A Brief History of the Collaboration between Dr Nathan Gillespie and Professor Nick Martin Including Personal Reflections.

This article describes Dr Nathan Gillespie's PhD training and supervision under Professor Nick Martin and their ongoing collaborations. Drs Gillespie and Martin have collaborated on numerous biometrical genetic analyses applied to cross-sectional and longitudinal twin data, combined molecular and phenotypic modeling, as well as genomewide meta-analyses of psychoactive substance use and misuse. Dr Gillespie remains an active collaborator with Professor Martin, including ongoing data collection, analysis and publications related to the Brisbane Longitudinal Twin Study.

Statistical Power and the Classical Twin Design.

Dr Nick Martin has made enormous contributions to the field of behavior genetics over the past 50 years. Of his many seminal papers that have had a profound impact, we focus on his early work on the power of twin studies. He was among the first to recognize the importance of sample size calculation before conducting a study to ensure sufficient power to detect the effects of interest. The elegant approach he developed, based on the noncentral chi-squared distribution, has been adopted by subsequent researchers for other genetic study designs, and today remains a standard tool for power calculations in structural equation modeling and other areas of statistical analysis. The present brief article discusses the main aspects of his seminal paper, and how it led to subsequent developments, by him and others, as the field of behavior genetics evolved into the present era.

Nicholas (Nick) G. Martin and the Extended Twin Model.

The extended twin model is a unique design in the genetic epidemiology toolbox that allows to simultaneously estimate multiple causes of variation such as genetic and cultural transmission, genotype-environment covariance and assortative mating, among others. Nick Martin has played a key role in the conception of the model, the collection of substantially large data sets to test the model, the application of the model to a range of phenotypes, the publication of the results including cross-cultural comparisons, the evaluation of bias and power of the design and the further elaborations of the model, such as the children-of-twins design.

The Chinese National Twin Registry: a 'gold mine' for scientific research.

The Chinese National Twin Registry (CNTR) currently includes data from 61 566 twin pair from 11 provinces or cities in China. Of these, 31 705, 15 060 and 13 531 pairs are monozygotic, same-sex dizygotic and opposite-sex dizygotic pairs, respectively, determined by opposite sex or intrapair similarity. Since its establishment in 2001, the CNTR has provided an important resource for analysing genetic and environmental influences on chronic diseases especially cardiovascular diseases. Recently, the CNTR has focused on collecting biologic specimens from disease-concordant or disease-discordant twin pairs or from twin pairs reared apart. More than 8000 pairs of these twins have been registered, and blood samples have been collected from more than 1500 pairs. In this review, we summarize the main findings from univariate and multivariate genetic effects analyses, gene-environment interaction studies, omics studies exploring DNA methylation and metabolomic markers associated with phenotypes. There remains further scope for CNTR research and data mining. The plan for future development of the CNTR is described. The CNTR welcomes worldwide collaboration.

Type I Error Rates and Parameter Bias in Multivariate Behavioral Genetic Models.

For many multivariate twin models, the numerical Type I error rates are lower than theoretically expected rates using a likelihood ratio test (LRT), which implies that the significance threshold for statistical hypothesis tests is more conservative than most twin researchers realize. This makes the numerical Type II error rates higher than theoretically expected. Furthermore, the discrepancy between the observed and expected error rates increases as more variables are included in the analysis and can have profound implications for hypothesis testing and statistical inference. In two simulation studies, we examine the Type I error rates for the Cholesky decomposition and Correlated Factors models. Both show markedly lower than nominal Type I error rates under the null hypothesis, a discrepancy that increases with the number of variables in the model. In addition, we observe slightly biased parameter estimates for the Cholesky decomposition and Correlated Factors models. By contrast, if the variance-covariance matrices for variance components are estimated directly (without constraints), the numerical Type I error rates are consistent with theoretical expectations and there is no bias in the parameter estimates regardless of the number of variables analyzed. We call this the direct symmetric approach. It appears that each model-implied boundary, whether explicit or implicit, increases the discrepancy between the numerical and theoretical Type I error rates by truncating the sampling distributions of the variance components and inducing bias in the parameters. The direct symmetric approach has several advantages over other multivariate twin models as it corrects the Type I error rate and parameter bias issues, is easy to implement in current software, and has fewer optimization problems. Implications for past and future research, and potential limitations associated with direct estimation of genetic and environmental covariance matrices are discussed.

The heritability of human disease: estimation, uses and abuses.

Relatives provide the basic material for the study of inheritance of human disease. However, the methodologies for the estimation of heritability and the interpretation of the results have been controversial. The debate arises from the plethora of methods used, the validity of the methodological assumptions and the inconsistent and sometimes erroneous genetic interpretations made. We will discuss how to estimate disease heritability, how to interpret it, how biases in heritability estimates arise and how heritability relates to other measures of familial disease aggregation.

The latent class twin method.

The twin method refers to the use of data from same-sex identical and fraternal twins to estimate the genetic and environmental contributions to a trait or outcome. The standard twin method is the variance component twin method that estimates heritability, the fraction of variance attributed to additive genetic inheritance. The latent class twin method estimates two quantities that are easier to interpret than heritability: the genetic prevalence, which is the fraction of persons in the genetic susceptibility latent class, and the heritability fraction, which is the fraction of persons in the genetic susceptibility latent class with the trait or outcome. We extend the latent class twin method in three important ways. First, we incorporate an additive genetic model to broaden the sensitivity analysis beyond the original autosomal dominant and recessive genetic models. Second, we specify a separate survival model to simplify computations and improve convergence. Third, we show how to easily adjust for covariates by extending the method of propensity scores from a treatment difference to zygosity. Applying the latent class twin method to data on breast cancer among Nordic twins, we estimated a genetic prevalence of 1%, a result with important implications for breast cancer prevention research.

Marginal genetic effects estimation in family and twin studies using random-effects models.

Random-effects models are often used in family-based genetic association studies to properly capture the within families relationships. In such models, the regression parameters have a conditional on the random effects interpretation and they measure, e.g., genetic effects for each family. Estimating parameters that can be used to make inferences at the population level is often more relevant than the family-specific effects, but not straightforward. This is mainly for two reasons: First the analysis of family data often requires high-dimensional random-effects vectors to properly model the familial relationships, for instance when members with a different degree of relationship are considered, such as trios, mix of monozygotic and dizygotic twins, etc. The second complication is the biased sampling design, such as the multiple cases families design, which is often employed to enrich the sample with genetic information. For these reasons deriving parameters with the desired marginal interpretation can be challenging. In this work we consider the marginalized mixed-effects models, we discuss challenges in applying them in ascertained family data and propose penalized maximum likelihood methodology to stabilize the parameter estimation by using external information on the disease prevalence or heritability. The performance of our methodology is evaluated via simulation and is illustrated on data from Rheumatoid Arthritis patients, where we estimate the marginal effect of HLA-DRB1*13 and shared epitope alleles across three different study designs and combine them using meta-analysis.

VEGAS2: Software for More Flexible Gene-Based Testing.

Gene-based tests such as versatile gene-based association study (VEGAS) are commonly used following per-single nucleotide polymorphism (SNP) GWAS (genome-wide association studies) analysis. Two limitations of VEGAS were that the HapMap2 reference set was used to model the correlation between SNPs and only autosomal genes were considered. HapMap2 has now been superseded by the 1,000 Genomes reference set, and whereas early GWASs frequently ignored the X chromosome, it is now commonly included. Here we have developed VEGAS2, an extension that uses 1,000 Genomes data to model SNP correlations across the autosomes and chromosome X. VEGAS2 allows greater flexibility when defining gene boundaries. VEGAS2 offers both a user-friendly, web-based front end and a command line Linux version. The online version of VEGAS2 can be accessed through https://vegas2.qimrberghofer.edu.au/. The command line version can be downloaded from https://vegas2.qimrberghofer.edu.au/zVEGAS2offline.tgz. The command line version is developed in Perl, R and shell scripting languages; source code is available for further development.

Operating Characteristics of Statistical Methods for Detecting Gene-by-Measured Environment Interaction in the Presence of Gene-Environment Correlation under Violations of Distributional Assumptions.

Accurately identifying interactions between genetic vulnerabilities and environmental factors is of critical importance for genetic research on health and behavior. In the previous work of Van Hulle et al. (Behavior Genetics, Vol. 43, 2013, pp. 71-84), we explored the operating characteristics for a set of biometric (e.g., twin) models of Rathouz et al. (Behavior Genetics, Vol. 38, 2008, pp. 301-315), for testing gene-by-measured environment interaction (GxM) in the presence of gene-by-measured environment correlation (rGM) where data followed the assumed distributional structure. Here we explore the effects that violating distributional assumptions have on the operating characteristics of these same models even when structural model assumptions are correct. We simulated N = 2,000 replicates of n = 1,000 twin pairs under a number of conditions. Non-normality was imposed on either the putative moderator or on the ultimate outcome by ordinalizing or censoring the data. We examined the empirical Type I error rates and compared Bayesian information criterion (BIC) values. In general, non-normality in the putative moderator had little impact on the Type I error rates or BIC comparisons. In contrast, non-normality in the outcome was often mistaken for or masked GxM, especially when the outcome data were censored.

Estimating heritability for cause specific mortality based on twin studies.

There has been considerable interest in studying the magnitude and type of inheritance of specific diseases. This is typically derived from family or twin studies, where the basic idea is to compare the correlation for different pairs that share different amount of genes. We here consider data from the Danish twin registry and discuss how to define heritability for cancer occurrence. The key point is that this should be done taking censoring as well as competing risks due to e.g.  death into account. We describe the dependence between twins on the probability scale and show that various models can be used to achieve sensible estimates of the dependence within monozygotic and dizygotic twin pairs that may vary over time. These dependence measures can subsequently be decomposed into a genetic and environmental component using random effects models. We here present several novel models that in essence describe the association in terms of the concordance probability, i.e., the probability that both twins experience the event, in the competing risks setting. We also discuss how to deal with the left truncation present in the Nordic twin registries, due to sampling only of twin pairs where both twins are alive at the initiation of the registries.

How meaningful are heritability estimates of liability?

It is commonly acknowledged that estimates of heritability from classical twin studies have many potential shortcomings. Despite this, in the post-GWAS era, these heritability estimates have come to be a continual source of interest and controversy. While the heritability estimates of a quantitative trait are subject to a number of biases, in this article we will argue that the standard statistical approach to estimating the heritability of a binary trait relies on some additional untestable assumptions which, if violated, can lead to badly biased estimates. The ACE liability threshold model assumes at its heart that each individual has an underlying liability or propensity to acquire the binary trait (e.g., disease), and that this unobservable liability is multivariate normally distributed. We investigated a number of different scenarios violating this assumption such as the existence of a single causal diallelic gene and the existence of a dichotomous exposure. For each scenario, we found that substantial asymptotic biases can occur, which no increase in sample size can remove. Asymptotic biases as much as four times larger than the true value were observed, and numerous cases also showed large negative biases. Additionally, regions of low bias occurred for specific parameter combinations. Using simulations, we also investigated the situation where all of the assumptions of the ACE liability model are met. We found that commonly used sample sizes can lead to biased heritability estimates. Thus, even if we are willing to accept the meaningfulness of the liability construct, heritability estimates under the ACE liability threshold model may not accurately reflect the heritability of this construct. The points made in this paper should be kept in mind when considering the meaningfulness of a reported heritability estimate for any specific disease.

Visual analysis of geocoded twin data puts nature and nurture on the map.

Twin studies allow us to estimate the relative contributions of nature and nurture to human phenotypes by comparing the resemblance of identical and fraternal twins. Variation in complex traits is a balance of genetic and environmental influences; these influences are typically estimated at a population level. However, what if the balance of nature and nurture varies depending on where we grow up? Here we use statistical and visual analysis of geocoded data from over 6700 families to show that genetic and environmental contributions to 45 childhood cognitive and behavioral phenotypes vary geographically in the United Kingdom. This has implications for detecting environmental exposures that may interact with the genetic influences on complex traits, and for the statistical power of samples recruited for genetic association studies. More broadly, our experience demonstrates the potential for collaborative exploratory visualization to act as a lingua franca for large-scale interdisciplinary research.

Sibling comparison designs: bias from non-shared confounders and measurement error.

Twins, full siblings, and half-siblings are increasingly used as comparison groups in matched cohort and matched case-control studies. The "within-pair" estimates acquired through these comparisons are free from confounding from all factors that are shared by the siblings. This has made sibling comparisons popular in studying associations thought likely to suffer confounding from socioeconomic or genetic factors. Despite the wide application of these designs in epidemiology, they have received little scrutiny from a statistical or methodological standpoint. In this paper we show, analytically and through a series of simulations, that the standard interpretation of the models is subject to several limitations that are rarely acknowledged.Although within-pair estimates will not be confounded by factors shared by the siblings, such estimates are more severely biased by non-shared confounders than the unpaired estimate. If siblings are less similar with regard to confounders than to the exposure under study, the within-pair estimate will always be more biased than the ordinary unpaired estimate. Attenuation of associations due to random measurement error in exposure will also be higher in the within-pair estimate, leading within-pair associations to be weaker than corresponding unpaired associations, even in the absence of confounding. Implications for the interpretation of sibling comparison results are discussed.

Adjusted confidence intervals for a bounded parameter.

It is well known that the regular likelihood ratio test of a bounded parameter is not valid if the boundary value is being tested. This is the case for testing the null value of a scalar variance component. Although an adjusted test of variance component has been suggested to account for the effect of its lower bound of zero, no adjustment of its interval estimate has ever been proposed. If left unadjusted, the confidence interval of the variance may still contain zero when the adjusted test rejects the null hypothesis of a zero variance, leading to conflicting conclusions. In this research, we propose two ways to adjust the confidence interval of a parameter subject to a lower bound, one based on the Wald test and the other on the likelihood ratio test. Both are compatible to the adjusted test and parametrization-invariant. A simulation study and two examples are given in the framework of ACDE models in twin studies.

Detecting specific genotype by environment interactions using marginal maximum likelihood estimation in the classical twin design.

Considerable effort has been devoted to the analysis of genotype by environment (G × E) interactions in various phenotypic domains, such as cognitive abilities and personality. In many studies, environmental variables were observed (measured) variables. In case of an unmeasured environment, van der Sluis et al. (2006) proposed to study heteroscedasticity in the factor model using only MZ twin data. This method is closely related to the Jinks and Fulker (1970) test for G × E, but slightly more powerful. In this paper, we identify four challenges to the investigation of G × E in general, and specifically to the heteroscedasticity approaches of Jinks and Fulker and van der Sluis et al. We propose extensions of these approaches purported to solve these problems. These extensions comprise: (1) including DZ twin data, (2) modeling both A × E and A × C interactions; and (3) extending the univariate approach to a multivariate approach. By means of simulations, we study the power of the univariate method to detect the different G × E interactions in varying situations. In addition, we study how well we could distinguish between A × E, A × C, and C × E. We apply a multivariate version of the extended model to an empirical data set on cognitive abilities.

Likelihood-based concordance tests for analysis of ascertained twin pair multinomial data.

The classic twin model design has a wide application in human genetics. Under the assumption that nongenetic effects are shared to the same degree by monozygotic (MZ) and dizygotic (DZ) twin pairs, a test of the equality of casewise concordances between MZ and DZ twins provides a clue to the influence of genetic and environmental factors on a disease. The casewise concordance is the conditional probability that given that one member of a twin pair is affected, the other is also affected. When disease prevalence is low or cost-effectiveness is considered, collection of twin pairs by ascertainment for performing casewise concordance analysis is required. In this article, by defining an overall casewise concordance parameter, several likelihood-based tests, such as likelihood ratio test LR, score test Score, the usual Wald test Wald and an alternative Wald test WaldA are investigated for a test of the equality of concordances between ascertained MZ and DZ twin pairs under multinomial models. Simulation studies were conducted for data with small sample sizes. The results show that the type I error rates and power of LR and Score are stable only when the overall casewise concordances are not extremely small or large. The Wald has higher power performance in most cases but would slightly inflate type I error rates; the WaldA is the most robust and recommended approach.

Parametric model-based statistics for possible genotyping errors and sample stratification in sibling-pair SNP data.

The detection of genotyping errors, based on apparent Mendelian incompatibilities in a sample of sib-pairs, is a complicated problem. In the case of a single marker and unknown parental genotypes, all combinations of sib-pair genotypes are self-consistent. Moreover, the observed deviation from equilibrium genotype frequencies may result from genotyping errors as well as from the sample's stratification. This in turn, may profoundly affect the results of association and linkage analyses, and therefore an estimation of these factors should be done beforehand. Here we present several parametric models, and using likelihood ratio statistics, we suggest a method of combined analysis of genotyping errors and a sample stratification for randomly ascertained sib-pair single nucleotide polymorphism (SNP) data. Specifically, we implemented two models of genotyping errors in either heterozygotes or homozygotes, and two models of sample stratification resulting from either the presence of families of different ethnic origin (e.g., a population admixture) or from a different ethnic origin of the parents in the family (e.g., intermarriage). The power of this method was established by Monte Carlo data simulation. The results clearly suggest that the proposed method is most efficient for detecting genotyping errors in heterozygotes, a common error caused by incorrect SNP data interpretation. We also provide an example of its application to real data.

Using non-normal SEM to resolve the ACDE model in the classical twin design.

One of the biggest problems in classical twin studies is that it cannot estimate additive genetic (A), non-additive genetic (D), shared environmental (C), and non-shared environmental (E) effects, simultaneously, because the model, referred to as the ACDE model, has negative degrees of freedom when using Structural Equation Modeling (SEM). Therefore, instead of the ACDE model, the ACE model or the ADE model is actually used. However, using the ACE or ADE models almost always leads to biased estimates. In the present paper, the univariate ACDE model is developed using non-normal Structural Equation Modeling (nnSEM). In SEM, (1st- and) 2nd-order moments, namely, (means and) covariances are used as information. However, nnSEM uses higher-order moments as well as (1st- and) 2nd-order moments. nnSEM has a number of advantages over SEM. One of which is that nnSEM can specify models that cannot be specified using SEM because of the negative degrees of freedom. Simulation studies have shown that the proposed method can decrease the biases. There are other factors that have possible effects on phenotypes, such as higher-order epistasis. Since the proposed method cannot estimate these effects, further research on developing a more exhaustive model is needed.

Statistical inference in mixed models and analysis of twin and family data.

Analysis of data from twin and family studies provides the foundation for studies of disease inheritance. The development of advanced theory and computational software for general linear models has generated considerable interest for using mixed-effect models to analyze twin and family data, as a computationally more convenient and theoretically more sound alternative to the classical structure equation modeling. Despite the long history of twin and family data analysis, some fundamental questions remain unanswered. We addressed two important issues. One is to determine the necessary and sufficient conditions for the identifiability in the mixed-effects models for twin and family data. The other is to derive the asymptotic distribution of the likelihood ratio test, which is novel due to the fact that the standard regularity conditions are not satisfied. We considered a series of specific yet important examples in which we demonstrated how to formulate mixed-effect models to appropriately reflect the data, and our key idea is the use of the Cholesky decomposition. Finally, we applied our method and theory to provide a more precise estimate of the heritability of two data sets than the previously reported estimate.