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1.  Spatial Scan Statistics for Models with Excess Zeros and Overdispersion 
To propose a more realistic model for disease cluster detection, through a modification of the spatial scan statistic to account simultaneously for inflated zeros and overdispersion.
Spatial Scan Statistics [1] usually assume Poisson or Binomial distributed data, which is not adequate in many disease surveillance scenarios. For example, small areas distant from hospitals may exhibit a smaller number of cases than expected in those simple models. Also, underreporting may occur in underdeveloped regions, due to inefficient data collection or the difficulty to access remote sites. Those factors generate excess zero case counts or overdispersion, inducing a violation of the statistical model and also increasing the type I error (false alarms). Overdispersion occurs when data variance is greater than the predicted by the used model. To accommodate it, an extra parameter must be included; in the Poisson model, one makes the variance equal to the mean.
Tools like the Generalized Poisson (GP) and the Double Poisson [2] may be a better option for this kind of problem, modeling separately the mean and variance, which could be easily adjusted by covariates. When excess zeros occur, the Zero Inflated Poisson (ZIP) model is used, although ZIP’s estimated parameters may be severely biased if nonzero counts are too dispersed, compared to the Poisson distribution. In this case the Inflated Zero models for the Generalized Poisson (ZIGP), Double Poisson (ZIDP) and Negative Binomial (ZINB) could be good alternatives to the joint modeling of excess zeros and overdispersion. By one hand, Zero Inflated Poisson (ZIP) models were proposed using the spatial scan statistic to deal with the excess zeros [3]. By the other hand, another spatial scan statistic was based on a Poisson-Gamma mixture model for overdispersion [4]. In this work we present a model which includes inflated zeros and overdispersion simultaneously, based on the ZIDP model. Let the parameter p indicate the zero inflation. As the the remaining parameters of the observed cases map and the parameter p are not independent, the likelihood maximization process is not straightforward; it becomes even more complicated when we include covariates in the analysis. To solve this problem we introduce a vector of latent variables in order to factorize the likelihood, and obtain a facilitator for the maximization process using the E-M (Expectation-Maximization) algorithm. We derive the formulas to maximize iteratively the likelihood, and implement a computer program using the E-M algorithm to estimate the parameters under null and alternative hypothesis. The p-value is obtained via the Fast Double Bootstrap Test [5].
Numerical simulations are conducted to assess the effectiveness of the method. We present results for Hanseniasis surveillance in the Brazilian Amazon in 2010 using this technique. We obtain the most likely spatial clusters for the Poisson, ZIP, Poisson-Gamma mixture and ZIDP models and compare the results.
The Zero Inflated Double Poisson Spatial Scan Statistic for disease cluster detection incorporates the flexibility of previous models, accounting for inflated zeros and overdispersion simultaneously.
The Hanseniasis study case map, due to excess of zero cases counts in many municipalities of the Brazilian Amazon and the presence of overdispersion, was a good benchmark to test the ZIDP model. The results obtained are easier to understand compared to each of the previous spatial scan statistic models, the Zero Inflated Poisson (ZIP) model and the Poisson-Gamma mixture model for overdispersion, taken separetely. The E-M algorithm and the Fast Double Bootstrap test are computationally efficient for this type of problem.
PMCID: PMC3692937
Scan statistics; Zero inflated; Overdispersion; Expectation-Maximization algorithm
2.  A random effect multiplicative heteroscedastic model for bacterial growth 
BMC Bioinformatics  2010;11:77.
Predictive microbiology develops mathematical models that can predict the growth rate of a microorganism population under a set of environmental conditions. Many primary growth models have been proposed. However, when primary models are applied to bacterial growth curves, the biological variability is reduced to a single curve defined by some kinetic parameters (lag time and growth rate), and sometimes the models give poor fits in some regions of the curve. The development of a prediction band (from a set of bacterial growth curves) using non-parametric and bootstrap methods permits to overcome that problem and include the biological variability of the microorganism into the modelling process.
Absorbance data from Listeria monocytogenes cultured at 22, 26, 38, and 42°C were selected under different environmental conditions of pH (4.5, 5.5, 6.5, and 7.4) and percentage of NaCl (2.5, 3.5, 4.5, and 5.5). Transformation of absorbance data to viable count data was carried out. A random effect multiplicative heteroscedastic model was considered to explain the dynamics of bacterial growth. The concept of a prediction band for microbial growth is proposed. The bootstrap method was used to obtain resamples from this model. An iterative procedure is proposed to overcome the computer intensive task of calculating simultaneous prediction intervals, along time, for bacterial growth. The bands were narrower below the inflection point (0-8 h at 22°C, and 0-5.5 h at 42°C), and wider to the right of it (from 9 h onwards at 22°C, and from 7 h onwards at 42°C). A wider band was observed at 42°C than at 22°C when the curves reach their upper asymptote. Similar bands have been obtained for 26 and 38°C.
The combination of nonparametric models and bootstrap techniques results in a good procedure to obtain reliable prediction bands in this context. Moreover, the new iterative algorithm proposed in this paper allows one to achieve exactly the prefixed coverage probability for the prediction band. The microbial growth bands reflect the influence of the different environmental conditions on the microorganism behaviour, helping in the interpretation of the biological meaning of the growth curves obtained experimentally.
PMCID: PMC2829529  PMID: 20141635
3.  Measurement error in two-stage analyses, with application to air pollution epidemiology 
Environmetrics  2014;24(8):501-517.
Public health researchers often estimate health effects of exposures (e.g., pollution, diet, lifestyle) that cannot be directly measured for study subjects. A common strategy in environmental epidemiology is to use a first-stage (exposure) model to estimate the exposure based on covariates and/or spatio-temporal proximity and to use predictions from the exposure model as the covariate of interest in the second-stage (health) model. This induces a complex form of measurement error. We propose an analytical framework and methodology that is robust to misspecification of the first-stage model and provides valid inference for the second-stage model parameter of interest.
We decompose the measurement error into components analogous to classical and Berkson error and characterize properties of the estimator in the second-stage model if the first-stage model predictions are plugged in without correction. Specifically, we derive conditions for compatibility between the first- and second-stage models that guarantee consistency (and have direct and important real-world design implications), and we derive an asymptotic estimate of finite-sample bias when the compatibility conditions are satisfied. We propose a methodology that (1) corrects for finite-sample bias and (2) correctly estimates standard errors. We demonstrate the utility of our methodology in simulations and an example from air pollution epidemiology.
PMCID: PMC3994141  PMID: 24764691
measurement error; spatial statistics; two-stage estimation; air pollution; environmental epidemiology
4.  The use of bootstrapping when using propensity-score matching without replacement: a simulation study 
Statistics in Medicine  2014;33(24):4306-4319.
Propensity-score matching is frequently used to estimate the effect of treatments, exposures, and interventions when using observational data. An important issue when using propensity-score matching is how to estimate the standard error of the estimated treatment effect. Accurate variance estimation permits construction of confidence intervals that have the advertised coverage rates and tests of statistical significance that have the correct type I error rates. There is disagreement in the literature as to how standard errors should be estimated. The bootstrap is a commonly used resampling method that permits estimation of the sampling variability of estimated parameters. Bootstrap methods are rarely used in conjunction with propensity-score matching. We propose two different bootstrap methods for use when using propensity-score matching without replacementand examined their performance with a series of Monte Carlo simulations. The first method involved drawing bootstrap samples from the matched pairs in the propensity-score-matched sample. The second method involved drawing bootstrap samples from the original sample and estimating the propensity score separately in each bootstrap sample and creating a matched sample within each of these bootstrap samples. The former approach was found to result in estimates of the standard error that were closer to the empirical standard deviation of the sampling distribution of estimated effects.
PMCID: PMC4260115  PMID: 25087884
propensity score; propensity-score matching; bootstrap; variance estimation; Monte Carlo simulations; matching
5.  The use of bootstrap methods for analysing health-related quality of life outcomes (particularly the SF-36) 
Health-Related Quality of Life (HRQoL) measures are becoming increasingly used in clinical trials as primary outcome measures. Investigators are now asking statisticians for advice on how to analyse studies that have used HRQoL outcomes.
HRQoL outcomes, like the SF-36, are usually measured on an ordinal scale. However, most investigators assume that there exists an underlying continuous latent variable that measures HRQoL, and that the actual measured outcomes (the ordered categories), reflect contiguous intervals along this continuum.
The ordinal scaling of HRQoL measures means they tend to generate data that have discrete, bounded and skewed distributions. Thus, standard methods of analysis such as the t-test and linear regression that assume Normality and constant variance may not be appropriate. For this reason, conventional statistical advice would suggest that non-parametric methods be used to analyse HRQoL data. The bootstrap is one such computer intensive non-parametric method for analysing data.
We used the bootstrap for hypothesis testing and the estimation of standard errors and confidence intervals for parameters, in four datasets (which illustrate the different aspects of study design). We then compared and contrasted the bootstrap with standard methods of analysing HRQoL outcomes. The standard methods included t-tests, linear regression, summary measures and General Linear Models.
Overall, in the datasets we studied, using the SF-36 outcome, bootstrap methods produce results similar to conventional statistical methods. This is likely because the t-test and linear regression are robust to the violations of assumptions that HRQoL data are likely to cause (i.e. non-Normality). While particular to our datasets, these findings are likely to generalise to other HRQoL outcomes, which have discrete, bounded and skewed distributions. Future research with other HRQoL outcome measures, interventions and populations, is required to confirm this conclusion.
PMCID: PMC543443  PMID: 15588308
Health Related Quality of Life; SF-36; Bootstrap Simulation; Statistical Analysis.
6.  Measurement error caused by spatial misalignment in environmental epidemiology 
Biostatistics (Oxford, England)  2008;10(2):258-274.
In many environmental epidemiology studies, the locations and/or times of exposure measurements and health assessments do not match. In such settings, health effects analyses often use the predictions from an exposure model as a covariate in a regression model. Such exposure predictions contain some measurement error as the predicted values do not equal the true exposures. We provide a framework for spatial measurement error modeling, showing that smoothing induces a Berkson-type measurement error with nondiagonal error structure. From this viewpoint, we review the existing approaches to estimation in a linear regression health model, including direct use of the spatial predictions and exposure simulation, and explore some modified approaches, including Bayesian models and out-of-sample regression calibration, motivated by measurement error principles. We then extend this work to the generalized linear model framework for health outcomes. Based on analytical considerations and simulation results, we compare the performance of all these approaches under several spatial models for exposure. Our comparisons underscore several important points. First, exposure simulation can perform very poorly under certain realistic scenarios. Second, the relative performance of the different methods depends on the nature of the underlying exposure surface. Third, traditional measurement error concepts can help to explain the relative practical performance of the different methods. We apply the methods to data on the association between levels of particulate matter and birth weight in the greater Boston area.
PMCID: PMC2733173  PMID: 18927119
Air pollution; Measurement error; Predictions; Spatial misalignment
7.  Combining test statistics and models in bootstrapped model rejection: it is a balancing act 
BMC Systems Biology  2014;8:46.
Model rejections lie at the heart of systems biology, since they provide conclusive statements: that the corresponding mechanistic assumptions do not serve as valid explanations for the experimental data. Rejections are usually done using e.g. the chi-square test (χ2) or the Durbin-Watson test (DW). Analytical formulas for the corresponding distributions rely on assumptions that typically are not fulfilled. This problem is partly alleviated by the usage of bootstrapping, a computationally heavy approach to calculate an empirical distribution. Bootstrapping also allows for a natural extension to estimation of joint distributions, but this feature has so far been little exploited.
We herein show that simplistic combinations of bootstrapped tests, like the max or min of the individual p-values, give inconsistent, i.e. overly conservative or liberal, results. A new two-dimensional (2D) approach based on parametric bootstrapping, on the other hand, is found both consistent and with a higher power than the individual tests, when tested on static and dynamic examples where the truth is known. In the same examples, the most superior test is a 2D χ2vsχ2, where the second χ2-value comes from an additional help model, and its ability to describe bootstraps from the tested model. This superiority is lost if the help model is too simple, or too flexible. If a useful help model is found, the most powerful approach is the bootstrapped log-likelihood ratio (LHR). We show that this is because the LHR is one-dimensional, because the second dimension comes at a cost, and because LHR has retained most of the crucial information in the 2D distribution. These approaches statistically resolve a previously published rejection example for the first time.
We have shown how to, and how not to, combine tests in a bootstrap setting, when the combination is advantageous, and when it is advantageous to include a second model. These results also provide a deeper insight into the original motivation for formulating the LHR, for the more general setting of nonlinear and non-nested models. These insights are valuable in cases when accuracy and power, rather than computational speed, are prioritized.
PMCID: PMC4022267  PMID: 24742065
Model rejection; Bootstrapping; Combining information; 2D; Insulin signaling; Model Mimicry; Likelihood ratio
8.  Standard error estimation using the EM algorithm for the joint modeling of survival and longitudinal data 
Biostatistics (Oxford, England)  2014;15(4):731-744.
Joint modeling of survival and longitudinal data has been studied extensively in the recent literature. The likelihood approach is one of the most popular estimation methods employed within the joint modeling framework. Typically, the parameters are estimated using maximum likelihood, with computation performed by the expectation maximization (EM) algorithm. However, one drawback of this approach is that standard error (SE) estimates are not automatically produced when using the EM algorithm. Many different procedures have been proposed to obtain the asymptotic covariance matrix for the parameters when the number of parameters is typically small. In the joint modeling context, however, there may be an infinite-dimensional parameter, the baseline hazard function, which greatly complicates the problem, so that the existing methods cannot be readily applied. The profile likelihood and the bootstrap methods overcome the difficulty to some extent; however, they can be computationally intensive. In this paper, we propose two new methods for SE estimation using the EM algorithm that allow for more efficient computation of the SE of a subset of parametric components in a semiparametric or high-dimensional parametric model. The precision and computation time are evaluated through a thorough simulation study. We conclude with an application of our SE estimation method to analyze an HIV clinical trial dataset.
PMCID: PMC4173103  PMID: 24771699
EM algorithm; HIV clinical trial; Numerical differentiation; Observed information matrix; Profile likelihood; Semiparametric joint modeling
9.  A Spatio-Temporal Downscaler for Output From Numerical Models 
Often, in environmental data collection, data arise from two sources: numerical models and monitoring networks. The first source provides predictions at the level of grid cells, while the second source gives measurements at points. The first is characterized by full spatial coverage of the region of interest, high temporal resolution, no missing data but consequential calibration concerns. The second tends to be sparsely collected in space with coarser temporal resolution, often with missing data but, where recorded, provides, essentially, the true value. Accommodating the spatial misalignment between the two types of data is of fundamental importance for both improved predictions of exposure as well as for evaluation and calibration of the numerical model. In this article we propose a simple, fully model-based strategy to downscale the output from numerical models to point level. The static spatial model, specified within a Bayesian framework, regresses the observed data on the numerical model output using spatially-varying coefficients which are specified through a correlated spatial Gaussian process.
As an example, we apply our method to ozone concentration data for the eastern U.S. and compare it to Bayesian melding (Fuentes and Raftery 2005) and ordinary kriging (Cressie 1993; Chilès and Delfiner 1999). Our results show that our method outperforms Bayesian melding in terms of computing speed and it is superior to both Bayesian melding and ordinary kriging in terms of predictive performance; predictions obtained with our method are better calibrated and predictive intervals have empirical coverage closer to the nominal values. Moreover, our model can be easily extended to accommodate for the temporal dimension. In this regard, we consider several spatio-temporal versions of the static model. We compare them using out-of-sample predictions of ozone concentration for the eastern U.S. for the period May 1–October 15, 2001. For the best choice, we present a summary of the analysis. Supplemental material, including color versions of Figures 4, 5, 6, 7, and 8, and MCMC diagnostic plots, are available online.
PMCID: PMC2990198  PMID: 21113385
Bayesian melding; Calibration; Markov chain Monte Carlo; Ordinary kriging; Spatial misalignment; Spatially varying coefficient model
10.  Assessing the Reliability to Detect Cerebral Hypometabolism in Probable Alzheimer's Disease and Amnestic Mild Cognitive Impairment 
Journal of neuroscience methods  2010;192(2):277-285.
Fluorodeoxyglucose positron emission tomography (FDG-PET) studies report characteristic patterns of cerebral hypometabolism in probable Alzheimer's disease (pAD) and amnestic mild cognitive impairment (aMCI). This study aims to characterize the consistency of regional hypometabolism in pAD and aMCI patients enrolled in the AD Neuroimaging Initiative (ADNI) using statistical parametric mapping (SPM) and bootstrap resampling, and to compare bootstrap based reliability index to the commonly used type-I error approach with or without correction for multiple comparisons. Batched SPM5 was run for each of 1,000 bootstrap iterations to compare FDG-PET images from 74 pAD and 142 aMCI patients, respectively, to 82 normal controls. Maps of the hypometabolic voxels detected for at least a specific percentage of times over the 1000 runs were examined and compared to an overlap of the hypometabolic maps obtained from 3 randomly partitioned independent sub-datasets. The results from the bootstrap derived reliability of regional hypometabolism in the overall data set were similar to that observed in each of the three non-overlapping sub-sets using family-wise error. Strong but non-linear association was found between the bootstrap based reliability index and the type-I error. For threshold p=0.0005, pAD was associated with extensive hypometabolic voxels in the posterior cingulate/precuneus and parietotemporal regions with reliability between 90% and 100%. Bootstrap analysis provides an alternative to the parametric family-wise error approach used to examine consistency of hypometabolic brain voxels in pAD and aMCI patients. These results provide a foundation for the use of bootstrap analysis characterize statistical ROIs or search regions in both cross-sectional and longitudinal FDG PET studies. This approach offers promise in the early detection and tracking of AD, the evaluation of AD-modifying treatments, and other biologically or clinical important measurements using brain images and voxel-based data analysis techniques.
PMCID: PMC2952503  PMID: 20678521
Alzheimer's Disease; MCI; FDG PET; Reproducibility of Results; Reliability; Bootstrap Resampling; Familywise Error; SPM
11.  Adaptive Choice of the Number of Bootstrap Samples in Large Scale Multiple Testing 
It is a common practice to use resampling methods such as the bootstrap for calculating the p-value for each test when performing large scale multiple testing. The precision of the bootstrap p-values and that of the false discovery rate (FDR) relies on the number of bootstraps used for testing each hypothesis. Clearly, the larger the number of bootstraps the better the precision. However, the required number of bootstraps can be computationally burdensome, and it multiplies the number of tests to be performed. Further adding to the computational challenge is that in some applications the calculation of the test statistic itself may require considerable computation time. As technology improves one can expect the dimension of the problem to increase as well. For instance, during the early days of microarray technology, the number of probes on a cDNA chip was less than 10,000. Now the Affymetrix chips come with over 50,000 probes per chip. Motivated by this important need, we developed a simple adaptive bootstrap methodology for large scale multiple testing, which reduces the total number of bootstrap calculations while ensuring the control of the FDR. The proposed algorithm results in a substantial reduction in the number of bootstrap samples. Based on a simulation study we found that, relative to the number of bootstraps required for the Benjamini-Hochberg (BH) procedure, the standard FDR methodology which was the proposed methodology achieved a very substantial reduction in the number of bootstraps. In some cases the new algorithm required as little as 1/6th the number of bootstraps as the conventional BH procedure. Thus, if the conventional BH procedure used 1,000 bootstraps, then the proposed method required only 160 bootstraps. This methodology has been implemented for time-course/dose-response data in our software, ORIOGEN, which is available from the authors upon request.
PMCID: PMC2752392  PMID: 18384266
12.  Time-space Kriging to address the spatiotemporal misalignment in the large datasets 
This paper presents a Bayesian hierarchical spatiotemporal method of interpolation, termed as Markov Cube Kriging (MCK). The classical Kriging methods become computationally prohibitive, especially for large datasets due to the O(n3) matrix decomposition. MCK offers novel and computationally efficient solutions to address spatiotemporal misalignment, mismatch in the spatiotemporal scales and missing values across space and time in large spatiotemporal datasets. MCK is flexible in that it allows for non-separable spatiotemporal structure and nonstationary covariance at the hierarchical spatiotemporal scales. Employing MCK we developed estimates of daily concentration of fine particulates matter ≤2.5 μm in aerodynamic diameter (PM2.5) at 2.5 km spatial grid for the Cleveland Metropolitan Statistical Area, 2000 to 2009. Our validation and cross-validation suggest that MCK achieved robust prediction of spatiotemporal random effects and underlying hierarchical and nonstationary spatiotemporal structure in air pollution data. MCK has important implications for environmental epidemiology and environmental sciences for exposure quantification and collocation of data from different sources, available at different spatiotemporal scales.
PMCID: PMC3768020  PMID: 24039539
Time-space Kriging; Spatiotemporal hierarchical model; Gaussian Markov Random Fields; Nonstationarity; Bayesian computation; Fine particulate matter PM2.5
13.  A National Prediction Model for PM2.5 Component Exposures and Measurement Error–Corrected Health Effect Inference 
Environmental Health Perspectives  2013;121(9):1017-1025.
Background: Studies estimating health effects of long-term air pollution exposure often use a two-stage approach: building exposure models to assign individual-level exposures, which are then used in regression analyses. This requires accurate exposure modeling and careful treatment of exposure measurement error.
Objective: To illustrate the importance of accounting for exposure model characteristics in two-stage air pollution studies, we considered a case study based on data from the Multi-Ethnic Study of Atherosclerosis (MESA).
Methods: We built national spatial exposure models that used partial least squares and universal kriging to estimate annual average concentrations of four PM2.5 components: elemental carbon (EC), organic carbon (OC), silicon (Si), and sulfur (S). We predicted PM2.5 component exposures for the MESA cohort and estimated cross-sectional associations with carotid intima-media thickness (CIMT), adjusting for subject-specific covariates. We corrected for measurement error using recently developed methods that account for the spatial structure of predicted exposures.
Results: Our models performed well, with cross-validated R2 values ranging from 0.62 to 0.95. Naïve analyses that did not account for measurement error indicated statistically significant associations between CIMT and exposure to OC, Si, and S. EC and OC exhibited little spatial correlation, and the corrected inference was unchanged from the naïve analysis. The Si and S exposure surfaces displayed notable spatial correlation, resulting in corrected confidence intervals (CIs) that were 50% wider than the naïve CIs, but that were still statistically significant.
Conclusion: The impact of correcting for measurement error on health effect inference is concordant with the degree of spatial correlation in the exposure surfaces. Exposure model characteristics must be considered when performing two-stage air pollution epidemiologic analyses because naïve health effect inference may be inappropriate.
Citation: Bergen S, Sheppard L, Sampson PD, Kim SY, Richards M, Vedal S, Kaufman JD, Szpiro AA. 2013. A national prediction model for PM2.5 component exposures and measurement error–corrected health effect inference. Environ Health Perspect 121:1017–1025;
PMCID: PMC3764074  PMID: 23757600
14.  Bayesian spatial modeling of disease risk in relation to multivariate environmental risk fields 
Statistics in medicine  2010;29(1):142-157.
The relationship between exposure to environmental chemicals during pregnancy and early childhood development is an important issue which has a spatial risk component. In this context, we have examined mental retardation and developmental delay (MRDD) outcome measures for children in a Medicaid population in South Carolina and sampled measures of soil chemistry (e.g. As, Hg, etc.) on a network of sites which are misaligned to the outcome residential addresses during pregnancy. The true chemical concentration at the residential addresses is not observed directly and must be interpolated from soil samples. In this study, we have developed a Bayesian joint model which interpolates soil chemical fields and estimates the associated MRDD risk simultaneously. Having multiple spatial fields to interpolate, we have considered a low-rank Kriging method for the interpolation which requires less computation than Bayesian Kriging. We performed a sensitivity analysis for a bivariate smoothing, changing the number of knots and the smoothing parameter. These analyses show that a low-rank Kriging method can be used as an alternative to a full-rank Kriging, reducing computational burden. However, the number of knots for the low-rank Kriging model need to be selected with caution as a bivariate surface estimation can be sensitive to the choice of the number of knots.
PMCID: PMC3004226  PMID: 19904772
environmental exposure; logistic; spatial; low-rank Kriging; Bayesian
15.  An Automatic Approach for Satisfying Dose-Volume Constraints in Linear Fluence Map Optimization for IMPT 
Journal of cancer therapy  2014;5(2):198-207.
Prescriptions for radiation therapy are given in terms of dose-volume constraints (DVCs). Solving the fluence map optimization (FMO) problem while satisfying DVCs often requires a tedious trial-and-error for selecting appropriate dose control parameters on various organs. In this paper, we propose an iterative approach to satisfy DVCs using a multi-objective linear programming (LP) model for solving beamlet intensities. This algorithm, starting from arbitrary initial parameter values, gradually updates the values through an iterative solution process toward optimal solution. This method finds appropriate parameter values through the trade-off between OAR sparing and target coverage to improve the solution. We compared the plan quality and the satisfaction of the DVCs by the proposed algorithm with two nonlinear approaches: a nonlinear FMO model solved by using the L-BFGS algorithm and another approach solved by a commercial treatment planning system (Eclipse 8.9). We retrospectively selected from our institutional database five patients with lung cancer and one patient with prostate cancer for this study. Numerical results show that our approach successfully improved target coverage to meet the DVCs, while trying to keep corresponding OAR DVCs satisfied. The LBFGS algorithm for solving the nonlinear FMO model successfully satisfied the DVCs in three out of five test cases. However, there is no recourse in the nonlinear FMO model for correcting unsatisfied DVCs other than manually changing some parameter values through trial and error to derive a solution that more closely meets the DVC requirements. The LP-based heuristic algorithm outperformed the current treatment planning system in terms of DVC satisfaction. A major strength of the LP-based heuristic approach is that it is not sensitive to the starting condition.
PMCID: PMC4261934  PMID: 25506501
Fluence Map Optimization (FMO); Linear Programming (LP); Nonlinear Programming (NLP); Dose-Volume Constraint (DVC); Intensity-Modulated Proton Therapy (IMPT)
16.  Standard Error Computations for Uncertainty Quantification in Inverse Problems: Asymptotic Theory vs. Bootstrapping 
Mathematical and computer modelling  2010;52(9-10):1610-1625.
We computationally investigate two approaches for uncertainty quantification in inverse problems for nonlinear parameter dependent dynamical systems. We compare the bootstrapping and asymptotic theory approaches for problems involving data with several noise forms and levels. We consider both constant variance absolute error data and relative error which produces non-constant variance data in our parameter estimation formulations. We compare and contrast parameter estimates, standard errors, confidence intervals, and computational times for both bootstrapping and asymptotic theory methods.
PMCID: PMC2935305  PMID: 20835347
Uncertainty quantification; parameter estimation; nonlinear dynamic models; bootstrapping; asymptotic theory standard errors; ordinary least squares vs. generalized least squares; computational examples
17.  Inference about the expected performance of a data-driven dynamic treatment regime 
Clinical trials (London, England)  2014;11(4):408-417.
A dynamic treatment regime (DTR) comprises a sequence of decision rules, one per stage of intervention, that recommends how to individualize treatment to patients based on evolving treatment and covariate history. These regimes are useful for managing chronic disorders, and fit into the larger paradigm of personalized medicine. The Value of a DTR is the expected outcome when the DTR is used to assign treatments to a population of interest.
The Value of a data-driven DTR, estimated using data from a sequential multiple assignment randomized trial, is both a data-dependent parameter and a non-smooth function of the underlying generative distribution. These features introduce additional variability that is not accounted for by standard methods for conducting statistical inference, e.g., the bootstrap or normal approximations, if applied without adjustment. Our purpose is to provide a feasible method for constructing valid confidence intervals for this quantity of practical interest.
We propose a conceptually simple and computationally feasible method for constructing valid confidence intervals for the Value of an estimated DTR based on subsampling. The method is self-tuning by virtue of an approach called the double bootstrap. We demonstrate the proposed method using a series of simulated experiments.
The proposed method offers considerable improvement in terms of coverage rates of the confidence intervals over the standard bootstrap approach.
In this paper, we have restricted our attention to Q-learning for estimating the optimal DTR. However, other methods can be employed for this purpose; to keep the discussion focused, we have not explored these alternatives.
Subsampling-based confidence intervals provide much better performance compared to standard bootstrap for the Value of an estimated DTR.
PMCID: PMC4265005  PMID: 24925083
18.  Gene–Environment Interactions on Growth Trajectories 
Genetic epidemiology  2012;36(3):206-213.
It has been suggested that children with larger brains tend to perform better on IQ tests or cognitive function tests. Prenatal head growth and head growth in infancy are two crucial periods for subsequent intelligence. Studies have shown that environmental exposure to air pollutants during pregnancy is associated with fetal growth reduction, developmental delay, and reduced IQ. Meanwhile, genetic polymorphisms may modify the effect of environment on head growth. However, studies on gene–environment or gene–gene interactions on growth trajectories have been quite limited partly due to the difficulty to quantitatively measure interactions on growth trajectories. Moreover, it is known that assessing the significance of gene–environment or gene–gene interactions on cross-sectional outcomes empirically using the permutation procedures may bring substantial errors in the tests. We proposed a score that quantitatively measures interactions on growth trajectories and developed an algorithm with a parametric bootstrap procedure to empirically assess the significance of the interactions on growth trajectories under the likelihood framework. We also derived a Wald statistic to test for interactions on growth trajectories and compared it to the proposed parametric bootstrap procedure. Through extensive simulation studies, we demonstrated the feasibility and power of the proposed testing procedures. We applied our method to a real dataset with head circumference measures from birth to age 7 on a cohort currently being conducted by the Columbia Center for Children's Environmental Health (CCCEH) in Krakow, Poland, and identified several significant gene–environment interactions on head circumference growth trajectories.
PMCID: PMC3380164  PMID: 22311237
gene–environment interactions; growth curves; Wald test; parametric bootstrap
19.  On validation and invalidation of biological models 
BMC Bioinformatics  2009;10:132.
Very frequently the same biological system is described by several, sometimes competing mathematical models. This usually creates confusion around their validity, ie, which one is correct. However, this is unnecessary since validity of a model cannot be established; model validation is actually a misnomer. In principle the only statement that one can make about a system model is that it is incorrect, ie, invalid, a fact which can be established given appropriate experimental data. Nonlinear models of high dimension and with many parameters are impossible to invalidate through simulation and as such the invalidation process is often overlooked or ignored.
We develop different approaches for showing how competing ordinary differential equation (ODE) based models of the same biological phenomenon containing nonlinearities and parametric uncertainty can be invalidated using experimental data. We first emphasize the strong interplay between system identification and model invalidation and we describe a method for obtaining a lower bound on the error between candidate model predictions and data. We then turn to model invalidation and formulate a methodology for discrete-time and continuous-time model invalidation. The methodology is algorithmic and uses Semidefinite Programming as the computational tool. It is emphasized that trying to invalidate complex nonlinear models through exhaustive simulation is not only computationally intractable but also inconclusive.
Biological models derived from experimental data can never be validated. In fact, in order to understand biological function one should try to invalidate models that are incompatible with available data. This work describes a framework for invalidating both continuous and discrete-time ODE models based on convex optimization techniques. The methodology does not require any simulation of the candidate models; the algorithms presented in this paper have a worst case polynomial time complexity and can provide an exact answer to the invalidation problem.
PMCID: PMC2704209  PMID: 19422679
20.  Semiparametric Bayesian analysis of gene-environment interactions with error in measurement of environmental covariates and missing genetic data 
Statistics and its interface  2011;4(3):305-316.
Case-control studies are widely used to detect gene-environment interactions in the etiology of complex diseases. Many variables that are of interest to biomedical researchers are difficult to measure on an individual level, e.g. nutrient intake, cigarette smoking exposure, long-term toxic exposure. Measurement error causes bias in parameter estimates, thus masking key features of data and leading to loss of power and spurious/masked associations. We develop a Bayesian methodology for analysis of case-control studies for the case when measurement error is present in an environmental covariate and the genetic variable has missing data. This approach offers several advantages. It allows prior information to enter the model to make estimation and inference more precise. The environmental covariates measured exactly are modeled completely nonparametrically. Further, information about the probability of disease can be incorporated in the estimation procedure to improve quality of parameter estimates, what cannot be done in conventional case-control studies. A unique feature of the procedure under investigation is that the analysis is based on a pseudo-likelihood function therefore conventional Bayesian techniques may not be technically correct. We propose an approach using Markov Chain Monte Carlo sampling as well as a computationally simple method based on an asymptotic posterior distribution. Simulation experiments demonstrated that our method produced parameter estimates that are nearly unbiased even for small sample sizes. An application of our method is illustrated using a population-based case-control study of the association between calcium intake with the risk of colorectal adenoma development.
PMCID: PMC3178196  PMID: 21949562
Bayesian inference; Errors in variables; Gene-environment interactions; Markov Chain Monte Carlo sampling; Missing data; Pseudo-likelihood; Semiparametric methods
21.  Linear theory for filtering nonlinear multiscale systems with model error 
In this paper, we study filtering of multiscale dynamical systems with model error arising from limitations in resolving the smaller scale processes. In particular, the analysis assumes the availability of continuous-time noisy observations of all components of the slow variables. Mathematically, this paper presents new results on higher order asymptotic expansion of the first two moments of a conditional measure. In particular, we are interested in the application of filtering multiscale problems in which the conditional distribution is defined over the slow variables, given noisy observation of the slow variables alone. From the mathematical analysis, we learn that for a continuous time linear model with Gaussian noise, there exists a unique choice of parameters in a linear reduced model for the slow variables which gives the optimal filtering when only the slow variables are observed. Moreover, these parameters simultaneously give the optimal equilibrium statistical estimates of the underlying system, and as a consequence they can be estimated offline from the equilibrium statistics of the true signal. By examining a nonlinear test model, we show that the linear theory extends in this non-Gaussian, nonlinear configuration as long as we know the optimal stochastic parametrization and the correct observation model. However, when the stochastic parametrization model is inappropriate, parameters chosen for good filter performance may give poor equilibrium statistical estimates and vice versa; this finding is based on analytical and numerical results on our nonlinear test model and the two-layer Lorenz-96 model. Finally, even when the correct stochastic ansatz is given, it is imperative to estimate the parameters simultaneously and to account for the nonlinear feedback of the stochastic parameters into the reduced filter estimates. In numerical experiments on the two-layer Lorenz-96 model, we find that the parameters estimated online, as part of a filtering procedure, simultaneously produce accurate filtering and equilibrium statistical prediction. In contrast, an offline estimation technique based on a linear regression, which fits the parameters to a training dataset without using the filter, yields filter estimates which are worse than the observations or even divergent when the slow variables are not fully observed. This finding does not imply that all offline methods are inherently inferior to the online method for nonlinear estimation problems, it only suggests that an ideal estimation technique should estimate all parameters simultaneously whether it is online or offline.
PMCID: PMC4032560  PMID: 25002829
filtering multi-scale systems; covariance inflation; stochastic parameterization; uncertainty quantification; model error; parameter estimation
22.  Satellite-based PM concentrations and their application to COPD in Cleveland, OH 
A hybrid approach is proposed to estimate exposure to fine particulate matter (PM2.5) at a given location and time. This approach builds on satellite-based aerosol optical depth (AOD), air pollution data from sparsely distributed Environmental Protection Agency (EPA) sites and local time–space Kriging, an optimal interpolation technique. Given the daily global coverage of AOD data, we can develop daily estimate of air quality at any given location and time. This can assure unprecedented spatial coverage, needed for air quality surveillance and management and epidemiological studies. In this paper, we developed an empirical relationship between the 2 km AOD and PM2.5 data from EPA sites. Extrapolating this relationship to the study domain resulted in 2.3 million predictions of PM2.5 between 2000 and 2009 in Cleveland Metropolitan Statistical Area (MSA). We have developed local time–space Kriging to compute exposure at a given location and time using the predicted PM2.5. Daily estimates of PM2.5 were developed for Cleveland MSA between 2000 and 2009 at 2.5 km spatial resolution; 1.7 million (~79.8%) of 2.13 million predictions required for multiyear and geographic domain were robust. In the epidemiological application of the hybrid approach, admissions for an acute exacerbation of chronic obstructive pulmonary disease (AECOPD) was examined with respect to time–space lagged PM2.5 exposure. Our analysis suggests that the risk of AECOPD increases 2.3% with a unit increase in PM2.5 exposure within 9 days and 0.05° (~5 km) distance lags. In the aggregated analysis, the exposed groups (who experienced exposure to PM2.5 >15.4 μg/m3) were 54% more likely to be admitted for AECOPD than the reference group. The hybrid approach offers greater spatiotemporal coverage and reliable characterization of ambient concentration than conventional in situ monitoring-based approaches. Thus, this approach can potentially reduce exposure misclassification errors in the conventional air pollution epidemiology studies.
PMCID: PMC3980441  PMID: 24045428
PM2.5 exposure; local time–space Kriging; aerosol optical depth; times–pace lagged exposure; COPD
23.  Performance of statistical models to predict mental health and substance abuse cost 
Providers use risk-adjustment systems to help manage healthcare costs. Typically, ordinary least squares (OLS) models on either untransformed or log-transformed cost are used. We examine the predictive ability of several statistical models, demonstrate how model choice depends on the goal for the predictive model, and examine whether building models on samples of the data affects model choice.
Our sample consisted of 525,620 Veterans Health Administration patients with mental health (MH) or substance abuse (SA) diagnoses who incurred costs during fiscal year 1999. We tested two models on a transformation of cost: a Log Normal model and a Square-root Normal model, and three generalized linear models on untransformed cost, defined by distributional assumption and link function: Normal with identity link (OLS); Gamma with log link; and Gamma with square-root link. Risk-adjusters included age, sex, and 12 MH/SA categories. To determine the best model among the entire dataset, predictive ability was evaluated using root mean square error (RMSE), mean absolute prediction error (MAPE), and predictive ratios of predicted to observed cost (PR) among deciles of predicted cost, by comparing point estimates and 95% bias-corrected bootstrap confidence intervals. To study the effect of analyzing a random sample of the population on model choice, we re-computed these statistics using random samples beginning with 5,000 patients and ending with the entire sample.
The Square-root Normal model had the lowest estimates of the RMSE and MAPE, with bootstrap confidence intervals that were always lower than those for the other models. The Gamma with square-root link was best as measured by the PRs. The choice of best model could vary if smaller samples were used and the Gamma with square-root link model had convergence problems with small samples.
Models with square-root transformation or link fit the data best. This function (whether used as transformation or as a link) seems to help deal with the high comorbidity of this population by introducing a form of interaction. The Gamma distribution helps with the long tail of the distribution. However, the Normal distribution is suitable if the correct transformation of the outcome is used.
PMCID: PMC1636059  PMID: 17067394
24.  Weighted bootstrapping: a correction method for assessing the robustness of phylogenetic trees 
Non-parametric bootstrapping is a widely-used statistical procedure for assessing confidence of model parameters based on the empirical distribution of the observed data [1] and, as such, it has become a common method for assessing tree confidence in phylogenetics [2]. Traditional non-parametric bootstrapping does not weigh each tree inferred from resampled (i.e., pseudo-replicated) sequences. Hence, the quality of these trees is not taken into account when computing bootstrap scores associated with the clades of the original phylogeny. As a consequence, traditionally, the trees with different bootstrap support or those providing a different fit to the corresponding pseudo-replicated sequences (the fit quality can be expressed through the LS, ML or parsimony score) contribute in the same way to the computation of the bootstrap support of the original phylogeny.
In this article, we discuss the idea of applying weighted bootstrapping to phylogenetic reconstruction by weighting each phylogeny inferred from resampled sequences. Tree weights can be based either on the least-squares (LS) tree estimate or on the average secondary bootstrap score (SBS) associated with each resampled tree. Secondary bootstrapping consists of the estimation of bootstrap scores of the trees inferred from resampled data. The LS and SBS-based bootstrapping procedures were designed to take into account the quality of each "pseudo-replicated" phylogeny in the final tree estimation. A simulation study was carried out to evaluate the performances of the five weighting strategies which are as follows: LS and SBS-based bootstrapping, LS and SBS-based bootstrapping with data normalization and the traditional unweighted bootstrapping.
The simulations conducted with two real data sets and the five weighting strategies suggest that the SBS-based bootstrapping with the data normalization usually exhibits larger bootstrap scores and a higher robustness compared to the four other competing strategies, including the traditional bootstrapping. The high robustness of the normalized SBS could be particularly useful in situations where observed sequences have been affected by noise or have undergone massive insertion or deletion events. The results provided by the four other strategies were very similar regardless the noise level, thus also demonstrating the stability of the traditional bootstrapping method.
PMCID: PMC2939571  PMID: 20716358
25.  Effect of Misalignment between Successive Corneal Videokeratography Maps on the Repeatability of Topography Data 
PLoS ONE  2015;10(11):e0139541.
To improve the reliability of corneal topographic data through the development of a method to estimate the magnitude of misalignment between successive corneal videokeratography (VK) maps and eliminate the effect of misalignment on the repeatability of topography data.
Anterior and posterior topography maps were recorded twice for 124 healthy eyes of 124 participants using a Pentacam, and the repeatability of measurements was assessed by calculating the differences in elevation between each two sets of data. The repeatability of measurements was re-assessed following the determination of the magnitude of misalignment components (translational displacements: x0, y0 and z0, and rotational displacements: α, β and γ) between each two data sets and using them to modify the second data set within each pair based on an Iterative Closest Point (ICP) algorithm. The method simultaneously considered the anterior and posterior maps taken for the same eye since they were assumed to have the same set of misalignment components. A new parameter, named Combined Misalignment parameter (CM), has been developed to combine the effect of all six misalignment components on topography data and so enable study of the association between misalignment and the data repeatability test results.
The repeatability tests resulted in average root mean square (RMS) differences in elevation data of 8.46±2.75 μm before ICP map matching when simultaneously considering anterior and posterior surfaces. With map matching and misalignment correction, the differences decreased to 7.28±2.58 μm (P = 0.00). When applied to only the anterior maps, misalignment correction led to a more pronounced reduction in elevation data differences from 4.58±1.84 μm to 2.97±1.29 μm (P = 0.00). CM was found to be associated with the repeatability error (P = 0.00), with posterior maps being responsible for most of the error due to their relatively lower accuracy compared to anterior maps.
The ICP algorithm can be used to estimate, and effectively correct for, the potential misalignment between successive corneal videokeratography maps.
PMCID: PMC4658180  PMID: 26599442

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