Showing papers in "Biometrika in 1969"

Estimating the components of a mixture of normal distributions

Abstract: SUMMARY The problem of estimating the components of a mixture of two normal distributions, multivariate or otherwise, with common but unknown covariance matrices is examined. The maximum likelihood equations are shown to be not unduly laborious to solve and the sampling properties of the resulting estimates are investigated, mainly by simulation. Moment estimators, minimum x2 and Bayes estimators are discussed but they appear greatly inferior to maximum likelihood except in the univariate case, the inferiority lying either in the sampling properties of the estimates or in the complexity of the computation. The wider problems obtained by allowing the components in the mixture to have different covariance matrices, or by having more than two components in the mixture, are briefly discussed, as is the relevance of this problem to cluster analysis.

On the theory of screening for chronic diseases

Abstract: SIUMMARY It is assumed that a chronic disease progresses from a pre-clinical state to a clinical state. If an individual, having pre-clinical disease, participates in an early detection programme, the disease may be detected in the pre-clinical state. The potential benefit of a screening programme is related to the lead time gained by early diagnosis. A stochastic model is developed for early detection programmes which leads to an estimate of the mean lead time as a function of observable variables. An investigation is also made of a non-progressive disease model in which individuals in a pre-clinical state may not necessarily advance to the clinical state. At the present time special diagnostic procedures are available for early detection of some chronic diseases. For example, chest X-rays have long been used to detect tuberculosis. Currently, there are many public health programmes to detect women having cancer of the uterine cervix by using Papanicolaou smears; other programmes designed to test for glaucoma and diabetes are in wide use. An especially interesting programme for early detection of breast cancer using soft tissue X-rays, mammography, is now being conducted by the Health Insurance Plan of Greater New York; see Shapiro, Strax & Venet (1967). The aim of all such programmes is to detect the disease earlier than it normally would be detected, the motivation being that earlier detection may result in a cure or better prognosis. Unfortunately, with only a few exceptions we know of no chronic disease in which unambiguous evidence has been coLlected showing that early detection has resulted in significantly improved prognosis. Even in cancer of the uterine cervix, the results are not without question, because the survival rate had been increasing before the widespread introduction of the Papanicolaou smear. It is the purpose of this paper to discuss statistical considerations associated with the evaluation of such early detection programmes. Attention is confined to screening programmes where an individual is examined only once. In a future paper, we shall examine problems associated with screening programmes where an individual is examined periodically. It will be assumed that a person having a particular chronic disease can be regarded as

Inference about the intersection in two-phase regression

Abstract: In recent years increasing interest has been shown in problems of discontinuity in models for sequences of random variables. Essentially two distinct problems arise. The first is that of testing a single model hypothesis Ho against the hypothesis H1 that a shift occurs from one model to another at some unknown value of t in the sequence of random variables {Xt: t = 1, ..., T}. The second problem is that of estimating and making inference about the point at which the shift occurs. Situations where a shift takes place in the location parameter of the distribution of xt under the two-model hypothesis have been studied by Page (1955), Chernoff & Zacks (1964) and Bhattacharyya & Johnson (1968). In the present paper the estimation and inference problem for a linear regression situation is discussed. That is, we assume the model under H1 to be x Q aO+/0out+et (t= 1,...,T), (1.1) aOC181lft+6t (t =T+1, ...*, T),

Tests for serial correlation in regression analysis based on the periodogram of least-squares residuals

Abstract: A well-known procedure for testing for serial correlation is to plot out the sample path of the cumulated periodogram and to compare the resulting graph with the Kolmogorov-Smirnov limits. The paper considers small-sample aspects of this procedure when the periodogram is calculated from the residuals from least-squares regression. It is shown that for a test against an excess of low-frequency relative to high-frequency variation in the errors of the regression model, a pair of lines can be drawn on the graph such that if the path crosses the upper line the hypothesis of serial independence is definitely rejected, while if the path fails to cross the lower line the hypothesis is definitely accepted. In the intermediate case the test is inconclusive. Similar procedures are given for tests against an excess of high-frequency variation and for two-sided tests. To facilitate the tests a table of significance values of the appropriate modified Kolmogorov-Smirnov statistics is given. A further test based on the mean of the ordinates of the cumulated periodogram is considered. It is shown that bounding significance values are easily obtainable from significance values of the mean of a uniform distribution.

An exact test for comparing matched proportions in crossover designs

Abstract: SUMMARY An exact test for comparing proportions in matched samples is derived for the situation where order within pairs may be important. The analysis is based on an extension of the logistic model of Cox (1958 c). The test is formally equivalent to Fisher's exact test for 2 x 2 tables. The McNemar test (1947) has long been used in comparing matched or paired responses each of which take only two values, 0 or 1. The test consists of counting only those pairs which are unlike, (0, 1) or (1, 0) and testing whether the distribution of these two types is consistent with a binomial distribution with a probability of 2. Cox (1958c) showed that this test may be derived from a logistic model, and found a confidence interval for the relevant parameter of non-centrality. Cochran (1950) extended the test to more than two proportions. In many experimental situations, matching entails an ordering within the pairs. For instance, in testing certain drugs the matching involves a patient receiving the two drugs in two different time periods. Sometimes the order of the drug administration has appreciable effect on the patient's response; see, for instance, Meier, Free & Jackson (1958). In such instances, the McNemar test may ignore pertinent and important information on order within pairs. To fix ideas, we recall an example cited by Mosteller (1952). Two drugs, A and B, are used on each of 100 subjects, the response being either 'nausea' or 'not nausea'. Of these subjects, 81 never had nausea, denoted by (0, 0); 9 subjects had nausea with A but not with B, (1, 0); 1 had nausea with B but not with A, (0, 1); and 9 had nausea with both drugs, (1, 1). Mosteller considered only the unlike pairs, and calculated from the binomial distribution the onetailed significance level,

The behaviour of some significance tests under experimental randomization

Abstract: 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.

Cyclic change-over designs

Abstract: SUMMARY A class of cyclic change-over designs, existing for any number of treatments and periods, is defined as a simple extension of the cyclic incomplete block designs. The analysis is presented for direct and first-order residual effects of treatments but the method generalizes for residual effects of higher order. Cyclic change-over designs may be analysed after any number of periods and extra periods may be added. A table of selected designs is given for 6 to 20 treatments in 3, 4 or 5 periods. The efficiencies of these designs compare favourably with existing designs and in general the cyclic designs require fewer units. The purpose of this paper is to show that these features carry over when caI designs are interpreted as change-over designs, the blocks now being considered as treatment sequences applied to the experimental units and the rows constituting periods. We thus obtain the class of cyclic change-over (cco) designs. These are constructed by the cyclic development of one or more generating sequences of treatments, corresponding to the initial blocks of the CIB designs, and a general method of analysis is developed for this class of designs. An attractive feature of cco designs is that they may be analysed after any number of periods, and further periods may be added if required, thus enabling change-over trials to be con- ducted sequentially. Only first-order residual effects will be considered in this paper, although the methods used generalize directly to higher order effects. Balanced (Bco) and partially balanced change-over (PBco) designs have been developed in detail by Patterson & Lucas (1962). In Table 4 of the present paper selected cco designs are presented requiring fewer units but nevertheless possessing average efficiencies which compare favourably with those of Patterson & Lucas. 2. PRELIMINARIES CIB designs in t treatments (labelled 0, 1, ..., t - 1) and block size p are obtained by developing some number, b, of initial blocks mod t. Each such design is a partially balanced incomplete block design with up to (It) associate classes, but the analysis is straightforward since the reduced normal matrix for the estimation of treatment effects is circulant. Hence there may exist up to (It) distinct efficiencies for the differences between treatment effects, although John (1966) has shown that the range of efficiencies may be quite small. Fractional

Asymptotic properties of spectral estimates of second order

Abstract: Let X(t) (t = 0, ± 1,... ) be a zero mean, r vector-valued, strictly stationary time series satisfying a particular assumption about the near-independence of widely separated values.

Statistical inference with bivariate gamma distributions

Abstract: SUMMARY In some experimental situations where the use of a covariate could increase the precision of the experiment the distributions of the test variate and the covariate are highly nonnormal. Some of these cases can be analysed by using a bivariate P distribution which is here defined. Methods of testing for a change in the scale parameter in one margin, using the other variate as a control, are described and the corresponding extension to cross-over experiments briefly outlined.

On the exact distribution of Wilks's criterion

Abstract: 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.

Round-robin tournament scores

Abstract: SUMMARY Some methods of scoring players in a round-robin tournament are proposed, based on a concept of fair allocation of rewards to each player. The scores are related to those determined by a previously suggested principle of 'consistency'. The scoring systems lead to the 7rr's of the Bradley-Terry model, and they have desirable ranking properties in the case of a general linear model. Various other properties of the scores are described.

Derivation of approximants to the inverse distribution function of a continuous univariate population from the order statistics of a sample

Abstract: SUMMARY Approximations to the mathematical relation between the variate value z and the unknown distribution function F of the continuous, univariate population from which a sample is available are obtained in which z is expressed as a series of polynomials in F; the coefficients are the expectations of linear combinations of the order statistics of the sample. A particular system of orthogonal polynomials and a particular system of n linearly independent linear systematic statistics of a sample of n emerge naturally as apt for the purpose. General relations are found for the variances and covariances of these statistics, thus enabling them to be used as a vector basis in terms of which other linear combinations of the order statistics can be expressed and their variances and mutual covariances investigated.

On the inverse of the covariance matrix of a first-order moving average.

Abstract: : 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)

On looking at large correlation matrices

Abstract: SUMMARY Two graphical techniques, familiar in other contexts, are applied to a correlation matrix. The method of half-normal plotting is used to determine which coefficients are numerically too large to have come from zero population values. A visual clustering method is used to select clusters of variables which have high positive correlations with each other. The first and sometimes only impression gained from looking at a large correlation matrix is its largeness. This note describes the application of two graphical techniques to the problem of spotting some structure in the matrix. They are illustrated by a matrix taken from a study of the physiological effects of examination strain on 48 medical students. The variables of interest were differences in levels between a normal period of time and the period of the examination. Differences were obtained for 13 physiological measurements including blood pressure, pulse rate and various substances secreted in the urine and blood. The matrix of correlations betlween the 13 differences based on 48 subjects is shown in Table 1. The first technique is an aid to answering the question: Which of these coefficients is large enough, either positive or negative, to be considered as arising from a population coefficient different from zero? It is convenient to use the z transform of the correlations, r, given by Z = 2log {(1 + r)|(l-r)}. The standard error of any z is then approximately 1/145 which is 1/6 708 = 0-1491. The 2x 13 x 12 = 78 values of z in half the symmetric matrix are not statistically independent, but if this fact is ignored the values of z may be plotted in a half-normal plot to see which are too large numerically to have come from a random sample from a normal distribution with mean zero and standard deviation 01491. The half-normal plot is familiar in the analysis of factorial experiments (Daniel, 1959) and more recently has been applied to multidimensional contingency tables (Cox & Lauh, 1966), an application which suggested that it might also be useful with correlation matrices. The plot is very simple to do using tables of the transformations r -> z and

Maximum likelihood paired comparison ranking by linear programming

Abstract: SUMMARY The problem of ranking n objects from 'best' to 'worst', using the results of a paired comparison experiment, is formulated as a linear programming problem with the logarithm of the likelihood as the objective function. The constraints are determined by the transitivity relationships implied in a ranking. The linear program will determine maximum likelihood rankings for the k fold replicated paired comparison experiment with or without the possibility of ties. The method is demonstrated with two examples from recent literature.

A test for discriminating between models

Abstract: SUMMARY Suppose that two or more theoretical formulae are available for predicting the value of a single valued observable response. A method is described for testing whether each formula describes the data equally well. The method is compared with other tests.

Tests for the von Mises distribution

Abstract: SUMMARY This paper provides tests in connexion with the von Mises distribution for the circle. The tests are for the concentration parameter and the modal vector of the distribution; they may be adapted to give confidence intervals, and may also be used for a bimodal distribution. New approximations are given for the distributions of the likelihood ratio statistics on which the tests depend.

The use of residuals as a concomitant variable

Abstract: SUMMARY Suppose that there is correlation between the yields of successive or adjacent experimental units. An estimator of treatment effects using the residuals of adjacent plots as a concomitant variable is investigated. It is shown to be very close to the maximum likelihood estimator when the errors form a first-order autoregressive series. In the analysis of experimental data it is quite commonly assumed that observations on successive or adjacent units have independent errors. In randomized experiments this assumption is justified by the randomization. But sometimes an analysis making explicit allowance for correlation between adjacent units will be valuable, either because higher precision can be obtained, or because non-randomized data are under analysis. One procedure for allowing for correlation is to introduce a specific model for the error variability, for example, a first-order autoregressive process where the units are arranged in time or along a line. Williams (1952) suggested appropriate designs for this case and developed the method of analysis. Another, apparently quite different, method (Papadakis, 1937; Bartlett, 1938) proceeds as follows. The yield of each unit is corrected for the mean effect over all units receiving the same treatment. The average of the corrected yields of adjacent units is then used as a concomitant variable in the analysis of covariance. No explicit probability model is assumed in forming the adjusted estimate of the treatment effects. The conditions under which the estimates of precision so obtained are meaningful seem never to have been defined. The object of the present paper is to consider Papadakis's procedure in more detail and, in particular, to show its connexion with the analysis based on an autoregressive process. Variation is considered in only one dimension, although the method was originally proposed for spatial variation in two dimensions.

Multivariate paired comparisons: The extension of a univariate model and associated estimation and test procedures

Abstract: : The study is concerned with the development of an extension of the Bradley-Terry Model for paired comparisons to situations in which responses to paired comparisons are obtained on each of several characteristics. At the outset a probability model for multivariate paired comparisons is proposed that may be represented in several distinct ways. An iterative procedure for obtaining maximum likelihood estimates of the parameters introduced into the model is given and its behavior examined in some detail. It is observed that the procedure generally performed well, the key to this conclusion being the relative stability of the initial estimates of the preference parameters.

The role of significance testing: some data with a message

Abstract: SUMMARY In the context of experimental data, allowance for data-snooping in significance testing is found to conflict with power considerations. Appendices give (a) specific results for the nonparametric statistics employed, (b) general theorems concerning expected significance level and (c) general theorems concerning allowance for data-snooping. In (b), a correction is made of a lacuna of Dempster and Schatzoff. 1. EXPERIMENTAL DATA Table 1 gives experimentally ascertained estimates of the half-lives in minutes of the exponential decay of the potentiation of the response of the depolarized rat uterus to vasopressin by magnesium and cobalt.

On evaluation of warranty assurance when life has a Weibull distribution

Abstract: : A two parameter Weibull model with neither parameter known is assumed for failure time. A value is specified for the probability that no failures, in a lot of identically distributed components yet to be manufactured, will occur before the expiration of the warranty period. The warranty period, which must satisfy an assurance criterion at the prescribed probability level regardless of the true parameter values within the distribution, is to be calculated from a small preliminary sample of life lengths. An expression for the warranty period is derived as a function of the ordered observations and is determined only by the lot size, the sample size and the prescribed assurance level. (Author)

The performance of some two-sample tests in small samples with and without censoring

Abstract: The power curves of three two-sample tests are compared in small samples from the exponential and four forms of the Weibull distribution with and without censoring. The tests considered are: F ratio, modified F ratio (F' test), and a generalized Wilcoxon test (W test). Monte Carlo methods are used to obtain power curves for each test with sample sizes 20 and 50. The F test is known to be most efficient when sampling is from exponential distributions. If sampling is from exponential distributions and either there are no censored observations or all censored observations equal T, the F' and W tests are nearly as powerful as F, with F' being more powerful than W. If the times to censoring are not equal, the F' test is not appropriate and the W test is nearly as powerful as F. When sampling is from Weibull distributions, with or without censoring, the F test is not robust; for example, its size is too small when the coefficient of variation is less than one. When sampling is from Weibull distributions and either there is no censoring or all censored observations have the same value, the F' test is more powerful than the W test. If the times to censoring are not equal, the F' test is not appropriate; the W test has the proper size and is a valid test.

Estimation of the parameters of the extreme value distribution by use of two or three order statistics

Abstract: 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.

Multi-sample tests for the Fisher distribution for directions

Abstract: SUMMARY Fisher (1953) introduced a unimodal distribution for vectors denoting directions in three dimensions. Exact and approximate tests are given for the equality of modal vectors or of concentration parameters, for two or more samples drawn from the Fisher distribution.

The use of random allocation for the control of selection bias

Abstract: 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