Stochastic search for semiparametric linear regression models

This paper introduces and analyzes a stochastic search method for parameter estimation in linear regression models in the spirit of Beran and Millar [Ann. Statist. 15(3) (1987) 1131–1154]. The idea is to generate a random finite subset of a parameter space which will automatically contain points which are very close to an unknown true parameter. The motivation for this procedure comes from recent work of Dümbgen et al. [Ann. Statist. 39(2) (2011) 702–730] on regression models with log-concave error distributions.

Design and analysis for three level hierarchical structures with count response

In numerous intervention studies and education field trials, random assignment to treatment occurs in clusters rather than at the level of observation. This departure of random assignment of units may be due to logistics, political feasibility, or ecological validity. Data within the same cluster or grouping are often correlated. Application of traditional regression techniques, which assume independence between observations, to clustered data produce consistent parameter estimates. However such estimators are often inefficient as compared to methods which incorporate the clustered nature of the data into the estimation procedure (Neuhaus 1993).1 Multilevel models, also known as random effects or random components models, can be used to account for the clustering of data by estimating higher level, or group, as well as lower level, or individual variation. Designing a study, in which the unit of observation is nested within higher level groupings, requires the determination of sample sizes at each level. This study investigates the design and analysis of various sampling strategies for a 3-level repeated measures design on the parameter estimates when the outcome variable of interest follows a Poisson distribution. ^ Results study suggest that second order PQL estimation produces the least biased estimates in the 3-level multilevel Poisson model followed by first order PQL and then second and first order MQL. The MQL estimates of both fixed and random parameters are generally satisfactory when the level 2 and level 3 variation is less than 0.10. However, as the higher level error variance increases, the MQL estimates become increasingly biased. If convergence of the estimation algorithm is not obtained by PQL procedure and higher level error variance is large, the estimates may be significantly biased. In this case bias correction techniques such as bootstrapping should be considered as an alternative procedure. For larger sample sizes, those structures with 20 or more units sampled at levels with normally distributed random errors produced more stable estimates with less sampling variance than structures with an increased number of level 1 units. For small sample sizes, sampling fewer units at the level with Poisson variation produces less sampling variation, however this criterion is no longer important when sample sizes are large. ^ 1Neuhaus J (1993). “Estimation efficiency and Tests of Covariate Effects with Clustered Binary Data”. Biometrics , 49, 989–996^

Optimizing Models of the North Atlantic Spring Bloom Using Physical, Chemical and Bio-Optical Observations from a Lagrangian Float

The North Atlantic spring bloom is one of the main events that lead to carbon export to the deep ocean and drive oceanic uptake of CO(2) from the atmosphere. Here we use a suite of physical, bio-optical and chemical measurements made during the 2008 spring bloom to optimize and compare three different models of biological carbon export. The observations are from a Lagrangian float that operated south of Iceland from early April to late June, and were calibrated with ship-based measurements. The simplest model is representative of typical NPZD models used for the North Atlantic, while the most complex model explicitly includes diatoms and the formation of fast sinking diatom aggregates and cysts under silicate limitation. We carried out a variational optimization and error analysis for the biological parameters of all three models, and compared their ability to replicate the observations. The observations were sufficient to constrain most phytoplankton-related model parameters to accuracies of better than 15 %. However, the lack of zooplankton observations leads to large uncertainties in model parameters for grazing. The simulated vertical carbon flux at 100 m depth is similar between models and agrees well with available observations, but at 600 m the simulated flux is larger by a factor of 2.5 to 4.5 for the model with diatom aggregation. While none of the models can be formally rejected based on their misfit with the available observations, the model that includes export by diatom aggregation has a statistically significant better fit to the observations and more accurately represents the mechanisms and timing of carbon export based on observations not included in the optimization. Thus models that accurately simulate the upper 100 m do not necessarily accurately simulate export to deeper depths.

An Early-Warning System for Hypo-/Hyperglycemic Events Based on Fusion of Adaptive Prediction Models

Introduction: Early warning of future hypoglycemic and hyperglycemic events can improve the safety of type 1 diabetes mellitus (T1DM) patients. The aim of this study is to design and evaluate a hypoglycemia / hyperglycemia early warning system (EWS) for T1DM patients under sensor-augmented pump (SAP) therapy. Methods: The EWS is based on the combination of data-driven online adaptive prediction models and a warning algorithm. Three modeling approaches have been investigated: (i) autoregressive (ARX) models, (ii) auto-regressive with an output correction module (cARX) models, and (iii) recurrent neural network (RNN) models. The warning algorithm performs postprocessing of the models′ outputs and issues alerts if upcoming hypoglycemic/hyperglycemic events are detected. Fusion of the cARX and RNN models, due to their complementary prediction performances, resulted in the hybrid autoregressive with an output correction module/recurrent neural network (cARN)-based EWS. Results: The EWS was evaluated on 23 T1DM patients under SAP therapy. The ARX-based system achieved hypoglycemic (hyperglycemic) event prediction with median values of accuracy of 100.0% (100.0%), detection time of 10.0 (8.0) min, and daily false alarms of 0.7 (0.5). The respective values for the cARX-based system were 100.0% (100.0%), 17.5 (14.8) min, and 1.5 (1.3) and, for the RNN-based system, were 100.0% (92.0%), 8.4 (7.0) min, and 0.1 (0.2). The hybrid cARN-based EWS presented outperforming results with 100.0% (100.0%) prediction accuracy, detection 16.7 (14.7) min in advance, and 0.8 (0.8) daily false alarms. Conclusion: Combined use of cARX and RNN models for the development of an EWS outperformed the single use of each model, achieving accurate and prompt event prediction with few false alarms, thus providing increased safety and comfort.

Models for intensive care training. A European perspective.

The diversity of European culture is reflected in its healthcare training programs. In intensive care medicine (ICM), the differences in national training programs were so marked that it was unlikely that they could produce specialists of equivalent skills. The Competency-Based Training in Intensive Care Medicine in Europe (CoBaTrICE) program was established in 2003 as a Europe-based worldwide collaboration of national training organizations to create core competencies for ICM using consensus methodologies to establish common ground. The group's professional and research ethos created a social identity that facilitated change. The program was easily adaptable to different training structures and incorporated the voice of patients and relatives. The CoBaTrICE program has now been adopted by 15 European countries, with another 12 countries planning to adopt the training program, and is currently available in nine languages, including English. ICM is now recognized as a primary specialty in Spain, Switzerland, and the UK. There are still wide variations in structures and processes of training in ICM across Europe, although there has been agreement on a set of common program standards. The combination of a common "product specification" for an intensivist, combined with persisting variation in the educational context in which competencies are delivered, provides a rich source of research inquiry. Pedagogic research in ICM could usefully focus on the interplay between educational interventions, healthcare systems and delivery, and patient outcomes, such as including whether competency-based program are associated with lower error rates, whether communication skills training is associated with greater patient and family satisfaction, how multisource feedback might best be used to improve reflective learning and teamworking, or whether increasing the proportion of specialists trained in acute care in the hospital at weekends results in better patient outcomes.

Functional error modeling for uncertainty quantification in hydrogeology

Approximate models (proxies) can be employed to reduce the computational costs of estimating uncertainty. The price to pay is that the approximations introduced by the proxy model can lead to a biased estimation. To avoid this problem and ensure a reliable uncertainty quantification, we propose to combine functional data analysis and machine learning to build error models that allow us to obtain an accurate prediction of the exact response without solving the exact model for all realizations. We build the relationship between proxy and exact model on a learning set of geostatistical realizations for which both exact and approximate solvers are run. Functional principal components analysis (FPCA) is used to investigate the variability in the two sets of curves and reduce the dimensionality of the problem while maximizing the retained information. Once obtained, the error model can be used to predict the exact response of any realization on the basis of the sole proxy response. This methodology is purpose-oriented as the error model is constructed directly for the quantity of interest, rather than for the state of the system. Also, the dimensionality reduction performed by FPCA allows a diagnostic of the quality of the error model to assess the informativeness of the learning set and the fidelity of the proxy to the exact model. The possibility of obtaining a prediction of the exact response for any newly generated realization suggests that the methodology can be effectively used beyond the context of uncertainty quantification, in particular for Bayesian inference and optimization.

Reconstruction of 3D Vertebral Models from a Single 2D Lateral Fluoroscopic Image

Accurate three-dimensional (3D) models of lumbar vertebrae are required for image-based 3D kinematics analysis. MRI or CT datasets are frequently used to derive 3D models but have the disadvantages that they are expensive, time-consuming or involving ionizing radiation (e.g., CT acquisition). In this chapter, we present an alternative technique that can reconstruct a scaled 3D lumbar vertebral model from a single two-dimensional (2D) lateral fluoroscopic image and a statistical shape model. Cadaveric studies are conducted to verify the reconstruction accuracy by comparing the surface models reconstructed from a single lateral fluoroscopic image to the ground truth data from 3D CT segmentation. A mean reconstruction error between 0.7 and 1.4 mm was found.

Improved star camera attitude data and their effect on the gravity field

Efforts are ongoing to decrease the noise of the GRACE gravity field models and hence to arrive closer to the GRACE baseline. The most significant error sources belong the untreated errors in the observation data and the imperfections in the background models. The recent study (Bandikova&Flury,2014) revealed that the current release of the star camera attitude data (SCA1B RL02) contain noise systematically higher than expected by about a factor 3-4. This is due to an incorrect implementation of the algorithms for quaternion combination in the JPL processing routines. Generating improved SCA data requires that valid data from both star camera heads are available which is not always the case because the Sun and Moon at times blind one camera. In the gravity field modeling, the attitude data are needed for the KBR antenna offset correction and to orient the non-gravitational linear accelerations sensed by the accelerometer. Hence any improvement in the SCA data is expected to be reflected in the gravity field models. In order to quantify the effect on the gravity field, we processed one month of observation data using two different approaches: the celestial mechanics approach (AIUB) and the variational equations approach (ITSG). We show that the noise in the KBR observations and the linear accelerations has effectively decreased. However, the effect on the gravity field on a global scale is hardly evident. We conclude that, at the current level of accuracy, the errors seen in the temporal gravity fields are dominated by errors coming from sources other than the attitude data.

Evaluation of 3-dimensional superimposition techniques on various skeletal structures of the head using surface models.

OBJECTIVES To test the applicability, accuracy, precision, and reproducibility of various 3D superimposition techniques for radiographic data, transformed to triangulated surface data. METHODS Five superimposition techniques (3P: three-point registration; AC: anterior cranial base; AC + F: anterior cranial base + foramen magnum; BZ: both zygomatic arches; 1Z: one zygomatic arch) were tested using eight pairs of pre-existing CT data (pre- and post-treatment). These were obtained from non-growing orthodontic patients treated with rapid maxillary expansion. All datasets were superimposed by three operators independently, who repeated the whole procedure one month later. Accuracy was assessed by the distance (D) between superimposed datasets on three form-stable anatomical areas, located on the anterior cranial base and the foramen magnum. Precision and reproducibility were assessed using the distances between models at four specific landmarks. Non parametric multivariate models and Bland-Altman difference plots were used for analyses. RESULTS There was no difference among operators or between time points on the accuracy of each superimposition technique (p>0.05). The AC + F technique was the most accurate (D<0.17 mm), as expected, followed by AC and BZ superimpositions that presented similar level of accuracy (D<0.5 mm). 3P and 1Z were the least accurate superimpositions (0.790.05), the detected structural changes differed significantly between different techniques (p<0.05). Bland-Altman difference plots showed that BZ superimposition was comparable to AC, though it presented slightly higher random error. CONCLUSIONS Superimposition of 3D datasets using surface models created from voxel data can provide accurate, precise, and reproducible results, offering also high efficiency and increased post-processing capabilities. In the present study population, the BZ superimposition was comparable to AC, with the added advantage of being applicable to scans with a smaller field of view.

An intercontinental comparison of chironomid palaeotemperature inference models: Europe vs North America

Chironomid-temperature inference models based on North American, European and combined surface sediment training sets were compared to assess the overall reliability of their predictions. Between 67 and 76 of the major chironomid taxa in each data set showed a unimodal response to July temperature, whereas between 5 and 22 of the common taxa showed a sigmoidal response. July temperature optima were highly correlated among the training sets, but the correlations for other taxon parameters such as tolerances and weighted averaging partial least squares (WA-PLS) and partial least squares (PLS) regression coefficients were much weaker. PLS, weighted averaging, WA-PLS, and the Modern Analogue Technique, all provided useful and reliable temperature inferences. Although jack-knifed error statistics suggested that two-component WA-PLS models had the highest predictive power, intercontinental tests suggested that other inference models performed better. The various models were able to provide good July temperature inferences, even where neither good nor close modern analogues for the fossil chironomid assemblages existed. When the models were applied to fossil Lateglacial assemblages from North America and Europe, the inferred rates and magnitude of July temperature changes varied among models. All models, however, revealed similar patterns of Lateglacial temperature change. Depending on the model used, the inferred Younger Dryas July temperature decrease ranged between 2.5 and 6°C.

Multiple imputation for missing continuous and dichotomous data: The effects of patterns of missingness and variable correlations on type I error with small samples

The purpose of this study is to investigate the effects of predictor variable correlations and patterns of missingness with dichotomous and/or continuous data in small samples when missing data is multiply imputed. Missing data of predictor variables is multiply imputed under three different multivariate models: the multivariate normal model for continuous data, the multinomial model for dichotomous data and the general location model for mixed dichotomous and continuous data. Subsequent to the multiple imputation process, Type I error rates of the regression coefficients obtained with logistic regression analysis are estimated under various conditions of correlation structure, sample size, type of data and patterns of missing data. The distributional properties of average mean, variance and correlations among the predictor variables are assessed after the multiple imputation process. ^ For continuous predictor data under the multivariate normal model, Type I error rates are generally within the nominal values with samples of size n = 100. Smaller samples of size n = 50 resulted in more conservative estimates (i.e., lower than the nominal value). Correlation and variance estimates of the original data are retained after multiple imputation with less than 50% missing continuous predictor data. For dichotomous predictor data under the multinomial model, Type I error rates are generally conservative, which in part is due to the sparseness of the data. The correlation structure for the predictor variables is not well retained on multiply-imputed data from small samples with more than 50% missing data with this model. For mixed continuous and dichotomous predictor data, the results are similar to those found under the multivariate normal model for continuous data and under the multinomial model for dichotomous data. With all data types, a fully-observed variable included with variables subject to missingness in the multiple imputation process and subsequent statistical analysis provided liberal (larger than nominal values) Type I error rates under a specific pattern of missing data. It is suggested that future studies focus on the effects of multiple imputation in multivariate settings with more realistic data characteristics and a variety of multivariate analyses, assessing both Type I error and power. ^

Optimal Test for Parameter Instability When the Unstable Process and the Error Distribution are Unknown

This paper proposes asymptotically optimal tests for unstable parameter process under the feasible circumstance that the researcher has little information about the unstable parameter process and the error distribution, and suggests conditions under which the knowledge of those processes does not provide asymptotic power gains. I first derive a test under known error distribution, which is asymptotically equivalent to LR tests for correctly identified unstable parameter processes under suitable conditions. The conditions are weak enough to cover a wide range of unstable processes such as various types of structural breaks and time varying parameter processes. The test is then extended to semiparametric models in which the underlying distribution in unknown but treated as unknown infinite dimensional nuisance parameter. The semiparametric test is adaptive in the sense that its asymptotic power function is equivalent to the power envelope under known error distribution.

Bayesian generalized linear models for meta-analysis of diagnostic tests

With the recognition of the importance of evidence-based medicine, there is an emerging need for methods to systematically synthesize available data. Specifically, methods to provide accurate estimates of test characteristics for diagnostic tests are needed to help physicians make better clinical decisions. To provide more flexible approaches for meta-analysis of diagnostic tests, we developed three Bayesian generalized linear models. Two of these models, a bivariate normal and a binomial model, analyzed pairs of sensitivity and specificity values while incorporating the correlation between these two outcome variables. Noninformative independent uniform priors were used for the variance of sensitivity, specificity and correlation. We also applied an inverse Wishart prior to check the sensitivity of the results. The third model was a multinomial model where the test results were modeled as multinomial random variables. All three models can include specific imaging techniques as covariates in order to compare performance. Vague normal priors were assigned to the coefficients of the covariates. The computations were carried out using the 'Bayesian inference using Gibbs sampling' implementation of Markov chain Monte Carlo techniques. We investigated the properties of the three proposed models through extensive simulation studies. We also applied these models to a previously published meta-analysis dataset on cervical cancer as well as to an unpublished melanoma dataset. In general, our findings show that the point estimates of sensitivity and specificity were consistent among Bayesian and frequentist bivariate normal and binomial models. However, in the simulation studies, the estimates of the correlation coefficient from Bayesian bivariate models are not as good as those obtained from frequentist estimation regardless of which prior distribution was used for the covariance matrix. The Bayesian multinomial model consistently underestimated the sensitivity and specificity regardless of the sample size and correlation coefficient. In conclusion, the Bayesian bivariate binomial model provides the most flexible framework for future applications because of its following strengths: (1) it facilitates direct comparison between different tests; (2) it captures the variability in both sensitivity and specificity simultaneously as well as the intercorrelation between the two; and (3) it can be directly applied to sparse data without ad hoc correction. ^

Selection bias and the cross-validation of regression models for prediction

Strategies are compared for the development of a linear regression model with stochastic (multivariate normal) regressor variables and the subsequent assessment of its predictive ability. Bias and mean squared error of four estimators of predictive performance are evaluated in simulated samples of 32 population correlation matrices. Models including all of the available predictors are compared with those obtained using selected subsets. The subset selection procedures investigated include two stopping rules, C$\sb{\rm p}$ and S$\sb{\rm p}$, each combined with an 'all possible subsets' or 'forward selection' of variables. The estimators of performance utilized include parametric (MSEP$\sb{\rm m}$) and non-parametric (PRESS) assessments in the entire sample, and two data splitting estimates restricted to a random or balanced (Snee's DUPLEX) 'validation' half sample. The simulations were performed as a designed experiment, with population correlation matrices representing a broad range of data structures.^ The techniques examined for subset selection do not generally result in improved predictions relative to the full model. Approaches using 'forward selection' result in slightly smaller prediction errors and less biased estimators of predictive accuracy than 'all possible subsets' approaches but no differences are detected between the performances of C$\sb{\rm p}$ and S$\sb{\rm p}$. In every case, prediction errors of models obtained by subset selection in either of the half splits exceed those obtained using all predictors and the entire sample.^ Only the random split estimator is conditionally (on $\\beta$) unbiased, however MSEP$\sb{\rm m}$ is unbiased on average and PRESS is nearly so in unselected (fixed form) models. When subset selection techniques are used, MSEP$\sb{\rm m}$ and PRESS always underestimate prediction errors, by as much as 27 percent (on average) in small samples. Despite their bias, the mean squared errors (MSE) of these estimators are at least 30 percent less than that of the unbiased random split estimator. The DUPLEX split estimator suffers from large MSE as well as bias, and seems of little value within the context of stochastic regressor variables.^ To maximize predictive accuracy while retaining a reliable estimate of that accuracy, it is recommended that the entire sample be used for model development, and a leave-one-out statistic (e.g. PRESS) be used for assessment. ^

Information bias and lifetime mortality risks of radiation-induced cancer

Additive and multiplicative models of relative risk were used to measure the effect of cancer misclassification and DS86 random errors on lifetime risk projections in the Life Span Study (LSS) of Hiroshima and Nagasaki atomic bomb survivors. The true number of cancer deaths in each stratum of the cancer mortality cross-classification was estimated using sufficient statistics from the EM algorithm. Average survivor doses in the strata were corrected for DS86 random error ($\sigma$ = 0.45) by use of reduction factors. Poisson regression was used to model the corrected and uncorrected mortality rates with covariates for age at-time-of-bombing, age at-time-of-death and gender. Excess risks were in good agreement with risks in RERF Report 11 (Part 2) and the BEIR-V report. Bias due to DS86 random error typically ranged from $-$15% to $-$30% for both sexes, and all sites and models. The total bias, including diagnostic misclassification, of excess risk of nonleukemia for exposure to 1 Sv from age 18 to 65 under the non-constant relative projection model was $-$37.1% for males and $-$23.3% for females. Total excess risks of leukemia under the relative projection model were biased $-$27.1% for males and $-$43.4% for females. Thus, nonleukemia risks for 1 Sv from ages 18 to 85 (DRREF = 2) increased from 1.91%/Sv to 2.68%/Sv among males and from 3.23%/Sv to 4.02%/Sv among females. Leukemia excess risks increased from 0.87%/Sv to 1.10%/Sv among males and from 0.73%/Sv to 1.04%/Sv among females. Bias was dependent on the gender, site, correction method, exposure profile and projection model considered. Future studies that use LSS data for U.S. nuclear workers may be downwardly biased if lifetime risk projections are not adjusted for random and systematic errors. (Supported by U.S. NRC Grant NRC-04-091-02.) ^