Showing papers in "Journal of the American Statistical Association in 1987"

Statistical Methods for Meta-Analysis.

Abstract: Preface. Introduction. Data Sets. Tests of Statistical Significance of Combined Results. Vote-Counting Methods. Estimation of a Single Effect Size: Parametric and Nonparametric Methods. Parametric Estimation of Effect Size from a Series of Experiments. Fitting Parametric Fixed Effect Models to Effect Sizes: Categorical Methods. Fitting Parametric Fixed Effect Models to Effect Sizes: General Linear Models. Random Effects Models for Effect Sizes. Multivariate Models for Effect Sizes. Combining Estimates of Correlation Coefficients. Diagnostic Procedures for Research Synthesis Models. Clustering Estimates of Effect Magnitude. Estimation of Effect Size When Not All Study Outcomes Are Observed. Meta-Analysis in the Physical and Biological Sciences. Appendix. References. Index.

Stochastic differential equations : an introduction with applications

Abstract: Some Mathematical Preliminaries.- Ito Integrals.- The Ito Formula and the Martingale Representation Theorem.- Stochastic Differential Equations.- The Filtering Problem.- Diffusions: Basic Properties.- Other Topics in Diffusion Theory.- Applications to Boundary Value Problems.- Application to Optimal Stopping.- Application to Stochastic Control.- Application to Mathematical Finance.

The calculation of posterior distributions by data augmentation

Abstract: The idea of data augmentation arises naturally in missing value problems, as exemplified by the standard ways of filling in missing cells in balanced two-way tables. Thus data augmentation refers to a scheme of augmenting the observed data so as to make it more easy to analyze. This device is used to great advantage by the EM algorithm (Dempster, Laird, and Rubin 1977) in solving maximum likelihood problems. In situations when the likelihood cannot be approximated closely by the normal likelihood, maximum likelihood estimates and the associated standard errors cannot be relied upon to make valid inferential statements. From the Bayesian point of view, one must now calculate the posterior distribution of parameters of interest. If data augmentation can be used in the calculation of the maximum likelihood estimate, then in the same cases one ought to be able to use it in the computation of the posterior distribution. It is the purpose of this article to explain how this can be done. The basic idea ...

Better Bootstrap Confidence Intervals

Abstract: We consider the problem of setting approximate confidence intervals for a single parameter θ in a multiparameter family. The standard approximate intervals based on maximum likelihood theory, , can be quite misleading. In practice, tricks based on transformations, bias corrections, and so forth, are often used to improve their accuracy. The bootstrap confidence intervals discussed in this article automatically incorporate such tricks without requiring the statistician to think them through for each new application, at the price of a considerable increase in computational effort. The new intervals incorporate an improvement over previously suggested methods, which results in second-order correctness in a wide variety of problems. In addition to parametric families, bootstrap intervals are also developed for nonparametric situations.

Asymptotic Properties of Maximum Likelihood Estimators and Likelihood Ratio Tests under Nonstandard Conditions

Abstract: Large sample properties of the likelihood function when the true parameter value may be on the boundary of the parameter space are described. Specifically, the asymptotic distribution of maximum likelihood estimators and likelihood ratio statistics are derived. These results generalize the work of Moran (1971), Chant (1974), and Chernoff (1954). Some of Chant's results are shown to be incorrect. The approach used in deriving these results follows from comments made by Moran and Chant. The problem is shown to be asymptotically equivalent to the problem of estimating the restricted mean of a multivariate Gaussian distribution from a sample of size 1. In this representation the Gaussian random variable corresponds to the limit of the normalized score statistic and the estimate of the mean corresponds to the limit of the normalized maximum likelihood estimator. Thus the limiting distribution of the maximum likelihood estimator is the same as the distribution of the projection of the Gaussian random v...

Summing Up: The Science of Reviewing Research.

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Fine-Tuning Some Resistant Rules for Outlier Labeling

Abstract: A previous study examined the performance of a standard rule from Exploratory Data Analysis, which uses the sample fourths, FL and FU , and labels as “outside” any observations below FL – k(FU – FL ) or above FU + k(FU – FL ), customarily with k = 1.5. In terms of the order statistics X (1) ≤ X (2) ≤ X (n) the standard definition of the fourths is FL = X(f) and FU = X (n + 1 − f), where f = ½[(n + 3)/2] and [·] denotes the greatest-integer function. The results of that study suggest that finer interpolation for the fourths might yield smoother behavior in the face of varying sample size. In this article we show that using f i = n/4 + (5/12) to define the fourths produces the desired smoothness. Corresponding to a common definition of quartiles, fQ = n/4 + (1/4) leads to similar results. Instead of allowing the some-outside rate per sample (the probability that a sample contains one or more outside observations, analogous to the experimentwise error rate in simultaneous inference) to vary, some us...

Testing a Point Null Hypothesis: The Irreconcilability of P Values and Evidence

Abstract: The problem of testing a point null hypothesis (or a “small interval” null hypothesis) is considered. Of interest is the relationship between the P value (or observed significance level) and conditional and Bayesian measures of evidence against the null hypothesis. Although one might presume that a small P value indicates the presence of strong evidence against the null, such is not necessarily the case. Expanding on earlier work [especially Edwards, Lindman, and Savage (1963) and Dickey (1977)], it is shown that actual evidence against a null (as measured, say, by posterior probability or comparative likelihood) can differ by an order of magnitude from the P value. For instance, data that yield a P value of .05, when testing a normal mean, result in a posterior probability of the null of at least .30 for any objective prior distribution. (“Objective” here means that equal prior weight is given the two hypotheses and that the prior is symmetric and nonincreasing away from the null; other definiti...

Model-Based Direct Adjustment

Abstract: Direct adjustment or standardization applies population weights to subclass means in an effort to estimate population quantities from a sample that is not representative of the population. Direct adjustment has several attractive features, but when there are many subclasses it can attach large weights to small quantities of data, often in a fairly erratic manner. In the extreme, direct adjustment can attach infinite weight to nonexistent data, a noticeable inconvenience in practice. This article proposes a method of model-based direct adjustment that preserves the attractive features of conventional direct adjustment while stabilizing the weights attached to small subclasses. The sample mean and conventional direct adjustment are both special cases of model-based direct adjustment under two different extreme models for the subclass-specific selection probabilities. The discussion of this method provides some insights into the behavior of true and estimated propensity scores: the estimated scores ...

Non-Gaussian State—Space Modeling of Nonstationary Time Series

Abstract: A non-Gaussian state—space approach to the modeling of nonstationary time series is shown. The model is expressed in state—space form, where the system noise and the observational noise are not necessarily Gaussian. Recursive formulas of prediction, filtering, and smoothing for the state estimation and identification of the non-Gaussian state—space model are given. Also given is a numerical method based on piecewise linear approximation to the density functions for realizing these formulas. Significant merits of non-Gaussian modeling and the wide range of applicability of the method are illustrated by some numerical examples. A typical application of this non-Gaussian modeling is the smoothing of a time series that has mean value function with both abrupt and gradual changes. Simple Gaussian state—space modeling is not adequate for this situation. Here the model with small system noise variance cannot detect jump, whereas the one with large system noise variance yields unfavorable wiggle. To work...

Nonparametric density estimation : the L[1] view

Abstract: Differentiation of Integrals Consistency Lower Bounds for Rates of Convergence Rates of Convergence in L1 The Automatic Kernel Estimate: L1 and Pointwise Convergence Estimates Related to the Kernel Estimate and the Histogram Estimate Simulation, Inequalities, and Random Variate Generation The Transformed Kernel Estimate Applications in Discrimination Operations on Density Estimates Estimators Based on Orthogonal Series Index.

Probabilistic Solution of Ill-Posed Problems in Computational Vision

Abstract: We formulate several problems in early vision as inverse problems. Among the solution methods we review standard regularization theory, discuss its limitations, and present new stochastic (in particular, Bayesian) techniques based on Markov Random Field models for their solution. We derive efficient algorithms and describe parallel implementations on digital parallel SIMD architectures, as well as a new class of parallel hybrid computers that mix digital with analog components.

Exploratory Projection Pursuit

Abstract: A new projection pursuit algorithm for exploring multivariate data is presented that has both statistical and computational advantages over previous methods. A number of practical issues concerning its application are addressed. A connection to multivariate density estimation is established, and its properties are investigated through simulation studies and application to real data. The goal of exploratory projection pursuit is to use the data to find low- (one-, two-, or three-) dimensional projections that provide the most revealing views of the full-dimensional data. With these views the human gift for pattern recognition can be applied to help discover effects that may not have been anticipated in advance. Since linear effects are directly captured by the covariance structure of the variable pairs (which are straightforward to estimate) the emphasis here is on the discovery of nonlinear effects such as clustering or other general nonlinear associations among the variables. Although arbitrary ...

Variance Function Estimation

Abstract: Heteroscedastic regression models are used in fields including economics, engineering, and the biological and physical sciences. Often, the heteroscedasticity is modeled as a function of the covariates or the regression and other structural parameters. Standard asymptotic theory implies that how one estimates the variance function, in particular the structural parameters, has no effect on the first-order properties of the regression parameter estimates; there is evidence, however, both in practice and higher-order theory to suggest that how one estimates the variance function does matter. Further, in some settings, estimation of the variance function is of independent interest or plays an important role in estimation of other quantities. In this article, we study variance function estimation in a unified way, focusing on common methods proposed in the statistical and other literature, to make both general observations and compare different estimation schemes. We show that there are significant di...

K-Sample Anderson–Darling Tests

Abstract: Two k-sample versions of an Anderson–Darling rank statistic are proposed for testing the homogeneity of samples. Their asymptotic null distributions are derived for the continuous as well as the discrete case. In the continuous case the asymptotic distributions coincide with the (k – 1)-fold convolution of the asymptotic distribution for the Anderson–Darling one-sample statistic. The quality of this large sample approximation is investigated for small samples through Monte Carlo simulation. This is done for both versions of the statistic under various degrees of data rounding and sample size imbalances. Tables for carrying out these tests are provided, and their usage in combining independent one- or k-sample Anderson–Darling tests is pointed out. The test statistics are essentially based on a doubly weighted sum of integrated squared differences between the empirical distribution functions of the individual samples and that of the pooled sample. One weighting adjusts for the possibly different s...

Generalized Additive Models: Some Applications

Abstract: Generalized additive models have the form η(x) = α + σ fj (x j ), where η might be the regression function in a multiple regression or the logistic transformation of the posterior probability Pr(y = 1 | x) in a logistic regression. In fact, these models generalize the whole family of generalized linear models η(x) = β′x, where η(x) = g(μ(x)) is some transformation of the regression function. We use the local scoring algorithm to estimate the functions fj (xj ) nonparametrically, using a scatterplot smoother as a building block. We demonstrate the models in two different analyses: a nonparametric analysis of covariance and a logistic regression. The procedure can be used as a diagnostic tool for identifying parametric transformations of the covariates in a standard linear analysis. A variety of inferential tools have been developed to aid the analyst in assessing the relevance and significance of the estimated functions: these include confidence curves, degrees of freedom estimates, and approximat...

Stochastic Blockmodels for Directed Graphs

Abstract: Holland and Leinhardt (1981) proposed the p 1 model for the analysis of binary directed graph data in network studies. Such a model provides information about the “attractiveness” and “expansiveness” of the individual nodes in the network, as well as the tendency of a pair of nodes to reciprocate relational ties. When the nodes are a priori partitioned into subgroups based on attributes such as race and sex, the density of ties from one subgroup to another can differ considerably from that relating another pair of subgroups, thus creating a situation called blocking in social networks. The p 1 model completely ignores this extra piece of information and is, therefore, unable to explain the block structure. Blockmodels that are simple extensions of the p 1 model are proposed specifically for such data. An iterative scaling algorithm is presented for fitting the model parameters by maximum likelihood. The methodology is illustrated in detail on two empirical examples.

Local Likelihood Estimation

Abstract: A scatterplot smoother is applied to data of the form {(x 1, y 1), (x 2, y 2, …, (xn, yn )} and uses local fitting to estimate the dependence of Y on X. A simple example is the running lines smoother, which fits a least squares line to the y values falling in a window around each x value. The value of the estimated function at x is given by the value of the least squares line at x. A smoother generalizes the least squares line, which assumes that the dependence of Y on X is linear. In this article, we extend the idea of local fitting to likelihood-based regression models. One such application is to the class of generalized linear models (Nelder and Wedderburn 1972). We enlarge this class by replacing the covariate form β0 + xβ1 with an unspecified smooth function s(x). This function is estimated from the data by a technique we call local likelihood estimation. The method consists of maximum likelihood estimation for β0 and β1, applied in a window around each x value. Multiple covariates are incor...

Biased and Unbiased Cross-Validation in Density Estimation

Abstract: Nonparametric density estimation requires the specification of smoothing parameters. The demands of statistical objectivity make it highly desirable to base the choice on properties of the data set. In this article we introduce some biased cross-validation criteria for selection of smoothing parameters for kernel and histogram density estimators, closely related to one investigated in Scott and Factor (1981). These criteria are obtained by estimating L 2 norms of derivatives of the unknown density and provide slightly biased estimates of the average squared L 2 error or mean integrated squared error. These criteria are roughly the analog of Wahba's (1981) generalized cross-validation procedure for orthogonal series density estimators. We present the relationship of the biased cross-validation procedure to the least squares cross-validation procedure, which provides unbiased estimates of the mean integrated squared error. Both methods are shown to be based on U statistics. We compare the two metho...

The Geometric Interpretation of Correspondence Analysis

Abstract: Correspondence analysis is an exploratory multivariate technique that converts a data matrix into a particular type of graphical display in which the rows and columns are depicted as points. The method has a long and varied history and has appeared in different forms in the psychometric and ecological literature, among others. In this article we review the geometry of correspondence analysis and its geometric interpretation. We also discuss various extensions of correspondence analysis to multivariate categorical data (multiple correspondence analysis) and a variety of other data types.

Maximum likelihood computations with repeated measures: application of the EM algorithm

Abstract: The purpose of this article is to consider the use of the EM algorithm (Dempster, Laird, and Rubin 1977) for both maximum likelihood (ML) and restricted maximum likelihood (REML) estimation in a general repeated measures setting using a multivariate normal data model with linear mean and covariance structure (Anderson 1973). Several models and methods of analysis have been proposed in recent years for repeated measures data; Ware (1985) presented an overview. Because the EM algorithm is a general-purpose, iterative method for computing ML estimates with incomplete data, it has often been used in this particular setting (Dempster et al. 1977; Andrade and Helms 1984; Jennrich and Schluchter 1985). There are two apparently different approaches to using the EM algorithm in this setting. In one application, each experimental unit is observed under a standard protocol specifying measurements at each of n occasions (or under n conditions), and incompleteness implies that the number of measurements actua...

Reconciling Bayesian and Frequentist Evidence in the One-Sided Testing Problem

Abstract: For the one-sided hypothesis testing problem it is shown that it is possible to reconcile Bayesian evidence against H 0, expressed in terms of the posterior probability that H 0 is true, with frequentist evidence against H 0, expressed in terms of the p value. In fact, for many classes of prior distributions it is shown that the infimum of the Bayesian posterior probability of H 0 is equal to the p value; in other cases the infimum is less than the p value. The results are in contrast to recent work of Berger and Sellke (1987) in the two-sided (point null) case, where it was found that the p value is much smaller than the Bayesian infimum. Some comments on the point null problem are also given.

Plane answers to complex questions: the theory of linear models

Abstract: Introduction.- Estimation.- Testing.- One-Way ANOVA.- Multiple Comparison Techniques.- Regression Analysis.- Multifactor Analysis of Variance.- Experimental Design Models.- Analysis of Covariance.- General Gauss-Markov Models.- Split Plot Models.- Mixed Models and Variance Components.- Model Diagnostics.- Variable Selection.- Collinearity and Alternative Estimates.-

Regression Methods for Poisson Process Data

Abstract: This article is directed toward situations where individuals can experience repeated events, and data on an individual consist of the number and occurrence times of events, along with concomitant variables. Methods of regression analysis are presented, based on Poisson process and proportional intensity assumptions. These include parametric and semi-parametric approaches to model fitting, model assessment, and the treatment of random effects. In addition, insight is gained as to the central role of Poisson and mixed Poisson regression analysis of counts in these methods, and as to the effects of unobserved heterogeneity on semi-parametric analyses. The methods in the article are based on the proportional intensity Poisson process model, for which an individual with given fixed covariate vector x has repeated events occur according to a nonhomogeneous Poisson process with intensity function λx(t) = λ0(t)exp(x′β). Estimation of β and the baseline intensity λ0(t) are considered when λ0(t) is specifi...

Scatterplot Matrix Techniques for Large N

Abstract: High-performance interaction with scatterplot matrices is a powerful approach to exploratory multivariate data analysis. For a small number of data points, real-time interaction is possible and overplotting is usually not a major problem. When the number of plotted points is large, however, display techniques that deal with overplotting and slow production are important. This article addresses these two problems. Topics include density representation by gray scale or by symbol area, alternatives to brushing, and animation sequences. We also discuss techniques that are generally applicable, including interactive graphical subset selection from any plot in a collection of scatterplots and comparison of scatterplot matrices.

Plots, transformations, and regression : an introduction to graphical methods of diagnostic regression analysis

Abstract: This handbook provides a detailed, down-to-earth introduction to regression diagnostic analysis, a technique of growing importance for work in applied statistics. Heavily illustrated, with numerous examples to illuminate the discussion, this timely volume outlines methods for regression models, stressing detection of outliers and inadequate models; describes the transformation of variables in an equation, particularly the response; and considers such advanced topics as generalized linear models. A useful guide that combines lucid explanations with up-to-date research findings.