Two-component Gaussian Mixture Model Research Articles

Deep Learning for Delineation of the Spinal Canal in Whole-Body Diffusion-Weighted Imaging: Normalising Inter- and Intra-Patient Intensity Signal in Multi-Centre Datasets.

Whole-Body Diffusion-Weighted Imaging (WBDWI) is an established technique for staging and evaluating treatment response in patients with multiple myeloma (MM) and advanced prostate cancer (APC). However, WBDWI scans show inter- and intra-patient intensity signal variability. This variability poses challenges in accurately quantifying bone disease, tracking changes over follow-up scans, and developing automated tools for bone lesion delineation. Here, we propose a novel automated pipeline for inter-station, inter-scan image signal standardisation on WBDWI that utilizes robust segmentation of the spinal canal through deep learning. We trained and validated a supervised 2D U-Net model to automatically delineate the spinal canal (both the spinal cord and surrounding cerebrospinal fluid, CSF) in an initial cohort of 40 patients who underwent WBDWI for treatment response evaluation (80 scans in total). Expert-validated contours were used as the target standard. The algorithm was further semi-quantitatively validated on four additional datasets (three internal, one external, 207 scans total) by comparing the distributions of average apparent diffusion coefficient (ADC) and volume of the spinal cord derived from a two-component Gaussian mixture model of segmented regions. Our pipeline subsequently standardises WBDWI signal intensity through two stages: (i) normalisation of signal between imaging stations within each patient through histogram equalisation of slices acquired on either side of the station gap, and (ii) inter-scan normalisation through histogram equalisation of the signal derived within segmented spinal canal regions. This approach was semi-quantitatively validated in all scans available to the study (N = 287). The test dice score, precision, and recall of the spinal canal segmentation model were all above 0.87 when compared to manual delineation. The average ADC for the spinal cord (1.7 × 10-3 mm2/s) showed no significant difference from the manual contours. Furthermore, no significant differences were found between the average ADC values of the spinal cord across the additional four datasets. The signal-normalised, high-b-value images were visualised using a fixed contrast window level and demonstrated qualitatively better signal homogeneity across scans than scans that were not signal-normalised. Our proposed intensity signal WBDWI normalisation pipeline successfully harmonises intensity values across multi-centre cohorts. The computational time required is less than 10 s, preserving contrast-to-noise and signal-to-noise ratios in axial diffusion-weighted images. Importantly, no changes to the clinical MRI protocol are expected, and there is no need for additional reference MRI data or follow-up scans.

A robust intelligent fault diagnosis method for rotating machinery under noisy labels

Despite achieving considerable success, the fault diagnosis methods will still be disturbed by noisy labels, this causes the model’s degradation and reduced diagnostic precision. Focused on solving the above issues, a robust intelligent fault diagnosis approach for rotating machinery under noisy labels is proposed. Firstly, we maintain two deep neural networks (DNNs) and alternatively execute parameters updating and models optimization by referring to the Co-teaching strategy, which can maximize filtering different error types and implement pre-training of DNNs. Secondly, adopting a two-component Gaussian mixture model (GMM) to fit training dataset’s cross-entropy (CE) loss and realize the clean and noisy labels division according to the threshold. Then, a data augmentation method called Mixup operation is employed in semi-supervised learning (SSL) to increase noise robustness and avoid error accumulation, subsequently, performing fine-tuning and correction for clean and noisy samples. Challenging experiments on a transmission gearbox dataset under different noisy labels levels show that the proposed method has robustness to noise and significantly surpasses other approaches, which provides an important reference value for accurate fault diagnosis for rotating equipment parts with noisy labels.

Period–colour and amplitude–colour relations for OGLE-δ Scuti stars in the Galactic bulge and LMC

ABSTRACT We present an analysis on the behaviour of the Galactic bulge and the Large Magellanic Cloud (LMC) δ Scuti stars in terms of period–colour and amplitude–colour (PCAC) relations at maximum, mean, and minimum light. The publicly available Optical Gravitational Lensing Experiment-IV (OGLE-IV) light curves for Galactic bulge and OGLE-III light curves for LMC δ Scuti stars are exploited for the analysis. It has been found that the Galactic bulge δ Scuti stars obey flat PC relations at maximum/mean/minimum light, while the LMC δ Scutis have sloped, sloped, and flat PC relations at maximum, mean, and minimum light, respectively. Both the Galactic bulge and the LMC δ Scutis have sloped, flat, and sloped AC relations at maximum, mean, and minimum light, respectively. These relations also show that Galactic δ Scutis are hotter as compared to their LMC counterparts. The period–amplitude (PA) relations for δ Scutis exhibit different behaviour in the Galactic bulge and the LMC. The LMC variables are found to have higher amplitudes at a given period. The amplitude of the Galactic bulge δ Scuti shows a bimodal distribution which can be modelled using a two-component Gaussian Mixture Model: one component with a lower amplitude and another with a higher amplitude. The observed behaviour of the δ Scuti PCAC relations can be explained using the theory of the interaction of hydrogen ionization front (HIF) and stellar photosphere as well as the PA diagram. We use mesa-rsp to calculate theoretical non-linear hydrodynamical pulsation models for δ Scuti stars with input metallicities of Z = 0.02 and Z = 0.008 appropriate for the Galactic bulge and LMC, respectively. The observed PCAC relations and theoretical calculations support the HIF–photosphere interaction theory.

TWO-DIMENSIONAL GMM-BASED CLUSTERING IN THE PRESENCE OF QUANTIZATION NOISE

In this paper, unlike to the commonly considered clustering, wherein data attributes are accurately presented, it is researched how successful clustering can be performed when data attributes are represented with smaller accuracy, i.e. by using the small number of bits. In particular, the effect of data attributes quantization on the two-dimensional two-component Gaussian mixture model (GMM)-based clustering by using expectation–maximization (EM) algorithm is analyzed. An independent quantization of data attributes by using uniform quantizers with the support limits adjusted to the minimal and maximal attribute values is assumed. The analysis makes it possible to determine the number of bits for data presentation that provides the accurate clustering. These findings can be useful in clustering wherein before being grouped the data have to be represented with a finite small number of bits due to their transmission through the bandwidth-limited channel.

Outdoor thermal stress changes in South Korea: Increasing inter-annual variability induced by different trends of heat and cold stresses

Changes of thermal environment can lead to unfavorable impacts such as a decrease of thermal stratification, increase of energy consumption, and increase of thermal health risk. Investigating changes in outdoor thermal environments can provide meaningful information for addressing economic and social issues and related challenges. In this study, thermal environment changes in South Korea were investigated using a nonstationary two-component Gaussian mixture model (NSGMM) for air temperature and two thermal comfort indices. For this, the perceived temperature (PT) and universal thermal climate index (UTCI) were employed as the thermal comfort index. Thermal comfort indices were computed using observed meteorological data at 26 weather stations for 37 years in South Korea. Meanwhile, trends of thermal comforts in the warm and cool seasons were simultaneously modeled by the NSGMM. The results indicate significant increasing trends in thermal comfort indices for South Korea. The increasing trends in thermal comfort indices both the warm and cool seasons were detected while the magnitudes of the trends are significantly different. This difference between the magnitude of trends led to an increase in mean and inter-annual variability of thermal comfort indices based on PT, while an increase of mean and decrease of inter-annual variability were observed based on the UTCI. Moreover, the annual proportion of the category referring to days in comfort based on the results of PT has decreased due to the different trends of thermal comfort indices in the warm and cool seasons. This decrease may lead to an increase of thermal health risk that is larger than what would be expected from the results considering the increasing trend of the annual mean temperature in South Korea. From this result, it can be inferred that the thermal health risk in South Korea may be more adverse than what we originally expected from the current temperature trend.

Pulse pileup rejection methods using a two-component Gaussian Mixture Model for fast neutron detection with pulse shape discriminating scintillator

Pulse shape discriminating scintillator materials in many cases allow the user to identify two basic kinds of pulses arising from two kinds of particles: neutrons and gammas, respectively. An uncomplicated solution for building a classifier consists of a two-component mixture model learned from mixtures of pulses from neutrons and gammas at a range of energies. Depending on the conditions of data gathered to be classified, multiple classes of events besides neutrons and gammas may occur, most notably pileup events. All these kinds of events that are neither neutron nor gamma are anomalous and, in cases where the class of the particle is in doubt, it is preferable to remove them from the analysis. This study compares the performance of two analytical methods for using the scores from the two-component model to identify anomalous events and in particular to remove pileup events. This study further benchmarks the analytical methods against supervised machine learning methods. This study also presents a means of assessing performance of pileup removal using ROC curves and precision–recall curves. A specific outcome of this study is to propose a novel anomaly score, denoted by G, from an unsupervised two-component model that is conveniently distributed on the interval [−1,1].

Identification of blowing snow particles in images from a Multi-Angle Snowflake Camera

Abstract. A new method to automatically discriminate between hydrometeors and blowing snow particles on Multi-Angle Snowflake Camera (MASC) images is introduced. The method uses four selected descriptors related to the image frequency, the number of particles detected per image, and their size and geometry to classify each individual image. The classification task is achieved with a two-component Gaussian mixture model fitted on a subset of representative images of each class from field campaigns in Antarctica and Davos, Switzerland. The performance is evaluated by labeling the subset of images on which the model was fitted. An overall accuracy and a Cohen kappa score of 99.4 % and 98.8 %, respectively, are achieved. In a second step, the probabilistic information is used to flag images composed of a mix of blowing snow particles and hydrometeors, which turns out to occur frequently. The percentage of images belonging to each class from an entire austral summer in Antarctica and during a winter in Davos, respectively, is presented. The capability to distinguish precipitation, blowing snow and a mix of those in MASC images is highly relevant to disentangle the complex interactions between wind, snowflakes and snowpack close to the surface.

Epileptic high-frequency oscillations: detection and classification

High-frequency oscillations (HFOs) in intracranial electroencephalograms of patients with epilepsy are regarded as promising biomarkers of epileptogenic zones. Their detection and classification can be achieved by visual assessment or automated approaches, although manual processing of large recordings can be laborious. As a result, an automated analysis scheme is indispensable to enable the clinical use of HFOs. In this paper, we present a two-stage strategy to detect and classify HFOs, which starts with a threshold-based approach to detect plausible HFO events followed by an event classification to discriminate different oscillations. Unlike existing approaches, the detection process in the proposed schemes starts by calculating various multi-channel features that allow interrelations among electrodes to be exploited for detection. On this basis, the detection thresholds are set epoch-by-epoch, relying on a two-component Gaussian mixture model to avoid threshold overestimation. The events deemed to be plausible HFOs are then subjected to classification. By simultaneously examining the raw data and time-frequency maps of these events, they are ultimately sorted into the following categories: HFOs, spikes, and spikes with HFOs, so that the oscillations solely caused by filtering sharp transients can be discriminated. Experimental results using simulated data and intracranial recordings from three epileptic patients demonstrate that our proposed schemes achieve promising sensitivity and precision, especially when the noise level is high.

Detection of movement onset using EMG signals for upper-limb exoskeletons in reaching tasks

BackgroundTo assist people with disabilities, exoskeletons must be provided with human-robot interfaces and smart algorithms capable to identify the user’s movement intentions. Surface electromyographic (sEMG) signals could be suitable for this purpose, but their applicability in shared control schemes for real-time operation of assistive devices in daily-life activities is limited due to high inter-subject variability, which requires custom calibrations and training. Here, we developed a machine-learning-based algorithm for detecting the user’s motion intention based on electromyographic signals, and discussed its applicability for controlling an upper-limb exoskeleton for people with severe arm disabilities.MethodsTen healthy participants, sitting in front of a screen while wearing the exoskeleton, were asked to perform several reaching movements toward three LEDs, presented in a random order. EMG signals from seven upper-limb muscles were recorded. Data were analyzed offline and used to develop an algorithm that identifies the onset of the movement across two different events: moving from a resting position toward the LED (Go-forward), and going back to resting position (Go-backward). A set of subject-independent time-domain EMG features was selected according to information theory and their probability distributions corresponding to rest and movement phases were modeled by means of a two-component Gaussian Mixture Model (GMM). The detection of movement onset by two types of detectors was tested: the first type based on features extracted from single muscles, whereas the second from multiple muscles. Their performances in terms of sensitivity, specificity and latency were assessed for the two events with a leave one-subject out test method.ResultsThe onset of movement was detected with a maximum sensitivity of 89.3% for Go-forward and 60.9% for Go-backward events. Best performances in terms of specificity were 96.2 and 94.3% respectively. For both events the algorithm was able to detect the onset before the actual movement, while computational load was compatible with real-time applications.ConclusionsThe detection performances and the low computational load make the proposed algorithm promising for the control of upper-limb exoskeletons in real-time applications. Fast initial calibration makes it also suitable for helping people with severe arm disabilities in performing assisted functional tasks.

Galactic double neutron star total masses and Gaussian mixture model selection

Huang et al. [arXiv:1804.03101] have analysed the population of 15 known galactic Double Neutron Stars (DNSs) regarding the total masses of these systems. They suggest the existence of two sub-populations, and report likelihood-based preference for a two-component Gaussian mixture model over a single Gaussian distribution. This note offers a cautionary perspective on model selection for this data set: Especially for such a small sample size, a pure likelihood ratio test can encourage overfitting. This can be avoided by penalising models with a higher number of free parameters. Re-examining the DNS total mass data set within the class of Gaussian mixture models, this can be achieved through several simple and well-established statistical tests, including information criteria (AICc, BIC), cross-validation, Bayesian evidence ratios and a penalised EM-test. While this re-analysis confirms the basic finding that a two-component mixture is consistent with the data, the model selection criteria consistently indicate that there is no robust preference for it over a single-component fit. Additional DNS discoveries will be needed to settle the question of sub-populations.

Automatic Quantitative Segmentation of Myotubes Reveals Single-cell Dynamics of S6 Kinase Activation.

Automatic cell segmentation is a powerful method for quantifying signaling dynamics at single-cell resolution in live cell fluorescence imaging. Segmentation methods for mononuclear and round shape cells have been developed extensively. However, a segmentation method for elongated polynuclear cells, such as differentiated C2C12 myotubes, has yet to be developed. In addition, myotubes are surrounded by undifferentiated reserve cells, making it difficult to identify background regions and subsequent quantification. Here we developed an automatic quantitative segmentation method for myotubes using watershed segmentation of summed binary images and a two-component Gaussian mixture model. We used time-lapse fluorescence images of differentiated C2C12 cells stably expressing Eevee-S6K, a fluorescence resonance energy transfer (FRET) biosensor of S6 kinase (S6K). Summation of binary images enhanced the contrast between myotubes and reserve cells, permitting detection of a myotube and a myotube center. Using a myotube center instead of a nucleus, individual myotubes could be detected automatically by watershed segmentation. In addition, a background correction using the two-component Gaussian mixture model permitted automatic signal intensity quantification in individual myotubes. Thus, we provide an automatic quantitative segmentation method by combining automatic myotube detection and background correction. Furthermore, this method allowed us to quantify S6K activity in individual myotubes, demonstrating that some of the temporal properties of S6K activity such as peak time and half-life of adaptation show different dose-dependent changes of insulin between cell population and individuals.Key words: time lapse images, cell segmentation, fluorescence resonance energy transfer, C2C12, myotube.

THE ECCENTRICITY DISTRIBUTION OF SHORT-PERIOD PLANET CANDIDATES DETECTED BY KEPLER IN OCCULTATION

ABSTRACT We characterize the eccentricity distribution of a sample of ∼50 short-period planet candidates using transit and occultation measurements from NASA’s Kepler Mission. First, we evaluate the sensitivity of our hierarchical Bayesian modeling and test its robustness to model misspecification using simulated data. When analyzing actual data assuming a Rayleigh distribution for eccentricity, we find that the posterior mode for the dispersion parameter is &sgr; = 0.081 ± 0.003 0.014 ?> . We find that a two-component Gaussian mixture model for e cos ω and e sin ω provides a better model than either a Rayleigh or Beta distribution. Based on our favored model, we find that ∼90% of planet candidates in our sample come from a population with an eccentricity distribution characterized by a small dispersion (∼0.01), and ∼10% come from a population with a larger dispersion (∼0.22). Finally, we investigate how the eccentricity distribution correlates with selected planet and host star parameters. We find evidence that suggests systems around higher metallicity stars and planet candidates with smaller radii come from a more complex eccentricity distribution.

Robust muscle activity onset detection using an unsupervised electromyogram learning framework.

Accurate muscle activity onset detection is an essential prerequisite for many applications of surface electromyogram (EMG). This study presents an unsupervised EMG learning framework based on a sequential Gaussian mixture model (GMM) to detect muscle activity onsets. The distribution of the logarithmic power of EMG signal was characterized by a two-component GMM in each frequency band, in which the two components respectively correspond to the posterior distribution of EMG burst and non-burst logarithmic powers. The parameter set of the GMM was sequentially estimated based on maximum likelihood, subject to constraints derived from the relationship between EMG burst and non-burst distributions. An optimal threshold for EMG burst/non-burst classification was determined using the GMM at each frequency band, and the final decision was obtained by a voting procedure. The proposed novel framework was applied to simulated and experimental surface EMG signals for muscle activity onset detection. Compared with conventional approaches, it demonstrated robust performance for low and changing signal to noise ratios in a dynamic environment. The framework is applicable for real-time implementation, and does not require the assumption of non EMG burst in the initial stage. Such features facilitate its practical application.

Temporal trend, clinicopathologic and sociodemographic characterization of age at diagnosis of breast cancer among US women diagnosed from 1990 to 2009.

This paper investigates the distribution of age at diagnosis of female breast cancer and its association with temporal trend, clinicopathologic and sociodemographic variables in the presence of two latent clusters that are directly unobservable. Such clusters help to identify two subpopulations of either young or old patients whose etiologies are thought to be different. A large sample drawn from registry data from the National Cancer Institute’s Surveillance, Epidemiology, and End Results program from 1990 to 2009 was analyzed using a two-component Gaussian mixture model. Evidence of a steady delay of age at diagnosis and an increasing proportion of young patients being diagnosed during the 20-year period was found. Histopathologic effects indicate that duct and lobular carcinomas differ significantly in regard to subpopulation membership, which confirms that they represent different etiologies. While the presence of estrogen receptor status in the model overlaps the effects of other important variables it is highly correlated with, it is found that the grade, extension and size of the tumor along with lymph node involvement status, race and marital status are important predictors of age at diagnosis. The results highlight the significant impacts that such features can have on breast cancer control efforts, and point to the importance of ensuring that medical decision making should use them along with an indicator of the age subpopulation a patient may belong to.

Estimating the Statistical Characteristics of Remote Sensing Big Data in the Wavelet Transform Domain

Since it is difficult to deal with big data using traditional models and algorithms, predicting and estimating the characteristics of big data is very important. Remote sensing big data consist of many large-scale images that are extremely complex in terms of their structural, spectral, and textual features. Based on multiresolution analysis theory, most of the natural images are sparse and have obvious clustering and persistence characters when they are transformed into another domain by a group of basic special functions. In this paper, we use a wavelet transform to represent remote sensing big data that are large scale in the space domain, correlated in the spectral domain, and continuous in the time domain. We decompose the big data set into approximate multiscale detail coefficients based on a wavelet transform. In order to determine whether the density function of wavelet coefficients in a big data set are peaky at zero and have a heavy tailed shape, a two-component Gaussian mixture model (GMM) is employed. For the first time, we use the expectation-maximization likelihood method to estimate the model parameters for the remote sensing big data set in the wavelet domain. The variance of the GMM with changing of bands, time, and scale are comprehensively analyzed. The statistical characteristics of different textures are also compared. We find that the cluster characteristics of the wavelet coefficients are still obvious in the remote sensing big data set for different bands and different scales. However, it is not always precise when we model the long-term sequence data set using the GMM. We also found that the scale features of different textures for the big data set are obviously reflected in the probability density function and GMM parameters of the wavelet coefficients.

A GMM-IG framework for selecting genes as expression panel biomarkers

The limitation of small sample size of functional genomics experiments has made it necessary to integrate DNA microarray experimental data from different sources. However, experimentation noises and biases of different microarray platforms have made integrated data analysis challenging. In this work, we propose an integrative computational framework to identify candidate biomarker genes from publicly available functional genomics studies. We developed a new framework, Gaussian Mixture Modeling-Coupled Information Gain (GMM-IG). In this framework, we first apply a two-component Gaussian mixture model (GMM) to estimate the conditional probability distributions of gene expression data between two different types of samples, for example, normal versus cancer. An expectation-maximization algorithm is then used to estimate the maximum likelihood parameters of a mixture of two Gaussian models in the feature space and determine the underlying expression levels of genes. Gene expression results from different studies are discretized, based on GMM estimations and then unified. Significantly differentially-expressed genes are filtered and assessed with information gain (IG) measures. DNA microarray experimental data for lung cancers from three different prior studies was processed using the new GMM-IG method. Target gene markers from a gene expression panel were selected and compared with several conventional computational biomarker data analysis methods. GMM-IG showed consistently high accuracy for several classification assessments. A high reproducibility of gene selection results was also determined from statistical validations. Our study shows that the GMM-IG framework can overcome poor reliability issues from single-study DNA microarray experiment while maintaining high accuracies by combining true signals from multiple studies. We present a conceptually simple framework that enables reliable integration of true differential gene expression signals from multiple microarray experiments. This novel computational method has been shown to generate interesting biomarker panels for lung cancer studies. It is promising as a general strategy for future panel biomarker development, especially for applications that requires integrating experimental results generated from different research centers or with different technology platforms.

CADrx for GBM Brain Tumors: Predicting Treatment Response from Changes in Diffusion-Weighted MRI

The goal of this study was to develop a computer-aided therapeutic response (CADrx) system for early prediction of drug treatment response for glioblastoma multiforme (GBM) brain tumors with diffusion weighted (DW) MR images. In conventional Macdonald assessment, tumor response is assessed nine weeks or more post-treatment. However, we will investigate the ability of DW-MRI to assess response earlier, at five weeks post treatment. The apparent diffusion coefficient (ADC) map, calculated from DW images, has been shown to reveal changes in the tumor’s microenvironment preceding morphologic tumor changes. ADC values in treated brain tumors could theoretically both increase due to the cell kill (and thus reduced cell density) and decrease due to inhibition of edema. In this study, we investigated the effectiveness of features that quantify changes from pre- and post-treatment tumor ADC histograms to detect treatment response. There are three parts to this study: first, tumor regions were segmented on T1w contrast enhanced images by Otsu’s thresholding method, and mapped from T1w images onto ADC images by a 3D region of interest (ROI) mapping tool using DICOM header information; second, ADC histograms of the tumor region were extracted from both pre- and five weeks post-treatment scans, and fitted by a two-component Gaussian mixture model (GMM). The GMM features as well as standard histogram-based features were extracted. Finally, supervised machine learning techniques were applied for classification of responders or non-responders. The approach was evaluated with a dataset of 85 patients with GBM under chemotherapy, in which 39 responded and 46 did not, based on tumor volume reduction. We compared adaBoost, random forest and support vector machine classification algorithms, using ten-fold cross validation, resulting in the best accuracy of 69.41% and the corresponding area under the curve (Az) of 0.70.

Update to Gage's "Birth-Weight-Specific Infant and Neonatal Mortality" (2002)

Update to Gage's "Birth-Weight-Specific Infant and Neonatal Mortality" (2002) Timothy B. Gage Keywords Birth Weight, Infant Mortality, Gestational Age, pediatric paradox, maternal age Fryer et al. (1984), Gage and Therriault (1998), and Gage (2000) show that a two-component Gaussian mixture model describes the birth-weight distribution very well and gestational age distribution reasonably well. This finding led to the hypothesis that one of the subpopulations, the subpopulation that accounted for low-birth-weight and high-birth-weight infants (generally considered "compromised" births), might provide a useful alternative to the standard definition of low birth weight, that is, an infant weighing less than 2,500 grams. The standard definition of low birth weight had been widely criticized as arbitrary, and its applicability to populations with widely varying mean birth weights has been questioned (Gage and Therriault 1998). A mixture model fitted to the population at hand would immediately resolve this issue. The problem was how to incorporate this new approach into an analysis of infant mortality. This was the objective of my 2002 paper (Gage 2002a). The approach I proposed was to simultaneously fit two logistic regressions of infant mortality, one to each of the components. The regressions were weighted on the basis of the proportion of the total population accounted for by each subpopulation at each birth weight, so that summed, the two logistic regressions represented total birth-weight-specific infant mortality. In the 2002 paper the model was fitted in two stages. First, the mixture model was fitted to the birth-weight distribution using the maximum-likelihood method, and then the two logistics were fitted based on the mixture model results. Thus the full model did not necessarily represent a full maximum-likelihood result. Applications were carried out on the 1988 birth cohorts of New York state by sex and race. The results of this analysis showed that the mixture model divided the population into a subpopulation with a mean in the "normal" birth-weight range with a relatively small standard deviation and a second subpopulation with a significantly lower mean and larger standard deviation. Because of the large standard deviation, the second subpopulation accounted for the majority of births in both tails of the birth-weight distribution. The first subpopulation was considered a "normal" [End Page 773] fetal development subpopulation, whereas the second subpopulation was considered a "compromised" fetal development subpopulation, given its association with low and macrosomic births. The logistic regressions indicated that the birth-weight-specific infant mortality of both subpopulations was reverse J-shaped and surprisingly that the birth-weight-specific infant mortality of the "compromised" subpopulation was lower at every birth weight than the corresponding birth-weight-specific infant mortality of the "normal" subpopulation. On the other hand, the "compromised" subpopulation, as might be expected, had the higher overall infant mortality. This is due to Simpson's paradox and is a result of the different birth-weight distributions of the two subpopulations; that is, although birth-weight-specific mortality was lower among "compromised" births, more "compromised" births occurred at the lower (and higher) birth weights, where mortality was generally higher. The analysis also showed that the difference between the two logistic regressions was significant, indicating that the mixture model subdivided the birth cohort in a way that identified some hidden heterogeneity in the birth cohort. In particular, the birth-weight-specific infant mortality curve was not precisely reverse J-shaped (a second-degree polynomial; Fryer et al. 1984) but had a shoulder in the curve that occurred at about 2,000–3,000 grams. A second paper, Gage et al. (2004), extended this approach to a full likelihood method. In this case the mixture model was not fitted separately; instead, the mixture model and both regressions were all fitted simultaneously. This allowed infant mortality to inform the mixture submodel as well as the regression models. In addition, this method ensured that the statistical advantages of the maximum likelihood method applied to the resulting estimates. On the other hand, the results were basically identical with the two-stage fitting procedure. The 2004 paper left several issues unresolved: (1) the number of subpopulations, (2) the exact functional form of the subpopulations, and...