Showing papers in "Biometrika in 1974"
TL;DR: In this paper, the Gauss-Newton method for calculating nonlinear least squares estimates generalizes easily to deal with maximum quasi-likelihood estimates, and a rearrangement of this produces a generalization of the method described by Nelder & Wedderburn (1972).
Abstract: SUMMARY To define a likelihood we have to specify the form of distribution of the observations, but to define a quasi-likelihood function we need only specify a relation between the mean and variance of the observations and the quasi-likelihood can then be used for estimation. For a one-parameter exponential family the log likelihood is the same as the quasi-likelihood and it follows that assuming a one-paramet'er exponential family is the weakest sort of distributional assumption t'hat can be made. The Gauss-Newton method for calculating nonlinear least squares estimates generalizes easily to deal wit'h maximum quasi-likelihood estimates, and a rearrangement of this produces a generalization of the method described by Nelder & Wedderburn (1972). This paper is mainly concerned wit'h fitting regression models, linear or nonlinear, in which the variance of each observation is specified to be either equal to, or proportional to, some function of its expectat'ion. If the form of distribution of the observations were specified, t'he method of maximum likelihood would give estimates of t'he parameters in t'he model. For instance, if it is specified that the observations have normally distributed errors with constant variance, then the method of least squares provides expressions for the variances and covariances of the estimates, exact for linear models and approximate for nonlinear ones, and these estimates and the expressions for their errors remain valid even if the observations are not normally distributed but merely have a fixed variance; thus, with linear models and a given error variance, the variance of least squares estimat'es is not affected by the distribution of the errors, and the same holds approximately for nonlinear ones. A more general situation is considered in this paper, namely the situation when there is a given relation between the variance and mean of the observations, possibly with an unknown constant of proportionality. A similar problem was considered from a Bayesian viewpoint by Hartigan (1969).We define a quasi-likelihood function, which can be used for estimation in the same way as a likelihood function. With constant variance this again leads to least squares estimation. When other mean-variance relationships are specified, the quasilikelihood sometimes turns out to be a recognizable likelihood function; for instance, for a constant coefficient of variation the quasi-likelihood function is the same as the likelihood obtained by treating the observations as if they had a gamma distribution.
2,063 citations
TL;DR: In this paper, two simple estimators are derived for the means of the random effect model by means of predictive sample reuse, which are applied to two sets of data in the literature and compared with several other procedures.
Abstract: SUMMARY Two simple estimators are derived here for the means of the random effect model by means of predictive sample reuse. They are applied to two sets of data in the literature and compared with several other procedures. The mixed model is also discussed.
1,859 citations
TL;DR: In this paper, a review of the literature on the use of the jackknife technique in bias reduction and robust interval estimation is presented, and speculations and suggestions about future research are made.
Abstract: SUMMARY Research on the jackknife technique since its introduction by Quenouille and Tukey is reviewed. Both its role in bias reduction and in robust interval estimation are treated. Some speculations and suggestions about future research are made. The bibliography attempts to include all published work on jackknife methodology.
1,620 citations
TL;DR: In this article, the authors considered a wide class of latent structure models, which can serve as possible explanations of the observed relationships among a set of m manifest polytomous variables.
Abstract: SUMMARY This paper considers a wide class of latent structure models. These models can serve as possible explanations of the observed relationships among a set of m manifest polytomous variables. The class of models considered here includes both models in which the parameters are identifiable and also models in which the parameters are not. For each of the models considered here, a relatively simple method is presented for calculating the maximum likeli- hood estimate of the frequencies in the m-way contingency table expected under the model, and for determining whether the parameters in the estimated model are identifiable. In addition, methods are presented for testing whether the model fits the observed data, and for replacing unidentifiable models that fit by identifiable models that fit. Some illus- trative applications to data are also included. This paper deals with the relationships among m polytomous variables, i.e. with the analysis of an m-way contingency table. These m variables are manifest variables in that, for each observed individual in a sample, his class with respect to each of the m variables is observed. We also consider here polytomous variables that are latent in that an individ- ual's class with respect to these variables is not observed. The classes of a latent variable will be called latent classes. Consider first a 4-way contingency table which cross-classifies a sample of n individuals with respect to four manifest polytomous variables A, B, C and D. If there is, say, some latent dichotomous variable X, so that each of the n individuals is in one of the two latent classes with respect to this variable, and within the tth latent class the manifest variables (A, B, C, D) are mutually independent, then this two-class latent structure would serve as a simple explanation of the observed relationships among the variables in the 4-way con- tingency table for the n individuals. There is a direct generalization when the latent variable has T classes. We shall present some relatively simple methods for determining whether the observed relationships among the variables in the m-way contingency table can be explained by a T-class structure, or by various modifications and extensions of this latent structure. To illustrate the methods we analyze Table 1, a 24 contingency table presented earlier by Stouffer & Toby (1951, 1962, 1963), which cross-classifies 216 respondents with respect to whether they tend towards universalistic values ( + ) or particularistic values (-) when confronted by each of four different situations of role conflict. The letters A, B, C and D in
1,583 citations
562 citations
TL;DR: In this paper, the effect of nonnormality on the Welch approximate degrees of freedom t test is demonstrated and a two-sample trimmed t statistic for unequal population variances is proposed and its performance is also evaluated in comparison to the Welch t test under normality and under long-tailed distributions.
Abstract: SUMMARY The effect of nonnormality on the Welch approximate degrees of freedom t test is demonstrated. A two-sample trimmed t statistic for unequal population variances is proposed and its performance is also evaluated in comparison to the Welch t test under normality and under long-tailed distributions. If the underlying distribution is long-tailed or contaminated with outliers, the trimmed t is strongly recommended. Some key word8: Behrens-Fisher problem; Robust test; t distribution; Trimming; Winsorization. Testing the equality of two means from independent samples is a common statistical problem. If the underlying distributions are normally distributed with equal population variances, it is well known that one would use the Student's t test. Unfortunately, this test statistic is sensitive to some nonnormal situations and that leads us to consider other more robust alternatives. Among the alternatives is the two-sample trimmed t, proposed and evaluated by Yuen & Dixon (1973); definitions of trimming will be given later. This statistic can be easily computed and its distribution is satisfactorily approximated by that of a Student's t with the degrees of freedom corresponding to the reduced sample. Results show that the loss of power efficiency in using trimmed t is small under exact normality, while the gain may be appreciable for long-tailed distributions. In cases where distributions are normal but population variances are unknown, the first exact solution was given by Behrens (1929) and later extended by Fisher (1939) as the fiducial solution. Among many others who have worked on this problem, Welch (1938, 1949) provided an approximate degrees of freedom t solution to his asymptotic series solution. Wang (1971) studied the probabilities of the type I errors of these two Welch solutions for selected cases and concluded that in practice, one can use the Welch approximate degrees of freedom t test without much loss of accuracy. For ease of reference in the rest of this paper, we shall refer to this approximate degrees of freedom t simply as the Welch t test. In this paper, the lack of robustness of the Welch t test when the underlying distribution is nonnormal is demonstrated. A two-sample trimmed t statistic for unequal population variances is suggested and its performance is evaluated in comparison to the Welch t test under normality and under long-tailed distributions.
431 citations
TL;DR: Good and Fienberg as mentioned in this paper proposed smoothing multinomial frequencies as one of estimation of the probabilities in the traditional multiinomial probability model, but their approach was limited to the case when at least one of these frequencies is small, as small as 0 or 1.
Abstract: Good (1965) and Fienberg & Holland (1972, 1973) have addressed the problem of smoothing multinomial frequencies as one of estimation of the probabilities in the traditional multinomial probability model. Suppose the multinomial has t categories and that the corresponding observed frequencies are n1, ..., nt with In, = N. Concerned with the case when at least one of these frequencies is small, as small as 0 or 1, Good ( 1965, p. 23) criticizes the maximum likelihood estimator {nj/N} of the multinomial probabilities {pj}:
309 citations
TL;DR: In this paper, a three parameter model is developed by reparameterizing and extending the distribution of the logarithm of a generalized gamma variate, where extreme value distributions for maxima and minima are included while the normal distribution is central.
Abstract: SUMMARY A three parameter model is developed by reparameterizing and extending the distribution of the logarithm of a generalized gamma variate. The extreme value distributions for maxima and minima are included while the normal distribution is central. Asymptotic maximum likelihood theory is given and maximum likelihood distributions are studied, via simulation, in some special cases. A regression generalization is given.
280 citations
TL;DR: In this article, the effects of certain risk factors or treatments are assumed to be primary interest and such effects are written as regression parameters in a proportional hazards model, and techniques based on marginally sufficient statistics are proposed for their estimation.
Abstract: Several models for paired failure time data are considered. It is assumed that the effects of certain risk factors or treatments are of primary interest and such effects are written as regression parameters in a proportional hazards model. Techniques based on marginally sufficient statistics are proposed for their estimation. Efficiency comparisons are made among estimates based on the suggested models and special attention is given to the study of bias and efficiency in the presence of censoring. Some comparisons are made between the models and analyses presented here and analyses based on study period survival and a binary logistic model for regression parameters.
180 citations
147 citations
TL;DR: In this article, the null hypothesis of row-column independence in a two-way contingency table can be expressed as a constraint on the parameters in various standard statistical sampling models, and it is shown that the Bayes factors for independence will factorize and expose the evidence residing in the marginal row and column of the table.
Abstract: SUMMARY The null hypothesis of row-column independence in a two-way contingency table can be expressed as a constraint on the parameters in various standard statistical sampling models. Four sampling models are considered, which are related by nested conditioning. By having the prior distribution in any one model induce the prior distribution in each further conditioned model, it is shown that the Bayes factors for independence will factorize, and thereby expose the evidence residing in the marginal row and column of the table. Bounds on the marginal Bayes factors justify, in a weak sense, Fisher's practice of conditioning. A general theorem is given for factorized Bayes factors from a factorized likelihood function.
TL;DR: In this article, the least square solution for a block design is considered in some detail for both the intrablock and the interblock cases, and the identification of certain contrasts, here called basic, that are estimated independently and with efficiency factors that are readily determined.
Abstract: SUMMARY The least squares solution for a block design is considered in some detail for both the intrablock and the interblock cases. It leads to the identification of certain contrasts, here called basic, that are estimated independently and with efficiency factors that are readily determined. Basic contrasts indicate possible partitions of the treatment sum of squares and aid in the computation of the appropriate analysis of variance. Also they help to clarify the relationship between three methods of computing the analysis for the general case of block designs, namely those of (i) Tocher, (ii) Kuiper and Corsten and (iii) Wilkinson.
TL;DR: In this paper, an extension of the negative binomial model as a species frequency distribution is given by allowing the shape parameter k to take values between -1 and 0, and different estimation methods are assessed for this model and the logarithmic series model.
Abstract: SUMMARY An extension of the negative binomial model as a species frequency distribution is given by allowing the shape parameter k to take values between -1 and 0. Different estimation methods are assessed for this extended negative binomial model and the logarithmic series model. Tables for standard errors are given. Many successful attempts have been made to fit mathematical models to populations of many species. Fisher's logarithmic series model (Fisher, Corbet & Williams, 1943), Preston's log normal model (Preston, 1948), the negative binomial model (Brian, 1953) and McArthur's broken stick model (McArthur, 1957) have all proved useful, giving a reasonably good fit to biological data. The purpose of this paper is to give an extension of the negative binomial model by allowing the parameter, usually symbolized by k, to take values between - 1 and 0, and to assess different estimation methods for this extended negative binomial model and the logarithmic series model. The extended model seems to fit well in situations where the negative binomial fails and has usually been substituted by the log normal model, which is more complicated to handle mathematically.
TL;DR: The related conditional Poisson process discussed recently by D. G. Kendall and W. A. O'N. Waugh is also studied, and the results are shown to illustrate the modern theory of exponential families and conditional inference as discussed by the authors.
Abstract: SUMMARY as k -* oo, is studied. The related conditional Poisson process discussed recently by D. G. Kendall and W. A. O'N. Waugh is also studied, and the results are shown to illustrate the modern theory of exponential families and conditional inference.
TL;DR: In this paper, the multiplicative and additive definitions of nointeraction in contingency tables are compared according to whether they possess or fail to possess the properties of being partitionable, closest to independence, implied by independence, amalgamation invariant, subtable invariant and placing no constraints on the marginal probabilities.
Abstract: SUMMARY The 'multiplicative' and 'additive' definitions of no-interaction in contingency tables are compared according to whether they possess or fail to possess the properties of being partitionable, closest to independence, implied by independence, amalgamation invariant, subtable invariant and of placing no constraints on the marginal probabilities. It is shown that both definitions fall short of the ideal. The author believes that the multiplicative definition is preferable by a small margin.
TL;DR: This paper proposed a general regression model for the analysis of ordered categorical data which is based upon an extension of the log odds transformation, and some simple odds ratio statistics are developed to summarize the difference in location between two distributions of a categorical variable.
Abstract: SUMMARY Snell (1 964) has proposed a general regression model for the analysis of ordered categorical data which is based upon an extension of the log odds transformation. With this model, some simple odds ratio statistics are developed to summarize the difference in location between two distributions of an ordered categorical variable. Similar statistics for describing association between two such variables are also discussed.
TL;DR: In this paper, an estimate of the difference of means is obtained when sampling from a bivariate normal distribution with variances o-2 and o- and correlation p, where some observations on either of the variables are missing.
Abstract: SUMMARY An estimate of the difference of means is obtained when sampling from a bivariate normal distribution with variances o-2 and o- and correlation p, where some observations on either of the variables are missing. It is shown that this estimate has desirable properties. In this paper a test of the hypothesis of the equality of means is also considered. The above estimate is adopted and three new statistics based on the difference of sample means are proposed for the test. Their empirical powers are computed for different values of p and aw2/o-2
TL;DR: In this paper, the authors used the maximum likelihood and minimum variance unbiased estimator for the variance of the estimator of the parameter and the mean of the offspring distribution in a branching process.
Abstract: SUMMARY If the offspring distribution in a branching process is a power series distribution, then, conditional on extinction, the total number of descendants for the process also has a power series distribution. For this case the estimation of the parameter and of the mean of the offspring distribution is considered, using the method of maximum likelihood and minimum variance unbiasedness. The minimum variance unbiased estimator for the variance of the estimator of the parameter is also given. The results are applied to smallpox datafor Europe.
TL;DR: In this article, the authors apply the jackknife method to stratified samples from a multivariate population of finite size, where the data omitted are those for a group of sampling units that cut across all strata, thereby compacting the region where approximate relationships are assumed to hold.
Abstract: SUMMARY The jackknife method of investigating and reducing the bias in nonlinear estimates of parameters is applied to stratified samples from a multivariate population of finite size. Attention is focused on estimators that can be expressed as functions of sample means. As in other applications of the jackknife method, an approximate relationship between the bias in an estimator and the sample size is exploited to reduce this bias by employing a linear combination of an estimate computed from all the data and several estimates each computed after omitting part of the data. Unlike some other applications to stratified sampling, however, where the data omitted are those for a group of sampling units that cut across all strata, the present application involves omitting just one sampling unit at a time, thereby compacting the region where approximate relationships are assumed to hold, and increasing the stability of the variance estimates. The relationships are derived by an analytic approach. Results are carried out far enough for use in second-order jackknife estimates of parameters and variances so as to eliminate bias to third-order moments of the variables observed. When there is just one stratum, the first-order estimators defined by equations (4.3) and (6-1), which are unbiased to second-order moments, are asymptotically the same -as those proposed by Tukey (1958).
TL;DR: In this article, the relative efficiency of the rank analysis arising out of Cox's regression and life model versus the analogous analysis in the exponential case is examined, and it is found to compare favourably with the more model-dependent analysis in most cases of practical importance.
Abstract: Survival models which incorporate a regression variable are considered with a view to estimating the regression parameters. The relative efficiency of the rank analysis arising out of Cox's regression and life model versus the analogous analysis in the exponential case is examined. The rank analysis is found to compare favourably with the more model-dependent analysis in most cases of practical importance.
TL;DR: In this paper, approximate maximum likelihood estimators have been obtained for the normal and gamma distributions and their efficiencies compared to those for the best linear unbiased estimators for these distributions.
Abstract: SUMMARY Approximate maximum likelihood estimators have been obtained for the normal and gamma distributions and their efficiencies compared to those for the best linear unbiased estimators for these distributions.
TL;DR: A statistical analysis is given of the mutant accumulation method of estimating mutation rates, and Phenotypic delay, that is delay from the epoch at which a cell mutates until cells in resulting clone possess the property which distinguishes them from nonmutant cells, is determined.
Abstract: SUMMARY A number of mathematical models for cell populations in which mutations are occurring are studied and compared by viewing the models as filtered Poisson processes. Based upon these models, a statistical analysis is given of the mutant accumulation method of estimating mutation rates. Phenotypic delay, that is delay from the epoch at which a cell mutates until cells in resulting clone possess the property which distinguishes them from nonmutant cells, is incorporated into the models and its effect upon some estimation procedures is determined. A suggestion is also given for obtaining information about the duration of phenotypic delay.
TL;DR: In this article, the problem of estimating parameters when controllable variables are subject to errors is studied and strongly consistent estimators with an asymptotically normal distribution are proposed.
Abstract: SUMMARY The problem is studied of estimating parameters when controllable variables are subject to errors. Strongly consistent estimators with an asymptotically normal distribution are proposed and an iterative procedure for their calculation is suggested. A numerical example is given in illustration.
TL;DR: In this paper, a decision-theoretical model for allocation problems admitting cases of doubt is developed for multinormal distributions, which is illustrated with an example from medical dif- ferential diagnosis.
Abstract: SUMMARY Allocation problems can involve elements whose original population is not conclusively indicated by the observation. These elements are called 'cases of doubt'. In this paper a decision-theoretical model for allocation problems admitting cases of doubt is developed. The sampling situation is discussed both from a distribution-free point of view and for multinormal distributions. The theory is illustrated with an example from medical dif- ferential diagnosis. Consider a population A of elements that can be split into two populations A1 and A2, such that A = A1 U A2 and Al n A2 = 0. The frequency of occurrence of the two populations will be specified in the a priori probabilities nr(A1) and Ir(A2). For each element we measure a p-dimensional vector variable x = (xl, ..., xp)'. The random variation of x in the two popula- tions will be described by the distribution functions F(xJAj) and F(xJA2). Suppose that we have an element randomly selected from A, but we do not know whether it comes from A1 or A2. The allocation problem we now consider is how to reach a decision about the origin of this element on the basis of its observation x. There is an extensive literature on this problem if one is forced to allocate the element to either A1 or A2. If one is not forced to allocate to either A1 or A2, one would like to introduce for inconclusive observations an 'allocation of doubt'. In this situation, for every possible observation x, we must choose one of three allocations:
TL;DR: In this article, three types of estimator of the second spectral moment of a stationary Gaussian process are considered: integral estimator, crossing estimator and nonzero level estimator.
Abstract: Three types of estimator of the second spectral moment of a stationary Gaussian process are considered. The integral estimator is based on the integral of the squared derivative of the process, while crossing estimators make use of the number of upcrossings of zero or nonzero levels. It is shown that the zero-crossing estimator can often compete with the integral estimator in efficiency and that it can be considerably improved by the additional use of nonzero levels.
TL;DR: In this paper, a design procedure involving the minimization of the generalized variance of estimates of parameters was adopted involving a changeover procedure for four treatments due to Williams (1949), in which every treatment follows every other treatment, was found to minimize the generalized variances for randomized block and Latin square arrangements for both positive and negative firstorder autocorrelative alternatives to the null hypothesis of independent errors.
Abstract: SUMMARY Design problems are considered in the context of possible autocorrelations between observations. A design procedure is adopted involving the minimization of the generalized variance of estimates of parameters. The changeover designs for four treatments due to Williams (1949), in which every treatment follows every other treatment, are found to minimize the generalized variances for randomized block and Latin square arrangements for both positive and negative first-order autocorrelative alternatives to the null hypothesis of independent errors. An explicit design is produced for the optimum settings of quantitative factors, under reasonably mild restrictions, in block designs and time sequences. The method is compared with that proposed by Box & Guttman (1966). Two particular cases are studied in detail.
Abstract: SUMMARY The problem of comparing several ordered dose levels with a control when a larger sample size is taken on the control is considered. The distributions of Bartholomew's tests are determined for the limiting case where the control mean is known and an approximation is given for the problem. The existing tables for Bartholomew's tests are extended. It is considered that these tests are superior in all situations where the sample size for the control is greater than the sample sizes for the nonzero dose levels.
TL;DR: In this article, the maximum likelihood estimator of an exponential polynomial rate function has moments equal to the corresponding sums of powers of the observed event times, and a goodness-of-fit test is derived from the relation between sums of power of event times and moments of the rate function.
Abstract: SUMMARY This paper presents a numerical method of statistical inference which overcomes some mathematical difficulties encountered in the nonstationary Poisson process by taking full advantage of modern computing equipment. The maximum likelihood estimator of an exponential polynomial rate function has moments equal to the corresponding sums of powers of the observed event times. A numerical determination of this function is demonstrated. The information matrix, a simple function of the moments of the rate function, can also be estimated by the sums of powers. Finally, a goodness-of-fit test is derived from the relation between sums of powers of event times and moments of the rate function. Computer programs which perform all the necessary calculations have been prepared and are available from the author.