Evaluation of Risk Prediction with Hierarchical Data: Dependency Adjusted Confidence Intervals for the AUC.

The area under the true ROC curve (AUC) is routinely used to determine how strongly a given model discriminates between the levels of a binary outcome. Standard inference with the AUC requires that outcomes be independent of each other. To overcome this limitation, a method was developed for the estimation of the variance of the AUC in the setting of two-level hierarchical data using probit-transformed prediction scores generated from generalized estimating equation models, thereby allowing for the application of inferential methods. This manuscript presents an extension of this approach so that inference for the AUC may be performed in a three-level hierarchical data setting (e.g., eyes nested within persons and persons nested within families). A method that accounts for the effect of tied prediction scores on inference is also described. The performance of 95% confidence intervals around the AUC was assessed through the simulation of three-level clustered data in multiple settings, including ones with tied data and variable cluster sizes. Across all settings, the actual 95% confidence interval coverage varied from 0.943 to 0.958, and the ratio of the theoretical variance to the empirical variance of the AUC varied from 0.920 to 1.013. The results are better than those from existing methods. Two examples of applying the proposed methodology are presented.

Semiparametric predictive inference for failure data using first-hitting-time threshold regression.

The progression of disease for an individual can be described mathematically as a stochastic process. The individual experiences a failure event when the disease path first reaches or crosses a critical disease level. This happening defines a failure event and a first hitting time or time-to-event, both of which are important in medical contexts. When the context involves explanatory variables then there is usually an interest in incorporating regression structures into the analysis and the methodology known as threshold regression comes into play. To date, most applications of threshold regression have been based on parametric families of stochastic processes. This paper presents a semiparametric form of threshold regression that requires the stochastic process to have only one key property, namely, stationary independent increments. As this property is frequently encountered in real applications, this model has potential for use in many fields. The mathematical underpinnings of this semiparametric approach for estimation and prediction are described. The basic data element required by the model is a pair of readings representing the observed change in time and the observed change in disease level, arising from either a failure event or survival of the individual to the end of the data record. An extension is presented for applications where the underlying disease process is unobservable but component covariate processes are available to construct a surrogate disease process. Threshold regression, used in combination with a data technique called Markov decomposition, allows the methods to handle longitudinal time-to-event data by uncoupling a longitudinal record into a sequence of single records. Computational aspects of the methods are straightforward. An array of simulation experiments that verify computational feasibility and statistical inference are reported in an online supplement. Case applications based on longitudinal observational data from The Osteoarthritis Initiative (OAI) study are presented to demonstrate the methodology and its practical use.

Estimation and testing for clustered interval-censored bivariate survival data with application using the semi-parametric version of the Clayton-Oakes model.

The Kaplan-Meier estimator is ubiquitously used to estimate survival probabilities for time-to-event data. It is nonparametric, and thus does not require specification of a survival distribution, but it does assume that the risk set at any time t consists of independent observations. This assumption does not hold for data from paired organ systems such as occur in ophthalmology (eyes) or otolaryngology (ears), or for other types of clustered data. In this article, we estimate marginal survival probabilities in the setting of clustered data, and provide confidence limits for these estimates with intra-cluster correlation accounted for by an interval-censored version of the Clayton-Oakes model. We develop a goodness-of-fit test for general bivariate interval-censored data and apply it to the proposed interval-censored version of the Clayton-Oakes model. We also propose a likelihood ratio test for the comparison of survival distributions between two groups in the setting of clustered data under the assumption of a constant between-group hazard ratio. This methodology can be used both for balanced and unbalanced cluster sizes, and also when the cluster size is informative. We compare our test to the ordinary log rank test and the Lin-Wei (LW) test based on the marginal Cox proportional Hazards model with robust standard errors obtained from the sandwich estimator. Simulation results indicate that the ordinary log rank test over-inflates type I error, while the proposed unconditional likelihood ratio test has appropriate type I error and higher power than the LW test. The method is demonstrated in real examples from the Sorbinil Retinopathy Trial, and the Age-Related Macular Degeneration Study. Raw data from these two trials are provided.

Accounting for delayed entry into observational studies and clinical trials: length-biased sampling and restricted mean survival time.

Individuals in many observational studies and clinical trials for chronic diseases are enrolled well after onset or diagnosis of their disease. Times to events of interest after enrollment are therefore residual or left-truncated event times. Individuals entering the studies have disease that has advanced to varying extents. Moreover, enrollment usually entails probability sampling of the study population. Finally, event times over a short to moderate time horizon are often of interest in these investigations, rather than more speculative and remote happenings that lie beyond the study period. This research report looks at the issue of delayed entry into these kinds of studies and trials. Time to event for an individual is modelled as a first hitting time of an event threshold by a latent disease process, which is taken to be a Wiener process. It is emphasized that recruitment into these studies often involves length-biased sampling. The requisite mathematics for this kind of sampling and delayed entry are presented, including explicit formulas needed for estimation and inference. Restricted mean survival time (RMST) is taken as the clinically relevant outcome measure. Exact parametric formulas for this measure are derived and presented. The results are extended to settings that involve study covariates using threshold regression methods. Methods adapted for clinical trials are presented. An extensive case illustration for a clinical trial setting is then presented to demonstrate the methods, the interpretation of results, and the harvesting of useful insights. The closing discussion covers a number of important issues and concepts.

Group sequential design for randomized trials using "first hitting time" model.

Group sequential design (GSD) has become a popular choice in recent clinical trials as it improves trial efficiency by providing options for early termination. The implementation of traditional tests for survival analysis (eg, the log-rank test and the Cox proportional hazard (PH) model) in the GSD setting has been widely discussed. The PH assumption is required for conventional (sequential) design, it is, however, often violated in practice. As an alternative, some generalized tests have been proposed (eg, the Max-Combo test) and their efficacies have been established. In this article, we explore the application of a more flexible, "first hitting time" based threshold regression (TR) model to GSD. TR assumes that subjects' health status is a latent (unobservable) process, and the clinical event of interest occurs when the latent health process hits a pre-specified boundary. The simulation results supported our findings that, in most cases, this comparable new method can successfully control type I error while providing higher early stopping opportunities in the sequential design, even when non-proportional hazard presents.

Drug overdose mortality is associated with employment status and occupation in the National Longitudinal Mortality Study.

BACKGROUND: Since 1999, over 702,000 people in the US have died of a drug overdose, and the drug overdose death rate has increased from 6.2 to 21.8 per 100,000. Employment status and occupation may be important social determinants of overdose deaths. OBJECTIVES: Estimate the risk of drug overdose death by employment status and occupation, controlling for other social and demographic factors known to be associated with overdose deaths. METHODS: Proportional hazard models were used to study US adults in the National Longitudinal Mortality Study with baseline measurements taken in the early 2000s and up to 6 years of follow-up (n = 438,739, 53% female, 47% male). Comparisons were made between adults with different employment statuses (employed, unemployed, disabled, etc.) and occupations (sales, construction, service occupations, etc.). Models were adjusted for age, sex, race/ethnicity, education, income and marital status. RESULTS: Adults who were disabled (hazard ratio (HR) = 6.96 (95% CI = 6.81-7.12)), unemployed (HR = 4.20, 95% CI = 4.09-4.32) and retired (HR = 2.94, 95% CI = 2.87-3.00) were at higher risk of overdose death relative to those who were employed. By occupation, those working in service (HR = 2.05, 95% CI = 1.97-2.13); construction and extraction (HR = 1.69, 95% CI = 1.64-1.76); management, business and financial (HR = 1.39, 95% CI = 1.33-1.44); and installation, maintenance and repair (HR = 1.32, 95% CI = 1.25-1.40) occupations displayed higher risk relative to professional occupations. CONCLUSIONS: In a large national cohort followed prospectively for up to 6 years, several employment statuses and occupations are associated with overdose deaths, independent of a range of other factors. Efforts to prevent overdose deaths may benefit from focusing on these high-risk groups.

Variable Selection in Threshold Regression Model with Applications to HIV Drug Adherence Data.

The threshold regression model is an effective alternative to the Cox proportional hazards regression model when the proportional hazards assumption is not met. This paper considers variable selection for threshold regression. This model has separate regression functions for the initial health status and the speed of degradation in health. This flexibility is an important advantage when considering relevant risk factors for a complex time-to-event model where one needs to decide which variables should be included in the regression function for the initial health status, in the function for the speed of degradation in health, or in both functions. In this paper, we extend the broken adaptive ridge (BAR) method, originally designed for variable selection for one regression function, to simultaneous variable selection for both regression functions needed in the threshold regression model. We establish variable selection consistency of the proposed method and asymptotic normality of the estimator of non-zero regression coefficients. Simulation results show that our method outperformed threshold regression without variable selection and variable selection based on the Akaike information criterion. We apply the proposed method to data from an HIV drug adherence study in which electronic monitoring of drug intake is used to identify risk factors for non- adherence.

Order-restricted inference for clustered ROC data with application to fingerprint matching accuracy.

Receiver operating characteristic (ROC) curve is commonly used to evaluate and compare the accuracy of classification methods or markers. Estimating ROC curves has been an important problem in various fields including biometric recognition and diagnostic medicine. In real applications, classification markers are often developed under two or more ordered conditions, such that a natural stochastic ordering exists among the observations. Incorporating such a stochastic ordering into estimation can improve statistical efficiency (Davidov and Herman, 2012). In addition, clustered and correlated data arise when multiple measurements are gleaned from the same subject, making estimation of ROC curves complicated due to within-cluster correlations. In this article, we propose to model the ROC curve using a weighted empirical process to jointly account for the order constraint and within-cluster correlation structure. The algebraic properties of resulting summary statistics of the ROC curve such as its area and partial area are also studied. The algebraic expressions reduce to the ones by Davidov and Herman (2012) for independent observations. We derive asymptotic properties of the proposed order-restricted estimators and show that they have smaller mean-squared errors than the existing estimators. Simulation studies also demonstrate better performance of the newly proposed estimators over existing methods for finite samples. The proposed method is further exemplified with the fingerprint matching data from the National Institute of Standards and Technology Special Database 4.

Two-sample test for correlated data under outcome-dependent sampling with an application to self-reported weight loss data.

Standard methods for two-sample tests such as the t-test and Wilcoxon rank sum test may lead to incorrect type I errors when applied to longitudinal or clustered data. Recent alternatives of two-sample tests for clustered data often require certain assumptions on the correlation structure and/or noninformative cluster size. In this paper, based on a novel pseudolikelihood for correlated data, we propose a score test without knowledge of the correlation structure or assuming data missingness at random. The proposed score test can capture differences in the mean and variance between two groups simultaneously. We use projection theory to derive the limiting distribution of the test statistic, in which the covariance matrix can be empirically estimated. We conduct simulation studies to evaluate the proposed test and compare it with existing methods. To illustrate the usefulness proposed test, we use it to compare self-reported weight loss data in a friends' referral group, with the data from the Internet self-joining group.

RNASeqDesign: A framework for RNA-Seq genome-wide power calculation and study design issues.

Massively parallel sequencing (a.k.a. next-generation sequencing, NGS) technology has emerged as a powerful tool in characterizing genomic profiles. Among many NGS applications, RNA sequencing (RNA-Seq) has gradually become a standard tool for global transcriptomic monitoring. Although the cost of NGS experiments has dropped constantly, the high sequencing cost and bioinformatic complexity are still obstacles for many biomedical projects. Unlike earlier fluorescence-based technologies such as microarray, modelling of NGS data should consider discrete count data. In addition to sample size, sequencing depth also directly relates to the experimental cost. Consequently, given total budget and pre-specified unit experimental cost, the study design issue in RNA-Seq is conceptually a more complex multi-dimensional constrained optimization problem rather than one-dimensional sample size calculation in traditional hypothesis setting. In this paper, we propose a statistical framework, namely "RNASeqDesign", to utilize pilot data for power calculation and study design of RNA-Seq experiments. The approach is based on mixture model fitting of p-value distribution from pilot data and a parametric bootstrap procedure based on approximated Wald test statistics to infer genome-wide power for optimal sample size and sequencing depth. We further illustrate five practical study design tasks for practitioners. We perform simulations and three real applications to evaluate the performance and compare to existing methods.

Generational trends and patterns in readmission within a statewide cohort of clients receiving heroin use disorder treatment in Maryland, 2007-2013.

The recent rise in opioid-related overdose deaths stresses the importance of understanding how heroin use disorders persist and what interventions are best suited for treating these illnesses. Trends show that there are diverse pathways leading to heroin use disorder that span multiple generations, but little is known about how different generations utilize and respond to treatment. This study provides insight into treatment utilization for young, middle-aged, and older adults by examination of an unusually rich longitudinal dataset of substance use disorder clients in Maryland who were treated for heroin use. Results show that clear patterns of treatment readmission emerge across generations in treatment-naïve clients with regard to gender, ethnicity, employment, geographical region, and treatment type/intensity. In particular, Millennials comprise the majority of the clients receiving heroin use disorder treatment and are the largest contributor to these readmission patterns. Millennials are also given opioid maintenance therapy (OMT) more frequently than other generations, while exhibiting a strong avoidance to treatment. Generational differences in treatment decisions and outcomes over the course of a treatment career are important for understanding the nature of the current opioid epidemic, and can play an important role in directing heroin use disorder treatment efforts and improving models of care.

Using threshold regression to analyze survival data from complex surveys: With application to mortality linked NHANES III Phase II genetic data.

The Cox proportional hazards (PH) model is a common statistical technique used for analyzing time-to-event data. The assumption of PH, however, is not always appropriate in real applications. In cases where the assumption is not tenable, threshold regression (TR) and other survival methods, which do not require the PH assumption, are available and widely used. These alternative methods generally assume that the study data constitute simple random samples. In particular, TR has not been studied in the setting of complex surveys that involve (1) differential selection probabilities of study subjects and (2) intracluster correlations induced by multistage cluster sampling. In this paper, we extend TR procedures to account for complex sampling designs. The pseudo-maximum likelihood estimation technique is applied to estimate the TR model parameters. Computationally efficient Taylor linearization variance estimators that consider both the intracluster correlation and the differential selection probabilities are developed. The proposed methods are evaluated by using simulation experiments with various complex designs and illustrated empirically by using mortality-linked Third National Health and Nutrition Examination Survey Phase II genetic data.

Association between community socioeconomic factors, animal feeding operations, and campylobacteriosis incidence rates: Foodborne Diseases Active Surveillance Network (FoodNet), 2004-2010.

BACKGROUND: Campylobacter is a leading cause of foodborne illness in the United States. Campylobacter infections have been associated with individual risk factors, such as the consumption of poultry and raw milk. Recently, a Maryland-based study identified community socioeconomic and environmental factors that are also associated with campylobacteriosis rates. However, no previous studies have evaluated the association between community risk factors and campylobacteriosis rates across multiple U.S. states. METHODS: We obtained Campylobacter case data (2004-2010; n = 40,768) from the Foodborne Diseases Active Surveillance Network (FoodNet) and socioeconomic and environmental data from the 2010 Census of Population and Housing, the 2011 American Community Survey, and the 2007 U.S. Census of Agriculture. We linked data by zip code and derived incidence rate ratios using negative binomial regression models. RESULTS: Community socioeconomic and environmental factors were associated with both lower and higher campylobacteriosis rates. Zip codes with higher percentages of African Americans had lower rates of campylobacteriosis (incidence rate ratio [IRR]) = 0.972; 95 % confidence interval (CI) = 0.970,0.974). In Georgia, Maryland, and Tennessee, three leading broiler chicken producing states, zip codes with broiler operations had incidence rates that were 22 % (IRR = 1.22; 95 % CI = 1.03,1.43), 16 % (IRR = 1.16; 95 % CI = 0.99,1.37), and 35 % (IRR = 1.35; 95 % CI = 1.18,1.53) higher, respectively, than those of zip codes without broiler operations. In Minnesota and New York FoodNet counties, two top dairy producing areas, zip codes with dairy operations had significantly higher campylobacteriosis incidence rates (IRR = 1.37; 95 % CI = 1.22, 1.55; IRR = 1.19; 95 % CI = 1.04,1.36). CONCLUSIONS: Community socioeconomic and environmental factors are important to consider when evaluating the relationship between possible risk factors and Campylobacter infection.

A threshold regression model to predict return to work after traumatic limb injury.

OBJECTIVE: The study aims to examine the severity of initial impairment and recovery rate of return-to-work (RTW) predictors among workers with traumatic limb injury. METHODS: This 2-year prospective cohort study recruited 1124 workers with traumatic limb injury during the first 2 weeks of hospital admission. Baseline data were obtained by questionnaire and chart review. Patient follow-up occurred at 1, 3, 6, 12, 18, and 24 months post injury. The primary outcome was the time of first RTW. The impact of potential predictors on initial impairment and rate of recovery towards RTW was estimated by threshold regression (TR). RESULTS: A total of 846 (75.27%) participants returned to work during the follow-up period. Our model revealed that the initial impairment level in elderly workers and lower limb injuries were 33% and 35% greater than their counterparts, respectively. Workers with >12 years of education, part-time job, and moderate and higher self-efficacy were less impaired at initial injury compared with their counterparts. In terms of the rate of recovery leading to RTW, workers with older age, part-time jobs, lower limbs, or combined injuries had a significantly slower recovery rate, while workers with 9-12 years of education and >12 years of education had a significantly faster recovery rate. CONCLUSIONS: Our study provides researchers and clinicians with evidence to understand the baseline impairment and rate of recovery towards RTW by explaining the predictors of RTW among workers with traumatic limb injuries.

Transition outcomes for young adults with disabilities.

PURPOSE: Transition to adulthood can be very challenging for children with special health care needs (CSHCN) especially for those with disabilities who experience functional limitations in activities at home, in school, and in the community. The study examined the transition outcomes in areas of health, education, and independent living for young adult with special health care needs (YASHCN) with disabilities. METHOD: The study is a secondary data analysis of the 2007 Survey of Adult Transition and Health (SATH). Multivariate logistic regression analysis assessed the association between having disabilities and the transition outcomes. RESULTS: Overall, YASHCN with disabilities reported favorable health related transition outcomes with improved access to primary care, care coordination, and physician engagement in transition discussions and connection to mentors. Furthermore, YASCHN with disabilities had higher odds of receiving Medicaid or other insurance for low income or disabilities as an adult (AOR=5.26, 95% CI=3.74, 7.04). However, they were less likely to report having control over personal finances, making friends, and obtaining a high school diploma. CONCLUSION: The findings suggest that YASHCN with disabilities may be among the small proportion of CSHCNs who had a positive transition to adult health care services. However, transition outcomes related to independent living still need more improvements.

Acceptability and performance of the menstrual cup in South Africa: a randomized crossover trial comparing the menstrual cup to tampons or sanitary pads.

BACKGROUND: In low-income settings, many women and girls face activity restrictions during menses, owing to lack of affordable menstrual products. The menstrual cup (MC) is a nonabsorbent reusable cup that collects menstrual blood. We assessed the acceptability and performance of the MPower® MC compared to pads or tampons among women in a low-resource setting. METHODS: We conducted a randomized two-period crossover trial at one site in Durban, South Africa, between January and November 2013. Participants aged 18-45 years with regular menstrual cycles were eligible for inclusion if they had no intention of becoming pregnant, were using an effective contraceptive method, had water from the municipal system as their primary water source, and had no sexually transmitted infections. We used a computer-generated randomization sequence to assign participants to one of two sequences of menstrual product use, with allocation concealed only from the study investigators. Participants used each method over three menstrual cycles (total 6 months) and were interviewed at baseline and monthly follow-up visits. The product acceptability outcome compared product satisfaction question scores using an ordinal logistic regression model with individual random effects. This study is registered on the South African Clinical Trials database: number DOH-27-01134273. RESULTS: Of 124 women assessed, 110 were eligible and randomly assigned to selected menstrual products. One hundred and five women completed all follow-up visits. By comparison to pads/tampons (usual product used), the MC was rated significantly better for comfort, quality, menstrual blood collection, appearance, and preference. Both of these comparative outcome measures, along with likelihood of continued use, recommending the product, and future purchase, increased for the MC over time. CONCLUSION: MC acceptance in a population of novice users, many with limited experience with tampons, indicates that there is a pool of potential users in low-resource settings.

A model for time to fracture with a shock stream superimposed on progressive degradation: the Study of Osteoporotic Fractures.

Osteoporotic hip fractures in the elderly are associated with a high mortality in the first year following fracture and a high incidence of disability among survivors. We study first and second fractures of elderly women using data from the Study of Osteoporotic Fractures. We present a new conceptual framework, stochastic model, and statistical methodology for time to fracture. Our approach gives additional insights into the patterns for first and second fractures and the concomitant risk factors. Our modeling perspective involves a novel time-to-event methodology called threshold regression, which is based on the plausible idea that many events occur when an underlying process describing the health or condition of a person or system encounters a critical boundary or threshold for the first time. In the parlance of stochastic processes, this time to event is a first hitting time of the threshold. The underlying process in our model is a composite of a chronic degradation process for skeletal health combined with a random stream of shocks from external traumas, which taken together trigger fracture events.

Semiparametric varying-coefficient model for interval censored data with a cured proportion.

Varying-coefficient models have claimed an increasing portion of statistical research and are now applied to censored data analysis in medical studies. We incorporate such flexible semiparametric regression tools for interval censored data with a cured proportion. We adopted a two-part model to describe the overall survival experience for such complicated data. To fit the unknown functional components in the model, we take the local polynomial approach with bandwidth chosen by cross-validation. We establish consistency and asymptotic distribution of the estimation and propose to use bootstrap for inference. We constructed a BIC-type model selection method to recommend an appropriate specification of parametric and nonparametric components in the model. We conducted extensive simulations to assess the performance of our methods. An application on a decompression sickness data illustrates our methods.

Logic minimization and rule extraction for identification of functional sites in molecular sequences.

BACKGROUND: Logic minimization is the application of algebraic axioms to a binary dataset with the purpose of reducing the number of digital variables and/or rules needed to express it. Although logic minimization techniques have been applied to bioinformatics datasets before, they have not been used in classification and rule discovery problems. In this paper, we propose a method based on logic minimization to extract predictive rules for two bioinformatics problems involving the identification of functional sites in molecular sequences: transcription factor binding sites (TFBS) in DNA and O-glycosylation sites in proteins. TFBS are important in various developmental processes and glycosylation is a posttranslational modification critical to protein functions. METHODS: In the present study, we first transformed the original biological dataset into a suitable binary form. Logic minimization was then applied to generate sets of simple rules to describe the transformed dataset. These rules were used to predict TFBS and O-glycosylation sites. The TFBS dataset is obtained from the TRANSFAC database, while the glycosylation dataset was compiled using information from OGLYCBASE and the Swiss-Prot Database.We performed the same predictions using two standard classification techniques, Artificial Neural Networks (ANN) and Support Vector Machines (SVM), and used their sensitivities and positive predictive values as benchmarks for the performance of our proposed algorithm. SVM were also used to reduce the number of variables included in the logic minimization approach. RESULTS: For both TFBS and O-glycosylation sites, the prediction performance of the proposed logic minimization method was generally comparable and, in some cases, superior to the standard ANN and SVM classification methods with the advantage of providing intelligible rules to describe the datasets. In TFBS prediction, logic minimization produced a very small set of simple rules. In glycosylation site prediction, the rules produced were also interpretable and the most popular rules generated appeared to correlate well with recently reported hydrophilic/hydrophobic enhancement values of amino acids around possible O-glycosylation sites. Experiments with Self-Organizing Neural Networks corroborate the practical worth of the logic minimization method for these case studies. CONCLUSIONS: The proposed logic minimization algorithm provides sets of rules that can be used to predict TFBS and O-glycosylation sites with sensitivity and positive predictive value comparable to those from ANN and SVM. Moreover, the logic minimization method has the additional capability of generating interpretable rules that allow biological scientists to correlate the predictions with other experimental results and to form new hypotheses for further investigation. Additional experiments with alternative rule-extraction techniques demonstrate that the logic minimization method is able to produce accurate rules from datasets with large numbers of variables and limited numbers of positive examples.

Constructing normalcy and discrepancy indexes for birth weight and gestational age using a threshold regression mixture model.

Birth weight and gestational age are important measures of a newborn's intrinsic health, serving both as outcome measures and explanatory variables in health studies. The measures are highly correlated but occasionally inconsistent. We anticipate that health researchers and other scientists would be helped by summary indexes of birth weight and gestational age that give more precise indications of whether the birth outcome is healthy or not. We propose a pair of indexes that we refer to as the birth normalcy index or BNI and birth discrepancy index or BDI. Both indexes are simple functions of birth weight and gestational age and in logarithmic form are orthogonal by construction. The BNI gauges whether the birth weight and gestational age combination are in a normal range. The BDI gauges whether birth weight and gestational age are consistent. We present a three-component mixture model for BNI, with the components representing premature, at-risk, and healthy births. The BNI distribution is derived from a stochastic model of fetal development proposed by Whitmore and Su (2007, Lifetime Data Analysis 13, 161-190) and takes the form of a mixture of inverse Gaussian distributions. We present a noncentral t-distribution as a model for BDI. BNI and BDI are also well suited for making comparisons of birth outcomes in different reference populations. A simple z-score and t-score are proposed for such comparisons. The BNI and BDI distributions can be estimated for births in any reference population of interest using threshold regression.