Biased estimators of quantitative trait locus heritability and location in interval mapping

In many empirical studies, it has been observed that genome scans yield biased estimates of heritability, as well as genetic effects. It is widely accepted that quantitative trait locus (QTL) mapping is a model selection procedure, and that the overestimation of genetic effects is the result of using the same data for model selection as estimation of parameters. There are two key steps in QTL modeling, each of which biases the estimation of genetic effects. First, test procedures are employed to select the regions of the genome for which there is significant evidence for the presence of QTL. Second, and most important for this demonstration, estimates of the genetic effects are reported only at the locations for which the evidence is maximal. We demonstrate that even when we know there is just one QTL present (ignoring the testing bias), and we use interval mapping to estimate its location and effect, the estimator of the effect will be biased. As evidence, we present results of simulations investigating the relative importance of the two sources of bias and the dependence of bias of heritability estimators on the true QTL heritability, sample size, and the length of the investigated part of the genome. Moreover, we present results of simulations demonstrating the skewness of the distribution of estimators of QTL locations and the resulting bias in estimation of location. We use computer simulations to investigate the dependence of this bias on the true QTL location, heritability, and the sample size.     

Estimation of Epistatic Variance Components and Heritability in Founder Populations and Crosses

Genetic association studies have explained only a small proportion of the estimated heritability of complex traits, leaving the remaining heritability “missing.” Genetic interactions have been proposed as an explanation for this, because they lead to overestimates of the heritability and are hard to detect. Whether this explanation is true depends on the proportion of variance attributable to genetic interactions, which is difficult to measure in outbred populations. Founder populations exhibit a greater range of kinship than outbred populations, which helps in fitting the epistatic variance. We extend classic theory to founder populations, giving the covariance between individuals due to epistasis of any order. We recover the classic theory as a limit, and we derive a recently proposed estimator of the narrow sense heritability as a corollary. We extend the variance decomposition to include dominance. We show in simulations that it would be possible to estimate the variance from pairwise interactions with samples of a few thousand from strongly bottlenecked human founder populations, and we provide an analytical approximation of the standard error. Applying these methods to 46 traits measured in a yeast (Saccharomyces cerevisiae) cross, we estimate that pairwise interactions explain 10% of the phenotypic variance on average and that third- and higher-order interactions explain 14% of the phenotypic variance on average. We search for third-order interactions, discovering an interaction that is shared between two traits. Our methods will be relevant to future studies of epistatic variance in founder populations and crosses.     

Closed-Form Approximations to the REML Estimator of a Variance Ratio (or Heritability) in a Mixed Linear Model

In this article, we estimate heritability or intraclass correlation in a mixed linear model having two sources of variation. In most applications, the commonly used restricted maximum likelihood (REML) estimator can only be obtained via an iterative approach. In some cases, the algorithm used to compute REML estimates may be slow or may even fail to converge. We develop a set of closed-form approximations to the REML estimator, and the performance of these estimators is compared with that of the REML estimator. We provide guidelines regarding how to choose the estimator that best approximates the REML estimator. Examples presented in the article suggest that the closed-form estimators compete with and, in some cases, outperform the REML estimator.     

A novel penalized inverse-variance weighted estimator for Mendelian randomization with applications to COVID-19 outcomes

Mendelian randomization utilizes genetic variants as instrumental variables (IVs) to estimate the causal effect of an exposure variable on an outcome of interest even in the presence of unmeasured confounders. However, the popular inverse-variance weighted (IVW) estimator could be biased in the presence of weak IVs, a common challenge in MR studies. In this article, we develop a novel penalized inverse-variance weighted (pIVW) estimator, which adjusts the original IVW estimator to account for the weak IV issue by using a penalization approach to prevent the denominator of the pIVW estimator from being close to zero. Moreover, we adjust the variance estimation of the pIVW estimator to account for the presence of balanced horizontal pleiotropy. We show that the recently proposed debiased IVW (dIVW) estimator is a special case of our proposed pIVW estimator. We further prove that the pIVW estimator has smaller bias and variance than the dIVW estimator under some regularity conditions. We also conduct extensive simulation studies to demonstrate the performance of the proposed pIVW estimator. Furthermore, we apply the pIVW estimator to estimate the causal effects of five obesity-related exposures on three coronavirus disease 2019 (COVID-19) outcomes. Notably, we find that hypertensive disease is associated with an increased risk of hospitalized COVID-19; and peripheral vascular disease and higher body mass index are associated with increased risks of COVID-19 infection, hospitalized COVID-19, and critically ill COVID-19.     

Anatomic brain asymmetry in vervet monkeys

Asymmetry is a prominent feature of human brains with important functional consequences. Many asymmetric traits show population bias, but little is known about the genetic and environmental sources contributing to inter-individual variance. Anatomic asymmetry has been observed in Old World monkeys, but the evidence for the direction and extent of asymmetry is equivocal and only one study has estimated the genetic contributions to inter-individual variance. In this study we characterize a range of qualitative and quantitative asymmetry measures in structural brain MRIs acquired from an extended pedigree of Old World vervet monkeys (n = 357), and implement variance component methods to estimate the proportion of trait variance attributable to genetic and environmental sources. Four of six asymmetry measures show pedigree-level bias and one of the traits has a significant heritability estimate of about 30%. We also found that environmental variables more significantly influence the width of the right compared to the left prefrontal lobe.     

The estimation of genetic relationships using molecular markers and their efficiency in estimating heritability in natural populations

Molecular marker data collected from natural populations allows information on genetic relationships to be established without referencing an exact pedigree. Numerous methods have been developed to exploit the marker data. These fall into two main categories: method of moment estimators and likelihood estimators. Method of moment estimators are essentially unbiased, but utilise weighting schemes that are only optimal if the analysed pair is unrelated. Thus, they differ in their efficiency at estimating parameters for different relationship categories. Likelihood estimators show smaller mean squared errors but are much more biased. Both types of estimator have been used in variance component analysis to estimate heritability. All marker-based heritability estimators require that adequate levels of the true relationship be present in the population of interest and that adequate amounts of informative marker data are available. I review the different approaches to relationship estimation, with particular attention to optimizing the use of this relationship information in subsequent variance component estimation.     

Application of approximate variances of variance components and their ratios in genetic tests

The application and underlying assumptions of formulae used to estimate the variance of variance components and ratios of variance components are fully described for (1) variance components estimated using Henderson's Method 3 (HM3) and Restricted Maximum Likelihood (REML) and (2) ratios of variance components commonly used in genetic tests — biased and unbiased heritabilities. A first-order Taylor series approximation is often used to estimate the variance of a ratio of two random variables (e.g., heritability), however the formula is complicated, thus making calculations prone to errors. Dickerson's approximation is considerably simpler, though relatively rarely used. In case studies using data from 148 slash pine full-sib progeny tests, Dickerson's method was found to be slightly more conservative than the Taylor series approximation when estimating the variance of heritability estimates, regardless of test size, age, or the trait (volume, which is a continuous trait, and rust resistance, which is a bernoulli trait). Both the Taylor series and Dickerson approximations compared favorably with an empirical estimate of the variance of heritability estimates, however there is some evidence of small-sample bias associated with the use of the asymptotic variance-covariances from REML variance component estimation.This is Florida Agricultural Experiment Station Journal Series No. R-03964 of the Institute of Food and Agricultural Sciences, University of Florida, Gainesville, FL 32611, USA     

Incorporating Estimates of Group Size in Sightability Models for Wildlife

ABSTRACT Sightability models have been used to estimate population size of many wildlife species; however, a limitation of these models is an assumption that groups of animals observed and counted during aerial surveys are enumerated completely. Replacing these unknown counts with maximum observed counts, as is typically done, produces population size estimates that are negatively biased. This bias can be substantial depending on the degree of undercounting occurring. We first investigated a method-of-moments estimator of group sizes. We then defined a population size estimator using the method-of-moments estimator of group sizes in place of maximum counts in the traditional sightability models, thereby correcting for bias associated with undercounting group size. We also provide associated equations for calculating the variance of our estimator. This estimator is an improvement over existing sightability model techniques because it significantly reduces bias, and variance estimates provide near nominal confidence interval coverage. The data needed for this estimator can be easily collected and implemented by wildlife managers with a field crew of only 3 individuals and little additional flight or personnel time beyond the normal requirements for developing sightability models.     

Assumptions and Properties of Limiting Pathway Models for Analysis of Epistasis in Complex Traits

For most complex traits, results from genome-wide association studies show that the proportion of the phenotypic variance attributable to the additive effects of individual SNPs, that is, the heritability explained by the SNPs, is substantially less than the estimate of heritability obtained by standard methods using correlations between relatives. This difference has been called the “missing heritability”. One explanation is that heritability estimates from family (including twin) studies are biased upwards. Zuk et al. revisited overestimation of narrow sense heritability from twin studies as a result of confounding with non-additive genetic variance. They propose a limiting pathway (LP) model that generates significant epistatic variation and its simple parametrization provides a convenient way to explore implications of epistasis. They conclude that over-estimation of narrow sense heritability from family data (‘phantom heritability’) may explain an important proportion of missing heritability. We show that for highly heritable quantitative traits large phantom heritability estimates from twin studies are possible only if a large contribution of common environment is assumed. The LP model is underpinned by strong assumptions that are unlikely to hold, including that all contributing pathways have the same mean and variance and are uncorrelated. Here, we relax the assumptions that underlie the LP model to be more biologically plausible. Together with theoretical, empirical, and pragmatic arguments we conclude that in outbred populations the contribution of additive genetic variance is likely to be much more important than the contribution of non-additive variance.     

Bias in the prediction of genetic gain due to mass and half-sib selection in random mating populations

The prediction of gains from selection allows the comparison of breeding methods and selection strategies, although these estimates may be biased. The objective of this study was to investigate the extent of such bias in predicting genetic gain. For this, we simulated 10 cycles of a hypothetical breeding program that involved seven traits, three population classes, three experimental conditions and two breeding methods (mass and half-sib selection). Each combination of trait, population, heritability, method and cycle was repeated 10 times. The predicted gains were biased, even when the genetic parameters were estimated without error. Gain from selection in both genders is twice the gain from selection in a single gender only in the absence of dominance. The use of genotypic variance or broad sense heritability in the predictions represented an additional source of bias. Predictions based on additive variance and narrow sense heritability were equivalent, as were predictions based on genotypic variance and broad sense heritability. The predictions based on mass and family selection were suitable for comparing selection strategies, whereas those based on selection within progenies showed the largest bias and lower association with the realized gain.     

Frequentist estimation of coalescence times from nucleotide sequence data using a tree-based partition

This article proposes a method of estimating the time to the most recent common ancestor (TMRCA) of a sample of DNA sequences. The method is based on the molecular clock hypothesis, but avoids assumptions about population structure. Simulations show that in a wide range of situations, the point estimate has small bias and the confidence interval has at least the nominal coverage probability. We discuss conditions that can lead to biased estimates. Performance of this estimator is compared with existing methods based on the coalescence theory. The method is applied to sequences of Y chromosomes and mtDNAs to estimate the coalescent times of human male and female populations.     

Bias in the heritability of preference and its potential impact on the evolution of mate choice

The evolution of mate choice is a function of the heritability of preference. Estimation in the laboratory is typically made by presenting a female with a limited number of males. We show that such an approach produces a downwardly biased estimate, which we term the heritability of choice. When preference is treated as a threshold trait then less biased estimates are obtained particularly for preferences based on the relative value of the preferred trait. Because females in the wild typically survey on average less than five males we argue that the heritability of choice may be more meaningful than the heritability of preference. The restricted number of males surveyed can lead to a reduction in the phenotypic variance of the preferred trait in the group of males selected by the females if the phenotypic variance in preference is equal to or less than the phenotypic variance in the referred trait. If the phenotypic variance in preference exceeds that of the preferred trait then the opposite occurs. A second effect of the restricted number of males sampled is that females are likely to mate initially with males that are not the most preferred. The failure to find the most preferred male may account for the common observation of multiple matings and extra-pair copulations. We suggest that current explanations for polyandry need to take this failure into account.     

Using maximum likelihood to estimate population size from temporal changes in allele frequencies.

We develop a maximum-likelihood framework for using temporal changes in allele frequencies to estimate the number of breeding individuals in a population. We use simulations to compare the performance of this estimator to an F-statistic estimator of variance effective population size. The maximum-likelihood estimator had a lower variance and smaller bias. Taking advantage of the likelihood framework, we extend the model to include exponential growth and show that temporal allele frequency data from three or more sampling events can be used to test for population growth.     

Methods for estimating the parameters of a linear model for ordered categorical data.

In many empirical analyses, the response of interest is categorical with an ordinal scale attached. Many investigators prefer to formulate a linear model, assigning scores to each category of the ordinal response and treating it as continuous. When the covariates are categorical, Haber (1985, Computational Statistics and Data Analysis 3, 1-10) has developed a method to obtain maximum likelihood (ML) estimates of the parameters of the linear model using Lagrange multipliers. However, when the covariates are continuous, the only method we found in the literature is ordinary least squares (OLS), performed under the assumption of homogeneous variance. The OLS estimates are unbiased and consistent but, since variance homogeneity is violated, the OLS estimates of variance can be biased and may not be consistent. We discuss a variance estimate (White, 1980, Econometrica 48, 817-838) that is consistent for the true variance of the OLS parameter estimates. The possible bias encountered by using the naive OLS variance estimate is discussed. An estimated generalized least squares (EGLS) estimator is proposed and its efficiency relative to OLS is discussed. Finally, an empirical comparison of OLS, EGLS, and ML estimators is made.     

Estimation of Heritability in Heteroscedastic One‐Way Unbalanced Random Model

A modified estimator of heritability is proposed under heteroscedastic one way unbalanced random model. The distribution, moments and probability of permissible values (PPV) for conventional and modified estimators are derived. The behaviour of two estimators has been investigated, numerically, to devise a suitable estimator of heritability under variance heterogeneity. The numerical results reveal that under balanced case the heteroscedasticity affects the bias, MSE and PPV of conventional estimator, marginally. In case of unbalanced situations, the conventional estimator underestimates the parameter when more variable group has more observations and overestimates when more variable group has less observations, MSE of the conventional estimator decreases when more variable group has more observations and increases when more variable group has less observations and PPV is marginally decreased. The MSE and PPV are comparable for two estimators while the bias of modified estimator is less than the conventional estimator particularly for small and medium values of the parameter. These results suggest the use of modified estimator with equal or more observations for more variable group in presence of variance heterogeneity.     

Blinded versus unblinded estimation of a correlation coefficient to inform interim design adaptations

Regulatory authorities require that the sample size of a confirmatory trial is calculated prior to the start of the trial. However, the sample size quite often depends on parameters that might not be known in advance of the study. Misspecification of these parameters can lead to under‐ or overestimation of the sample size. Both situations are unfavourable as the first one decreases the power and the latter one leads to a waste of resources. Hence, designs have been suggested that allow a re‐assessment of the sample size in an ongoing trial. These methods usually focus on estimating the variance. However, for some methods the performance depends not only on the variance but also on the correlation between measurements. We develop and compare different methods for blinded estimation of the correlation coefficient that are less likely to introduce operational bias when the blinding is maintained. Their performance with respect to bias and standard error is compared to the unblinded estimator. We simulated two different settings: one assuming that all group means are the same and one assuming that different groups have different means. Simulation results show that the naïve (one‐sample) estimator is only slightly biased and has a standard error comparable to that of the unblinded estimator. However, if the group means differ, other estimators have better performance depending on the sample size per group and the number of groups.     

Testing marker-based estimates of heritability in the wild

Marker-based estimates of heritability are an attractive alternative to pedigree-based methods for estimating quantitative genetic parameters in field studies where it is difficult or impossible to determine relationships and pedigrees. Here I test the ability of the marker-based method to estimate heritability of a suite of traits in a wild population of bighorn sheep (Ovis canadensis) using marker data from 32 microsatellite loci. I compared marker-based estimates with estimates obtained using a pedigree and the animal model. Marker-based estimates of heritability were imprecise and downwardly biased. The high degree of uncertainty in marker-based estimates suggests that the method may be sufficient to detect the presence of genetic variance for highly heritable traits, but not sufficiently reliable to estimate genetic parameters.     

The accuracy of a heritability estimator using molecular information

The heritability of a quantitative trait is a key parameter to quantify the genetic variation present in a population. Although estimates of heritability require accurate information on the genetic relationship among individuals, pedigree data is generally lacking in natural populations. Nowadays, the increasing availability of DNA markers is making possible the estimation of coancestries from neutral molecular information. In 1996, K. Ritland developed an approach to estimate heritability from the regression of the phenotypic similarity on the marker-based coancestry. We carried out simulations to analyze the accuracy of the estimates of heritability obtained by this method using information from a variable number of neutral codominant markers. Because the main application of the estimator is on populations with no family structure, such as natural populations, its accuracy was tested under this scenario. However, the method was also investigated under other scenarios, in order to test the influence of different factors (family structure, assortative mating and phenotypic selection) on the precision. Our results suggest that the main factor causing a directional bias in the estimated heritability is the presence of phenotypic selection, and that very noisy estimates are obtained in the absence of a familiar structure and for small population sizes. The estimated heritabilities from marker-based coancestries showed lower accuracy than the estimated heritabilities from genealogical coancestries. However, a large amount of bias occurred even in the most favourable situation where genealogical coancestries are known. The results also indicate that the molecular markers are more suitable to infer coancestry than inbreeding.     

Empirical Bayes Estimation and Prediction Using Summary-Level Information From External Big Data Sources Adjusting for Violations of Transportability

Large external data sources may be available to augment studies that collect data to address a specific research objective. In this article we consider the problem of building regression models for prediction based on individual-level data from an “internal” study while incorporating summary information from an “external” big data source. We extend the work of Chatterjee et al. (J Am Stat Assoc 111(513):107–117, 2006) by introducing an adaptive empirical Bayes shrinkage estimator that uses the external summary-level information and the internal data to trade bias with variance for protection against departures in the conditional probability distribution of the outcome given a set of covariates between the two populations. We use simulation studies and a real data application using external summary information from the Prostate Cancer Prevention Trial to assess the performance of the proposed methods in contrast to maximum likelihood estimation and the constrained maximum likelihood (CML) method developed by Chatterjee et al. (J Am Stat Assoc 111(513):107–117, 2006). Our simulation studies show that the CML method can be biased and inefficient when the assumption of a transportable covariate distribution between the external and internal populations is violated, and our empirical Bayes estimator provides protection against bias and loss of efficiency.     

On the Estimation of Heritability from Unbalanced Data

The distribution and moments, of ANOVA estimator of heritability are given under unbalanced random model. These expressions are used to investigate the effect of unbalancedness on the bias and variance/MSE of the estimator and also the validity of certain approximations for its variance, numerically. The computed results reveal that the unbalancedness increases both the bias and variance/MSE of the estimator and the Smith-approximation for the variance of the estimator provides better accuracy.