Notes in Case-Control Studies with Matched Pairs under Inverse Sampling

In case-control studies with matched pairs, the traditional point estimator of odds ratio (OR) is well-known to be biased with no exact finite variance under binomial sampling. In this paper, we consider use of inverse sampling in which we continue to sample subjects to form matched pairs until we obtain a pre-determined number (>0) of index pairs with the case unexposed but the control exposed. In contrast to use of binomial sampling, we show that the uniformly minimum variance unbiased estimator (UMVUE) of OR does exist under inverse sampling. We further derive an exact confidence interval of OR in closed form. Finally, we develop an exact test and an asymptotic test for testing the null hypothesis H₀: OR = 1, as well as discuss sample size determination on the minimum required number of index pairs for a desired power at α-level.     

Testing Average Equivalence — Finding a Compromise Between Theory and Practice

Recently, Brown , Hwang , and Munk (1998) proposed and unbiased test for the average equivalence problem which improves noticeably in power on the standard two one‐sided tests procedure. Nevertheless, from a practical point of view there are some objections against the use of this test which are mainly adressed to the ‘unusual’ shape of the critical region. We show that every unbiased test has a critical region with such an ‘unusual’ shape. Therefore, we discuss three (biased) modifications of the unbiased test. We come to the conclusion that a suitable modification represents a good compromise between a most powerful test and a test with an appealing shape of its critical region. In order to perform these tests figures are given containing the rejection region. Finally, we compare all tests in an example from neurophysiology. This shows that it is beneficial to use these improved tests instead of the two one‐sided tests procedure.     

Exact tests for association between alleles at arbitrary numbers of loci

Associations between allelic frequencies, within and between loci, can be tested for with an exact test. The probability of the set of multi-locus genotypes in a sample, conditional on the allelic counts, is calculated from multinomial theory under the hypothesis of no association. Alleles are then permuted and the conditional probability calculated for the permuted genotypic array. The proportion of arrays no more probable than the original sample provides the significance level for the test. An algorithm is provided for counting genotypes efficiently in the arrays, and the powers of the test presented for various kinds of association. The powers for the case when associations are generated by admixture of several populations suggest that exact tests are capable of detecting levels of association that would affect forensic calculations to a significant extent.     

Estimating Genetic Effects and Quantifying Missing Heritability Explained by Identified Rare-Variant Associations

Next-generation sequencing has led to many complex-trait rare-variant (RV) association studies. Although single-variant association analysis can be performed, it is grossly underpowered. Therefore, researchers have developed many RV association tests that aggregate multiple variant sites across a genetic region (e.g., gene), and test for the association between the trait and the aggregated genotype. After these aggregate tests detect an association, it is only possible to estimate the average genetic effect for a group of RVs. As a result of the "winner’s curse," such an estimate can be biased. Although for common variants one can obtain unbiased estimates of genetic parameters by analyzing a replication sample, for RVs it is desirable to obtain unbiased genetic estimates for the study where the association is identified. This is because there can be substantial heterogeneity of RV sites and frequencies even among closely related populations. In order to obtain an unbiased estimate for aggregated RV analysis, we developed bootstrap-sample-split algorithms to reduce the bias of the winner’s curse. The unbiased estimates are greatly important for understanding the population-specific contribution of RVs to the heritability of complex traits. We also demonstrate both theoretically and via simulations that for aggregate RV analysis the genetic variance for a gene or region will always be underestimated, sometimes substantially, because of the presence of noncausal variants or because of the presence of causal variants with effects of different magnitudes or directions. Therefore, even if RVs play a major role in the complex-trait etiologies, a portion of the heritability will remain missing, and the contribution of RVs to the complex-trait etiologies will be underestimated.     

The estimator of the optimal measure of allelic association: mean, variance and probability distribution when the sample size tends to infinity

The allelic association or linkage disequilibrium between two loci is a parameter of fundamental interest in modern population genetics for evolutionary inference and association mapping studies. Among the many measures available, the optimal measure of allelic association rho presents a strong evolutionary theory basis and is modeled on the physical distance along the chromosome with the Malécot equation for isolation by distance. Moreover, rho is equal to the absolute value of D', the standardized measure of gametic disequilibrium. We studied here the statistical properties of the rho sample estimator. We derived its asymptotic probability distribution and showed that it is neither asymptotically normal nor unbiased when rho=0 or when allelic frequencies are equal at both loci, in contrast to previous claims. This asymptotic study leads to propose a new test for absence of linkage disequilibrium. We compared it to Pearson's Chi2 test for independence in a contingency table and showed by simulations that the range in power of these two tests depends on the sign of D'. The new test outperformed slightly the Chi2 test, when D', polarized with respect to major alleles, is negative. Finally, we derived the asymptotic bias and information of the rho estimator that are due to the experimental sampling and showed by simulation that its bias is large in small samples. The consequences of these findings on applications using the rho measure are then discussed in particular for constructing LD unit maps, and call for a revised statistical treatment.     

Novel case-control test in a founder population identifies P-selectin as an atopy-susceptibility locus

To avoid problems related to unknown population substructure, association studies may be conducted in founder populations. In such populations, however, the relatedness among individuals may be considerable. Neglecting such correlations among individuals can lead to seriously spurious associations. Here, we propose a method for case-control association studies of binary traits that is suitable for any set of related individuals, provided that their genealogy is known. Although we focus here on large inbred pedigrees, this method may also be used in outbred populations for case-control studies in which some individuals are relatives. We base inference on a quasi-likelihood score (QLS) function and construct a QLS test for allelic association. This approach can be used even when the pedigree structure is far too complex to use an exact-likelihood calculation. We also present an alternative approach to this test, in which we use the known genealogy to derive a correction factor for the case-control association chi2 test. We perform analytical power calculations for each of the two tests by deriving their respective noncentrality parameters. The QLS test is more powerful than the corrected chi2 test in every situation considered. Indeed, under certain regularity conditions, the QLS test is asymptotically the locally most powerful test in a general class of linear tests that includes the corrected chi2 test. The two methods are used to test for associations between three asthma-associated phenotypes and 48 SNPs in 35 candidate genes in the Hutterites. We report a highly significant novel association (P=2.10-6) between atopy and an amino acid polymorphism in the P-selectin gene, detected with the QLS test and also, but less significantly (P=.0014), with the transmission/disequilibrium test.     

A Note on Combining Dependent Tests of Significance

In combining several tests of significance the individual test statistics are allowed to be stochastically dependent. By choosing the weighted inverse normal method for the combination, the dependency of the original test statistics is then characterized by a correlation of the transformed statistics. For this correlation a confidence region, an unbiased estimator and an unbiased estimate of its variance are derived. The combined test statistic is extended to include the case of possibly dependent original test statistics. Simulation studies show the performance of the actual significance level.     

Cautions on direct gene flow estimation in plant populations

Through simulations we have investigated the statistical properties of two of the main approaches for directly estimating pollen gene flow (m) in plant populations: genotypic exclusion and mating models. When the assumptions about accurately known background pollen pool allelic frequencies are met, both methods provide unbiased results with comparable variances across a range of true m values. However, when presumed allelic frequencies differ from actual ones, which is more likely in research practice, both estimators are biased. We demonstrate that the extent and direction of bias largely depend on the difference (measured as genetic distance) between the presumed and actual pollen pools, and on the degree of genetic differentiation between the local population and the actual background pollen sources. However, one feature of the mating model is its ability to estimate pollen gene flow simultaneously with background pollen pool allelic frequencies. We have found that this approach gives nearly unbiased pollen gene flow estimates, and is practical because it eliminates the necessity of providing independent estimates of background pollen pool allelic frequencies. Violations of the mating model assumptions of random mating within local population affect the precision of the estimates only to a limited degree.     

UMPU and Alternative Tests for Association in 2 × 2 Tables

The use of the uniformly most powerful among the unbiased (UMPU) test was recently suggested for the study of gametic association between two polymorphic loci as an alternative to the Fisher's exact test (Zapata and Alvarez, 1997, Annals of Human Genetics 61, 71-77). However, the proposed test is not UMPU for two-sided alternatives. In this study, we present the UMPU test, discuss criticisms against the use of randomized tests, and compare the power of several tests. We show that, in many practical cases, the use of the UMPU test is less than desirable and propose the alternative adjusted-more extreme tables (A-MET) and the equal-tails (ET) tests. We suggest that some of the general arguments against the use of randomized tests can be alleviated by a newly proposed extended p-value definition.     

Class I and II HLA genes are associated with susceptibility and age at onset in Finnish families with type 1 diabetes

OBJECTIVES: We explored the properties of the long-term survivor model (LTS) in the genetic association studies and studied allelic and haplotypic associations between the age at onset and partially latent susceptibility of type 1 diabetes (T1DM) and Human Leucocyte Antigen (HLA) A, B and DR loci. METHODS: The authors applied the long-term survivor model (LTS) for sibships collected in a population-based registry during a calendar time period. The method uses sibs that could not become probands and includes the proband's age at onset during the recruitment period. Association between the candidate gene and the partially latent susceptibility is modeled with logistic regression and the age at onset with a two-parameter gamma distribution, where a scale parameter depends on the candidate genotypes. We also performed a simulation study of nuclear families to compare the power of the likelihood ratio tests of the genetic association based on the LTS model with those obtained using family-based association method (FBAT) and bias of the case-pseudo control design. In addition, we analysed allele and haplotype associations between HLA A, B and DR loci (IDDM1) with T1DM, using population-based ascertainment of 705 sibships with complete HLA information. RESULTS: A simulation study showed that the estimates of the genetic association using an ascertainment-corrected LTS model are virtually unbiased and that the relative risk estimates obtained from case-pseudo control design (TDT) are negatively biased. In the analysis of the Finnish T1DM families we found that only B62 (p < 0.05) is positively significantly associated with susceptibility after adjusting for the haplotype effects. Five alleles were significantly associated with age at onset (B8 and DR3, p < 0.01; A2, B60 and DR6, p < 0.05). No significant three-locus haplotype associations with the susceptibility were found, but A3B18DR4 (p < 0.001) haplotype was associated with older age at onset than average. CONCLUSIONS: Estimates of genetic relative risk obtained from the case-pseudo control design are negatively biased and the prospective LTS model is an appropriate choice, when there are non-susceptible subjects in the population with variable age at onset. Based on the analysis of T1DM, we conclude that there are gene(s) in the HLA region that are associated with susceptibility and/or age at onset of T1DM, and this should be taken into account in future studies.     

A test for genetic association that incorporates information about deviation from Hardy-Weinberg proportions in cases

For assessment of genetic association between single-nucleotide polymorphisms (SNPs) and disease status, the logistic-regression model or generalized linear model is typically employed. However, testing for deviation from Hardy-Weinberg proportion in a patient group could be another approach for genetic-association studies. The Hardy-Weinberg proportion is one of the most important principles in population genetics. Deviation from Hardy-Weinberg proportion among cases (patients) could provide additional evidence for the association between SNPs and diseases. To develop a more powerful statistical test for genetic-association studies, we combined evidence about deviation from Hardy-Weinberg proportion in case subjects and standard regression approaches that use case and control subjects. In this paper, we propose two approaches for combining such information: the mean-based tail-strength measure and the median-based tail-strength measure. These measures integrate logistic regression and Hardy-Weinberg-proportion tests for the study of the association between a binary disease outcome and an SNP on the basis of case- and control-subject data. For both mean-based and median-based tail-strength measures, we derived exact formulas to compute p values. We also developed an approach for obtaining empirical p values with the use of a resampling procedure. Results from simulation studies and real-disease studies demonstrate that the proposed approach is more powerful than the traditional logistic-regression model. The type I error probabilities of our approach were also well controlled.     

The Exact Test for Cytonuclear Disequilibria

We extend the analysis of the statistical properties of cytonuclear disequilibria in two major ways. First, we develop the asymptotic sampling theory for the nonrandom associations between the alleles at a haploid cytoplasmic locus and the alleles and genotypes at a diploid nuclear locus, when there are an arbitrary number of alleles at each marker. This includes the derivation of the maximum likelihood estimators and their sampling variances for each disequilibrium measure, together with simple tests of the null hypothesis of no disequilibrium. In addition to these new asymptotic tests, we provide the first implementation of Fisher's exact test for the genotypic cytonuclear disequilibria and some approximations of the exact test. We also outline an exact test for allelic cytonuclear disequilibria in multiallelic systems. An exact test should be used for data sets when either the marginal frequencies are extreme or the sample size is small. The utility of this new sampling theory is illustrated through applications to recent nuclear-mtDNA and nuclear-cpDNA data sets. The results also apply to population surveys of nuclear loci in conjunction with markers in cytoplasmically inherited microorganisms.     

A simple and robust TDT-type test against genotyping error with error rates varying across families

The transmission/disequilibrium test (TDT), a family based test of linkage and association, is a popular test for studies of complex inheritance, as it is nonparametric and robust against spurious conclusions induced by hidden genetic structure, such as stratification or admixture. However, the TDT may be biased by genotyping errors. Undetected genotyping errors may be contributing to an inflated type I error rate among reported TDT-derived associations. To adjust for bias, a popular approach is to assume a genotype error model for describing the pattern of errors and propose association tests using likelihood method. However, all model-based approaches tend to perform unsatisfactorily if the related genotyping error rates are not identical across all families. In this paper, we propose a TDT-type association test which is not only simple, robust against population stratification (and hence the assumption of Hardy-Weinberg equilibrium is not required), but also robust against genotyping error with error rates varying across families. Simulation studies confirm that the new test has very reasonable performance.     

Estimation of a parameter and its exact confidence interval following sequential sample size reestimation trials

For confirmatory trials of regulatory decision making, it is important that adaptive designs under consideration provide inference with the correct nominal level, as well as unbiased estimates, and confidence intervals for the treatment comparisons in the actual trials. However, naive point estimate and its confidence interval are often biased in adaptive sequential designs. We develop a new procedure for estimation following a test from a sample size reestimation design. The method for obtaining an exact confidence interval and point estimate is based on a general distribution property of a pivot function of the Self-designing group sequential clinical trial by Shen and Fisher (1999, Biometrics55, 190-197). A modified estimate is proposed to explicitly account for futility stopping boundary with reduced bias when block sizes are small. The proposed estimates are shown to be consistent. The computation of the estimates is straightforward. We also provide a modified weight function to improve the power of the test. Extensive simulation studies show that the exact confidence intervals have accurate nominal probability of coverage, and the proposed point estimates are nearly unbiased with practical sample sizes.     

Product limit estimation for infectious disease data when the diagnostic test for the outcome is measured with uncertainty

Low sensitivity and/or specificity of a diagnostic test for outcome results in biased estimates of the time to first event using product limit estimation. For example, if a test has low specificity, estimates of the cumulative distribution function (cdf) are biased towards time zero, while estimates of the cdf are biased away from time zero if a test has low sensitivity. In the context of discrete time survival analysis for infectious disease data, we develop self-consistent algorithms to obtain unbiased estimates of the time to first event when the sensitivity and/or specificity of the diagnostic test for the outcome is less than 100%. Two examples are presented. The first involves estimating time to first detection of HIV-1 infection in infants in a randomized clinical trial, and the second involves estimating time to first Neisseria gonorrhoeae infection in a cohort of Kenyan prostitutes.     

The problems of using the transmission/disequilibrium test to infer tight linkage

Family-based association methods such as the transmission/disequilibrium test (TDT) have become very popular during the past few years, often being preferred to case-control studies because family-based approaches avoid the difficulties of ascertainment of appropriate populations of cases and controls for case-control studies. Significant TDT results indicate both linkage and allelic association. However, significant TDT results are often interpreted as implying tight linkage of marker and disease locus, and we shall argue here that, in general, this interpretation is not justified.     

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.     

Multi‐allelic exact tests for Hardy–Weinberg equilibrium that account for gender

Statistical tests for Hardy–Weinberg equilibrium are important elementary tools in genetic data analysis. X‐chromosomal variants have long been tested by applying autosomal test procedures to females only, and gender is usually not considered when testing autosomal variants for equilibrium. Recently, we proposed specific X‐chromosomal exact test procedures for bi‐allelic variants that include the hemizygous males, as well as autosomal tests that consider gender. In this study, we present the extension of the previous work for variants with multiple alleles. A full enumeration algorithm is used for the exact calculations of tri‐allelic variants. For variants with many alternate alleles, we use a permutation test. Some empirical examples with data from the 1,000 genomes project are discussed.     

The Knox method and other tests for space-time interaction

The Knox method, as well as other tests for space-time interaction, are biased when there are geographical population shifts, i.e., when there are different percent population growths in different regions. In this paper, the size of the population shift bias is investigated for the Knox test, and it is shown that it can be a considerable problem. A Monte Carlo method for constructing unbiased space-time interaction tests is then presented and illustrated on the Knox test as well as for a combined Knox test. Practical implications are discussed in terms of the interpretation of past results and the design of future studies.     

Selection against hybrids in mixed populations of Brassica rapa and Brassica napus: model and synthesis

Pollen of the crop oilseed rape (Brassica napus, AACC) can cross-fertilize ovules of Brassica rapa (AA), which leads to an influx of unpaired C-chromosomes into wild B. rapa populations. The presence of such extra chromosomes is thought to be an indicator of introgression. Backcrosses and F(1) hybrids were found in Danish populations but, surprisingly, only F(1) hybrids were found in the UK and the Netherlands. Here, a model tests how the level of selection and biased vs unbiased transmission affect the population frequency of C-chromosomes. In the biased-transmission scenario the experimental results of the first backcross are extrapolated to estimate survival of gametes with different numbers of C-chromosomes from all crosses in the population. With biased transmission, the frequency of C-chromosomes always rapidly declines to zero. With unbiased transmission, the continued presence of plants with extra C-chromosomes depends on selection in the adult stage and we argue that this is the most realistic option for modeling populations. We suggest that selection in the field against plants with unpaired C-chromosomes is strong in Dutch and UK populations. The model highlights what we do not know and makes suggestions for further research on introgression.