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Cross-trait polygenic risk score (PRS) method has gained popularity for assessing genetic correlation of complex traits using summary statistics from biobank-scale genome-wide association studies (GWAS). However, empirical evidence has shown a common bias phenomenon that highly significant cross-trait PRS can only account for a very small amount of genetic variance (R2 can be < 1 % ) in independent testing GWAS. The aim of this paper is to investigate and address the bias phenomenon of cross-trait PRS in numerous GWAS applications. We show that the estimated genetic correlation can be asymptotically biased toward zero. A consistent cross-trait PRS estimator is then proposed to correct such asymptotic bias. In addition, we investigate whether or not SNP screening by GWAS p-values can lead to improved estimation and show the effect of overlapping samples among GWAS. We analyze GWAS summary statistics of reaction time and brain structural magnetic resonance imaging-based features measured in the Pediatric Imaging, Neurocognition, and Genetics study. We find that the raw cross-trait PRS estimators heavily underestimate the genetic similarity between cognitive function and human brain structures (mean R 2 = 1.32 % ), whereas the bias-corrected estimators uncover the moderate degree of genetic overlap between these closely related heritable traits (mean R 2 = 22.42 % ). Supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement. [ABSTRACT FROM AUTHOR]

In statistical prediction, classical approaches for model selection and model evaluation based on covariance penalties are still widely used. Most of the literature on this topic is based on what we call the "Fixed-X" assumption, where covariate values are assumed to be nonrandom. By contrast, it is often more reasonable to take a "Random-X" view, where the covariate values are independently drawn for both training and prediction. To study the applicability of covariance penalties in this setting, we propose a decomposition of Random-X prediction error in which the randomness in the covariates contributes to both the bias and variance components. This decomposition is general, but we concentrate on the fundamental case of ordinary least-squares (OLS) regression. We prove that in this setting the move from Fixed-X to Random-X prediction results in an increase in both bias and variance. When the covariates are normally distributed and the linear model is unbiased, all terms in this decomposition are explicitly computable, which yields an extension of Mallows' Cp that we call RCp. RCp also holds asymptotically for certain classes of nonnormal covariates. When the noise variance is unknown, plugging in the usual unbiased estimate leads to an approach that we call RCp ^ , which is closely related to Sp, and generalized cross-validation (GCV). For excess bias, we propose an estimate based on the "shortcut-formula" for ordinary cross-validation (OCV), resulting in an approach we call RCp+. Theoretical arguments and numerical simulations suggest that RCp+ is typically superior to OCV, though the difference is small. We further examine the Random-X error of other popular estimators. The surprising result we get for ridge regression is that, in the heavily regularized regime, Random-X variance is smaller than Fixed-X variance, which can lead to smaller overall Random-X error. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

Literature on change point analysis mostly requires a sudden change in the data distribution, either in a few parameters or the distribution as a whole. We are interested in the scenario, where the variance of data may make a significant jump while the mean changes in a smooth fashion. The motivation is a liver procurement experiment monitoring organ surface temperature. Blindly applying the existing methods to the example can yield erroneous change point estimates since the smoothly changing mean violates the sudden-change assumption. We propose a penalized weighted least-squares approach with an iterative estimation procedure that integrates variance change point detection and smooth mean function estimation. The procedure starts with a consistent initial mean estimate ignoring the variance heterogeneity. Given the variance components the mean function is estimated by smoothing splines as the minimizer of the penalized weighted least squares. Given the mean function, we propose a likelihood ratio test statistic for identifying the variance change point. The null distribution of the test statistic is derived together with the rates of convergence of all the parameter estimates. Simulations show excellent performance of the proposed method. Application analysis offers numerical support to non invasive organ viability assessment by surface temperature monitoring. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

Direct regression modeling of the subdistribution has become popular for analyzing data with multiple, competing event types. All general approaches so far are based on nonlikelihood-based procedures and target covariate effects on the subdistribution. We introduce a novel weighted likelihood function that allows for a direct extension of the Fine–Gray model to a broad class of semiparametric regression models. The model accommodates time-dependent covariate effects on the subdistribution hazard. To motivate the proposed likelihood method, we derive standard nonparametric estimators and discuss a new interpretation based on pseudo risk sets. We establish consistency and asymptotic normality of the estimators and propose a sandwich estimator of the variance. In comprehensive simulation studies, we demonstrate the solid performance of the weighted nonparametric maximum likelihood estimation in the presence of independent right censoring. We provide an application to a very large bone marrow transplant dataset, thereby illustrating its practical utility. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

Error variance estimation plays an important role in statistical inference for high-dimensional regression models. This article concerns with error variance estimation in high-dimensional sparse additive model. We study the asymptotic behavior of the traditional mean squared errors, the naive estimate of error variance, and show that it may significantly underestimate the error variance due to spurious correlations that are even higher in nonparametric models than linear models. We further propose an accurate estimate for error variance in ultrahigh-dimensional sparse additive model by effectively integrating sure independence screening and refitted cross-validation techniques. The root n consistency and the asymptotic normality of the resulting estimate are established. We conduct Monte Carlo simulation study to examine the finite sample performance of the newly proposed estimate. A real data example is used to illustrate the proposed methodology. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

Replication procedures have proven useful for variance estimation for large scale complex surveys. As an extension of bootstrap procedures to rejective samples, we define a bootstrap sample that is a rejective, unequal probability, replacement sample selected from the original sample. A modification of the bootstrap with improved performance is suggested for stratified samples with small stratum sizes. Simulations for Poisson and stratified rejective samples support the use of replicates in estimating the variance of the regression estimator for rejective samples. [ABSTRACT FROM AUTHOR]

Motivated by an observational study of the effect of hospital ward versus intensive care unit admission on severe sepsis mortality, we develop methods to address two common problems in observational studies: (1) when there is a lack of covariate overlap between the treated and control groups, how to define an interpretable study population wherein inference can be conducted without extrapolating with respect to important variables; and (2) how to use randomization inference to form confidence intervals for the average treatment effect with binary outcomes. Our solution to problem (1) incorporates existing suggestions in the literature while yielding a study population that is easily understood in terms of the covariates themselves, and can be solved using an efficient branch-and-bound algorithm. We address problem (2) by solving a linear integer program to use the worst-case variance of the average treatment effect among values for unobserved potential outcomes that are compatible with the null hypothesis. Our analysis finds no evidence for a difference between the 60-day mortality rates if all individuals were admitted to the ICU and if all patients were admitted to the hospital ward among less severely ill patients and among patients with cryptic septic shock. We implement our methodology in R, providing scripts in the supplementary material. [ABSTRACT FROM PUBLISHER]

When the empirical likelihood (EL) of a parameter θ is constructed with right censored data, literature shows that − 2log (empirical likelihood ratio) typically has an asymptotic scaled chi-squared distribution, where the scale parameter is a function of some unknown asymptotic variances. Therefore, the EL construction of confidence intervals for θ requires an additional estimation of the scale parameter. Additional estimation would reduce the coverage accuracy for θ. By using a special influence function as an estimating function, we prove that under very general conditions, − 2log (empirical likelihood ratio) has an asymptotic standard chi-squared distribution with one degree of freedom. This eliminates the need for estimating the scale parameter as well as eases some of the often demanding computations of the EL method. Our estimating function yields a smaller asymptotic variance than those of Wang and Jing (2001) and Qin and Zhao (2007). Thus, it is not surprising that confidence intervals using the special influence functions give a better coverage accuracy as demonstrated by simulations. [ABSTRACT FROM PUBLISHER]

We develop nonparametric smoothing for regression when both the predictor and the response variables are defined on a sphere of whatever dimension. A local polynomial fitting approach is pursued, which retains all the advantages in terms of rate optimality, interpretability, and ease of implementation widely observed in the standard setting. Our estimates have a multi-output nature, meaning that each coordinate is separately estimated, within a scheme of a regression with a linear response. The main properties include linearity and rotational equivariance. This research has been motivated by the fact that very few models describe this kind of regression. Such current methods are surely not widely employable since they have a parametric nature, and also require the same dimensionality for prediction and response spaces, along with nonrandom design. Our approach does not suffer these limitations. Real-data case studies and simulation experiments are used to illustrate the effectiveness of the method. [ABSTRACT FROM AUTHOR]

Using the von Mises expansion, we study the higher-order infinitesimal robustness of a generalM-functional and characterize its second-order properties. We show that second-order robustness is equivalent to the boundedness of both the estimator’s estimating function and its derivative with respect to the parameter. It implies, at the same time, (i) variance robustness and (ii) robustness of higher-order saddlepoint approximations to the estimator’s finite sample density. The proposed construction of second-order robustM-estimators is fairly general and potentially useful in a variety of relevant settings. Besides the theoretical contributions, we discuss the main computational issues and provide an algorithm for the implementation of second-order robustM-estimators. Finally, we illustrate our theory by Monte Carlo simulation and in a real-data estimation of the maximal losses of Nikkei 225 index returns. Our findings indicate that second-order robust estimators can improve on other widely applied robust estimators, in terms of efficiency and robustness, for moderate to small sample sizes and in the presence of deviations from ideal parametric models. Supplementary materials for this article are available online. [ABSTRACT FROM PUBLISHER]

In many applications, time series exhibit nonstationary behavior that might reasonably be modeled as a time-varying autoregressive (AR) process. In the context of such a model, we discuss the problem of testing for modality of the variance function. We propose a test of modality that is local and, when used iteratively, can be used to identify the total number of modes in a given series. This problem is closely related to peak detection and identification, which has applications in many fields. We propose a test that, under appropriate assumptions, is asymptotically distribution free under the null hypothesis, even though nonparametric estimation of the AR parameter functions is involved. Simulation studies and applications to real datasets illustrate the behavior of the test. [ABSTRACT FROM PUBLISHER]

In longitudinal studies, mixed-effects models are important for addressing subject-specific effects. However, most existing approaches assume a normal distribution for the random effects, and this could affect the bias and efficiency of the fixed-effects estimator. Even in cases where the estimation of the fixed effects is robust with a misspecified distribution of the random effects, the estimation of the random effects could be invalid. We propose a new approach to estimate fixed and random effects using conditional quadratic inference functions (QIFs). The new approach does not require the specification of likelihood functions or a normality assumption for random effects. It can also accommodate serial correlation between observations within the same cluster, in addition to mixed-effects modeling. Other advantages include not requiring the estimation of the unknown variance components associated with the random effects, or the nuisance parameters associated with the working correlations. We establish asymptotic results for the fixed-effect parameter estimators that do not rely on the consistency of the random-effect estimators. Real data examples and simulations are used to compare the new approach with the penalized quasi-likelihood (PQL) approach, and SAS GLIMMIX and nonlinear mixed-effects model (NLMIXED) procedures. Supplemental materials including technical details are available online. [ABSTRACT FROM AUTHOR]

We propose a general strategy for variable selection in semiparametric regression models by penalizing appropriate estimating functions. Important applications include semiparametric linear regression with censored responses and semiparametric regression with missing predictors. Unlike the existing penalized maximum likelihood estimators, the proposed penalized estimating functions may not pertain to the derivatives of any objective functions and may be discrete in the regression coefficients. We establish a general asymptotic theory for penalized estimating functions and present suitable numerical algorithms to implement the proposed estimators. In addition, we develop a resampling technique to estimate the variances of the estimated regression coefficients when the asymptotic variances cannot be evaluated directly. Simulation studies demonstrate that the proposed methods perform well in variable selection and variance estimation. We illustrate our methods using data from the Paul Coverdell Stroke Registry. [ABSTRACT FROM AUTHOR]

Optimal incomplete block designs are pursued through the E criterion of minimizing maximal variance. Methodology is developed for design choice based on graphs and an extensive catalog of downloadable designs is complied. Along with the general methodology, a near complete solution for E-optimal block designs is provided for up to 15 treatments. E optimality is revealed to be a flexible criterion that, depending on the application, can offer many choices for good designs. [ABSTRACT FROM AUTHOR]

Microarray technology allows a scientist to study genomewide patterns of gene expression. Thousands of individual genes are measured with a relatively small number of replications, which poses challenges to traditional statistical methods. In particular, the gone-specific estimators of variances are not reliable and gene-by-gene tests have low powers. In this article we propose a family of shrinkage estimators for variances raised to a fixed power. We derive optimal shrinkage parameters under both Stein and squared loss functions. Our results show that the standard sample variance is inadmissible under either loss function. We propose several estimators for the optimal shrinkage parameters and investigate their asymptotic properties under two scenarios: large number of replications and large number of genes. We conduct simulations to evaluate the finite sample performance of the data-driven optimal shrinkage estimators and compare them with some existing methods. We construct F-like statistics using these shrinkage variance estimators and apply them to detect differentially expressed genes in a microarray experiment. We also conduct simulations to evaluate performance of these F-like statistics and compare them with some existing methods. [ABSTRACT FROM AUTHOR]

We present two models for the short-term prediction of the number of deaths arising from common cancers in the United States. The first is a local linear model, in which the slope of the segment joining the number of deaths for any two consecutive time period is assumed to be random with a nonparametric distribution, which has a Dirichlet process prior. For slightly longer prediction periods, we present a local quadratic model. This extension of the local linear model includes an additional "acceleration" term that allows it to quickly adjust to sudden changes in the time series. The proposed models can be used to obtain the predictive distributions of the future number of deaths. as well their meals and variances through Markov chain Monte Carlo techniques. We illustrate our methods by runs on data from selected cancer sites. [ABSTRACT FROM AUTHOR]

A jackknife or balanced repeated replication variance estimator in a large survey typically requires a large number of replicates and replicate weights. Reducing the number of replicates may have important advantages for computations and for limiting the risk of data disclosure from public use data files. This article proposes algorithms adapted from scheduling theory to combine variance strata and, thus, reduce the number of replicates. The algorithms are simple and efficient and can be adapted to easily account for vector characteristics and analytic domains. An important concern with combining strata is that the resulting variance estimators may be inconsistent. We establish conditions for the consistency of the combined variance estimator and show that the proposed algorithms ensure they are met. We also derive bounds on the degrees of freedom that the algorithms will assure. The algorithms are applied both to a real sample survey and to samples from simulated populations, and the algorithms perform very well, attaining variance estimators with precision levels close to the upper bounds. [ABSTRACT FROM AUTHOR]

In survey sampling there is often a need to coordinate the selection of pairs of samples drawn from two overlapping populations so as to maximize or minimize their expected overlap, subject to constraints on the marginal probabilities determined by the respective designs. For instance, maximizing the expected overlap between repeated samples can stabilize the resulting estimates of change and reduce the costs of first contacts; minimizing the expected overlap can avoid overburdening respondents with multiple surveys. We focus on the important special case in which both samples are selected by simple random sampling without replacement (SRSWOR) conducted independently within each stratum. Optimizing the expected sample overlap can be formulated as a linear programming problem known as a transportation problem (TP). We show that by appropriately grouping and ordering the possible samples in each survey, one can reduce the initial TP to a much smaller TP amenable to solution by an algorithm known as the Northwest Corner Rule (NWCR). The proposed NWCR method proceeds in two easily implemented steps: first selecting the numbers of births (new units) and deaths (deleted units) by a random selection from a hypergeometric distribution, and then selecting the births and deaths by SRSWOR. We formally prove properties of the NWCR solutions, including a minimal variance property of the minimal overlap solution. In a simulation study, the NWCR method compares favorably with a popular method based on assignment of permanent random numbers to each sampling unit. [ABSTRACT FROM AUTHOR]

The study of water waves has generated a wealth of sophisticated modeling developments in applied mathematics. Empirical observation capabilities have created a need for novel data analysis tools. This article is motivated by consideration of wind-generated wave field image data from a wave tank facility. A quantitative measure of wave field regularity is developed. The methodology is based on decomposition of the wave field into simple plane waves (an adaptation of projection pursuit). The percent variance explained as a function of the number of terms in the plane wave representation is used to define a variogram, and regularity of the wave field is assessed in terms of the weighted difference between the observed variogram and the expected variogram for a completely random field. The proposed regularity measure is illustrated by application to image data from wind-generated waves. The results suggest that the regularity measure is a function of parameters describing the generation of the wave field (wind speed and evolution). An analysis of the statistical behavior of the regularity measure as a function of sample size (image resolution) is carried out. Even though the regularity measure is based on the nonparametric estimation of functions, provided that this estimation is carried out in a consistent fashion, the error in estimation of regularity has a parametric dependence on image resolution. [ABSTRACT FROM AUTHOR]

In this article we study the semiparametric proportional odds model with random effects for correlated, right-censored failure time data. We establish that the maximum likelihood estimators for the parameters of this model are consistent and asymptotically Gaussian. Furthermore, the limiting variances achieve the semiparametric efficiency bounds and can be consistently estimated. Simulation studies show that the asymptotic approximations are accurate for practical sample sizes and that the efficiency gains of the proposed estimators over those of Cai, Cheng, and Wei can be substantial. A real example is provided to illustrate the proposed methods. [ABSTRACT FROM AUTHOR]

The article presents a rejoinder by the author on his article "Sieve Bootstrap With Variable-Length Markov Chains (VLMC) for Stationary Categorical Time Series." Scholar D.N. Politis suggests that we should compare the VLMC sieve bootstrap with a full Markov sieve bootstrap that resamples from an estimated full Markov model of suitable order p. This is a natural first idea in the setting of categorical time series and it is certainly easier to implement than its VLMC cousin. Some problems should he pointed out, however. The full Markov sieve bootstrap will often be faced with the problem of how to estimate transition probabilities for states that never occurred in the data. The implementation derived in an equation puts some parsimony on estimating never-occurring states, using an ad hoc rule. VLMCs deal in a formalized way not only with never-occurring states, but also with putting parsimony on rarely occurring states. Then why should we prefer ad hoc rules over the much better-formalized VLMC fitting technique, except that the latter requires much more sophisticated computation.

A bootstrap for stationary categorical time series based on the method of sieves is studied here. The data generating process is approximated by the so-called variable length Markov chain (VLMC), a flexible class of Markov models that allows for parsimonious structure. Then the resampling is given by simulating from the fitted model. It is shown that for a whole class of stationary categorical time series that is more general than VLMC, the VLMC sieve has faster rate of convergence for variance estimation than the more general block bootstrap. Results are illustrated from a theoretical and empirical perspective. Foe the latter, a real data application about (in-) homogeneity classification of a DNA stand is also presented. Finally, the VLMC sieve scheme enjoys an implementational advantage of using the plug-in rule for bootstrapping a statistical procedure, which generally is not the case for the block method. [ABSTRACT FROM AUTHOR]

The article presents comments of the author on the article "Sieve Bootstrap With Variable-Length Markov Chains (VLMC) Stationary Categorical Time Series," by Peter Bühlmann. This article is an interesting alternative to the "block subsampling" principle for data arising from temporally or spatially correlated processes. An important novel feature here is the semiparametric nature of his model and subsampling algorithm. The extension of dependent data subsampling in this direction is technically challenging and possibly of great practical value. I briefly discuss the principle behind subsampling and its historical development, suggest possible alternatives to the VLMC, make some comments on the comparison between the block bootstrap and the VLMC and finally suggest possible extensions to spatial data. The usual objective in subsampling, or resampling, is to construct an estimator of the unknown sampling distribution of some random quantity. One example of a random quantity is a standardized statistic to he used as a pivot for confidence intervals. More simply, the user may wish to estimate the variance or other moments of a complicated statistic.

Nearest-neighbor imputation is a popular hot deck imputation method used to compensate for nonresponse in sample surveys. Although this method has a long history of application, the problem of variance estimation after nearest-neighbor imputation has not been fully investigated. Because nearest-neighbor imputation is a nonparametric method, a nonparametric variance estimation technique, such as the jackknife, is desired. We show that the naive jackknife that treats imputed values as observed data produces serious underestimation. We also show that Rao and Shao's adjusted jackknife, or the jackknife with each pseudoreplicate reimputed, which produces asymptotically unbiased and consistent jackknife variance estimators for other imputation methods (such as mean imputation, random hot deck imputation, and ratio or regression imputation), produces serious overestimation in the case of nearest-neighbor imputation. Two partially reimputed and a partially adjusted jackknife variance estimators are proposed and shown to be asymptotically unbiased and consistent. Some empirical results are provided to examine finite-sample properties of these jackknife variance estimators. [ABSTRACT FROM AUTHOR]

Various aspects of the wavelet approach to nonparametric regression are considered, with the overall aim of extending the scope of wavelet techniques to irregularly spaced data, to regularly spaced datasets of arbitrary size, to heteroscedastic and correlated data, and to data that contain outliers. The core of the methodology is an algorithm for finding all of the variances and within-level covariances in the wavelet table of a sequence with given covariance structure. If the original covariance matrix is band-limited, then the algorithm is linear in the length of the sequence. The variance calculation algorithm allows data on any set of independent variable values to be treated, by first interpolating to a fine regular grid of suitable length, and then constructing a wavelet expansion of the gridded data. Various thresholding methods are discussed and investigated. Exact risk formulas for the mean square error of the methodology for given design are derived. Good performance is obtained by noise-proportional thresholding, with thresholds somewhat smaller than the classical universal threshold. Outliers in the data can be removed or downweighted, and aspects of such robust techniques are developed and demonstrated in an example. Another natural application is to correlated data, where the covariance of the wavelet coefficients is not due to an initial grid transform but rather is an intrinsic feature. The use of the method in these circumstances is demonstrated by an application to data synthesized in the study of ion channel gating. Our basic approach has many other potential applications, some of which are discussed briefly. [ABSTRACT FROM AUTHOR]

One commonly observed response pattern in a one-factor design with ordered treatment levels is the umbrella or unimodal ordering, in which the response variable increases with an increase in the treatment level up to a point, then decreases with further increase in the treatment level. The turning point is called the peak of an umbrella ordering. In many practical settings, there are two or more different populations with umbrella orderings, and it is of interest to compare these populations. In this article, likelihood ratio tests are developed for testing whether two groups with umbrella orderings have the same peaks. Three cases are considered: variances known, variances unknown and different for each group, and variances unknown and the same for both groups. Extensions to randomized block designs and to a more general class of testing problems are discussed. [ABSTRACT FROM AUTHOR]

This article extends Royall's (1992) results on optimal and robust sampling strategies to cluster sampling. It also gives the lower bound on the model-based variances of best linear unbiased predictors of finite population totals under certain classes of superpopulation models. [ABSTRACT FROM AUTHOR]

The Theil-Sen estimator of the slope parameter in simple linear regression is extended to data with both the response and the covariate subject to censoring. Based on inverting a suitable version of Kendall's r statistic, this estimator requires weak assumptions and is simple to compute, and a simple estimate of its asymptotic variance is obtained. A second extension of the Theil-Sen estimator, based on a direct estimation of the median of pairwise slopes, is given. These estimators are compared numerically with versions of Schmitt's estimator and applied to two data sets from the recent astronomical literature. [ABSTRACT FROM AUTHOR]

We study the possibility of identifying general linear and nonlinear time series models using nonparametric methods. The kernel estimators of the conditional mean and variance are used as a basis, and the properties of these quantities as model indicators are briefly discussed. Some drawbacks are pointed out, and motivated by these we introduce projections as tools of identification. The projections are especially useful for additive modeling. Expressions for the asymptotic bias and variance are obtained. The projection of the conditional variance is suggested as a tool for identifying heteroscedastic time series. The results are illustrated by simulations for both the estimators of the projections and the estimators of the conditional mean and variance. [ABSTRACT FROM AUTHOR]

In this article we suggest a nonparametric procedure for selecting significant lags in the model description of a general nonlinear stationary time series The procedure can be applied to both the conditional mean and the conditional variance and is valid for heteroscedastic series The procedure is illustrated by simulations and sunspot data, lynx data, and blowfly data are analyzed It is indicated that projectors can be used in conjunction with the procedure for selecting significant tags to check the adequacy of an additive time series model. [ABSTRACT FROM AUTHOR]

The comparison of primary interest in a 2 x 2 crossover trial typically concerns the effect of the treatments, say A and B, on the mean response level. This article deals with another important aspect, namely the within-subject response variability under A and B. Differences in drug formulation and/or administration may lead to considerable differences in within-subject variability, whatever the difference in terms of mean level. Consideration of both these aspects may, therefore, be of considerable importance for the judgement of the treatments. It is shown that although there are no within-subject treatment replications, it is possible to make various exact references about the A/B ratio of within-subject variances and about the (A - B)-difference in mean level, simultaneously and marginally. These inferences are semiparametric in that no distributional assumption is made about the between-subject variability, whereas a normality assumption is used for the within-subject variability. The references include tests, confidence regions, and a multiple test procedure. A power approximation is also given. The results are illustrated numerically. [ABSTRACT FROM AUTHOR]

Consider an experiment where the factors are measured on a continuous scale, and suppose that the experimenter is permitted to augment the existing observations with one or more new data points Bondar's universal optimality criterion (U optimality) suggests that the problem is best studied in the eigenvector coordinate system. We proceed by showing how to construct new points that first equate and then jointly increase the smaller eigenvalues of the crossproduct of the independent variables. We discuss the limitations of design augmentation strategies based solely on the crossproduct matrix Our goals are to equate and minimize the variances of the estimated regression coefficients, keep the new points constrained in a prespecified experimental region, use as much of the information in the original points as possible, and keep the number of required new points as small as possible. We offer several data augmentation strategies to meet these requirements. As U optimality subsumes each of the D, A, E, and (M, S) optimality criteria, our strategies guarantee the U, D, A, E, and (M, S) optimality of the set of new points. The recommended number of additional points depends on how nearly optimal is the layout based on the existing observations and on the tightness of the regional constraints. We illustrate our strategies with a well-known experimental data set. For an additive model, our U-optimal solutions to the data augmentation problem are superior to other solutions available in the literature. [ABSTRACT FROM AUTHOR]

In many applications available data from several independent studies address the same question, and it is essential to have statistical methods for combining the results from the different studies. This article addresses this issue in two setups. (1) a testing hypothesis concerning the common mean vector of two independent linear models having different variances, and (2) a testing hypothesis concerning a common variance component in linear models revolving two variance components The interblock analysis of a balanced incomplete block design (BIBD) is a special case of (1) when we are interested in testing the equality of the treatment effects Testing the significance of the treatment variance component in a BIBD with random effects is a special case of (2). We suggest some new test procedures for the testing problems in (1) and (2) and also give a review of the various existing tests. We numerically compare the powers of the various tests and make specific recommendations regarding the choice of the test to be used in practical applications. [ABSTRACT FROM AUTHOR]

The definition of a confidence interval is generalized so that problems such as constructing exact confidence regions for the difference in two normal means can be tackled without the assumption of equal variances. Under certain conditions, the extended definition is shown to preserve a repeated sampling property that a practitioner expects from exact confidence intervals. The proposed procedure is also applied to the problem of constructing confidence intervals for the difference in two exponential means and for variance components in mixed models. A repeated sampling property of generalized p values is also given. With this characterization one can carry out fixed level tests of parameters of continuous distributions on the basis of generalized p values. Finally, Pratt's paradox is revisited, and a procedure that resolves the paradox is given. [ABSTRACT FROM AUTHOR]

We derive the asymptotic distribution of the maximum partial likelihood estimator beta for the vector of regression coefficients beta under a possibly misspecified Cox proportional hazards model. As in the parametric setting, this estimator beta converges to a well-defined constant vector beta. In addition, the random vector n[sup 1/2](beta - beta[sup *]) is asymptotically normal with mean 0 and with a covariance matrix that can be consistently estimated. The newly proposed robust covariance matrix estimator is similar to the so-called "sandwich" variance estimators that have been extensively studied for parametric cases. For many misspecified Cox models, the asymptotic limit fl* or part of it can be interpreted meaningfully. In those circumstances, valid statistical inferences about the corresponding covariate effects can be drawn based on the aforementioned asymptotic theory of beta and the related results for the score statistics. Extensive studies demonstrate that the proposed robust tests and interval estimation procedures are appropriate for practical use. In particular, the robust score tests perform quite well even for small samples. In contrast, the conventional model-based inference procedures often lead to tests with supranominal size and confidence intervals with rather poor coverage probability. [ABSTRACT FROM AUTHOR]

Dose--response studies are frequently used to study the, effects of an experimental compound on various responses. One characteristic of the compound that is frequently of interest is the minimum effective dose (MED). Generally, inference about the MED is made by comparing various dose groups with a control (Dunnett 1955; Williams 1971). This article examines two families of contrasts used for related problems and proposes a new family of contrasts (called basin contrasts) designed specifically for identifying the MED. Monte Carlo studies are used to demonstrate the superiority of the contrast procedures and make recommendations for their use. [ABSTRACT FROM AUTHOR]

Statistical methods for fittings nonlinear functions to data generated by correlated response variates are discussed. Estimation of the mode] parameters is performed with an iterative two-stage scheme. Thc estimation procedure accommodates both within unit and between-unit variability in fitting a response surface. Under regularity conditions the procedure yields asymptotically normal, strongly consistent estimators. If desired a patterned variance-covariance matrix can be assumed md incorporated into thc model. The methods arc illustrated by an analysis, is of data from a study of the combined effects of hepatotoxins in which between and within subject measurements are recorded. [ABSTRACT FROM AUTHOR]

We consider two-stage models of the kind used in parametric empirical Bayes (PEB) methodology, calling them conditionally independent hierarchical models. We suppose that there are k "units," which may be experimental subjects, cities, study centers, etcetera. At the first stage, the observation vectors Y, for units i = 1,..., k are independently distributed with densities p(y, | theta[sub l]), or more generally, p(y, | theta[sub l] lambda). At the second stage, the unit-specific parameter vectors g are iid with densities p(theta[sub l] | lambda). The PEB approach proceeds by regarding the second-stage distribution as a prior and noting that, if lambda were known, inference about theta could be based on its posterior. Since lambda is not known, the simplest PEB methods estimate the parameter lambda by maximum likelihood or some variant, and then treat lambda as if it were known to be equal to this estimate. Although this procedure is sometimes satisfactory, a well-known defect is that it neglects the uncertainty due to the estimation of lambda. In this article we suggest that approximate Bayesian inference can provide simple and manageable solutions to this problem. In Bayesian inferences, a prior density pi(.) on lambda is introduced, the posterior p(lambda | y) is calculated, and the posterior density of theta[sub l] is then equal to the expectation, with respect to p(lambda | y), of the conditional posterior p(theta[sub l] | y[sub l], lambda). From the Bayesian point of view, the PEB estimate is of interest because it is a first-order approximation to the posterior mean [having an error of order O(k[sup -1])]. Letting E[sub lambda] and V[sub lambda] denote the expectation and variance with respect to p(lambda | y), we may write the posterior variance of theta[sub l] as V(theta[sub l] | y) = E[sub l]{V(theta[sub l] | y[sub l], lambda)} + V[sub lambda ]{E(theta[sub l] | y[sub l], lambda)}. The conditional posterior variance V(theta[sub l | y[sub l ], lambda ), whe [ABSTRACT FROM AUTHOR]

Tierney and Kadane (1986) presented a simple second-order approximation for posterior expectations of positive functions. They used Laplace's method for asymptotic evaluation of integrals, in which the integrand is written as f(theta)exp(- nh(theta)) and the function h is approximated by a quadratic. The form in which they applied Laplace's method, however, was fully exponential: The integrand was written instead as exp[-nh(theta) + log f(theta)]; this allowed first-order approximations to be used in the numerator and denominator of a ratio of integrals to produce a second-order expansion for the ratio. Other second-order expansions (Hartigan 1965; Johnson 1970; Lindley 1961, 1980; Mosteller and Wallace 1964) require computation of more derivatives of the log-likelihood function. In this article we extend the fully exponential method to apply to expectations and variances of nonpositive functions. To obtain a second-order approximation to an expectation E(g(theta)), we use the fully exponential method to approximate the moment-generating function E(exp(sg(theta))), whose integrand is positive, and then differentiate the result. This method is formally equivalent to that of Lindley and that of Mosteller and Wallace, yet does not require third derivatives of the likelihood function. It is also equivalent to another alternative approach to the approximation of E(g(theta)): We may add a large constant c to g(theta), apply the fully exponential method to E(c + g(theta)), and subtract c; on passing to the limit as c tends to infinity we regain the approximation based on the moment-generating function. Furthermore, the second derivative of the logarithm of the approximation E(exp(sg(theta))), which is an approximate cumulant-generating function, yields a simple second-order approximation... [ABSTRACT FROM AUTHOR]

A commonly occurring statistical inference problem in practice is that of making simultaneous comparisons among three or more treatment means mu[sub l] (1 is less than or equal to i is less than or equal to k) for certain experimental designs. One way to do this, the T method proposed by Tukey (1953), constructs simultaneous confidence intervals for all pairwise differences of the treatment means mu[sub 1] - mu[sub 1] (1 is less than or equal to l, j is less than or equal to k, i is not equal to j). The joint confidence level of these confidence intervals, however, depends in a very complicated fashion on the covariance structure of the treatment mean estimates mu[sub ,] and, therefore, for designs where the covariance structure is not "simple," the joint confidence level is not readily apparent. In general, this is the case for any unbalanced design or for designs in which the treatment mean estimates mu[sub l] have unequal correlations. Tukey conjectured in 1953 that whatever the correlation structure of the treatment mean estimates mu[sub ] the T method would always provide a conservative set of confidence intervals, that is, that the actual joint confidence level of the confidence intervals would always be at least as great as the nominal joint confidence level 1 - alpha. In this article a discussion is undertaken of the evidence that the T method in general provides a conservative set of simultaneous confidence intervals. It is shown that the coverage probability of the simultaneous confidence intervals depends on the covariance structure only through the k(k - 1)/2 variances of the pairwise differences of the treatment mean estimates. A set of sufficient, but not necessary, conditions on these variances is given, which ensures that the T-method confidence intervals are conservative. In addition, the application of the T method to various common experimental designs that produce correlated treatment... [ABSTRACT FROM AUTHOR]

The problem considered is the computation reduction for general delete-d jackknife variance estimators. The delete-d jackknife estimator was proved consistent (Shao and Wu 1986), and in this article its mean squared error is shown to have order o(n[sup -2]), where n is the sample size. These properties are not shared by the traditional delete-1 jackknife in some situations. Use of the delete-d jackknife, however, requires ((Multiple lines cannot be cannot be converted in ASCII text)) recomputations of a point estimate theta, which increases rapidly as n and d increase. Using techniques from survey sampling, a shortcut can be taken with theta evaluated only tn times, m ≪ (Multiple lines cannot be cannot be converted in ASCII text)). The efficiency and consistency of the resulting jackknife-sampling (hybrid) variance estimators (JSVE's) are studied. If m is chosen so that n/m arrow right 0, the increase in mean squared error by using the JSVE is relatively negligible. For the consistency of JSVE, m arrow right Infinity is sufficient. Hence the JSVE with m < n can also be used to alleviate the computational burden for the delete-1 jackknife in the case where n is large and evaluating theta needs large computations. The performance of JSVE is also studied in a simulation study. [ABSTRACT FROM AUTHOR]

The loss of precision in estimates of means due to variability among interviewers can be substantial for some questionnaire items and survey designs. The most commonly used methods for estimating the magnitude of this loss are inappropriate for binary items in complex surveys. This article shows how parameters from a model for variance components in binary variables (Anderson and Aitkin 1985) are related to the increased variance of population estimates. It is suggested that one of these parameters, a measure of correlation between interviewer observations on a latent variable, may be more appropriate than the intrainterviewer correlation p for measuring the magnitude of interviewer effects. This is because it is unaffected by the level of the attribute in the population, which is not true of p. Interviewer effects for a household respondent's recorded labor-force status in a random digit dial telephone survey are examined using the new model and estimation process. Small but positive effects are present for at least one of the labor-force categories. [ABSTRACT FROM AUTHOR]

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 telephone-call load. I also discuss the relationship of the proposed technique to the Box-Cox and ACE procedures. Efron's work on transformations provides some of the theoretical basis for the methodology. [ABSTRACT FROM AUTHOR]

Because they require very little storage and can be computationally quite efficient, gradient algorithms are attractive methods for fitting large nonorthogonal analysis of variance (ANOVA) models. A coordinate-free approach is used to provide very simple definitions for a number of well-known gradient algorithms and insights into their similarities and differences. The key to finding a good algorithm is finding an algorithm metric that leads to easily computed gradients and that is as close as possible to the metric defined by the ANOVA problem. This leads to the proposal of a new class of algorithms based on a proportional subclass metric. Several new theoretical results on convergence are derived, and some empirical comparisons are made. A similar, but much briefer, treatment of analysis of covariance is given. On theoretical convergence of the methods it is shown, for example, that the Golub and Nash (1982) algorithm requires at most d + 1 iterations if all but d of the cells in the model have the same cell count, that the proportional subclass algorithm converges in one step for proportional subclass problems, and that it demands at most 2 min(a, b) - 1 iterations when fitting the two-way additive ANOVA model of size a by b. This can, for example, lead to large savings for models with many more rows than columns. For empirical comparisons a two-way ANOVA model is fitted to some artificial and nonartificial data. For the problems considered, the proportional subclass algorithm requires the fewest iterations followed by the Golub and Nash, optimized steepest descent, Hemmerle, and Yates algorithms, in that order. Some of the differences are quite substantial, involving factors of 10 or more. [ABSTRACT FROM AUTHOR]

The present article examines the robustness of sampling strategies, that is, estimators and sample designs, from the point of view of attaining the minimum value of the Godambe-Joshi lower bound to the expected variance of a strategy. It is shown that asymptotic design unbiasedness is necessary but insufficient for robustness defined in the above sense. The concepts of weak and strong robustness of estimators, for given sampling strategies, are introduced. The article establishes that a sufficient condition for a sampling strategy to be robust is that the estimator is model unbiased and weakly robust. Sufficient conditions are also given for an estimator to be weakly robust. These conditions are then used to examine the robustness of strategies when the design matrix or the covariance matrix of the superpopulation model is misspecified. The robustness of model-based predictors is also examined. [ABSTRACT FROM AUTHOR]

The assumption of iid observations that underlies many statistical procedures is called into question when analyzing complex survey data. The population structure--particularly the existence of clusters in two-stage samples that usually exhibit positive intracluster correlation--invalidates the independence assumption. Kish and Frankel (1974) investigated the impact of this fact on regression analysis by using the standard sample-survey-theory framework; Campbell (1977) and Scott and Holt (1982) used the linear model framework. In general, although ordinary least squares (OLS) procedures are unbiased but not fully efficient for estimation of the regression coefficients, serious difficulties can arise when using OLS estimators for second-order terms. Variances of the OLS estimators for the regression coefficients can be larger (sometimes much larger) than the usual OLS variance expression would indicate. Failure to consider this possibility leads to underestimation of variances, with consequences for confidence intervals. This article follows this effect through to the F statistic, because of its importance to hypothesis tests and confidence ellipsoids. Our major aim is to investigate the effect of intracluster correlation on the F statistic. We propose a diagnostic measure identifying when the ordinary F statistic is likely to be affected and give decomposition in terms of the contributions of the individual regressors and their cross-products, based on a similar decomposition for the projection matrix in Appendix A. We establish numerically and theoretically the effectiveness of this measure in understanding the degree of distortion of F by intracluster correlation. The measure leads to a correction for the F test for unknown intracluster correlation. This is a slightly simpler numerical procedure than the generalized least squares (GLS), since it does not require iteration. The correction is shown to perform at least as well as the GLS in a simulation study. [ABSTRACT FROM AUTHOR]

The Kalman filter is generalized to cover state-space models in which the variance of the observation error depends on the state vector. Derivations of the filter yielding minimum mean squared error linear estimators and associated error covariance matrices are obtained from two differing viewpoints: linear Bayes theory and Gauss-Markov theory. The results are applied to a model for which {y[sub t]: = 1, 2, ... , n} follow a Poisson distribution with corresponding intensities {theta[sub t]: t = 1, 2, ... , n) that are assumed to follow an autoregressive process of order 1, namely theta[sub t] - theta = rho(theta[sub t-1] - theta) + w[sub t]. The steady-state generalized Kalman filter algorithm in the case for which rho = 1 gives a generalization of exponential smoothing for a Poisson process with time-varying intensity. [ABSTRACT FROM AUTHOR]

Interval estimation for individual parameters in nonlinear regression is examined in relation to curvature of the solution locus. Criteria are developed to define severe curvature and distinguish cases in which linear methods may be used from those in which parameter transformations are necessary. A formula for improving the linear approximation in cases of relatively mild curvature is proposed. For example, suppose that interest is focused on one specific parameter, theta[sub j], from among theta = {theta[sub 1], theta[sub 2], ..., theta[sub p]} in a nonlinear regression model. A crude approach to setting 95% confidence limits to theta[sub j] might proceed as follows: First calculate the standard error of the estimate [SE(theta[sub j])] as the square root of its asymptotic variance. Then call theta[sub j] +/- 1.96 SE(theta[sub j]) the confidence limits. In this article, higher-order correction terms GAMMA and beta are proposed that enable us to develop a power series expansion for the confidence limits, as theta[sub j] - 1.96 SE(theta[sub j]){1 + 1/2(1.96)GAMMA sigma + 1/2(1.96)[sup 2]beta sigma[sup 2] + ... and theta[sub l] + 1.96 SE(theta[sub l){1 - 1/2(1.96)GAMMA sigma + 1/2(1.96)[sup 2] beta sigma[sup 2] + .... respectively. Here sigma[sup 2]: denotes some estimate of sigma[sup 2] the variance of a single observation. [ABSTRACT FROM AUTHOR]

The minimum norm quadratic unbiased estimator type (MINQUE type) of estimates considered in this article are obtained by requiring identical values for the ratios of the a priori variances to the a priori error variance and letting this common value tend to infinity. The resulting estimates are invariant quadratic unbiased estimators with certain parametric and nonparametric optimality properties: assuming normally distributed random effects the efficiency of the proposed estimates to the minimum variance quadratic unbiased estimates (MIVQUE's) approaches unity when the true variance ratios are identical and tend to infinity. Assuming nonnormal effect distributions in the model with two variance components, the estimates are asymptotically efficient: in a sequence of designs where the number of classes and the number of observations on each class approach infinity, it is shown that the asymptotic variances of the estimates are equivalent to the theoretical minimum variances for invariant quadratic unbiased estimators. The result is interesting and useful since the usual analysis of variance (ANOVA) estimate of between-classes variance has strictly larger asymptotic variance for the unbalanced one-way model. Commonly considered estimates result from this procedure; the usual residual mean square (assuming that all non-error effects are fixed) is the resulting estimate of the error variance. In the one-way model the resulting estimates coincide with estimates considered by Thomas and Hultquist (1978), Burdick and Graybill (1984), Ahrens, Kleffe, and Tenzler (1981), and Kaplan (1982). In particular, the convergence of the MINQUE estimates was proved in the latter two papers in the context of the unbalanced one-way model. The procedure yields computationally convenient estimates in the general mixed ANOVA model. Computing formulas are... [ABSTRACT FROM AUTHOR]

The regression estimator and the ratio estimator are commonly used in survey practice. In the past more attention has been given to the ratio estimator because of its computational ease and applicability for general sampling designs. The ratio estimator is appropriate for populations whose regression line passes close to the origin. If the intercept of the regression line is significantly nonzero, however, it is much less efficient than the regression estimator (Deng 1984). In general, apart from n[sup -2] terms, the mean squared error (MSE) of the former is bigger than that of the latter (Cochran 1977, p. 196). Given the present computing capacity, the computational advantage of the ratio estimator should be less of a concern and the regression estimator will gain wider popularity. The main purpose of this article is to provide a theoretical and empirical comparison of several variance estimators for the regression estimator in simple random sampling without replacement. The companion problem for the ratio estimator has been studied in the literature (see Wu and Deng 1983). Under comparison are several design-based and model-based estimators and a new class of estimators. Their second-order expressions and biases are derived and compared. Empirical results on the biases and MSE's of the variance estimators and the conditional and unconditional coverage probabilities of their associated t intervals are obtained. They lend support to the theoretical results and suggest questions for further investigation. Our empirical and theoretical study provides a guide to the use of these estimators in practice. [ABSTRACT FROM AUTHOR]