Biometrika - recent issues

Nonparametric Bayes local partition models for random effects

Dunson, D. B. — 2009-05-20

This paper focuses on the problem of choosing a prior for an unknown random effects distribution within a Bayesian hierarchical model. The goal is to obtain a sparse representation by allowing a combination of global and local borrowing of information. A local partition process prior is proposed, which induces dependent local clustering. Subjects can be clustered together for a subset of their parameters, and one learns about similarities between subjects increasingly as parameters are added. Some basic properties are described, including simple two-parameter expressions for marginal and conditional clustering probabilities. A slice sampler is developed which bypasses the need to approximate the countably infinite random measure in performing posterior computation. The methods are illustrated using simulation examples, and an application to hormone trajectory data.

Mixtures of Polya trees for flexible spatial frailty survival modelling

Zhao, L., Hanson, T. E., Carlin, B. P. — 2009-05-20

Mixtures of Polya trees offer a very flexible nonparametric approach for modelling time-to-event data. Many such settings also feature spatial association that requires further sophistication, either at the point level or at the lattice level. In this paper, we combine these two aspects within three competing survival models, obtaining a data analytic approach that remains computationally feasible in a fully hierarchical Bayesian framework using Markov chain Monte Carlo methods. We illustrate our proposed methods with an analysis of spatially oriented breast cancer survival data from the Surveillance, Epidemiology and End Results program of the National Cancer Institute. Our results indicate appreciable advantages for our approach over competing methods that impose unrealistic parametric assumptions, ignore spatial association or both.

Gamma frailty transformation models for multivariate survival times

Zeng, D., Chen, Q., Ibrahim, J. G. — 2009-05-20

We propose a class of transformation models for multivariate failure times. The class of transformation models generalize the usual gamma frailty model and yields a marginally linear transformation model for each failure time. Nonparametric maximum likelihood estimation is used for inference. The maximum likelihood estimators for the regression coefficients are shown to be consistent and asymptotically normal, and their asymptotic variances attain the semiparametric efficiency bound. Simulation studies show that the proposed estimation procedure provides asymptotically efficient estimates and yields good inferential properties for small sample sizes. The method is illustrated using data from a cardiovascular study.

Generalized method of moments estimation for linear regression with clustered failure time data

Li, H., Yin, G. — 2009-05-20

We propose a generalized method of moments approach to the accelerated failure time model with correlated survival data. We study the semiparametric rank estimator using martingale-based moments. We circumvent direct estimation of correlation parameters by concatenating the moments and minimizing a quadratic objective function. We establish the consistency and asymptotic normality of the parameter estimators, and derive the limiting distribution of the objective function. We carry out simulation studies to examine the finite-sample properties of the method, and demonstrate its substantial efficiency gain over the conventional method. Finally, we illustrate the new proposal with an example from a diabetic retinopathy study.

Hierarchically penalized Cox regression with grouped variables

Wang, S., Nan, B., Zhu, N., Zhu, J. — 2009-05-20

In many biological and other scientific applications, predictors are often naturally grouped. For example, in biological applications, assayed genes or proteins are grouped by biological roles or biological pathways. When studying the dependence of survival outcome on these grouped predictors, it is desirable to select variables at both the group level and the within-group level. In this article, we develop a new method to address the group variable selection problem in the Cox proportional hazards model. Our method not only effectively removes unimportant groups, but also maintains the flexibility of selecting variables within the identified groups. We also show that the new method offers the potential for achieving the asymptotic oracle property.

A generalized Dantzig selector with shrinkage tuning

James, G. M., Radchenko, P. — 2009-05-20

The Dantzig selector performs variable selection and model fitting in linear regression. It uses an L₁ penalty to shrink the regression coefficients towards zero, in a similar fashion to the lasso. While both the lasso and Dantzig selector potentially do a good job of selecting the correct variables, they tend to overshrink the final coefficients. This results in an unfortunate trade-off. One can either select a high shrinkage tuning parameter that produces an accurate model but poor coefficient estimates or a low shrinkage parameter that produces more accurate coefficients but includes many irrelevant variables. We extend the Dantzig selector to fit generalized linear models while eliminating overshrinkage of the coefficient estimates, and develop a computationally efficient algorithm, similar in nature to least angle regression, to compute the entire path of coefficient estimates. A simulation study illustrates the advantages of our approach relative to others. We apply the methodology to two datasets.

A group bridge approach for variable selection

Huang, J., Ma, S., Xie, H., Zhang, C.-H. — 2009-05-20

In multiple regression problems when covariates can be naturally grouped, it is important to carry out feature selection at the group and within-group individual variable levels simultaneously. The existing methods, including the lasso and group lasso, are designed for either variable selection or group selection, but not for both. We propose a group bridge approach that is capable of simultaneous selection at both the group and within-group individual variable levels. The proposed approach is a penalized regularization method that uses a specially designed group bridge penalty. It has the oracle group selection property, in that it can correctly select important groups with probability converging to one. In contrast, the group lasso and group least angle regression methods in general do not possess such an oracle property in group selection. Simulation studies indicate that the group bridge has superior performance in group and individual variable selection relative to several existing methods.

Covariate-adjusted generalized linear models

Senturk, D., Muller, H.-G. — 2009-05-20

We propose covariate adjustment methodology for a situation where one wishes to study the dependence of a generalized response on predictors while both predictors and response are distorted by an observable covariate. The distorting covariate is thought of as a size measurement that affects predictors in a multiplicative fashion. The generalized response is modelled by means of a random threshold, where the subject-specific thresholds are affected by a multiplicative factor that is a function of the distorting covariate. While the various factors are modelled as smooth unknown functions of the distorting covariate, the underlying relationship between response and covariates is assumed to be governed by a generalized linear model with a known link function. This model provides an extension of a covariate-adjusted regression approach to the case of a generalized linear model. We demonstrate that this contamination model leads to a semiparametric varying-coefficient model. Numerical implementation is straightforward by combining binning, quasilikelihood, and smoothing steps. The asymptotic distribution of the proposed estimators for the regression coefficients of the latent generalized linear model is derived by means of a martingale central limit theorem. Combining this result with consistent estimators for the asymptotic variance makes it then possible to obtain asymptotic inference for the targeted parameters. Both real and simulated data are used in illustrating the proposed methodology.

Adjusting for covariate effects on classification accuracy using the covariate-adjusted receiver operating characteristic curve

Janes, H., Pepe, M. S. — 2009-05-20

Recent scientific and technological innovations have produced an abundance of potential markers that are being investigated for their use in disease screening and diagnosis. In evaluating these markers, it is often necessary to account for covariates associated with the marker of interest. Covariates may include subject characteristics, expertise of the test operator, test procedures or aspects of specimen handling. In this paper, we propose the covariate-adjusted receiver operating characteristic curve, a measure of covariate-adjusted classification accuracy. Nonparametric and semiparametric estimators are proposed, asymptotic distribution theory is provided and finite sample performance is investigated. For illustration we characterize the age-adjusted discriminatory accuracy of prostate-specific antigen as a biomarker for prostate cancer.

Nonparametric additive regression for repeatedly measured data

Carroll, R. J., Maity, A., Mammen, E., Yu, K. — 2009-05-20

We develop an easily computed smooth backfitting algorithm for additive model fitting in repeated measures problems. Our methodology easily copes with various settings, such as when some covariates are the same over repeated response measurements. We allow for a working covariance matrix for the regression errors, showing that our method is most efficient when the correct covariance matrix is used. The component functions achieve the known asymptotic variance lower bound for the scalar argument case. Smooth backfitting also leads directly to design-independent biases in the local linear case. Simulations show our estimator has smaller variance than the usual kernel estimator. This is also illustrated by an example from nutritional epidemiology.

Optimal testing of multiple hypotheses with common effect direction

Bittman, R. M., Romano, J. P., Vallarino, C., Wolf, M. — 2009-05-20

We present a theoretical basis for testing related endpoints. Typically, it is known how to construct tests of the individual hypotheses, but not how to combine them into a multiple test procedure that controls the familywise error rate. Using the closure method, we emphasize the role of consonant procedures, from an interpretive as well as a theoretical viewpoint. Surprisingly, even if each intersection test has an optimality property, the overall procedure obtained by applying closure to these tests may be inadmissible. We introduce a new procedure, which is consonant and has a maximin property under the normal model. The results are then applied to PROactive, a clinical trial designed to investigate the effectiveness of a glucose-lowering drug on macrovascular outcomes among patients with type 2 diabetes.

Non-finite Fisher information and homogeneity: an EM approach

Li, P., Chen, J., Marriott, P. — 2009-05-20

Even simple examples of finite mixture models can fail to fulfil the regularity conditions that are routinely assumed in standard parametric inference problems. Many methods have been investigated for testing for homogeneity in finite mixture models, for example, but all rely on regularity conditions including the finiteness of the Fisher information and the space of the mixing parameter being a compact subset of some Euclidean space. Very simple examples where such assumptions fail include mixtures of two geometric distributions and two exponential distributions, and, more generally, mixture models in scale distribution families. To overcome these difficulties, we propose and study an em-test statistic, which has a simple limiting distribution for examples in this paper. Simulations show that the em-test has accurate Type I errors and is more efficient than existing methods when they are applicable. A real example is included.

Double block bootstrap confidence intervals for dependent data

Lee, S. M. S., Lai, P. Y. — 2009-05-20

The block bootstrap confidence interval for dependent data can outperform the conventional normal approximation only with nontrivial studentization which, in the case of complicated statistics, calls for specialist treatment and often results in unstable endpoints. We propose two double block bootstrap approaches for improving the accuracy of the block bootstrap confidence interval under very general conditions. The first approach calibrates the nominal coverage level and the second calculates studentizing factors directly from a block bootstrap series without the need for nontrivial analytical treatment. We prove that the two approaches reduce the coverage error of the block bootstrap interval by an order of magnitude with simple tuning of block lengths at the two block bootstrapping levels. Empirical properties of the procedures are investigated by simulations and application to an econometric time series.

Marginal analysis of panel counts through estimating functions

Hu, X. J., Lagakos, S. W., Lockhart, R. A. — 2009-05-20

We develop nonparametric estimation procedures for the marginal mean function of a counting process based on periodic observations, using two types of self-consistent estimating equations. The first is derived from the likelihood studied by Wellner & Zhang (2000), assuming a Poisson counting process. It gives a nondecreasing estimator, which equals the nonparametric maximum likelihood estimator of Wellner & Zhang and is consistent without the Poisson assumption. Motivated by the construction of parametric generalized estimating equations, the second type is a set of data-adaptive quasi-score functions, which are likelihood estimating functions under a mixed-Poisson assumption. We evaluate the procedures using simulation, and illustrate them with the data from a bladder cancer study.

Jackknife estimation of mean squared error of small area predictors in nonlinear mixed models

Lohr, S. L., Rao, J. N. K. — 2009-05-20

Empirical Bayes predictors of small area parameters of interest are often obtained under a linear mixed model for continuous response data or a generalized linear mixed model for binary responses or count data. However, estimation of the unconditional mean squared error of prediction is complicated, particularly for a nonlinear mixed model. Jiang et al. (2002) proposed a jackknife method for estimating the unconditional mean squared error and showed that the resulting estimator is nearly unbiased. The leading term of this estimator does not depend on the area-specific responses in the nonlinear case, whereas the posterior variance of the small area parameter given the model parameters is area-specific. Rao (2003) proposed an alternative method that leads to a computationally simpler jackknife estimator with an area-specific leading term. We show that a modification of Rao's method leads to a nearly unbiased area-specific jackknife estimator, which is also nearly unbiased for the conditional mean squared error given the area-specific responses. We examine the relative performances of the jackknife estimators, conditionally as well as unconditionally, in a simulation study, and apply the proposed method to estimate small area mean squared errors in disease mapping problems.

Scale adjustments for classifiers in high-dimensional, low sample size settings

Chan, Y.-B., Hall, P. — 2009-05-20

Distance-based classifiers are generally considered to be effective at discriminating between populations that differ in location. Indeed, nearest-neighbour methods and the support vector machine are frequently used in very high-dimensional problems involving gene expression data, where it is believed that elevated levels of expression convey much of the information for classification. However, one problem inherent to distance-based classifiers is that scale differences can mask location differences. In consequence, such classifiers can have poor performance if the information for classification accumulates through a large number of relatively small location differences in data components, rather than via large differences. In this paper, we show that a simple adjustment for scale, applicable to a variety of distance-based classifiers, can remedy the problem. For some classifiers, such as those based on the support vector machine or the centroid method, scale corrections are important primarily in the case of small training-sample sizes. However, for other classifiers, including those based on nearest-neighbour and average-distance methods, scale adjustments are helpful more generally.

Saddlepoint approximation for mixture models

Davison, A. C., Mastropietro, D. — 2009-05-20

Two-component mixture distributions with one component a point mass and the other a continuous density may be used as priors for Bayesian inference when sparse representation of an underlying signal is required. We show how saddlepoint approximation in such models can yield highly accurate quantiles for posterior distributions, and illustrate this numerically, using wavelet regression with point mass/Laplace and point mass/normal prior distributions.

Some results on D-optimal designs for nonlinear models with applications

Li, G., Majumdar, D. — 2009-05-20

Sufficient conditions are established for the locally D$-optimal design for a nonlinear model to have a minimal number of support points. The conditions are applied to obtain locally D-optimal designs for a one-compartment pharmacokinetic model and a Poisson regression model.

Dimension reduction in time series and the dynamic factor model

Pena, D. — 2009-05-20

This note shows that the dimension reduction method proposed by Li & Shedden (2002) is equivalent to the dynamic factor model introduced by Peña & Box (1987).

Modelling pairwise dependence of maxima in space

Naveau, P., Guillou, A., Cooley, D., Diebolt, J. — 2009-02-13

We model pairwise dependence of temporal maxima, such as annual maxima of precipitation, that have been recorded in space, either on a regular grid or at irregularly spaced locations. The construction of our estimators stems from the variogram concept. The asymptotic properties of our pairwise dependence estimators are established through properties of empirical processes. The performance of our approach is illustrated by simulations and by the treatment of a real dataset. In addition to bringing new results about the asymptotic behaviour of copula estimators, the latter being linked to first-order variograms, one main advantage of our approach is to propose a simple connection between extreme value theory and geostatistics.

Efficient nonparametric estimation of causal effects in randomized trials with noncompliance

Cheng, J., Small, D. S., Tan, Z., Ten Have, T. R. — 2009-02-13

Causal approaches based on the potential outcome framework provide a useful tool for addressing noncompliance problems in randomized trials. We propose a new estimator of causal treatment effects in randomized clinical trials with noncompliance. We use the empirical likelihood approach to construct a profile random sieve likelihood and take into account the mixture structure in outcome distributions, so that our estimator is robust to parametric distribution assumptions and provides substantial finite-sample efficiency gains over the standard instrumental variable estimator. Our estimator is asymptotically equivalent to the standard instrumental variable estimator, and it can be applied to outcome variables with a continuous, ordinal or binary scale. We apply our method to data from a randomized trial of an intervention to improve the treatment of depression among depressed elderly patients in primary care practices.

Partial and latent ignorability in missing-data problems

Harel, O., Schafer, J. L. — 2009-02-13

When an assumption of missing at random is untenable, it becomes necessary to model missing-data indicators, which carry information about the parameters of the complete-data population. Within a given application, however, researchers may believe that some aspects of missingness are ignorable but others are not. We argue that there are two different ways to formalize the notion that only part of the missingness is ignorable. These approaches correspond to assumptions that we call partially missing at random and latently missing at random. We explain these concepts and apply them in a latent-class analysis of survey questions with item nonresponse.

Orthogonal and nearly orthogonal designs for computer experiments

Bingham, D., Sitter, R. R., Tang, B. — 2009-02-13

We introduce a method for constructing a rich class of designs that are suitable for use in computer experiments. The designs include Latin hypercube designs and two-level fractional factorial designs as special cases and fill the vast vacuum between these two familiar classes of designs. The basic construction method is simple, building a series of larger designs based on a given small design. If the base design is orthogonal, the resulting designs are orthogonal; likewise, if the base design is nearly orthogonal, the resulting designs are nearly orthogonal. We present two generalizations of our basic construction method. The first generalization improves the projection properties of the basic method; the second generalization gives rise to designs that have smaller correlations. Sample constructions are presented and properties of these designs are discussed.

D-optimal design of split-split-plot experiments

Jones, B., Goos, P. — 2009-02-13

In industrial experimentation, there is growing interest in studies that span more than one processing step. Convenience often dictates restrictions in randomization in passing from one processing step to another. When the study encompasses three processing steps, this leads to split-split-plot designs. We provide an algorithm for computing D-optimal split-split-plot designs and several illustrative examples.

Optimal two-level regular fractional factorial block and split-plot designs

Cheng, C.-S., Tsai, P.-W. — 2009-02-13

We propose a general and unified approach to the selection of regular fractional factorial designs, which can be applied to experiments that are unblocked, blocked or have a split-plot structure. Our criterion is derived as a good surrogate for the model-robustness criterion of information capacity. In the case of random block effects, it takes the ratio of intra- and interblock variances into account. In most of the cases we have examined, there exist designs that are optimal for all values of that ratio. Examples of optimal designs that depend on the ratio are provided. We also demonstrate that our criterion can further discriminate designs that cannot be distinguished by the existing minimum-aberration criteria.

Bayesian-inspired minimum aberration two- and four-level designs

Joseph, V. R., AI, M., Wu, C. F. J. — 2009-02-13

Motivated by a Bayesian framework, we propose a new minimum aberration-type criterion for designing experiments with two- and four-level factors. The Bayesian approach helps in overcoming the ad hoc nature of effect ordering in the existing minimum aberration-type criteria. The approach is also capable of distinguishing between qualitative and quantitative factors. Numerous examples are given to demonstrate its advantages.

Confidence intervals for spectral mean and ratio statistics

Shao, X. — 2009-02-13

We propose a new method, to construct confidence intervals for spectral mean and related ratio statistics of a stationary process, that avoids direct estimation of their asymptotic variances. By introducing a bandwidth, a self-normalization procedure is adopted and the distribution of the new statistic is asymptotically nuisance-parameter free. The bandwidth is chosen using information criteria and a moving average sieve approximation. Through a simulation study, we demonstrate good finite sample performance of our method when the sample size is moderate, while a comparison with an empirical likelihood-based method for ratio statistics is made, confirming a wider applicability of our method.

Tapered empirical likelihood for time series data in time and frequency domains

Nordman, D. J. — 2009-02-13

We investigate data tapering in two formulations of empirical likelihood for time series. One empirical likelihood is formed from tapered data blocks in the time domain and a second is based on the tapered periodogram in the frequency domain. Limiting distributions are provided for both empirical likelihood versions under tapering. Theoretical and simulation evidence indicates that a data taper improves the coverage accuracy of empirical likelihood confidence intervals for time series parameters, such as means and correlations.

Model checking in regression via dimension reduction

Xia, Y. — 2009-02-13

Lack-of-fit checking for parametric and semiparametric models is essential in reducing misspecification. The efficiency of most existing model-checking methods drops rapidly as the dimension of the covariates increases. We propose to check a model by projecting the fitted residuals along a direction that adapts to the systematic departure of the residuals from the desired pattern. Consistency of the method is proved for parametric and semiparametric regression models. A bootstrap implementation is also discussed. Simulation comparisons with several existing methods are made, suggesting that the proposed methods are more efficient than the existing methods when the dimension increases. Air pollution data from Chicago are used to illustrate the procedure.

Bayesian nonparametric functional data analysis through density estimation

Rodriguez, A., Dunson, D. B., Gelfand, A. E. — 2009-02-13

In many modern experimental settings, observations are obtained in the form of functions and interest focuses on inferences about a collection of such functions. We propose a hierarchical model that allows us simultaneously to estimate multiple curves nonparametrically by using dependent Dirichlet process mixtures of Gaussian distributions to characterize the joint distribution of predictors and outcomes. Function estimates are then induced through the conditional distribution of the outcome given the predictors. The resulting approach allows for flexible estimation and clustering, while borrowing information across curves. We also show that the function estimates we obtain are consistent on the space of integrable functions. As an illustration, we consider an application to the analysis of conductivity and temperature at depth data in the north Atlantic.

Wilcoxon-type generalized Bayesian information criterion

Wang, L. — 2009-02-13

We develop a generalized Bayesian information criterion for regression model selection. The new criterion relaxes the usually strong distributional assumption associated with Schwarz's bic by adopting a Wilcoxon-type dispersion function and appropriately adjusting the penalty term. We establish that the Wilcoxon-type generalized bic preserves the consistency of Schwarz's bic without the need to assume a parametric likelihood. We also show that it outperforms Schwarz's bic with heavier-tailed data in the sense that asymptotically it can yield substantially smaller L₂ risk. On the other hand, when the data are normally distributed, both criteria have similar L₂ risk. The new criterion function is convex and can be conveniently computed using existing statistical software. Our proposal provides a flexible yet highly efficient alternative to Schwarz's bic; at the same time, it broadens the scope of Wilcoxon inference, which has played a fundamental role in classical nonparametric analysis.

Reducing variability of crossvalidation for smoothing-parameter choice

Hall, P., Robinson, A. P. — 2009-02-13

One of the attractions of crossvalidation, as a tool for smoothing-parameter choice, is its applicability to a wide variety of estimator types and contexts. However, its detractors comment adversely on the relatively high variance of crossvalidatory smoothing parameters, noting that this compromises the performance of the estimators in which those parameters are used. We show that the variability can be reduced simply, significantly and reliably by employing bootstrap aggregation or bagging. We establish that in theory, when bagging is implemented using an adaptively chosen resample size, the variability of crossvalidation can be reduced by an order of magnitude. However, it is arguably more attractive to use a simpler approach, based for example on half-sample bagging, which can reduce variability by approximately 50%.

Dealing with limited overlap in estimation of average treatment effects

Crump, R. K., Hotz, V. J., Imbens, G. W., Mitnik, O. A. — 2009-02-13

Estimation of average treatment effects under unconfounded or ignorable treatment assignment is often hampered by lack of overlap in the covariate distributions between treatment groups. This lack of overlap can lead to imprecise estimates, and can make commonly used estimators sensitive to the choice of specification. In such cases researchers have often used ad hoc methods for trimming the sample. We develop a systematic approach to addressing lack of overlap. We characterize optimal subsamples for which the average treatment effect can be estimated most precisely. Under some conditions, the optimal selection rules depend solely on the propensity score. For a wide range of distributions, a good approximation to the optimal rule is provided by the simple rule of thumb to discard all units with estimated propensity scores outside the range [0.1,0.9].

On fuzzy familywise error rate and false discovery rate procedures for discrete distributions

Kulinskaya, E., Lewin, A. — 2009-02-13

Fuzzy multiple comparisons procedures are introduced as a solution to the problem of multiple comparisons for discrete test statistics. The critical function of the randomized p-values is proposed as a measure of evidence against the null hypotheses. The classical concept of randomized tests is extended to multiple comparisons. This approach makes all theory of multiple comparisons developed for continuously distributed statistics automatically applicable to the discrete case. Examples of familywise error rate and false discovery rate procedures are discussed and an application to linkage disequilibrium testing is given. Software for implementing the procedures is available.

Fast block variance estimation procedures for inhomogeneous spatial point processes

Guan, Y. — 2009-02-13

We introduce two new variance estimation procedures that use non-overlapping and overlapping blocks, respectively. The non-overlapping blocks estimator can be viewed as the limit of the thinned block bootstrap estimator recently proposed in Guan Loh (2007), by letting the number of thinned processes and bootstrap samples therein both increase to infinity. The non-overlapping blocks estimator can be obtained quickly since it does not require any thinning or bootstrap steps, and it is more stable. The overlapping blocks estimator further improves the performance of the non-overlapping blocks with a modest increase in computation time. A simulation study demonstrates the superiority of the proposed estimators over the thinned block bootstrap estimator.

A note on semiparametric efficient inference for two-stage outcome-dependent sampling with a continuous outcome

Song, R., Zhou, H., Kosorok, M. R. — 2009-02-13

Outcome-dependent sampling designs have been shown to be a cost-effective way to enhance study efficiency. We show that the outcome-dependent sampling design with a continuous outcome can be viewed as an extension of the two-stage case-control designs to the continuous-outcome case. We further show that the two-stage outcome-dependent sampling has a natural link with the missing-data and biased-sampling frameworks. Through the use of semiparametric inference and missing-data techniques, we show that a certain semiparametric maximum-likelihood estimator is computationally convenient and achieves the semiparametric efficient information bound. We demonstrate this both theoretically and through simulation.

A note on profile likelihood for exponential tilt mixture models

Tan, Z. — 2009-02-13

Suppose that independent observations are drawn from multiple distributions, each of which is a mixture of two component distributions such that their log density ratio satisfies a linear model with a slope parameter and an intercept parameter. Inference for such models has been studied using empirical likelihood, and mixed results have been obtained. The profile empirical likelihood of the slope and intercept has an irregularity at the null hypothesis so that the two component distributions are equal. We derive a profile empirical likelihood and maximum likelihood estimator of the slope alone, and obtain the usual asymptotic properties for the estimator and the likelihood ratio statistic regardless of the null. Furthermore, we show the maximum likelihood estimator of the slope and intercept jointly is consistent and asymptotically normal regardless of the null. At the null, the joint maximum likelihood estimator falls along a straight line through the origin with perfect correlation asymptotically to the first order.

A note on cause-specific residual life

Jeong, J.-H., Fine, J. P. — 2009-02-13

In medical research, investigators often wish to characterize the distributions of remaining lifetimes. While nonparametric analyses of residual life distributions have been widely studied with independently right-censored data, residual life analysis has not been examined in the competing risks setting, with multiple, potentially dependent, failure types. We define the cause-specific residual life distribution as the residual cumulative incidence function conditionally on survival to a given time. Because of the improper form of the cause-specific distribution, the mean cause-specific residual lifetime does not exist, theoretically. We develop nonparametric inferences for the cause-specific residual life function and its corresponding quantiles, which may exist. Theoretical justification, including uniform consistency and weak convergence, is established. Simulation studies and a breast cancer data analysis demonstrate the practical utility of the methods.

Construction of orthogonal and nearly orthogonal Latin hypercubes

Lin, C. D., Mukerjee, R., Tang, B. — 2009-02-13

We propose a method for constructing orthogonal or nearly orthogonal Latin hypercubes. The method yields a large Latin hypercube by coupling an orthogonal array of index unity with a small Latin hypercube. It is shown that the large Latin hypercube inherits the exact or near orthogonality of the small Latin hypercube. Thus, effort for searching for large Latin hypercubes, that are exactly or nearly orthogonal, can be focussed on finding small Latin hypercubes with the same property. We obtain a useful collection of orthogonal or nearly orthogonal Latin hypercubes, which have a large factor-to-run ratio and the results are often much more economical than existing methods.

A note on time-ordered classification

He, H. — 2009-02-13

A multi-dimensional scaling approach to shape analysis

Dryden, I. L., Kume, A., Le, H., Wood, A. T. A. — 2008-11-25

We propose an alternative to Kendall's shape space for reflection shapes of configurations in with k labelled vertices, where reflection shape consists of all the geometric information that is invariant under compositions of similarity and reflection transformations. The proposed approach embeds the space of such shapes into the space of (k – 1) x (k – 1) real symmetric positive semidefinite matrices, which is the closure of an open subset of a Euclidean space, and defines mean shape as the natural projection of Euclidean means in on to the embedded copy of the shape space. This approach has strong connections with multi-dimensional scaling, and the mean shape so defined gives good approximations to other commonly used definitions of mean shape. We also use standard perturbation arguments for eigenvalues and eigenvectors to obtain a central limit theorem which then enables the application of standard statistical techniques to shape analysis in two or more dimensions.

Covariance reducing models: An alternative to spectral modelling of covariance matrices

Cook, R. D., Forzani, L. — 2008-11-25

We introduce covariance reducing models for studying the sample covariance matrices of a random vector observed in different populations. The models are based on reducing the sample covariance matrices to an informational core that is sufficient to characterize the variance heterogeneity among the populations. They possess useful equivariance properties and provide a clear alternative to spectral models for covariance matrices.

Testing the covariance structure of multivariate random fields

Li, B., Genton, M. G., Sherman, M. — 2008-11-25

There is an increasing wealth of multivariate spatial and multivariate spatio-temporal data appearing. For such data, an important part of model building is an assessment of the properties of the underlying covariance function describing variable, spatial and temporal correlations. In this paper, we propose a methodology to evaluate the appropriateness of several types of common assumptions on multivariate covariance functions in the spatio-temporal context. The methodology is based on the asymptotic joint normality of the sample space-time cross-covariance estimators. Specifically, we address the assumptions of symmetry, separability and linear models of coregionalization. We conduct simulation experiments to evaluate the sizes and powers of our tests and illustrate our methodology on a trivariate spatio-temporal dataset of pollutants over California.

A goodness-of-fit test for inhomogeneous spatial Poisson processes

Guan, Y. — 2008-11-25

We introduce a formal testing procedure to assess the fit of an inhomogeneous spatial Poisson process model, based on a discrepancy measure function that is constructed from residuals obtained from the fitted model. We derive the asymptotic distributional properties of and develop a test statistic based on them. Our test statistic has a limiting standard normal distribution, so that the test can be performed by simply comparing the test statistic with readily available critical values. We perform a simulation study to assess the performance of the proposed method and apply it to a real data example.

Estimating equations for spatially correlated data in multi-dimensional space

Lin, P.-S. — 2008-11-25

We use the quasilikelihood concept to propose an estimating equation for spatial data with correlation across the study region in a multi-dimensional space. With appropriate mixing conditions, we develop a central limit theorem for a random field under various L_p metrics. The consistency and asymptotic normality of quasilikelihood estimators can then be derived. We also conduct simulations to evaluate the performance of the proposed estimating equation, and a dataset from East Lansing Woods is used to illustrate the method.

Bayesian nonparametric inference on stochastic ordering

Dunson, D. B., Peddada, S. D. — 2008-11-25

We consider Bayesian inference about collections of unknown distributions subject to a partial stochastic ordering. To address problems in testing of equalities between groups and estimation of group-specific distributions, we propose classes of restricted dependent Dirichlet process priors. These priors have full support in the space of stochastically ordered distributions, and can be used for collections of unknown mixture distributions to obtain a flexible class of mixture models. Theoretical properties are discussed, efficient methods are developed for posterior computation using Markov chain Monte Carlo simulation and the methods are illustrated using data from a study of DNA damage and repair.

Pairwise curve synchronization for functional data

Tang, R., Muller, H.-G. — 2008-11-25

Data collected by scientists are increasingly in the form of trajectories or curves. Often these can be viewed as realizations of a composite process driven by both amplitude and time variation. We consider the situation in which functional variation is dominated by time variation, and develop a curve-synchronization method that uses every trajectory in the sample as a reference to obtain pairwise warping functions in the first step. These initial pairwise warping functions are then used to create improved estimators of the underlying individual warping functions in the second step. A truncated averaging process is used to obtain robust estimation of individual warping functions. The method compares well with other available time-synchronization approaches and is illustrated with Berkeley growth data and gene expression data for multiple sclerosis.

Model diagnostic tests for selecting informative correlation structure in correlated data

Qu, A., Lee, J. J., Lindsay, B. G. — 2008-11-25

In the generalized method of moments approach to longitudinal data analysis, unbiased estimating functions can be constructed to incorporate both the marginal mean and the correlation structure of the data. Increasing the number of parameters in the correlation structure corresponds to increasing the number of estimating functions. Thus, building a correlation model is equivalent to selecting estimating functions. This paper proposes a chi-squared test to choose informative unbiased estimating functions. We show that this methodology is useful for identifying which source of correlation it is important to incorporate when there are multiple possible sources of correlation. This method can also be applied to determine the optimal working correlation for the generalized estimating equation approach.

On the asymptotics of marginal regression splines with longitudinal data

Zhu, Z., Fung, W. K., He, X. — 2008-11-25

There have been studies on how the asymptotic efficiency of a nonparametric function estimator depends on the handling of the within-cluster correlation when nonparametric regression models are used on longitudinal or cluster data. In particular, methods based on smoothing splines and local polynomial kernels exhibit different behaviour. We show that the generalized estimation equations based on weighted least squares regression splines for the nonparametric function have an interesting property: the asymptotic bias of the estimator does not depend on the working correlation matrix, but the asymptotic variance, and therefore the mean squared error, is minimized when the true correlation structure is specified. This property of the asymptotic bias distinguishes regression splines from smoothing splines.

Small area estimation when auxiliary information is measured with error

Ybarra, L. M. R., Lohr, S. L. — 2008-11-25

Small area estimation methods typically combine direct estimates from a survey with predictions from a model in order to obtain estimates of population quantities with reduced mean squared error. When the auxiliary information used in the model is measured with error, using a small area estimator such as the Fay–Herriot estimator while ignoring measurement error may be worse than simply using the direct estimator. We propose a new small area estimator that accounts for sampling variability in the auxiliary information, and derive its properties, in particular showing that it is approximately unbiased. The estimator is applied to predict quantities measured in the U.S. National Health and Nutrition Examination Survey, with auxiliary information from the U.S. National Health Interview Survey.

Multiple imputation when records used for imputation are not used or disseminated for analysis

Reiter, J. P. — 2008-11-25

When some of the records used to estimate the imputation models in multiple imputation are not used or available for analysis, the usual multiple imputation variance estimator has positive bias. We present an alternative approach that enables unbiased estimation of variances and, hence, calibrated inferences in such contexts. First, using all records, the imputer samples m values of the parameters of the imputation model. Second, for each parameter draw, the imputer simulates the missing values for all records n times. From these mn completed datasets, the imputer can analyse or disseminate the appropriate subset of records. We develop methods for interval estimation and significance testing for this approach. Methods are presented in the context of multiple imputation for measurement error.

Semiparametric maximum likelihood estimation in normal transformation models for bivariate survival data

Li, Y., Prentice, R. L., Lin, X. — 2008-11-25

We consider a class of semiparametric normal transformation models for right-censored bivariate failure times. Nonparametric hazard rate models are transformed to a standard normal model and a joint normal distribution is assumed for the bivariate vector of transformed variates. A semiparametric maximum likelihood estimation procedure is developed for estimating the marginal survival distribution and the pairwise correlation parameters. This produces an efficient estimator of the correlation parameter of the semiparametric normal transformation model, which characterizes the dependence of bivariate survival outcomes. In addition, a simple positive-mass-redistribution algorithm can be used to implement the estimation procedures. Since the likelihood function involves infinite-dimensional parameters, empirical process theory is utilized to study the asymptotic properties of the proposed estimators, which are shown to be consistent, asymptotically normal and semiparametric efficient. A simple estimator for the variance of the estimates is derived. Finite sample performance is evaluated via extensive simulations.

Estimating the false discovery rate using the stochastic approximation algorithm

Liang, F., Zhang, J. — 2008-11-25

Testing of multiple hypotheses involves statistics that are strongly dependent in some applications, but most work on this subject is based on the assumption of independence. We propose a new method for estimating the false discovery rate of multiple hypothesis tests, in which the density of test scores is estimated parametrically by minimizing the Kullback–Leibler distance between the unknown density and its estimator using the stochastic approximation algorithm, and the false discovery rate is estimated using the ensemble averaging method. Our method is applicable under general dependence between test statistics. Numerical comparisons between our method and several competitors, conducted on simulated and real data examples, show that our method achieves more accurate control of the false discovery rate in almost all scenarios.

Identification of the age-period-cohort model and the extended chain-ladder model

Kuang, D., Nielsen, B., Nielsen, J. P. — 2008-11-25

We consider the identification problem that arises in the age-period-cohort models as well as in the extended chain-ladder model. We propose a canonical parameterization based on the accelerations of the trends in the three factors. This parameterization is exactly identified and eases interpretation, estimation and forecasting. The canonical parameterization is applied to a class of index sets which have trapezoidal shapes, including various Lexis diagrams and the insurance-reserving triangles.

Forecasting with the age-period-cohort model and the extended chain-ladder model

Kuang, D., Nielsen, B., Nielsen, J. P. — 2008-11-25

We consider forecasting from age-period-cohort models, as well as from the extended chain-ladder model. The parameters of these models are known only to be identified up to linear trends. Forecasts from such models may therefore depend on arbitrary linear trends. A condition for invariant forecasts is proposed. A number of standard forecast models are analysed.

On the consequences of overstratification

De Stavola, B. L., Cox, D. R. — 2008-11-25

It is common, in particular in observational studies in epidemiology, to impose stratification to adjust for possible effects of age and other variables on the binary outcome of interest. Overstratification may lower the precision of the estimated effects of interest. Understratification risks bias. These issues are studied analytically. Asymptotic results show that loss of efficiency depends on the true effect and on a measure of the average imbalance across strata between exposed and unexposed individuals. Bias depends on the correlation between stratum-specific size imbalances and event rates in the unexposed. Approximate results are also given. An example is used.

On consistency of Kendall's tau under censoring

Oakes, D. — 2008-11-25

Necessary and sufficient conditions for consistency of a simple estimator of Kendall's tau under bivariate censoring are presented. The results are extended to data subject to bivariate left truncation as well as right censoring.

On an internal method for deriving a summary measure

Cox, D. R. — 2008-11-25

Some preliminary comments are made about the reasons for combining component observations into composite or derived variables. A method for forming derived variables sensitive to specified changes in the underlying multivariate distribution is described and illustrated by an issue in a study of animal pathology.

A note on nonparametric quantile inference for competing risks and more complex multistate models

Beyersmann, J., Schumacher, M. — 2008-11-25

Nonparametric quantile inference for competing risks has recently been studied by Peng & Fine (2007). Their key result establishes uniform consistency and weak convergence of the inverse of the Aalen–Johansen estimator of the cumulative incidence function, using the representation of the cumulative incidence estimator as a sum of independent and identically distributed random variables. The limit process is of a form similar to that of the standard survival result, but with the cause-specific hazard of interest replacing the all-causes hazard. We show that this fact is not a coincidence, but can be derived from a general Hadamard differentiation result. We discuss a simplified proof and extensions of the approach to more complex multistate models. As a further consequence, we find that the bootstrap works.