Showing papers in "Journal of the American Statistical Association in 1988"

A Test of Missing Completely at Random for Multivariate Data with Missing Values

Abstract: A common concern when faced with multivariate data with missing values is whether the missing data are missing completely at random (MCAR); that is, whether missingness depends on the variables in the data set. One way of assessing this is to compare the means of recorded values of each variable between groups defined by whether other variables in the data set are missing or not. Although informative, this procedure yields potentially many correlated statistics for testing MCAR, resulting in multiple-comparison problems. This article proposes a single global test statistic for MCAR that uses all of the available data. The asymptotic null distribution is given, and the small-sample null distribution is derived for multivariate normal data with a monotone pattern of missing data. The test reduces to a standard t test when the data are bivariate with missing data confined to a single variable. A limited simulation study of empirical sizes for the test applied to normal and nonnormal data suggests th...

Locally Weighted Regression: An Approach to Regression Analysis by Local Fitting

Abstract: Locally weighted regression, or loess, is a way of estimating a regression surface through a multivariate smoothing procedure, fitting a function of the independent variables locally and in a moving fashion analogous to how a moving average is computed for a time series With local fitting we can estimate a much wider class of regression surfaces than with the usual classes of parametric functions, such as polynomials The goal of this article is to show, through applications, how loess can be used for three purposes: data exploration, diagnostic checking of parametric models, and providing a nonparametric regression surface Along the way, the following methodology is introduced: (a) a multivariate smoothing procedure that is an extension of univariate locally weighted regression; (b) statistical procedures that are analogous to those used in the least-squares fitting of parametric functions; (c) several graphical methods that are useful tools for understanding loess estimates and checking the a

Elements of Econometrics.

Abstract: This classic text has proven its worth in university classrooms and as a tool kit in research--selling over 40,000 copies in the United States and abroad in its first edition alone. Users have included undergraduate and graduate students of economics and business, and students and researchers in political science, sociology, and other fields where regression models and their extensions are relevant. The book has also served as a handy reference in the "real world" for people who need a clear and accurate explanation of techniques that are used in empirical research.Throughout the book the emphasis is on simplification whenever possible, assuming the readers know college algebra and basic calculus. Jan Kmenta explains all methods within the simplest framework, and generalizations are presented as logical extensions of simple cases. And while a relatively high degree of rigor is preserved, every conflict between rigor and clarity is resolved in favor of the latter. Apart from its clear exposition, the book's strength lies in emphasizing the basic ideas rather than just presenting formulas to learn and rules to apply.The book consists of two parts, which could be considered jointly or separately. Part one covers the basic elements of the theory of statistics and provides readers with a good understanding of the process of scientific generalization from incomplete information. Part two contains a thorough exposition of all basic econometric methods and includes some of the more recent developments in several areas.As a textbook, "Elements of Econometrics" is intended for upper-level undergraduate and master's degree courses and may usefully serve as a supplement for traditional Ph.D. courses in econometrics. Researchers in the social sciences will find it an invaluable reference tool.A solutions manual is also available for teachers who adopt the text for coursework.Jan Kmenta is Professor Emeritus of Economics and Statistics, University of Michigan.

Testing for Common Trends

Abstract: Cointegrated multiple time series share at least one common trend. Two tests are developed for the number of common stochastic trends (i.e., for the order of cointegration) in a multiple time series with and without drift. Both tests involve the roots of the ordinary least squares coefficient matrix obtained by regressing the series onto its first lag. Critical values for the tests are tabulated, and their power is examined in a Monte Carlo study. Economic time series are often modeled as having a unit root in their autoregressive representation, or (equivalently) as containing a stochastic trend. But both casual observation and economic theory suggest that many series might contain the same stochastic trends so that they are cointegrated. If each of n series is integrated of order 1 but can be jointly characterized by k > n stochastic trends, then the vector representation of these series has k unit roots and n — k distinct stationary linear combinations. Our proposed tests can be viewed alterna...

Statistical Models for Earthquake Occurrences and Residual Analysis for Point Processes

Abstract: This article discusses several classes of stochastic models for the origin times and magnitudes of earthquakes. The models are compared for a Japanese data set for the years 1885–1980 using likelihood methods. For the best model, a change of time scale is made to investigate the deviation of the data from the model. Conventional graphical methods associated with stationary Poisson processes can be used with the transformed time scale. For point processes, effective use of such residual analysis makes it possible to find features of the data set that are not captured in the model. Based on such analyses, the utility of seismic quiescence for the prediction of a major earthquake is investigated.

Bayesian Variable Selection in Linear Regression

Abstract: This article is concerned with the selection of subsets of predictor variables in a linear regression model for the prediction of a dependent variable. It is based on a Bayesian approach, intended to be as objective as possible. A probability distribution is first assigned to the dependent variable through the specification of a family of prior distributions for the unknown parameters in the regression model. The method is not fully Bayesian, however, because the ultimate choice of prior distribution from this family is affected by the data. It is assumed that the predictors represent distinct observables; the corresponding regression coefficients are assigned independent prior distributions. For each regression coefficient subject to deletion from the model, the prior distribution is a mixture of a point mass at 0 and a diffuse uniform distribution elsewhere, that is, a “spike and slab” distribution. The random error component is assigned a normal distribution with mean 0 and standard deviation ...

Alternative Transformations to Handle Extreme Values of the Dependent Variable

Abstract: Transformations that could be used to reduce the influence of extreme observations of dependent variables, which can assume either sign, on regression coefficient estimates are studied in this article. Two that seem reasonable on a priori grounds—the extended Box—Cox (BC) and the inverse hyperbolic sine (IHS)—are evaluated in detail. One feature is that the log-likelihood function for IHS is defined for zero values of the dependent variable, which is not true of BC. The double-length regression technique (Davidson and MacKinnon 1984) is used to perform hypothesis tests of one transformation against the other using Canadian data on household net worth. These tests support the use of IHS instead of BC for this data set. Empirical investigators in economics often work with a logged dependent variable (taking the natural logarithm of a data series is, of course, a special case of BC) to reduce the weight their particular estimation procedure might otherwise attach to extreme values of the dependent v...

Newton-Raphson and EM Algorithms for Linear Mixed-Effects Models for Repeated-Measures Data

Abstract: We develop an efficient and effective implementation of the Newton—Raphson (NR) algorithm for estimating the parameters in mixed-effects models for repeated-measures data. We formulate the derivatives for both maximum likelihood and restricted maximum likelihood estimation and propose improvements to the algorithm discussed by Jennrich and Schluchter (1986) to speed convergence and ensure a positive-definite covariance matrix for the random effects at each iteration. We use matrix decompositions to develop efficient and computationally stable implementations of both the NR algorithm and an EM algorithm (Laird and Ware 1982) for this model. We compare the two methods (EM vs. NR) in terms of computational order and performance on two sample data sets and conclude that in most situations a well-implemented NR algorithm is preferable to the EM algorithm or EM algorithm with Aitken's acceleration. The term repeated measures refers to experimental designs where there are several individuals and several...

Logistic Regression, Survival Analysis, and the Kaplan-Meier Curve

Abstract: We discuss the use of standard logistic regression techniques to estimate hazard rates and survival curves from censored data. These techniques allow the statistician to use parametric regression modeling on censored data in a flexible way that provides both estimates and standard errors. An example is given that demonstrates the increased structure that can be seen in a parametric analysis, as compared with the nonparametric Kaplan-Meier survival curves. In fact, the logistic regression estimates are closely related to Kaplan-Meier curves, and approach the Kaplan-Meier estimate as the number of parameters grows large.

An Error-Components Model for Prediction of County Crop Areas Using Survey and Satellite Data

Abstract: Knowledge of the area under different crops is important to the U.S. Department of Agriculture. Sample surveys have been designed to estimate crop areas for large regions, such as crop-reporting districts, individual states, and the United States as a whole. Predicting crop areas for small areas such as counties has generally not been attempted, due to a lack of available data from farm surveys for these areas. The use of satellite data in association with farm-level survey observations has been the subject of considerable research in recent years. This article considers (a) data for 12 Iowa counties, obtained from the 1978 June Enumerative Survey of the U.S. Department of Agriculture and (b) data obtained from land observatory satellites (LANDSAT) during the 1978 growing season. Emphasis is given to predicting the area under corn and soybeans in these counties. A linear regression model is specified for the relationship between the reported hectares of corn and soybeans within sample segments in...

Families of Multivariate Distributions

Abstract: For many years there has been an interest in families of bivariate distributions with marginals as parameters. Genest and MacKay (1986a,b) showed that several such families that appear in the literature can be derived by a unified method. A similar conclusion is obtained in this article through the use of mixture models. These models might be regarded as multivariate proportional hazards models with random constants of proportionality. The mixture models are useful for two purposes. First, they make some properties of the derived distributions more transparent; the positive-dependency property of association is sometimes exposed, and a method for simulation of data from the distributions is suggested. But the mixture models also allow derivation of several new families of bivariate distributions with marginals as parameters, and they indicate obvious multivariate extensions. Some of the new families of bivariate distributions given in this article extend known distributions by adding a parameter ...

Optimal Rates of Convergence for Deconvolving a Density

Abstract: Suppose that the sum of two independent random variables X and Z is observed, where Z denotes measurement error and has a known distribution, and where the unknown density f of X is to be estimated. One application is the estimation of a prior density for a sequence of location parameters. A second application arises in the errors-in-variables problem for nonlinear and generalized linear models, when one attempts to model the distribution of the true but unobservable covariates. This article shows that if Z is normally distributed and f has k bounded derivatives, then the fastest attainable convergence rate of any nonparametric estimator of f is only (log n)–k/2. Therefore, deconvolution with normal errors may not be a practical proposition. Other error distributions are also treated. Stefanski—Carroll (1987a) estimators achieve the optimal rates. The results given have versions for multiplicative errors, where they imply that even optimal rates are exceptionally slow.

Prepivoting Test Statistics: A Bootstrap View of Asymptotic Refinements

Abstract: Approximate tests for a composite null hypothesis about a parameter θ may be obtained by referring a test statistic to an estimated critical value. Either asymptotic theory or bootstrap methods can be used to estimate the desired quantile. The simple asymptotic test ϕA refers the test statistic to a quantile of its asymptotic null distribution after unknown parameters have been estimated. The bootstrap approach used here is based on the concept of prepivoting. Prepivoting is the transformation of a test statistic by the cdf of its bootstrap null distribution. The simple bootstrap test ϕB refers the prepivoted test statistic to a quantile of the uniform (0, 1) distribution. Under regularity conditions, the bootstrap test ϕB has a smaller asymptotic order of error in level than does the asymptotic test ϕA , provided that the asymptotic null distribution of the test statistic does not depend on unknown parameters. In the contrary case, both ϕA and ϕB have the same order of level error. Certain class...

Concomitant-Variable Latent-Class Models

Abstract: This article introduces and illustrates a new type of latent-class model in which the probability of latent-class membership is functionally related to concomitant variables with known distribution. The function (or so-called submodel) may be logistic, exponential, or another suitable form. Concomitant-variable models supplement latent-class models incorporating grouping by providing more parsimonious representations of data for some cases. Also, concomitant-variable models are useful when grouping models involve a greater number of parameters than can be meaningfully fit to existing data sets. Although parameter estimates may be calculated using standard iterative procedures such as the Newton—Raphson method, sample analyses presented here employ a derivative-free approach known as the simplex method. A general procedure for imposing linear constraints on the parameter estimates is introduced. A data set involving arithmetic test items in a mastery testing context is used to illustrate fitting a...

Resampling Inference with Complex Survey Data

Abstract: Methods for standard errors and confidence intervals for nonlinear statistics —such as ratios, regression, and correlation coefficients—have been extensively studied for stratified multistage designs in which the clusters are sampled with replacement, in particular, the important special case of two sampled clusters per stratum. These methods include the customary linearization (or Taylor) method and resampling methods based on the jackknife and balanced repeated replication (BRR). Unlike the jackknife or the BRR, the linearization method is applicable to general sampling designs, but it involves a separate variance formula for each nonlinear statistic, thereby requiring additional programming efforts. Both the jackknife and the BRR use a single variance formula for all nonlinear statistics, but they are more computing-intensive. The resampling methods developed here retain these features of the jackknife and the BRR, yet permit extension to more complex designs involving sampling without replace...

How Far are Automatically Chosen Regression Smoothing Parameters from their Optimum

Abstract: We address the problem of smoothing parameter selection for nonparametric curve estimators in the specific context of kernel regression estimation. Call the “optimal bandwidth” the minimizer of the average squared error. We consider several automatically selected bandwidths that approximate the optimum. How far are the automatically selected bandwidths from the optimum? The answer is studied theoretically and through simulations. The theoretical results include a central limit theorem that quantifies the convergence rate and gives the differences asymptotic distribution. The convergence rate turns out to be excruciatingly slow. This is not too disappointing, because this rate is of the same order as the convergence rate of the difference between the minimizers of the average squared error and the mean average squared error. In some simulations by John Rice, the selectors considered here performed quite differently from each other. We anticipated that these differences would be reflected in differ...

Telephone survey methods : sampling, selection, and supervision

Abstract: INTRODUCTION Total Survey Error Telephone Surveying in Perspective The Telephone Phenomenon in the United States Basic Steps in the Telephone Survey Process Computer-Assisted Telephone Interviewing (CATI) Contents and Organization of This Text Additional Sources of Instruction on Survey Methodologies Exercises GENERATING TELEPHONE SURVEY SAMPLING POOLS Choosing a Valid Sampling Design Random Digit Dialing (RDD) Directory and List-Based Sampling Pools Mixed Mode Sampling Pools Sampling Pool Size Exercises PROCESSING TELEPHONE SURVEY SAMPLING POOLS Issues in Controlling the Sampling Pool Using the Call-Sheet to Control the Sampling Pool Refusal Report Form, Refusal Conversions, and Nonresponse Error Expectations in Processing Sampling Pools Sampling Pool Dispositions and Survey Response and Nonresponse Rates Sampling Pools in Mixed Mode Surveys Processing Sampling Pools with CATI Exercises SELECTING RESPONDENTS AND SECURING COOPERATION Starting Off on a Solid Footing Introducing the Survey Respondent Selection and Screening Techniques Advance Contract of Respondents Exercises SUPERVISION I STRUCTURING INTERVIEWERS' WORK Quality Control in Telephone Versus Personal Interviewing Recruitment and Hiring of Interviewers Training Sessions Presurvey Practice and On-the-Job Training Total Survey Error and the Interviewer Exercises SUPERVISION II STRUCTURING SUPERVISORY WORK Staffing and Scheduling Interviewing Sessions Interviewing Session Setup Supervising Interviewing Sessions in PAPI and CATI Surveys Verifying Completed Interviews Conclusion Exercises

High Breakdown-Point Estimates of Regression by Means of the Minimization of an Efficient Scale

Abstract: A new class of robust estimates, τ estimates, is introduced. The estimates have simultaneously the following properties: (a) they are qualitatively robust, (b) their breakdown point is .5, and (c) they are highly efficient for regression models with normal errors. They are defined by minimizing a new scale estimate, τ, applied to the residuals. Asymptotically, a τ estimate is equivalent to an M estimate with a ψ function given by a weighted average of two ψ functions, one corresponding to a very robust estimate and the other to a highly efficient estimate. The weights are adaptive and depend on the underlying error distribution. We prove consistency and asymptotic normality and give a convergent iterative computing algorithm. Finally, we compare the biases produced by gross error contamination in the τ estimates and optimal bounded-influence estimates.

Tree-Structured Classification via Generalized Discriminant Analysis.

Abstract: The problem of constructing classification rules that can be represented as decision trees is considered. Each object to be classified has an associated x vector containing possibly incomplete covariate information. Tree construction is based on the information provided in a “learning sample” of objects with known class identities. The x vectors in the learning sample may have missing values as well. Procedures are proposed for each of the components of classifier construction, such as split selection, tree-size determination, treatment of missing values, and ranking of variables. The main idea is recursive application of linear discriminant analysis, with the variables at each stage being appropriately chosen according to the data and the type of splits desired. Standard statistical techniques used as basic building blocks include analysis of variance, linear and canonical discriminant analysis, and principal component analysis. A new method of tree-structured classification is obtained by assem...

Randomized Response : Theory and Techniques

Abstract: Offering a concise account of the most appropriate and efficient procedures for analyzing data from queries dealing with sensitive and confidential issues- including the first book-length treatment of infinite and finite population set-ups - this volume begins with the simplest problems, complete with their properties and solutions, and proceeds to incrementally more difficult topics. Randomized Response is mandatory reading for statisticians and biostatisticians, market researchers, operations researchers, pollsters, sociologists, political scientists, economists and advanced undergraduate and graduate students in these areas.

A Method for Obtaining Short-Term Projections and Lower Bounds on the Size of the AIDS Epidemic

Abstract: A methodology is proposed for obtaining short-term projections of the acquired immunodeficiency syndrome (AIDS) epidemic by projecting the number of cases from those already infected with the AIDS virus. This is a lower bound on the size of the AIDS epidemic, because even if future infections could be prevented, one could still anticipate this number of cases. The methodology is novel in that no assumptions are required about either the number of infected individuals in the population or the probability of an infected individual eventually developing AIDS. The methodology presupposes knowledge of the incubation distribution, however, among those destined to develop AIDS. Although the method does not account for new infections, it may produce accurate short-term projections because of the relatively long incubation period from infection to clinically diagnosed AIDS. The estimation procedure “back-calculates” from AIDS incidence data to numbers previously infected. The number of cases diagnosed in ...

Identification of Binary Response Models

Abstract: This article studies identification of the threshold-crossing model of binary response. Most research on binary response has considered specific estimators and tests. The study of identification exposes the foundations of binary response analysis by making explicit the assumptions needed to justify different methods. It also clarifies the connections between reduced-form and structural analyses of binary response data. Assume that the binary outcome z is determined by an observable random vector x and by an unobservable scalar u through a model z = 1[xβ + u ≤ 0]. Also assume that Fu|x , the probability distribution of u conditional on x, is continuous and strictly increasing. Given these maintained assumptions, we investigate the identifiability of β given the following restrictions on the distributions (Fu|x, x ∈ X): mean independence, quantile independence, index sufficiency, statistical independence, and statistical independence with the distribution known. We find that mean independence has n...

Bayesian Confidence Intervals for Smoothing Splines

Abstract: The frequency properties of Wahba's Bayesian confidence intervals for smoothing splines are investigated by a large-sample approximation and by a simulation study. When the coverage probabilities for these pointwise confidence intervals are averaged across the observation points, the average coverage probability (ACP) should be close to the nominal level. From a frequency point of view, this agreement occurs because the average posterior variance for the spline is similar to a consistent estimate of the average squared error and because the average squared bias is a modest fraction of the total average squared error. These properties are independent of the Bayesian assumptions used to derive this confidence procedure, and they explain why the ACP is accurate for functions that are much smoother than the sample paths prescribed by the prior. This analysis accounts for the choice of the smoothing parameter (bandwidth) using cross-validation. In the case of natural splines an adaptive method for avo...

Characterization of a Ranked-Set Sample with Application to Estimating Distribution Functions

Abstract: Ranked-set sampling has been shown to provide improved estimators of the mean and variance when actual measurement of the observations is difficult but ranking of the elements in a sample is relatively easy. This result holds even if the ranking is imperfect. In this article, we provide a characterization of a ranked-set sample that makes the source of this additional information intuitively clear. It is applied to show that the empirical distribution function of a ranked-set sample is unbiased and has greater precision than that from a random sample. The null distribution of a Kolmogorov-Smirnov statistic based on this empirical distribution function is derived for the case in which perfect ranking is possible. It is seen to be stochastically smaller than the usual Kolmogorov-Smirnov statistic based on a simple random sample, resulting in a smaller simultaneous confidence interval for the cumulative distribution function.

The Bias of Autoregressive Coefficient Estimators

Abstract: This article presents simple expressions for the bias of estimators of the coefficients of an autoregressive model of arbitrary, but known, finite order. The results include models both with and without a constant term. The effects of overspecification of the model order on the bias are described. The emphasis is on least-squares and Yule-Walker estimators, but the methods extend to other estimators of similar design. Although only the order T -1 component of the bias is captured, where T is the series length, this asymptotic approximation is shown to be very accurate for least-squares estimators through some numerical simulations. The simulations examine fourth-order autoregressions chosen to resemble some data series from the literature. The order T -1 bias approximations for Yule-Walker estimators need not be accurate, especially if the zeros of the associated polynomial have moduli near 1. Examples are given where the approximation is accurate and where it is useless. The bias expressions are...

Generalized Logistic Models

Abstract: A class of models indexed by two shape parameters is introduced, both to extend the scope of the standard logistic model to asymmetric probability curves and improve the fit in the noncentral probability regions. One-parameter subclasses can be used to examine symmetric or asymmetric deviations from the logistic model. The delta algorithm is adapted to obtain maximum likelihood estimates of the parameters. A review is made of other proposed generalizations. The standard linear logistic model is widely used for modeling the dependence of binary data on explanatory variables. Its success is due to its broad applicability, simplicity of form, and ease of interpretation. This model works well for many common applications; however, it assumes that the expected probability curve μ(η) is skew-symmetric about μ = ½ and that the shape of μ(η) is the cumulative distribution function of the logistic distribution. Symmetric data with a shallower or steeper slope of ascent may not be fitted well by this model...

Bootstrapping in Nonparametric Regression: Local Adaptive Smoothing and Confidence Bands

Abstract: The operation of the bootstrap in the context of nonparametric regression is considered. Bootstrap samples are taken from estimated residuals to study the distribution of a suitably recentered kernel estimator. The application of this principle to the problem of local adaptive choice of bandwidth and to the construction of confidence bands is investigated and compared with a direct method based on asymptotic means and variances. The technique of the bootstrap is to replace any occurrence of the unknown distribution in the definition of the statistical function of interest by the empirical distribution function of the observed errors. In a regression context these errors are not directly observed, although their role can be played by the residuals from the fitted model. In this article the fitted model is a kernel nonparametric regression estimator. Since nonparametric smoothing is involved, an additional difficulty is created by the bias incurred in smoothing. This bias, however, can be estimated...

Regression Analysis for Categorical Variables with Outcome Subject to Nonignorable Nonresponse

Abstract: We develop a log-linear model for categorical response subject to nonignorable nonresponse. The paper differs from Fay (1986) in its focus on estimation and hypothesis testing in a regression setting, as opposed to imputation in a multivariate setting. We present several new results concerning the existence of solutions on the boundary of the parameter space and the construction of confidence intervals for estimates. We illustrate the method by estimating the proportion of voters preferring Truman in a 1948 preelection poll (Mosteller, Hyman, McCarthy, Marks, and Truman 1949). Results may depend strongly on the model assumed for nonresponse; goodness-of-fit tests are available for comparing alternative models.

Estimating Transformations for Regression via Additivity and Variance Stabilization

Abstract: I propose a method for the nonparametric estimation of transformations for regression. It is much more flexible than the familiar Box-Cox procedure, allowing general smooth transformations of the variables, and is similar to the ACE (alternating conditional expectation) algorithm of Breiman and Friedman (1985). The ACE procedure uses scatterplot smoothers in an iterative fashion to find the maximally correlated transformations of the variables. Like ACE, my proposal can incorporate continuous, categorical, or periodic variables, or any mixture of these types. The method differs from ACE in that it uses a (nonparametric) variance-stabilizing transformation for the response variable. The technique seems to alleviate many of the anomalies that ACE suffers with regression data, including the inability to reproduce model transformations and sensitivity to the marginal distribution of the predictors. I provide several examples, including an analysis of the “brain and body weight” data and some data on ...