Interval estimation of genetic susceptibility for retrospective case-control studies

Background

This article describes classical and Bayesian interval estimation of genetic susceptibility based on random samples with pre-specified numbers of unrelated cases and controls.

Results

Frequencies of genotypes in cases and controls can be estimated directly from retrospective case-control data. On the other hand, genetic susceptibility defined as the expected proportion of cases among individuals with a particular genotype depends on the population proportion of cases (prevalence). Given this design, prevalence is an external parameter and hence the susceptibility cannot be estimated based on only the observed data. Interval estimation of susceptibility that can incorporate uncertainty in prevalence values is explored from both classical and Bayesian perspective. Similarity between classical and Bayesian interval estimates in terms of frequentist coverage probabilities for this problem allows an appealing interpretation of classical intervals as bounds for genetic susceptibility. In addition, it is observed that both the asymptotic classical and Bayesian interval estimates have comparable average length. These interval estimates serve as a very good approximation to the "exact" (finite sample) Bayesian interval estimates. Extension from genotypic to allelic susceptibility intervals shows dependency on phenotype-induced deviations from Hardy-Weinberg equilibrium.

Conclusions

The suggested classical and Bayesian interval estimates appear to perform reasonably well. Generally, the use of exact Bayesian interval estimation method is recommended for genetic susceptibility, however the asymptotic classical and approximate Bayesian methods are adequate for sample sizes of at least 50 cases and controls.     

Interval estimation for mark-recapture studies of closed populations.

Textbooks continue to recommend the use of an asymptotic normal distribution to provide an interval estimate for the unknown size, N, of a closed population studied by a mark-recapture experiment or multiple-record system. A likelihood interval approach is proposed and its implementation demonstrated for a range of models for such studies, including all main effect and interaction models for incomplete contingency tables.     

Tests of homogeneity for the relative risk in multiply-matched case-control studies

We examine several tests of homogeneity of the odds ratio in the analysis of 2 x 2 tables arising from epidemiologic 1:R matched case-control studies. The T4 and T5 statistics proposed by Liang and Self (1985, Biometrika 72, 353-358) are unable to detect obvious inhomogeneity in two numerical examples and in simulation studies. The null hypothesis is rejected by the chi-square statistic of Ejigou and McHugh (1984, Biometrika 71, 408-411) and by a new proposed method whose significance level must be simulated.     

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.     

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.     

Semiparametric methods for evaluating risk prediction markers in case-control studies

The performance of a well-calibrated risk model for a binary disease outcome can be characterized by the population distribution of risk and displayed with the predictiveness curve. Better performance is characterized by a wider distribution of risk, since this corresponds to better risk stratification in the sense that more subjects are identified at low and high risk for the disease outcome. Although methods have been developed to estimate predictiveness curves from cohort studies, most studies to evaluate novel risk prediction markers employ case-control designs. Here we develop semiparametric methods that accommodate case-control data. The semiparametric methods are flexible, and naturally generalize methods previously developed for cohort data. Applications to prostate cancer risk prediction markers illustrate the methods.     

Using pooled exposure assessment to improve efficiency in case-control studies

Assays can be so expensive that interesting hypotheses become impractical to study epidemiologically. One need not, however, perform an assay for everyone providing a biological specimen. We propose pooling equal-volume aliquots from randomly grouped sets of cases and randomly grouped sets of controls, and then assaying the smaller number of pooled samples. If an effect modifier is of concern, the pooling can be done within strata defined by that variable. For covariates assessed on individuals (e.g., questionnaire data), set-based counterparts are calculated by adding the values for the individuals in each set. The pooling set then becomes the unit of statistical analysis. We show that, with appropriate specification of a set-based logistic model, standard software yields a valid estimated exposure odds ratio, provided the multiplicative formulation is correct. Pooling minimizes the depletion of irreplaceable biological specimens and can enable additional exposures to be studied economically. Statistical power suffers very little compared with the usual, individual-based analysis. In settings where high assay costs constrain the number of people an investigator can afford to study, specimen pooling can make it possible to study more people and hence improve the study's statistical power with no increase in cost.     

Bayesian semiparametric modeling for matched case-control studies with multiple disease states

We present a Bayesian approach to analyze matched "case-control" data with multiple disease states. The probability of disease development is described by a multinomial logistic regression model. The exposure distribution depends on the disease state and could vary across strata. In such a model, the number of stratum effect parameters grows in direct proportion to the sample size leading to inconsistent MLEs for the parameters of interest even when one uses a retrospective conditional likelihood. We adopt a semiparametric Bayesian framework instead, assuming a Dirichlet process prior with a mixing normal distribution on the distribution of the stratum effects. We also account for possible missingness in the exposure variable in our model. The actual estimation is carried out through a Markov chain Monte Carlo numerical integration scheme. The proposed methodology is illustrated through simulation and an example of a matched study on low birth weight of newborns (Hosmer, D. A. and Lemeshow, S., 2000, Applied Logistic Regression) with two possible disease groups matched with a control group.     

Semiparametric estimation of marginal hazard function from case-control family studies

Estimating marginal hazard function from the correlated failure time data arising from case-control family studies is complicated by noncohort study design and risk heterogeneity due to unmeasured, shared risk factors among the family members. Accounting for both factors in this article, we propose a two-stage estimation procedure. At the first stage, we estimate the dependence parameter in the distribution for the risk heterogeneity without obtaining the marginal distribution first or simultaneously. Assuming that the dependence parameter is known, at the second stage we estimate the marginal hazard function by iterating between estimation of the risk heterogeneity (frailty) for each family and maximization of the partial likelihood function with an offset to account for the risk heterogeneity. We also propose an iterative procedure to improve the efficiency of the dependence parameter estimate. The simulation study shows that both methods perform well under finite sample sizes. We illustrate the method with a case-control family study of early onset breast cancer.     

Combining association tests across multiple genetic markers in case-control studies

In the search to detect genetic associations between complex traits and DNA variants, a practice is to select a subset of Single Nucleotide Polymorphisms (tag SNPs) in a gene or chromosomal region of interest. This allows study of untyped polymorphisms in this region through the phenomenon of linkage disequilibrium (LD). However, it is crucial in the analysis to utilize such multiple SNP markers efficiently. In this study, we present a robust testing approach (T(C)) that combines single marker association test statistics or p values. This combination is based on the summation of single test statistics or p values, giving greater weight to those with lower p values. We compared the powers of T(C) in identifying common trait loci, using tag SNPs within the same haplotype block that the trait loci reside, with competing published tests, in case-control settings. These competing tests included the Bonferroni procedure (T(B)), the simple permutation procedure (T(P)), the permutation procedure proposed by Hoh et al. (T(P-H)) and its revised version using 'deflated' statistics (T(P-H_def)), the traditional chi(2) procedure (T(CHI)), the regression procedure (Hotelling T(2) test) (T(R)) and the haplotype-based test (T(H)). Results of these comparisons show that our proposed combining procedure (T(C)) is preferred in all scenarios examined. We also apply this new test to a data set from a previously reported association study on airway responsiveness to methacholine.     

A score test for determining sample size in matched case-control studies with categorical exposure

The paper considers the problem of determining the number of matched sets in 1 : M matched case-control studies with a categorical exposure having k + 1 categories, k > or = 1. The basic interest lies in constructing a test statistic to test whether the exposure is associated with the disease. Estimates of the k odds ratios for 1 : M matched case-control studies with dichotomous exposure and for 1 : 1 matched case-control studies with exposure at several levels are presented in Breslow and Day (1980), but results holding in full generality were not available so far. We propose a score test for testing the hypothesis of no association between disease and the polychotomous exposure. We exploit the power function of this test statistic to calculate the required number of matched sets to detect specific departures from the null hypothesis of no association. We also consider the situation when there is a natural ordering among the levels of the exposure variable. For ordinal exposure variables, we propose a test for detecting trend in disease risk with increasing levels of the exposure variable. Our methods are illustrated with two datasets, one is a real dataset on colorectal cancer in rats and the other a simulated dataset for studying disease-gene association.     

Efficient semiparametric estimation of haplotype-disease associations in case-cohort and nested case-control studies

Estimating the effects of haplotypes on the age of onset of a disease is an important step toward the discovery of genes that influence complex human diseases. A haplotype is a specific sequence of nucleotides on the same chromosome of an individual and can only be measured indirectly through the genotype. We consider cohort studies which collect genotype data on a subset of cohort members through case-cohort or nested case-control sampling. We formulate the effects of haplotypes and possibly time-varying environmental variables on the age of onset through a broad class of semiparametric regression models. We construct appropriate nonparametric likelihoods, which involve both finite- and infinite-dimensional parameters. The corresponding nonparametric maximum likelihood estimators are shown to be consistent, asymptotically normal, and asymptotically efficient. Consistent variance-covariance estimators are provided, and efficient and reliable numerical algorithms are developed. Simulation studies demonstrate that the asymptotic approximations are accurate in practical settings and that case-cohort and nested case-control designs are highly cost-effective. An application to a major cardiovascular study is provided.     

Attributable risk estimation from case-control data via logistic regression.

By fitting an unconditional logistic regression model to unmatched case-control data, an estimate of the joint population attributable risk for the factor included is obtained. This estimate and its asymptotic variance can easily be computed from the intercept parameter and its asymptotic variance. A generalization to the analysis of stratified data with large strata enables the calculation of stratum-specific attributable risks and their variances via stratum-specific intercept parameters. If sampling of cases is independent of strata, an estimate of the summary attributable risk and its asymptotic variance may be obtained as a weighted sum of the stratum-specific attributable risks.     

Past exposure to sun,skin phenotype,and risk of multiple sclerosis: case-control study

Objective To examine whether past high sun exposure is associated with a reduced risk of multiple sclerosis.Design Population based case-control study.Setting Tasmania, latitudes 41-3°S.Participants 136 cases with multiple sclerosis and 272 controls randomly drawn from the community and matched on sex and year of birth.Main outcome measure Multiple sclerosis defined by both clinical and magnetic resonance imaging criteria.Results Higher sun exposure when aged 6-15 years (average 2-3 hours or more a day in summer during weekends and holidays) was associated with a decreased risk of multiple sclerosis (adjusted odds ratio 0.31, 95% confidence interval 0.16 to 0.59). Higher exposure in winter seemed more important than higher exposure in summer. Greater actinic damage was also independently associated with a decreased risk of multiple sclerosis (0.32, 0.11 to 0.88 for grades 4-6 disease). A dose-response relation was observed between multiple sclerosis and decreasing sun exposure when aged 6-15 years and with actinic damage.Conclusion Higher sun exposure during childhood and early adolescence is associated with a reduced risk of multiple sclerosis. Insufficient ultraviolet radiation may therefore influence the development of multiple sclerosis.