A bias-reduced generalized estimating equation approach for proportional odds models with small-sample longitudinal ordinal data.

BACKGROUND: Longitudinal ordinal data are commonly analyzed using a marginal proportional odds model for relating ordinal outcomes to covariates in the biomedical and health sciences. The generalized estimating equation (GEE) consistently estimates the regression parameters of marginal models even if the working covariance structure is misspecified. For small-sample longitudinal binary data, recent studies have shown that the bias of regression parameters may result from the GEE and have addressed the issue by applying Firth's adjustment for the likelihood score equation to the GEE as if generalized estimating functions were likelihood score functions. In this manuscript, for the proportional odds model for longitudinal ordinal data, the small-sample properties of the GEE were investigated, and a bias-reduced GEE (BR-GEE) was derived. METHODS: By applying the adjusted function originally derived for the likelihood score function of the proportional odds model to the GEE, we produced the BR-GEE. We investigated the small-sample properties of both GEE and BR-GEE through simulation and applied them to a clinical study dataset. RESULTS: In simulation studies, the BR-GEE had a bias closer to zero, smaller root mean square error than the GEE with coverage probability of confidence interval near or above the nominal level. The simulation also showed that BR-GEE maintained a type I error rate near or below the nominal level. CONCLUSIONS: For the analysis of longitudinal ordinal data involving a small number of subjects, the BR-GEE is advantageous for obtaining estimates of the regression parameters of marginal proportional odds models.

Tutorial on Firth's Logistic Regression Models for Biomarkers in Preclinical Space.

Preclinical studies are broad and can encompass cellular research, animal trials, and small human trials. Preclinical studies tend to be exploratory and have smaller datasets that often consist of biomarker data. Logistic regression is typically the model of choice for modeling a binary outcome with explanatory variables such as genetic, imaging, and clinical data. Small preclinical studies can have challenging data that may include a complete separation or quasi-complete separation issue that will result in logistic regression inflated coefficient estimates and standard errors. Penalized regression approaches such as Firth's logistic regression are a solution to reduce the bias in the estimates. In this tutorial, a number of examples with separation (complete or quasi-complete) are illustrated and the results from both logistic regression and Firth's logistic regression are compared to demonstrate the inflated estimates from the standard logistic regression model and bias-reduction of the estimates from the penalized Firth's approach. R code and datasets are provided in the supplement.

Reparametrized Firth's Logistic Regressions for Dose-Finding Study With the Biased-Coin Design.

Finding an adequate dose of the drug by revealing the dose-response relationship is very crucial and a challenging problem in the clinical development. The main concerns in dose-finding study are to identify a minimum effective dose (MED) in anesthesia studies and maximum tolerated dose (MTD) in oncology clinical trials. For the estimation of MED and MTD, we propose two modifications of Firth's logistic regression using reparametrization, called reparametrized Firth's logistic regression (rFLR) and ridge-penalized reparametrized Firth's logistic regression (RrFLR). The proposed methods are designed by directly reducing the small-sample bias of the maximum likelihood estimate for the parameter of interest. In addition, we develop a method on how to construct confidence intervals for rFLR and RrFLR using profile penalized likelihood. In the up-and-down biased-coin design, numerical studies confirm the superior performance of the proposed methods in terms of the mean squared error, bias, and coverage accuracy of confidence intervals.

Firth-Type Penalized Methods of the Modified Poisson and Least-Squares Regression Analyses for Binary Outcomes.

The modified Poisson and least-squares regression analyses for binary outcomes have been widely used as effective multivariable analysis methods to provide risk ratio and risk difference estimates in clinical and epidemiological studies. However, there is no certain evidence that assessed their operating characteristics under small and sparse data settings and no effective methods have been proposed for these regression analyses to address this issue. In this article, we show that the modified Poisson regression provides seriously biased estimates under small and sparse data settings. In addition, the modified least-squares regression provides unbiased estimates under these settings. We further show that the ordinary robust variance estimators for both of the methods have certain biases under situations that involve small or moderate sample sizes. To address these issues, we propose the Firth-type penalized methods for the modified Poisson and least-squares regressions. The adjustment methods lead to a more accurate and stable risk ratio estimator under small and sparse data settings, although the risk difference estimator is not invariant. In addition, to improve the inferences of the effect measures, we provide an improved robust variance estimator for these regression analyses. We conducted extensive simulation studies to assess the performances of the proposed methods under real-world conditions and found that the accuracies of the point and interval estimations were markedly improved by the proposed methods. We illustrate the effectiveness of these methods by applying them to a clinical study of epilepsy.

Exploring the Linkages between farm efficiency, farm environmental performance, and agri-environmental scheme adoption: Lessons from France.

The agricultural detrimental effects on the environment are a source of concern. Public mea-sures, such as agri-environmental schemes (AES), have been designed to incentivize farmers to adopt more sound environmental practices on the farm. In this study, we examine the effects of past initial economic and environmental performances on AES adoption by focusing on crop farms. Using Firth's logistic regression to address small sample bias with French FADN data from 1997 to 2007, we mainly find that technical efficiency has heterogeneous effects on AES adoption, depending on environmental indexes. This result suggests the presence of windfall effects. We also show complex interactions (antagonism or synergy) between economic and environmental performances in adoption decisions, and heterogeneous effects depending on the type of farming. The agricultural detrimental effects on the environment are a source of concern. Public mea-sures, such as agri-environmental schemes (AES), have been designed to incentivize farmers to adopt more sound environmental practices on the farm. In this study, we examine the effects of past initial economic and environmental performances on AES adoption by focusing on crop farms. Using Firth's logistic regression to address small sample bias with French FADN data from 1997 to 2007, we mainly find that technical efficiency has heterogeneous effects on AES adoption, depending on environmental indexes. This result suggests the presence of windfall effects. We also show complex interactions (antagonism or synergy) between economic and environmental performances in adoption decisions, and heterogeneous effects depending on the type of farming.

Bias reduction for semi-competing risks frailty model with rare events: application to a chronic kidney disease cohort study in South Korea.

In a semi-competing risks model in which a terminal event censors a non-terminal event but not vice versa, the conventional method can predict clinical outcomes by maximizing likelihood estimation. However, this method can produce unreliable or biased estimators when the number of events in the datasets is small. Specifically, parameter estimates may converge to infinity, or their standard errors can be very large. Moreover, terminal and non-terminal event times may be correlated, which can account for the frailty term. Here, we adapt the penalized likelihood with Firth's correction method for gamma frailty models with semi-competing risks data to reduce the bias caused by rare events. The proposed method is evaluated in terms of relative bias, mean squared error, standard error, and standard deviation compared to the conventional methods through simulation studies. The results of the proposed method are stable and robust even when data contain only a few events with the misspecification of the baseline hazard function. We also illustrate a real example with a multi-centre, patient-based cohort study to identify risk factors for chronic kidney disease progression or adverse clinical outcomes. This study will provide a better understanding of semi-competing risk data in which the number of specific diseases or events of interest is rare.

A comparison of bias-adjusted generalized estimating equations for sparse binary data in small-sample longitudinal studies.

Using a generalized estimating equation (GEE) can lead to a bias in regression coefficients for a small sample or sparse data. The bias-corrected GEE (BCGEE) and penalized GEE (PGEE) were proposed to resolve the small-sample bias. Moreover, the standard sandwich covariance estimator leads to a bias of standard error for small samples; several modified covariance estimators have been proposed to address this issue. We review the modified GEEs and modified covariance estimators, and evaluate their performance in sparse binary data from small-sample longitudinal studies. The simulation results showed that GEE and BCGEE often failed to achieve convergence, whereas the convergence proportion for PGEE was quite high. The bias for the regression coefficients was generally in the ascending order of PGEE < $$ < $$ BCGEE < $$ < $$ GEE. However, PGEE and BCGEE did not sufficiently remove the bias involving 20-30 subjects with unequal exposure levels with a 5% response rate. The coverage probability (CP) of the confidence interval for BCGEE was relatively poor compared with GEE and PGEE. The CP with the sandwich covariance estimator deteriorated regardless of the GEE methods under the small sample size and low response rate, whereas the CP with the modified covariance estimators-such as Morel's method-was relatively acceptable. PGEE will be the reasonable way for analyzing sparse binary data in small-sample studies. Instead of using the standard sandwich covariance estimator, one should always apply the modified covariance estimators for analyzing these data.

Penalized maximum likelihood inference under the mixture cure model in sparse data.

INTRODUCTION: When a study sample includes a large proportion of long-term survivors, mixture cure (MC) models that separately assess biomarker associations with long-term recurrence-free survival and time to disease recurrence are preferred to proportional-hazards models. However, in samples with few recurrences, standard maximum likelihood can be biased. OBJECTIVE AND METHODS: We extend Firth-type penalized likelihood (FT-PL) developed for bias reduction in the exponential family to the Weibull-logistic MC, using the Jeffreys invariant prior. Via simulation studies based on a motivating cohort study, we compare parameter estimates of the FT-PL method to those by ML, as well as type 1 error (T1E) and power obtained using likelihood ratio statistics. RESULTS: In samples with relatively few events, the Firth-type penalized likelihood estimates (FT-PLEs) have mean bias closer to zero and smaller mean squared error than maximum likelihood estimates (MLEs), and can be obtained in samples where the MLEs are infinite. Under similar T1E rates, FT-PL consistently exhibits higher statistical power than ML in samples with few events. In addition, we compare FT-PL estimation with two other penalization methods (a log-F prior method and a modified Firth-type method) based on the same simulations. DISCUSSION: Consistent with findings for logistic and Cox regressions, FT-PL under MC regression yields finite estimates under stringent conditions, and better bias-and-variance balance than the other two penalizations. The practicality and strength of FT-PL for MC analysis is illustrated in a cohort study of breast cancer prognosis with long-term follow-up for recurrence-free survival.

Bias in Odds Ratios From Logistic Regression Methods With Sparse Data Sets.

BACKGROUND: Logistic regression models are widely used to evaluate the association between a binary outcome and a set of covariates. However, when there are few study participants at the outcome and covariate levels, the models lead to bias of the odds ratio (OR) estimated using the maximum likelihood (ML) method. This bias is known as sparse data bias, and the estimated OR can yield impossibly large values because of data sparsity. However, this bias has been ignored in most epidemiological studies. METHODS: We review several methods for reducing sparse data bias in logistic regression. The primary aim is to evaluate the Bayesian methods in comparison with the classical methods, such as the ML, Firth's, and exact methods using a simulation study. We also apply these methods to a real data set. RESULTS: Our simulation results indicate that the bias of the OR from the ML, Firth's, and exact methods is considerable. Furthermore, the Bayesian methods with hyper-É¡ prior modeling of the prior covariance matrix for regression coefficients reduced the bias under the null hypothesis, whereas the Bayesian methods with log F-type priors reduced the bias under the alternative hypothesis. CONCLUSION: The Bayesian methods using log F-type priors and hyper-É¡ prior are superior to the ML, Firth's, and exact methods when fitting logistic models to sparse data sets. The choice of a preferable method depends on the null and alternative hypothesis. Sensitivity analysis is important to understand the robustness of the results in sparse data analysis.

Smallholder milk-quality awareness in Indonesian dairy farms.

In most low- and middle-income countries, milk is produced by smallholders, thereby contributing to the livelihood of their households. With the increasing importance of milk production in these countries, it is essential that milk quality is of a high level to ensure a safe product for consumers. It is, however, unclear whether smallholder dairy farmers are aware of the quality of their milk. The aim of this cross-sectional study was to gain insight on Indonesian smallholder dairy farmer awareness of milk quality parameters and to identify factors associated with the total plate count (TPC) and somatic cell count (SCC). A stratified sampling method was used to select smallholder farms in 4 districts in West Java, Indonesia, that were interviewed between August and September 2017. Factors putatively associated with awareness of TPC were investigated with multinomial regression models, whereas a Firth-type logistic regression was applied to identify factors associated with SCC awareness. Of the total 600 farmers surveyed, 264 (44%), 109 (18%), 170 (28%), 111 (19%), and 23 (4%) farmers were aware of TPC, total solid, fat content, milk density, and SCC, respectively, but did not know its value. Those that were conceptually aware of these quality parameters were generally unaware of their value. Furthermore, this study revealed that the following variables were significantly associated with dairy farmers' awareness of TPC: cooperative to which the farmer belonged, distance to neighboring dairy farmer, technology adoption index, TPC as the most important quality factor for the buyer, milk production information from cooperatives, and cow health information from veterinarians. Similarly, cooperative, dairy business experience, and milk quality test adoption were significantly associated with dairy farmers' awareness of SCC. Cooperative was the only variable that was significant in both final statistical models. This indicates that cooperatives play an important role in increasing farmer awareness of milk quality parameters in these smallholder dairies. This may be valid for other regions in the world also where milk production is dominated by smallholder dairy farmers.

An investigation of penalization and data augmentation to improve convergence of generalized estimating equations for clustered binary outcomes.

BACKGROUND: In binary logistic regression data are 'separable' if there exists a linear combination of explanatory variables which perfectly predicts the observed outcome, leading to non-existence of some of the maximum likelihood coefficient estimates. A popular solution to obtain finite estimates even with separable data is Firth's logistic regression (FL), which was originally proposed to reduce the bias in coefficient estimates. The question of convergence becomes more involved when analyzing clustered data as frequently encountered in clinical research, e.g. data collected in several study centers or when individuals contribute multiple observations, using marginal logistic regression models fitted by generalized estimating equations (GEE). From our experience we suspect that separable data are a sufficient, but not a necessary condition for non-convergence of GEE. Thus, we expect that generalizations of approaches that can handle separable uncorrelated data may reduce but not fully remove the non-convergence issues of GEE. METHODS: We investigate one recently proposed and two new extensions of FL to GEE. With 'penalized GEE' the GEE are treated as score equations, i.e. as derivatives of a log-likelihood set to zero, which are then modified as in FL. We introduce two approaches motivated by the equivalence of FL and maximum likelihood estimation with iteratively augmented data. Specifically, we consider fully iterated and single-step versions of this 'augmented GEE' approach. We compare the three approaches with respect to convergence behavior, practical applicability and performance using simulated data and a real data example. RESULTS: Our simulations indicate that all three extensions of FL to GEE substantially improve convergence compared to ordinary GEE, while showing a similar or even better performance in terms of accuracy of coefficient estimates and predictions. Penalized GEE often slightly outperforms the augmented GEE approaches, but this comes at the cost of a higher burden of implementation. CONCLUSIONS: When fitting marginal logistic regression models using GEE on sparse data we recommend to apply penalized GEE if one has access to a suitable software implementation and single-step augmented GEE otherwise.

Modified score function for monotone likelihood in the semiparametric mixture cure model.

The cure fraction models are intended to analyze lifetime data from populations where some individuals are immune to the event under study, and allow a joint estimation of the distribution related to the cured and susceptible subjects, as opposed to the usual approach ignoring the cure rate. In situations involving small sample sizes with many censored times, the detection of nonfinite coefficients may arise via maximum likelihood. This phenomenon is commonly known as monotone likelihood (ML), occurring in the Cox and logistic regression models when many categorical and unbalanced covariates are present. An existing solution to prevent the issue is based on the Firth correction, originally developed to reduce the estimation bias. The method ensures finite estimates by penalizing the likelihood function. In the context of mixture cure models, the ML issue is rarely discussed in the literature; therefore, this topic can be seen as the first contribution of our paper. The second major contribution, not well addressed elsewhere, is the study of the ML issue in cure mixture modeling under the flexibility of a semiparametric framework to handle the baseline hazard. We derive the modified score function based on the Firth approach and explore finite sample size properties of the estimators via a Monte Carlo scheme. The simulation results indicate that the performance of coefficients related to the binary covariates are strongly affected to the imbalance degree. A real illustration, in the melanoma dataset, is discussed using a relatively novel data set collected in a Brazilian university hospital.

To tune or not to tune, a case study of ridge logistic regression in small or sparse datasets.

BACKGROUND: For finite samples with binary outcomes penalized logistic regression such as ridge logistic regression has the potential of achieving smaller mean squared errors (MSE) of coefficients and predictions than maximum likelihood estimation. There is evidence, however, that ridge logistic regression can result in highly variable calibration slopes in small or sparse data situations. METHODS: In this paper, we elaborate this issue further by performing a comprehensive simulation study, investigating the performance of ridge logistic regression in terms of coefficients and predictions and comparing it to Firth's correction that has been shown to perform well in low-dimensional settings. In addition to tuned ridge regression where the penalty strength is estimated from the data by minimizing some measure of the out-of-sample prediction error or information criterion, we also considered ridge regression with pre-specified degree of shrinkage. We included 'oracle' models in the simulation study in which the complexity parameter was chosen based on the true event probabilities (prediction oracle) or regression coefficients (explanation oracle) to demonstrate the capability of ridge regression if truth was known. RESULTS: Performance of ridge regression strongly depends on the choice of complexity parameter. As shown in our simulation and illustrated by a data example, values optimized in small or sparse datasets are negatively correlated with optimal values and suffer from substantial variability which translates into large MSE of coefficients and large variability of calibration slopes. In contrast, in our simulations pre-specifying the degree of shrinkage prior to fitting led to accurate coefficients and predictions even in non-ideal settings such as encountered in the context of rare outcomes or sparse predictors. CONCLUSIONS: Applying tuned ridge regression in small or sparse datasets is problematic as it results in unstable coefficients and predictions. In contrast, determining the degree of shrinkage according to some meaningful prior assumptions about true effects has the potential to reduce bias and stabilize the estimates.

Cervical cancer screening prevalence and its correlates in Cameroon: secondary data analysis of the 2018 demographic and health surveys.

BACKGROUND: Cervical cancer, although preventable, is the fourth most common cancer among women globally, and the second most common and deadliest gynaecological cancer in low-and-middle-income countries. Screening is key to the prevention and early detection of the disease for treatment. A few studies estimated the prevalence of cervical cancer screening and its correlates in Cameroon but relied on data that were limited to certain regions of the country. Therefore, this study sought to examine the prevalence and correlates of cervical cancer screening among Cameroonian women using current data that is nationally representative of reproductive-age women. METHODS: We used secondary data from the 2018 Cameroon Demographic and Health Survey. Summary statistics were used for the sample description. We employed the Firth logistic regression using the "firthlogit" command in STATA-14 to perform the bivariate analyses between the outcome variable and each of the explanatory variables. Given that all the explanatory variables were statistically significant correlates, they were all adjusted for in a multivariable analysis. All analyses were performed in STATA version 14. RESULTS: The proportion of Cameroonian women who have ever screened for cervical cancer continue to remain low at approximately 4%. In the adjusted model, women with the following sociodemographic characteristics have a higher likelihood of undergoing cervical cancer screening: ever undergone HIV screening (AOR = 4.446, 95% CI: 2.475, 7.986), being 24-34 years (AOR = 2.233, 95% CI: 1.606, 3.103) or 35-44 years (AOR = 4.008, 95% CI: 2.840, 5.657) or at least 45 years old (AOR = 5.895, 95% CI: 3.957, 8.784), having attained a post-secondary education (AOR = 1.849, 95% CI: 1.032, 3.315), currently (AOR = 1.551, 95% CI: 1.177, 2.043) or previously married (AOR = 1.572, 95% CI: 1.073, 2.302), dwelling in the richest household (AOR = 4.139, 95% CI: 1.769, 9.682), and residing in an urban area (AOR = 1.403, 95% CI: 1.004,1.960). Except for the North-West region, residing in some five regions, compared to Yaounde, was negatively associated with cervical cancer screening. CONCLUSION: Cervical cancer screening programs and policies should target Cameroonian women who are younger, less educated, and those in poor households and rural areas.

Firth adjusted score function for monotone likelihood in the mixture cure fraction model.

Models for situations where some individuals are long-term survivors, immune or non-susceptible to the event of interest, are extensively studied in biomedical research. Fitting a regression can be problematic in situations involving small sample sizes with high censoring rate, since the maximum likelihood estimates of some coefficients may be infinity. This phenomenon is called monotone likelihood, and it occurs in the presence of many categorical covariates, especially when one covariate level is not associated with any failure (in survival analysis) or when a categorical covariate perfectly predicts a binary response (in the logistic regression). A well known solution is an adaptation of the Firth method, originally created to reduce the estimation bias. The method provides a finite estimate by penalizing the likelihood function. Bias correction in the mixture cure model is a topic rarely discussed in the literature and it configures a central contribution of this work. In order to handle this point in such context, we propose to derive the adjusted score function based on the Firth method. An extensive Monte Carlo simulation study indicates good inference performance for the penalized maximum likelihood estimates. The analysis is illustrated through a real application involving patients with melanoma assisted at the Hospital das Clínicas/UFMG in Brazil. This is a relatively novel data set affected by the monotone likelihood issue and containing cured individuals.

Multi-subject Stochastic Blockmodels for adaptive analysis of individual differences in human brain network cluster structure.

There is considerable interest in elucidating the cluster structure of brain networks in terms of modules, blocks or clusters of similar nodes. However, it is currently challenging to handle data on multiple subjects since most of the existing methods are applicable only on a subject-by-subject basis or for analysis of an average group network. The main limitation of per-subject models is that there is no obvious way to combine the results for group comparisons, and of group-averaged models that they do not reflect the variability between subjects. Here, we propose two new extensions of the classical Stochastic Blockmodel (SBM) that use a mixture model to estimate blocks or clusters of connected nodes, combined with a regression model to capture the effects of subject-level covariates on individual differences in cluster structure. The proposed Multi-Subject Stochastic Blockmodels (MS-SBMs) can flexibly account for between-subject variability in terms of homogeneous or heterogeneous covariate effects on connectivity using subject demographics such as age or diagnostic status. Using synthetic data, representing a range of block sizes and cluster structures, we investigate the accuracy of the estimated MS-SBM parameters as well as the validity of inference procedures based on the Wald, likelihood ratio and permutation tests. We show that the proposed multi-subject SBMs recover the true cluster structure of synthetic networks more accurately and adaptively than standard methods for modular decomposition (i.e. the Fast Louvain and Newman Spectral algorithms). Permutation tests of MS-SBM parameters were more robustly valid for statistical inference and Type I error control than tests based on standard asymptotic assumptions. Applied to analysis of multi-subject resting-state fMRI networks (13 healthy volunteers; 12 people with schizophrenia; n=268 brain regions), we show that Heterogeneous Stochastic Blockmodel (Het-SBM) identifies a range of network topologies simultaneously, including modular and core structures.

Spatio-temporal variability of tidal-stream energy in north-western Europe.

Initial selection of tidal stream energy sites is primarily based on identifying areas with the maximum current speeds. However, optimal design and deployment of turbines requires detailed investigations of the temporal variability of the available resource, focusing on areas with reduced variability, and hence the potential for more continuous energy supply. These aspects are investigated here for some of the most promising sites for tidal array development across the north-western European shelf seas: the Alderney Race, the Fromveur Strait, the Pentland Firth and the channels of Orkney. Particular attention was dedicated to asymmetry between the flood and ebb phases of the tidal cycle (due to the phase relationship between M2 and M4 constituents), and spring-neap variability of the available resource (due to M2 and S2 compound tides). A series of high-resolution models were exploited to (i) produce a detailed harmonic database of these three components, and (ii) characterize, using energy resource metrics, temporal variability of the available power density. There was a clear contrast between the Alderney Race, with reduced temporal variability over semi-diurnal and fortnightly time scales, and sites in western Brittany and North Scotland which, due to increased variability, appeared less attractive for optimal energy conversion. This article is part of the theme issue 'New insights on tidal dynamics and tidal energy harvesting in the Alderney Race'.

Investigation of Predictors of Newborn Screening Refusal in a Large Birth Cohort in North Dakota, USA.

Objectives The objective of this study was to identify maternal and provider predictors of newborn screening (NBS) refusal in North Dakota between 2011 and 2014. Methods Records of 40,440 live resident births occurring in North Dakota between 2011 and 2014 were obtained from the North Dakota Department of Health and included in the study. Factor-specific percentages of NBS refusals and 95% confidence intervals were computed for each predictor. Since the outcome is rare, multivariable Firth logistic regression was used to investigate maternal and provider predictors of NBS refusal. Model goodness-of-fit test was evaluated using the Hosmer-Lemeshow test. All analyses were conducted in SAS 9.4. Results Of the 40,440 live births, 135 (0.33%) were NBS refusals. 97% of the refusals were to white women, 94% were homebirths, and 93% utilized state non-credentialed birth attendants. The odds of NBS refusals were significantly higher among non-credentialed birth attendants (p < 0.0001), homebirths (p < 0.0001), and among those that refused Hepatitis B vaccination (HBV) at birth (p = 0.047). On the other hand, odds of NBS refusals were significantly (p < 0.0001) lower among women that had more prenatal visits. Conclusions for Practice This study provides preliminary evidence of association between NBS refusal and provider type, home births, and HBV refusal. Additional studies of obstetric providers, home births and women are needed to improve our understanding of the reasons for NBS refusal to better deliver preventive services to newborns.

Estimation of log-odds ratio from group testing data using Firth correction.

We consider the estimation of the prevalence of a rare disease, and the log-odds ratio for two specified groups of individuals from group testing data. For a low-prevalence disease, the maximum likelihood estimate of the log-odds ratio is severely biased. However, Firth correction to the score function leads to a considerable improvement of the estimator. Also, for a low-prevalence disease, if the diagnostic test is imperfect, the group testing is found to yield more precise estimate of the log-odds ratio than the individual testing.

Observation of a porbeagle shark Lamna nasus aggregation at a North Sea oil platform.

A minimum of 20 porbeagles Lamna nasus were observed circling the Alba oil platform (UK North Sea) over several days in July 2014. Although schools of fish often aggregate around oil platforms, less is known about their ability to aggregate pelagic sharks and L. nasus are rarely seen at the surface. This unusual observation provides new insights into L. nasus behaviour and habitat use and the potential role of offshore artificial structures on pelagic predators.