predict accurately the purity of a chemical resulting from a given reaction. A psychologist might use the same procedure to decide which of a battery of test scores are necessary to predict an individual's 'success' in a given job. Many examples could be quoted of possible applications of the above-mentioned adjustment of the usual regression technique. Similarly, many different types of adjustments (that is, adapting regression techniques to answer different types of questions) could also be mentioned. (See, for example, Fireman & Wadleigh, 1951; Hollingsworth, 1959; Peperzak, 1956; Summerfield & Lubin, 1951.) Since this paper is concerned only with the particular technique already mentioned, no reference will be made to other possible adaptations. Bancroft (1944) was one of the first to consider the impact of preliminary tests of significance on subsequent estimation. His work was concerned with the bias introduced into an estimate of a regression coefficient, if a second coefficient is tested to be zero. Mosteller (1948) investigated the problem of pooling two means in estimating a population mean, using a preliminary test of significance to make the decision. Kitagawa (1951) obtained the distribution function and moments of the estimator studied by Bancroft and of a pooled estimator of a mean. He later (1959) studied the effect of certain preliminary tests of significance, to decide how many predictors should be included in a predictand, on the

In the Neyman-Pearson theory of testing statistical hypotheses, the efficiency of a statistical test is to be judged by its power of detecting departures from the null hypothesis. Thus besides knowing the random sampling distribution of a given statistic T under this hypothesis, say Ho, it is also necessary to know the distribution of T under admissible hypotheses alternative to Ho. Hence the power function of the test is obtained. In the case of the wellknown tests using X2, t and F, the evaluation of their power functions involves the use of what have been called non-central distributions. For example, if we are applying the t-test to examine if a sample has come from a normal population with mean # = 0(HO), we know that under Ho, t has a 5 % chance of exceeding the 5 % point of its distribution. But in order to compute the power of the test we wish to know the chance that t exceeds this point when M has alternative values, not equal to zero. This chance is given by the non-central t-integral. This distribution has been studied by Fisher (1931), Neyman (1935), Neyman & Tokarska (1936) and Johnson & Welch (1939). In a similar way, the non-central X2and F-distributions arise in consideration of the power functions of the X2and variance-ratio tests. The power function may be used either to determine the extent of the departures from Ho in a given direction, which will be detected as significant (at a prescribed level) with a given probability, or it may be used to determine in advance the size of experiment necessary to ensure that a worth-while difference will be established as significant, if it exists. But apart from its value in this connexion, the study of non-central distributions is of considerable interest. The mathematical forms of these distributions of t, x2 and F have been long known, but their use without extensive tabling has not been easy. The present paper is therefore concerned with two lines of investigation: (a) The derivation of certain approximations to the probability integrals of (i) non-central X2, and (ii) the ratio of non-central X2 to an independent central X2, which we have termed noncentral F. These approximations, depending on tabled functions, permit easy calculation. (b) Discussion of the ways in which these distributions may be used in connexion with the power functions of statistical tests.

This paper presents a general mathematical theory for quantitative investigation of the properties of a dilutely distributed particulate phase. If the process by which the specimen is sampled and the process by which particles in the sample are measured can be appropriately modelled, then this theory provides unbiased estimates of any linear property of the particle size cumulative frequency function under the assumption that particle centres are uniformly distributed within the specimen. Uniqueness of the estimate is discussed and variance formulae given. Some classical results on spherical particles and new results on cylindrical particles are presented as specific examples. The theory unifies a large body of current literature and also provides the basis for the inclusion of the effects of sample preparation and any anomalies encountered in the measurement process.

SUMMARY This paper is devoted to the early history of the law of large numbers. An outline of the prehistory of this law is given in ? 1. The algebraic part of J. Bernoulli's theorem is presented in a logarithmic form and the lesser known role of N. Bernoulli is described in ? 2. Comments on the derivation of the De Moivre-Laplace limit theorems by De Moivre, in particular, on the inductive character of his work, on the priority of De Moivre as to the continuous uniform distribution, on the unaccomplished possibility of Simpson having arrived at the normal distribution and on the role of Laplace are presentedin ? 3. The historical role of J. Bernoulli's form of the law of large numbers is discussed in ? 4.

SUMMARY In this paper, a procedure is developed which can be used to decide if a doubtful variable should be included in an experiment to fit a linear model to a set of yields. The criterion of success of the experiment is based on the mean square error of the fitted linear model, averaged over a specified region of the independent variables. A study is made of the gains to be expected when a variable is correctly excluded.

This paper describes and lists certain block designs for two non-interacting sets of treatments. The designs have the following properties: (a) each set of treatments is arranged relative to blocks in a non-symmetrical balanced incomplete block design, the same balanced incomplete block solution being used for both sets of treatments; (b) each set of treatments is totally balanced with respect to the other, i.e. each is disposed relative to the other in the same way that treatments are disposed relative to blocks in a balanced incomplete block design; (c) the number of replications of each treatment of each set is either one more than or one fewer than the number of treatments in the set; and (d) the designs are balanced, in the sense that the variance of an estimated difference in effect between two treatments from the same set is independent of the treatments compared.

SUMMARY In comparing two treatments, suppose the suitable subjects arrive sequentially and must be treated at once. In such situations, if the experiment calls for fixed treatment numbers, the experimenter can, using his knowledge of the number of treatments that have been assigned, bias the experiment by his selection of subjects. If we consider the method of assigning treatments as an experimental design, Blackwell & Hodges (1957) have shown that the minimax design is the truncated binomial. In this paper we show that random allocation is a restricted Bayes design within the class of Markov designs, and is in many senses preferable to the minimax design. In particular, it is possible for the random allocation design effectively to eliminate the bias asymptotically when the minimax design does not, and in no case will random allocation have a much worse performance than the minimax. In the problem of comparing the effectiveness of two treatments, it is common for the statistician to have the experimenter select 2n subjects suitable for treatment and treat n subjects with each treatment. Clearly, if the experimenter is aware of or guesses which treatment a subject will receive before he selects the subject, then he can, consciously or unconsciously, bias the experiment by his choice; therefore the assignment of treatments to subjects is usually done 'randomly'. Ideally, the 2n subjects will be selected in advance, and the treatment assignments made by random sampling without replacement. This is, however, not always possible. In many cases, suitable subjects arrive sequentially and must be treated immediately or not at all. Two examples which have been cited are clinical trials and cloud seeding experiments. In clinical trials a patient often must be treated as soon as the disease has been diagnosed; in cloud seeding it is physically impossible to collect the subjects (storm clouds) for simultaneous assignment of treatments. In some experiments it may be possible to eliminate this bias by having the selection of suitable subjects done by a third party who is ignorant of the past assignment of treatments, or by defining 'suitability' in a sufficiently objective manner to permit selection without an exercise of judgment. There remain many cases in which the best or only judges available are those involved in administering the treatments. For such situations it is reasonable to ask what strategy the statistician should adopt in the assignment of treatments in order to reduce the expected bias as much as possible. This problem was first considered by Blackwell & Hodges (1957), who have proposed the following model. In order to compare two treatments it is decided to treat 2n subjects, n with treatment A and n with treatment B. Candidates for treatment arrive sequentially, and as each arrives, the experimenter E decides whether or not it is a suitable subject. If it is

application of the iteration process. The use of the iteration process is justified in large samples, at least, if we can show that the above probability tends to one as n, the sample size, tends to infinity. In this paper we consider the multi-parametric case of the Newton-Raphson process and the usual method of scoring for parameters. It is shown that these methods are justifiable in large samples at least.

SUMMARY Given a sample from each of three populations, it is sometimes of interest to decide whether the distribution function of one of the populations is a linear mixture of the distribution functions of the other two populations. This paper discusses distribution free procedures for testing such a hypothesis.

SUMMARY Sequential sampling is considered wherein, at each trial, the experimenter can choose to stop and take some action or else to perform one of two related experiments. An approximation to an asymptotically optimum decision rule is shown to be easily defined in practice and to be a Bayes rule in some cases. Average risks of this and other sequential design rules are compared. The technique of sequential design of experiments generalizes that of sequential analysis by allowing the experimenter, when he decides not to stop sampling, to choose his next experiment from among a class g of possible experiments. Much recent attention has been devoted to finding sequential designs which are optimum in some sense. For examples, see the references. In most of these, optimality means asymptotic optimality, as defined by Kiefer & Sacks (1963, Theorem 1). Whittle (1965) criticizes the crude nature of asymptotic optimality as a criterion and develops approximate, non-asymptotic results. In the present paper, yet another asymptotically optimum rule is considered in the special case wherein there are two possible states of nature, E = {O, 1}, and two possible experiments, = {O, 1}, with the density fg,e of the observable random variable given by fg,e = gl-el, where g0 and g, are known. It is shown that an approximate form of the rule is simply specified. In a special case, the proposed rule is proved to be a Bayes rule and its riskis compared with approximate risks of other proposed rules. Proof of results which are stated but not proved herein are given in my University of N. Carolina Ph.D. thesis (1965).

SUMMARY This paper discusses the use of order statistics in estimating the parameters of the extreme value distribution. The best linear unbiased estimator of the location and scale parameter is given, based on two or three order statistics, for samples up to size 20. For very large samples, the analogous estimators based on sample quantiles are obtained.

SUMMARY In this paper, the exact distribution of Wilks's likelihood ratio criterion, A, is obtained, and explicit expressions for A are given for p = 3, 4, 5 and 6, where p is the number of variables. The distribution is expressed as a finite series except where p and f2, the degrees of freedom, are both odd, in which case it is given in infinite series form. Lower percentage points are tabulated for selected values off2> 10, extending the tabulations of Schatzoff (1966) for the above values of p.

The use of 'range' as a short-cut substitute for variance has recently been considered in the more elementary problems of analysis of variance. Patnaik (1950) dealt with a simple classification with equal cell frequencies and Hartley (1950) with a double classification. In this paper these methods are extended to cover a number of further orthogonal designs including: (a) a double classification with cell replication; (b) a double classification with two-factor 'treatment combinations'; (c) the split-plot experiment. A procedure for the general three-way cross-classification is indicated. Approximate methods to deal with a simple classification with unequal cell frequencies are also considered. In each of the main cases an example is given as well as a summary of procedure, which it is hoped will be adequate for the reader interested only in the practical applications of the methods developed.

SUMMARY Often a positive correlation exists between the quality of sequentially produced batches of items. The present paper considers whether, for a model where the mean batch quality is assumed to have a normal prior distribution, an estimate of this correlation and information on some or all of the preceding batches can be used advantageously when designing an acceptance sampling scheme.

SUMMARY In this paper we present stochastic models to describe the changes that take place with time in the numbers of females and males in the various age-intervals in the population. We first consider the case where females, or males, are marriage dominant, i.e. where the joint distribution of the numbers of boys and girls born at time t depends upon the numbers of females, or males, in the various age-intervals in the immediately preceding time-period. We then consider the case where neither females nor males are marriage-dominant, i.e. where the joint distribution of the numbers of boys and girls born at time t depends upon both the numbers of females and males in the various age-intervals in the immediately preceding time-period. Methods are given for calculating the expected values, variances, and covariances of the number of females and males in each age-interval at time t, and asymptotic results are presented for the case where t -* so. The application of these results is illustrated with data on the fertility and mortality conditions in the U.S. in 1965. The literature on two-sex stochastic models of population growth starts with articles by Kendall (1949) and Goodman (1953). Kendall outlined some of the analytic difficulties of a study of the stochastic aspects of two-sex population growth, and for simplicity limited his discussion of two-sex stochastic models to the special case where (a) each birth is equally likely to add a new female or a new male to the population, (b) the death-rate for females is equal to that for males, and (c) the birth- and death-rates per person are constants that are independent of the age of the person, and also independent of other relevant variables. Although Goodman's analysis was not limited by (a) and (b), it was limited by (c), as was all subsequent analytic work by other authors. One of the models of Goodman (1953) was a stochastic model where one sex is marriage-dominant, i.e. where the joint distribution of the numbers of girls and boys born at time t depends only upon the number of the dominant sex in the population in the immediately preceding time-period; he calculated the means, variances, and covariances of the numbers of females and males in the population at time t. A similar analysis of this model was made by Joshi (1954); see also Barucha-Reid (1960, pp. 175-9), Bailey (1964, pp. 119-20), and Keyfitz (1968). Lamens (1957) supposed that the birth- and death-rates may be functions of time, but he did not allow for possible dependence of these rates on the age-composition of the population, nor for the possibility that neither females nor males are marriage-dominant. Here we shall deal with stochastic models where the birth- and death-rates for females and for males may depend upon the age-composition of the population. First, say, the females will be assumed marriage-dominant, and then the more general situation will be considered. The above articles dealt mainly with some of the stochastic aspects of two-sex populationgrowth. Some of the deterministic aspects have been discussed in the demographic and bio

SUMMARY Some problems in the design of change-over trials when (i) the treatments are four equally spaced levels of a single treatment factor; (ii) estimates are required of linear direct x linear first residual interaction as well as direct effects and residual effects; and (iii) second, third, etc., residual effects are negligible are reviewed. Designs considered include (a) designs for eight periods and sixteen subjects constructed by the method described by Berenblut (1964), (b) additional designs for eight periods and sixteen subjects related to designs (a) but constructed by a different method, (c) designs as in (a) and (b) but with the subjects in blocks of four or eight, and (d) the design for four periods and eight subjects described by Berenblut (1967 b). These designs are shown to vary widely in efficiency of estimation of linear direct x linear first residual interaction. The most efficient designs are listed. Berenblut (1964) introduced change-over designs in which first residual effects are orthogonal to direct effects and direct effects are orthogonal to subject differences. These properties result in improved estimates of direct and residual effects relative to ordinary change-over designs. The designs require 2t periods and t2 subjects for comparison of t treatments. An example with t = 4 is given in Table 1. Later Berenblut (1967a) described the analysis appropriate when direct effects and residual effects are additive. He also pointed out, without giving details, that all components of the direct x first residual interaction are orthogonal to direct effects (Berenblut, 1968). This property allows interaction to be tested without loss of information on direct effects. The present paper is concerned with designs for testing a single quantitative factor at four equally spaced levels. Our main purpose is to show that, although the designs described by Berenblut (1964) are suitable for the estimation of direct effects and residual effects, other designs may be preferable for the estimation of linear direct x linear residual interaction.

Armitage (1957) introduced a class of closed sequential procedures, called 'restricted', which have been proved useful in medical trials (Armitage, 1960) where the risk of a long sequential experiment is not easily borne. Restricted plans can be regarded as procedures for testing a null hypothesis, Ho, that the mean, It, of a random variate is zero. For two-sided alternatives there are three decisions, D1 (desirable when It > 0), D-1 (desirable when It < 0) and Do (desirable when It = 0). For one-sided alternatives, the choice is between two decisions, Do and, say, D1. In using restricted plans one sacrifices some of the reduction in average sample number achieved by the open plans of Wald (1947). For two-sided alternatives restricted plans have the same 'outer' boundaries leading to D1 and D-1 as when two Wald plans are used in conjunction (as proposed by Armitage (1947) and Sobel & Wald (1949)). Instead of the wedge-shaped 'inner' boundary of the Sobel-Wald plans, the restricted plans require a truncating boundary drawn after a certain number of observations, N1, the value of which is chosen to preserve certain specified type I and type II errors. It is the substitution of this boundary in place of the Sobel-Wald wedge that causes the increase in average sample number in the restricted plans. In this paper we describe a family of closed sequential plans for normal variates, which retain the same outer boundaries as the Sobel-Wald and the restricted plans. The limiting cases, at opposite extremes, of the family of plans considered here are (a) a restricted plan, and (b) an open plan similar to the Sobel-Wald plan. The new plans, which we call 'wedge' plans, have better average sample number characteristics than restricted plans. The wedge plans appear to have average sample number functions similar to those of the equivalent open plans. It is difficult to make a precise comparison of the average sample number functions because of the difficulty in finding comparable plans with exactly the same operating characteristic. There is, however, some evidence that the wedge plan may have rather smaller average sample numbers, for some values of Itt between Ito and #,, than the equivalent open plans. If this were so, the wedge plans would in this respect resemble the test procedures described by Anderson (1960), although the methods used here, and the resulting boundaries, are different from those of Anderson. In particular Anderson considers only two-decision procedures, whereas we consider also three-decision procedures. Results based on Monte Carlo trials for the twodecision problem ('one-sided') and the three-decision problem ('two-sided') are given in Tables 6-9. These are discussed in detail later.

SUMMARY In this paper we determine, for an individual of age x existing at some specified point in time, the probability p(x) that his line of descendants will eventually become extinct, for the age-dependent birth-and-death process studied by D. G. Kendall (1949) and Bartlett (1955) in which the time-scale and age-scale are continuous, and also for the analogous agedependent birth-and-death process earlier studied by Bartlett and by Pollard (1966) in which the time-scale and age-scale are discrete. The extinction probabilities are also determined for generalizations of these processes in which, in a unit time-interval in the discretetime, discrete-age process (or in a small time-interval in the continuous-time, continuous-age process), a given individual may produce more than one new individual (multiple births), and his chances of producing new individuals may depend, in ways other than those specified for the original processes, on whether he does or does not live through the time-interval. In addition, the extinction probabilities p(x) are determined for a multiphase birth-and-death process, which is a generalization of the multiphase birth process studied by Kendall (1948 a), in which an individual may move through a number of different phases during his lifetime, and the phase in which he is at a specified point in time may affect his chances of producing new individuals and his chances of dying at that time. For each of these processes, and for any value of x > 0, we present an explicit expression for the extinction probability p(x) in terms of p(O), and we determine p(O) from a single equation.

SUMMARY For tests of two values of a parameter 0 from a family of distributions indexed by 0, the class of sequential probability ratio tests is considered, including procedures in which no observations are taken. With a simple loss structure, the Bayes risk for given error probabilities is minimized in this class; in this paper, the nominal risk is minimized without constraints on the error probabilities, and the results compared numerically with approximations of Chernoff. This is done for 0 in a general class of distributions, in more detail when 0 is the mean of a multinormal distribution and losses for wrong decision are equal, and also for tests in which the error probabilities are equal.

In this paper we obtain some results concerning the optimum allocation of sampling effort among k strata at the second phase of a two phase sampling procedure, using information obtained from the first phase. Two different approaches are employed; a Bayesian posterior analysis and a Bayesian preposterior analysis. Two different allocation methods are derived and illustrated with some numerical examples, for cases where some or all of the nuisance parameters are unknown. The problem when all nuisance parameters are known has been discussed by Ericson (1965). (Author)

The generalized probit analysis which we will discuss in this paper arose from a problem in entomology. We will suggest a solution to this particular problem and then consider a more general situation where this method of solution might be appropriate. The particular problem is as follows. In the course of its lifetime the Petrobius Leach (Thysanura, Machilidae), which is a primitive wingless insect allied to the domestic silverfish, passes through various stages, technically referred to as 'instars'. A problem of biological interest is to estimate the mean time spent by such insects in each instar (stage). Since there are difficulties involved in keeping these insects alive in the laboratory (and consequently in keeping them under effectively continuous observation during their lifetime), they were sampled at various dates in the field, and those observed were classified according to instar. The actual set of data to be analysed is given in Table 1.* The insects started hatching on 30 April and times will be measured from this date. The problem is to estimate from these data the mean time spent in each stage.

Let (x sub t) be a first-order moving-average process; that is, x sub t = epsilon sub t + beta (epsilon sub t-1), where the sequence (epsilon sub t, t = 0, plus or minus 1,...) consists of uncorrelated random variables with mean 0 and variance v, and the absolute value of beta is < 1. Another parameterization which is useful involves sigma squared = v(1 + beta squared) and sigma squared rho = v(beta). This paper discusses the problem of inverting the covariance matrix Sigma sub T of x = (x sub 1,..., x sub T)'. (Author)

SUMMARY This paper is concerned with protocols for sequential clinical trials which meet the classical statistical criteria of significance and power and which tend to assign the majority of patients to the superior method of treatment. The object is to test the null hypothesis that there is no difference in mean survival times associated with two treatments. It is assumed that survival times are exponential with parameters, death rates, which depend upon the treatment and that patients arrive sequentially over a period of time and are assigned immediately to a single treatment. The assignment rules utilize the data collected about patients already treated to choose the treatment for the next patient. The trials are terminated when the likelihood ratio crosses a given boundary. It is shown that the operating characteristic is insensitive to the assignment rule, and that the expected number of patients receiving the inferior treatment can be markedly reduced by the use of data-dependent rules.

SUMMARY This paper is concerned with the relative merits of statistical procedures for the classical Behrens-Fisher problem in thep-dimensional case. There are three such methods considered here: (i)the approximate degrees-of-freedom solution put forward byYao (1965), (ii) James's (1954) asymptotic series solution and (iii) Bennett's (1951) extension of Scheff6's (1943) solution to the univariate problem. Monte-Carlo studies are made of the level of significance and power function.

The comparison of the means of two populations on the basis of two independent samples is one of the oldest problems in statistics. Indeed, it has been a testing ground for many methods of inference as well as for a variety of analytic approaches to practical problems. The univariate problem was first studied by Behrens (1929) and his solution was presented by Fisher (1935) in terms of the fiducial theory. Welch studied it in the confidence theory framework and provided an 'approximate degrees of freedom' solution as well as an asymptotic series solution (1936, 1947). Many others have investigated this topic and various methods of approach were also suggested by Jeifreys (1940), Scheff6 (1943), McCullough, Gurland & Rosenberg (1960), Banerjee (1961), and Savage (1961). In the multivariate extension of the Behrens-Fisher problem, Bennett (1951) has extended the Scheff6 solution, and James (1954) the Welch series solution. The present paper studies an extension of the Welch 'approximate degrees of freedom' (APDF) solution provided by Tukey (1959), and discusses the results of a Monte Carlo sampling study on this new APDF solution and its comparison with the James series solution.