Variance of Stratified Survey Estimators With Probability of Detection Adjustments

Abstract: Estimates of wildlife population sizes are frequently constructed by combining counts of observed animals from a stratified survey of aerial sampling units with an estimated probability of detecting animals. Unlike traditional stratified survey designs, stratum-specific estimates of population size will be correlated if a common detection model is used to adjust counts for undetected animals in all strata. We illustrate this concept in the context of aerial surveys, considering 2 cases: 1) a single-detection parameter is estimated under the assumption of constant detection probabilities, and 2) a logistic-regression model is used to estimate heterogeneous detection probabilities. Naïve estimates of variance formed by summing stratum-specific estimates of variance may result in significant bias, particularly if there are a large number of strata, if detection probabilities are small, or if estimates of detection probabilities are imprecise. (JOURNAL OF WILDLIFE MANAGEMENT 72(3):837–844; 2008)     

Comparison of confidence intervals for adjusted attributable risk estimates under multinomial sampling

The epidemiologic concept of the adjusted attributable risk is a useful approach to quantitatively describe the importance of risk factors on the population level. It measures the proportional reduction in disease probability when a risk factor is eliminated from the population, accounting for effects of confounding and effect-modification by nuisance variables. The computation of asymptotic variance estimates for estimates of the adjusted attributable risk is often done by applying the delta method. Investigations on the delta method have shown, however, that the delta method generally tends to underestimate the standard error, leading to biased confidence intervals. We compare confidence intervals for the adjusted attributable risk derived by applying computer intensive methods like the bootstrap or jackknife to confidence intervals based on asymptotic variance estimates using an extensive Monte Carlo simulation and within a real data example from a cohort study in cardiovascular disease epidemiology. Our results show that confidence intervals based on bootstrap and jackknife methods outperform intervals based on asymptotic theory. Best variants of computer intensive confidence intervals are indicated for different situations.     

Background stratified Poisson regression analysis of cohort data

Background stratified Poisson regression is an approach that has been used in the analysis of data derived from a variety of epidemiologically important studies of radiation-exposed populations, including uranium miners, nuclear industry workers, and atomic bomb survivors. We describe a novel approach to fit Poisson regression models that adjust for a set of covariates through background stratification while directly estimating the radiation-disease association of primary interest. The approach makes use of an expression for the Poisson likelihood that treats the coefficients for stratum-specific indicator variables as ‘nuisance’ variables and avoids the need to explicitly estimate the coefficients for these stratum-specific parameters. Log-linear models, as well as other general relative rate models, are accommodated. This approach is illustrated using data from the Life Span Study of Japanese atomic bomb survivors and data from a study of underground uranium miners. The point estimate and confidence interval obtained from this ‘conditional’ regression approach are identical to the values obtained using unconditional Poisson regression with model terms for each background stratum. Moreover, it is shown that the proposed approach allows estimation of background stratified Poisson regression models of non-standard form, such as models that parameterize latency effects, as well as regression models in which the number of strata is large, thereby overcoming the limitations of previously available statistical software for fitting background stratified Poisson regression models.     

Variance calculations and confidence intervals for estimates of the attributable risk based on logistic models

The attributable risk (AR), defined as AR = [Pr(disease) - Pr(disease/no exposure)]/Pr(disease), measures the proportion of disease risk that is attributable to an exposure. Recently Bruzzi et al. (1985, American Journal of Epidemiology 122, 904-914) presented point estimates of AR based on logistic models for case-control data to allow for confounding factors and secondary exposures. To produce confidence intervals, we derived variance estimates for AR under the logistic model and for various designs for sampling controls. Calculations for discrete exposure and confounding factors require covariances between estimates of the risk parameters of the logistic model and the proportions of cases with given levels of exposure and confounding factors. These covariances are estimated from Taylor series expansions applied to implicit functions. Similar calculations for continuous exposures are derived using influence functions. Simulations indicate that those asymptotic procedures yield reliable variance estimates and confidence intervals with near nominal coverage. An example illustrates the usefulness of variance calculations in selecting a logistic model that is neither so simplified as to exhibit systematic lack of fit nor so complicated as to inflate the variance of the estimate of AR.     

A method for analysing case-control studies with ordinal exposure variables

A method is proposed for analysing case-control studies with ordinal or continuous, but unruly, exposure levels. A test is proposed which is an appropriate linear combination of stratum-specific Wilcoxon tests. An estimator for the mean percentile shift between the case and control exposure levels is given. Methods for assessing heterogeneity of the mean percentile shift across strata are also discussed. The performance of the test as a function of the number of controls per case is studied for both local and distant alternatives. The asymptotic relative efficiency compared to the best parametric test with the same number of controls is evaluated and its performance is compared to tests based on a dichotomized exposure variable. Finally, the method is illustrated by two numerical examples.     

Indoor tanning and skin cancer in Canada: A meta-analysis and attributable burden estimation

BackgroundConsistent epidemiologic and experimental studies have demonstrated that UV-emitting tanning devices cause melanoma and non-melanoma skin cancer. The purpose of this study was to estimate the relative risk of skin cancer associated with the use of indoor tanning devices relevant to Canada, to estimate the proportion and number of skin cancers in Canada in 2015 that were attributable to indoor tanning, and to explore differences by age and sex.MethodsSkin cancer cases attributable to the use of an indoor tanning devices were estimated using Levin’s population attributable risk (PAR) formula. Relative risks for skin cancer subtypes that were relevant to Canada were estimated through meta-analyses and prevalence of indoor tanning was estimated from the 2006 National Sun Survey. Age- and sex-specific melanoma data for 2015 were obtained from the Canadian Cancer Registry, while estimated NMSC incidence data were obtained from the 2015 Canadian Cancer Statistics report.ResultsEver use of indoor tanning devices was associated with relative risks of 1.38 (95% CI 1.22–1.58) for melanoma, 1.39 (1.10–1.76) for basal cell carcinoma (BCC), and 1.49 (1.23–1.80) for squamous cell carcinoma (SCC). Overall, 7.0% of melanomas, 5.2% of BCCs, and 7.5% of SCCs in 2015 were attributable to ever of indoor tanning devices. PARs were higher for women and decreased with age.ConclusionIndoor tanning contributes to a considerable burden of skin cancer in Canada. Strategies aimed at reducing use should be increased and a total ban or restrictions on use and UV-intensity should be considered by health regulators.     

Estimation of Attributable Risk in the Presence of Confounding

LEVIN 's attributable risk due to a risk factor is estimated in the presence of confounding in the retrospective study context. The crude attributable risk is decomposed into the attributable risk due to a risk factor and a bias factor which is shown to vanish only when the confounding factor is unassociated to either the risk factor or the Disease or to both. Asymptotic tests are provided for testing interaction, and an appropriate pooled estimate is derived when constancy of partial attributable risks prevails over the levels of the confounding factor.     

Weight calibration to improve the efficiency of pure risk estimates from case-control samples nested in a cohort

Cohort studies provide information on relative hazards and pure risks of disease. For rare outcomes, large cohorts are needed to have sufficient numbers of events, making it costly to obtain covariate information on all cohort members. We focus on nested case-control designs that are used to estimate relative hazard in the Cox regression model. In 1997, Langholz and Borgan showed that pure risk can also be estimated from nested case-control data. However, these approaches do not take advantage of some covariates that may be available on all cohort members. Researchers have used weight calibration to increase the efficiency of relative hazard estimates from case-cohort studies and nested cased-control studies. Our objective is to extend weight calibration approaches to nested case-control designs to improve precision of estimates of relative hazards and pure risks. We show that calibrating sample weights additionally against follow-up times multiplied by relative hazards during the risk projection period improves estimates of pure risk. Efficiency improvements for relative hazards for variables that are available on the entire cohort also contribute to improved efficiency for pure risks. We develop explicit variance formulas for the weight-calibrated estimates. Simulations show how much precision is improved by calibration and confirm the validity of inference based on asymptotic normality. Examples are provided using data from the American Association of Retired Persons Diet and Health Cohort Study.     

Hierarchical Modeling for Estimating Relative Risks of Rare Genetic Variants: Properties of the Pseudo‐Likelihood Method

Summary Many major genes have been identified that strongly influence the risk of cancer. However, there are typically many different mutations that can occur in the gene, each of which may or may not confer increased risk. It is critical to identify which specific mutations are harmful, and which ones are harmless, so that individuals who learn from genetic testing that they have a mutation can be appropriately counseled. This is a challenging task, since new mutations are continually being identified, and there is typically relatively little evidence available about each individual mutation. In an earlier article, we employed hierarchical modeling ( Capanu et al., 2008 , Statistics in Medicine 27 , 1973–1992) using the pseudo‐likelihood and Gibbs sampling methods to estimate the relative risks of individual rare variants using data from a case–control study and showed that one can draw strength from the aggregating power of hierarchical models to distinguish the variants that contribute to cancer risk. However, further research is needed to validate the application of asymptotic methods to such sparse data. In this article, we use simulations to study in detail the properties of the pseudo‐likelihood method for this purpose. We also explore two alternative approaches: pseudo‐likelihood with correction for the variance component estimate as proposed by Lin and Breslow (1996, Journal of the American Statistical Association 91 , 1007–1016) and a hybrid pseudo‐likelihood approach with Bayesian estimation of the variance component. We investigate the validity of these hierarchical modeling techniques by looking at the bias and coverage properties of the estimators as well as at the efficiency of the hierarchical modeling estimates relative to that of the maximum likelihood estimates. The results indicate that the estimates of the relative risks of very sparse variants have small bias, and that the estimated 95% confidence intervals are typically anti‐conservative, though the actual coverage rates are generally above 90%. The widths of the confidence intervals narrow as the residual variance in the second‐stage model is reduced. The results also show that the hierarchical modeling estimates have shorter confidence intervals relative to estimates obtained from conventional logistic regression, and that these relative improvements increase as the variants become more rare.     

Attributable risk estimation for adjusted disability multistate models: Application to nosocomial infections

Attributable risk has become an important concept in clinical epidemiology. In this paper, we suggest to estimate the attributable risk of nosocomial infections using a multistate approach. Recently, a multistate model (called progressive disability model in the literature) has been developed in order to take into consideration both the time‐dependency of the risk factor (e.g., nosocomial infections) and the presence of competing risks (e.g., death and discharge) at each time point. However, this approach does not take into account the possible heterogeneity of the study population. In this paper, we investigate an extension of this model and suggest an adjusted disability multistate model including covariates in each transition. This new multistate model has led us to define the concepts of overall and profiled attributable risk. We use a classical semiparametric approach to estimate the model and the new attributable risk. A simulation study is investigated and we show, in particular, that neglecting the presence of covariates when estimating the model can lead to an important bias. The methodology developed in this paper is applied to data on ventilator‐associated pneumonia in 12 French intensive care units.     

On pooling across strata when frequency matching has been followed in a cohort study

In a study designed to assess the relationship between a dichotomous exposure and the eventual occurrence of a dichotomous outcome, frequency matching has been proposed as a way to balance the exposure cohorts with respect to the sampling distribution of potential confounding factors. This paper discusses the pooled estimator for the log relative risk, and provides an estimator for its variance which takes into account the dependency in the pooled outcomes induced by frequency matching. The pooled estimator has asymptotic relative efficiency less than but close to 1, relative to the usual, inverse variance weighted, stratified estimator. Simulations suggest, however, that the pooled estimator is likely to outperform the stratified estimator when samples are of moderate size. This estimator carries the added advantage that it consistently estimates a meaningful population parameter under heterogeneity of the relative risk across strata.     

Attributable risk estimation from matched case-control data

A methodology is proposed for obtaining summary estimators, variances, and confidence intervals for attributable risk measures from data obtained through a case-control study design where one or more controls have been matched to each case. The sampling design for obtaining these data is conceptualized as a simple random sample of cases being equivalent to a simple random sample of matched sets. By combining information across the strata determined by the matched sets, this approach provides all of the benefits associated with the Mantel-Haenszel procedure for the estimators of attributable risk among the exposed and population attributable risk. Asymptotic variances are derived under the assumption that the frequencies of the unique response patterns follow the multinomial distribution. Simulation results indicate that these methods fare very well with respect to bias and coverage probability.     

Estimating the global distribution and disease burden of intestinal nematode infections: Adding up the numbers - A review

Intestinal nematode infections are among the most common infections of humans in developing countries, but precise estimates of the populations at risk of infection, morbidity and mortality are difficult to derive. Careful evaluation of the global distribution and disease burden of nematodes is essential to determine the cost-effectiveness of control and ensure that control programmes are focused appropriately. In turn, understanding the disease burden depends on a summary measure of health as well as reliable data on risks of infection, morbidity and mortality. This review provides an overview of data sources and methods adopted in the Global Burden of Disease study to estimate the burden of intestinal nematodes, including the empirical and modelling challenges in its estimation. Particular attention is paid to efforts to improve our ability to define at-risk populations, based on a Global Atlas of Helminth Infection, and to better estimate attributable morbidity.     

Competing risks regression for stratified data

For competing risks data, the Fine-Gray proportional hazards model for subdistribution has gained popularity for its convenience in directly assessing the effect of covariates on the cumulative incidence function. However, in many important applications, proportional hazards may not be satisfied, including multicenter clinical trials, where the baseline subdistribution hazards may not be common due to varying patient populations. In this article, we consider a stratified competing risks regression, to allow the baseline hazard to vary across levels of the stratification covariate. According to the relative size of the number of strata and strata sizes, two stratification regimes are considered. Using partial likelihood and weighting techniques, we obtain consistent estimators of regression parameters. The corresponding asymptotic properties and resulting inferences are provided for the two regimes separately. Data from a breast cancer clinical trial and from a bone marrow transplantation registry illustrate the potential utility of the stratified Fine-Gray model.     

Multiple Imputation Methods for Estimating Regression Coefficients in the Competing Risks Model with Missing Cause of Failure

We propose a method to estimate the regression coefficients in a competing risks model where the cause-specific hazard for the cause of interest is related to covariates through a proportional hazards relationship and when cause of failure is missing for some individuals. We use multiple imputation procedures to impute missing cause of failure, where the probability that a missing cause is the cause of interest may depend on auxiliary covariates, and combine the maximum partial likelihood estimators computed from several imputed data sets into an estimator that is consistent and asymptotically normal. A consistent estimator for the asymptotic variance is also derived. Simulation results suggest the relevance of the theory in finite samples. Results are also illustrated with data from a breast cancer study.     

Genetic Factors Are Not the Major Causes of Chronic Diseases

The risk of acquiring a chronic disease is influenced by a person’s genetics (G) and exposures received during life (the ‘exposome’, E) plus their interactions (G×E). Yet, investigators use genome-wide association studies (GWAS) to characterize G while relying on self-reported information to classify E. If E and G×E dominate disease risks, this imbalance obscures important causal factors. To estimate proportions of disease risk attributable to G (plus shared exposures), published data from Western European monozygotic (MZ) twins were used to estimate population attributable fractions (PAFs) for 28 chronic diseases. Genetic PAFs ranged from 3.4% for leukemia to 48.6% for asthma with a median value of 18.5%. Cancers had the lowest PAFs (median = 8.26%) while neurological (median = 26.1%) and lung (median = 33.6%) diseases had the highest PAFs. These PAFs were then linked with Western European mortality statistics to estimate deaths attributable to G for heart disease and nine cancer types. Of 1.53 million Western European deaths in 2000, 0.25 million (16.4%) could be attributed to genetics plus shared exposures. Given the modest influences of G-related factors on the risks of chronic diseases in MZ twins, the disparity in coverage of G and E in etiological research is problematic. To discover causes of disease, GWAS should be complemented with exposome-wide association studies (EWAS) that profile chemicals in biospecimens from incident disease cases and matched controls.     

Genetic steady-state under BLUP selection for an infinite and homogeneous population with discrete generations

A matrix derivation is proposed to analytically calculate the asymptotic genetic variance-covariance matrix under BLUP selection according to the initial genetic parameters in a large population with discrete generations. The asymptotic genetic evolution of a homogeneous population with discrete generations is calculated for a selection operating on an index including all information (pedigree and records) from a non-inbred and unselected base population (BLUP selection) or on an index restricted to records of a few ancestral generations. Under the first hypothesis, the prediction error variance of the selection index is independent of selection and is calculated from the genetic parameters of the base population. Under the second hypothesis, the prediction error variance depends on selection. Furthermore, records of several generations of ancestors of the candidates for selection must be used to maintain a constant prediction error variance over time. The number of ancestral generations needed depends on the population structure and on the occurrence of fixed effects. Without fixed effects to estimate, accounting for two generations of ancestors is sufficient to estimate the asymptotic prediction error variance. The amassing of information from an unselected base population proves to be important in order not to overestimate the asymptotic genetic gains and not to underestimate the asymptotic genetic variances.     

Genetic effects on variation in red-blood-cell folate in adults: implications for the familial aggregation of neural tube defects.

Recent studies have implicated folic acid as an important determinant of normal human growth, development, and function. Insufficient folate levels appear to be a risk factor for neural tube defects (NTD), as well as for several chronic diseases of adulthood. However, relatively little is known about the factors that influence folate status in the general population. To estimate the relative contribution of genetic and nongenetic factors to variation in folate, we have evaluated red blood cell (RBC) folate levels in 440 pairs of MZ twins and in 331 pairs of DZ twins. The data were best described by a model in which 46% of the variance in RBC folate was attributable to additive genetic effects, 16% of the variance was due to measured phenotypic covariates, and 38% of the variance was due to random environmental effects. Moreover, the correlations for RBC folate in MZ co-twins (r = .46) and in repeat measures from the same individual (r = .51) were very similar, indicating that virtually all repeatable variation in RBC folate is attributable to genetic factors. On the basis of these results, it would seem reasonable to initiate a search for the specific genes that influence RBC folate levels in the general population. Such genes ultimately may be used to identify individuals at increased risk for NTD and other folate-related diseases.     

Multiple outputation: inference for complex clustered data by averaging analyses from independent data

This article applies a simple method for settings where one has clustered data, but statistical methods are only available for independent data. We assume the statistical method provides us with a normally distributed estimate, theta, and an estimate of its variance sigma. We randomly select a data point from each cluster and apply our statistical method to this independent data. We repeat this multiple times, and use the average of the associated theta's as our estimate. An estimate of the variance is given by the average of the sigma2's minus the sample variance of the theta's. We call this procedure multiple outputation, as all "excess" data within each cluster is thrown out multiple times. Hoffman, Sen, and Weinberg (2001, Biometrika 88, 1121-1134) introduced this approach for generalized linear models when the cluster size is related to outcome. In this article, we demonstrate the broad applicability of the approach. Applications to angular data, p-values, vector parameters, Bayesian inference, genetics data, and random cluster sizes are discussed. In addition, asymptotic normality of estimates based on all possible outputations, as well as a finite number of outputations, is proven given weak conditions. Multiple outputation provides a simple and broadly applicable method for analyzing clustered data. It is especially suited to settings where methods for clustered data are impractical, but can also be applied generally as a quick and simple tool.