Exact Maximum Likelihood Estimates of Cox's Regression Parameters Based on Categorized Factorial Data

This paper describes how Cox's Proportional Hazards model may be used to analyze dichotomized factorial data obtained from a right-censored epidemiological study where time to response is of interest. Exact maximum likelihood estimates of the relative mortality rates are obtained for any number of prognostic factors, as well as their joint asymptotic sampling distribution. These rates represent excess mortality due to the various levels of the prognostic factors. The results are used to discuss the effect of the factors on the survival probability distribution of a cohort of industrial workers who have been exposed to a carcinogen. Kaplan-Meier estimates of the survival function of the internal control population are used to determine the expected number of deaths in the study population. This method differs from the usual lite-table procedure. Asymptotic tests are proposed for some simultaneous and conditional statistical hypotheses.     

A Preliminary Continental Risk Map for Malaria Mortality among African Children

Approaches to global public health are increasingly driven by an understanding of regional patterns of disease-specific mortality and disability. Current estimates of disease risks associated with Plasmodium falciparum in sub-Saharan Africa remain poorly defined. Through the integration of high-resolution population and climate probability models of P. falciparum transmission, geographical information systems have been used to define the spatial limits of populations exposed to the risk of infection in Africa. These estimates were combined with a range of annual malaria-specific mortality rates, derived from a variety of epidemiological approaches, among children aged 0–4 years. The best estimates of malaria-attributable mortality using this approach ranged between 0.43 million and 0.68 million deaths per annum among an exposed population of ∼66 million children in 1990. Despite the limitations of modelled transmission and population distributions, these empirical approaches to probabilities of infection risk and epidemiological data on mortality provide a novel approach to present and projected burdens of malaria mortality, as discussed here by Bob Snow, Marlies Craig, Uwe Deichmann and Dave le Sueur.     

A model for analyzing insect stage-frequency data when mortality varies with time

A model is developed for the analysis of insect stage-frequency data which may be applied to populations with age-dependent mortality. The analysis of stage-frequency data is divided into two steps. In the first step, the number of different mortality rates and their values are estimated. The second step provides estimates of developmental rates and variances for each developmental stage and in addition provides estimates of the number of recruits to each stage. The model may be used both in analysis and prediction of insect stage frequencies. Hence, in addition to estimating developmental and mortality rates from stage-frequency data, it may also be used as a simulation model for an insect population. The model is applied to two populations of Hemileuca oliviaeCockerell , a lepidopterous pest of New Mexico grasslands. The model identifies, in the two populations, different mortality rates that are related to plant productivity.     

Disparities in COVID-19 mortality by county racial composition and the role of spring social distancing measures

Available COVID-19 data shows higher shares of cases and deaths occur among Black Americans, but reporting of data by race is poor. This paper investigates disparities in county-level mortality rates across counties with higher and lower than national average Black population shares using nonlinear regression decomposition and estimates potential differential impact of social distancing measures. I find counties with Black population shares above the national share have mortality rates 2 to 3 times higher than in other counties. Observable differences in living conditions, health, and work characteristics reduce the disparity to approximately 1.25 to 1.65 overall, and explain 100% of the disparity at 21 days after the first case. Though higher rates of comorbidities in counties with higher Black population shares are an important predictor, living situation factors like single parenthood and population density are just as important. Higher rates of co-residence with grandchildren explain 11% of the 21 day disparity but do not appear important by 42 days, suggesting families may have been better able to protect vulnerable family members later in the epidemic. To analyze differential effects of social distancing measures use two approaches. First, I exploit the timing of interventions relative to the first case among counties that began their epidemic at the same time. Second, I use event study analysis to analyze within-county changes in mortality. Findings for social distancing measures are not always consistent across approaches. Overall, I find no evidence that school closures were less effective in counties with larger Black population shares, and some estimates suggest closures may have disproportionately helped more diverse counties and counties with high rates of grandparent and grandchild co-residence. Conversely, stay at home orders are less clearly associated with mortality in any counties, reaching peak unemployment did not reduce mortality in any models, and some estimates indicate reaching peak unemployment before the first case was associated with higher mortality rates, especially in more diverse counties.     

The adequacy of aging techniques in vertebrates for rapid estimation of population mortality rates from age distributions

As a key parameter in population dynamics, mortality rates are frequently estimated using mark–recapture data, which requires extensive, long‐term data sets. As a potential rapid alternative, we can measure variables correlated to age, allowing the compilation of population age distributions, from which mortality rates can be derived. However, most studies employing such techniques have ignored their inherent inaccuracy and have thereby failed to provide reliable mortality estimates. In this study, we present a general statistical model linking birth rate, mortality rate, and population age distributions. We next assessed the reliability and data needs (i.e., sample size) for estimating mortality rate of eight different aging techniques. The results revealed that for half of the aging techniques, correlations with age varied considerably, translating into highly variable accuracies when used to estimate mortality rate from age distributions. Telomere length is generally not sufficiently correlated to age to provide reliable mortality rate estimates. DNA methylation, signal‐joint T‐cell recombination excision circle (sjTREC), and racemization are generally more promising techniques to ultimately estimate mortality rate, if a sufficiently high sample size is available. Otolith ring counts, otolithometry, and age‐length keys in fish, and skeletochronology in reptiles, mammals, and amphibians, outperformed all other aging techniques and generated relatively accurate mortality rate estimation with a sample size that can be feasibly obtained. Provided the method chosen is minimizing and estimating the error in age estimation, it is possible to accurately estimate mortality rates from age distributions. The method therewith has the potential to estimate a critical, population dynamic parameter to inform conservation efforts within a limited time frame as opposed to mark–recapture analyses.     

Bayesian detection of clusters and discontinuities in disease maps

An interesting epidemiological problem is the analysis of geographical variation in rates of disease incidence or mortality. One goal of such an analysis is to detect clusters of elevated (or lowered) risk in order to identify unknown risk factors regarding the disease. We propose a nonparametric Bayesian approach for the detection of such clusters based on Green's (1995, Biometrika 82, 711-732) reversible jump MCMC methodology. The prior model assumes that geographical regions can be combined in clusters with constant relative risk within a cluster. The number of clusters, the location of the clusters, and the risk within each cluster is unknown. This specification can be seen as a change-point problem of variable dimension in irregular, discrete space. We illustrate our method through an analysis of oral cavity cancer mortality rates in Germany and compare the results with those obtained by the commonly used Bayesian disease mapping method of Besag, York, and Mollié (1991, Annals of the Institute of Statistical Mathematics, 43, 1-59).     

Child Mortality Estimation: Consistency of Under-Five Mortality Rate Estimates Using Full Birth Histories and Summary Birth Histories

Background

Given the lack of complete vital registration data in most developing countries, for many countries it is not possible to accurately estimate under-five mortality rates from vital registration systems. Heavy reliance is often placed on direct and indirect methods for analyzing data collected from birth histories to estimate under-five mortality rates. Yet few systematic comparisons of these methods have been undertaken. This paper investigates whether analysts should use both direct and indirect estimates from full birth histories, and under what circumstances indirect estimates derived from summary birth histories should be used.

Methods and Findings

Usings Demographic and Health Surveys data from West Africa, East Africa, Latin America, and South/Southeast Asia, I quantify the differences between direct and indirect estimates of under-five mortality rates, analyze data quality issues, note the relative effects of these issues, and test whether these issues explain the observed differences. I find that indirect estimates are generally consistent with direct estimates, after adjustment for fertility change and birth transference, but don''t add substantial additional insight beyond direct estimates. However, choice of direct or indirect method was found to be important in terms of both the adjustment for data errors and the assumptions made about fertility.

Conclusions

Although adjusted indirect estimates are generally consistent with adjusted direct estimates, some notable inconsistencies were observed for countries that had experienced either a political or economic crisis or stalled health transition in their recent past. This result suggests that when a population has experienced a smooth mortality decline or only short periods of excess mortality, both adjusted methods perform equally well. However, the observed inconsistencies identified suggest that the indirect method is particularly prone to bias resulting from violations of its strong assumptions about recent mortality and fertility. Hence, indirect estimates of under-five mortality rates from summary birth histories should be used only for populations that have experienced either smooth mortality declines or only short periods of excess mortality in their recent past. Please see later in the article for the Editors'' Summary.     

A simple method of removing the effect of a bottleneck and unequal population sizes on pairwise genetic distances

In this paper, we derive the expectation of two popular genetic distances under a model of pure population fission allowing for unequal population sizes. Under the model, we show that conventional genetic distances are not proportional to the divergence time and generally overestimate it due to unequal genetic drift and to a bottleneck effect at the divergence time. This bias cannot be totally removed even if the present population sizes are known. Instead, we present a method to estimate the divergence times between populations which is based on the average number of nucleotide differences within and between populations. The method simultaneously estimates the divergence time, the ancestral population size and the relative sizes of the derived populations. A simulation study revealed that this method is essentially unbiased and that it leads to better estimates than traditional approaches for a very wide range of parameter values. Simulations also indicated that moderate population growth after divergence has little effect on the estimates of all three estimated parameters. An application of our method to a comparison of humans and chimpanzee mitochondrial DNA diversity revealed that common chimpanzees have a significantly larger female population size than humans.     

Moose (Alces alces) calf survival rates in the presence of wolves (Canis lupus) in southeast Norway

Wolves (Canis lupus) are recolonizing Scandinavia and becoming a new limiting factor that should be taken into account in the management of moose (Alces alces). However, there is a lack of empirical estimates of moose survival after wolf recolonization. We investigated the effects of wolf abundance, moose litter size (single/twin calves), and climatic factors on annual and seasonal calf survival rates in a moose population in southeast Norway. We used data that were obtained over 7?years by radio-tracking and regular visual observations of 68 moose cows to determine the presence or absence of calves at heel. Annual and winter calf survival rates were 20–40 percentage points lower in the wolf territory compared with previous estimates of moose calf survival in similar areas that lacked wolves. Cause-specific studies of mortality would further enhance our ability to determine the relative role of various limiting factors. Our study suggests that moose managers should regulate quotas to buffer the lower survival rates after wolf recolonization.     

Population growth rates of reef sharks with and without fishing on the great barrier reef: robust estimation with multiple models

Overfishing of sharks is a global concern, with increasing numbers of species threatened by overfishing. For many sharks, both catch rates and underwater visual surveys have been criticized as indices of abundance. In this context, estimation of population trends using individual demographic rates provides an important alternative means of assessing population status. However, such estimates involve uncertainties that must be appropriately characterized to credibly and effectively inform conservation efforts and management. Incorporating uncertainties into population assessment is especially important when key demographic rates are obtained via indirect methods, as is often the case for mortality rates of marine organisms subject to fishing. Here, focusing on two reef shark species on the Great Barrier Reef, Australia, we estimated natural and total mortality rates using several indirect methods, and determined the population growth rates resulting from each. We used bootstrapping to quantify the uncertainty associated with each estimate, and to evaluate the extent of agreement between estimates. Multiple models produced highly concordant natural and total mortality rates, and associated population growth rates, once the uncertainties associated with the individual estimates were taken into account. Consensus estimates of natural and total population growth across multiple models support the hypothesis that these species are declining rapidly due to fishing, in contrast to conclusions previously drawn from catch rate trends. Moreover, quantitative projections of abundance differences on fished versus unfished reefs, based on the population growth rate estimates, are comparable to those found in previous studies using underwater visual surveys. These findings appear to justify management actions to substantially reduce the fishing mortality of reef sharks. They also highlight the potential utility of rigorously characterizing uncertainty, and applying multiple assessment methods, to obtain robust estimates of population trends in species threatened by overfishing.     

Parasites, predators and the duration of developmental periods

Nest predation has been suggested to be the most important environmental factor determining the evolution of incubation and nestling periods in birds. This paradigm is based on a number of comparative analyses that have shown correlations between rates of nest predation and the relative duration of the nestling period. A second important natural selection pressure on reproduction is caused by parasitism. Here, I investigated the independent effects of daily nest predation rates and daily nest mortality rates due to parasites to assess the relative importance of these two environmental factors as determinants of the relative duration of incubation and nestling periods in birds, using the 43 species for which information on parasite-induced nestling mortality was available. The relative duration of the incubation period was negatively related to daily mortality rate due to parasitism, while nest predation had no significant effect. The relative duration of the nestling period was not significantly related to daily rate of mortality due to parasitism or nest predation. Daily nest predation rate was significantly negatively correlated with daily mortality rate due to parasitism in the subsample of species, for which some mortality due to parasitism was recorded. These findings are mainly based on hole nesters and colonially breeding species, because estimates of parasite-induced mortality generally is unavailable from open nesting and solitarily breeding species, precluding any generalizations to these species. However, the conclusions did not change after statistically controlling for nest site and sociality. These results suggest that developmental rates may be more influenced by the effects of parasites rather than nest predators, and that parasites constitute a neglected selective force affecting developmental rates.     

Population genetic analysis identifies source-sink dynamics for two sympatric garter snake species (Thamnophis elegans and Thamnophis sirtalis)

Population genetic structure can be shaped by multiple ecological and evolutionary factors, but the genetic consequences of these factors for multiple species inhabiting the same environment remain unexplored. We used microsatellite markers to examine the population structures of two coexisting species of garter snake, Thamnophis elegans and Thamnophis sirtalis, to determine if shared landscape and biology imposed similar population genetic structures. These snakes inhabit a series of ponds, lakes and flooded meadows in northern California and tend to converge on prey type wherever they coexist. Both garter snakes had comparable effective population sizes and bidirectional migration rates (estimated using a maximum-likelihood method based on the coalescent) with low but significant levels of genetic differentiation (F(ST) = 0.024 for T. elegans and 0.035 for T. sirtalis). Asymmetrical gene flow revealed large source populations for both species as well as potential sinks, suggesting frequent extinction-recolonization and metapopulation dynamics. In addition, we found a significant correlation between their genetic structures based on both pairwise F(ST)s for shared populations (P = 0.009) and for bidirectional migration rates (P = 0.024). Possible ecological and evolutionary factors influencing similarities and differences in genetic structure for the two species are discussed. Genetic measures of effective population size and migration rates obtained in this study are also compared with estimates obtained from mark-recapture data.     

A comparison of breeding population estimators using nest and brood monitoring data

For many species, breeding population size is an important metric for assessing population status. A variety of simple methods are often used to estimate this metric for ground‐nesting birds that nest in open habitats (e.g., beaches, riverine sandbars). The error and bias associated with estimates derived using these methods vary in relation to differing monitoring intensities and detection rates. However, these errors and biases are often difficult to obtain, poorly understood, and largely unreported. A method was developed to estimate the number of breeding pairs using counts of nests and broods from monitoring data where multiple surveys were made throughout a single breeding season (breeding pair estimator; BPE). The BPE method was compared to two commonly used estimation methods using simulated data from an individual‐based model that allowed for the comparison of biases and accuracy. The BPE method underestimated the number of breeding pairs, but generally performed better than the other two commonly used methods when detection rates were low and monitoring frequency was high. As detection rates and time between surveys increased, the maximum nest and brood count method performs similar to the BPE. The BPE was compared to four commonly used methods to estimate breeding pairs for empirically derived data sets on the Platte River. Based on our simulated data, we expect our BPE to be closest to the true number of breeding pairs as compared to other methods. The methods tested resulted in substantially different estimates of the numbers of breeding pairs; however, coefficients from trend analyses were not statistically different. When data from multiple nest and brood surveys are available, the BPE appears to result in reasonably precise estimates of numbers of breeding pairs. Regardless of the estimation method, investigators are encouraged to acknowledge whether the method employed is likely to over‐ or underestimate breeding pairs. This study provides a means to recognize the potential biases in breeding pair estimates.     

Overestimates of maternity and population growth rates in multi-annual breeders

There has been limited attention to estimating maternity rate because it appears to be relatively simple. However, when used for multi-annual breeder species, such as the largest carnivores, the most common estimators introduce an upward bias by excluding unproductive females. Using a simulated dataset based on published data, we compare the accuracy of maternity estimates derived from standard methods against estimates derived from an alternative method. We show that standard methods overestimate maternity rates in the presence of unsuccessful pregnancies. Importantly, population growth rates derived from a matrix model parameterized with the biased estimates may indicate increasing populations although the populations are stable or even declining. We recommend the abandonment of the biased standard methods and to instead use the unbiased alternative method for population projections and assessments of population viability.     

A stochastic catch-effort method for estimating animal abundance

A method for estimating the size of a heavily exploited animal population from catch data and relative-harvest-effort data is presented. The method assumes a competing-risk model of adult deaths and captures that is similar to the hazard-regression model of Cox (1972, Journal of the Royal Statistical Society, Series B 34, 187-220). This model avoids making any assumptions about birth rates or juvenile mortality rates, and allows the user to incorporate an arbitrary number of time-dependent covariates into the natural and catch hazard functions. Estimates of the population's size, together with asymptotic error bounds and predictions of subsequent catches, are derived from maximum likelihood estimates of the parameters of the model. A simulation study is presented which indicates that this method is far more accurate than previously available catch-effort techniques. The method is illustrated with some fisheries data. A series of models is fitted to the data with the objective of improving the goodness of fit while maintaining biologic plausibility of the model. In this example a 68% reduction in the mean sum of squares for error is obtained and the accuracy of future catch predictions is greatly improved. This method is particularly appropriate for estimating the sizes of commercially exploited aquatic populations whose sizes are too large to make mark-recapture techniques feasible, and which are not amenable to line-transect techniques.     

Calculation of the morbid risk in genetic-epidemiologic studies of age-dependent diseases

Distributions of age at onset are widely used in the genetic epidemiology of age-dependent diseases. Examples are estimation of recurrent risks in genetic counselling and testing genetic hypotheses in segregation and linkage analyses. In this study, morbidity parameters are defined, including age-specific morbidity rates, morbidity net risk (incidence), and cumulative incidence (population risk, an integrated measure of population susceptibility to the disease at the moment of the study). Age-specific morbidity risks are calculated from the respective morbidity rates, which are analogous to mortality rates used in demography. Population data typically used for calculation of morbidity rates are discussed. Methods of calculation of morbidity rates based on the data of single and interval epidemiological studies are described. Methods for calculating standard errors of these parameters, estimating their statistical reliability, and testing statistical hypotheses are discussed.     

Population Viability and Vital Rate Sensitivity of an Endangered Avian Cooperative Breeder,the White-Breasted Thrasher (Ramphocinclus brachyurus)

Social behaviors can significantly affect population viability, and some behaviors might reduce extinction risk. We used population viability analysis to evaluate effects of past and proposed habitat loss on the White-breasted Thrasher (Ramphocinclus brachyurus), a cooperatively breeding songbird with a global population size of <2000 individuals. We used an individual-based approach to build the first demographic population projection model for this endangered species, parameterizing the model with data from eight years of field study before and after habitat loss within the stronghold of the species’ distribution. The recent habitat loss resulted in an approximately 18% predicted decline in population size; this estimate was mirrored by a separate assessment using occupancy data. When mortality rates remained close to the pre-habitat loss estimate, quasi-extinction probability was low under extant habitat area, but increased with habitat loss expected after current plans for resort construction are completed. Post-habitat loss mortality rate estimates were too high for projected populations to persist. Vital rate sensitivity analyses indicated that population growth rate and population persistence were most sensitive to juvenile mortality. However, observed values for adult mortality were closest to the threshold value above which populations would crash. Adult mortality, already relatively low, may have the least capacity to change compared to other vital rates, whereas juvenile mortality may have the most capacity for improvement. Results suggest that improving mortality estimates and determining the cause(s) of juvenile mortality should be research priorities. Despite predictions that aspects of cooperative systems may result in variation in reproduction or juvenile mortality being the most sensitive vital rates, adult mortality was the most sensitive in half of the demographic models of other avian cooperative breeders. Interestingly, vital rate sensitivity differed by model type. However, studies that explicitly modeled the species’ cooperative breeding system found reproduction to be the most sensitive rate.     

A New Method for Assessing How Sensitivity and Specificity of Linkage Studies Affects Estimation

Background

While the importance of record linkage is widely recognised, few studies have attempted to quantify how linkage errors may have impacted on their own findings and outcomes. Even where authors of linkage studies have attempted to estimate sensitivity and specificity based on subjects with known status, the effects of false negatives and positives on event rates and estimates of effect are not often described.

Methods

We present quantification of the effect of sensitivity and specificity of the linkage process on event rates and incidence, as well as the resultant effect on relative risks. Formulae to estimate the true number of events and estimated relative risk adjusted for given linkage sensitivity and specificity are then derived and applied to data from a prisoner mortality study. The implications of false positive and false negative matches are also discussed.

Discussion

Comparisons of the effect of sensitivity and specificity on incidence and relative risks indicate that it is more important for linkages to be highly specific than sensitive, particularly if true incidence rates are low. We would recommend that, where possible, some quantitative estimates of the sensitivity and specificity of the linkage process be performed, allowing the effect of these quantities on observed results to be assessed.     

Collision mortality has no discernible effect on population trends of North American birds

Avian biodiversity is threatened by numerous anthropogenic factors and migratory species are especially at risk. Migrating birds frequently collide with manmade structures and such losses are believed to represent the majority of anthropogenic mortality for North American birds. However, estimates of total collision mortality range across several orders of magnitude and effects on population dynamics remain unknown. Herein, we develop a novel method to assess relative vulnerability to anthropogenic threats, which we demonstrate using 243,103 collision records from 188 species of eastern North American landbirds. After correcting mortality estimates for variation attributable to population size and geographic overlap with potential collision structures, we found that per capita vulnerability to collision with buildings and towers varied over more than four orders of magnitude among species. Species that migrate long distances or at night were much more likely to be killed by collisions than year-round residents or diurnal migrants. However, there was no correlation between relative collision mortality and long-term population trends for these same species. Thus, although millions of North American birds are killed annually by collisions with manmade structures, this source of mortality has no discernible effect on populations.     

When Are We Most Vulnerable to Temperature Variations in a Day?

Daily temperature measures are commonly used when examining the association between temperature and mortality. In fact, temperature measures are available 24 hours a day and more detailed records may provide a better prediction of mortality compared to daily statistics. In this article, monthly stratified analysis modeling for mortality is conducted for the total population as well as the stratified elderly and younger subgroups. We identified the most significant time during the day that is associated with daily mortality. Surprisingly, the estimates of relative risk and magnitude of associations derived from the hourly temperature measures are similar or even stronger compared to those modeled by the daily statistics. This phenomenon remains true for lagged hourly temperature measures and the changing patterns of associations from January through December are revealed. In summary, people are the most vulnerable to temperature variations in the early morning around 5 am and the night time around 8 pm.