Estimation of attributable risk for case-control studies with multiple matching

Kuritz and Landis considered case-control studies with multiple matching and proposed an asymptotic interval estimator of the attributable risk based on Wald's statistic. Using Monte Carlo simulation, Kuritz and Landis demonstrated that their interval estimator could perform well when the number of matched sets was large (>or=100). However, the number of matched sets may often be moderate or small in practice. In this paper, we evaluate the performance of Kuritz and Landis' interval estimator in small or moderate number of matched sets and compare it with four other interval estimators. We note that the coverage probability of Kuritz and Landis' interval estimator tends to be less than the desired confidence level when the probability of exposure among cases is large. In these cases, the interval estimator using the logarithmic transformation and the two interval estimators derived from the quadratic equations developed here can generally improve the coverage probability of Kuritz and Landis' interval estimator, especially for the case of a small number of matched sets. Furthermore, we find that an interval estimator derived from a quadratic equation is consistently more efficient than Kuritz and Landis' interval estimator. The interval estimator using the logit transformation, although which performs poorly when the underlying odds ratio (OR) is close to 1, can be useful when both the probability of exposure among cases and the underlying OR are moderate or large.     

Interval estimation of the attributable risk in case-control studies with matched pairs

OBJECTIVE: The attributable risk (AR), which represents the proportion of cases who can be preventable when we completely eliminate a risk factor in a population, is the most commonly used epidemiological index to assess the impact of controlling a selected risk factor on community health. The goal of this paper is to develop and search for good interval estimators of the AR for case-control studies with matched pairs. METHODS: This paper considers five asymptotic interval estimators of the AR, including the interval estimator using Wald's statistic suggested elsewhere, the two interval estimators using the logarithmic transformations: log(x) and log(1-x), the interval estimator using the logit transformation log(x/(1-x)), and the interval estimator derived from a simple quadratic equation developed in this paper. This paper compares the finite sample performance of these five interval estimators by calculation of their coverage probability and average length in a variety of situations. RESULTS: This paper demonstrates that the interval estimator derived from the quadratic equation proposed here can not only consistently perform well with respect to the coverage probability, but also be more efficient than the interval estimator using Wald's statistic in almost all the situations considered here. This paper notes that although the interval estimator using the logarithmic transformation log(1-x) may also perform well with respect to the coverage probability, using this estimator is likely to be less efficient than the interval estimator using Wald's statistic. Finally, this paper notes that when both the underlying odds ratio (OR) and the prevalence of exposure (PE) in the case group are not large (OR < or =2 and PE < or =0.10), the application of the two interval estimators using the transformations log(x) and log(x/(1-x)) can be misleading. However, when both the underlying OR and PE in the case group are large (OR > or =4 and PE > or =0.50), the interval estimator using the logit transformation can actually outperform all the other estimators considered here in terms of efficiency. CONCLUSIONS: When there is no prior knowledge of the possible range for the underlying OR and PE, the interval estimator derived from the quadratic equation developed here for general use is recommended. When it is known that both the OR and PE in the case group are large (OR > or =4 and PE > or =0.50), it is recommended that the interval estimator using the logit transformation is used.     

Interval estimation of the proportion ratio under multiple matching

The discussions on interval estimation of the proportion ratio (PR) of responses or the relative risk (RR) of a disease for multiple matching have been generally focused on the odds ratio (OR) based on the assumption that the latter can approximate the former well. When the underlying proportion of outcomes is not rare, however, the results for the OR would be inadequate for use if the PR or RR was the parameter of our interest. In this paper, we develop five asymptotic interval estimators of the common PR (or RR) for multiple matching. To evaluate and compare the finite sample performance of these estimators, we apply Monte Carlo simulation to calculate the coverage probability and the average length of the resulting confidence intervals in a variety of situations. We note that when we have a constant number of matching, the interval estimator using the logarithmic transformation of the Mantel-Haenszel estimator, the interval estimator derived from the quadratic inequality given in this paper, and the interval estimator using the logarithmic transformation of the ratio estimator can consistently perform well. When the number of matching varies between matched sets, we find that the interval estimator using the logarithmic transformation of the ratio estimator is probably the best among the five interval estimators considered here in the case of a small number (=20) of matched sets. To illustrate the use of these interval estimators, we employ the data studying the supplemental ascorbate in the supportive treatment of terminal cancer patients.     

Bayesian methods of confidence interval construction for the population attributable risk from cross‐sectional studies

Population attributable risk measures the public health impact of the removal of a risk factor. To apply this concept to epidemiological data, the calculation of a confidence interval to quantify the uncertainty in the estimate is desirable. However, because perhaps of the confusion surrounding the attributable risk measures, there is no standard confidence interval or variance formula given in the literature. In this paper, we implement a fully Bayesian approach to confidence interval construction of the population attributable risk for cross‐sectional studies. We show that, in comparison with a number of standard Frequentist methods for constructing confidence intervals (i.e. delta, jackknife and bootstrap methods), the Bayesian approach is superior in terms of percent coverage in all except a few cases. This paper also explores the effect of the chosen prior on the coverage and provides alternatives for particular situations. Copyright © 2016 John Wiley & Sons, Ltd.     

Estimating risk and rate levels,ratios and differences in case-control studies

Classic (or 'cumulative') case-control sampling designs do not admit inferences about quantities of interest other than risk ratios, and then only by making the rare events assumption. Probabilities, risk differences and other quantities cannot be computed without knowledge of the population incidence fraction. Similarly, density (or 'risk set') case-control sampling designs do not allow inferences about quantities other than the rate ratio. Rates, rate differences, cumulative rates, risks, and other quantities cannot be estimated unless auxiliary information about the underlying cohort such as the number of controls in each full risk set is available. Most scholars who have considered the issue recommend reporting more than just risk and rate ratios, but auxiliary population information needed to do this is not usually available. We address this problem by developing methods that allow valid inferences about all relevant quantities of interest from either type of case-control study when completely ignorant of or only partially knowledgeable about relevant auxiliary population information.     

Interval estimation of the mean response in a log-regression model

A standard approach to the analysis of skewed response data with concomitant information is to use a log-transformation to normalize the distribution of the response variable and then conduct a log- regression analysis. However, the mean response at original scale is often of interest. El-Shaarawi and Viveros developed an interval estimation of the mean response of a log-regression model based on large sample theory. There is however very little information available in the literature on constructing such estimates when the sample size is small. In this paper, we develop a small-sample corrected interval by using the likelihood-based inference method developed by Barndorff-Nielson and Fraser et al. Simulation results show that the proposed interval provides almost exact coverage probability, even for small samples.     

The effects of joint misclassification of exposure and disease on the attributable risk

While there is extensive methodological literature analysing the effects of misclassification on the relative risk under various misclassification scenarios, for the attributable risk only the effects of non-differential misclassification either of exposure or disease, and the effects of non-differential independent misclassification of exposure and disease have been discussed for the 2 x 2-situation. The paper investigates the effects of non-differential correlated misclassification of exposure and disease on the attributable risk taking possible correlations of both types of misclassification into account. Furthermore, a comparison with the corresponding effects on the relative risk is drawn. We propose a matrix-based approach to describe the underlying structure of non-differential misclassification. The bias arising from non-differential misclassification in the attributable risk and relative risk is evaluated in four examples assuming under- or overreporting of exposure and disease. In each of the four examples we found scenarios where pronounced differences in degree and, more importantly, in direction of bias occurred. Our results clearly demonstrate the danger lying in the stereotype transfer of findings regarding misclassification effects on the relative risk to other epidemiologic risk measures and underline the necessity of specific analyses of the effects of misclassification on the attributable risk.     

Systematically missing confounders in individual participant data meta‐analysis of observational cohort studies

One difficulty in performing meta‐analyses of observational cohort studies is that the availability of confounders may vary between cohorts, so that some cohorts provide fully adjusted analyses while others only provide partially adjusted analyses. Commonly, analyses of the association between an exposure and disease either are restricted to cohorts with full confounder information, or use all cohorts but do not fully adjust for confounding. We propose using a bivariate random‐effects meta‐analysis model to use information from all available cohorts while still adjusting for all the potential confounders. Our method uses both the fully adjusted and the partially adjusted estimated effects in the cohorts with full confounder information, together with an estimate of their within‐cohort correlation. The method is applied to estimate the association between fibrinogen level and coronary heart disease incidence using data from 154 012 participants in 31 cohorts.^? One hundred and ninety‐nine participants from the original 154 211 withdrew their consent and have been removed from this analysis. Copyright © 2009 John Wiley & Sons, Ltd.     

Interval estimation of the risk difference in non-compliance randomized trials with repeated binary measurements

In a randomized clinical trial (RCT), we often encounter non-compliance with the treatment protocol for a subset of patients. The intention-to-treat (ITT) analysis is probably the most commonly used method in a RCT with non-compliance. However, the ITT analysis estimates 'the programmatic effectiveness' rather than 'the biological efficacy'. In this paper, we focus attention on the latter index and consider use of the risk difference (RD) to measure the effect of a treatment. Based on a simple additive risk model proposed elsewhere, we develop four asymptotic interval estimators of the RD for repeated binary measurements in a RCT with non-compliance. We apply Monte Carlo simulation to evaluate and compare the finite-sample performance of these interval estimators in a variety of situations. We find that all interval estimators considered here can perform well with respect to the coverage probability. We further find that the interval estimator using a tanh(-1)(x) transformation is probably more precise than the others, while the interval estimator derived from a randomization-based approach may cause a slight loss of precision. When the number of patients per treatment is large and the probability of compliance to an assigned treatment is high, we find that all interval estimators discussed here are essentially equivalent. Finally, we illustrate use of these interval estimators with data simulated from a trial of using macrophage colony-stimulating factor to reduce febrile neutropenia incidence in acute myeloid leukaemia patients.     

Resampling-based multiple hypothesis testing procedures for genetic case-control association studies

In case-control studies of unrelated subjects, gene-based hypothesis tests consider whether any tested feature in a candidate gene--single nucleotide polymorphisms (SNPs), haplotypes, or both--are associated with disease. Standard statistical tests are available that control the false-positive rate at the nominal level over all polymorphisms considered. However, more powerful tests can be constructed that use permutation resampling to account for correlations between polymorphisms and test statistics. A key question is whether the gain in power is large enough to justify the computational burden. We compared the computationally simple Simes Global Test to the min P test, which considers the permutation distribution of the minimum p-value from marginal tests of each SNP. In simulation studies incorporating empirical haplotype structures in 15 genes, the min P test controlled the type I error, and was modestly more powerful than the Simes test, by 2.1 percentage points on average. When disease susceptibility was conferred by a haplotype, the min P test sometimes, but not always, under-performed haplotype analysis. A resampling-based omnibus test combining the min P and haplotype frequency test controlled the type I error, and closely tracked the more powerful of the two component tests. This test achieved consistent gains in power (5.7 percentage points on average), compared to a simple Bonferroni test of Simes and haplotype analysis. Using data from the Shanghai Biliary Tract Cancer Study, the advantages of the newly proposed omnibus test were apparent in a population-based study of bile duct cancer and polymorphisms in the prostaglandin-endoperoxide synthase 2 (PTGS2) gene.     

Effect of exposure misclassification on the mean squared error of population attributable risk and prevented fraction estimates

Previous work has considered the effect of exposure misclassification on the bias of population attributable risk (AR) estimates, but little is known about the corresponding effects on their precision or mean squared error (MSE). This paper considers AR estimation in typical scenarios for case-control and cohort studies. The analogous index used when exposure reduces the risk--the prevented fraction (PF)--is also investigated. It is shown, through both theoretical and simulation results, that even with quite modest levels of exposure misclassification, the MSE can increase substantially, relative to the variance of AR estimated without measurement error. When exposure assessment is perfectly sensitive, there is no bias in AR but lack of measurement specificity can still cause a substantial loss of precision. In a few cases, the AR or PF with misclassified exposure can actually have smaller MSE; these exceptional cases arise when sensitivity is poor and the bias in AR or PF is relatively large. We conclude that while bias can be reduced by defining exposure on a highly sensitive basis, one must also consider the deleterious effect on precision by doing so. Loss of precision in the AR and PF estimates can be safely ignored only when the exposure measure is very accurate.     

Interval estimation of binomial proportion in clinical trials with a two-stage design

Generally, a two-stage design is employed in Phase II clinical trials to avoid giving patients an ineffective drug. If the number of patients with significant improvement, which is a binomial response, is greater than a pre-specified value at the first stage, then another binomial response at the second stage is also observed. This paper considers interval estimation of the response probability when the second stage is allowed to continue. Two asymptotic interval estimators, Wald and score, as well as two exact interval estimators, Clopper-Pearson and Sterne, are constructed according to the two binomial responses from this two-stage design, where the binomial response at the first stage follows a truncated binomial distribution. The mean actual coverage probability and expected interval width are employed to evaluate the performance of these interval estimators. According to the comparison results, the score interval is recommended for both Simon's optimal and minimax designs.     

Interval estimation of the difference in proportions under m-to-one matching

When the number of potential controls is large relative to the number of available cases, or when little effort needs to be expended in collecting the relevant information on the controls, we often apply multiple matching to attain the validity or increase the efficiency of our inference in epidemiological studies. In this paper, we focus interval estimation on the difference in proportions for m-to-one matching. We consider four asymptotic interval estimators, including the estimator directly using the Mantel-Haenszel (MH) point estimator, the estimator using the tanh(-1)(x) transformation, the estimator derived from the Cochran-Mantel-Haenszel (CMH) test statistic, and the estimator derived from the quadratic inequality developed in this paper. To evaluate and compare the performance of these estimators, we employ Monte Carlo simulation. We find that the estimator directly using the MH estimator can have the coverage probability less than the desired confidence level when the number of matched sets is small. We note that the estimator derived from the quadratic inequality can perform well when the underlying difference is close to 0 even for a small number of matched sets. However, this estimator tends to have the coverage probability less than the desired confidence level as well when the underlying difference in proportions is large. By contrast, the estimator using the CMH statistic tends to have the coverage probability larger than the desired confidence level when the underlying difference is small. We also find that the estimator using the tanh(-1)(x) transformation consistently outperforms the interval estimator directly using the MH estimator. We use the data regarding the association between induced abortions and ectopic pregnancy to illustrate the use of these estimators.     

Exploiting gene-environment independence in family-based case-control studies: increased power for detecting associations, interactions and joint effects

Family-based case-control studies are popularly used to study the effect of genes and gene-environment interactions in the etiology of rare complex diseases. We consider methods for the analysis of such studies under the assumption that genetic susceptibility (G) and environmental exposures (E) are independently distributed of each other within families in the source population. Conditional logistic regression, the traditional method of analysis of the data, fails to exploit the independence assumption and hence can be inefficient. Alternatively, one can estimate the multiplicative interaction between G and E more efficiently using cases only, but the required population-based G-E independence assumption is very stringent. In this article, we propose a novel conditional likelihood framework for exploiting the within-family G-E independence assumption. This approach leads to a simple and yet highly efficient method of estimating interaction and various other risk parameters of scientific interest. Moreover, we show that the same paradigm also leads to a number of alternative and even more efficient methods for analysis of family-based case-control studies when parental genotype information is available on the case-control study participants. Based on these methods, we evaluate different family-based study designs by examining their relative efficiencies to each other and their efficiencies compared to a population-based case-control design of unrelated subjects. These comparisons reveal important design implications. Extensions of the methodologies for dealing with complex family studies are also discussed.     

Population‐attributable risk of low birthweight related to PM10 pollution in seven Korean cities

Seo J‐H, Leem J‐H, Ha E‐H, Kim O‐J, Kim B‐M, Lee J‐Y, Park H‐S, Kim H‐C, Hong Y‐C, Kim Y‐J. Population‐attributable risk of low birthweight related to PM₁₀ pollution in seven Korean cities. Paediatric and Perinatal Epidemiology 2010; 24: 140–148. To understand the preventable fraction of low birthweight (LBW) deliveries due to maternal exposure to air pollution during pregnancy in Korea, it is important to quantify the population‐attributable risk (PAR). Thus, we investigated the association between maternal exposure to air pollution during pregnancy and LBW, and calculated the PAR for air pollution and LBW in seven Korean cities. We used birth records from the Korean National Birth Register for 2004. A geographic information system and kriging methods were used to construct exposure models. Associations between air pollution and LBW were evaluated using univariable and multivariable logistic regression, and the PAR for LBW due to air pollution was calculated. Of 177 660 full‐term singleton births, 1.4% were LBW. When only spatial variation of air pollution was considered in each city, the adjusted odds ratios unit of particulate matter <10 µm in diameter (PM₁₀) for LBW were 1.08 [95% confidence interval [CI] 0.99, 1.18] in Seoul, 1.24 [95% CI 1.02, 1.52] in Pusan, 1.19 [95% CI 1.04, 1.37] in Daegu, 1.12 [95% CI 0.98, 1.28] in Incheon, 1.22 [95% CI 0.98, 1.52] in Kwangju, 1.05 [95% CI 1.00, 1.11] in Daejeon and 1.19 [95% CI 1.03, 1.38] in Ulsan. The PARs for LBW attributable to maternal PM₁₀ exposure during pregnancy were 7%, 19%, 16%, 11%, 18%, 5% and 16% respectively. Because a large proportion of pregnant women in Korea are exposed to PM₁₀– which is associated with LBW – a substantial proportion of LBW could be prevented in Korea if air pollution was reduced.     

Double-robust estimation of an exposure-outcome odds ratio adjusting for confounding in cohort and case-control studies

Modern epidemiologic studies often aim to evaluate the causal effect of a point exposure on the risk of a disease from cohort or case-control observational data. Because confounding bias is of serious concern in such non-experimental studies, investigators routinely adjust for a large number of potential confounders in a logistic regression analysis of the effect of exposure on disease outcome. Unfortunately, when confounders are not correctly modeled, standard logistic regression is likely biased in its estimate of the effect of exposure, potentially leading to erroneous conclusions. We partially resolve this serious limitation of standard logistic regression analysis with a new iterative approach that we call ProRetroSpective estimation, which carefully combines standard logistic regression with a logistic regression analysis in which exposure is the dependent variable and the outcome and confounders are the independent variables. As a result, we obtain a correct estimate of the exposure-outcome odds ratio, if either thestandard logistic regression of the outcome given exposure and confounding factors is correct, or the regression model of exposure given the outcome and confounding factors is correct but not necessarily both, that is, it is double-robust. In fact, it also has certain advantadgeous efficiency properties. The approach is general in that it applies to both cohort and case-control studies whether the design of the study is matched or unmatched on a subset of covariates. Finally, an application illustrates the methods using data from the National Cancer Institute's Black/White Cancer Survival Study.     

Robust estimation and testing of haplotype effects in case-control studies

Haplotype-based analyses are thought to play a major role in the study of common complex diseases. This has led to the development of a variety of statistical methods for detecting disease-haplotype associations from case-control study data. However, haplotype phase is often uncertain when only genotype data is available. Methods that account for haplotype ambiguity by modeling the distribution of haplotypes can, if this distribution is misspecified, lead to substantial bias in parameter estimates even when complete genotype data is available. Here we study estimators that can be derived from score functions of appropriate likelihoods. We use the efficient score approach to estimation in the presence of nuisance parameters to a derive novel estimators that are robust to the haplotype distribution. We establish key relationships between estimators and study their empirical performance via simulation.     

Analysis of case-control studies of genetic and environmental factors with missing genetic information and haplotype-phase ambiguity

Case-control studies of unrelated subjects are now widely used to study the role of genetic susceptibility and gene-environment interactions in the etiology of complex diseases. Exploiting an assumption of gene-environment independence, and treating the distribution of environmental exposures as completely nonparametric, Chatterjee and Carroll recently developed an efficient retrospective maximum-likelihood method for analysis of case-control studies. In this article, we develop an extension of the retrospective maximum-likelihood approach to studies where genetic information may be missing on some study subjects. In particular, special emphasis is given to haplotype-based studies where missing data arise due to linkage-phase ambiguity of genotype data. We use a profile likelihood technique and an appropriate expectation-maximization (EM) algorithm to derive a relatively simple procedure for parameter estimation, with or without a rare disease assumption, and possibly incorporating information on the marginal probability of the disease for the underlying population. We also describe two alternative robust approaches that are less sensitive to the underlying gene-environment independence and Hardy-Weinberg-equilibrium assumptions. The performance of the proposed methods is studied using simulation studies in the context of haplotype-based studies of gene-environment interactions. An application of the proposed method is illustrated using a case-control study of ovarian cancer designed to investigate the interaction between BRCA1/2 mutations and reproductive risk factors in the etiology of ovarian cancer.     

Methods of adjustment for estimating the attributable risk in case-control studies: a review.

In the 1980's, progress was made in adjusting estimates of the attributable risk (AR) for confounding factors and in calculating associated confidence intervals. In this paper, methods of adjustment for estimation of the AR in case-control studies are reviewed. The limitations and problems associated with two methods based on stratification, the weighted-sum approach and the Mantel-Haenszel approach, are discussed. They include small-sample bias with the weighted-sum approach and the difficulty of taking interaction into account with the Mantel-Haenszel approach. A third method based on logistic regression is reviewed. It is argued that this latter method has the greatest generality and flexibility, and includes the two other approaches as special cases. Throughout the paper, an example of a case-control study of oesophageal cancer illustrates the use of the methods described.