Showing papers in "Biometrika in 1981"

Nonparametric estimates of standard error: The jackknife, the bootstrap and other methods

Abstract: SUMMARY We discuss several nonparametric methods for attaching a standard error to a point estimate: the jackknife, the bootstrap, half-sampling, subsampling, balanced repeated replications, the infinitesimal jackknife, influence function techniques and the delta method. The last three methods are shown to be identical. All the methods derive from the same basic idea, which is also the idea underlying the common parametric methods. Extended numerical comparisons are made for the special case of the correlation coefficient.

On the regression analysis of multivariate failure time data

Abstract: SUMMARY The paper is concerned with the analysis of regression effects when individual study subjects may experience multiple failures. The proposed methods are most likely to be useful when there are a fairly large number of study subjects. Two general classes of regression models are considered in order to relate the hazard or intensity function to covariates and to preceding failure time history. Both models are of a stratified proportional hazards type. One model includes baseline intensity functions that are arbitrary as a function of time from the beginning of study, while the other includes baseline intensity functions that are arbitrary as a function of time from the study subject's immediately preceding failure. Partial likelihood functions are derived for the regression coefficients. Generalizations and illustrations are given.

An optimal selection of regression variables

Abstract: SUMMARY An asymptotically optimal selection of regression variables is proposed. The key assumption is that the number of control variables is infinite or increases with the sample size. It is also shown that Mallows's Qp, Akaike's FPE -and AIC methods are all asymptotically equivalent to this method.

Some matrix-variate distribution theory: Notational considerations and a Bayesian application

Abstract: SUMMARY We introduce and justify a convenient notation for certain matrix-variate distributions which, by its emphasis on the important underlying parameters, and the theory on which it is based, eases greatly the task of manipulating such distributions. Important examples include the matrix-variate normal, t, F and beta, and the Wishart and inverse Wishart distributions. The theory is applied to compound matrix distributions and to Bayesian prediction in the multivariate linear model.

Modelling nonlinear random vibrations using an amplitude-dependent autoregressive time series model

Abstract: SUMMARY The behaviour of nonlinear deterministic vibrations has been studied by many authors, and may typically include such features as jump phenomena and limit cycles. Nonlinear random vibrations in continuous time have also been studied and these may commonly give rise to the phenomenon of amplitude-dependent frequency. A discrete time series model is introduced, which may be demonstrated to have properties similar to those of nonlinear random vibrations. This model is of autoregressive form with amplitudedependent coefficients and may be estimated using an extension of a method for estimating linear time series models. The model is fitted to the Canadian lynx data and demonstrates that it may be possible to regard the periodic behaviour of this series as being generated by some underlying self-exciting mechanism.

The asymptotic properties of nonparametric tests for comparing survival distributions

Abstract: The asymptotic distribution under alternative hypotheses is derived for a class of statistics used to test the equality of two survival distributions in the presence of arbitrary, and possibly unequal, right censoring. The test statistics include equivalents to the log rank statistic, the modified Wilcoxon statistic and the class of rank invariant test procedures introduced by Peto & Peto. When there are equal censoring distributions and the hazard functions are proportional the sample size formula for the F test used to compare exponential samples is shown to be valid for the log rank test. In certain situations the power of the log rank test falls as the amount of censoring decreases.

A note on the estimation of a distribution function and quantiles by a kernel method

Abstract: There has recently been extensive work on the estimation by kernel methods of probability densities and their derivatives; for a review, see Wertz (1978). For estimating the cumulative distribution function, the empirical distribution function, F,, is in a sense already quite smooth, although it is known that further smoothing can be an advantage. The object of the present note is to report briefly an investigation of kernel estimation applied to distribution function, and quantiles. Further details are available from the author. For independent and identically distributed random variables X1, ...,X the usual kernel estimator of the density f (.) is

On two families of transformations to additivity for binary response data

Abstract: SUMMARY Two families of power transformations for probabilities are introduced to model symmetric and asymmetric departures from the logistic model. The aim is to provide, in a comprehensive way, a representation of alternative scales in terms of which to carry out the analysis of binary response data. Maximum likelihood methods are employed to estimate an appropriate transformation using GLIM. Tests for detection of symmetric and asymmetric departures are proposed.

Two graphical displays for outlying and influential observations in regression

Abstract: SUMMARY The paper describes two procedures for detecting observations with outlying values either in the response variable or in the explanatory variables in multiple regression. These procedures are presented as half normal plots with envelopes derived from simulation in order to avoid overinterpretation of the data. Analysis of a well-known data set leads to the use of a data transformation, a simple test for which is commended, and to some comments on the relationship with robust regression. The widespread availability of sophisticated computer software has made the fitting of multiple regression equations painless, but the very ease of these procedures may cause insufficient care to be given to checking and scrutiny of the data. Transcription, punching and data manipulation errors may lead to bad values both of the observations y and of the independent or explanatory variables x. It is the purpose of the present paper to describe and exemplify two plots which provide checks against such bad values. It is argued that these plots and a test for transformations should accompany any thorough regression analysis. Bad values of the response, or outliers, have long been detected by a variety of plots of residuals. Bad values of the explanatory variables are less easily detected even when they lead to an extreme point in the design space. Since the fitted equation will pass close to such an influential observation the residual may not be especially large even after allowance has been made for the small variance of the fitted value at such a point. To detect such behaviour a quantity is needed which exhibits the dependence of the fitted model on each point or group of points. Discussions of such quantities are given by Hoaglin & Welsch (1978) and, more recently, by Belsley, Kuh & Welsch (1980, ? 2 1), by Cook & Weisberg (1980) and by Pregibon (1981). The quantity used here is a modification of a statistic proposed by Cook (1977), which is derived in ? 2. Two half normal plots are presented in ? 3. As an aid to interpretation envelopes to the plots are generated by simulation. Examples of the plots are given in ? 4, both for simulated data and for the stack loss data given by Brownlee (1965, p. 454). Transformations of this data set are discussed in ? 5. The paper concludes with some comments on the relationship with robust

A class of smooth estimators for discrete distributions

Abstract: SUMMARY This paper presents a class of smooth weight function estimators for discrete distributions. Any estimator in the class depends on choosing a parameterized set of weights. The resulting estimators are strongly consistent and asymptotically normal under mild regularity conditions. A general procedure for choosing the weight function smoothing parameter is given along with specific solutions in some cases. Mean squared error comparisons with the maximum likelihood estimator based on large-sample theory and small-sample simulations are obtained. Typically, the weight function estimates yield significantly smaller mean squared error in these comparisons.

Likelihood calculations for matched case-control studies and survival studies with tied death times

Abstract: SUMMARY A recursion is presented which permits rapid computation of a conditional likelihood function which arises in matched case-control studies and survival studies with tied death times. An example with seven parameters and ten large strata, the largest of which contains 29 cases and 52 controls, illustrates the feasibility of the conditional maximum likelihood solution. These methods may be specialized to permit rapid calculation of the conditional maximum likelihood estimate of the common odds ratio for several 2 x 2 tables.

The asymptotic joint distribution of the efficient scores test for the proportional hazards model calculated over time

Abstract: The log rank test derived by Mantel (1966) and Peto (1972) is a special case of the efficient scores test for the proportional hazards model. It is used frequently in the analysis of survival duration in clinical trials where patients entering the study are assigned at random to different treatments and are followed until they either fail or the study is terminated. Most survival studies are designed so that the log rank test is calculated at the termination of the study; however it is not unusual for the investigator to monitor the experiment periodically. It would be very useful to know the joint distribution of the test statistic at different points in time in order to adjust for the effect of repeated significance testing. Although the joint distribution has not been derived previously, sequential procedures for the analysis of survival data in clinical trials have been given by several authors including Armitage (1975, pp. 140-6), Jones & Whitehead (1979) and Nagelkerke & Hart (1980). In this paper, we establish the asymptotic joint distribution of the more general efficient scores test for the proportional hazards model calculated at different points in time, and indicate how this can be used for repeated significance testing.

Estimating the median survival time

Abstract: SUMMARY Two methods are proposed for setting confidence intervals for the median survival time. It is shown how these are related to bootstrap methods for censored data, and some numerical comparisons are presented. In many medical follow-up studies, the survival experience of a group of patients is summarized by the product-limit, or Kaplan-Meier, estimate of the survival distribution. This may follow an analysis of the relation between survival time and some explanatory variables, by, for example, the proportional hazards model. The median of the estimated survival distribution is sometimes a convenient summary statistic, and may be preferred to the mean, because it is less sensitive to long tails in the estimated survival distribution. This paper is concerned with estimating the distribution of the estimated median, and providing confidence intervals for the true median. Two methods are considered in ? 2; they are both based, in different ways, on generalizations of the solution for the uncensored case. These methods turn out to be connected to two ways of extending bootstrap methods for use with censored data; this connexion is discussed in ? 3. Thinking about extensions of the bootstrap to censored data highlights the need for careful consideration of the role of the censoring mechanism and how it should be incorporated into the data analysis. Section 4 provides some examples and Monte Carlo simulations. The work is closely related to that of Efron (1981) and Emerson (1981).

Association models and the bivariate normal for contingency tables with ordered categories

Abstract: SUMMARY Association models considered by Goodman (1979) for contingency tables with ordered categories are presented here in a somewhat different form to facilitate comparison with the bivariate normal. Association models can be applied when an underlying bivariate normal is assumed and also under more general conditions, and they provide alternatives to tetrachoric and polychoric correlation. These models agree more closely with the bivariate normal than does Plackett's model (1965), when the row and column classifications arise from underlying univariate normal distributions. The general utility of the association models is illustrated here.

On prediction and the power transformation family

Abstract: SUMMARY The power transformation family is often used for transforming to a normal linear model The variance of the regression parameter estimators can be much larger when the transformation parameter is unknown and must be estimated, compared to when the transformation parameter is known We consider prediction of future untransformed observations when the data can be transformed to a linear model When the transformation must be estimated, the prediction error is not much larger than when the parameter is known

Nonparametric bivariate estimation with randomly censored data

Abstract: SUMMARY The estimation of a bivariate distribution function with randomly censored data is considered. It is assumed that the censoring occurs independently of the lifetimes, and that deaths and losses which occur simultaneously can be separated. Two estimators are developed: a reduced-sample estimator and a self-consistent one. It is shown that the latter estimator satisfies a nonparametric likelihood function and is unique up to the final uncensored values in any dimension; it jumps at the points of double deaths in both dimensions. Some key word8: Censored data; Kaplan-Meier estimator; Life table; Product-limit estimator; Survival estimation.

The two-dimensional hyperbolic distribution and related distributions, with an application to Johannsen's bean data

Abstract: SUMMARY The contours of equal density of the two-dimensional hyperbolic distribution indicate that this distribution is capable of describing a very specific and simple form for departure from the two-dimensional normal distribution. One of the classical examples of twodimensional data showing nonnormnal variation is Johannsen's bean data. After a brief discussion of elementary properties of the two-dimensional hyperbolic distribution, the possibility of fitting this distribution to a set of data, obtained from Johannsen's data by a simple transformation, is considered. The two-dimensional hyperbolic distribution belongs to the class of generalized hyperbolic distributions which is shown to be closed under the formation of marginal distributions, under conditioning and under affine transformation.

A class of nonparametric tests based on multiresponse permutation procedures

Abstract: SUMMARY A class of univariate rank tests based on multiresponse permutation procedures is introduced. Asymptotic equivalence is established between one member of this class and the Kruskal-Wallis test. Simulated power comparisons for location shift detection indicate that another member of the class provides superior detection efficiency for location shifts of bimodal distributions and of heavy tailed unimodal distributions. Furthermore, both tests possess substantial computational advantages over other members of the class.

Estimation of the average hazard ratio

Abstract: SUMMARY Simple estimators of the average hazard ratio are obtained for the two sample problem with censored failure time data These estimators are compared with estimators arising out of the partial likelihood within the proportional hazards class Efficiency results are found to be generally quite favourable provided the hazard ratio is not too large Some comments are made on extensions to the k sample problem

Inhomogeneous and modulated gamma processes

Abstract: SUMMARY Two classes of point processes, the inhomogeneous and modulated gamma processes, are defined. Asymptotic test statistics of stationarity are derived for modulated gamma processes. Some examples are studied in detail and applied to two sets of data.

A Bayesian analysis of multiple-recapture sampling for a closed population

Abstract: SUMMARY This paper considers from a Bayesian viewpoint inferences about the size of a closed animal population from data obtained by a multiple-recapture sampling scheme. The method developed enables prior information about the population size and the catch probabilities to be utilized to produce considerable improvements in certain cases on ordinary maximum likelihood methods. Several ways of expressing such prior information are explored and a practical example of the uses of these ways is given. The main result of the paper is an approximation to the posterior distribution of sample size that exhibits the contributions made by the likelihood and the prior ideas. The multiple-recapture sampling scheme involves taking samples from a population of animals, at each stage counting the number of marked animals in the sample, marking the previously unmarked animals and returning the sample to the population. The literature is reviewed by Cormack (1968) and the papers most relevant to our problem are those of Chapman (1952) and Darroch (1958). In these papers, maximum likelihood estimates for the population parameters and variances for these estimates are given. Our approach differs from previous ones by introducing prior information about the population size, and about the propensities of the animals to be captured, called 'catch probabilities' by some previous authors. The incorporation of these propensities is a feature not shared by many previous approaches. Darroch (1958) found their introduction into his model does not change the maximum likelihood estimates of the population size. Our approach will show that strong prior knowledge about the catch probabilities can greatly affect inference about the population size, and it is in this respect that the greatest difference from previous approaches lies. Although it is recognized that models of an open population incorporating death and immigration such as those of Jolly (1965) and Seber (1965) are more practically realistic, we feel that our method, which deals only with a closed population, is worth investigating as a step towards providing a Bayesian treatment of the open population problem.

A class of estimators of the population mean in survey sampling using auxiliary information

Abstract: SUMMARY For estimating the mean of a finite population using information on an auxiliary variable, a class of estimators is defined as a function of the ratio of sample mean to population mean and the ratio of sample variance to population variance of the auxiliary variable. Asymptotic expressions for bias and mean squared error are obtained. A condition is given for the minimum mean squared error of estimators of this class to be smaller than those which use only the ratio of sample mean to population mean of the auxiliary variable.

Nonparametric analysis of an accelerated failure time model

Abstract: SUMMARY Survival distributions can be characterized by and compared through their hazard functions. Tests using a proportional hazards model have good power if the two hazards do not cross, but without time-dependent covariates can have low power if they do. The method contained herein is designed to provide a complement to the proportional hazards model. Differences in survival distributions are parameterized by a scale change and the log rank statistic is used to generate an estimate of the change and a confidence interval based test. This approach is fully efficient for the Weibull family with no censoring. In general it compares favourably with the proportional hazards approach, although neither method dominates.

On the Koziol-Green model for random censorship

Abstract: SUMMARY In the Koziol-Green (1976) model of random censorship the survival distribution of the censoring variables is some power, the censoring parameter, of the survival distribution of the lifetimes. Using a strong approximation result of Burke, Csorgo and Horvath, we construct empirical exact confidence bands for the life distribution of the censored variable, and a randomly Efron-transformed variant of the usual product-limit process is shown to converge weakly to the Brownian motion process on any interval [0, T] when the censoring parameter is also estimated from the sample. The technique automatically gives rates of convergence for a wide class of functionals including the Cramer-von Mises functional whose limit theory is worked out in detail. The latter goodness-of-fit statistic is applied to reexamine the hospital data of Koziol & Green (1976) to test whether oestrogen treatment for prostatic cancer was effective or not for the cancer itself while it caused censoring deaths of cardiovascular diseases.

The graphical evaluation of explanatory variables in proportional hazard regression models

Abstract: SUMMARY A graphical technique for assessing explanatory variables in Cox's proportional hazards regression model is examined. The method arises from consideration of the cumulative hazard transformation and the partial likelihood score function, and consists of permuting the observed rank statistic to account for the effects of explanatory variables fitted to the model. An example is provided.

The Fisher information matrix for log linear models arguing conditionally on observed explanatory variable

Abstract: SUMMARY For Poisson or multinomial contingency table data the conditional distribution is product multinomial when conditioning on observed values of explanatory variables. Birch (1963) showed that under the restriction formed by keeping the marginal totals of one margin fixed at their observed values the Poisson, multinomial and product multinomial likelihoods are proportional and give the same estimates for common parameters in the log linear model. Here the inverses of the Fisher information matrices are shown to be identical over common parameters so that the asymptotic covariance matrices of the estimates correspond.

Estimation of survival in marked populations with possibly dependent sighting probabilities

Abstract: SUMMARY A simply parameterized class of models for the estimation of survival probability in a marked population is introduced which allows tests of dependence between sighting probabilities and constancy of survival or sighting probabilities. Estimation and hypothesis testing based on the likelihood function are discussed. By simulation, the asymptotic properties of the test for dependence are shown to hold for quite modest sample sizes. The degree of bias in estimation of survival probabilities when dependence between sightings is not taken into account is shown to increase as the probability of a nonsighting following a nonsighting increases. A comprehensive analysis of the data from a mark-recapture experiment dealing with survival of abalone is given.

On nonparametric multivariate binary discrimination

Abstract: Aitchison & Aitken (1976) introduced a novel and ingenious nonparametric method for estimating probabilities in a multidimensional binary space. The technique is designed for use in multivariate binary discrimination. Their estimator depends crucially on an unknown smoothing parameter A, and Aitchison & Aitken proposed a maximum likelihood method for determining A from the sample. Unfortunately this leads to an adaptive estimator which can behave very erratically when there are a number of empty or near empty cells present. We demonstrate this both theoretically and by example. To overcome these difficulties we introduce another method of estimating A which is designed to minimize a global function of the mean squared error.

Normalizing transformations of some statistics in multivariate analysis

Abstract: SUMMARY A general procedure for finding normalizing transformations is applied to various statistics in multivariate analysis. It is shown that Fisher's z transformation for a sample correlation coefficient in a normal sample and Wilson & Hilferty's approximation for a chi-squared variate can be derived by the same line of approach.