Parametric Distributional Assumptions Research Articles

The biomarker revolution

The biomarker revolution

Likelihood-based Imprecise Regression

We introduce a new approach to regression with imprecisely observed data, combining likelihood inference with ideas from imprecise probability theory, and thereby taking different kinds of uncertainty into account. The approach is very general: it provides a uniform theoretical framework for regression analysis with imprecise data, where all kinds of relationships between the variables of interest may be considered and all types of imprecisely observed data are allowed.Furthermore, we propose a regression method based on this approach, where no parametric distributional assumption is needed and likelihood-based interval estimates of quantiles of the residuals distribution are used to identify a set of plausible descriptions of the relationship of interest. Thus, the proposed regression method is very robust and yields a set-valued result, whose extent is determined by the amounts of both kinds of uncertainty involved in the regression problem with imprecise data: statistical uncertainty and indetermination.In addition, we apply our robust regression method to an interesting question in the social sciences by analyzing data from a social survey. As result we obtain a large set of plausible relationships, reflecting the high uncertainty inherent in the analyzed data set.

A nonparametric fault isolation approach through one-class classification algorithms

Multivariate control charts provide control limits for the monitoring of processes and detection of abnormal events so that processes can be improved. However, these multivariate control charts provide limited information about the contribution of any specific variable to the out-of-control alarm. Although many fault isolation methods have been developed to address this deficiency, most of these methods require a parametric distributional assumption that restricts their applicability to specific problems of process control and thus limits their broader usefulness. This study proposes a nonparametric fault isolation method based on a one-class classification algorithm that overcomes the limitation posed by the parametric assumption in existing fault isolation methods. The proposed approach decomposes the monitoring statistics obtained from a one-class classification algorithm into components that reflect the contribution of each variable to the out-of-control signal. A bootstrap approach is used to determine the significance of each variable. A simulation study is presented that examines the performance of the proposed method under various scenarios and to results are compared with those obtained using the T 2 decomposition method. The simulation results reveal that the proposed method outperforms the T 2 decomposition method in non-normal distribution cases.

Bayesian Semiparametric Frailty Selection in Multivariate Event Time Data

Biomedical studies often collect multivariate event time data from multiple clusters (either subjects or groups) within each of which event times for individuals are correlated and the correlation may vary in different classes. In such survival analyses, heterogeneity among clusters for shared and specific classes can be accommodated by incorporating parametric frailty terms into the model. In this article, we propose a Bayesian approach to relax the parametric distribution assumption for shared and specific-class frailties by using a Dirichlet process prior while also allowing for the uncertainty of heterogeneity for different classes. Multiple cluster-specific frailty selections rely on variable selection-type mixture priors by applying mixtures of point masses at zero and inverse gamma distributions to the variance of log frailties. This selection allows frailties with zero variance to effectively drop out of the model. A reparameterization of log-frailty terms is performed to reduce the potential bias of fixed effects due to variation of the random distribution and dependence among the parameters resulting in easy interpretation and faster Markov chain Monte Carlo convergence. Simulated data examples and an application to a lung cancer clinical trial are used for illustration.

Spatial Inference of Nitrate Concentrations in Groundwater

We develop a method for multiscale estimation of pollutant concentrations, based on a nonparametric spatial statistical model. We apply this method to estimate nitrate concentrations in groundwater over the mid-Atlantic states, using measurements gathered during a period of 10 years. A map of the fine-scale estimated nitrate concentration is obtained, as well as maps of the estimated county-level average nitrate concentration and similar maps at the level of watersheds and other geographic regions. The fine-scale and coarse-scale estimates arise naturally from a single model, without refitting or ad hoc aggregation. As a result, the uncertainty associated with each estimate is available, without approximations relying on high spatial density of measurements or parametric distributional assumptions. Several risk measures are also obtained, including the probability of the pollutant concentration exceeding a particular threshold. These risk measures can be obtained at the fine scale, or at the level of counties or other regions. The nonparametric Bayesian statistical model allows for this flexibility in estimation while avoiding strong assumptions. This method can be applied directly to estimate ozone concentrations in air, pesticide concentrations in groundwater, or any other quantity that varies over a geographic region, based on approximate measurements at some locations and perhaps of associated covariates. An S-PLUS package with this capability is provided as supplemental material.

Nonparametric Identification of Multinomial Choice Demand Models with Heterogeneous Consumers

We consider identification of nonparametric random utility models of multinomial choice using micro data, i.e., observation of the characteristics and choices of individual consumers. Our model of preferences nests random coefficients discrete choice models widely used in practice with parametric functional form and distributional assumptions. However, the model is nonparametric and distribution free. It allows choice-specific unobservables, endogenous choice characteristics, unknown heteroskedasticity, and high-dimensional correlated taste shocks. Under standard and instrumental variables assumptions, we show identifiability of the random utility model. We demonstrate robustness of these results to relaxation of the large support condition and show that when it is replaced with a weaker common choice probability condition, the demand structure is still identified. We show that key maintained hypotheses are testable.

Bayesian Analysis of Cancer Rates From SEER Program Using Parametric and Semiparametric Joinpoint Regression Models

Cancer is the second leading cause of death in the United States. Cancer incidence and mortality rates measure the progress against cancer; these rates are obtained from the Surveillance, Epidemiology, and End Results (SEER) Program of the National Cancer Institute (NCI). Lung cancer has the highest mortality rate among all cancers, whereas prostate cancer has the highest number of new cases among males. In this article, we analyze the incidence rates of these two cancers, as well as colon and rectal cancer. The NCI reports trends in cancer age-adjusted mortality and incidence rates in its annual report to the nation and analyzes them using the Joinpoint software. The location of the joinpoints signifies changes in cancer trends, whereas changes in the regression slope measure the degree of change. The Joinpoint software uses a numerical search to detect the joinpoints, fits regression within two consecutive joinpoints by least squares, and finally selects the number of joinpoints by either a series of permutation tests or the Bayesian information criterion. We propose Bayesian joinpoint models and provide statistical estimates of the joinpoints and the regression slopes. While the Joinpoint software and other work in this area assumes that the joinpoints occur on the discrete time grid, we allow a continuous prior for the joinpoints induced by the Dirichlet distribution on the spacings in between. This prior further allows the user to impose prespecified minimum gaps in between two consecutive joinpoints. We develop parametric as well as semiparametric Bayesian joinpoint models; the semiparametric framework relaxes parametric distributional assumptions by modeling the distribution of regression slopes and error variances using Dirichlet process mixtures. These Bayesian models provide statistical inference with finite sample validity. Through a simulation study, we demonstrate the performance of the proposed parametric and semiparametric joinpoint models and compare the results with the ones from the Joinpoint software. We analyze age-adjusted cancer incidence rates from the SEER Program using these Bayesian models with different numbers of joinpoints by employing the deviance information criterion and the cross-validated predictive criterion. In addition, we model the lung cancer incidence rates and the smoking rates jointly and explore the relation between the two longitudinal processes.

Reference Bayesian methods for recapture models with heterogeneity

In the context of capture–recapture experiments heterogeneous capture probabilities are often perceived as one of the most challenging features to be incorporated in statistical models. In this paper we propose within a Bayesian framework a new modeling strategy for inference on the unknown population size in the presence of heterogeneity of subject characteristics. Our approach is attractive in that parameters are easily interpretable. Moreover, no parametric distributional assumptions are imposed on the latent distribution of individual heterogeneous propensities to be captured. Bayesian inference based on marginal likelihood by-passes some common identifiability issues, and a formal default prior distribution can be derived. Alternative default prior choices are considered and compared. Performance of our formal default approach is favorably evaluated with two real data sets and with a small simulation study.

Semiparametric Bayesian approaches to joinpoint regression for population-based cancer survival data

According to the American Cancer Society report (1999), cancer surpasses heart disease as the leading cause of death in the United States of America (USA) for people of age less than 85. Thus, medical research in cancer is an important public health interest. Understanding how medical improvements are affecting cancer incidence, mortality and survival is critical for effective cancer control. In this paper, we study the cancer survival trend on the population level cancer data. In particular, we develop a parametric Bayesian joinpoint regression model based on a Poisson distribution for the relative survival. To avoid identifying the cause of death, we only conduct analysis based on the relative survival. The method is further extended to the semiparametric Bayesian joinpoint regression models wherein the parametric distributional assumptions of the joinpoint regression models are relaxed by modeling the distribution of regression slopes using Dirichlet process mixtures. We also consider the effect of adding covariates of interest in the joinpoint model. Three model selection criteria, namely, the conditional predictive ordinate (CPO), the expected predictive deviance (EPD), and the deviance information criteria (DIC), are used to select the number of joinpoints. We analyze the grouped survival data for distant testicular cancer from the Surveillance, Epidemiology, and End Results (SEER) Program using these Bayesian models.

Quantal models: a review with additional methodological development

Analysis of quantal models is a particular aspect of the general problem of investigating multimodality. The distinction is that the spacings between modes are integral multiples of some unspecified fundamental unit and that the number of modes is not defined. Such semi-structured models arise in a wide variety of contexts such as biology, cosmology, archaeology and molecular physics. This paper presents a brief review of their historical development in such areas as an aid to their recognition in other contexts as well as giving guidance to their analysis from the statistical viewpoint. The available methodology for their analysis is collated into a coherent and self-contained account, establishing various optimality properties under particular parametric distributional assumptions. An illustrative power study shows how dependence on sample size and failure of assumptions such as underlying distribution, origin of measurements and independence affect the power of various analyses. These aspects are illustrated by an example from developmental biology.

Efficient nonparametric estimation of causal effects in randomized trials with noncompliance

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.

Worm plot to diagnose fit in quantile regression

The worm plot is a series of detrended Q-Q plots, split by covariate levels. The worm plot is a diagnostic tool for visualizing how well a statistical model fits the data, for finding locations at which the fit can be improved, and for comparing the fit of different models. This paper shows how the worm plot can be used in conjunction with quantile regression. No parametric distributional assumptions are needed to create the worm plot. We fitted both an LMS and a quantile regression model on Dutch height data. The worm plot shows that the quantile regression model is superior to the LMS model in terms of fit. At the same time, it also contains a warning that the particular quantile model used may actually overfit the data. The resulting quantile curves are wiggly at the extremes, and appear less well suited for drawing growth diagrams. The paper concludes that the worm plot is a natural diagnostic tool for quantile regression.

Graphical Estimators From Probability Plots With Right-Censored Data

Probability plots are popular graphical tools used by reliability engineers and other practitioners for assessing parametric distributional assumptions. They are particularly well suited for location-scale families or those that can be transformed to such families. When the plot indicates a reasonable conformity to the assumed family, it is common to estimate the underlying location and scale parameters by fitting a line through the plot. This quick-and-easy method is especially useful with censored data. Indeed, the current version of a popular statistical software package uses this as the default estimation method. In this article we investigate the properties of graphical estimators with multiply right-censored data and compare their performance with that of maximum likelihood estimators. Large-sample results on consistency, asymptotic normality, and asymptotic variance expressions are obtained. Small-sample properties are studied through simulation for selected distributions and censoring patterns. The results presented in this article extend the work of Nair (1984) to right-censored data.

Dummy Endogenous Variables in Weakly Separable Models

In this paper, we consider the nonparametric identification and estimation of the average effect of a dummy endogenous regressor in models where the regressors are weakly but not additively separable from the error term. The model is not required to be strictly increasing in the error term, and the class of models considered includes limited dependent variable models such as discrete choice models. Conditions are established conditions under which it is possible to identify the average effect of the dummy endogenous regressor in a weakly separable model without relying on parametric functional form or distributional assumptions and without the use of large support conditions.

A semiparametric likelihood approach to joint modeling of longitudinal and time-to-event data.

Joint models for a time-to-event (e.g., survival) and a longitudinal response have generated considerable recent interest. The longitudinal data are assumed to follow a mixed effects model, and a proportional hazards model depending on the longitudinal random effects and other covariates is assumed for the survival endpoint. Interest may focus on inference on the longitudinal data process, which is informatively censored, or on the hazard relationship. Several methods for fitting such models have been proposed, most requiring a parametric distributional assumption (normality) on the random effects. A natural concern is sensitivity to violation of this assumption; moreover, a restrictive distributional assumption may obscure key features in the data. We investigate these issues through our proposal of a likelihood-based approach that requires only the assumption that the random effects have a smooth density. Implementation via the EM algorithm is described, and performance and the benefits for uncovering noteworthy features are illustrated by application to data from an HIV clinical trial and by simulation.

A Pooled Response Strategy for Combining Multiple Lines of Evidence to Quantitatively Estimate Impact

The impacts of sediment contaminants can be evaluated by different lines of evidence, including toxicity tests and ecological community studies. Responses from 10 different toxicity assays/tests were combined to arrive at a “site score.” We employed a relatively simple summary measure, pooled P-values where we quantify a potential decrement in response in a contaminated site relative to nominally clean reference sites. The response-specific P-values were defined relative to a “null” distribution of responses in reference sites, and were then pooled using standard meta-analytic methods. Ecological community data were also evaluated using an analogous strategy. A distribution of distances of the reference sites from thecentroid of the reference sites was obtained. The distance from each of the test sites from the centroid of the reference sites was then calculated, and the proportion of reference distances that exceed the test site difference was used to define an empirical P-value for that test site. A plot of the toxicity P-value versus the community P-value was used to identify sites based on both alteration in community structure and toxicity, that is, by weight-of-evidence. This approach provides a useful strategy for examining multiple lines of evidence that should be accessible to the broader scientific community. The use of a large collection of reference sites to empirically define P-values is appealing in that parametric distribution assumptions are avoided, although this does come at the cost of assuming the reference sites provide an appropriate comparison group for test sites.

Parametric modeling for survival with competing risks and masked failure causes.

We consider a life testing situation in which systems are subject to failure from independent competing risks. Following a failure, immediate (stage-1) procedures are used in an attempt to reach a definitive diagnosis. If these procedures fail to result in a diagnosis, this phenomenon is called masking. Stage-2 procedures, such as failure analysis or autopsy, provide definitive diagnosis for a sample of the masked cases. We show how stage-1 and stage-2 information can be combined to provide statistical inference about (a) survival functions of the individual risks, (b) the proportions of failures associated with individual risks and (c) probability, for a specified masked case, that each of the masked competing risks is responsible for the failure. Our development is based on parametric distributional assumptions and the special case for which the failure times for the competing risks have a Weibull distribution is discussed in detail.

Genetic loci controlling breast cancer susceptibility in the Wistar-Kyoto rat.

In this study, the Wistar-Kyoto (WKy) rat was genetically characterized for loci that modify susceptibility to mammary carcinogenesis. We used a genetic backcross between resistant WKy and susceptible Wistar-Furth (WF) rats as a panel for linkage mapping to genetically identify mammary carcinoma susceptibility (Mcs) loci underlying the resistance of the WKy rat. Rats were phenotyped for DMBA-induced mammary carcinomas and genotyped using microsatellite markers. To detect quantitative trait loci (QTL), we analyzed the genome scan data under both parametric and nonparametric distributional assumptions and used permutation tests to calculate significance thresholds. A generalized linear model analysis was also performed to test for interactions between significant QTL. This methodology was extended to identify interactions between the significant QTL and other genome locations. Chromosomes 5, 7, 10, and 14 were found to contain significant QTL, termed Mcs5, Mcs6, Mcs7, and Mcs8, respectively. The WKy alleles of Mcs5, -6, and -8 are associated with mammary carcinoma resistance; the WKy allele of Mcs7 is associated with an increased incidence of mammary cancer. In addition, we identified an interaction between Mcs8 and a region on chromosome 6 termed Mcsm1 (modifier of Mcs), which had no significant main effect on mammary cancer susceptibility in this genetic analysis.

Resampling as a cluster validation technique in fMRI.

Exploratory, data-driven analysis approaches such as cluster analysis, principal component analysis, independent component analysis, or neural network-based techniques are complementary to hypothesis-led methods. They may be considered as hypothesis generating methods. The representative time courses they produce may be viewed as alternative hypotheses to the null hypothesis, i.e., "no activation." We present here a resampling technique to validate the results of exploratory fuzzy clustering analysis. In this case an alternative hypothesis is represented by a cluster centroid. For both simulated and in vivo functional magnetic resonance imaging data, we show that by permutation-based resampling, statistical significance may be computed for each voxel belonging to a cluster of interest without parametric distributional assumptions.

Semiparametric estimation of major gene effects for age of onset

Analysis of age of onset is a key factor in the segregation and linkage analysis of some complex genetic traits. Previous work in the genetics literature has used parametric distributional assumptions on age of onset. In this paper, a Cox model with latent major gene effects is used: a semiparametric model with unspecified baseline hazard. A Monte Carlo EM procedure is used to obtain maximum likelihood estimates. Markov chain Monte Carlo is used to realize genotypic configurations from the posterior distribution given the current model and the observed data, and these genotypic configurations are used to estimate the expectations in the EM algorithm. Simulated data sets indicate that the parameters can be estimated well, and one real data set shows the practical applicability of the proposed method.