Showing papers in "Biometrics in 1988"

Comparing the areas under two or more correlated receiver operating characteristic curves: a nonparametric approach.

Abstract: Methods of evaluating and comparing the performance of diagnostic tests are of increasing importance as new tests are developed and marketed. When a test is based on an observed variable that lies on a continuous or graded scale, an assessment of the overall value of the test can be made through the use of a receiver operating characteristic (ROC) curve. The curve is constructed by varying the cutpoint used to determine which values of the observed variable will be considered abnormal and then plotting the resulting sensitivities against the corresponding false positive rates. When two or more empirical curves are constructed based on tests performed on the same individuals, statistical analysis on differences between curves must take into account the correlated nature of the data. This paper presents a nonparametric approach to the analysis of areas under correlated ROC curves, by using the theory on generalized U-statistics to generate an estimated covariance matrix.

Models for longitudinal data: a generalized estimating equation approach.

Abstract: This article discusses extensions of generalized linear models for the analysis of longitudinal data. Two approaches are considered: subject-specific (SS) models in which heterogeneity in regression parameters is explicitly modelled; and population-averaged (PA) models in which the aggregate response for the population is the focus. We use a generalized estimating equation approach to fit both classes of models for discrete and continuous outcomes. When the subject-specific parameters are assumed to follow a Gaussian distribution, simple relationships between the PA and SS parameters are available. The methods are illustrated with an analysis of data on mother's smoking and children's respiratory disease.

Principles of multivariate analysis: a user's perspective

Abstract: Part I: Looking at multivariate data Part II: Samples, populations, and models Part III: Analysing ungrouped data Part IV: Analysing grouped data Part V: Analysing association among variables Appendix: some basic matrix theory A1 Definitions A2 Elementary arithmetic operations A3 Determinants and inverses A4 Quadratic forms A5 Latent roots and vectors A6 Matrix square root A7 Partitioned matrices A8 Vector differentiation References Index

Correlated binary regression with covariates specific to each binary observation.

Abstract: Regression methods are considered for the analysis of correlated binary data when each binary observation may have its own covariates. It is argued that binary response models that condition on some or all binary responses in a given "block" are useful for studying certain types of dependencies, but not for the estimation of marginal response probabilities or pairwise correlations. Fully parametric approaches to these latter problems appear to be unduly complicated except in such special cases as the analysis of paired binary data. Hence, a generalized estimating equation approach is advocated for inference on response probabilities and correlations. Illustrations involving both small and large block sizes are provided.

Statistical Analysis of Spherical Data.

Abstract: This is the first comprehensive, yet clearly presented, account of statistical methods for analysing spherical data. The analysis of data, in the form of directions in space or of positions of points on a spherical surface, is required in many contexts in the earth sciences, astrophysics and other fields, yet the methodology required is disseminated throughout the literature. Statistical Analysis of Spherical Data aims to present a unified and up-to-date account of these methods for practical use. The emphasis is on applications rather than theory, with the statistical methods being illustrated throughout the book by data examples.

Model selection and validation for yield trials with interaction

Abstract: SUMMARY The additive main effects and multiplicative interaction (AMMI) model first applies the additive analysis of variance (ANOVA) model to two-way data, and then applies the multiplicative principal components analysis (PCA) model to the residual from the additive model, that is, to the interaction. AMMI analysis of yield trial data is a useful extension of the more familiar ANOVA, PCA, and linear regression procedures, particularly given a large genotype-by-environment interaction. Model selection and validation are considered from both predictive and postdictive perspectives, using data splitting and F-tests, respectively. A New York soybean yield trial serves as an example. Yield trials generate observations of yield, ordinarily replicated, for a number of genotypes grown in a number of environments (site-year combinations). Often the data are rather noisy, with a standard deviation for plot yields in excess of 25 % of the mean. An additional challenging feature of these data is the frequent presence of important and complex genotype-by-environment (GE) interactions. Plant breeders use yield trials to identify promising genotypes, and agronomists use them to make recommendations for farmers. The level of success in meeting these goals depends critically on two factors: (i) the accuracy of yield estimates, and (ii) the magnitudes of genotype-by-site, genotype-by-year, and genotype-by-site-by-year interactions (Talbot, 1984). In essence, these two factors reflect within-trial accuracy and between-trial predictability. This paper addresses only the first concern. Nothing is said regarding between-trial predictability, or agrotechnology transfer, other than to observe that success with withintrial accuracy is a necessary prelude to success with between-trial predictability. The within-trial accuracy of a statistical model may be assessed by two fundamentally different criteria: Postdictive success concerns a model's fit to its own data, whereas predictive success concerns the fit between a model constructed using part of the data and validation data not used in modelling. In either case, the statistical setting is that of an incompletely specified model, where empirical considerations enter into the decision to include a given' potential source in the model, or alternatively to relegate it to the model's residual (Bancroft, 1964). Whenever the data are noisy, postdiction and prediction are different tasks, and in general the model chosen by predictive criteria will be different and simpler than the model chosen by postdictive criteria. Because the several replicates of a yield trial constitute a noisy sample, a model chosen by predictive criteria may be expected to provide yield estimates that are closer to the true means (and to validation observations) than will a model chosen by postdictive criteria. This claim is readily subjected to experimental test. In what follows, the relative performances of statistical models chosen

A Criterion for Determining the Number of Groups in a Data Set Using Sum-of-Squares Clustering

Abstract: Marriott (1971, Biometrics 27, 501-514) used a heuristic argument to derive the criterion g2 l W I for determining the number of groups in a data set when the clustering objective function is the withingroup determinant I W 1. An analogous argument is employed to derive a criterion for use with the within-group sum-of-squares objective function trace (W). The behaviour of both Marriott's criterion and the new criterion is investigated by Monte Carlo methods. For homogeneous data based on uniform and independent variables, the performance of the new criterion is close to expectation while Marriott's criterion shows much more extreme behaviour. For grouped data, the new criterion correctly identifies the number of groups in 85% of data sets under a wide range of conditions, while Marriott's criterion shows a success rate of less than 40%. The new criterion is illustrated on the wellknown Iris data, and some cautionary comments are made about its use.

An approach to the analysis of repeated measurements.

Abstract: SUMMARY A linear model for repeated measurements is proposed in which the correlation structure within each time sequence of measurements includes parameters for measurement error, variation between experimental units, and serial correlation within units. An approach to data analysis is presented which involves preliminary analysis by ordinary least squares, use of the empirical semi-variogram of residuals to suggest a suitable correlation structure, and formal inference using likelihood-based methods. Applications to two biological data sets are described.

Regression Trees for Censored Data

Abstract: The regression-tree methodology is extended to right-censored response variables by replacing the conventional splitting rules with rules based on the Tarone-Ware or Harrington-Fleming classes of two-sample statistics. New pruning strategies for determining desirable tree size are also devised. Properties of this approach are developed and comparisons with existing procedures, in terms of practical problems, are discussed. Illustrative, real-world performances of the technique are presented.

Markov regression models for time series: a quasi-likelihood approach.

Abstract: This paper discusses a quasi-likelihood (QL) approach to regression analysis with time series data. We consider a class of Markov models, referred to by Cox (1981, Scandinavian Journal of Statistics 8, 93-115) as "observation-driven" models in which the conditional means and variances given the past are explicit functions of past outcomes. The class includes autoregressive and Markov chain models for continuous and categorical observations as well as models for counts (e.g., Poisson) and continuous outcomes with constant coefficient of variation (e.g., gamma). We focus on Poisson and gamma data for illustration. Analogous to QL for independent observations, large-sample properties of the regression coefficients depend only on correct specification of the first conditional moment.

The B-value: a tool for monitoring data.

Abstract: This paper considers the problem of monitoring slowly accruing data from a nonsequentially designed experiment. We describe the use of the B-value, which is a transformed Z-value, for the calculation of conditional power. In data monitoring, interim Z-values do not allow simple projections to the end of the study. Moreover, because of their popular association with P-values, Z-values are often misinterpreted. If observed trends are viewed as the realization of a Brownian motion process, the B-value and its decomposition allow simple extrapolations to the end of the study under a variety of hypotheses. Applications are presented to one- and two-sample Z-tests, the two-sample Wilcoxon rank sum test, and the log-rank test.

Sample sizes based on the log-rank statistic in complex clinical trials.

Abstract: The log-rank test is frequently used to compare survival curves. While sample size estimation for comparison of binomial proportions has been adapted to typical clinical trial conditions such as noncompliance, lag time, and staggered entry, the estimation of sample size when the log-rank statistic is to be used has not been generalized to these types of clinical trial conditions. This paper presents a method of estimating sample sizes for the comparison of survival curves by the log-rank statistic in the presence of unrestricted rates of noncompliance, lag time, and so forth. The method applies to stratified trials in which the above conditions may vary across the different strata, and does not assume proportional hazards. Power and duration, as well as sample sizes, can be estimated. The method also produces estimates for binomial proportions and the Tarone-Ware class of statistics.

Unimodal models to relate species to environment

Abstract: To assess the impact of environmental change on biological communities knowledge about species-environment relationships is indispensable. Ecologists attempt to uncover the relationships between species and environment from data obtained from field surveys. In the survey, species are scored on their presence or their abundance at each of several sampling sites and environmental variables that ecologists believe to be important are measured. The research that led to this thesis aimed to unravel the assumptions required for the application of statistical methods that are popular among ecologists to analyse such data. From a statistical point of view, species data are difficult to analyse: - there are many species involved (10 - 500), - many species occur at a few sites only. So the data contain numerous zeroes, - relations between species and environmental variables are not linear, but unimodal: a plant, for example, preferably grows under for that species optimal moisture conditions and is encountered less frequently at drier or wetter sites. A mathematical model for a unimodal relationship is the Gaussian response model. Standard statistical methods such as linear regression, principal components analysis and canonical correlation analysis are often inappropriate for analysing species data because they are based on linear relationships. One of the methods that ecologists use instead is correspondence analysis. This thesis contributes to the understanding of the underlying response model. With correspondence analysis, species and sites are arranged to discover the structure in the data (ordination) and the arrangement is subsequently related to environmental variables. It is an indirect method to detect relations between species and environment, hence R.H. Whittaker's term "indirect gradient analysis". Correspondence analysis has been invented around 1935 but did not receive interest from ecologists before 1973 when M.O. Hill derived the technique once more as the repeated application of "weighted averaging" - a method that was familiar to ecologists ever since 1930. Weighted averaging has the advantage of being simple to apply. The method can be used for two different aims: (1) to estimate the optimum of a species for an environmental variable and (2) to estimate the value of an environmental variable at a site from known optima of the species present (calibration). In chapter 2, estimating optima by weighted averaging is compared with the results of non-linear regression on the basis of the Gaussian response model. Under particular conditions, both methods agree precisely. In other cases, weighted averaging gives a biased estimate of the optimum and non-linear regression is the method to be preferred. An additional advantage of non-linear regression is that it can also be used to fit response models with more than one environmental variable. In chapter 3, weighted averaging to estimate the value of an environmental variable is compared with calibration on the basis of the Gaussian response model. Also in this context the techniques are sometimes equivalent. Chapter 4 deals with correspondence analysis. It is shown that, under particular conditions, correspondence analysis approximates ordination on the basis of the Gaussian response model, which is computationally much more complicated. To detect relations, indirect methods have an important disadvantage. The impact of some environmental variables on the species composition can be so large that the impact of other interesting environmental variables may fail to be detected. This problem can be overcome by using non-linear regression, but with many species and environmental variables this is laborious. In chapter 5, a simpler "direct" method is proposed, canonical correspondence analysis. In chapter 6, canonical correspondence analysis turns out to be a multivariate extension of weighted averaging. The results can be displayed graphically. In chapter 7, an extension with "covariables" is discussed, which leads to partial canonical correspondence analysis. Chapter 7 also shows that Gaussian models and, hence, canonical correspondence analysis are relevant to the analysis of contingency tables. Chapter 8 describes a study to estimate ecological amplitudes of plant species with respect to Ellenberg's moisture scale from species data alone. The question that is addressed as well, is how consequent Ellenberg's moisture indicator values are. Finally, chapter 9 cross-tabulates various gradient-analysis techniques by the type of problem (regression, calibration, ordination, etc.) and the response model (linear or unimodal). Furthermore, improvements are proposed for detrended correspondence analysis. A computer program, named MOM is written which can perform most of the methods discussed.

Power calculations for matched case-control studies.

Abstract: Power calculations are derived for matched case-control studies in terms of the probability po of exposure among the control patients, the correlation coefficient phi for exposure between matched case and control patients, and the odds ratio psi for exposure in case and control patients. For given Type I and Type II error probabilities alpha and beta, the odds ratio that can be detected with a given sample size is derived as well as the sample size needed to detect a specified value of the odds ratio. Graphs are presented for paired designs that show the relationship between sample size and power for alpha = .05, beta = .2, and different values of po, phi, and psi. The sample size needed for designs involving M matched control patients can be derived from these graphs by means of a simple equation. These results quantify the loss of power associated with increasing correlation between the exposure status of matched case and control patients. Sample size requirements are also greatly increased for values of po near 0 or 1. The relationship between sample size, psi, phi, and po is discussed and illustrated by examples.

Effects of treatment-induced mortality and tumor-induced mortality on tests for carcinogenicity in small samples

Abstract: Statistical tests of carcinogenicity are shown to have varying degrees of robustness to the effects of mortality. Mortality induced by two different mechanisms is studied--mortality due to the tumor of interest, and mortality due to treatment independent of the tumor. The two most commonly used tests, the life-table test and the Cochran-Armitage linear trend test, are seen to be highly sensitive to increases in treatment lethality using small-sample simulations. Increases in tumor lethality are seen to affect the performance of commonly used prevalence tests such as logistic regression. A simple survival-adjusted quantal response test appears to be the most robust of all the procedures considered.

Approximate interval estimation of the ratio of binomial parameters: a review and corrections for skewness.

Abstract: Various methods for finding confidence intervals for the ratio of binomial parameters are reviewed and evaluated numerically. It is found that the method based on likelihood scores (Koopman, 1984, Biometrics 40, 513-517; Miettinen and Nurminen, 1985, Statistics in Medicine 4, 213-226) performs best in achieving the nominal confidence coefficient, but it may distribute the tail probabilities quite disparately. Using general theory of Bartlett (1953, Biometrika 40, 306-317; 1955, Biometrika 42, 201-203), we correct this method for asymptotic skewness. Following Gart (1985, Biometrika 72, 673-677), we extend this correction to the case of estimating the common ratio in a series of two-by-two tables. Computing algorithms are given and applied to numerical examples. Parallel methods for the odds ratio and the ratio of Poisson parameters are noted.

Power/Sample Size Calculations for Generalized Linear Models

Abstract: An approach for estimating power/sample size is described within the framework of generalized linear models. This approach is based on an asymptotic approximation to the power of the score test under contiguous alternatives and is applicable to tests of composite null hypotheses. An implementation is described for the special case of logistic regression models. Simulation studies are presented which indicate that the asymptotic approximation to the finite-sample situation is good over a range of parameter configurations.

Fitting Mixture-Models to Grouped and Truncated Data Via the Em Algorithm

Abstract: The fitting of finite mixture models via the EM algorithm is considered for data which are available only in grouped form and which may also be truncated. A practical example is presented where a mixture of two doubly truncated log-normal distributions is adopted to model the distribution of the volume of red blood cells in cows during recovery from anemia.

Mixed Poisson likelihood regression models for longitudinal interval count data.

Abstract: SUMMARY In many longitudinal studies it is desired to estimate and test the rate over time of a particular recurrent event Often only the event counts corresponding to the elapsed time intervals between each subject's successive observation times, and baseline covariate data, are available The intervals may vary substantially in length and number between subjects, so that the corresponding vectors of counts are not directly comparable A family of Poisson likelihood regression models incorporating a mixed random multiplicative component in the rate function of each subject is proposed for this longitudinal data structure A related empirical Bayes estimate of random-effect parameters is also described These methods are illustrated by an analysis of dyspepsia data from the National Cooperative Gallstone Study

Simulated Percentage Points for the Null Distribution of the Likelihood Ratio Test for a Mixture of Two Normals

Abstract: SUMMARY We find the percentage points of the likelihood ratio test of the null hypothesis that a sample of n observations is from a normal distribution with unknown mean and variance against the alternative that the sample is from a mixture of two distinct normal distributions, each with unknown mean and unknown (but equal) variance. The mixing proportion ir is also unknown under the alternative hypothesis. For 2,500 samples of sizes n = 15, 20, 25, 40, 50, 70, 75, 80, 100, 150, 250, 500, and 1,000, we calculated the likelihood ratio statistic, and from these values estimated the percentage points of the null distributions. Our algorithm for the calculation of the maximum likelihood estimates of the unknown parameters included precautions against convergence of the maximization algorithm to a local rather than global maximum. Investigations for convergence to an asymptotic distribution indicated that convergence was very slow and that stability was not apparent for samples as large as 1,000. Comparisons of the percentage points to the commonly assumed chi-squared distribution with 2 degrees of freedom indicated that this assumption is too liberal; i.e., one's P-value is greater than that indicated by X2. We conclude then that one would need what is usually an unfeasibly large sample size (n > 1,000) for the use of large-sample approximations to be justified.

A lifetime asymptotic growth curve for human height.

Abstract: A new seven-parameter asymptotic growth curve has been applied to longitudinal data on the height of 13 boys and 14 girls from 1 month to 19 years of age. The residual sums of squares with this new curve were 7.5 times lower on the average than with the currently-used five-parameter curve No. 1 of Preece and Baines (1978, Annals of Human Biology 5, 1-24) and 2.4 times lower than with the recent six-parameter curve of Shohoji and Sasaki (1987, Growth 51, 432-450). The new curve is expressed with respect to total age, passes through the origin, and fits infants as satisfactorily as older children.

One-sided sequential stopping boundaries for clinical trials: a decision-theoretic approach.

Abstract: SUMMARY We address one-sided stopping rules for clinical trials, or more generally, drug development programs, from a decision-theoretic point of view. If efflcacy results are sufflciently negative then the trial will be stopped. But regardless of how positive the efflcacy results are, the trial will continue in order to demonstrate safety. We show how sequential decisions should be made by a pharmaceutical company attempting to maximize its expected profits. When should a drug development program be stopped for evidence that suggests that the drug has insufficient efficacy? When should a drug program be stopped for adverse experiences? These questions are of paramount concern to pharmaceutical companies. They can be addressed using the rather formal cost-benefit analysis provided by statistical decision theory. We focus on the first question here. While we do not address the second question explicitly, adverse events should be taken into account in assessing the loss structure of the problem we do consider. We show how to decide when to stop the development of an experimental drug as a function of accumulating information on the drug's performance. But to relate our approach to the more classical methods of interim analysis (Armitage, McPherson, and Rowe, 1969; Pocock, 1977; Lan and DeMets, 1983), we address the apparently smaller question of when efficacy results from a clinical trial are sufficiently negative that the trial should be stopped. The trial will not be stopped for positive results. This is consistent with the need for safety information regarding an experimental drug; the trial will continue as planned to obtain this information even if interim efficacy results are very favorable. There is no issue related to adjusting P-values for early stopping, for example, because the trial results would not then be used to convince anyone of the drug's efficacy: by stopping the trial the company effectively gives up on the possibility of obtaining marketing approval for the drug. We will take a Bayesian decision-theoretic approach (Raiffa and Schlaifer, 1961; DeGroot, 1970) in which consequences of the various decisions are considered explicitly in terms of the company's assessment of the various consequences. There are many types of consequences that are related to the objectives of the clinical trial. Some of these can be easily assessed and compared with each other, and some cannot. We consider only those consequences that are financial or can be easily converted to a financial scale.

Locally most powerful tests for detecting treatment effects when only a subset of patients can be expected to "respond" to treatment.

Abstract: Two two-parameter models are developed for testing the hypothesis of no treatment effect against the alternative that a subset of the treated patients will show an improvement. To keep the range of measurements the same for treated and control patients, Lehmann alternatives are used in both models. Locally most powerful rank tests are developed for each model and each parameter. The asymptotic relative efficiency leads to a test that uses the scores s(i) = [i/(N + 1)]4. Two examples that support the usefulness of this nonparametric test are presented.

A New Probability Model for Determining Exact P-Values for 2 x 2 Contingency Tables When Comparing Binomial Proportions

Abstract: A new probability model to determine exact P-values for 2 x 2 contingency tables and its computerized solution are described. This model can be used in place of Fisher's "exact test" when analyzing contingency tables that compare binomial proportions estimated from samples of larger populations. The model accommodates different levels of a priori information about the underlying probability of success. The computerized solution is interactive, written in Pascal, and specifically designed to run on small desktop microcomputers. Applications of the probability model to behavioral and evolutionary data are described.

Kalman Filter Analysis in Dendroclimatology

Abstract: SUMMARY The use of the Kalman filter in the analysis of tree-ring data is summarized. By use of this filter technique, the traditional multiple regression models can be modified to cover linear models with time-dependent coefficients. In that way changes in tree response to weather variations could be detected, possibly indicating anthropogenic influences. The filtering and smoothing operations of the filter are illustrated using simulated tree-ring series. A method for the selection of predictor variables is proposed, which takes into account the special features of time-dependent regression models. Furthermore, the consequences of highly correlated predictors are discussed. An application is given to a ring-width series of a European silver fir from Bad Herrenalb (F.R.G.). The method appears suitable for dealing with both gradual changes and sudden shocks in tree response.