Robust tests for matched case-control genetic association studies

Background

Infectious disease of livestock continues to be a cause of substantial economic loss and has adverse welfare consequences in both the developing and developed world. New solutions to control disease are needed and research focused on the genetic loci determining variation in immune-related traits has the potential to deliver solutions. However, identifying selectable markers and the causal genes involved in disease resistance and vaccine response is not straightforward. The aims of this study were to locate regions of the bovine genome that control the immune response post immunisation. 195 F2 and backcross Holstein Charolais cattle were immunised with a 40-mer peptide derived from foot-and-mouth disease virus (FMDV). T cell and antibody (IgG1 and IgG2) responses were measured at several time points post immunisation. All experimental animals (F0, F1 and F2, n = 982) were genotyped with 165 microsatellite markers for the genome scan.

Results

Considerable variability in the immune responses across time was observed and sire, dam and age had significant effects on responses at specific time points. There were significant correlations within traits across time, and between IgG1 and IgG2 traits, also some weak correlations were detected between T cell and IgG2 responses. The whole genome scan detected 77 quantitative trait loci (QTL), on 22 chromosomes, including clusters of QTL on BTA 4, 5, 6, 20, 23 and 25. Two QTL reached 5% genome wide significance (on BTA 6 and 24) and one on BTA 20 reached 1% genome wide significance.

Conclusions

A proportion of the variance in the T cell and antibody response post immunisation with an FDMV peptide has a genetic component. Even though the antigen was relatively simple, the humoral and cell mediated responses were clearly under complex genetic control, with the majority of QTL located outside the MHC locus. The results suggest that there may be specific genes or loci that impact on variation in both the primary and secondary immune responses, whereas other loci may be specifically important for early or later phases of the immune response. Future fine mapping of the QTL clusters identified has the potential to reveal the causal variations underlying the variation in immune response observed.     

Sensitivity analysis for matched case-control studies

A sensitivity analysis in an observational study indicates the degree to which conclusions would be altered by hidden biases of various magnitudes. A method of sensitivity analysis previously proposed for cohort studies is extended for use in matched case-control studies with multiple controls, where slightly different derivations and calculations are required. Also discussed is a sensitivity analysis for case-control studies that have two distinct types of controls, say hospital and neighborhood controls, where the two types may be affected by different biases. For illustration, the method is applied to five case-control studies, including a study of herniated lumbar disc in which there are three types of cases, and a study of breast cancer with two types of controls.     

Power calculations for matched case-control studies

Power calculations are derived for matched case-control studies in terms of the probability po of exposure among the control patients, the correlation coefficient phi for exposure between matched case and control patients, and the odds ratio psi for exposure in case and control patients. For given Type I and Type II error probabilities alpha and beta, the odds ratio that can be detected with a given sample size is derived as well as the sample size needed to detect a specified value of the odds ratio. Graphs are presented for paired designs that show the relationship between sample size and power for alpha = .05, beta = .2, and different values of po, phi, and psi. The sample size needed for designs involving M matched control patients can be derived from these graphs by means of a simple equation. These results quantify the loss of power associated with increasing correlation between the exposure status of matched case and control patients. Sample size requirements are also greatly increased for values of po near 0 or 1. The relationship between sample size, psi, phi, and po is discussed and illustrated by examples.     

Exact inference for matched case-control studies

In an epidemiological study with a small sample size or a sparse data structure, the use of an asymptotic method of analysis may not be appropriate. In this paper we present an alternative method of analyzing data for case-control studies with a matched design that does not rely on large-sample assumptions. A recursive algorithm to compute the exact distribution of the conditional sufficient statistics of the parameters of the logistic model for such a design is given. This distribution can be used to perform exact inference on model parameters, the methodology of which is outlined. To illustrate the exact method, and compare it with the conventional asymptotic method, analyses of data from two case-control studies are also presented.     

A fast, unbiased and exact allelic test for case-control association studies

Association studies are traditionally performed in the case-control framework. As a first step in the analysis process, comparing allele frequencies using the Pearson's chi-square statistic is often invoked. However such an approach assumes the independence of alleles under the hypothesis of no association, which may not always be the case. Consequently this method introduces a bias that deviates the expected type I error-rate. In this article we first propose an unbiased and exact test as an alternative to the biased allelic test. Available data require to perform thousands of such tests so we focused on its fast execution. Since the biased allelic test is still widely used in the community, we illustrate its pitfalls in the context of genome-wide association studies and particularly in the case of low-level tests. Finally, we compare the unbiased and exact test with the Cochran-Armitage test for trend and show it perfoms similarly in terms of power. The fast, unbiased and exact allelic test code is available in R, C++ and Perl at: http://stat.genopole.cnrs.fr/software/fueatest.     

Evaluation of exact and asymptotic interval estimators in logistic analysis of matched case-control studies.

We compare six methods for constructing confidence intervals for a single parameter in stratified logistic regression. Three of these are based on inversion of standard asymptotic tests--namely, the Wald, the score, and the likelihood ratio tests. The other three are based on the exact distribution of the sufficient statistic for the parameter of interest. These include the traditional exact method of constructing confidence intervals, and two others, the mid-P and mean-P methods, which are modifications of this procedure that aim at reducing the conservative bias of the exact method. Using efficient algorithms, the six methods are compared by determination of their exact coverage levels in a series of conditional sample spaces. An incident case-control study of lung cancer in women is used to further illustrate the differences among the various methods. Computation of coverage functions is seen as a useful graphical diagnostic tool for assessing the appropriateness of different methods. The mid-P and the score methods are seen to have better coverage properties than the other four.     

Data analytic methods for matched case-control studies

The recent introduction of complex multivariate statistical models in matched case-control studies is a mixed blessing. Their use can lead to a better understanding of the way in which many variables contribute to the risk of disease. On the other hand, these powerful methods can obscure salient features in the data that might have been detected by other, less sophisticated methods. This shortcoming is due to a lack of support methodology for the routine use of these models. Satisfactory computation of estimated relative risks and their standard errors is not sufficient justification for the fitted model. Goodness of fit must be examined if inferences are to be trusted. This paper is concerned with the analysis of matched case-control studies with logistic models. Analogies of these models to linear regression models are emphasized. In particular, basic concepts such as analysis of variance, multiple correlation coefficient, one-degree-of-freedom tests, and residual analysis are discussed. The fairly new field of regression diagnostics is also introduced. All procedures are illustrated on a study of bladder cancer in males.     

Sample size for individually matched case-control studies

The standard formulas used to calculate sample size for an individually matched case-control study assume a constant probability of exposure throughout the pool of possible controls. We propose new formulas that allow for heterogeneity in the probability of exposure among controls in different matched sets. Since matching factors are suspected of being confounders, they are expected to divide the total population into subgroups with different proportions exposed. Thus, the assumption of homogeneity of exposure among controls, made by the currently used formulas, is inconsistent with the assumptions used to design a matched study. The proposed formulas avoid this inconsistency. We present an example to illustrate how heterogeneity can affect the required sample size.     

Asymptotic distribution for epistatic tests in case-control studies

We propose a statistical model for dissecting a multilocus genotypic value into its main (additive and dominant) effects and epistatic effects between different loci in a case-control association study. The model can discern four different kinds of epistasis, additive × additive, additive × dominant, dominant × additive, and dominant × dominant interactions. To test each kind of epistasis, a χ² test statistic was computed for a two by two contingency table derived from combined genotypes in both case and control groups. We derived an analytical approach for estimating the asymptotic distribution of the χ² test statistic for epistatic tests under the null hypothesis, with the result being consistent with that from Monte Carlo simulations. The new model was used to analyze a case-control data set for candidate gene studies of stroke, leading to the identification of several significant interactions between causal SNPs on this disease.     

Investigation of spatial clustering from individually matched case-control studies

The paper demonstrates how existing theory to assess spatial clustering based on second-moment properties of a labelled point process can be adapted to matched case-control studies. The null hypothesis that cases are a random sample from the superposition of cases and controls is replaced by the hypothesis that each case is a random sample from the set consisting of itself and its k matched controls. We compare the proposed test with other tests of spatial clustering, and describe an application to data on childhood diabetes in Yorkshire, England.     

On the use of concordant pairs in matched case-control studies

A new estimator of the common odds ratio in one-to-one matched case-control studies is proposed. The connection between this estimator and the James-Stein estimating procedure is highlighted through the argument of estimating functions. Comparisons are made between this estimator, the conditional maximum likelihood estimator, and the estimator ignoring the matching in terms of finite sample bias, mean squared error, coverage probability, and length of confidence interval. In many situations, the new estimator is found to be more efficient than the conditional maximum likelihood estimator without being as biased as the estimator that ignores matching. The extension to multiple risk factors is also outlined.     

A permutation procedure to correct for confounders in case-control studies, including tests of rare variation

Many case-control tests of rare variation are implemented in statistical frameworks that make correction for confounders like population stratification difficult. Simple permutation of disease status is unacceptable for resolving this issue because the replicate data sets do not have the same confounding as the original data set. These limitations make it difficult to apply rare-variant tests to samples in which confounding most likely exists, e.g., samples collected from admixed populations. To enable the use of such rare-variant methods in structured samples, as well as to facilitate permutation tests for any situation in which case-control tests require adjustment for confounding covariates, we propose to establish the significance of a rare-variant test via a modified permutation procedure. Our procedure uses Fisher's noncentral hypergeometric distribution to generate permuted data sets with the same structure present in the actual data set such that inference is valid in the presence of confounding factors. We use simulated sequence data based on coalescent models to show that our permutation strategy corrects for confounding due to population stratification that, if ignored, would otherwise inflate the size of a rare-variant test. We further illustrate the approach by using sequence data from the Dallas Heart Study of energy metabolism traits. Researchers can implement our permutation approach by using the R package BiasedUrn.     

Trend tests for case-control studies of genetic markers: power,sample size and robustness

The Cochran-Armitage trend test is commonly used as a genotype-based test for candidate gene association. Corresponding to each underlying genetic model there is a particular set of scores assigned to the genotypes that maximizes its power. When the variance of the test statistic is known, the formulas for approximate power and associated sample size are readily obtained. In practice, however, the variance of the test statistic needs to be estimated. We present formulas for the required sample size to achieve a prespecified power that account for the need to estimate the variance of the test statistic. When the underlying genetic model is unknown one can incur a substantial loss of power when a test suitable for one mode of inheritance is used where another mode is the true one. Thus, tests having good power properties relative to the optimal tests for each model are useful. These tests are called efficiency robust and we study two of them: the maximin efficiency robust test is a linear combination of the standardized optimal tests that has high efficiency and the MAX test, the maximum of the standardized optimal tests. Simulation results of the robustness of these two tests indicate that the more computationally involved MAX test is preferable.     

Testing hypotheses in case-control studies--equivalence of Mantel-Haenszel statistics and logit score tests.

The two approaches in common use for the analysis of case-control studies are cross-classification by confounding variables, and modeling the logarithm of the odds ratio as a function of exposure and confounding variables. We show here that score statistics derived from the likelihood function in the latter approach are identical to the Mantel-Haenszel test statistics appropriate for the former approach. This identity holds in the most general situation considered, testing for marginal homogeneity in mK tables. This equivalence is demonstrated by a permutational argument which leads to a general likelihood expression in which the exposure variable may be a vector of discrete and/or continuous variables and in which more than two comparison groups may be considered. This likelihood can be used in analyzing studies in which there are multiple controls for each case or in which several disease categories are being compared. The possibility of including continuous variables makes this likelihood useful in situations that cannot be treated using the Mantel-Haenszel cross-classification approach.     

Exact tests for the analysis of case-control studies of genetic markers

OBJECTIVES: The association of a candidate gene with disease can be evaluated by a case-control study in which the genotype distribution is compared for diseased cases and unaffected controls. Usually, the data are analyzed with Armitage's test using the asymptotic null distribution of the test statistic. Since this test does not generally guarantee a type I error rate less than or equal to the significance level alpha, tests based on exact null distributions have been investigated. METHODS: An algorithm to generate the exact null distribution for both Armitage's test statistic and a recently proposed modification of the Baumgartner-Weiss-Schindler statistic is presented. I have compared the tests in a simulation study. RESULTS: The asymptotic Armitage test is slightly anticonservative whereas the exact tests control the type I error rate. The exact Armitage test is very conservative, but the exact test based on the modification of the Baumgartner-Weiss-Schindler statistic has a type I error rate close to alpha. The exact Armitage test is the least powerful test; the difference in power between the other two tests is often small and the comparison does not show a clear winner. CONCLUSION: Simulation results indicate that an exact test based on the modification of the Baumgartner-Weiss-Schindler statistic is preferable for the analysis of case-control studies of genetic markers.     

Robust genomic control and robust delta centralization tests for case-control association studies

The population-based case-control design is a powerful approach for detecting susceptibility markers of a complex disease. However, this approach may lead to spurious association when there is population substructure: population stratification (PS) or cryptic relatedness (CR). Two simple approaches to correct for the population substructure are genomic control (GC) and delta centralization (DC). GC uses the variance inflation factor to correct for the variance distortion of a test statistic, and the DC centralizes the non-central chi-square distribution of the test statistic. Both GC and DC have been studied for case-control association studies mainly under a specific genetic model (e.g. recessive, additive or dominant), under which an optimal trend test is available. The genetic model is usually unknown for many complex diseases. In this situation, we study the performance of three robust tests based on the GC and DC corrections in the presence of the population substructure. Our results show that, when the genetic model is unknown, the DC- (or GC-) corrected maximum and Pearson's association test are robust and have good control of Type I error and high power relative to the optimal trend tests in the presence of PS (or CR).