Comparison of 'model-free' and 'model-based' linkage statistics in the presence of locus heterogeneity: single data set and multiple data set applications

Earlier work [Knapp et al.: Hum Hered 1994;44:44-51] focusing on affected sib pair (ASP) data established the equivalence between the mean test and a test based on a simple recessive lod score, as well as equivalences between certain forms of the maximum likelihood score (MLS) statistic [Risch: Am J Hum Genet 1990;46:242-253] and particular forms of the lod score. Here we extend the results of Knapp et al. [1994] by reconsidering these equivalences for ASP data, but in the presence of locus heterogeneity. We show that Risch's MLS statistic under the possible triangle constraints [Holmans: Am J Hum Genet 1993;52:362-374] is locally equivalent to the ordinary heterogeneity lod score assuming a simple recessive model (HLOD/R); while the one-parameter MLS assuming no dominance variance is locally equivalent to the (homogeneity) recessive lod. The companion paper (this issue, pp 199-208) showed that when considering multiple data sets in the presence of locus heterogeneity, the HLOD can suffer appreciable losses in power. We show here that in ASP data, these equivalences ensure that this same loss in power is incurred by both forms of the MLS statistic as well. The companion paper also introduced an adaptation of the lod, the compound lod score (HLOD/C). We confirm that the HLOD/C maintains higher power than these 'model-free' methods when applied to multiple heterogeneous data sets, even when it is calculated assuming the wrong genetic model.     

Mapping quantitative trait loci using multiple phenotypes in general pedigrees

The use of correlated phenotypes may dramatically increase the power to detect the underlying quantitative trait loci (QTLs). Current approaches for multiple phenotypes include regression-based methods, the multivariate variance of components method, factor analysis and structural equations. Issues with these methods include: 1) They are computation intensive and are subject to problems of optimization algorithms; 2) Existing claims on the asymptotic distribution of the likelihood ratio statistic for the multivariate variance of components method are contradictory and erroneous; 3) The dimension reduction of the parameter space under the null hypothesis, a phenomenon that is unique to multivariate analyses, makes the asymptotic distribution of the likelihood ratio statistic more complicated than expected. In this article, three cases of varying complexity are considered. For each case, the efficient score statistic, which is asympotically equivalent to the likelihood ratio statistic, is derived, so is its asymptotic distribution [correction]. These methods are straightforward to calculate. Finite-sample properties of these score statistics are studied through extensive simulations. These score statistics are for use with general pedigrees.     

A Scan Statistic for Binary Outcome Based on Hypergeometric Probability Model,with an Application to Detecting Spatial Clusters of Japanese Encephalitis

As a useful tool for geographical cluster detection of events, the spatial scan statistic is widely applied in many fields and plays an increasingly important role. The classic version of the spatial scan statistic for the binary outcome is developed by Kulldorff, based on the Bernoulli or the Poisson probability model. In this paper, we apply the Hypergeometric probability model to construct the likelihood function under the null hypothesis. Compared with existing methods, the likelihood function under the null hypothesis is an alternative and indirect method to identify the potential cluster, and the test statistic is the extreme value of the likelihood function. Similar with Kulldorff’s methods, we adopt Monte Carlo test for the test of significance. Both methods are applied for detecting spatial clusters of Japanese encephalitis in Sichuan province, China, in 2009, and the detected clusters are identical. Through a simulation to independent benchmark data, it is indicated that the test statistic based on the Hypergeometric model outweighs Kulldorff’s statistics for clusters of high population density or large size; otherwise Kulldorff’s statistics are superior.     

Detection of linkage for heterogeneous disorders by using multipoint linkage analysis.

We have compared the efficiency of the lod score test which assumes heterogeneity (lod2) to the standard lod score test which assumes homogeneity (lod1) when three-point linkage analysis is used in successive map intervals. If it is assumed that a gene located midway between two linked marker loci is responsible for a proportion of disease cases, then the lod1 test loses power relative to the lod2 test, as the proportion of linked families decreases, as the flanking markers are more closely linked, and as more map intervals are tested. Moreover, when multipoint analysis is used, linkage for a disease gene is more likely to be incorrectly excluded from a complete and dense linkage map if true genetic heterogeneity is ignored. We thus conclude that, in general, the lod2 linkage test is more efficient for detecting a true linkage when a complete genetic marker map is screened for a heterogeneous disorder.     

Statistical inference for stochastic simulation models--theory and application

Statistical models are the traditional choice to test scientific theories when observations, processes or boundary conditions are subject to stochasticity. Many important systems in ecology and biology, however, are difficult to capture with statistical models. Stochastic simulation models offer an alternative, but they were hitherto associated with a major disadvantage: their likelihood functions can usually not be calculated explicitly, and thus it is difficult to couple them to well-established statistical theory such as maximum likelihood and Bayesian statistics. A number of new methods, among them Approximate Bayesian Computing and Pattern-Oriented Modelling, bypass this limitation. These methods share three main principles: aggregation of simulated and observed data via summary statistics, likelihood approximation based on the summary statistics, and efficient sampling. We discuss principles as well as advantages and caveats of these methods, and demonstrate their potential for integrating stochastic simulation models into a unified framework for statistical modelling.     

Numerical comparison between powers of maximum likelihood and analysis of variance methods for QTL detection in progeny test designs: the case of monogenic inheritance

Simulations are used to compare four statistics for the detection of a quantitative trait locus (QTL) in daughter and grand-daughter designs as defined by Soller and Genizi (1978) and Weller et al. (1990): (1) the Fisher test of a linear model including a marker effect within sire or grand-sire effect; (2) the likelihood ratio test of a segregation analysis without the information given by the marker; (3) the likelihood ratio test of a segregation analysis considering the information from the marker; and (4) the lod score which is the likelihood ratio test of absence of linkage between the marker and the QTL. In all cases the two segregation analyses are more powerful for QTL detection than are either the linear method or the lod score. The differences in power are generally limited but may be significant (in a ratio of 1 to 3 or 4) when the QTL has a small effect (0.2 standard deviations) and is not closely linked to the marker (recombination rate of 20% or more).     

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.     

Testing the Hardy-Weinberg equilibrium in the HLA system

A new method of testing the Hardy-Weinberg equilibrium in the human leukocyte antigen (HLA) system is proposed and applied to real data. The derivation is based on the maximum likelihood method and closely related to standard regression theory. The test statistic has a closed representation of residual sum of squares by a projection mapping of data onto the estimated regression plane. Under the Hardy-Weinberg law the noniterative estimates for the gene frequencies are suggested by the use of the projection mapping. The test statistic and gene frequency estimates are shown to be asymptotically equivalent to the maximum likelihood method and to be more efficient than the other suggested test statistic when there are more than two identified alleles.     

On the use of historical control data to estimate dose response trends in quantal bioassay.

New tests for trend in proportions, in the presence of historical control data, are proposed. One such test is a simple score statistic based on a binomial likelihood for the "current" study and beta-binomial likelihoods for each historical control series. A closely related trend statistic based on estimating equations is also proposed. Trend statistics that allow overdispersed proportions in the current study are also developed, including a version of Tarone's (1982, Biometrics 38, 215-220) test that acknowledges sampling variation in the beta distribution parameters, and a trend statistic based on estimating equations. Each such trend test is evaluated with respect to size and power under both binomial and beta-binomial sampling conditions for the current study, and illustrations are provided.     

A score test for linkage analysis of ordinal traits based on IBD sharing

Statistical methods for linkage analysis are well established for both binary and quantitative traits. However, numerous diseases including cancer and psychiatric disorders are rated on discrete ordinal scales. To analyze pedigree data with ordinal traits, we recently proposed a latent variable model which has higher power to detect linkage using ordinal traits than methods using the dichotomized traits. The challenge with the latent variable model is that the likelihood is usually very complicated, and as a result, the computation of the likelihood ratio statistic is too intensive for large pedigrees. In this paper, we derive a computationally efficient score statistic based on the identity-by-decent sharing information between relatives. Using simulation studies, we examined the asymptotic distribution of the test statistic and the power of our proposed test under various levels of heritability. We compared the computing time as well as power of the score test with the likelihood ratio test. We then applied our method for the Collaborative Study on the Genetics of Alcoholism and performed a genome scan to map susceptibility genes for alcohol dependence. We found a strong linkage signal on chromosome 4.     

A comparison of test statistics for assessing the effects of concomitant variables in survival analysis

In data analysis involving the proportional-hazards regression model due to Cox (1972, Journal of the Royal Statistical Society, Series B 34, 187-220), the test criteria commonly used for assessing the partial contribution to survival of subsets of concomitant variables are the classical likelihood ratio (LR) and Wald statistics. This paper presents an investigation of three other test criteria with potentially major computational advantages over the classical tests, especially for stepwise variable selection in moderate to large data sets. The alternative criteria considered are Rao's efficient score statistic and two other score statistics. Under the Cox model, the performance of these tests is examined empirically and compared with the performance of the LR and Wald statistics. Rao's test performs comparably to the LR test in all the cases considered. The performance of the other criteria is competitive in many cases. The use of these statistics is illustrated in a study of coronary artery disease.     

Efficient strategies for genomic searching using the affected-pedigree-member method of linkage analysis.

The affected-pedigree-member (APM) method of linkage analysis is a nonparametric statistic that tests for nonrandom cosegregation of a disease and marker loci. The APM statistic is based on the observation that if a marker locus is near a disease-susceptibility locus, then affected individuals within a family should be more similar at the marker locus than is expected by chance. The APM statistic measures marker similarity in terms of identity by state (IBS) of marker alleles; that is, two alleles are IBS if they are the same, regardless of their ancestral origin. Since the APM statistic measures increased marker similarity, it makes no assumptions concerning how the disease is inherited; this can be an advantage when dealing with complex diseases for which the mode of inheritance is difficult to determine. We investigate here the power of the APM statistic to detect linkage in the context of a genomewide search. In such a search, the APM statistic is evaluated at a grid of markers. Then regions with high APM statistics are investigated more thoroughly by typing more markers in the region. Using simulated data, we investigate various search strategies and recommend an optimal search strategy that maximizes the power to detect linkage while minimizing the false-positive rate and number of markers. We determine an optimal series of three increasing cut-points and an independent criterion for significance.     

Incorporating Functional Information in Tests of Excess De Novo Mutational Load

A number of recent studies have investigated the role of de novo mutations in various neurodevelopmental and neuropsychiatric disorders. These studies attempt to implicate causal genes by looking for an excess load of de novo mutations within those genes. Current statistical methods for assessing this excess are based on the implicit assumption that all qualifying mutations in a gene contribute equally to disease. However, it is well established that different mutations can have radically different effects on the ultimate protein product and, as a result, on disease risk. Here, we propose a method, fitDNM, that incorporates functional information in a test of excess de novo mutational load. Specifically, we derive score statistics from a retrospective likelihood that incorporates the probability of a mutation being damaging to gene function. We show that, under the null, the resulting test statistic is distributed as a weighted sum of Poisson random variables and we implement a saddlepoint approximation of this distribution to obtain accurate p values. Using simulation, we have shown that our method outperforms current methods in terms of statistical power while maintaining validity. We have applied this approach to four de novo mutation datasets of neurodevelopmental and neuropsychiatric disorders: autism spectrum disorder, epileptic encephalopathy, schizophrenia, and severe intellectual disability. Our approach also implicates genes that have been implicated by existing methods. Furthermore, our approach provides strong statistical evidence supporting two potentially causal genes: SUV420H1 in autism spectrum disorder and TRIO in a combined analysis of the four neurodevelopmental and neuropsychiatric disorders investigated here.     

The ML estimation and testing of generalized ABO-like data with no observed double recessives

The general method of the discrepancy or heterogeneity chi-square is applied to ABO-like data in which there are no observed double blanks in either the disease or the control group. When the recessive gene frequency is assumed zero, this method leads to an approximate chi-square test identical to that suggested by Smouse and Williams (1982, Biometrics 38, 757-768). When this assumption is relaxed, there arise two cases which are determined by whether the maximum likelihood estimate of this frequency is zero or not. It is shown that the value of the simple score statistic of Gart and Nam (1984, Biometrics 40, 887-894) discriminates between the two cases. The various omnibus test statistics for comparing groups are shown to differ little in several practical examples. However, under the more general assumption the appropriate degrees of freedom is one more than the number previously suggested.     

A multilocus extension of the affected-pedigree-member method of linkage analysis.

The affected-pedigree-member (APM) method of linkage analysis is designed to detect departures from independent segregation of disease and marker phenotypes. The underlying statistic of the APM method operates on the identity-by-state relations implied by the marker phenotypes of the affected within a pedigree. Here we generalize the APM statistic to multiple linked markers. This generalization relies on recursive computation of two-locus kinship coefficients by an algorithm of Thompson. The distributional properties of the extended APM statistic are investigated theoretically and by simulation in the context of one real and one artificial data set. In both examples, the multilocus statistic tends to reject, more strongly than the single-locus statistics do, the null hypothesis of independent segregation between the disease locus and the marker loci.     

A Score-Statistic Approach for the Mapping of Quantitative-Trait Loci with Sibships of Arbitrary Size

The Haseman-Elston method is widely used for the mapping of quantitative-trait loci. However, this method does not use all the information in the data, because it only considers the sib-pair trait-value difference. In addition, the Haseman-Elston method was developed for independent sib pairs; its generalization to nonindependent sib pairs is not straightforward. Here we introduce a score test statistic derived from a normal likelihood based on multiplex sibship data, conditional on identical-by-descent sharing statuses. This score test is asymptotically equivalent to the corresponding likelihood-ratio test, but it is much easier to implement. Because the proposed test uses all of the trait values, it makes more efficient use of the data than does the Haseman-Elston method. The proposed test is naturally applicable to sibships of arbitrary size. The finite-sample properties of the proposed score statistic are evaluated via simulations.     

Robust analysis with related samples under the presence of population substructure and its application to body mass index

Recent investigations such as a more powerful quasi-likelihoods score test (MQLS) statistic have enabled the efficient association analysis with related samples. Although those approaches are robust against the mis-specified phenotypic distribution and covariance structure, it has been shown that MQLS statistic becomes violated under the presence of the population substructure if the level of population substructure depends on the genomic location. In this report, we propose a new statistical method which combines EIGENSTRAT approach and MQLS-statistic. The proposed method was evaluated with simulation data under various scenarios and we found that proposed method performs better than the traditional methods such as transmission disequilibrium test. The proposed method was applied to genetic association analysis for body mass index with Framingham heart study, and we found that rs1121980 and rs9940128 in the linkage block in FTO gene are associated with the body mass index.     

Combining information within and between pedigrees for mapping complex traits.

This paper is concerned with efficient strategies for gene mapping using pedigrees containing small numbers of affecteds and identity-by-descent data from closely spaced markers throughout the genome. Particular attention is paid to additive traits involving phenocopies and/or locus heterogeneity. For a sample of pedigrees containing a particular configuration of affecteds, e.g., pairs of siblings together with a first cousin, we use a likelihood analysis to find 1-df statistics that are very efficient over a broad range of penetrances and allele frequencies. We identify configurations of affecteds that are particularly powerful for detecting linkage, and we show how pedigrees containing different numbers and configurations of affecteds can be efficiently combined in an overall test statistic.     

Homogeneity test of rate ratios in stratified matched-pair studies

This paper investigates homogeneity test of rate ratios in stratified matched-pair studies on the basis of asymptotic and bootstrap-resampling methods. Based on the efficient score approach, we develop a simple and computationally tractable score test statistic. Several other homogeneity test statistics are also proposed on the basis of the weighted least-squares estimate and logarithmic transformation. Sample size formulae are derived to guarantee a pre-specified power for the proposed tests at the pre-given significance level. Empirical results confirm that (i) the modified score statistic based on the bootstrap-resampling method performs better in the sense that its empirical type I error rate is much closer to the pre-specified nominal level than those of other tests and its power is greater than those of other tests, and is hence recommended, whilst the statistics based on the weighted least-squares estimate and logarithmic transformation are slightly conservative under some of the considered settings; (ii) the derived sample size formulae are rather accurate in the sense that their empirical powers obtained from the estimated sample sizes are very close to the pre-specified nominal powers. A real example is used to illustrate the proposed methodologies.     

Adjusted exponentially tilted likelihood with applications to brain morphology

Summary .  In this article, we develop a nonparametric method, called adjusted exponentially tilted (ET) likelihood, and apply it to the analysis of morphometric measures. The adjusted exponential tilting estimator is shown to have the same first-order asymptotic properties as that of the original ET likelihood. The adjusted ET likelihood ratio statistic is applied to test linear hypotheses of unknown parameters, such as the associations of brain measures (e.g., cortical and subcortical surfaces) with covariates of interest, such as age, gender, and gene. Simulation studies show that the adjusted exponential tilted likelihood ratio statistic performs as well as the  t -test when the imaging data are symmetrically distributed, while it is superior when the imaging data have skewed distribution. We demonstrate the application of our new statistical methods to the detection of statistically significant differences in the morphology of the hippocampus between two schizophrenia groups and healthy subjects.