Rare Disease Assumption Research Articles

Mendelian randomization analysis of case‐control data using structural mean models

'Instrumental Variable' (IV) methods provide a basis for estimating an exposure's causal effect on the risk of disease. In Mendelian randomization studies, where genetic information plays the role of the IV, IV analyses are routinely performed on case-control data, rather than prospectively collected observational data. Although it is a well-appreciated fact that ascertainment bias may invalidate such analyses, ad hoc assumptions and approximations are made to justify their use. In this paper we attempt to explain and clarify why they may fail and show how they can be adjusted for improved performance. In particular, we propose consistent estimators of the causal relative risk and odds ratio if a priori knowledge is available regarding either the population disease prevalence or the population distribution of the IV (e.g. population allele frequencies). We further show that if no such information is available, approximate estimators can be obtained under a rare disease assumption. We illustrate this with matched case-control data from the recently completed EPIC study, from which we attempt to assess the evidence for a causal relationship between C-reactive protein levels and the risk of Coronary Artery Disease.

Invited Commentary: Decomposing with a Lot of Supposing

In this issue of the Journal, VanderWeele and Vansteelandt (Am J Epidemiol. 2010;172(12):1339-1348) provide simple formulae for estimation of direct and indirect effects using standard logistic regression when the exposure and outcome are binary, the mediator is continuous, and the odds ratio is the chosen effect measure. They also provide concisely stated lists of assumptions necessary for estimation of these effects, including various conditional independencies and homogeneity of exposure and mediator effects over covariate strata. They further suggest that this will allow effect decomposition in case-control studies if the sampling fractions and population outcome prevalence are known with certainty. In this invited commentary, the author argues that, in a well-designed case-control study in which the sampling fraction is known, it should not be necessary to rely on the odds ratio. The odds ratio has well-known deficiencies as a causal parameter, and its use severely complicates evaluation of confounding and effect homogeneity. Although VanderWeele and Vansteelandt propose that a rare disease assumption is not necessary for estimation of controlled direct effects using their approach, collapsibility concerns suggest otherwise when the goal is causal inference rather than merely measuring association. Moreover, their clear statement of assumptions necessary for the estimation of natural/pure effects suggests that these quantities will rarely be viable estimands in observational epidemiology.

What Do Case-Control Studies Estimate? Survey of Methods and Assumptions in Published Case-Control Research

To evaluate strategies used to select cases and controls and how reported odds ratios are interpreted, the authors examined 150 case-control studies published in leading general medicine, epidemiology, and clinical specialist journals from 2001 to 2007. Most of the studies (125/150; 83%) were based on incident cases; among these, the source population was mostly dynamic (102/125; 82%). A minority (23/125; 18%) sampled from a fixed cohort. Among studies with incident cases, 105 (84%) could interpret the odds ratio as a rate ratio. Fifty-seven (46% of 125) required the source population to be stable for such interpretation, while the remaining 48 (38% of 125) did not need any assumptions because of matching on time or concurrent sampling. Another 17 (14% of 125) studies with incident cases could interpret the odds ratio as a risk ratio, with 16 of them requiring the rare disease assumption for this interpretation. The rare disease assumption was discussed in 4 studies but was not relevant to any of them. No investigators mentioned the need for a stable population. The authors conclude that in current case-control research, a stable exposure distribution is much more frequently needed to interpret odds ratios than the rare disease assumption. At present, investigators conducting case-control studies rarely discuss what their odds ratios estimate.

Statistical methods for haplotype‐based matched case‐control association studies

Kraft et al. [2005] proposed a method for matched haplotype-based association studies and compared the performances of six analytic strategies for estimating the odds ratio parameters using a conditional likelihood function. Zhang et al. [2006] modified the conditional likelihood and proposed a new method for matched haplotype-based association studies. The main assumptions of Zhang et al. were that the disease was rare, the population was in Hardy-Weinberg equilibrium (HWE), and the haplotypes were independent of the covariates and matching variable(s). In this article, we modify the estimation procedure proposed by Zhang et al. and introduce a fixation index so that the assumption of HWE is relaxed. Using the Wald test, we compare the current modified method with the procedure developed by Kraft et al. through simulations. The results show that the modified method is uniformly more powerful than that described in Kraft et al. Furthermore, the results indicate that the modified method is quite robust to the rare disease assumption.

Epidemiology of Tick Bites and Borreliosis in Children Attending Kindergarten or So-Called “Forest Kindergarten” in Southwest Germany

The so-called "forest kindergartens" have been increasingly popular in Germany since the beginning of the 1990s. These are nurseries located in forested areas where children spend all-season full-time outdoors. Fifty-three kindergartens in the state of Baden-Württemberg, Germany participated in this study. In a prospective clinical cohort study, the child's personal data, history, protective parental habits concerning tick bites, number of tick bites, and cases of borreliosis were recorded monthly (March-October 2004) using a questionnaire. Altogether, 1,707 children of 25 "forest kindergartens" (506 children) and 28 conventional kindergartens (1,201 children) were included. The response rate was 75% in "forest kindergartens" and 65% in conventional kindergartens. In the "forest kindergartens", 1,503 tick bites especially on the trunk and on the head were found, whereas 502 tick bites were registered in conventional kindergartens. Sixteen cases of borreliosis were diagnosed (10 in "forest kindergartens", six in conventional kindergarten), most frequently manifesting as erythema migrans. Children attending a "forest kindergarten" have a 2.8 times increased risk of experiencing tick bites and a 4.6 times increased risk of suffering from borreliosis compared to conventional kindergarten in Germany, although protective parental behavior in "forest kindergarten" children was significantly better than that in conventional kindergarten.

Haplotype‐Based Association Analysis in Cohort and Nested Case–Control Studies

Genetic epidemiologic studies often collect genotype data at multiple loci within a genomic region of interest from a sample of unrelated individuals. One popular method for analyzing such data is to assess whether haplotypes, i.e., the arrangements of alleles along individual chromosomes, are associated with the disease phenotype or not. For many study subjects, however, the exact haplotype configuration on the pair of homologous chromosomes cannot be derived with certainty from the available locus-specific genotype data (phase ambiguity). In this article, we consider estimating haplotype-specific association parameters in the Cox proportional hazards model, using genotype, environmental exposure, and the disease endpoint data collected from cohort or nested case-control studies. We study alternative Expectation-Maximization algorithms for estimating haplotype frequencies from cohort and nested case-control studies. Based on a hazard function of the disease derived from the observed genotype data, we then propose a semiparametric method for joint estimation of relative-risk parameters and the cumulative baseline hazard function. The method is greatly simplified under a rare disease assumption, for which an asymptotic variance estimator is also proposed. The performance of the proposed estimators is assessed via simulation studies. An application of the proposed method is presented, using data from the Alpha-Tocopherol, Beta-Carotene Cancer Prevention Study.

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.

Conditional likelihood score functions for mixed models in linkage analysis

In this paper, we develop a general strategy for linkage analysis, applicable for arbitrary pedigree structures and genetic models with one major gene, polygenes and shared environmental effects. Extending work of Whittemore (1996), McPeek (1999) and Hossjer (2003d), the efficient score statistic is computed from a conditional likelihood of marker data given phenotypes. The resulting semiparametric linkage analysis is very similar to nonparametric linkage based on affected individuals. The efficient score S depends not only on identical-by-descent sharing and phenotypes, but also on a few parameters chosen by the user. We focus on (1) weak penetrance models, where the major gene has a small effect and (2) rare disease models, where the major gene has a possibly strong effect but the disease causing allele is rare. We illustrate our results for a large class of genetic models with a multivariate Gaussian liability. This class incorporates one major gene, polygenes and shared environmental effects in the liability, and allows e.g. binary, Gaussian, Poisson distributed and life-length phenotypes. A detailed simulation study is conducted for Gaussian phenotypes. The performance of the two optimal score functions S(wpairs) and S(normdom) are investigated. The conclusion is that (i) inclusion of polygenic effects into the score function increases overall performance for a wide range of genetic models and (ii) score functions based on the rare disease assumption are slightly more powerful.

Risk of Haemophilus influenzae type b meningitis in Polish children varies directly with number of siblings: possible implications for vaccination strategies.

In the absence of an effective vaccine, Haemophilus influenzae type b (Hib) meningitis has frequently been found to be the most common bacterial meningitis among children less than or equal to 5 years of age. This study was conducted prior to the introduction of Hib vaccine in Poland, and is the first case-control study of Hib meningitis in this country. A 1:3 matched case-control study was nested within a population-based surveillance study of Hib meningitis in children less than or equal to 5 years old in Poland. Controls were matched on the case's age at onset of disease and immunization center. Having one or more siblings is a highly significant risk factor for Hib meningitis and, under the rare disease assumption, risk increases linearly with the increase in the number of siblings. The size of the living area of the home was not itself a significant risk factor for disease. Breast-feeding was not protective in contrast to previous studies. Childcare outside of the home was a significant risk factor for Hib meningitis especially among children greater than 16 months of age, whereas the effect of sibling number on Hib meningitis was much greater among the younger children. Risk of Hib meningitis approximately doubles for every unit increase in the number of siblings. Routine vaccination of Polish infants who have two or more siblings could potentially prevent half of the Hib meningitis cases.

Hazard rate ratio and prospective epidemiological studies

Analysis of prospective follow-up data usually includes a Cox regression model. When a hazard rate ratio, obtained as the exponential of an estimated regression coefficient from the Cox model, is greater than 1.0, it consistently exceeds relative risk, and is exceeded by the odds ratio. The divergence of these distinct epidemiologic measures increases with the product of three factors: (1) the length of follow-up, (2) the average rate of the end point occurence over the follow-up period, and (3) the magnitude of risk, either above or below 1. Cornfield's rare disease assumption is basically the product of the first two of these factors. However, risks in excess of 2.5 have a powerful effect on the divergence of these measures, and this point has received less emphasis. Conversely, and as seen frequently in applications, relative risk, hazard rate ratio, and odds ratio numerically approximate one another with shorter follow-up, rarer end points, and risks closer to 1. Although the hazard rate ratio is not always distinguished from relative risk, it is commonly close to, and is always between, relative risk and the odds ratio. Consistent and accurate terminology would have us use hazard rate ratio with Cox regression and odds ratio with logistic regression. The term “relative risk” seems to be a default choice, regardless of the model being used. However, when relative risk is the object of the model chosen, as in a Poisson regression approximation of two binomial proportions or an equivalent weighted least squares, then for us, relative risk is the accurate terminology.

Ascertainment bias in rate ratio estimation from case-sibling control studies of variable age-at-onset diseases.

Motivated by a Finnish case-control study of early onset diabetes in which diabetic children are matched to sibling controls, we investigate ascertainment bias of the usual rate ratio estimator from case-control data under simplex complete ascertainment of families during a fixed interval of time. Analytic results indicate that the assumptions necessary for valid estimation are that the disease is rare and the factors under study are exchangeable--essentially that the covariate distribution does not depend on calendar time or birth order. Further, we found that the rare disease assumption could be dropped by restricting to cases that were diagnosed during the enrollment period of the study or including all cases but eliminating the proband as a control for non-enrollment-period cases. An important consequence of this work is that standard family-based case-control studies are subject to ascertainment bias if exchangeability of the covariates under investigation does not hold.

A Modification of Stone's Test for Trend for Binary Outcome

Stone (1988) suggested the “first isotonic regression estimator” as a tool for drawing inferences on possibly increased cancer case counts among several subregions around a putative source. He assumed the case counts to be Poisson distributed and therefore introduced a rare disease assumption into his approach. However, when analyzing cross sectional data one would rather refer to prevalence estimates among these subregions around a point risk source (for example the origin of chemical fallout). Therefore we applied antitonic regression estimation in Binomial distributions to derive a test statistic and a p value to test for a possible trend in the observed prevalence data around the putative source. The computation of this p value will be illustrated as well as severe difficulties concerning its interpretation. Further the Maximum Likelihood Ratio approach will be used to derive an alternative test statistic.

Risk ratio estimation in case-cohort studies.

In traditional (cumulative-incidence) case-control studies, the exposure odds ratio can be used as an estimator of the risk ratio only when the disease under study is rare. The case-cohort study is a recently developed useful modification of the case-control study. This design allows direct estimation of the risk ratio from a fixed cohort, but does not require any rare-disease assumption. This article reviews recent developments in risk ratio estimation procedures for the analysis of case-cohort data. In the crude analysis, it is shown that the empirical risk ratio estimator is not fully efficient, and the maximum likelihood estimation of the crude risk ratio is discussed. In the stratified analysis, several common risk ratio estimation procedures and standardization methods have been proposed for large strata. However, the Mantel-Haenszel risk ratio and its variance estimator are the only available methods for sparse data.

Estimation of a common risk ratio in stratified case-cohort studies.

The case-cohort study design is a useful modification of the case-control design, which allows direct estimation of the risk ratio without the rare-disease assumption. While several risk ratio estimation procedures have been proposed under large-strata settings, only the Mantel-Haenszel point estimator is available in sparse stratifications. This paper provides simple confidence limits methods, based on the large-sample distribution of the Mantel-Haenszel risk ratio, that apply to both large-strata and sparse-data situations. For the Tarone risk ratio, I give a new large-strata variance estimate.

ERRATA

Drs. Philip Kass and Sander Greenland wish to point out an error in the second example in the paper by Greenland and Thomas, “On the need for the rare disease assumption in case studies” (Am J Epidemiol 1982; 116:547–53). In this example, B0 is given as 38.51 but should be 44.95, with the result that the bias in OR0 in this example is negligible. This correction does not alter the remainder of the paper

Case-Control Designs in the Study of Common Diseases: Updates on the Demise of the Rare Disease Assumption and the Choice of Sampling Scheme for Controls

In recent years the use of case-control designs has been extended to the study of common diseases. It has been shown that the rare disease assumption is not necessary, and that by a suitable choice of sampling scheme for controls, it is possible to obtain direct estimates of relative risk and relative rate, instead of relying on the odds ratio as an indirect estimate. The majority of papers addressing these issues are theoretical, and the arguments have been couched in mathematical terms. As such they are not readily accessible to many practising epidemiologists. This paper summarizes the discussion in a simplified manner. It describes the three different measures of relative incidence, namely the relative risk, the relative rate and the odds ratio, together with their corresponding case-control designs. The discussion is extended to show that the choice of the appropriate measure of relative incidence depends on the mode of action of the risk factor, as well as on characteristics of disease. We propose a classification scheme comprising five different categories of situation, and make recommendations regarding study designs for each.

A Graphic Method of Estimating the Specificity of Screening Programmes from Incomplete Follow-up Data

SummaryThe specificity of a screening test determines, on the one hand, the extent of inconvenience and risk involved for healthy participants of a screening programme. This is, therefore, a possible determinant of programme acceptance. On the other hand, follow-up costs generally represent a considerable proportion of the total screening programme cost and are determined by the specificity. The specificity is thus an essential determinant of the efficiency of such a programme.Under the rare disease assumption it is possible to assess the specificity of screening programmes from cross-sectional data. On the basis of this method, a simple graphic way of evaluating the specificity is also described and illustrated by data stratified by age and screening history from the German National Cancer Screening Programme.

The rare-disease assumption revisited. A critique of "estimators of relative risk for case-control studies".

The extension of case-control methods to the study of common outcomes has led to the development of several design and analysis techniques which do not employ the rare-disease assumption. Unfortunately, the principles underlying valid application of these techniques are more subtle than those first considered by Cornfield in the rare-disease setting, and appear to be easily misunderstood. We especially wish to caution that: The unrestricted inclusion of prevalent cases in the control group (as described by Hogue et al. for estimation of the risk ratio) will not make the odds ratio an unbiased estimate of the risk ratio (or anything else). In their response to our article, following, Hogue et al. describe restrictions on prevalence and duration necessary for the odds ratio from a case-exposure design to unbiasedly estimate the risk ratio in a stable population; these conditions were not mentioned in their original article, and in their new paper Hogue et al. do not provide mathematical proof that the conditions are sufficient to guarantee unbiasedness. Exclusion ("decontamination") of incident cases from the control group (as recommended by Hogue et al. for testing and test-based interval estimation) will result in improperly narrow risk-ratio confidence intervals whether or not the population is stable, and, in unstable populations, will generally lead to an invalid test. Methods that replace the rare-disease assumption with the stable-population assumption (such as case-exposure designs applied to open populations) will not yield unbiased results when the source population is a fixed cohort. (Of course, this will not be an issue for methods that are not based on either assumption, such as the case-base design applied to fixed cohorts, and the matched density design.) As each case-control design has certain practical implications for selection and interviewing, in choosing a design one should carefully consider practical issues (such as vulnerability to recall bias and ease of control selection) in addition to the statistical issues discussed here. In general, however, one should be wary of methods of studying incidence that involve the use of prevalent cases (such as the approach of Hogue et al.): prevalence is influenced by factors related to treatment, recovery, and fatality, and thus any etiologic study employing prevalent cases may be biased by such factors.(ABSTRACT TRUNCATED AT 400 WORDS)

On the need for the rare disease assumption in case-control studies.

The conditions under which matched and unmatched odds ratios are consistent estimators of the incidence-density ratio in case-control studies are examined. Under "incidence-density" sampling, in which controls are selected from those at risk at the time of onset of each case, the matched estimator is shown to be consistent. In contrast, the unmatched estimator is biased unless the proportion exposed in the population at risk is constant over the study period; the bias is, however, negligible unless there is very large variation in the proportion exposed. No assumption of rarity of the disease is needed for these results. On the other hand, when the risk ratio is the parameter of interest, the assumption of rarity is needed for the odds ratio to be a consistent estimator. In such situations, the odds ratio obtained under "incidence-density" sampling will in general provide a better approximation to the risk ratio than will the odds ratio obtained under "cumulative-incidence" sampling, in which controls are selected from those still unaffected at the end of the study period. Even if the disease is rare, however, the odds ratio obtained under cumulative sampling need not consistently estimate any parameter of interest unless the proportion exposed is approximately constant.

Estimating exposure-specific disease rates from case-control studies using Bayes' theorem.

The methods used for selecting subjects yield three types of case-control studies: 1) incident cases are compared to non-cases chosen to be representative of the exposure distribution among the person-years which produced the cases. In this type of study the exposure-odds ratio equals the incidence density ratio; 2) incident cases are compared to residual non-cases at the end of the risk period (exposure-odds ratio = cumulative incidence-odds ratio); 3) prevalent cases are compared to non-cases (exposure-odds ratio = prevalence odds ratio). In study type 1 the equivalence of odds ratio to rate ratio requires no "rare disease assumption;" this permits estimation of exposure-specific illness rates when the overall rate is known. In study types 2 and 3 the exposure-odds ratio equals the corresponding rate ratios only when exposure-specific rates are low. Nonetheless, exposure-specific rates can be calculated without making any rare disease assumption using Bayes' theorem and information on the overall disease rate. A method for obtaining approximate confidence limits around the exposure-specific rates is presented.