Showing papers in "Biometrika in 1967"

Estimation of the probability of an event as a function of several independent variables

Abstract: SUMMARY A method for estimating the probability of occurrence of an event from dichotomous or polychotomous data is developed, using a recursive approach. The method in the dichotomous case is applied to the data of a 10-year prospective study of coronary disease. Other areas of application are briefly indicated. The purpose of this paper is to develop a method for estimating from dichotomous (quantal) or polychotomous data, the probability of occurrence of an event as a function of a relatively large number of independent variables. A key feature of the method is a recursive approach based on Kalman's work (Kalman, 1960 and unpublished report) in linear dynamic filtering and prediction, derivable also from the work of Swerling (1959), which provides an example of many other possible uses of recursive techniques in nonlinear estimation and in related areas. The problem that motivated the investigation is a central one in the epidemiology of coronary heart disease, and it will be used to fix ideas and illustrate the method. Some indication of the range of applications will be given in the conclusion.

Maximum-likelihood estimation for the mixed analysis of variance model

Abstract: SUMMARY A procedure is developed for the maximum-likelihood estimation of the unknown constants and variances included in the general mixed analysis of variance model, involving fixed and random factors and interactions. The method applies to all cases where the design matrices satisfy certain conditions. The consistency and asymptotic efficiency of the estimates are discussed. Tests of hypotheses and confidence regions are derived. In this paper we develop a procedure for maximum-likelihood estimation for the general mixed analysis of variance model, defined in (1) below, involving any number of fixed and random factors and possibly interactions of any order. We do not specify 'equal numbers' or indeed any other experimental balance for our procedure, but we do require that our design matrices satisfy certain conditions of estimability for the parameters. In the case of balanced designs the estimation problem for the constants and variances involved in the linear model has been extensively treated: confining ourselves to just one reference on variance estimation, optimality properties of the classical analysis of variance procedures have already been demonstrated for various balanced designs (e.g. Graybill, 1961). However, results for unbalanced factorial and nested data are much more restricted: Henderson (1953) has suggested a method of unbiased estimation of variance components for the unbalanced two-way classification but his method is computationally cumbersome for a mixed model and when the number of classes is large. Searle & Henderson (1961) have suggested a simpler method also for the unbalanced two-way classification with one fixed factor containing a moderate number of levels and a random factor permitted to have quite a large number of levels. Bush & Anderson (1963) have investigated for the two-way classification random model the relative efficiency of Henderson's (1953) method and two other methods, A and B, based on the respective methods of fitting constants and weighted squares of means described by Yates (1934) for experiments based on a fixed effects model which also provide unbiased estimates of variance components. Possibilities of generalizations are indicated. In all the above methods the estimates of any constants in the model are computed from the 'Aitken Type' weighted least squares estimators based on the exact variance-covariance matrix of the experimental responses which involves the unknown variance ratios. The estimation of the latter is then based on various unbiased procedures so that little is known about any optimality properties of any of the resulting estimators. However, all these methods reduce to the well-known procedures based on minimal sufficient statistics in the special cases of balanced designs. The method of maximum-likelihood estimation here developed differs from the above in that maximum-likelihood equations are used and solved for both the estimates of constants

On the bias of various estimators of the logit and its variance with application to quantal bioassay.

Abstract: SUMMARY The bias of several logit estimators and their corresponding variance estimators is investigated in small samples. Their use in quantal bioassay is similarly explored. The logit transformation has been suggested in the analysis of higher dimensional contingency tables, by Woolf (1954) and many others more recently, and also in estimating the parameters of the logistic function in the quantal bioassay problem (Berkson, 1944, 1953). Various modifications of the logit have been suggested by Berkson (1953), Haldane (1955), Anscombe (1956), Tukey, mentioned by Anscombe (1956), and Hitchcock (1962). Modifications of its usual variance estimator have been proposed by Haldane, Goodman (1964) and Gart (1966). More recently Goodman, in an unpublished paper, has derived several further modifications of his estimator. In this paper we present a numerical comparison of the bias of these estimators and give conditions under which one or the other may be preferred. The use of logits in the quantal bioassay problem is briefly explored with particular reference to the asymptotic results of Hitchcock regarding the bias of the estimators.

Estimating the mean and standard deviation from a censored normal sample

Abstract: SUMMARY The relation g(x) -c- + fx, where g(x) is the ratio of the ordinate and the probability integral of a normal distribution, is used to derive estimators ,uc and oc of the population mean and standard deviation, from a censored normal sample. These estimators are easy to compute. For symmetrical censoring ,uc is unbiased and for moderate censoring the bias in both ,c and oc is small. In terms of efficiency, ,c and o-c seem to be as good as the maximum likelihood (M.L.) estimators and better than the best linear unbiased (B.L.U.) estimators. A pair of nearly unbiased estimators ,u and o-, more efficient than the B.L.U. estimators, is also obtained.

The discarding of variables in multivariate analysis.

Abstract: In many multivariate situations we are presented with more variables than we would like, and the question arises whether they are all necessary and if not which can be discarded. In this paper we consider two such situations. (a) Regression analysis. The problem here is whether any variables can be discarded as adding little or nothing to the accuracy with which the regression equation correlates with the dependent variable. (b) Interdependence analysis. The problem is whether a constellation in p dimensions collapses, exactly or approximately, into fewer dimensions, and if so whether any of the original variables can be discarded. We may define the best solution to (a) using any given number of variables as the one that maximizes the multiple correlation between the selected variables and the dependent variable, and similarly for (b) as the one that maximizes the smallest multiple correlation with any of the rejected variables. In practice it is usual to accept an approximate solution to (a) based on 'step-wise' multiple regression: we know of no standard program for (b). We have developed cut-off rules that enable us to find the best solution to both problems by partial enumeration. The paper discusses the details of this approach, and computational experience.

On sampling without replacement with unequal probabilities of selection

Abstract: A sample of n different units is to be drawn from a population or stratum in such a way that unit i has probability np,, assumed less than 1, of appearing in the sample. A mathematical solution of this problem is given by a formula from which the required probability of selection of any possible sample can be calculated: this formula is an extension of one, due to Durbin, for n = 2. The required npi can be achieved in practice in three ways: (a) by evaluating the required probabilities for all possible samples, and selecting one; (b) selecting units without replacement, with probabilities of selection that must be recalculated after each drawing; and (c) by selecting up to n units with replacement, the first drawing being made with probabilities pi, and all subsequent ones with probabilities proportional to p(l -npi), and rejecting completely any sample that does not contain n different units. Method (c) seems likely to be the most convenient in practice. The probability of the simultaneous appearance in the sample of any pair of units is relatively easily calculated, so that unbiased variance estimates can be obtained without undue labour.

Upper and lower probability inferences based on a sample from a finite univariate population.

Abstract: This paper provides detailed formulae for upper and lower probability inferences, based on a sample from a finite population with a discrete univariate observable characteristic, and directed towards the unknown parameters of the population or properties of a future sample. The model is set up in ? 2 and the detailed formulae are given in ? 3. Section 4 demonstrates a limited relationship with confidence statements and ? 5 explores some special cases.

Time series with periodic structure.

Abstract: Time series which are encountered in meteorology exhibit non-stationary behaviour. If a variable is observed at the same time each day, it appears to be stationary over a period of a few weeks, but there is a seasonal variation in structure (Monin, 1963). For example, the mean and spectral density are different in the summer and in the winter. The usual approach to this problem is to divide the year into parts and analyse the data separately for the same two or three months from several years. This paper is concerned with the problem of predicting time series with periodic structure, including the estimation of the necessary parameters and variances.

Discrimination between alternative binary response models

Abstract: SUMMARY The logistic and integrated normal binary response curves are known to agree closely except in the tails. For experiments based on three dose levels the power of a significance test is found for the null hypothesis that the response curve is logistic against the alternative that it is normal, and vice versa. From this an appropriate spacing of dose levels for discrimination is found. Approximately 1000 observations are necessary for even modest sensitivity.

Studies in the History of Probability and Statistics. XV The historical development of the Gauss linear model

Abstract: SUMMARY The linear regression model owes so much to Gauss that we believe it should bear his name. Other authors who made substantial contributions are: Cauchy who introduced the idea of orthogonality; Chebyshev who applied it to polynomial models; Pizzetti who found the distribution of the sum of squares of the residuals on the Normal assumption; Karl Pearson who linked the model with the multivariate Normal thereby broadening the field of applications; and R. A. Fisher whose extension of orthogonality to qualitative comparisons laid the foundations of the modern theory of experimental design.

Testing for correlation between non-negative variates

Abstract: In analysing point processes (Cox & Lewis, 1966) a commonly occurring requirement is to test whether successive intervals between points are statistically independent. A natural test for this purpose is the use of the ordinary serial correlation coefficient. Such tests should be chosen to have some form of optimal properties and in order to study this question it is necessary to be able to define an alternative hypothesis in which the successive intervals are not statistically independent. The lengths of the intervals are usually assumed to have negative exponential, or more generally gamma-type, distributions. We therefore begin by studying correlation between pairs of variates which are non-negative and which have non-symmetric distributions.

On methods of asymptotic approximation for multivariate distributions.

Abstract: Methods are discussed for producing Edgeworth expansions, and the validity of the expansions is examined Algorithms for perturbation approximations in multivariate statistics are presented, in general and for the cases of principal components, canonical analysis and discriminant analysis The algorithms are suitable for implementation in a system of computerized algebra

The use of prior distributions in the design of experiments for parameter estimation in non-linear situations

Abstract: SUMMARY When prior information is available on the parameters of a non-linear model, it should influence the choice of the appropriate experimental design. We here investigate, for the single response case, how prior information on the parameters in the form of a multivariate normal distribution can be used in selecting designs for parameter estimation. An example illustrates how the precision of the prior knowledge can affect the positioning of the design points.

Some contributions to contingency-type bivariate distributions.

Abstract: SUMMARY Plackett (1965) has given a class of contingency-type bivariate distributions which contains the boundary distributions and the member corresponding to independent random variables. This is the only bivariate class known to possess this important property. In addition, Plackett (1965) has given some interesting applications of these distributions. The class is determined by solving a quadratic equation and discarding one of the roots not satisfying a set of inequalities. In this paper, we obtain the required root. We provide a simple method of drawing random samples from this type of population. We derive the moment-formulae appropriate to this class and apply these to simplify moments of contingency-type normal and uniform distributions. We compare the bivariate normal distribution with the contingency-type bivariate normal, and demonstrate that this latter distribution provides a simple method of computing tetrachoric correlation. We give a new estimate of the coefficient of associatioli and show that the asymptotic efficiency of Plackett's estimate relative to ours, in the region of interest, lies between 46 and 56 %. Another estimate is also considered.

The capacity of an uncontrolled intersection

Abstract: A formula is derived for the capacity of a minor road at an intersection when the vehicles have to wait for simultaneous gaps in several streams of traffic on the major road.

The use of prior distributions in the design of experiments for parameter estimation in non-linear situations: multiresponse case.

Abstract: : When prior information is available on the parameters of a nonlinear model, it should influence the choice of the appropriate experimental design. The authors investigate, for the multiresponse case, how prior information on the parameters in the form of a multivariate normal distribution can be used in selecting designs for parameter estimation. An example illustrates how the precision of the prior knowledge can affect the positioning of the design points. (Author)

Some tests of separate families of hypotheses in time series analysis.

Abstract: 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.

Unlimited simultaneous discrimination intervals in regression.

Abstract: The discrimination problem can be described as follows: The statistician has n pairs of values (xl, YI), (x2, Y2), ... , (xn, YJ) from which he estimates the regression line ac +,fx. He now observes K additional observations Y*, 4Y*, ... , YK for which the corresponding independent variable values x1', 4,..., XK are unknown. The statistician wishes to estimate these values of x and bracket them by means of simultaneous confidence intervals. This problem was first treated by Mandel (1958) and another solution was given by Miller (1966). When K is unknown and possibly arbitrarily large, these results do not apply. A solution to this problem of arbitrary K is given in terms of unlimited simultaneous discrimination intervals. Unlimited simultaneous discrimination intervals [D(P), D+*(P)] are presented which are based upon the same estimated linear regression and which have the property that at least lOOP per cent of the discrimination intervals will contain the true x's with confidence 1-ca. In this paper two techniques for obtaining unlimited simultaneous discrimination intervals are given. The first method is a procedure obtained through the Bonferroni inequality, while the second technique is based upon an idea of Lieberman & Miller (1963). A numerical example is analyzed. A general discussion and comparison of the two methods for finding unlimited simultaneous discrimination intervals is given.

Rank sum multiple comparisons in one- and two-way classifications

Abstract: The F-test provides a test of the equality of several treatment means. Analogous nonparametric methods are the Kruskal-Wallis (1952) test for the one-way classification and Friedman's X2 or the concordance coefficient of Kendall-Babington Smith for the two-way classification. However, the findings of the F-test or one of its analogues may not be very interesting; we may wish to know why the null hypothesis of equal treatment effects has been rejected. A preliminary narrow characterization of a multiple comparison method might be that it is used to augment the F-test with a view to answering this question. A second line of thought is the following. In scanning the results of an experiment our attention will naturally be drawn to the largest contrasts and the smaller ones will not be examined. We become interested in calculating something very much like a conditional probability: what value will be exceeded by a sample contrast with a prescribed conditional probability, the condition being that the contrast was large enough to attract our attention in the first place from among a class of contrast effects of interest? The various multiple comparison methods are attempts to formulate and answer this imprecise question in different ways and especially for different classes of interesting contrasts. P. Nemenyi, in his unpublished 1963 Princeton University dissertation, 'Distributionfree multiple comparisons', surveys the entire subject of distribution free multiple comparisons. The present work treats two specific distribution free-multiple comparison techniques: (1) the Wilcoxon method and (2) the extreme rank sum test for outliers. The Wilcoxon method of multiple comparisons concentrates on detecting differences between treatment means. In the normal theory case we may ask for the value that will be exceeded by the largest absolute Studentized difference withl given probability. This leads to the Tukey method of multiple comparisons. The Wilcoxon method performs the same practical functions as that of Tukey but with weaker distributional assumptions. The desirability of using sums to detect significant differences in the m-ultiple comparison context was recognized by Kramer (1956). Kramer treated two criteria of classification, whichl we may call objects and judges; the objects are ranked by each of the judges and the sum of these ranks is then determined for each object. Objects with significantly large rank sum differences are then declared to have different means. Wilcoxon clarified the critical 'extreme value' character of this problem and in 1956, in an unpublished notebook, * Present address: Office of Naval Research, Washington, D.C.

The estimation of a lagged regression relation

Abstract: SUMMARY The paper discusses a technique for estimating the matrices of coefficients, B(j), in a regression relation relating a vector time series, z(n), to lagged values, y(n -j), -p < j < q, of a second vector time series. The technique depends upon calculation of spectra and crossspectra. Once these are computed the estimates B(j) are obtained successively without recalculation when an additional lag is introduced. When the residuals from the regression are generated by a linear process independent of y(n) it is shown that under some additional regularity conditions the estimates are asymptotically jointly normal, the variances and covariances of the elements of A(j) being independent of j and of p and q. The method .of estimation is not efficient unless the spectra of the y process and the residual process are the same. Some idea of the magnitude of B(k) for lags for which computations have not been done can be obtained without doing these computations.

Upper percentage points of the largest root of a matrix in multivariate analysis

Abstract: SUMMARY The general expressions obtained by Pillai (1965) for approximating at the upper end to the C.D.F. of the largest of s characteristic roots of a matrix jointly distributed according to the Fisher-Girshick-Hsu-Roy distribution, have been used to compute upper 5 and 1 % points of the largest root for s up to 20, of which those for s = 14, 16, 18 and 20 are presented in this paper. Methods of interpolation for obtaining such percentage points for intermediate values of s have been suggested and errors of interpolation and approximation discussed.