Asymptotic properties of likelihood ratio test statistics in affected‐sib‐pair analysis

In an affected‐sib‐pair genetic linkage analysis, identical by descent data for affected sib pairs are routinely collected at a large number of markers along chromosomes. Under very general genetic assumptions, the IBD distribution at each marker satisfies the possible triangle constraint. Statistical analysis of IBD data should thus utilize this information to improve efficiency. At the same time, this constraint renders the usual regularity conditions for likelihood‐based statistical methods unsatisfied. In this paper, the authors study the asymptotic properties of the likelihood ratio test (LRT) under the possible triangle constraint. They derive the limiting distribution of the LRT statistic based on data from a single locus. They investigate the precision of the asymptotic distribution and the power of the test by simulation. They also study the test based on the supremum of the LRT statistics over the markers distributed throughout a chromosome. Instead of deriving a limiting distribution for this test, they use a mixture of chi‐squared distributions to approximate its true distribution. Their simulation results show that this approach has desirable simplicity and satisfactory precision.     

Power and robustness of sib-pair linkage tests and extension to larger sibships

We investigate the power and robustness of Haseman and Elstonfs sib-pair test for genetic linkage between a marker locus and a locus affecting a quantitative trait, and compare the test to that of Penrose. The Haseman-Elston test is more powerful than Penrrose's test; its power is acceptable for cases of tight linkage and high heritability due to the hypothesized quantitative trait locus, but is quite low in other situations. Computer simulations indicate that both tests are valid for normally distributed trait values, and that the Haseman-Elston test is robust for a variety of continuous distributions of the trait values. Several linkage tests are developed for sib trios that are much more powerful , for the same total number of sibs, than the test on independent sib pairs. The Haseman-Elston test on all possible sib pairs is suggested for sibships of size larger than three and for samples including sibships of various sizes.     

Fisher information in randomly sampled sib pairs and extremely discordant sib pairs in genetic analysis for a quantitative trait locus

The conditional mixture likelihood method using the absolute difference of the trait values of a sib pair to estimate genetic parameters underlies commonly used method in linkage analysis. Here, the statistical properties of the model are examined. The marginal model with a pseudo-likelihood function based on a sample of the absolute difference of sib-traits is also studied. Both approaches are compared numerically. When genotyping is much more expensive than screening a quantitative trait, it is known that extremely discordant sib pairs provide more powerful linkage tests than randomly sampled sib pairs. The Fisher information about genetic parameters contained in extremely discordant sib pairs is calculated using the marginal mixture model. Our results supplement current research showing that extremely discordant sib pairs are powerful for the linkage detection by demonstrating they also contain more information about other genetic parameters.     

Locating genes involved in human diseases

The increasing amount of information that is becoming available about the structure and composition of the DNA constituting the human chromosomes has provided new opportunities to locate genes that affect susceptibilities to a range of diseases. The accurate location of these genes is important in genetic counselling and in understanding the effects of genes that may result in disease. Various methods of analysing the data when DNA information is available at a single marker locus for an affected child and his or her parents are reviewed and applied to data on insulin-dependent diabetes mellitus . The importance of distinguishing between the association of alleles at a marker locus and at a disease locus resulting from chromosomal linkage from that resulting from other causes is emphasized.     

Mapping quantitative trait loci using both extremely discordant and concordant sib pairs: a unified approach for meta-analysis

Extremely discordant (ED) and concordant (EC) sib pairs are recommended by Risch and Zhang ( 1995, 1996 ) and Zhang and Risch ( 1996 ) for the detection of linkage on the basis of power analysis. They did not, however, include both discordant and concordant sib pairs simultaneously in the test statistic. We construct a new test statistic including both types of sib pairs. The size of the sample needed to identify the sib pairs and parents, who will be genotyped, is reduced substantially by including a hybrid of discordant and concordant sib pairs in the analysis. This mixed design also provides a common parameter that simplifies meta-analysis of sib pair linkage studies.     

Sample sizes for the transmission disequilibrium tests: tdt,s-tdt and 1-tdt

The transmission/disequilibrium test (TDT) is widely used to detect the linkage disequilibrium between a candidate locus (a marker) and a disease locus. The TDT is a family-based design and has the advantage that it is a valid test when population stratification exist. The TDT requires the marker genotypes of affected individuals and their parents. For diseases with late age of onset, it is difficult or impossible to obtain the marker genotype of the parents. Therefore, when both parents marker genotypes are unavailable, Ewex and Spielman extended the TDT to the S-TDT for use in sibships with at least one affected individual and one unaffected individual. When only one of the parents' genotype is available. Sun et al. proposed a test the 1-TDT, for use with niarker genotypes of affected individuals and only one available parent. Here, we study the saniple sizes of TDT, S-TDT, and 1-TDT. We show that the sample size needed for the 1-TDT is rogghly the same as the sample size needed for the S-TDT with two sibs and is about twice the sample size for the TDT.     

Some Statistical Properties of Efficiency Robust Tests with Applications to Genetic Association Studies

Although efficiency robust tests are preferred for genetic association studies when the genetic model is unknown, their statistical properties have been studied for different study designs separately under special situations. We study some statistical properties of the maximin efficiency robust test and a maximum‐type robust test (MAX3) under a general setting and obtain unified results. The results can also be applied to testing hypothesis with a constrained two‐dimensional parameter space. The results are applied to genetic association studies using case–parents trio data.     

Carry‐over in cross‐over trials in bioequivalence: theoretical concerns and empirical evidence

There is now general agreement that pre‐testing for carry‐over in the AB/BA design is harmful and that efficient analysis of this design must proceed on the assumption that carry‐over has not affected the results to any appreciable degree. A general consensus has not been achieved in the case of higher‐order designs. Since particular forms of carry‐over can be estimated on a within‐patient basis and unbiased within‐patient treatment estimators are possible, some statisticians favour pre‐testing and some favour automatic adjustment for carry‐over. We present theoretical arguments that show that, just as in the AB/BA case, the strategy of pre‐testing is biased as a whole and also that the loss in terms of efficiency in adjusting is not negligible. We also present data from two large series of bioequivalence studies to provide empirical evidence that in this context carry‐over is either absent or rare. We conclude that adjusting or testing for carry‐over in bioequivalence studies is at worst harmful and at best pointless, and that this may also apply to other kinds of study. Copyright © 2004 John Wiley & Sons, Ltd.     

A review of statistical methods in imaging genetics

With the rapid growth of modern technology, many biomedical studies are being conducted to collect massive datasets with volumes of multi‐modality imaging, genetic, neurocognitive and clinical information from increasingly large cohorts. Simultaneously extracting and integrating rich and diverse heterogeneous information in neuroimaging and/or genomics from these big datasets could transform our understanding of how genetic variants impact brain structure and function, cognitive function and brain‐related disease risk across the lifespan. Such understanding is critical for diagnosis, prevention and treatment of numerous complex brain‐related disorders (e.g., schizophrenia and Alzheimer's disease). However, the development of analytical methods for the joint analysis of both high‐dimensional imaging phenotypes and high‐dimensional genetic data, a big data squared (BD²) problem, presents major computational and theoretical challenges for existing analytical methods. Besides the high‐dimensional nature of BD², various neuroimaging measures often exhibit strong spatial smoothness and dependence and genetic markers may have a natural dependence structure arising from linkage disequilibrium. We review some recent developments of various statistical techniques for imaging genetics, including massive univariate and voxel‐wise approaches, reduced rank regression, mixture models and group sparse multi‐task regression. By doing so, we hope that this review may encourage others in the statistical community to enter into this new and exciting field of research. The Canadian Journal of Statistics 47: 108–131; 2019 © 2019 Statistical Society of Canada     

An Additive Genetic Gamma Frailty Model for Linkage Analysis of Diseases with Variable Age of Onset Using Nuclear Families

Many late-onset complex diseases exhibit variable age of onset. Efficiently incorporating age of onset information into linkage analysis can potentially increase the power of dissecting complex diseases. In this paper, we treat age of onset as a genetic trait with censored observations. We use multiple markers to infer the inheritance vector at the disease susceptibility (DS) locus in order to extract information about the inheritance pattern of the disease allele in a pedigree. Given the inheritance distribution at the DS locus, we define the genetic frailty for each individual within a nuclear family as the sum of frailties due to a putative major disease gene and a polygenic effect due to any remaining DS loci. Conditioning on these frailties we use the proportional hazards model for the risk of developing disease. We show that a test of linkage can be formulated as a test of zero variance due to a specific locus of the additive gamma frailties. Maximum likelihood estimation, using the EM algorithm, and likelihood ratio tests are employed for parameter estimation and tests of linkage. A simulation study presented indicates that the proposed method is well behaved and can be more powerful than the currently available allele-sharing based linkage methods. A breast cancer data example is used for illustration.     

Loss functions for estimation of extrema with an application to disease mapping

It is often of interest to find the maximum or near maxima among a set of vector‐valued parameters in a statistical model; in the case of disease mapping, for example, these correspond to relative‐risk “hotspots” where public‐health intervention may be needed. The general problem is one of estimating nonlinear functions of the ensemble of relative risks, but biased estimates result if posterior means are simply substituted into these nonlinear functions. The authors obtain better estimates of extrema from a new, weighted ranks squared error loss function. The derivation of these Bayes estimators assumes a hidden‐Markov random‐field model for relative risks, and their behaviour is illustrated with real and simulated data.     

Cost‐efficient higher‐order crossover designs for two‐treatment clinical trials

Higher‐order crossover designs have drawn considerable attention in clinical trials, because of their ability to test direct treatment effects in the presence of carry‐over effects. The important question, when applying higher‐order crossover designs in practice, is how to choose a design with both statistical and cost efficiencies from various alternatives. In this paper, we propose a general cost function and compare five statistically optimal or near‐optimal designs with this cost function for a two‐treatment study under different carry‐over models. Based on our study, to achieve both statistical and cost efficiencies, a four‐period, four‐sequence crossover design is generally recommended under the simple carry‐over or no carry‐over models, and a three‐period, two‐sequence crossover design is generally recommended under the steady‐state carry‐over models. Copyright © 2005 John Wiley & Sons, Ltd.     

The role of reversals in order‐restricted inference

A statistical model is said to be an order‐restricted statistical model when its parameter takes its values in a closed convex cone C of the Euclidean space. In recent years, order‐restricted likelihood ratio tests and maximum likelihood estimators have been criticized on the grounds that they may violate a cone order monotonicity (COM) property, and hence reverse the cone order induced by C. The authors argue here that these reversals occur only in the case that C is an obtuse cone, and that in this case COM is an inappropriate requirement for likelihood‐based estimates and tests. They conclude that these procedures thus remain perfectly reasonable procedures for order‐restricted inference.     

On the Optimality of Multivariate S‐Estimators

Abstract. In this article, we maximize the efficiency of a multivariate S‐estimator under a constraint on the breakdown point. In the linear regression model, it is known that the highest possible efficiency of a maximum breakdown S‐estimator is bounded above by 33 per cent for Gaussian errors. We prove the surprising result that in dimensions larger than one, the efficiency of a maximum breakdown S‐estimator of location and scatter can get arbitrarily close to 100 per cent, by an appropriate selection of the loss function.     

Inference for domains under imputation for missing survey data

The authors study the estimation of domain totals and means under survey‐weighted regression imputation for missing items. They use two different approaches to inference: (i) design‐based with uniform response within classes; (ii) model‐assisted with ignorable response and an imputation model. They show that the imputed domain estimators are biased under (i) but approximately unbiased under (ii). They obtain a bias‐adjusted estimator that is approximately unbiased under (i) or (ii). They also derive linearization variance estimators. They report the results of a simulation study on the bias ratio and efficiency of alternative estimators, including a complete case estimator that requires the knowledge of response indicators.     

Score tests for heterogeneity and overdispersion in zero‐inflated Poisson and binomial regression models

Hall (2000) has described zero‐inflated Poisson and binomial regression models that include random effects to account for excess zeros and additional sources of heterogeneity in the data. The authors of the present paper propose a general score test for the null hypothesis that variance components associated with these random effects are zero. For a zero‐inflated Poisson model with random intercept, the new test reduces to an alternative to the overdispersion test of Ridout, Demério & Hinde (2001). The authors also examine their general test in the special case of the zero‐inflated binomial model with random intercept and propose an overdispersion test in that context which is based on a beta‐binomial alternative.     

A likelihood ratio test for goodness‐of‐fit of recessive and dominant models for case–control studies

Testing goodness‐of‐fit of commonly used genetic models is of critical importance in many applications including association studies and testing for departure from Hardy–Weinberg equilibrium. Case–control design has become widely used in population genetics and genetic epidemiology, thus it is of interest to develop powerful goodness‐of‐fit tests for genetic models using case–control data. This paper develops a likelihood ratio test (LRT) for testing recessive and dominant models for case–control studies. The LRT statistic has a closed‐form formula with a simple $\chi^{2}(1)$ null asymptotic distribution, thus its implementation is easy even for genome‐wide association studies. Moreover, it has the same power and optimality as when the disease prevalence is known in the population. The Canadian Journal of Statistics 41: 341–352; 2013 © 2013 Statistical Society of Canada     

Ascertainment-Adjusted Maximum Likelihood Estimation for the Additive Genetic Gamma Frailty Model

The additive genetic gamma frailty model has been proposed for genetic linkage analysis for complex diseases to account for variable age of onset and possible covariates effects. To avoid ascertainment biases in parameter estimates, retrospective likelihood ratio tests are often used, which may result in loss of efficiency due to conditioning. This paper considers when the sibships are ascertained by having at least two affected sibs with the disease before a given age and provides two approaches for estimating the parameters in the additive gamma frailty model. One approach is based on the likelihood function conditioning on the ascertainment event, the other is based on maximizing a full ascertainment-adjusted likelihood. Explicit forms for these likelihood functions are derived. Simulation studies indicate that when the baseline hazard function can be correctly pre-specified, both approaches give accurate estimates of the model parameters. However, when the baseline hazard function has to be estimated simultaneously, only the ascertainment-adjusted likelihood method gives an unbiased estimate of the parameters. These results imply that the ascertainment-adjusted likelihood ratio test in the context of the additive genetic gamma frailty may be used for genetic linkage analysis.     

Testing for order among K populations: theory and examples

Testing for stochastic order among K populations is a common and important problem in statistical practice. It arises in the analysis of both planned experiments and observational studies. The authors develop a new nonparametric test for order among K populations that can accommodate any stochastic ordering. The test is based on a maximally selected chi‐bar‐square statistic. The authors find its limiting distribution and use simulations to derive critical values. Three important examples are used to illustrate the applicability of the general method. The authors find that the new tests outperform the existing methods in many practical cases. The Canadian Journal of Statistics 38: 97–115; 2010 © 2009 Statistical Society of Canada     

On the Ghoudi,Khoudraji, and Rivest test for extreme‐value dependence

Ghoudi, Khoudraji & Rivest [The Canadian Journal of Statistics 1998;26:187–197] showed how to test whether the dependence structure of a pair of continuous random variables is characterized by an extreme‐value copula. The test is based on a U‐statistic whose finite‐ and large‐sample variance are determined by the present authors. They propose estimates of this variance which they compare to the jackknife estimate of Ghoudi, Khoudraji & Rivest ( 1998 ) through simulations. They study the finite‐sample and asymptotic power of the test under various alternatives. They illustrate their approach using financial and geological data. The Canadian Journal of Statistics © 2009 Statistical Society of Canada