Using Stochastic Approximation Techniques to Efficiently Construct Confidence Intervals for Heritability

2.1. A faster method for calculating CIs for heritability

CIs constructed from SEs, which are based on the assumption of a normal distribution for the heritability estimators, were previously shown to be inaccurate (Lohr and Divan, 1997; Burch, 2007; Burch, 2011; Kraemer, 2012; Kruijer et al., 2015; Schweiger et al., 2016). In this article, we introduce FIESTA, a method that generates accurate CIs for h², the true heritability value, given \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \hat h^2}$$ \end{document} , the REML estimator for h² (see section 3). FIESTA uses the principle of test inversion to construct accurate CIs, using a stochastic approximation method that directly estimates the CI boundaries. We review FIESTA hereunder; for a full description, see section 3.

The methodology of test inversion can be described as follows. The estimator \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \hat h^2}$$ \end{document} is a function of the phenotype, which is a random variable whose distribution depends on h², assuming a fixed kinship matrix. Therefore, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \hat h^2}$$ \end{document} is distributed differently for every value of h². For each true value of h², we select a subset of possible \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \hat h^2}$$ \end{document} values that has a sampling probability of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$1 - \alpha$$ \end{document} , where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \hat h^2}$$ \end{document} is distributed under the assumption of a true heritability value h². We define this subset to be the acceptance region for that value of h². The CI accompanying an estimate \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \hat h^2}$$ \end{document} is the interval containing all values of h² whose acceptance region includes \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \hat h^2}$$ \end{document} , namely, for which \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \hat h^2}$$ \end{document} does not imply the rejection of the null hypothesis that the true heritability value is h², with a significance level of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\alpha$$ \end{document} .

It remains to define suitable acceptance regions. In section 3, we review our scheme for defining acceptance regions. A basic ingredient of our construction of acceptance regions is inverting certain quantile functions of the distribution of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \hat h^2}$$ \end{document} , as a function of h². For example, finding the inverse of a value H² of the 95%-quantile function is finding a heritability value h² for which \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\mathop {{ \rm{P}}r} \nolimits_{{h^2}} ( { \hat h^2} \le {H^2} ) = 0.95$$ \end{document} , that is, the probability to get an heritability estimate of H² or below is precisely 95%, when \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \hat h^2}$$ \end{document} is distributed with the heritability value h².

Instead of carrying out this task by a full parametric bootstrap estimate of the distribution of the estimator, we employ a technique from the field of stochastic approximation to achieve the same results with a fraction of the computational cost. The modified Robbins–Monro procedure (Joseph, 2004), described in section 3, is an iterative method that finds the inverse of the quantile function of a one-parameter distribution. It operates by iteratively (1) drawing a sample with a true heritability value equal to our current guess for the required inverse value, (2) comparing its estimated heritability to H², (3) updating our current guess accordingly, by moving in the right direction, with a step size that decreases with the number of iterations. An additional speedup is acquired by using a fast method to calculate the derivative of likelihood of the sample, and using the derivative to compare its estimated heritability to H², instead of performing the full likelihood maximization.

We applied FIESTA to construct 95% CIs for the NFBC data set (Sabatti et al., 2009) and the WTCCC2 data set (Sawcer et al., 2011), as seen in Figure 1. We then turned to verify the accuracy of these CIs, which can be measured as follows. Draw multiple phenotype vectors from the distribution assumed by the LMM with parameters that correspond to a true heritability value h². From each such phenotype, construct a CI for its estimated heritability with a confidence level of, for example, 95%. If the constructed CIs are accurate, then they should cover the true underlying h² 95% of the time. Then, check the percentage of times in which the CI covered h², as a function of h². We measured the accuracy of FIESTA, with CIs designed to have a coverage of 95%. The results are shown in Figure 2, demonstrating that FIESTA accurately achieves the desired confidence levels.

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FIG. 1.

The 95% CIs for the NFBC and WTCCC2 data sets. Accurate 95% CIs constructed for the NFBC data set (Sabatti et al., 2009) (left) and the WTCCC2 (Sawcer et al., 2011) data set (right) by FIESTA. For each \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \hat h^2}$$ \end{document} on a fine grid of 1000 values (x axis), we constructed a CI, whose boundaries are shown (y axis). For example, for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \hat h^2} = 0.5$$ \end{document} (denoted by a dashed line), the CI for NFBC is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$[ 0.282 , 0.705 ]$$ \end{document} (denoted by a full line). CI, confidence interval; FIESTA, Fast confidence IntErvals using STochastic Approximation; NFBC, Northern Finland Birth Cohort; WTCCC2, Wellcome Trust Case Control Consortium 2.

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FIG. 2.

Accuracy of CIs for the NFBC and WTCCC2 data sets. The coverage probabilities of the FIESTA CIs. The coverage probabilities are shown for CIs designed to have coverage probabilities of 95%. The CIs achieve accurate coverage.

2.2. Benchmarks

We compared the speed of the stochastic approximation approach, implemented in FIESTA, with that of using the parametric bootstrap for estimating the distribution of heritability estimator. The latter was tested either as implemented naively by using either GCTA (Yang et al., 2011) and pylmm (Furlotte and Eskin, 2015) or by using ALBI (Schweiger et al., 2016). Both the stochastic approximation and parametric bootstrap approaches require the calculation of the eigendecomposition of the kinship matrix. As this is already often a part of the heritability estimation algorithm, its calculation time is separated in the benchmarks. In section 4, we discuss how this step could be avoided altogether.

One difference between the two approaches is that the bootstrap approach performs a lengthy preprocessing step that estimates many distributions. Once these distributions are estimated, constructing a CI is very rapid. In contrast, the stochastic approximation approach does not perform a preprocessing step, but performs a nontrivial calculation per CI.

The construction of a single CI with FIESTA consists of calculating six to eight values using the modified Robbins–Monro procedure (see section 3). The first four values depend only on the kinship matrix, but not on the heritability estimate for which we construct a CI, so they need to be calculated only once per kinship matrix, and can then be shared between several CIs. Each modified Robbins–Monro run has the complexity of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$O ( nT )$$ \end{document} , where n is the number of individuals in the sample and T is the number of iterations (in the order of 1000; see section 3). Therefore, in total, the time complexity to construct K CIs with FIESTA grows linearly with K, T and n.

We also compared FIESTA with the performance of the analytical SE approach. Although often inaccurate, analytical SEs are often the go-to method by many practitioners: first, their calculation is conceptually easy to understand, since a closed-form formula exists for the SEs (see section 5); second, using a closed-form expression is often perceived as faster than more involved algorithmic procedures. However, this is not the case for heritability estimation, as SEs are calculated using variants of the Fisher information matrix [e.g., the average information (AI) matrix, as in GCTA (Yang et al., 2011)], whose calculation requires matrix-by-vector multiplications, which are \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$O ( {n^2} )$$ \end{document} . In contrast, FIESTA is linear in n, giving it an advantage at larger data sets in particular.

We performed a benchmark to evaluate FIESTA, using the NFBC and WTCCC2 data sets. We estimated the distributions of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \hat h^2}$$ \end{document} for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${h^2} = 0 , \,0.01 , \ldots , \,1$$ \end{document} , with GCTA (Yang et al., 2011) and pylmm (Furlotte and Eskin, 2015), both of which perform full estimation, using 1000 random bootstrap samples. For the same task, we also used ALBI (Schweiger et al., 2016), at a grid resolution of 0.001. The accuracy of CIs constructed according to the full estimation approach, as implemented in ALBI, is shown in section 5. As already explained, the time of construction of CIs given these distributions is negligible relative to the time required for the estimation of the distributions. We also constructed analytical SEs for both data sets using the AI method (see section 5). These times are reported in Table 1.

As a comparison, we used FIESTA to construct varying number of CIs, using 1000 iterations in the modified Robbins–Monro procedure (see section 3). In Table 1, it can be seen that FIESTA is significantly faster, particularly when few CIs are needed. We also note that FIESTA is currently implemented in the Python language, using the numpy package; a significant additional speedup can be obtained by migrating to a compiled language, for example, C++.

We then continued to investigate the stability of CI construction and its dependency on the number of iterations. We ran FIESTA 100 times to construct CI for the NFBC and WTCCC2 data sets using 200, 500, 1000, or 2000 iterations. We measured the variance in the constructed CI endpoints (Table 2). As expected, the variance decreases with the number of iterations. In addition, we measured the mean and variance of the coverage of CIs under a grid of true heritability values. Here, also, we observed that variance of coverage decreases with the number of iterations. We note that 500 iterations are sufficient for reasonably accurate CIs for these data sets, and that the coverage of even 200 iterations is only slightly biased downwards.

3.1. The LMM and REML

We consider the following standard LMM (see Searle et al., 2009 for a detailed review). Let n be the number of individuals and m be the number of SNPs. Let y be an \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$n \times 1$$ \end{document} vector of phenotype measurements for each individual. Let X be an \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$n \times p$$ \end{document} matrix of p covariates (possibly including an intercept vector \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${{ \bf{1}}_n}$$ \end{document} as a first column, as well as other covariates such as gender and age). Let Z be the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$n \times m$$ \end{document} standardized genotype matrix, that is, columns have zero mean and unit variance. Let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\beta$$ \end{document} be a \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$p \times 1$$ \end{document} vector of fixed effects, s an \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$m \times 1$$ \end{document} vector of random effects, and e an \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$n \times 1$$ \end{document} vector of errors. Then, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \bf{y}} = { \bf{X}} \beta + { \bf{Zs}} + { \bf{e}}.$$ \end{document} We assume s and e are statistically independent and are distributed normally as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$ { \bf { s } } \sim { \cal N } \left( { { { \bf { 0 } } _m } , \frac { 1 } { m } \sigma _g^2 { { \bf { I } } _m } } \right)$$ \end{document} , \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \bf{e}} \sim { \cal N} \left( {{{ \bf{0}}_n} , \sigma _e^2{{ \bf{I}}_n}} \right)$$ \end{document} . The fixed effects \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\beta$$ \end{document} and the coefficients \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\sigma _g^2$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\sigma _e^2$$ \end{document} are the parameters of the model.

Define \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$ { \bf { K } } = \frac { 1 } { m } { \bf { Z } } { { \bf { Z } } ^ { \bf { T } } } $$ \end{document} . Typically, K is commonly called the kinship matrix, or the genetic relationship matrix. Under these conditions, it follows (Yang et al., 2010) that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { \bf{y}} \sim { \cal N} \left( {{ \bf{X}} \beta , \sigma _g^2{ \bf{K}} + \sigma _e^2{{ \bf{I}}_n}} \right) . \tag{1} \end{align*} \end{document}

The narrow-sense heritability due to genotyped common SNPs is defined as the proportion of total variance explained by genetic factors (Visscher et al., 2008): \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { h^2 } = { \frac { \sigma _g^2 } { \sigma _g^2 + \sigma _e^2 } } \;. \tag { 2 } \end{align*} \end{document}

Defining \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\sigma _p^2 = \sigma _g^2 + \sigma _e^2$$ \end{document} , Equation 1 becomes \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \bf{y}} \sim { \cal N} \left( {{ \bf{X}} \beta , \sigma _p^2{{ \bf{V}}_{{h^2}}}} \right) ,$$ \end{document} where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${{ \bf{V}}_{{h^2}}} = {h^2}{ \bf{K}} + ( 1 - {h^2} ) {{ \bf{I}}_n}$$ \end{document} .

The most common way of estimating h² is REML estimation. REML consists of maximizing the likelihood function associated with the projection of the phenotype onto the subspace orthongonal to that of the fixed effects of the model (Patterson and Thompson, 1971). In Schweiger et al. (2016), it is shown that the distribution of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \hat h^2}$$ \end{document} depends only on h², and is invariant under changes to \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\sigma _p^2$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\beta$$ \end{document} . We may, therefore, limit our study to the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \hat h^2}$$ \end{document} estimator alone, in the special case of fixed \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\sigma _p^2 = 1$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\beta = {{ \bf{0}}_p}$$ \end{document} , which substantially simplifies the problem; namely, we may focus on properties of the distribution \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \cal N} \left( {{{ \bf{0}}_n} , {{ \bf{V}}_{{h^2}}}} \right)$$ \end{document} instead of the more general \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \cal N} \left( {{ \bf{X}} \beta , \sigma _p^2{{ \bf{V}}_{{h^2}}}} \right)$$ \end{document} .

3.2. CIs for h²

We wish to build CIs with a coverage probability of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$1 - \alpha$$ \end{document} (e.g., 95%). The full derivation is developed in Schweiger et al. (2016), and is reviewed in section 5; we cite the results here.

Let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${c_ \beta } ( {h^2} )$$ \end{document} be the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\beta$$ \end{document} -th quantile function of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \hat h^2}$$ \end{document} , when the true heritability is h²; that is, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\mathop {{ \rm{P}}r} \nolimits_{{h^2}} ( { \hat h^2} \le {c_ \beta } ( {h^2} ) ) = \beta$$ \end{document} . Define s and t to be the values for which \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\mathop {{ \rm{P}}r} \nolimits_{{h^2} = s} ( { \hat h^2} = 0 ) = \alpha / 2$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\mathop {{ \rm{P}}r} \nolimits_{{h^2} = t} ( { \hat h^2} = 1 ) = \alpha / 2$$ \end{document} . In addition, let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${s^*} = {c_{1 - \alpha }} ( 0 ) , {t^*} = {c_ \alpha } ( 1 )$$ \end{document} . Then the lower and upper CI boundaries for an estimate H² are given, respectively, by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \begin{matrix} {{l_{{H^2}}} = \left( { \begin{matrix} 0 & {{ \kern 1pt} {\rm if} \;{ \kern 1pt} {H^2} \le {s^*}} \\ {c_{1 - \alpha }^{ - 1} ( {H^2} ) } & {{ \kern 1pt} {\rm if} \;{ \kern 1pt} c_{1 - \alpha }^{ - 1} ( {H^2} ) < s} \\ s & {{ \kern 1pt} {\rm if} \;{ \kern 1pt} s \in [ c_{1 - \alpha / 2}^{ - 1} ( {H^2} ) , c_{1 - \alpha }^{ - 1} ( {H^2} ) ] } \\ {c_{1 - \alpha / 2}^{ - 1} ( {H^2} ) } & {{ \kern 1pt} {\rm if} \;{ \kern 1pt} s < c_{1 - \alpha / 2}^{ - 1} ( {H^2} ) } {} \\ \end{matrix} } \right.} \\ \end{matrix} \tag{3} \end{align*} \end{document}

and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \begin{matrix} {{u_{{H^2}}} = \left( { \begin{matrix} {c_{1 - \alpha / 2}^{ - 1} ( {H^2} ) } & {{ \kern 1pt} {\rm if} \;{ \kern 1pt} c_{ \alpha / 2}^{ - 1} ( {H^2} ) < t} \\ t & {{ \kern 1pt} {\rm if} \;{ \kern 1pt} t \in [ c_ \alpha ^{ - 1} ( {H^2} ) , c_{ \alpha / 2}^{ - 1} ( {H^2} ) ] } \\ {c_ \alpha ^{ - 1} ( {H^2} ) } & {{ \kern 1pt} {\rm if} \;{ \kern 1pt} t < c_ \alpha ^{ - 1} ( {H^2} ) } \\ 1 & {{ \kern 1pt} {\rm if} \;{ \kern 1pt} {t^*} \le {H^2}{ \kern 1pt} .} {} \\ \end{matrix} } \right.} \\ \end{matrix} \tag{4} \end{align*} \end{document}

3.3. Using stochastic approximation to calculate CIs

3.4. Using the modified Robbins–Monro procedure to obtain CIs for heritability

We now describe how to rapidly construct CIs for heritability. As already described, the first step is to find \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$s , t , {s^*}$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${t^*}$$ \end{document} . To find s, we employ the modified Robbins–Monro procedure (Joseph, 2004), where the parameter of interest is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\theta : = {h^2}$$ \end{document} , the function is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$g ( \theta ) : = \mathop {{ \rm{P}}r} \nolimits_{{h^2} = \theta } ( { \hat h^2} = 0 )$$ \end{document} , and the inverse value we wish to find corresponds to \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$C = \alpha / 2$$ \end{document} . We note that we chose g here to be nonincreasing for the sake of clarity of presentation; to conform with the Robbins–Monro formulation, we would need to redefine \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$g \to 1 - g$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$C \to 1 - C$$ \end{document} . At a single iteration of the modified Robbins–Monro procedure, we have an estimate \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$h_n^2$$ \end{document} for s, and we need to sample from a distribution whose mean is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\mathop {{ \rm{P}}r} \nolimits_{h_n^2} ( { \hat h^2} = 0 )$$ \end{document} . To achieve that, we draw a sample from the distribution corresponding to \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$h_n^2$$ \end{document} , \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \cal N} \left( {{{ \bf{0}}_n} , {{ \bf{V}}_{h_n^2}}} \right)$$ \end{document} , and check whether the maximum likelihood estimate for it is 0 (or higher). This procedure can be done quickly in \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$O ( n )$$ \end{document} , as we now describe, circumventing the need to perform a full likelihood maximization for the sample.

As already detailed, we make repeated use of the following procedure: (1) draw a random sample y from the distribution corresponding to a given heritability value h², \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \cal N} \left( {{{ \bf{0}}_n} , {{ \bf{V}}_{{h^2}}}} \right)$$ \end{document} and (2) decide whether its heritability estimate, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \hat h^2} ( { \bf{y}} )$$ \end{document} , is larger than a given value, H². In Schweiger et al. (2016), it is shown that when \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \bf{X}} = {{ \bf{1}}_n}$$ \end{document} , these two steps may equivalently be performed by drawing a vector u of i.i.d. (independent and identically), standard normal variables \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \bf{u}} \sim { \cal N} ( {{ \bf{0}}_n} , {{ \bf{I}}_n} )$$ \end{document} , and checking whether \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \mathop \sum \limits_{i = 1}^n \xi _i^{{h^2} , {H^2}}u_i^2 > 0 \; , \tag{9} \end{align*} \end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \xi _i^ { { h^2 } , { H^2 } } = { \frac { { h^2 } ( { d_i } - 1 ) + 1 } { { H^2 } ( { d_i } - 1 ) + 1 } } \left( { { \frac { { d_i } - 1 } { { H^2 } ( { d_i } - 1 ) + 1 } } - \frac { 1 } { { n - 1 } } \mathop \sum \limits_ { j = 1 } ^ { n - 1 } { \frac { { d_j } - 1 } { { H^2 } ( { d_j } - 1 ) + 1 } } } \right) \tag { 10 } \end{align*} \end{document}

for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$i = 1 , \ldots , n - 1$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\xi _n^{{h^2} , {H^2}} = 0$$ \end{document} , with d_i being the eigenvalues of K. The sign of the expression in Equation (9) is equal to the sign of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$ { \frac { \partial { \ell _ { REML } } } { \partial { h^2 } } } ( { H^2 } )$$ \end{document} , the derivative of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \ell _{REML}}$$ \end{document} at the point H². Therefore, assuming the restricted likelihood function is well behaved, a positive derivative indicates that the REML heritability estimate is larger than H². Similar expressions are defined for a general X in Schweiger et al. (2016). Once the eigendecomposition of K is obtained, this procedure may be performed in a time complexity linear in n.

Similarly, for finding \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${s^*}$$ \end{document} , we define the function \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$g ( \theta ) : = \mathop {{ \rm{P}}r} \nolimits_{{h^2} = 0} ( { \hat h^2} \le \theta )$$ \end{document} , for which we want to find the inverse of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$C = 1 - \alpha$$ \end{document} . The procedures for finding t and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${t^*}$$ \end{document} are similar.

The second step involves calculating the quantities \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$c_{ \alpha / 2}^{ - 1} ( {H^2} ) , c_ \alpha ^{ - 1} ( {H^2} ) , c_{1 - \alpha }^{ - 1} ( {H^2} )$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$c_{1 - \alpha / 2}^{ - 1} ( {H^2} )$$ \end{document} as required. This can again be done by the modified Robbins–Monro procedure, by setting \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\theta : = {h^2}$$ \end{document} , \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$g ( \theta ) : = \mathop {{ \rm{P}}r} \nolimits_{{h^2}} ( { \hat h^2} \le {H^2} )$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$C = \alpha / 2 , \alpha , 1 - \alpha / 2$$ \end{document} or \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$1 - \alpha$$ \end{document} . To sample from a distribution whose mean is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\mathop {{ \rm{P}}r} \nolimits_{h_n^2} ( { \hat h^2} \le {H^2} )$$ \end{document} , we draw a sample from the distribution corresponding to \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$h_n^2$$ \end{document} , and check whether the maximum likelihood estimate for it is more than H². Again, this procedure can be done quickly in \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$O ( n )$$ \end{document} . Once these quantities have been calculated, the CI can be calculated as detailed in Equations (3) and (4).

In practice, we used the following choices in the modified Robbins–Monro procedure: (1) we used \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$T = 1000$$ \end{document} iterations; (2 we set the prior standard deviation to \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\tau = 0.4$$ \end{document} , used to derive a_n and C_n through the Bayesian analysis (see Joseph, 2004); (3) we used the midpoint between the estimate and relevant boundary (0 or 1, depending on the quantile required) as a starting point; and (4) we adaptively changed the step size constant, following the suggestion of Garthwaite and Buckland, by approximating the derivative with an expression proportional to the distance from \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\hat \theta$$ \end{document} : \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} g \prime ( { u_ \theta } ) \approx k ( h_n^2 - { H^2 } ) , \quad k = \frac { 2 } { { { z_ \beta } \cdot { { ( 2 \pi ) } ^ { - 1 / 2 } } \cdot { e^ { - z_ \beta ^2 / 2 } } } } , \tag { 11 } \end{align*} \end{document}

where z is the quantile function of the normal distribution and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\beta$$ \end{document} is the required quantile.

3.5. The NFBC data set

We analyzed 5236 individuals from the NFBC data set, which consists of genotypes at 331,476 genotyped SNPs and 10 phenotypes (Sabatti et al., 2009). From each pair of individuals with relatedness of >0.025, one was reserved, resulting in 2520 individuals.

3.6. The WTCCC2 data set

We analyzed the WTCCC2 data set (Sawcer et al., 2011). In the multiple sclerosis and ulcerative colitis data sets, we used the same data processing described in Yang et al. (2014) to ensure consistency. In brief, U.K. controls and cases from both United Kingdom and non-United Kingdom were used. SNPs were removed with \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$> 0.5 \%$$ \end{document} missing data, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$p < 0.01$$ \end{document} for allele frequency difference between two control groups, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$p < 0.05$$ \end{document} for deviation from Hardy–Weinberg equilibrium, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$p < 0.05$$ \end{document} for differential missingness between cases and controls, or minor allele frequency \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$< 1 \%$$ \end{document} . In all analyses, SNPs within 5 M base pairs of the human leukocyte antigen region were excluded, because they have large effect sizes and highly unusual linkage disequilibrium patterns, which can bias or exaggerate the results. Finally, from each pair of individuals with relatedness of >0.025, one was reserved, resulting in 13,950 individuals.

5.1. Variance of estimators

The main method of calculating the variance of the estimator, applied by all widely used linear mixed model methods, employs the Fisher information matrix, or a variant of which, possibly applying the delta method in addition Wasserman, L. (2013). All of statistics: A concise course in statistical inference. Springer Science & Business Media. The observed information matrix \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \cal J} ( \theta )$$ \end{document} of parameters \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\theta$$ \end{document} is the negative of the Hessian of the log-likelihood of the data y . Namely, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$ { \cal J } { ( \theta ) _ { i , j } } = - \frac { \partial } { { \partial { \theta _i } { \theta _j } } } \ell ( \theta ; { \bf { y } } )$$ \end{document} . The Fisher information matrix \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\mathcal{I} ( \theta )$$ \end{document} is the expectation of the observed information matrix. Namely, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\mathcal { I } { ( \theta ) _ { i , j } } = { \rm { E } } \left[ { - \frac { \partial } { { \partial { \theta _i } { \theta _j } } } \ell ( \theta ; { \bf { y } } ) } \right]$$ \end{document} . Asymptotically, under certain regularity conditions, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $\sqrt n ( \widehat \theta - \theta ) \buildrel d \over \longrightarrow { \cal N } ( { \bf { 0 } } , \mathcal { I } { ( \theta ) ^ { - 1 } } )$ \end{document} . According to the delta method, the asymptotic distribution of a function \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$f ( \theta )$$ \end{document} satisfies \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $\sqrt n ( f ( \widehat \theta ) - f ( \theta ) ) \buildrel d \over \longrightarrow { \cal N } ( { \bf { 0 } } , \nabla f { ( \theta ) ^T } \mathcal { I } { ( \theta ) ^ { - 1 } } \nabla f ( \theta ) )$ \end{document} .

GCTA uses the average information (Gilmour et al., 1995) matrix \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\cal A$$ \end{document} to calculate the variance of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\sigma _g^2$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\sigma _e^2$$ \end{document} , where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$ { \cal A } = \frac { 1 } { 2 } ( \mathcal { I } + { \cal J } )$$ \end{document} . For the REML method, this is the matrix: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { \cal A } = \frac { 1 } { 2 } \cdot \left( { \begin{matrix} { { { \bf { y } } ^ { \bf { T } } } { \bf { QKQKQy } } } & { { { \bf { y } } ^ { \bf { T } } } { \bf { QKQQy } } } \\ { { { \bf { y } } ^ { \bf { T } } } { \bf { QQKQy } } } & { { { \bf { y } } ^ { \bf { T } } } { \bf { QQQy } } } { } \\ \end{matrix}} \right) , \tag { 12 } \end{align*} \end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \bf{Q}} = { \Sigma ^{ - 1}} - { \Sigma ^{ - 1}}{ \bf{X}}{ \left( {{{ \bf{X}}^T}{ \Sigma ^{ - 1}}{{ \bf{X}}^T}} \right) ^{ - 1}}{{ \bf{X}}^T}{ \Sigma ^{ - 1}}$$ \end{document} , with \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\Sigma = \sigma _g^2{ \bf{K}} + \sigma _e^2{ \bf{I}}$$ \end{document} . Then, the delta method is used to calculate the variance of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \hat h^2}$$ \end{document} : \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { \rm{Var}} ( { \hat h^2} ) = ( \hat \sigma _g^2 + \hat \sigma _e^2{ ) ^{ - 4}} \left( { \begin{matrix} { \hat \sigma _e^2} { - \hat \sigma _g^2} \\ \end{matrix} } \right) {{ \cal A}^{ - 1}}{ \vert _{ \sigma _g^2 = \hat \sigma _g^2 , \sigma _e^2 = \hat \sigma _e^2}} \left( { \begin{matrix} { \hat \sigma _e^2} \\ { - \hat \sigma _g^2} \\ \end{matrix} } \right) . \tag{13} \end{align*} \end{document}

Given the eigendecomposition of K, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \Sigma ^{ - 1}}$$ \end{document} (and thus Q) can be calculated in O(n) (where n is the number of individuals), avoiding an expensive matrix inversion. Several other computational improvements may be carried out, depending on software implementation. However, we note that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$O ( {n^2} )$$ \end{document} matrix-by-vector multiplications cannot be avoided.

5.2. CIs for heritability

Our approach is based on the duality between hypothesis testing and CIs. As the distribution of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \hat h^2}$$ \end{document} depends solely on h², we may assume without loss of generality that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\sigma _p^2 = 1$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\beta = {{ \bf{0}}_p}$$ \end{document} . For a fixed value h², an acceptance region \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${A_{{h^2}}}$$ \end{document} is defined as the subset of values \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \hat h^2}$$ \end{document} for which a test does not reject the null hypothesis that the phenotype vector is drawn from \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \cal N} \left( {{{ \bf{0}}_n} , {{ \bf{V}}_{{h^2}}}} \right)$$ \end{document} . The probability of the event \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${A_{{h^2}}}$$ \end{document} under \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \cal N} \left( {{{ \bf{0}}_n} , {{ \bf{V}}_{{h^2}}}} \right)$$ \end{document} should be \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\ge 1 - \alpha$$ \end{document} . This region may be indirectly derived from an actual test (e.g., a generalized likelihood ratio test) or constructed explicitly. The corresponding CI for an estimate \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \hat h^2} = {H^2}$$ \end{document} , \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${C_{{H^2}}}$$ \end{document} , comprises the set of parameter values for which \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \hat h^2}$$ \end{document} does not imply the rejection of the null hypothesis that the true heritability value is h²: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} {C_{{H^2}}} = \left\{ {{h^2} \left\vert {{H^2} \in {A_{{h^2}}}} \right.} \right\} \;. \tag{14} \end{align*} \end{document}

Since the distribution of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \hat h^2}$$ \end{document} is bounded and generally asymmetric, the choice of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${A_{{h^2}}}$$ \end{document} is not unique. It remains to determine \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${A_{{h^2}}}$$ \end{document} for every h². We give here a general description of the construction; in Schweiger et al. (2016), we give a full description of the method, along with proofs.

Let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${c_ \beta } ( {h^2} )$$ \end{document} be the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\beta$$ \end{document} -th quantile function of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \hat h^2}$$ \end{document} , when the true heritability is h², that is, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\mathop {{ \rm{P}}r} \nolimits_{{h^2}} ( { \hat h^2} \le {c_ \beta } ( {h^2} ) ) = \beta$$ \end{document} . A natural choice for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${A_{{h^2}}}$$ \end{document} would be taking the interval obtained by removing a \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\alpha / 2$$ \end{document} tail from both sides of the distribution of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \hat h^2}$$ \end{document} given h², that is, choosing the two-sided \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${A_{{h^2}}} = [ {c_{ \alpha / 2}} ( {h^2} ) , {c_{1 - \alpha / 2}} ( {h^2} ) ]$$ \end{document} . If this were always possible, a succinct way of describing the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$1 - \alpha$$ \end{document} CI, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${C_{{H^2}}} = [ {l_{{H^2}}} , {h_{{H^2}}} ]$$ \end{document} , would be using the fact that its endpoints are exactly those following \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} {c_{1 - \alpha / 2}} ( {l_{{H^2}}} ) = {H^2} \rightarrow {l_{{H^2}}} = c_{1 - \alpha / 2}^{ - 1} ( {H^2} ) \tag{15} \end{align*} \end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} {c_{ \alpha / 2}} ( {h_{{H^2}}} ) = {H^2} \rightarrow {u_{{H^2}}} = c_{ \alpha / 2}^{ - 1} ( {H^2} ) . \tag{16} \end{align*} \end{document}

as described in Figure 3.

OPEN IN VIEWER

FIG. 3.

An illustration of acceptance regions and CIs. The diagonal lines are the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\alpha / 2$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$1 - \alpha / 2$$ \end{document} quantile functions, shown for values in the midrange of heritability values. Several example acceptance regions are denoted as horizontal lines, in parameter regions where simple two-sided acceptance regions can be defined. The CI for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \hat h^2} = 0.5$$ \end{document} is shown as a vertical line.

However, since the distribution is of a mixed type with discontinuity points, it may be the case that the probability of the interval \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$[ {c_{ \alpha / 2}} ( {h^2} ) , {c_{1 - \alpha / 2}} ( {h^2} ) ]$$ \end{document} might be greater than \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$( 1 - \alpha / 2 ) - \alpha / 2 = 1 - \alpha$$ \end{document} . For example, if \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\mathop {{ \rm{P}}r} \nolimits_{{h^2}} ( { \hat h^2} = 0 ) > \alpha / 2$$ \end{document} , then \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${c_{ \alpha / 2}} = 0$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\mathop {{ \rm{P}}r} \nolimits_{{h^2}} ( { \hat h^2} \in [ 0 , {c_{ \alpha / 2}} ) ) > \alpha / 2$$ \end{document} . In this case, we then instead choose to take the one-sided interval \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${A_{{h^2}}} = [ 0 , {c_{1 - \alpha }} ( {h^2} ) ]$$ \end{document} . Similarly, if \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\mathop {{ \rm{P}}r} \nolimits_{{h^2}} ( { \hat h^2} = 1 ) > \alpha / 2$$ \end{document} , then \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${c_{1 - \alpha }} = 1$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\mathop {{ \rm{P}}r} \nolimits_{{h^2}} ( { \hat h^2} \in ( {c_{1 - \alpha / 2}} , 1 ] ) > \alpha / 2$$ \end{document} . In this case, we similarly choose the one-sided interval \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$[ {c_ \alpha } ( {h^2} ) , 1 ]$$ \end{document} instead. We are, therefore, interested in the maximal value s for which \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\mathop {{ \rm{P}}r} \nolimits_s ( { \hat h^2} = 0 ) \ge \alpha / 2$$ \end{document} , and the minimal value t for which \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\mathop {{ \rm{P}}r} \nolimits_{{h^2}} ( { \hat h^2} = 1 ) \ge \alpha / 2$$ \end{document} , because in the range of values \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${h^2} \in [ s , t ]$$ \end{document} , it holds that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\mathop {{ \rm{P}}r} \nolimits_{{h^2}} ( { \hat h^2} \in [ {c_{ \alpha / 2}} ( {h^2} ) , {c_{1 - \alpha / 2}} ( {h^2} ) ] ) = 1 - \alpha$$ \end{document} . Equivalently, assuming \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\mathop {{ \rm{P}}r} \nolimits_{{h^2}} ( { \hat h^2} = 0 )$$ \end{document} (respectively, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\mathop {{ \rm{P}}r} \nolimits_{{h^2}} ( { \hat h^2} = 1 )$$ \end{document} ) is decreasing (respectively, increasing) in h², we may simply define s and t to be the values for which \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\mathop {{ \rm{P}}r} \nolimits_{{h^2} = s} ( { \hat h^2} = 0 ) = \alpha / 2$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\mathop {{ \rm{P}}r} \nolimits_{{h^2} = t} ( { \hat h^2} = 1 ) = \alpha / 2$$ \end{document} .

The following assumes s and t exist, and that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$s < t$$ \end{document} ; for the general case, see Schweiger et al. (2016). We divide our construction into distinct cases, by setting \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} {A_{{h^2}}} = \begin{cases} \begin{matrix} { [ 0 , {c_{1 - \alpha }} ( {h^2} ) ] } & {{ \kern 1pt} {\rm if} \;{ \kern 1pt} {h^2} \in [ 0 , s ) } \\ { [ {c_{ \alpha / 2}} ( {h^2} ) , {c_{1 - \alpha / 2}} ( {h^2} ) ] } & {{ \kern 1pt} {\rm if} \;{ \kern 1pt} {h^2} \in [ s , t ] } \\ { [ {c_ \alpha } ( {h^2} ) , 1 ] } & {{ \kern 1pt} {\rm if} \;{ \kern 1pt} {h^2} \in ( t , 1 ] .} {} \\ \end{matrix} \end{cases} \end{align*} \end{document}

The three region types are illustrated in Figure 4. Inverting the acceptance regions, we get the following definition for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${C_{{H^2}}} = [ {l_{{H^2}}} , {h_{{H^2}}} ]$$ \end{document} . For the lower endpoint, we have \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} {l_{{H^2}}} = \begin{cases} \begin{matrix} 0 & {{ \kern 1pt} {\rm if} \;{ \kern 1pt} {H^2} \in [ 0 , {c_{1 - \alpha }} ( 0 ) ) } \\ {c_{1 - \alpha }^{ - 1} ( {H^2} ) } & {{ \kern 1pt} { \kern 1pt} {\rm if} \;{ \kern 1pt} {H^2} \in [ {c_{1 - \alpha }} ( 0 ) , {c_{1 - \alpha }} ( s ) ) } \\ s & {{ \kern 1pt} { \kern 1pt} {\rm if} \;{ \kern 1pt} {H^2} \in [ {c_{1 - \alpha }} ( s ) , {c_{1 - \alpha / 2}} ( s ) ) } \\ {c_{1 - \alpha / 2}^{ - 1} ( {H^2} ) } & {{ \kern 1pt} {\rm if} \;{ \kern 1pt} {H^2} \in [ {c_{1 - \alpha / 2}} ( s ) , 1 ] } {} \\ \end{matrix} \end{cases}. \end{align*} \end{document}

For the higher endpoint, we have \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} {u_{{H^2}}} = \begin{cases} \begin{matrix} {c_{ \alpha / 2}^{ - 1} ( {H^2} ) } & {{ \kern 1pt} {\rm if} \;{ \kern 1pt} {H^2} \in [ 0 , {c_{ \alpha / 2}} ( t ) ) } \\ t & {{ \kern 1pt} { \kern 1pt} {\rm if} \;{ \kern 1pt} {H^2} \in [ {c_{ \alpha / 2}} ( t ) , {c_ \alpha } ( t ) ) } \\ {c_ \alpha ^{ - 1} ( {H^2} ) } & {{ \kern 1pt} { \kern 1pt} {\rm if} \;{ \kern 1pt} {H^2} \in [ {c_ \alpha } ( t ) , {c_ \alpha } ( 1 ) ) } \\ 1 & {{ \kern 1pt} {\rm if} \;{ \kern 1pt} {H^2} \in [ {c_ \alpha } ( 1 ) , 1 ] } {} \\ \end{matrix} \end{cases}. \end{align*} \end{document}

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FIG. 4.

An illustration of the three acceptance region types. The diagonal lines, from left to right, indicate the quantile functions for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\alpha / 2 , \alpha , 1 - \alpha$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$1 - \alpha / 2$$ \end{document} . The three region types are indicated as horizontal lines. The points s and t, where region types used are changed, are indicated as horizontal dashed lines. See section 3 for a full description.

These conditions, phrased in terms of the quantile functions \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${c_ \beta }$$ \end{document} , for example, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${H^2} \le {c_ \alpha } ( t )$$ \end{document} , can be equivalently written in terms of the value of inverse quantile functions of the estimate H², for example, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$c_ \alpha ^{ - 1} ( {H^2} ) \le t$$ \end{document} . In addition, let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${s^*} = {c_{1 - \alpha }} ( 0 ) , {t^*} = {c_ \alpha } ( 1 )$$ \end{document} . Explicitly, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \begin{matrix} {{l_{{H^2}}} = \begin{cases} \begin{matrix} 0 & {{ \kern 1pt} { \kern 1pt} {\rm if} \;{ \kern 1pt} {H^2} \le {s^*}} \\ {c_{1 - \alpha }^{ - 1} ( {H^2} ) } & {{ \kern 1pt} { \kern 1pt} {\rm if} \;{ \kern 1pt} c_{1 - \alpha }^{ - 1} ( {H^2} ) < s} \\ s & {{ \kern 1pt} { \kern 1pt} {\rm if} \;{ \kern 1pt} s \in [ c_{1 - \alpha / 2}^{ - 1} ( {H^2} ) , c_{1 - \alpha }^{ - 1} ( {H^2} ) ] } \\ {c_{1 - \alpha / 2}^{ - 1} ( {H^2} ) } & {{ \kern 1pt} { \kern 1pt} {\rm if} \;{ \kern 1pt} s < c_{1 - \alpha / 2}^{ - 1} ( {H^2} ) } {} \\ \end{matrix} \end{cases}} \hfill \\ \end{matrix} , \tag{17} \end{align*} \end{document}

and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \begin{matrix} {{u_{{H^2}}} = \begin{cases} \begin{matrix} {c_{1 - \alpha / 2}^{ - 1} ( {H^2} ) } & {{ \kern 1pt} { \kern 1pt} {\rm if} \;{ \kern 1pt} c_{ \alpha / 2}^{ - 1} ( {H^2} ) < t} \\ t & {{ \kern 1pt} { \kern 1pt} {\rm if} \;{ \kern 1pt} t \in [ c_ \alpha ^{ - 1} ( {H^2} ) , c_{ \alpha / 2}^{ - 1} ( {H^2} ) ] } \\ {c_ \alpha ^{ - 1} ( {H^2} ) } & {{ \kern 1pt} { \kern 1pt} {\rm if} \;{ \kern 1pt} t < c_ \alpha ^{ - 1} ( {H^2} ) } \\ 1 & {{ \kern 1pt} { \kern 1pt} {\rm if} \;{ \kern 1pt} {t^*} \le {H^2}} {} \\ \end{matrix} \end{cases}} \hfill \\ \end{matrix} . \tag{18} \end{align*} \end{document}

It follows from the mentioned discussion that to construct a CI for an heritability estimate H², we need to first find s, t as previously, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${s^*} = {c_{1 - \alpha }} ( 0 )$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${t^*} = {c_ \alpha } ( 1 )$$ \end{document} , and then we need only calculate \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$c_ \beta ^{ - 1} ( {H^2} )$$ \end{document} for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\beta = \alpha / 2 , \alpha , 1 - \alpha$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$1 - \alpha / 2$$ \end{document} . Therefore, the entire construction relies on inverting certain quantile functions.

5.3. Accuracy of ALBI CIs

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FIG. 5.

Accuracy of CIs for the NFBC and WTCCC2 data sets. The coverage probabilities of the ALBI CIs. The coverage probabilities are shown for CIs designed to have coverage probabilities of 95%.