Showing papers in "Biometrics in 1984"

Procedures for Comparing Samples with Multiple Endpoints

Abstract: SUMMARY Five procedures are considered for the comparison of two or more multivariate samples. These procedures include a newly proposed nonparametric rank-sum test and a generalized least squares test. Also considered are the following tests: ordinary least squares, Hotelling's P, and a Bonferroni per-experiment error-rate approach. Applications are envisaged in which each variable represents a qualitatively different measure of response to treatment. The null hypothesis of no treatment difference is tested with power directed towards alternatives in which at least one treatment is uniformly better than the others. In all simulations the nonparametric procedure provided relatively good power and accurate control over the size of the test, and is recommended for general use. Alternatively, the generalized least squares procedure may also be useful with normally distributed data in moderate or large samples. A convenient expression for this procedure is obtained and its asymptotic relative efficiency with respect to the ordinary least squares test is evaluated. Clinical trials are often conducted for the purpose of evaluating the relative efficacy of two or more modes of therapy. When the therapies are administered to independent groups and efficacy is measured on the basis of a single response variable, appropriate parametric and nonparametric one-way analysis-of-variance (ANOVA) procedures and their properties are well-known. Often, however, efficacy is measured by more than one variable. Although univariate methods for assessing each characteristic individually are useful in this setting, there is often the additional need for a single, overall, objective probability statement that addresses the question of whether or not the experimental therapy is efficacious. This need is particularly acute when the medical question is controversial and the sample size is small. Although a large literature exists on the comparison of multivariate samples, the procedures tend to be infrequently used in practice. As pointed out by Meier (1975, p. 523), the standard analysis for the comparison of two multivariate samples, which is based on Hotelling's T2 statistic, addresses somewhat the wrong question and consequently has very poor power for the alternatives of primary interest. Specifically, the T2 procedure addresses the question 'Are one or more of the treatments different?', making no distinction between variables that change favorably and variables that change unfavorably. A second approach would be to assign per-experiment error rates to each of the univariate tests by using Bonferroni's inequality (i.e. by multiplying each univariate P-value by the number of variables studied). The desired overall probability statement is then given by the minimum of the per-experiment univariate P-values. Although this approach may be useful in some settings, it may lack power for alternatives in which most or all measures of efficacy are improved. This will be of particular concern when the number, K, of endpoints studied is large relative to sample size. In fact, statistical significance will not be possible in some instances. This problem will be exacerbated when the measures of efficacy are highly

Timber Management-A Quantitative Approach.

Abstract: GROWTH AND YIELD PREDICTION. Estimating the Volumes and Weights of Individual Trees. Evaluating Site Quality. Growing Stock and Stand Density. Predicting Growth and Yield. FINANCIAL ASPECTS OF TIMBER MANAGEMENT. Forest Finance. Taxes and Risk in the Evaluation of Forest Investments. TIMBER MANAGEMENT PLANNING. Timber Management - Some Introductory Comments. Stand-Level Management Planning. Forest-Level Management Planning: Basic Concepts. Forest-Level Management Planning: Current Techniques. Appendices. Index.

Random-effects models for serial observations with binary response

Abstract: This paper presents a general mixed model for the analysis of serial dichotomous responses provided by a panel of study participants. Each subject's serial responses are assumed to arise from a logistic model, but with regression coefficients that vary between subjects. The logistic regression parameters are assumed to be normally distributed in the population. Inference is based upon maximum likelihood estimation of fixed effects and variance components, and empirical Bayes estimation of random effects. Exact solutions are analytically and computationally infeasible, but an approximation based on the mode of the posterior distribution of the random parameters is proposed, and is implemented by means of the EM algorithm. This approximate method is compared with a simpler two-step method proposed by Korn and Whittemore (1979, Biometrics 35, 795-804), using data from a panel study of asthmatics originally described in that paper. One advantage of the estimation strategy described here is the ability to use all of the data, including that from subjects with insufficient data to permit fitting of a separate logistic regression model, as required by the Korn and Whittemore method. However, the new method is computationally intensive.

Nonparametric estimation of species richness

Abstract: Two procedures, the jackknife and the bootstrap, are discussed as methods for estimating the number of species by the sampling of quadrats. Explicit formulas for both procedures are presented and evaluated under a model with a random distribution of individuals. The jackknife and bootstrap are shown to reduce the bias although they underestimate the actual number of species if there is a large number of rare species and the number of quadrats sampled is small. When a small number of quadrats is sampled, the jackknife is shown to give better estimates. When the number of quadrats is large, the jackknife tends to overestimate the number of species and the bootstrap performs better.

A mixed-model procedure for analyzing ordered categorical data

Abstract: A mixed-model procedure is developed for predicting the value of an ordered categorical response from knowledge of various predictor variables. This procedure resembles the best linear unbiased procedure of Henderson (1975, Biometrics 31, 423-447) for predicting the value of a quantitative response. The approach is based on a mixed-model version of the threshold model in which it is assumed that the observed category is determined by the value of an underlying unobservable continuous response that follows a mixed linear model. The results are illustrated by an application to the problem of predicting the degree of difficulty that will be experienced by a dairy cow in the birth of her calf.

Biased selection of controls for case-control analyses of cohort studies.

Abstract: It is known that unbiased estimates of the relative risk in a cohort study may be obtained by a matched case-control analysis that compares each case with a random sample of controls obtained from those at risk at the time of case incidence. Through inadvertence , or for practical or scientific reasons, a biased referent group may be selected instead. Three kinds of biasing restrictions on controls are commonly imposed: (i) the requirement that controls remain completely disease-free for a fixed time interval, (ii) the exclusion of all cases incident during observation as controls, and (iii) the exclusion, from the referent group, of subjects who develop other diseases, which may be related to the exposure of interest. The bias in estimation of the relative risk associated with each of these restrictions is evaluated under the proportional-hazards model. For several examples of cancer mortality data, the bias from (iii) appears quite small, whereas the bias from (i) can be appreciable and is mostly attributable to the bias from case exclusion (ii). The effect of random variation in the time of onset of exposure is to reduce these biases.

Exact significance testing to establish treatment equivalence with ordered categorical data.

Abstract: This communication concerns the problem of establishing the therapeutic equivalence of two treatments that are being compared on the basis of ordered categorical data. The problem is formulated as a significance test in which the null hypothesis specifies a treatment difference. An efficient numerical algorithm for computing the exact significance level is provided, along with a simple method for obtaining the asymptotic significance level. Both methods are applied to a clinical trial of a new agent versus an active control. Guidelines for when to use the exact procedure and when to rely on asymptotic theory are provided.

Simplified models of screening for chronic disease: estimation procedures from mass screening programmes.

Abstract: The performance of a screening programme depends on a number of parameters, which may need different types of information for their estimation. In this paper, two characteristics of interest, the sojourn-time distribution and the sensitivity, are considered, and estimation procedures, based on data which should be available from a mass screening programme, are proposed. The methods are applied to published data from the HIP breast cancer screening project, to which a negative exponential distribution for the sojourn-time distribution fitted better than the other families of distribution considered.

Multivariate methods in ophthalmology with application to other paired-data situations.

Abstract: Methods are presented for performing multiple regression analyses and multiple logistic regression analyses on ophthalmologic data with normally and binomially distributed outcome variables, while accounting for the intraclass correlation between eyes. These methods are extended to more general nested data structures where a variable number of subunits are available for each primary unit of analysis, as in familial data. These methods can also be applied to other types of paired data, as in matched studies with a variable matching ratio, where one has a continuous outcome variable and wishes to control for other confounding variables while maintaining the matching. Examples are given of these methods with a group of over 400 patients with retinitis pigmentosa, in which spherical refractive error and visual acuity are related to genetic type after the effects of age, sex and the presence of cataract, have been controlled.

Monte Carlo confidence intervals

Abstract: Des intervalles de confiance non parametriques obtenus par la methode de Monte-Carlo sont utiles quand les intervalles obtenus par les methodes analytiques sont indisponibles ou ne correspondent pas au niveau nominal. On fournit des formules permettant de generer un intervalle de niveau donne et de quantifier la variation due a la methode de Monte-Carlo

Confidence intervals for the ratio of two binomial proportions

Abstract: Sometimes the ratio of two binomial proportions is the parameter of major interest, for instance as the risk ratio in a two-group cohort study (Fleiss, 1973) or as the likelihood ratio for a diagnostic test procedure (McNeill, Keeler and Adelstein, 1975). Let X and Y be independent binomial variates based on sample sizes m and n and parameters p' and P2, respectively. Let 0 = Pm/P2. We consider the construction of confidence intervals for 0. Santner and Snell (1980) have proposed a nonanalytical procedure which guarantees a lower bound on the confidence coefficient of the 1 a interval for 0. Their algorithm is usable for small m and n but may produce conservative intervals. In ?2, an analytical method for obtaining approximate confidence intervals for 0 is proposed and compared to the one recommended by Katz et al. (1978).

Exact confidence intervals following a group sequential test.

Abstract: A numerical method is used to compute confidence intervals, which have exact coverage probabilities, for the mean of a normal distribution following a group sequential test This method, which uses an ordering of the sample space similar to that employed by Siegmund (1978, Biometrika 65, 341-349), is contrasted with the usual confidence interval for the mean

Covariances of relatives stemming from a population undergoing mixed self and random mating.

Abstract: We consider covariances of all parent and first-generation relatives from outcrossing or self-fertilization in a parent population that is in equilibrium with respect to these processes. The results, which are for any number of alleles and loci with additive and dominance effects, are phrased in terms of six quadratic genetic components whose coefficients are given by descent measures for equilibrium populations. Because of the variation in the inbreeding coefficients for this system of mating, the expressions include joint contributions of loci to the variances and covariances of relatives. By inclusion of the full complement of relatives, all quadratic components can be estimated. The findings of Ghai (1982, Biometrics 38, 87-92) for compound functions of the covariances with two alleles at a single locus are analyzed in terms of the more general model.

Properties of proportional-hazards score tests under misspecified regression models

Abstract: The effects are investigated of misspecifying a proportional-hazards regression model on the associated partial-likelihood score test for comparing two randomized treatments in the presence of covariates The asymptotic efficiency of the proportional-hazards score test, relative to the optimal partial-likelihood test, declines slowly as the hazard functions for the two treatments deviate from proportionality; the efficiency can be very low when the hazard functions cross or differ only at large survival times Misspecification of the functional form of the regression portion of a proportional-hazards model introduces a quantitative treatment-covariate interaction In the situations that we examine, based on a binary covariate, this misspecification usually results in only a minor drop in efficiency The omission of a covariate that is balanced across treatments has a negligible effect on the size of the score test, but can substantially reduce power when the covariate effect is strong The loss of power from mismodeling a balanced covariate is usually small

A cautionary example on the use of second-order methods for analyzing point patterns

Abstract: A popular approach to the analysis of mapped planar point patterns is through secondorder methods, where the object of attention is the variance of the number of points falling in a test set of a given size and shape, or the behaviour of all distances between pairs of points in the pattern. Standard references for the analysis of spatial data are works by Ripley (1981, Ch. 8) and Diggle (1983). Many of the methods for point-pattern analysis given in the literature are second-order in nature. The analysis of mapped point patterns is important in a variety of biological applications, ranging from ecology to physiology, and some examples are given in these books by Ripley and by Diggle. Suppose that an observed point pattern can be considered as a realization of a random point-process model which is stationary and isotropic. Of course this is a strong assumption which may not be justified in practice, but it is nevertheless the assumption under which much of the existing point-process methodology has been developed. The property of the underlying random process elicited, directly or indirectly, by second-order methods is the second-moment cumulative function K(t), which satisfies the following properties under suitable regularity conditions (see Ripley, 1977, p. 150): (i) if X is the intensity of the process, then XK(t) is the expected number of further points within distance t of a typical point of the process; (ii) specifying K(t) for all t is equivalent to specifying the variance of the number of points falling in any given set. The function K(t) defined by Ripley (1977) is an edge-corrected version of the empirical distribution function of all interpoint distances in the observed pattern, and provides an approximately unbiased estimator of K(t). The estimate K(t) can be used to construct tests of the hypothesis that an observed pattern is consistent with the 'completely random' Poisson point-process model, and, more importantly, to quantify the apparent deviation

Difficulties with Regression Analyses of Age-Adjusted Rates

Abstract: A common type of observational study compares population rates in several regions having differing policies in an effort to assess the effects of those policies. In many studies, particularly in public health and epidemiology, age-adjusted rates are regressed on predictor variables to give a covariance-adjusted estimate of effect; this estimate is shown to be generally biased for the appropriate regression coefficient. For familiar models, the analysis of crude rates with age as a covariate can lead to unbiased estimates, and therefore can be preferable. Several other regression methods are also considered.

Approximate case influence for the proportional hazards regression model with censored data.

Abstract: A method is presented for approximating the influence of individual cases upon regression coefficient estimates obtained from the Cox proportional hazards model. Observations can thus be identified which may greatly influence statistical inferences regarding the effects of prognostic factors upon survival time. An example from a cancer clinical trial is given.

Additive and multiplicative models for relative survival rates.

Abstract: The relative survival rate is the ratio of the overall survival rate to that 'expected' for demographically similar individuals in a reference population. It is commonly used to estimate the effect of a particular disease on mortality, when the cause of death is not reliably known. The effect of the disease on the underlying hazards may be multiplicative or additive; the former case has been considered elsewhere but seems biologically less plausible than the latter. Both models are examined in this paper. The effect of the disease is assumed to be constant throughout the follow-up period, or to be piecewise constant within K follow-up intervals. Maximum likelihood estimates and related statistics are presented, as are simpler statistics based on moment estimators of the disease effect. The moment-based statistic for testing the homogeneity of r groups may be expressed as a sum of individual scores, which are shown to be closely related to the logrank scores when follow-up intervals are made arbitrarily small.

The Use of Auxiliary Variables in Capture-Recapture and Removal Experiments

Abstract: The dependence of animal capture probabilities on auxiliary variables is an important practical problem which has not been considered in the development of estimation procedures for capturerecapture and removal experiments. In this paper the linear logistic binary regression model is used to relate the probability of capture to continuous auxiliary variables. The auxiliary variables could be environmental quantities such as air or water temperature, or characteristics of individual animals, such as body length or weight. Maximum likelihood estimators of the population parameters are considered for a variety of models which all assume a closed population. Testing between models is also considered. The models can also be used when one auxiliary variable is a measure of the effort expended in obtaining the sample.

A two-sample censored-data rank test for acceleration.

Abstract: A score test for the null hypothesis of proportional hazards against rank-regression alternatives is proposed as a complement to the logrank test for comparing censored survival curves. The test statistic has an asymptotic normal distribution that is independent of the logrank distribution under the null hypothesis, and its power is good against acceleration alternatives (i.e. with crossing hazards) where the logrank test fails. Monte Carlo studies indicate that its small-sample properties are comparable to those of the logrank and ranksum procedures.