Unconditional versions of several tests commonly used in the analysis of contingency tables

By focusing on a confidence interval for a nuisance parameter, Berger and Boos (1994, Journal of the American Statistical Association 89, 1012-1016) proposed new unconditional tests. In particular, they showed that, for a 2 x 2 table, this procedure generally was more powerful than Fisher's exact test. This paper utilizes and extends their approach to obtain unconditional tests for combining several 2 x 2 tables and testing for trend and homogeneity in a 2 x K table. The unconditional procedures are compared to the conditional ones by reanalyzing some published biomedical data.     

Normal approximation diagnostics for the Cox model

We discuss two diagnostic methods for assessing the accuracy of the normal approximated confidence region to the likelihood-based confidence region for the Cox proportional hazards model with censored data. The proposed diagnostic methods are extensions of the contour measures of Hodges (1987, Journal of the American Statistical Association 82, 149-154) and Cook and Tsai (1990, Journal of the American Statistical Association 85, 770-777) and the curvature measures of Jennings (1986, Journal of the American Statistical Association 81, 471-476) and Cook and Tsai (1990). These methods are also illustrated in a study of hepatocyte growth factor in patients with lung cancer and a Mayo Clinic randomized study of participants with primary biliary cirrhosis.     

Small-sample confidence regions in exponential families

This article presents an algorithm for small-sample conditional confidence regions for two or more parameters for any discrete regression model in the generalized linear interactive model family. Regions are constructed by careful inversion of conditional hypothesis tests. This method presupposes the use of approximate or exact techniques for enumerating the sample space for some components of the vector of sufficient statistics conditional on other components. Such enumeration may be performed exactly or by exact or approximate Monte Carlo, including the algorithms of Kolassa and Tanner (1994, Journal of the American Statistical Association 89, 697-702; 1999, Biometrics 55, 246-251). This method also assumes that one can compute certain conditional probabilities for a fixed value of the parameter vector. Because of a property of exponential families, one can use this set of conditional probabilities to directly compute the conditional probabilities associated with any other value of the vector of the parameters of interest. This observation dramatically reduces the computational effort required to invert the hypothesis test to obtain the confidence region. To construct a region with confidence level 1 - alpha, the algorithm begins with a grid of values for the parameters of interest. For each parameter vector on the grid (corresponding to the current null hypothesis), one transforms the initial set of conditional probabilities using exponential tilting and then calculates the p value for this current null hypothesis. The confidence region is the set of parameter values for which the p value is at least alpha.     

Robustified maximum likelihood estimation in generalized partial linear mixed model for longitudinal data

Summary .  In this article, we study the robust estimation of both mean and variance components in generalized partial linear mixed models based on the construction of robustified likelihood function. Under some regularity conditions, the asymptotic properties of the proposed robust estimators are shown. Some simulations are carried out to investigate the performance of the proposed robust estimators. Just as expected, the proposed robust estimators perform better than those resulting from robust estimating equations involving conditional expectation like Sinha (2004, Journal of the American Statistical Association 99, 451–460) and Qin and Zhu (2007, Journal of Multivariate Analysis 98, 1658–1683). In the end, the proposed robust method is illustrated by the analysis of a real data set.     

Approximate Monte Carlo conditional inference in exponential families

This article presents an algorithm for approximate frequentist conditional inference on two or more parameters for any regression model in the Generalized Linear Model (GLIM) family. We thereby extend highly accurate inference beyond the cases of logistic regression and contingency tables implimented in commercially available software. The method makes use of the double saddlepoint approximations of Skovgaard (1987, Journal of Applied Probability 24, 875-887) and Jensen (1992, Biometrika 79, 693-703) to the conditional cumulative distribution function of a sufficient statistic given the remaining sufficient statistics. This approximation is then used in conjunction with noniterative Monte Carlo methods to generate a sample from a distribution that approximates the joint distribution of the sufficient statistics associated with the parameters of interest conditional on the observed values of the sufficient statistics associated with the nuisance parameters. This algorithm is an alternate approach to that presented by Kolassa and Tanner (1994, Journal of the American Statistical Association 89, 697-702), in which a Markov chain is generated whose equilibrium distribution under certain regularity conditions approximates the joint distribution of interest. In Kolassa and Tanner (1994), the Gibbs sampler was used in conjunction with these univariate conditional distribution function approximations. The method of this paper does not require the construction and simulation of a Markov chain, thus avoiding the need to develop regularity conditions under which the algorithm converges and the need for the data analyst to check convergence of the particular chain. Examples involving logistic and truncated Poisson regression are presented.     

Isotonized CDF estimation from judgment poststratification data with empty strata

In applications that require cost efficiency, sample sizes are typically small so that the problem of empty strata may often occur in judgment poststratification (JPS), an important variant of balanced ranked set sampling. In this article, we consider estimation of population cumulative distribution functions (CDF) from JPS samples with empty strata. In the literature, the standard and restricted CDF estimators (Stokes and Sager, 1988, Journal of the American Statistical Association 83, 374381; Frey and Ozturk, 2011, Annals of the Institute of Statistical Mathematics, to appear) do not perform well when simply ignoring empty strata. In this article, we show that the original isotonized estimator (Ozturk, 2007, Journal of Nonparametric Statistics 19, 131-144) can handle empty strata automatically through two methods, MinMax and MaxMin. However, blindly using them can result in undesirable results in either tail of the CDF. We thoroughly examine MinMax and MaxMin and find interesting results about their behaviors and performance in the presence of empty strata. Motivated by these results, we propose modified isotonized estimators to improve estimation efficiency. Through simulation and empirical studies, we show that our estimators work well in different regions of the CDF, and also improve the overall performance of estimating the whole function.     

Survival analysis in clinical trials: past developments and future directions

The field of survival analysis emerged in the 20th century and experienced tremendous growth during the latter half of the century. The developments in this field that have had the most profound impact on clinical trials are the Kaplan-Meier (1958, Journal of the American Statistical Association 53, 457-481) method for estimating the survival function, the log-rank statistic (Mantel, 1966, Cancer Chemotherapy Report 50, 163-170) for comparing two survival distributions, and the Cox (1972, Journal of the Royal Statistical Society, Series B 34, 187-220) proportional hazards model for quantifying the effects of covariates on the survival time. The counting-process martingale theory pioneered by Aalen (1975, Statistical inference for a family of counting processes, Ph.D. dissertation, University of California, Berkeley) provides a unified framework for studying the small- and large-sample properties of survival analysis statistics. Significant progress has been achieved and further developments are expected in many other areas, including the accelerated failure time model, multivariate failure time data, interval-censored data, dependent censoring, dynamic treatment regimes and causal inference, joint modeling of failure time and longitudinal data, and Baysian methods.     

Survival estimation using splines

A nonparametric maximum likelihood procedure is given for estimating the survivor function from right-censored data. It approximates the hazard rate by a simple function such as a spline, with different approximations yielding different estimators. A special case is that proposed by Nelson (1969, Journal of Quality Technology 1, 27-52) and Altshuler (1970, Mathematical Biosciences 6, 1-11). The estimators are uniformly consistent and have the same asymptotic weak convergence properties as the Kaplan-Meier (1958, Journal of the American Statistical Association 53, 457-481) estimator. However, in small and in heavily censored samples, the simplest spline estimators have uniformly smaller mean squared error than do the Kaplan-Meier and Nelson-Altshuler estimators. The procedure is extended to estimate the baseline hazard rate and regression coefficients in the Cox (1972, Journal of the Royal Statistical Society, Series B 34, 187-220) proportional hazards model and is illustrated using experimental carcinogenesis data.     

Shrunken p-values for assessing differential expression with applications to genomic data analysis

In many scientific problems involving high-throughput technology, inference must be made involving several hundreds or thousands of hypotheses. Recent attention has focused on how to address the multiple testing issue; much focus has been devoted toward the use of the false discovery rate. In this article, we consider an alternative estimation procedure titled shrunken p-values for assessing differential expression (SPADE). The estimators are motivated by risk considerations from decision theory and lead to a completely new method for adjustment in the multiple testing problem. In addition, the decision-theoretic framework can be used to derive a decision rule for controlling the number of false positive results. Some theoretical results are outlined. The proposed methodology is illustrated using simulation studies and with application to data from a prostate cancer gene expression profiling study.     

Incorporating marginal covariate information in a nonparametric regression model for a sample of R x C tables

SUMMARY: Nonparametric regression models are proposed in the framework of ecological inference for exploratory modeling of disease prevalence rates adjusted for variables, such as age, ethnicity/race, and socio-economic status. Ecological inference is needed when a response variable and covariate are not available at the subject level because only summary statistics are available for the reporting unit, for example, in the form of R x C tables. In this article, only the marginal counts are assumed available in the sample of R x C contingency tables for modeling the joint distribution of counts. A general form for the ecological regression model is proposed, whereby certain covariates are included as a varying coefficient regression model, whereas others are included as a functional linear model. The nonparametric regression curves are modeled as splines fit by penalized weighted least squares. A data-driven selection of the smoothing parameter is proposed using the pointwise maximum squared bias computed from averaging kernels (explained by O'Sullivan, 1986, Statistical Science 1, 502-517). Analytic expressions for bias and variance are provided that could be used to study the rates of convergence of the estimators. Instead, this article focuses on demonstrating the utility of the estimators in a study of disparity in health outcomes by ethnicity/race.     

Exploring the information in p-values for the analysis and planning of multiple-test experiments

Summary A new methodology is proposed for estimating the proportion of true null hypotheses in a large collection of tests. Each test concerns a single parameter δ whose value is specified by the null hypothesis. We combine a parametric model for the conditional cumulative distribution function (CDF) of the p‐value given δ with a nonparametric spline model for the density g(δ) of δ under the alternative hypothesis. The proportion of true null hypotheses and the coefficients in the spline model are estimated by penalized least squares subject to constraints that guarantee that the spline is a density. The estimator is computed efficiently using quadratic programming. Our methodology produces an estimate of the density of δ when the null is false and can address such questions as “when the null is false, is the parameter usually close to the null or far away?” This leads us to define a falsely interesting discovery rate (FIDR), a generalization of the false discovery rate. We contrast the FIDR approach to Efron's (2004, Journal of the American Statistical Association 99, 96–104) empirical null hypothesis technique. We discuss the use of in sample size calculations based on the expected discovery rate (EDR). Our recommended estimator of the proportion of true nulls has less bias compared to estimators based upon the marginal density of the p‐values at 1. In a simulation study, we compare our estimators to the convex, decreasing estimator of Langaas, Lindqvist, and Ferkingstad (2005, Journal of the Royal Statistical Society, Series B 67, 555–572). The most biased of our estimators is very similar in performance to the convex, decreasing estimator. As an illustration, we analyze differences in gene expression between resistant and susceptible strains of barley.     

Inference in spline-based models for multiple time-to-event data, with applications to a breast cancer prevention trial

As part of the National Surgical Adjuvant Breast and Bowel Project, a controlled clinical trial known as the Breast Cancer Prevention Trial (BCPT) was conducted to assess the effectiveness of tamoxifen as a preventive agent for breast cancer. In addition to the incidence of breast cancer, data were collected on several other, possibly adverse, outcomes, such as invasive endometrial cancer, ischemic heart disease, transient ischemic attack, deep vein thrombosis and/or pulmonary embolism. In this article, we present results from an illustrative analysis of the BCPT data, based on a new modeling technique, to assess the effectiveness of the drug tamoxifen as a preventive agent for breast cancer. We extended the flexible model of Gray (1994, Spline-based test in survival analysis, Biometrics 50, 640-652) to allow inference on multiple time-to-event outcomes in the style of the marginal modeling setup of Wei, Lin, and Weissfeld (1989, Regression analysis of multivariate incomplete failure time data by modeling marginal distributions, Journal of the American Statistical Association 84, 1065-1073). This proposed model makes inference possible for multiple time-to-event data while allowing for greater flexibility in modeling the effects of prognostic factors with nonlinear exposure-response relationships. Results from simulation studies on the small-sample properties of the asymptotic tests will also be presented.     

N-mixture models for estimating population size from spatially replicated counts

Spatial replication is a common theme in count surveys of animals. Such surveys often generate sparse count data from which it is difficult to estimate population size while formally accounting for detection probability. In this article, I describe a class of models (N-mixture models) which allow for estimation of population size from such data. The key idea is to view site-specific population sizes, N, as independent random variables distributed according to some mixing distribution (e.g., Poisson). Prior parameters are estimated from the marginal likelihood of the data, having integrated over the prior distribution for N. Carroll and Lombard (1985, Journal of American Statistical Association 80, 423-426) proposed a class of estimators based on mixing over a prior distribution for detection probability. Their estimator can be applied in limited settings, but is sensitive to prior parameter values that are fixed a priori. Spatial replication provides additional information regarding the parameters of the prior distribution on N that is exploited by the N-mixture models and which leads to reasonable estimates of abundance from sparse data. A simulation study demonstrates superior operating characteristics (bias, confidence interval coverage) of the N-mixture estimator compared to the Caroll and Lombard estimator. Both estimators are applied to point count data on six species of birds illustrating the sensitivity to choice of prior on p and substantially different estimates of abundance as a consequence.     

Nonparametric models and methods for designs with dependent censored data: part I

We consider a nonparametric (NP) approach to the analysis of repeated measures designs with censored data. Using the NP model of Akritas and Arnold (1994, Journal of the American Statistical Association 89, 336-343) for marginal distributions, we present test procedures for the NP hypotheses of no main effects, no interaction, and no simple effects. This extends the existing NP methodology for such designs (Wei and Lachin, 1984, Journal of the American Statistical Association 79, 653-661). The procedures do not require any modeling assumptions and should be useful in cases where the assumptions of proportional hazards or location shift fail to be satisfied. The large-sample distribution of the test statistics is based on an i.i.d. representation for Kaplan-Meier integrals. The testing procedures apply also to ordinal data and to data with ties. Useful small-sample approximations are presented, and their performance is examined in a simulation study. Finally, the methodology is illustrated with two real life examples, one with censored and one with missing data. It is indicated that one of the data sets does not conform to any set of assumptions underlying the available methods and also that the present method provides a useful additional analysis even when data sets conform to modeling assumptions.     

On corrected score approach for proportional hazards model with covariate measurement error

In the presence of covariate measurement error with the proportional hazards model, several functional modeling methods have been proposed. These include the conditional score estimator (Tsiatis and Davidian, 2001, Biometrika 88, 447-458), the parametric correction estimator (Nakamura, 1992, Biometrics 48, 829-838), and the nonparametric correction estimator (Huang and Wang, 2000, Journal of the American Statistical Association 95, 1209-1219) in the order of weaker assumptions on the error. Although they are all consistent, each suffers from potential difficulties with small samples and substantial measurement error. In this article, upon noting that the conditional score and parametric correction estimators are asymptotically equivalent in the case of normal error, we investigate their relative finite sample performance and discover that the former is superior. This finding motivates a general refinement approach to parametric and nonparametric correction methods. The refined correction estimators are asymptotically equivalent to their standard counterparts, but have improved numerical properties and perform better when the standard estimates do not exist or are outliers. Simulation results and application to an HIV clinical trial are presented.     

Flexible hazard regression modeling for medical cost data

The modeling of lifetime (i.e. cumulative) medical cost data in the presence of censored follow-up is complicated by induced informative censoring, rendering standard survival analysis tools invalid. With few exceptions, recently proposed nonparametric estimators for such data do not extend easily to handle covariate information. We propose to model the hazard function for lifetime cost endpoints using an adaptation of the HARE methodology (Kooperberg, Stone, and Truong, Journal of the American Statistical Association, 1995, 90, 78-94). Linear splines and their tensor products are used to adaptively build a model that incorporates covariates and covariate-by-cost interactions without restrictive parametric assumptions. The informative censoring problem is handled using inverse probability of censoring weighted estimating equations. The proposed method is illustrated using simulation and also with data on the cost of dialysis for patients with end-stage renal disease.     

Monte Carlo EM for missing covariates in parametric regression models

We propose a method for estimating parameters for general parametric regression models with an arbitrary number of missing covariates. We allow any pattern of missing data and assume that the missing data mechanism is ignorable throughout. When the missing covariates are categorical, a useful technique for obtaining parameter estimates is the EM algorithm by the method of weights proposed in Ibrahim (1990, Journal of the American Statistical Association 85, 765-769). We extend this method to continuous or mixed categorical and continuous covariates, and for arbitrary parametric regression models, by adapting a Monte Carlo version of the EM algorithm as discussed by Wei and Tanner (1990, Journal of the American Statistical Association 85, 699-704). In addition, we discuss the Gibbs sampler for sampling from the conditional distribution of the missing covariates given the observed data and show that the appropriate complete conditionals are log-concave. The log-concavity property of the conditional distributions will facilitate a straightforward implementation of the Gibbs sampler via the adaptive rejection algorithm of Gilks and Wild (1992, Applied Statistics 41, 337-348). We assume the model for the response given the covariates is an arbitrary parametric regression model, such as a generalized linear model, a parametric survival model, or a nonlinear model. We model the marginal distribution of the covariates as a product of one-dimensional conditional distributions. This allows us a great deal of flexibility in modeling the distribution of the covariates and reduces the number of nuisance parameters that are introduced in the E-step. We present examples involving both simulated and real data.     

Augmented Inverse Probability Weighted Estimator for Cox Missing Covariate Regression

This article investigates an augmented inverse selection probability weighted estimator for Cox regression parameter estimation when covariate variables are incomplete. This estimator extends the Horvitz and Thompson (1952, Journal of the American Statistical Association 47, 663-685) weighted estimator. This estimator is doubly robust because it is consistent as long as either the selection probability model or the joint distribution of covariates is correctly specified. The augmentation term of the estimating equation depends on the baseline cumulative hazard and on a conditional distribution that can be implemented by using an EM-type algorithm. This method is compared with some previously proposed estimators via simulation studies. The method is applied to a real example.     

Test of marginal compatibility and smoothing methods for exchangeable binary data with unequal cluster sizes

Exchangeable binary data are often collected in developmental toxicity and other studies, and a whole host of parametric distributions for fitting this kind of data have been proposed in the literature. While these distributions can be matched to have the same marginal probability and intra-cluster correlation, they can be quite different in terms of shape and higher-order quantities of interest such as the litter-level risk of having at least one malformed fetus. A sensible alternative is to fit a saturated model (Bowman and George, 1995, Journal of the American Statistical Association 90, 871-879) using the expectation-maximization (EM) algorithm proposed by Stefanescu and Turnbull (2003, Biometrics 59, 18-24). The assumption of compatibility of marginal distributions is often made to link up the distributions for different cluster sizes so that estimation can be based on the combined data. Stefanescu and Turnbull proposed a modified trend test to test this assumption. Their test, however, fails to take into account the variability of an estimated null expectation and as a result leads to inaccurate p-values. This drawback is rectified in this article. When the data are sparse, the probability function estimated using a saturated model can be very jagged and some kind of smoothing is needed. We extend the penalized likelihood method (Simonoff, 1983, Annals of Statistics 11, 208-218) to the present case of unequal cluster sizes and implement the method using an EM-type algorithm. In the presence of covariate, we propose a penalized kernel method that performs smoothing in both the covariate and response space. The proposed methods are illustrated using several data sets and the sampling and robustness properties of the resulting estimators are evaluated by simulations.