Analysis of Dichotomized Factorial Data Using Cox's Proportional Hazards Model and Related Tests

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 derived for any number of prognostic factors, but for the sake of simplicity, the mathematical details are presented for the case of two factors. This method is not based on the life table procedure. Kaplan-Meier estimates are obtained for the survival function of the internal control population, Which are in turn used to determine the expected number of deaths in the study population. The asymptotic (large sample) joint sampling distribution of the relative mortality rates is derived and some relevant simultaneous and conditional statistical tests are discussed. The relative mortality rates of several prognostic factors may be jointly considered as the multivariate extension of the familiar standard mortality ratio (SMR) of epidemiological studies. A numerical example is discussed to illustrate the method.     

Calculation of narrower confidence intervals for tree mortality rates when we know nothing but the location of the death/survival events

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The Very High Premature Mortality Rate among Active Professional Wrestlers Is Primarily Due to Cardiovascular Disease

Purpose

Recently, much media attention has been given to the premature deaths in professional wrestlers. Since no formal studies exist that have statistically examined the probability of premature mortality in professional wrestlers, we determined survival estimates for active wresters over the past quarter century to establish the factors contributing to the premature mortality of these individuals.

Methods

Data including cause of death was obtained from public records and wrestling publications in wrestlers who were active between January 1, 1985 and December 31, 2011. 557 males were considered consistently active wrestlers during this time period. 2007 published mortality rates from the Center for Disease Control were used to compare the general population to the wrestlers by age, BMI, time period, and cause of death. Survival estimates and Cox hazard regression models were fit to determine incident premature deaths and factors associated with lower survival. Cumulative incidence function (CIF) estimates given years wrestled was obtained using a competing risks model for cause of death.

Results

The mortality for all wrestlers over the 26-year study period was.007 deaths/total person-years or 708 per 100,000 per year, and 16% of deaths occurred below age 50 years. Among wrestlers, the leading cause of deaths based on CIF was cardiovascular-related (38%). For cardiovascular-related deaths, drug overdose-related deaths and cancer deaths, wrestler mortality rates were respectively 15.1, 122.7 and 6.4 times greater than those of males in the general population. Survival estimates from hazard models indicated that BMI is significantly associated with the hazard of death from total time wrestling (p<0.0001).

Conclusion

Professional wrestlers are more likely to die prematurely from cardiovascular disease compared to the general population and morbidly obese wrestlers are especially at risk. Results from this study may be useful for professional wrestlers, as well as wellness policy and medical care implementation.     

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.     

Effects of sample size on estimates of population growth rates calculated with matrix models

Background

Matrix models are widely used to study the dynamics and demography of populations. An important but overlooked issue is how the number of individuals sampled influences estimates of the population growth rate (λ) calculated with matrix models. Even unbiased estimates of vital rates do not ensure unbiased estimates of λ–Jensen''s Inequality implies that even when the estimates of the vital rates are accurate, small sample sizes lead to biased estimates of λ due to increased sampling variance. We investigated if sampling variability and the distribution of sampling effort among size classes lead to biases in estimates of λ.

Methodology/Principal Findings

Using data from a long-term field study of plant demography, we simulated the effects of sampling variance by drawing vital rates and calculating λ for increasingly larger populations drawn from a total population of 3842 plants. We then compared these estimates of λ with those based on the entire population and calculated the resulting bias. Finally, we conducted a review of the literature to determine the sample sizes typically used when parameterizing matrix models used to study plant demography.

Conclusions/Significance

We found significant bias at small sample sizes when survival was low (survival = 0.5), and that sampling with a more-realistic inverse J-shaped population structure exacerbated this bias. However our simulations also demonstrate that these biases rapidly become negligible with increasing sample sizes or as survival increases. For many of the sample sizes used in demographic studies, matrix models are probably robust to the biases resulting from sampling variance of vital rates. However, this conclusion may depend on the structure of populations or the distribution of sampling effort in ways that are unexplored. We suggest more intensive sampling of populations when individual survival is low and greater sampling of stages with high elasticities.     

Survival,site fidelity,and the population dynamics of Piping Plovers in Saskatchewan

ABSTRACT To conserve threatened species, managers require predictions about the effects of natural and anthropogenic factors on population growth that in turn require accurate estimates of survival, birth, and dispersal rates, and their correlation with natural and anthropogenic factors. For Piping Plovers (Charadrius melodus), fledging rate is often more amenable to management than adult survival, and population models can be used to estimate the productivity (young produced per breeding female) necessary to maintain or increase populations for given levels of survival. We estimated true survival and site fidelity of adult and subadult (from fledging to second year) Piping Plovers breeding in Saskatchewan using mark‐resight data from 2002 to 2009. By estimating true survival rather than apparent survival (which is confounded with permanent emigration), we were able to provide more accurate projections of population trends. Average adult and subadult survival rates during our study were 0.80 and 0.57, respectively. Adult survival declined over time, possibly due in part to the loss of one breeding site to flooding. Average adult and subadult site fidelity were 0.86 and 0.46, respectively. Adult site fidelity declined during our study at two study sites, most strongly at the flooded site. Male and female Piping Plovers had similar survival rates, but males had greater site fidelity than females in some years. Based on our survival estimates, productivity needed for a stationary population was 0.75, a benchmark used for plover management on the Atlantic Coast, but not previously estimated for Prairie Canada. In stochastic simulations incorporating literature‐based variation in survival rates, productivity needed for a stationary population increased to 0.86, still lower than that previously estimated for western populations. Mean productivity for our study sites ranged from 0.87 to 0.96 fledged young per pair. Our results suggest that fledging rates of Piping Plovers in Saskatchewan were sufficient to ensure a stationary or increasing population during our study period. However, large‐scale habitat changes such as drought or anthropogenic flooding may lead to dispersal of breeding adults and possibly mortality that will increase the fledging rate needed for a stationary population.     

Point and interval estimation of baseline risk and treatment effect based on logistic model for observational studies

In observational studies with dichotomous outcome of a population, researchers usually report treatment effect alone, although both baseline risk and treatment effect are needed to evaluate the significance of the treatment effect to the population. In this article, we study point and interval estimates including confidence region of baseline risk and treatment effect based on logistic model, where baseline risk is the risk of outcome of the population under control treatment while treatment effect is measured by the risk difference between outcomes of the population under active versus control treatments. Using approximate normal distribution of the maximum‐likelihood (ML) estimate of the model parameters, we obtain an approximate joint distribution of the ML estimate of the baseline risk and the treatment effect. Using the approximate joint distribution, we obtain point estimate and confidence region of the baseline risk and the treatment effect as well as point estimate and confidence interval of the treatment effect when the ML estimate of the baseline risk falls into specified range. These interval estimates reflect nonnormality of the joint distribution of the ML estimate of the baseline risk and the treatment effect. The method can be easily implemented by using any software that generates normal distribution. The method can also be used to obtain point and interval estimates of baseline risk and any other measure of treatment effect such as risk ratio and the number needed to treat. The method can also be extended from logistic model to other models such as log‐linear model.     

Estimating Wind Turbine-Caused Bird Mortality

ABSTRACT Mortality estimates are needed of birds and bats killed by wind turbines because wind power generation is rapidly expanding worldwide. A mortality estimate is based on the number of fatalities assumed caused by wind turbines and found during periodic searches, plus the estimated number not found. The 2 most commonly used estimators adjust mortality estimates by rates of searcher detection and scavenger removal of carcasses. However, searcher detection trials can be biased by the species used in the trial, the number volitionally placed for a given fatality search, and the disposition of the carcass on the ground. Scavenger removal trials can be biased by the metric representing removal rate, the number of carcasses placed at once, the duration of the trial, species used, whether carcasses were frozen, whether carcasses included injuries consistent with wind turbine collisions, season, distance from the wind turbines, and general location. I summarized searcher detection rates among reported trials, and I developed models to predict the proportion of carcasses remaining since the last fatality search. The summaries I present can be used to adjust previous and future estimates of mortality to improve comparability. I also identify research directions to better understand these and other adjustments needed to compare mortality estimates among wind farms.     

SURVIVAL RATES OF PHOTOGRAPHICALLY IDENTIFIED HECTOR'S DOLPHINS FROM 1984 TO 1988

Re-sightings of photographically identified individuals were used to estimate survival rates for a free-living population of Hector's dolphins Cephalorhynchus hectori , a species endemic to New Zealand waters. Most individuals were identified from injuries to the dorsal fin. Consequently, the photographic catalog contained very few young individuals. Our analysis included no newborn calves or yearlings, and provided estimates of survival rates only after the first year of life. We used two complementary methods for calculating survival rates: a modified Jolly-Seber model, and a simpler method which corrects in a more explicit way for individual dolphins being alive but not sighted. Selection of the most reliable subset of the data had a greater effect on computed survival rates than did the difference between the two methods. We conclude that careful inspection of resighting data before analysis, and, if necessary, selection of a subset, is very important in studies of this kind. Survival rate estimates came from a population which was subject to relatively heavy mortality from gillnet entanglement. Standard errors of the survival rate estimates have been used to assess the conditional probability of population decline given three fertility scenarios. The high probability that the Banks Peninsula Hector's dolphin population was decreasing during the study period (0.78 to 0.99) suggests that gillnet entanglement constituted a serious risk to this population.     

A population estimate of Heaviside's dolphins, Cephalorhynchus heavisidii, at the southern end of their range

Heaviside's dolphins, Cephalorhynchus heavisidii , are endemic to southwestern Africa, where they are exposed to unknown levels of anthropogenic threats, including inshore set netting. Using photo-ID data collected over 3 yr on the west coast of South Africa, we calculated Chapman's-modified Petersen estimates of the number of distinctive individuals at three spatial scales. Sample sizes were small and recapture rates low resulting in high variance. Total population abundance was extrapolated from the proportion of well-marked animals in the population (14%–17%) with between-year estimates adjusted for mortality using data from Commerson's dolphin. The total population size was calculated as 527 animals (CV = 0.35, CI 272–1,020) in the 1999 study area (20 km of coastline, within season), 3,429 animals (CV = 0.36, CI 1,721–6,828) in the central study area (150 km of coastline, 3 yr), and 6,345 animals (CV = 0.26, CI 3,573–11,267) in the full study area (390 km of coastline, 2 yr). Dolphins fitted with satellite transmitters varied in their use of the inshore photographic study area from 39.5% to 94.7% of transmission days (38–51 total). Given the known or suspected biases in the data, these abundance estimates are likely to be biased downward.     

Geographic variation in the life history of the sagebrush lizard: the role of thermal constraints on activity

Thermal constraints on the time available for activity have been proposed as a proximate mechanism to explain variation in suites of life history traits. The longer that an ectotherm can maintain activity, the more time it has to forage and the greater chance that it will encounter a predator and be eaten. Thus, the thermal environment may produce a trade off between growth and survival when variation in the environment favors increased activity. I used mark-recapture data from a demographic study of three natural populations of the sagebrush lizard (Sceloporus graciosus) and estimates of thermal opportunity for each population to evaluate whether variation in the thermal environment can explain patterns of growth and survival that occur over an elevational gradient. Lizards from the highest elevation population exhibited higher individual growth rates than those of lizards from lower elevation, while mortality rates increased with elevation for these populations. The covariation of fast growth and high mortality with increased thermal opportunity is the opposite trend expected if the thermal environment alone is to explain patterns of life history in these lizards. Additional factors including thermal heterogeneity in the distribution of microhabitats of lizards, adaptation to local environmental conditions, and a potential trade-off between resource acquisition and predation risk need to be addressed to obtain a satisfactory explanation of the causative mechanisms producing life history variation.     

Simulation of the population dynamics and social structure of the Virunga mountain gorillas

An agent-based model was developed to simulate the growth rate, age structure, and social system of the endangered mountain gorillas (Gorilla beringei beringei) in the Virunga Volcanoes region. The model was used to compare two types of data: 1) estimates of the overall population size, age structure, and social structure, as measured by six censuses of the entire region that were conducted in 1971-2000; and 2) information about birth rates, mortality rates, dispersal patterns, and other life history events, as measured from three to five habituated research groups since 1967. On the basis of the research-group data, the "base simulation" predicted a higher growth rate than that observed from the census data (3% vs. 1%). This was as expected, because the research groups have indeed grown faster than the overall population. Additional simulations suggested that the research groups primarily have a lower mortality rate, rather than higher birth rates, compared to the overall population. Predictions from the base simulation generally fell within the range of census values for the average group size, the percentage of multimale groups, and the distribution of females among groups. However, other discrepancies predicted from the research-group data were a higher percentage of adult males than observed, an overestimation of the number of multimale groups with more than two silverbacks, and an overestimated number of groups with only two or three members. Possible causes for such discrepancies include inaccuracies in the census techniques used, and/or limitations with the long-term demographic data set obtained from only a few research groups of a long-lived species. In particular, estimates of mortality and male dispersal obtained from the research groups may not be representative of the entire population. Our final simulation addressed these discrepancies, and provided a better basis for further studies on the complex relationships among individual life history events, group composition, population age structure, and growth rate patterns.     

Estimating Survival for Elusive Juvenile Pond-Breeding Salamanders

Juvenile vital rates have important effects on population dynamics for many species, but this demographic is often difficult to locate and track. As such, we frequently lack reliable estimates of juvenile survival, which are necessary for accurately assessing population stability and potential management approaches to conserve biodiversity. We estimated survival rates for elusive juveniles of 3 species, the ringed salamander (Ambystoma annulatum), spotted salamander (A. maculatum), and small-mouthed salamander (A. texanum), using 2 approaches. First, we conducted an 11-month (2016–2017) mark-recapture study within semi-natural enclosures and used Bayesian Cormack-Jolly-Seber models to estimate survival and recapture probabilities. Second, we inferred the expected annual juvenile survival rate given published vital rates for pre-metamorphic and adult ambystomatids assuming stable population growth. For all 3 species, juvenile survival probabilities were constant across recapture occasions, whereas recapture probability estimates were time-dependent. Further, survival and recapture probabilities among study species were similar. Post-study sampling revealed that the initial study period median estimate of annual survival probability (0.39) underestimated the number of salamanders known alive at 11 months. We therefore appended approximately 1 year of opportunistic data, which produced a median annual survival probability of 0.50, encompassing salamanders that we knew to have been alive. Calculation from literature values suggested a mean annual terrestrial juvenile ambystomatid survival probability of 0.49. Similar results among our approaches indicated that juvenile survival estimates for the study species were robust and likely comparable to rates in nature. These estimates can now be confidently applied to research, monitoring, and management efforts for the study species and ecologically similar taxa. Our findings indicated that similarly robust vital rate estimates for subsets of ecologically and phylogenetically similar species can provide reasonable surrogate demographic information that can be used to reveal key factors influencing population viability for data-deficient species. © 2020 The Wildlife Society.     

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.     

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.     

Density-Dependent Multiplication and Survival Rates in Heterodera glycines

Seasonal multiplication and overwinter survival are density-dependent in Heterodera glycines. At low to moderate population densities, the nematode is capable of large population increases on susceptible soybean cultivars and high rates of oversummer or overwinter survival in the absence of a host. To improve estimates of H. glycines multiplication and survival rates, egg densities were monitored for 12 cropping sequences across 10 years. Log-linear regression analysis was used to describe and compare density-dependent relationships. Growing-season change in H. glycines egg densities was density-dependent for all crops (susceptible soybean, resistant soybean, and nonhost), with slope estimates for the density-dependent relationship greater for susceptible soybean compared with a non-host crop. Overwinter population change also was density-dependent, with similar declines in survival rates observed for all crops as population densities increased. Survival was greater following susceptible soybean compared with resistant soybean, with an intermediate rate of survival associated with non-host crops. Survival estimates greater than 100% frequently were obtained at low population densities, despite attempts to account for sampling error. Rates of growing-season multiplication and survival, when standardized for population density, declined with year of the study. Standardized overwinter survival rates were inversely related to average daily minimum temperature and monthly snow cover.     

Density‐dependent regulation of the critically endangered black rhinoceros population in Ithala Game Reserve,South Africa

Ensuring the persistence of populations of endangered species requires an understanding of, and response to, the causes of population declines. Species occurring in small populations are vulnerable to stochastic problems that are environmental, demographic, or genetic in nature and can reduce survival as much as the deterministic threats of habitat degradation. Critically endangered black rhinoceros (Diceros bicornis) populations declined throughout Africa since 1960, with numbers steadily increasing at a continental level, but remaining lower than three generations ago. However, size, demographics, trends, and factors affecting these, are poorly known. We used 18 years (1990–2008) of long‐term sightings data from Ithala Game Reserve, KwaZulu‐Natal, South Africa, to determine population estimates, growth rate and fecundity over time, as well as sex and age structure and age‐specific probabilities of survival. Calf survivorship between the ages of 0 and 1 year was 74% for females and 94% for males. Age‐specific survivorship for both sexes was significantly higher from yearling to subadult age‐classes (1–6 years) than for adults (7–30 years). The most frequent cause of mortality was attributed to unknown causes while fighting injuries was recorded as the second most common cause of mortality, particularly among subadult and adult males. There was no significant difference in the sex ratios at birth, although the proportion of females in the population was 0.58. There was strong evidence for density‐dependent regulation, with density in conception year a key driver of population performance (birth rate). The population does not appear to be at ecological carrying capacity; however, social effects are delaying conception. To mitigate density‐dependent social effects, we recommend an adaptive management strategy of pre‐selecting individuals for removal from the reserve, so as to maintain stability in the social organization of the population.     

Population dynamics of an expanding passerine at the distribution margin

Individuals may be maladapted to novel environments at the species’ distribution margin. We investigated population dynamics in a marginal habitat where reproduction has been proven poor. Survival, population growth rate (λ) and its components, breeding and natal dispersal were studied in great tits Parus major breeding at the northern margin of its distribution in northern Finland. We used long term capture–mark–recapture data sets. Study area size and population density were used to explain adult survival rates. The average annual estimates of adult survival rose from 0.371 to 0.388 between the periods of 1971–1984 and 1999–2009. The estimates are slightly lower than estimates of small passerines in Europe. Low local survival rate of fledglings (0.050–0.055) probably reflects intensified emigration from this low quality area. Temporal variation in λ was large (0.498–1.856). Despite of low adult survival and recruitment rates, the mean estimates of λ (1.008 and 1.033) indicate an overall stability in the population size. Indeed, our results suggest that the immigration has an important role in the population dynamics of northern great tits. Thus the population is demographically and genetically dependent on core habitats which may cause adaptive problems due to intensive gene flow. Given those limitations, options for evolution of local adaptations in northern distribution margins are discussed.     

Adjusting for the proportion of cancer deaths in the general population when using relative survival: a sensitivity analysis

Background: Relative survival is an extensively used method in population based cancer studies as it provides a measure of survival without the need for accurate cause of death information. It gives an estimate for the probability of dying from cancer in the absence of other causes by estimating the excess mortality in the study population when compared to an external group. The external group is usually the general population within a country or state and mortality estimates are taken from national life tables that are broken down by age, sex, calendar year and, where applicable, race/ethnicity. One potential bias when using relative survival that is most often overlooked occurs when there are a high proportion of deaths due to a specific cancer in the external group. Methods: This paper uses data from the Finnish Cancer Registry to illustrate, through the use of a simple sensitivity analysis, the impact that specific cancer deaths in the population mortality figures can have on the estimate of relative survival. Results: We found that when examining specific diseases such as breast cancer and colon cancer, the proportion of deaths due to these specific cancers in the general population is so small in comparison to the total mortality that they make little difference to the relative survival estimates. However, prostate cancer proved to be an exception to this. For all cancer sites combined the sensitivity analysis illustrates a major limitation for this type of analysis, particularly with the older age groups. Conclusion: We recommend that, with a classification of diseases as wide as all cancer sites, relative survival should not be used without appropriate adjustment.