Efficient Bayesian hierarchical functional data analysis with basis function approximations using Gaussian-Wishart processes.

Functional data are defined as realizations of random functions (mostly smooth functions) varying over a continuum, which are usually collected on discretized grids with measurement errors. In order to accurately smooth noisy functional observations and deal with the issue of high-dimensional observation grids, we propose a novel Bayesian method based on the Bayesian hierarchical model with a Gaussian-Wishart process prior and basis function representations. We first derive an induced model for the basis-function coefficients of the functional data, and then use this model to conduct posterior inference through Markov chain Monte Carlo methods. Compared to the standard Bayesian inference that suffers serious computational burden and instability in analyzing high-dimensional functional data, our method greatly improves the computational scalability and stability, while inheriting the advantage of simultaneously smoothing raw observations and estimating the mean-covariance functions in a nonparametric way. In addition, our method can naturally handle functional data observed on random or uncommon grids. Simulation and real studies demonstrate that our method produces similar results to those obtainable by the standard Bayesian inference with low-dimensional common grids, while efficiently smoothing and estimating functional data with random and high-dimensional observation grids when the standard Bayesian inference fails. In conclusion, our method can efficiently smooth and estimate high-dimensional functional data, providing one way to resolve the curse of dimensionality for Bayesian functional data analysis with Gaussian-Wishart processes.

Multivariate functional response regression, with application to fluorescence spectroscopy in a cervical pre-cancer study.

Many scientific studies measure different types of high-dimensional signals or images from the same subject, producing multivariate functional data. These functional measurements carry different types of information about the scientific process, and a joint analysis that integrates information across them may provide new insights into the underlying mechanism for the phenomenon under study. Motivated by fluorescence spectroscopy data in a cervical pre-cancer study, a multivariate functional response regression model is proposed, which treats multivariate functional observations as responses and a common set of covariates as predictors. This novel modeling framework simultaneously accounts for correlations between functional variables and potential multi-level structures in data that are induced by experimental design. The model is fitted by performing a two-stage linear transformation-a basis expansion to each functional variable followed by principal component analysis for the concatenated basis coefficients. This transformation effectively reduces the intra-and inter-function correlations and facilitates fast and convenient calculation. A fully Bayesian approach is adopted to sample the model parameters in the transformed space, and posterior inference is performed after inverse-transforming the regression coefficients back to the original data domain. The proposed approach produces functional tests that flag local regions on the functional effects, while controlling the overall experiment-wise error rate or false discovery rate. It also enables functional discriminant analysis through posterior predictive calculation. Analysis of the fluorescence spectroscopy data reveals local regions with differential expressions across the pre-cancer and normal samples. These regions may serve as biomarkers for prognosis and disease assessment.

A statistical model for removing inter-device differences in spectroscopy.

We are investigating spectroscopic devices designed to make in vivo cervical tissue measurements to detect pre-cancerous and cancerous lesions. All devices have the same design and ideally should record identical measurements. However, we observed consistent differences among them. An experiment was designed to study the sources of variation in the measurements recorded. Here we present a log additive statistical model that incorporates the sources of variability we identified. Based on this model, we estimated correction factors from the experimental data needed to eliminate the inter-device variability and other sources of variation. These correction factors are intended to improve the accuracy and repeatability of such devices when making future measurements on patient tissue.

Classifying tissue samples from measurements on cells with within-class tissue sample heterogeneity.

We consider here the problem of classifying a macro-level object based on measurements of embedded (micro-level) observations within each object, for example, classifying a patient based on measurements on a collection of a random number of their cells. Classification problems with this hierarchical, nested structure have not received the same statistical understanding as the general classification problem. Some heuristic approaches have been developed and a few authors have proposed formal statistical models. We focus on the problem where heterogeneity exists between the macro-level objects within a class. We propose a model-based statistical methodology that models the log-odds of the macro-level object belonging to a class using a latent-class variable model to account for this heterogeneity. The latent classes are estimated by clustering the macro-level object density estimates. We apply this method to the detection of patients with cervical neoplasia based on quantitative cytology measurements on cells in a Papanicolaou smear. Quantitative cytology is much cheaper and potentially can take less time than the current standard of care. The results show that the automated quantitative cytology using the proposed method is roughly equivalent to clinical cytopathology and shows significant improvement over a statistical model that does not account for the heterogeneity of the data.

Accuracy of optical spectroscopy for the detection of cervical intraepithelial neoplasia: Testing a device as an adjunct to colposcopy.

Testing emerging technologies involves the evaluation of biologic plausibility, technical efficacy, clinical effectiveness, patient satisfaction, and cost-effectiveness. The objective of this study was to select an effective classification algorithm for optical spectroscopy as an adjunct to colposcopy and obtain preliminary estimates of its accuracy for the detection of CIN 2 or worse. We recruited 1,000 patients from screening and prevention clinics and 850 patients from colposcopy clinics at two comprehensive cancer centers and a community hospital. Optical spectroscopy was performed, and 4,864 biopsies were obtained from the sites measured, including abnormal and normal colposcopic areas. The gold standard was the histologic report of biopsies, read 2 to 3 times by histopathologists blinded to the cytologic, histopathologic, and spectroscopic results. We calculated sensitivities, specificities, receiver operating characteristic (ROC) curves, and areas under the ROC curves. We identified a cutpoint for an algorithm based on optical spectroscopy that yielded an estimated sensitivity of 1.00 [95% confidence interval (CI) = 0.92-1.00] and an estimated specificity of 0.71 [95% CI = 0.62-0.79] in a combined screening and diagnostic population. The positive and negative predictive values were 0.58 and 1.00, respectively. The area under the ROC curve was 0.85 (95% CI = 0.81-0.89). The per-patient and per-site performance were similar in the diagnostic and poorer in the screening settings. Like colposcopy, the device performs best in a diagnostic population. Alternative statistical approaches demonstrate that the analysis is robust and that spectroscopy works as well as or slightly better than colposcopy for the detection of CIN 2 to cancer.

Likelihood of achieving air quality targets under model uncertainties.

Regulatory attainment demonstrations in the United States typically apply a bright-line test to predict whether a control strategy is sufficient to attain an air quality standard. Photochemical models are the best tools available to project future pollutant levels and are a critical part of regulatory attainment demonstrations. However, because photochemical models are uncertain and future meteorology is unknowable, future pollutant levels cannot be predicted perfectly and attainment cannot be guaranteed. This paper introduces a computationally efficient methodology for estimating the likelihood that an emission control strategy will achieve an air quality objective in light of uncertainties in photochemical model input parameters (e.g., uncertain emission and reaction rates, deposition velocities, and boundary conditions). The method incorporates Monte Carlo simulations of a reduced form model representing pollutant-precursor response under parametric uncertainty to probabilistically predict the improvement in air quality due to emission control. The method is applied to recent 8-h ozone attainment modeling for Atlanta, Georgia, to assess the likelihood that additional controls would achieve fixed (well-defined) or flexible (due to meteorological variability and uncertain emission trends) targets of air pollution reduction. The results show that in certain instances ranking of the predicted effectiveness of control strategies may differ between probabilistic and deterministic analyses.

A bayesian hierarchical model for classification with selection of functional predictors.

In functional data classification, functional observations are often contaminated by various systematic effects, such as random batch effects caused by device artifacts, or fixed effects caused by sample-related factors. These effects may lead to classification bias and thus should not be neglected. Another issue of concern is the selection of functions when predictors consist of multiple functions, some of which may be redundant. The above issues arise in a real data application where we use fluorescence spectroscopy to detect cervical precancer. In this article, we propose a Bayesian hierarchical model that takes into account random batch effects and selects effective functions among multiple functional predictors. Fixed effects or predictors in nonfunctional form are also included in the model. The dimension of the functional data is reduced through orthonormal basis expansion or functional principal components. For posterior sampling, we use a hybrid Metropolis-Hastings/Gibbs sampler, which suffers slow mixing. An evolutionary Monte Carlo algorithm is applied to improve the mixing. Simulation and real data application show that the proposed model provides accurate selection of functional predictors as well as good classification.

Robust Smoothing: Smoothing Parameter Selection and Applications to Fluorescence Spectroscopy.

Fluorescence spectroscopy has emerged in recent years as an effective way to detect cervical cancer. Investigation of the data preprocessing stage uncovered a need for a robust smoothing to extract the signal from the noise. Various robust smoothing methods for estimating fluorescence emission spectra are compared and data driven methods for the selection of smoothing parameter are suggested. The methods currently implemented in R for smoothing parameter selection proved to be unsatisfactory, and a computationally efficient procedure that approximates robust leave-one-out cross validation is presented.

Analyzing single-molecule manipulation experiments.

Single-molecule manipulation studies can provide quantitative information about the physical properties of complex biological molecules without ensemble artifacts obscuring the measurements. We demonstrate computational techniques which aim at more fully utilizing the wealth of information contained in noisy experimental time series. The "noise" comes from multiple sources e.g., inherent thermal motion, instrument measurement error, etc. The primary focus of this paper is a methodology that uses time domain based methods to extract the effective molecular friction from single-molecule pulling data. We studied molecules composed of eight tandem repeat titin I27 domains, but the modeling approaches have applicability to other single-molecule mechanical studies. The merits and challenges associated with applying such a computational approach to existing single-molecule manipulation data are also discussed.

Quantifying multiscale noise sources in single-molecule time series.

When analyzing single-molecule data, a low-dimensional set of system observables typically serves as the observational data. We calibrate stochastic dynamical models from time series that record such observables. Numerical techniques for quantifying noise from multiple time scales in a single trajectory, including experimental instrument and inherent thermal noise, are demonstrated. The techniques are applied to study time series coming from both simulations and experiments associated with the nonequilibrium mechanical unfolding of titin's I27 domain. The estimated models can be used for several purposes, (1) detect dynamical signatures of "rare events" by analyzing the effective diffusion and force as a function of the monitored observable, (2) quantify the influence that conformational degrees of freedom, which are typically difficult to directly monitor experimentally, have on the dynamics of the monitored observable, (3) quantitatively compare the inherent thermal noise to other noise sources, for example, instrument noise, variation induced by conformational heterogeneity, and so forth, (4) simulate random quantities associated with repeated experiments, and (5) apply pathwise, that is, trajectory-wise, hypothesis tests to assess the goodness-of-fit of the models and even detect conformational transitions in noisy signals. These items are all illustrated with several examples.

Accuracy of colposcopy in the diagnostic setting compared with the screening setting.

OBJECTIVE: To estimate the accuracy of colposcopy to identify cervical precancer in screening and diagnostic settings. METHODS: As part of a larger clinical trial to evaluate the diagnostic accuracy of optical spectroscopy, we recruited 1,850 patients into a diagnostic or a screening group depending on their history of abnormal findings on Papanicolaou tests. Colposcopic examinations were performed and biopsies specimens obtained from abnormal and normal colposcopic sites for all patients. The criterion standard of test accuracy was the histologic report of biopsies. We calculated sensitivities, specificities, likelihood ratios, receiver operating characteristic curves, and areas under the receiver operating characteristic curves. RESULTS: The prevalence of high-grade squamous intraepithelial lesions (HSIL) or cancer was 29.0% for the diagnostic group and 2.2% for the screening group. Using a disease threshold of HSIL, colposcopy had a sensitivity of 0.983 and a specificity of 0.451 in the diagnostic group when the test threshold was low-grade squamous intraepithelial lesions (LSIL), and a sensitivity of 0.714 and a specificity of 0.813 when the test threshold was HSIL. Using the same HSIL disease threshold, in the screening group, colposcopy had a sensitivity of 0.286 and a specificity of 0.877 when the test threshold was LSIL, and a sensitivity of 0.191 and a specificity of 0.961 when the threshold was HSIL. The colposcopy area under the receiver operating characteristic curve was 0.821 (95% confidence interval 0.79-0.85) in the diagnostic setting compared with 0.587 (95% confidence interval 0.56-0.62) in the screening setting. Changing the disease threshold to LSIL demonstrated similar patterns in the tradeoff of sensitivity and specificity and measure of accuracy. CONCLUSION: Colposcopy performs well in the diagnostic setting and poorly in the screening setting. Colposcopy should not be used to screen for cervical intraepithelial neoplasia. LEVEL OF EVIDENCE: II.

Model-based analysis of reflectance and fluorescence spectra for in vivo detection of cervical dysplasia and cancer.

Development, validation, and implementation of an analytical model to extract biologically and diagnostically relevant parameters from measured cervical tissue reflectance and fluorescence spectra are presented. Monte Carlo simulations of tissue reflectance are used to determine the relative contribution of the signal from the epithelium and stroma. The results indicate that the clinical probe used collects a majority of its reflectance signal from the stroma; therefore, a one-layer analytical model of reflectance is used. Two analytical approaches to calculate reflectance spectra are compared to Monte Carlo simulations, and a diffusion theory-based model is implemented. The model is validated by fitting spectra generated from Monte Carlo simulations and comparing the input and output parameters. Median agreement between extracted optical properties and input parameters is 10.6%. The reflectance model is used together with an analytical model of tissue fluorescence to extract optical properties and fluorophore concentrations from 748 clinical measurements of cervical tissue. A diagnostic algorithm based on these extracted parameters is developed and evaluated using cross-validation. The sensitivity/specificity of this algorithm relative to the gold standard of histopathology per measurement are 8551%; this is comparable to accuracy reported in other studies of optical technologies for detection of cervical cancer and its precursors.

Fluorescence and reflectance device variability throughout the progression of a phase II clinical trial to detect and screen for cervical neoplasia using a fiber optic probe.

Large phase II trials of fluorescence and reflectance spectroscopy using a fiber optic probe in the screening and diagnostic settings for detecting cervical neoplasia have been conducted. We present accrual and histopathology data, instrumentation, data processing, and the preliminary results of interdevice consistencies throughout the progression of a trial. Patients were recruited for either a screening trial (no history of abnormal Papanicolaou smears) or a diagnostic trial (a history of abnormal Papanicolaou smears). Colposcopy identified normal and abnormal squamous, columnar, and transformation zone areas that were subsequently measured with the fiber probe and biopsied. In the course of the clinical trial, two generations of spectrometers (FastEEM2 and FastEEM3) were designed and utilized as optical instrumentation for in vivo spectroscopic fluorescence and reflectance measurements. Data processing of fluorescence and reflectance data is explained in detail and a preliminary analysis of the variability across each device and probe combination is explored. One thousand patients were recruited in the screening trial and 850 patients were recruited in the diagnostic trial. Three clinical sites attracted a diverse range of patients of different ages, ethnicities, and menopausal status. The fully processed results clearly show that consistencies exist across all device and probe combinations throughout the diagnostic trial. Based on the stratification of the data, the results also show identifiable differences in mean intensity between normal and high-grade tissue diagnosis, pre- and postmenopausal status, and squamous and columnar tissue type. The mean intensity values of stratified data show consistent separation across each of the device and probe combinations. By analyzing trial spectra, we provide more evidence that biographical variables such as menopausal status as well as tissue type and diagnosis significantly affect the data. Understanding these effects will lead to better modeling parameters when analyzing the performance of fluorescence and reflectance spectroscopy.

Instrumentation as a source of variability in the application of fluorescence spectroscopic devices for detecting cervical neoplasia.

We report on a study designed to assess variability among three different fluorescence spectroscopy devices, four fiber optic probes, and three sets of optical calibration standards to better understand the reproducibility of measurements and interdevice comparisons of fluorescence spectroscopic data intended for clinical diagnostic use. Multiple measurements are acquired from all sets of standards using each combination of spectrometer, fiber optic probe, and optical standard. Data are processed using standard calibration methods to remove instrument-dependant responses. Processed spectra are analyzed using an analysis of variance to assess the percent variance explained by each factor that was statistically significant. Analysis of processed data confirms statistically significant differences among the spectrometers and fiber optic probes. However, no differences are found when varying calibration standards or measurement date and time. The spectrometers and fiber optic probes are significant sources of variability, but appropriate data processing substantially reduces these effects. Studies of inter- and intradevice variability are important methodological issues for optical device trials and must be included in the quality assurance studies for the clinical trial design.

Smoothing and Mean-Covariance Estimation of Functional Data with a Bayesian Hierarchical Model.

Functional data, with basic observational units being functions (e.g., curves, surfaces) varying over a continuum, are frequently encountered in various applications. While many statistical tools have been developed for functional data analysis, the issue of smoothing all functional observations simultaneously is less studied. Existing methods often focus on smoothing each individual function separately, at the risk of removing important systematic patterns common across functions. We propose a nonparametric Bayesian approach to smooth all functional observations simultaneously and nonparametrically. In the proposed approach, we assume that the functional observations are independent Gaussian processes subject to a common level of measurement errors, enabling the borrowing of strength across all observations. Unlike most Gaussian process regression models that rely on pre-specified structures for the covariance kernel, we adopt a hierarchical framework by assuming a Gaussian process prior for the mean function and an Inverse-Wishart process prior for the covariance function. These prior assumptions induce an automatic mean-covariance estimation in the posterior inference in addition to the simultaneous smoothing of all observations. Such a hierarchical framework is flexible enough to incorporate functional data with different characteristics, including data measured on either common or uncommon grids, and data with either stationary or nonstationary covariance structures. Simulations and real data analysis demonstrate that, in comparison with alternative methods, the proposed Bayesian approach achieves better smoothing accuracy and comparable mean-covariance estimation results. Furthermore, it can successfully retain the systematic patterns in the functional observations that are usually neglected by the existing functional data analyses based on individual-curve smoothing.

An alternative approach for estimating the accuracy of colposcopy in detecting cervical precancer.

INTRODUCTION: Since colposcopy helps to detect cervical cancer in its precancerous stages, as new strategies and technologies are developed for the clinical management of cervical neoplasia, precisely determining the accuracy of colposcopy is important for characterizing its continued role. Our objective was to employ a more precise methodology to estimate of the accuracy of colposcopy to better reflect clinical practice. STUDY DESIGN: For each patient, we compared the worst histology result among colposcopically positive sites to the worst histology result among all sites biopsied, thereby more accurately determining the number of patients that would have been underdiagnosed by colposcopy than previously estimated. MATERIALS AND METHODS: We utilized data from a clinical trial in which 850 diagnostic patients had been enrolled. Seven hundred and ninety-eight of the 850 patients had been examined by colposcopy, and biopsy samples were taken at colposcopically normal and abnormal sites. Our endpoints of interest were the percentages of patients underdiagnosed, and sensitivity and specificity of colposcopy. RESULTS: With the threshold of low-grade squamous intraepithelial lesions for positive colposcopy and histology diagnoses, the sensitivity of colposcopy decreased from our previous assessment of 87.0% to 74.0%, while specificity remained the same. The drop in sensitivity was the result of histologically positive sites that were diagnosed as negative by colposcopy. Thus, 28.4% of the 798 patients in this diagnostic group would have had their condition underdiagnosed by colposcopy in the clinic. CONCLUSIONS: In utilizing biopsies at multiple sites of the cervix, we present a more precise methodology for determining the accuracy of colposcopy. The true accuracy of colposcopy is lower than previously estimated. Nevertheless, our results reinforce previous conclusions that colposcopy has an important role in the diagnosis of cervical precancer.

Prediction using hierarchical data: Applications for automated detection of cervical cancer.

Although the Papanicolaou smear has been successful in decreasing cervical cancer incidence in the developed world, there exist many challenges for implementation in the developing world. Quantitative cytology, a semi-automated method that quantifies cellular image features, is a promising screening test candidate. The nested structure of its data (measurements of multiple cells within a patient) provides challenges to the usual classification problem. Here we perform a comparative study of three main approaches for problems with this general data structure: a) extract patient-level features from the cell-level data; b) use a statistical model that accounts for the hierarchical data structure; and c) classify at the cellular level and use an ad hoc approach to classify at the patient level. We apply these methods to a dataset of 1,728 patients, with an average of 2,600 cells collected per patient and 133 features measured per cell, predicting whether a patient had a positive biopsy result. The best approach we found was to classify at the cellular level and count the number of cells that had a posterior probability greater than a threshold value, with estimated 61% sensitivity and 89% specificity on independent data. Recent statistical learning developments allowed us to achieve high accuracy.

Detecting the signal of the menstrual cycle in fluorescence spectroscopy of the cervix.

Fluorescence spectroscopy of the cervix has been shown to be an effective noninvasive diagnostic tool for cervical intraepithelial neoplasia (precancer). To assess the effect of the menstrual cycle on fluorescence spectroscopy, daily measurements were made on ten subjects for the length of their cycle. These measurements were analyzed to determine if there was a statistically significant signal associated with the menstrual cycle. A signal was found for emission wavelengths between 425 and 445 nm inclusive--near the main hemoglobin absorption band, the Soret band, at 420 nm. We suspect that the slight displacement of the Soret band is due to the nearby dominant NAD(P)H peak, which increases the signal-to-noise ratio and affects statistical significance. The signal consists of a reduction in fluorescence intensity for the first few days of the cycle. This analysis indicates that hemoglobin absorption is the main menstrual-cycle effect on the use of fluorescence spectroscopy on the cervix. The effect is confined to a small set of excitation/emission wavelengths and to approximately the first 8 days of the cycle. This suggests that any problems from the menstrual cycle can be avoided with a simple requirement that the device not be used during the period of menstrual bleeding.

Accuracy of optical spectroscopy for the detection of cervical intraepithelial neoplasia without colposcopic tissue information; a step toward automation for low resource settings.

Optical spectroscopy has been proposed as an accurate and low-cost alternative for detection of cervical intraepithelial neoplasia. We previously published an algorithm using optical spectroscopy as an adjunct to colposcopy and found good accuracy (sensitivity=1.00 [95% confidence interval (CI)=0.92 to 1.00], specificity=0.71 [95% CI=0.62 to 0.79]). Those results used measurements taken by expert colposcopists as well as the colposcopy diagnosis. In this study, we trained and tested an algorithm for the detection of cervical intraepithelial neoplasia (i.e., identifying those patients who had histology reading CIN 2 or worse) that did not include the colposcopic diagnosis. Furthermore, we explored the interaction between spectroscopy and colposcopy, examining the importance of probe placement expertise. The colposcopic diagnosis-independent spectroscopy algorithm had a sensitivity of 0.98 (95% CI=0.89 to 1.00) and a specificity of 0.62 (95% CI=0.52 to 0.71). The difference in the partial area under the ROC curves between spectroscopy with and without the colposcopic diagnosis was statistically significant at the patient level (p=0.05) but not the site level (p=0.13). The results suggest that the device has high accuracy over a wide range of provider accuracy and hence could plausibly be implemented by providers with limited training.

Optical technologies and molecular imaging for cervical neoplasia: a program project update.

There is an urgent global need for effective and affordable approaches to cervical cancer screening and diagnosis. In developing nations, cervical malignancies remain the leading cause of cancer-related deaths in women. This reality may be difficult to accept given that these deaths are largely preventable; where cervical screening programs have been implemented, cervical cancer-related deaths have decreased dramatically. In developed countries, the challenges of cervical disease stem from high costs and overtreatment. The National Cancer Institute-funded Program Project is evaluating the applicability of optical technologies in cervical cancer. The mandate of the project is to create tools for disease detection and diagnosis that are inexpensive, require minimal expertise, are more accurate than existing modalities, and can be feasibly implemented in a variety of clinical settings. This article presents the status and long-term goals of the project.