Multi-marker genetic association and interaction tests with interval-censored survival outcomes

Abstract

The development of set-based genetic-survival association tests has been focusing on right-censored survival outcomes. However, interval-censored failure time data arise widely from health science studies, especially those on the development of chronic diseases. In this paper, we proposed a suite of set-based genetic association and interaction tests for interval-censored survival outcomes under a unified weighted-V-statistic framework. Besides dealing with interval censoring, the new tests can account for genetic effect heterogeneity and accommodate left truncation of survival outcomes. Simulation studies showed that the new tests perform well in terms of size and power under various scenarios and that the new interaction test is more powerful than the standard likelihood ratio test for testing gene-gene/gene-environment interactions. The practical utility of the developed tests was illustrated by a genome-wide association study of age to early childhood caries.

Introduction

With the development of genotyping and sequencing technologies and the increasing size of genome databases, the genetic predisposition to human diseases has been studied extensively in the past two decades, using agnostic approaches such as genome-wide association study (GWAS) and whole-genome sequence association study. However, the hypothesis-free approaches face the issues of multiple testing and small effect sizes of individual variants, which motivated the development of multi-marker tests. Multi-marker tests test the joint association of multiple genetic markers within a biological unit, e.g., a gene or a biological pathway. The most popular multi-marker tests are based on kernel machine regressions, including Liu, Lin, and Ghosh (2007), Liu, Ghosh, and Lin (2008) and Wu et al. (2011) among others. The kernel association tests increase the statistical power of association studies by not only aggregating the signals of individual markers but also capturing their potentially non-linear and/or interactive effects through the use of non-linear kernels.

Most of the existing kernel association tests were developed for completely observable quantitative/categorical phenotypes. Less effort was devoted to developing kernel association tests for censored survival traits, albeit survival traits are more informative than case-control phenotypes about the genetic contribution to disease development/progression. Existing kernel association tests for censored survival traits include the test based on Cox kernel machine regression (coxKM) (Cai, Tonini, & Lin, 2011), the test based on accelerated failure time kernel machine regression (aftKM) (Sinnott & Cai, 2013), the global test (gt) (Goeman, Oosting, Cleton-Jansen, Anninga, & van Houwelingen, 2005), and the sequence kernel association test in a Cox regression framework (coxSKAT) (Chen et al., 2014). The latter two use the same test statistic as coxKM with linear kernel but different methods to obtain its null distribution. All the four tests have note-worthy drawbacks as shown in Li, Wu, and Lu (2021). Specifically, aftKM, gt and coxSKAT do not have the nominal size when the sample size is not large, nor does coxKM when adjusting for covariates correlated with genetic markers of interest. To address those drawbacks, Li et al. (2021) developed a multi-marker test based on a weighted V statistic. The weighted V test (WV) inherits the advantage of the survival kernel association tests that it can be powerful to detect complex effects (e.g., nonlinear or interactive effects) of a SNP set or a gene set on survival outcomes. In addition, Li et al. (2021) demonstrated empirically that WV has the nominal size in moderate-sized samples and showed theoretically and empirically that the WV with linear kernel maintains the nominal size when adjusting for covariates that are linearly correlated with the markers of interest. Furthermore, WV can handle left truncation of survival traits and account for genetic effect heterogeneity (genetic heterogeneity) to increase power.

Survival traits of chronic diseases are usually interval censored due to the intermittent examinations of the disease status. Examples of such traits are age to early childhood caries (ECC), age to Alzheimer’s disease, age to type 2 diabetes, and time from menopause to osteoporosis in women, to name a few. To the best of our knowledge, multi-marker tests for interval-censored survival outcomes are lacking, which hampers the knowledge discovery of genetic components in chronic diseases. In this article, we develop the first set of multi-marker tests for genetic association and gene-gene/gene-environment (G–G/G–E) interaction with interval-censored survival traits. The tests are developed under the weight-V-statistic framework as adopted by Li et al. (2021) and thus are called the weighted V tests in the sequel. The proposed tests are fully nonparametric when not adjusting for covariates, and are semiparametric with covariate adjustment in the sense that only the effects of adjustment covariates on the survival outcome need to be specified through a semiparametric survival model. Besides dealing with interval censoring, another innovation of the proposed tests compared to Li et al. (2021) is that they can use any semiparametric transformation model (Zeng & Lin, 2006) to adjust for covariates, whereas Li et al. (2021) only uses the Cox model. We conducted an extensive set of simulation studies, which showed the excellent finite-sample performances in terms of size and power for the new tests. We also applied the proposed tests to a dbGaP dataset, Dental Caries: Whole Genome Association and Gene × Environment Studies (Accession No.: phs000095.v3.p1), to perform a gene-based association analysis of early childhood caries.

Methods

Set-up

We jointly test the effects of p genetic markers (e.g., expression values of a set of genes, or a set of SNPs, each coded as 0, 1 or 2, representing the number of minor alleles), G = (G₁, …, G_p)^T, on a survival time T, possibly adjusting for q covariates, Z = (Z₁, …, Z_q)^T, which could be confounders or predictors that are independent of G. The survival time T is subject to interval censoring and possible left truncation. Therefore, the survival data of a subject can be summarized by ($\mathcal{L}$, L, R), where $\mathcal{L}$ is the observed left truncation time, L is the last failure inspection time before the failure occurrence, and R is the first inspection time after the occurrence. L = 0 and R = ∞ when T is left- and right-censored respectively. All the data of a random sample of n subjects are then denoted by ${\left\{\left(G_i,Z_i,{\mathcal{L}}_i,L_i,R_i\right)\right\}}_{i=1}^n$. We assume that the number and times of failure inspections are independent of T given Z and that the left-truncation time is independent of T given Z as well.

Weighted V tests for genetic association without considering genetic heterogeneity

The null hypothesis of our covariate-adjusted association test is that G is independent of T conditioning on Z. The test assumes that under the null, T given Z follows a semiparametric transformation model, $\Lambda (t\mid Z)=G\left[{\Lambda }_0(t)e^{{\gamma }^{T}Z}\right]$, where Λ(t|Z) is the cumulative hazard function of T given Z, G(x) = log(1 + rx)/r with a specified r (r ≥ 0), and Λ₀(t) is an unknown non-decreasing function with Λ₀(0) = 0. Correspondingly, we use a working semiparametric transformation frailty model to construct the test. The model is $\Lambda (t\mid Z,h)=G\left[{\Lambda }_0(t)e^{{\gamma }^{T}Z+h}\right]$, where e^h is a subject-level frailty. The covariate-adjusted association test is based on the following weighted V statistic,

$$V_{Z,IC}^{*}=n^{-2}\sum _{i=1}^n\sum _{j=1}^n{\tilde{f}}_Z\left(G_i,G_j\right)S_{Z,i}{S}_{Z,j},$$ (1)

where $S_{Z,i}=\partial \text{log}{L}_i\left({\Lambda }_0,\mathit{\gamma },h_i\right)/{\partial h_i|}_{h_i=0}$ and $L_i\left({\Lambda }_0,\gamma ,h_i\right)=\left\{\text{exp}\left(-G\left[{\Lambda }_0\left(L_i\right)e^{{\gamma }^T{Z}_i+h_i}\right]\right)-\text{exp}\left(-G\left[{\Lambda }_0\left(R_i\right)e^{{\gamma }^T{Z}_i+h_i}\right]\right)\right\}/\text{exp}\left(-G\left[{\Lambda }_0\left({\mathcal{L}}_i\right)e^{{\gamma }^T{Z}_i+h_i}\right]\right)$ is the conditional likelihood (given frailty $e^{h_i}$ and left-truncation time) of the i-th subject’s data based on the semiparametric transformation frailty model. S_Z,iS_Z,j is considered as the V-statistic kernel and is a covariate-adjusted phenotype similarity measure. $\tilde{f}\left(G_i,G_j\right)$ is considered as the weight function and is a covariate-centered genetic similarity measure defined by

$${\tilde{f}}_Z\left(G_i,G_j\right)=f\left(G_i,G_j\right)-E\left[f\left(G_i,G_k\right)u\left(Z_k,Z_j\right)\mid G_i,Z_j\right]-E\left[u\left(Z_i,Z_k\right)f\left(G_k,G_j\right)\mid Z_i,G_j\right]+E\left[u\left(Z_i,Z_m\right)f\left(G_m,G_k\right)u\left(Z_k,Z_j\right)\mid Z_i,Z_j\right],$$

where $u\left(Z_i,Z_j\right)=\left(1,Z_i^T\right){\left[E\left\{{\left(1,Z^T\right)}^T\left(1,Z^T\right)\right\}\right]}^{-1}{\left(1,Z_j^T\right)}^T$ and f(G_i, G_j) is a genetic similarity function. It can be shown that $E\left[{\tilde{f}}_Z\left(G_i,G_j\right)\left(1,Z_j^T\right)\mid G_i,Z_i\right]=0$. The choice of f(G_i, G_j) depends on the type of G and the expected form of G’s effect. If the effect of G is expected to be linear, we use cross-product kernel $f\left(G_i,G_j\right)=G_i^T{G}_j$, a.k.a. linear kernel. For SNP covariates, we use IBS kernel $f\left(G_i,G_j\right)={\sum }_{k=1}^p\left(2-\left|G_{ik}-G_{jk}\right|\right)/(2p)$, if the effect of G is not expected to be linear. For gene expression covariates, we use polynomial kernels or Gaussian kernel if non-linear and/or interactive effects of G are expected. As suggested by Wei and Lu (2017), we can also choose a unified Laplacian-kernel-based genetic similarity, $f\left(G_i,G_j\right)=\text{exp}\left(-{\sum }_{k=1}^p{\omega }_k\left|G_{i,k}-G_{j,k}\right|/{\rm Y}\right)$, where G_i,k can be discrete or continuous variables, ${\rm Y}={\sum }_{k=1}^p{\omega }_k$, and ω_k is a function of variance ${\sigma }_k^2$ of G_k defined by ω_k = 1/σ_k for k = 1…, p. This kernel is particularly useful for association analyses with sequencing data, where a large portion of genetic variants are rare.

Under the null hypothesis, S_Z,i has conditional mean zero given Z_i. When Z is independent of G, following the proof of Theorem 1 in Li et al. (2021), the null limiting distribution of $n{V}_{Z,IC}^{*}$ can be shown to be

$$\zeta \sum _{t=1}^{\infty }{\nu }_t{\chi }_{1t}^2,$$ (2)

where $\zeta =E\left(S_{Z,i}^2\right)$, ${\chi }_{1t}^2$ are independent chi-square variables with degree 1, and ν_t’s are the eigenvalues of ${\tilde{f}}_Z\left(g_i,g_j\right)$ (i.e., ${\tilde{f}}_Z\left(g_i,g_j\right)={\sum }_{t=1}^{\infty }{\nu }_t{\varphi }_t\left(g_i,z_i\right){\varphi }_t\left(g_j,z_j\right)$, with E(ϕ_s(G, Z)ϕ_t(G, Z)) = I(s = t)). In this case, the unadjusted test introduced below and the test based on $V_{Z,IC}^{*}$ (adjusted test) are both asymptotically correct, but the latter is more powerful than the former in finite samples if Z affects the survival outcome. When Z affects both G and the survival outcome, the unadjusted test is likely to yield false positives, whereas the adjusted test is less likely to yield false positives in general. When the cross-product kernel is used for constructing f(G_i, G_j), the adjusted test is still asymptotically correct in the presence of linear confounding, i.e., G = a + B^TZ + e, where a and B are respectively a constant vector and a constant matrix, and e is a zero-mean random error vector that is independent of Z. It can be shown, following the proof of Theorem 2 in Li et al. (2021), that the null limiting distribution of $n{V}_{Z,IC}^{*}$ with the cross-product kernel for f(G_i, G_j) is (2) under linear confounding. Under the alternative hypothesis that G has effects on the survival outcome adjusting for Z, the covariate-adjusted phenotype similarity S_Z,iS_Z,j is concordant with the covariate-centered genetic similarity ${\tilde{f}}_{\text{Z}}\left(G_i,G_j\right)$. In other words, larger phenotype similarity is weighted heavier and smaller phenotype similarity is weighted lighter, leading to a large value of $V_{Z,IC}^{*}$.

S_Z,i involves unknown (Λ₀, γ) and ${\tilde{f}}_Z\left(G_i,G_j\right)$ involves unknown conditional expectations. In real applications, we replace (Λ₀, γ) by the estimates $\left({\widehat{\Lambda }}_0,\widehat{\mathit{\gamma }}\right)$ obtained from a nonparametric maximum likelihood estimation (Li, Pak, & Todem, 2020) for the model $\Lambda (t\mid Z,h=0)=G\left[{\Lambda }_0(t)e^{{\gamma }^{T}Z}\right]$, and replace the expectations in ${\tilde{f}}_Z\left(G_i,G_j\right)$ by the corresponding sample averages. The resulting weighted V statistic can be shown to be $V_{Z,IC}={\widehat{S}}_Z^T{(I-H)}^{T}F(I-H){\widehat{S}}_Z$, where ${\widehat{S}}_Z={\left({\widehat{S}}_{Z,1},\dots ,{\widehat{S}}_{Z,n}\right)}^T$, ${\widehat{S}}_{Z,i}=\partial \text{log}L\left({\Lambda }_0,\mathit{\gamma },h\right)/{\partial h_i|}_{{\Lambda }_0={\widehat{\Lambda }}_0,\gamma =\widehat{\gamma },h_i=0}$, F = {f(G_i, G_j)}_n×n, I is an n × n identity matrix, and $H=\tilde{Z}{\left({\tilde{Z}}^T\tilde{Z}\right)}^{-1}{\tilde{Z}}^T$ with $\tilde{Z}$ being a n × (q + 1) matrix whose i-th row is $\left(1,Z_i^T\right)$ (i = 1, …, n). We approximate ζ by $\widehat{\zeta }={\widehat{S}}_Z^T{\widehat{S}}_Z/n$ and use a matrix eigen-decomposition of (I − H)^T F(I − H) to obtain a finite-dimensional approximation for ν_t’s. Denote the eigen-values and eigen-vectors of (I − H)^T F(I − H) by ${\tilde{\nu }}_t(t=1,\dots ,n)$ and ${\tilde{\varphi }}_t(t=1,\dots ,n)$, respectively. Because ${\tilde{\varphi }}_t$ satisfies ${\sum }_{i=1}^n{\tilde{\varphi }}_{t,i}^2=1$, instead of $n^{-1}{\sum }_{i=1}^n{\tilde{\varphi }}_{t,i}^2=1$, a finite-dimentional approximation for ν_t is ${\widehat{\nu }}_t={\tilde{\nu }}_t/n$. With $\widehat{\zeta }$ and ${\widehat{\nu }}_t$, the large sample null distribution of nV_Z,IC is approximately

$$n{V}_{Z,IC}~\widehat{\zeta }\sum _{t=1}^n{\widehat{\nu }}_t{\chi }_{1t}^2.$$ (3)

Based on this distribution, we use Davies’ method (Davies, 1980) to compute the p-value P(nV_Z,IC ≥ nV_Z,IC,obs). The test without covariate adjustment can be developed in the same manner. The corresponding weighted V statistic is $V_{IC}={\widehat{S}}^T{(I-J)}^{T}F(I-J)\widehat{S}$, where J is a n × n matrix with all elements being 1/n and $\widehat{S}$ is the same as ${\widehat{S}}_Z$ except that $\widehat{\gamma }$ is fixed as zero and that G is taken to be the identity transformation. Note that this is a nonparametric test, i.e., it has no model assumption, although we exploited a frailty model to derive $\widehat{\text{S}}$. In the sequel, the tests based on V_Z,IC and V_IC are collectively called WV-IC.

Weighted V tests for genetic association considering genetic heterogeneity

The effects of a set of genetic markers on a survival outcome could vary across subpopulations (e.g., different sexes, races or genetic backgrounds). To jointly test the effects of multiple genetic markers considering heterogeneous effects across subpopulations, we extend the weighted V statistics, V_IC and V_Z,IC, in the previous section by replacing f(G_i, G_j) with a heterogeneity weighted kernel function to consider the (latent) population structure. Specifically, f(G_i, G_j) is replaced by w_ij = (1 + k_ij)f(G_i, G_j), where k_ij is a measurement of subpopulation similarity between two individuals. The resulting weighted V statistics are

$$V_{IC}^H={\widehat{S}}^T{(I-J)}^{T}W(I-J)\widehat{S}$$

and

$$V_{Z,IC}^H={\widehat{S}}_Z^T{(I-H)}^{T}W(I-H){\widehat{S}}_Z,$$

where W = (1 + K) ⊙ F, 1 = {1}_n×n, K = {k_ij}_n×n and ⊙ is the element-wise matrix product operator. The tests based on these heterogeneity weighted V statistics are collectively called HWV-IC. They can accommodate the population structure that is completely determined by a vector of observable variables X = (X₁, …, X_D)^T or the latent population structure (e.g., subpopulations with different ancestry backgrounds) that cannot be completely determined but can be inferred by a vector of observable variables X (e.g., a large number of SNPs from the GWAS data). In either case, we can choose k_ij to be $k_{ij}=\text{exp}\left\{-D^{-1}{{\sum }_{d=1}^D\left(X_{i,d}^s-X_{j,d}^s\right)}^2\right\}$, where $X_{i,d}^s$ is the standardized X_i,d, i.e., ${\sum }_{i=1}^n{X}_{i,d}^s/n=0$ and ${{\sum }_{i=1}^n\left(X_{i,d}^s\right)}^2/n=1$. If X is a vector of dummy variables coding several subpopulations (e.g. males and females), we can also use the identity kernel I(X_i = X_j) for k_ij. If X is a set of SNPs, we can use the IBS kernel for k_ij as well. Under the null hypothesis that G has no effect on the survival outcome in any subpopulation adjusting for Z, we use (3) to approximate the distribution of $n{V}_{Z,IC}^H$, replacing F by W to compute ${\widehat{\nu }}_t{}^{\prime }\text{s}$. The null limiting distribution of $n{V}_{IC}^H$ can be obtained similarly. Based on these distributions, we use Davies’ method (Davies, 1980) to compute the p-values $P\left(n{V}_{IC}^H\ge n{V}_{IC,obs}^H\right)$ and $P\left(n{V}_{\text{Z},IC}^H\ge n{V}_{Z,IC,obs}^H\right)$.

Weighted V tests for G–G/G–E interactions

We denote a gene under testing by G = (G₁, …, G_M)^T, where G_m codes the genotype (0, 1 or 2) of the m-th SNP in the gene. In the G–E interaction analysis, the environmental variable is represented by another vector X = (X₁, …, X_D)^T, whose dimension is greater than one if the variable is categorical with more than two levels. In the G–G interaction analysis, we denote the other gene under testing by H = (H₁, …, H_Q)^T.

For testing G–G interactions, we first construct a phenotype similarity measure ${\widehat{S}}_i{\widehat{S}}_j$ based on a semiparametric transformation model with G and H as two covariates, assuming additive genetic effects of G and H under the null hypothesis of no interaction effects. That is, ${\widehat{S}}_i=\partial \text{log}{L}_i\left({\Lambda }_0,\alpha ,\mathit{\gamma },h_i\right)/{\partial h_i|}_{{\Lambda }_0={\widehat{\Lambda }}_0,\mathit{\alpha }=\widehat{\mathit{\alpha }},\gamma =\widehat{\gamma },h_i=0}$,

$$L_i\left({\Lambda }_0,\mathit{\alpha },\mathit{\gamma },h_i\right)=\frac{\text{exp}\left(-G\left[{\Lambda }_0\left(L_i\right)e^{{\mathit{\alpha }}^T{G}_i+{\gamma }^T{H}_i+h_i}\right]\right)-\text{exp}\left(-G\left[{\Lambda }_0\left(R_i\right)e^{{\mathit{\alpha }}^T{G}_i+{\gamma }^T{H}_i+h_i}\right]\right)}{\text{exp}\left(-G\left[{\Lambda }_0\left({\mathcal{L}}_i\right)e^{{\mathit{\alpha }}^T{G}_i+{\gamma }^T{H}_i+h_i}\right]\right)}$$

and (${\widehat{\Lambda }}_0$, $\widehat{\alpha }$, $\widehat{\gamma }$) is the nonparametric maximum likelihood estimator for (Λ₀, α, γ) under the model $\Lambda (t\mid G,H,h=0)=G\left[{\Lambda }_0(t)e^{{\alpha }^{T}G+{\gamma }^{T}H}\right]$. Define $\tilde{G}={\left(G_1,\dots ,G_n\right)}^T$ and $\tilde{H}={\left(H_1,\dots ,H_n\right)}^T$. Let f(G_i, G_j) and k(H_i, H_j) respectively represent the similarities in the genes G and H between subjects i and j. The choice of kernel functions f(·, ·) and k(·, ·) depends on the mode of inheritance. For example, we can use the cross-product kernel if the effect of G is expected to be additive. We propose a G–G interaction test based on the following weighted V statistic

$$V_{IC,I}={\widehat{S}}^T{(I-O)}^T(F\odot K)(I-O)\widehat{S},$$

where $\widehat{S}={\left({\widehat{S}}_1,\dots ,{\widehat{S}}_n\right)}^T$, $O=(1\tilde{G}\tilde{H}){\left\{{(1\tilde{G}\tilde{H})}^T(1\tilde{G}\tilde{H})\right\}}^{-1}{(1\tilde{G}\tilde{H})}^T$. where 1 denotes an n-dim column vector of ones, F = {f(G_i, G_j)}_n×n, K = {k(H_i, H_j)}_n×n. The null limiting distribution of nV_IC,I can be approximated by a mixture of chi-square distributions computed in the same way as for nV_Z,IC. Based on the null distribution, we use Davies’s method to compute the p-value, P(nV_IC,I ≥ nV_IC,I,obs), where V_IC,I,obs is the observed value of V_IC,I. When M + Q is large, we multiply the null distribution by a factor n/(n − M − Q − 1) to take into account the projection operator I − O for finite samples, as Li et al. (2021) does. If the model of inheritance is unspecified for G and H, we propose a similar G–G interaction test to the above with the following modifications:

$$\Lambda (t\mid G,H,h)=G\left[{\Lambda }_0(t)\text{exp}\left({\sum }_{m=1}^M\left\{I\left(G_m=1\right){\alpha }_{1m}+I\left(G_m=2\right){\alpha }_{2m}\right\}+{\sum }_{q=1}^Q\left\{I\left(H_q=1\right){\gamma }_{1q}+I\left(H_q=2\right){\gamma }_{2q}\right\}+h\right)\right],f\left(G_i,G_j\right)={\sum }_{m=1}^{M}I\left(G_{im}=G_{jm}\right),$$

$k\left(H_i,H_j\right)={\sum }_{q=1}^{Q}I\left(H_{iq}=H_{jq}\right)$, $\tilde{G}$ becomes a n × 2M matrix with the i-th row being (I(G_i1 = 1), I(G_i1 = 2), …, I(G_iM = 1), I(G_iM = 2)), and $\tilde{H}$ is defined similarly.

The G–E interaction test is the same as the G–G interaction test except that the genetic variable H is replaced by the environment variable X. The G–G and G–E interaction tests can both adjust for covariates by incorporating them into the projection operator I − O and as additive covariates into the null model.

The weighted V interaction tests (WVI-IC) are more powerful than the regular tests (e.g., likelihood Ratio and Wald tests) for detecting interactions in a semiparametric transformation model when G and/or H contains correlated SNPs, because our test has an effective degree of freedom (DF) lower than the DF of the classical tests. Here we define the effective DF to be the DF d₀ of a rescaled chi-square distribution $c_0{\chi }_{d_0}^2$ that matches the first two moments of nV_IC,I.

Simulations

We performed Monte Carlo simulations to assess the finite-sample performance of the (heterogeneity) weighted V tests for interval-censored and possibly left-truncated survival outcomes. In all simulation scenarios, subjects’ baseline covariates were adjusted, and survival times were generated from the proportion hazards and proportional odds models, two special cases of semiparametric transformation models (r = 0 and r = 1). Due to the lack of available estimation methods for the proportional odds model with left-truncated and interval-censored data, left truncation was considered only under the proportional hazards model. For each subject, three follow-up times, {V₁, V₂, V₃}, were generated as follows: V₁ ~ Unif(L, R), V₂ ~ V₁ + Unif(L, R), and V₃ ~ V₂ + Unif(L, R), where L and R were two positive numbers chosen in a way that left and right censoring rates were controlled within 20%−30% and 30%−40%, respectively. In all simulations, unless otherwise specified, two sample sizes, n = 500, 1000, and two marker-set sizes, p = 4, 8, were considered. Also, in the presence of adjustment covariates, a binary covariate Z₁ ~ Bernoulli(0.5) and a continuous covariate Z₂ ~ Unif(0, 2) were generated for each subject. The genetic markers under testing were SNPs, except that when considering linear confounding, the markers were gene expressions. To mimic linkage disequilibrium (LD) between SNPs within a SNP set, genotypes were generated via a two-step procedure: 1) sample n vectors independently from a multivariate normal distribution with zero mean and a covariance matrix Σ_p×p = {0.5^|k−l|} (unless otherwise specified); 2) categorize each component of every multivariate normal vector into three levels labeled with 0, 1 and 2 using the cut-off values that were selected to achieve the Hardy–Weinberg equilibrium (HWE) and set the minor allele frequency (MAF) at 0.2 (unless otherwise specified). In all scenarios, 1000 Monte Carlo samples were generated. Unless otherwise specified, the significance level of a test was set at 0.05. Simulation results for the proportional hazards model in the absence of left truncation are presented below. Additional simulation results for the proportional hazards model under left truncation and simulation results for the proportional odds model in the absence of left truncation are given in the Supplementary Materials. The proposed tests without covariate adjustment performed well in simulations, but the corresponding simulation results are not presented to save space.

Testing genetic association in the absence of genetic heterogeneity

In this series of simulations, we investigated the performance of WV-IC in detecting the association of a SNP set with a survival outcome subject to interval censoring under no genetic heterogeneity. The survival time for subject i (i = 1, …, n) was generated using the inverse CDF method from the proportional hazard model with the following hazard function,

$$\lambda (t\mid G,Z)=\text{exp}\left(\sum _{j=1}^p{\beta }_{1j}{G}_{ij}+\sum _{k=1}^2{\beta }_{2k}{Z}_{ik}\right),$$ (4)

where the values of β_1j and β_2k vary depending on the simulation scenario.

Empirical size and power of WV-IC under various n’s and p’s.

In this simulation, we set the regression coefficients in (4) to be β_1j = β_2k = 0.05 and β_1j = 0, β_2k = 0.05 (j = 1, …, p and k = 1, 2) in power assessment and size assessment, respectively. We set L = 0.1 and R = 0.6 for the exam time generation. The IBS kernel was used to measure genetic similarity in WV-IC. Table 1 shows that the empirical sizes of WV-IC are close to the nominal level under various n and p. The power of the test increases with the sample size.

Table 1.

Empirical size and power of WV-IC with r = 0 in testing genetic effects in the absence of left truncation.

	Empirical Size (Power)
WV-IC	p=4, n=500	p=4, n=1000
	0.047 (0.231)	0.057 (0.410)
WV-IC	p=8, n=500	p=8, n=1000
	0.048 (0.415)	0.054 (0.738)

Empirical size and power of WV-IC under linear confounding.

In this simulation, gene expressions were considered as the genetic markers under testing. G_i = 1Z_i1 + 1Z_i2 + e_i, where G_i = (G_i1, …, G_ip)^T are the expression levels of p genes in subject i, 1 is a p-dimensional vector of ones, and e_i = (e_i1, …, e_ip)^T follows multivariate normal distribution with zero mean and covariance matrix Σ_p×p = {0.5^|k−l|}. We set L = 0.1, R = 0.6 and coefficient parameters in (4) to be β_1j = 0, β_2k = 0.05 for empirical size assessment and β_1j = β_2k = 0.02 for power evaluation (j = 1, …, p; k = 1, 2). We used the cross-product kernel to measure gene expression similarity in WV-IC. Table 2 shows that the empirical sizes of WV-IC were close to the nominal level under various choices of n and p, which supports the validity of WV-IC under linear confounding. The power of the test increases with the sample size.

Table 2.

Empirical size and power of WV-IC with r = 0 in testing genetic effects under linear confounding and no left truncation.

	Empirical Size (Power)
WV-IC	p=4, n=500	p=4, n=1000
	0.050 (0.158)	0.055 (0.313)
WV-IC	p=8, n=500	p=8, n=1000
	0.041 (0.320)	0.056 (0.574)

Empirical size and power of WV-IC for testing rare variants.

In this simulation, We assessed the empirical size and power of WV-IC for testing the effects of SNP sets consisting of rare variants. The rare variants’ MAFs were generated from Unif(0.005, 0.05). In the power assessment, the effect of a rare variant was given by the formula: $\beta =0.035*\sqrt{1/\{2*MAF*(1-MAF)\}}$. All the other settings were the same as for Table 1. The simulation results are shown in Table 3. The empirical sizes of WV-IC with the IBS and the unified Laplacian kernels are both close to the nominal level. Since the rarer variants had larger effects, WV-IC with the unified Laplacian kernel had higher power than WV-IC with the IBS kernel.

Table 3.

Empirical size and power of WV-IC with r = 0 for testing genetic effects of rare variants in the absence of left truncation.

	Genetic Similarity Kernel	Empirical Size (Power)
		p=4, n=500	p=4, n=1000
WV-IC	IBS	0.043 (0.206)	0.053 (0.413)
WV-IC	Laplacian	0.046 (0.234)	0.048 (0.476)
		p=8, n=500	p=8, n=1000
WV-IC	IBS	0.044 (0.307)	0.050 (0.660)
WV-IC	Laplacian	0.041 (0.382)	0.047 (0.766)

Empirical size and power of WV-IC for testing a large SNP set.

We also performed simulations to assess the empirical size and power of WV-IC for testing the effect of a set of 54 SNPs with a sample size of 1000. The SNP-set size of 54 was chosen because it is the 95% quantile of the sizes of the 23008 genes tested in the real application section. The sample size of 1000 was chosen because the sample size in the real application presented later is 1125. The simulation settings are the same as in the previous section of testing rare variants, except p = 54 and $\beta =0.02*\sqrt{1/\{2*MAF*(1-MAF)\}}$. The simulation results are shown in Table 4. One can see that the empirical sizes of WV-IC with the IBS and the unified Laplacian kernels are both close to the nominal level, but WV-IC with the unified Laplacian kernel had higher power due to the rarer variants having larger effects.

Table 4.

Empirical size and power of WV-IC with r = 0 for testing genetic effects of a large set of SNPs in the absence of left truncation.

	Genetic Similarity Kernel	Empirical Size (Power)
		p=54, n=1000
WV-IC	IBS	0.045 (0.760)
WV-IC	Laplacian	0.047 (0.919)

Testing genetic association in the presence of genetic heterogeneity

In this series of simulations, we investigated the performance of HWV-IC in testing the association of a SNP set in the presence of genetic heterogeneity across 1) observable subpopulations, 2) two latent subpopulations, 3) twenty latent subpopulations, and 4) individual genome profiles, respectively.

Empirical size and power of HWV-IC in the presence of genetic heterogeneity across observable subpopulations.

In this simulation, we investigated the empirical size and power of HWV-IC in the presence of genetic heterogeneity across four observable subpopulations with equal proportions. We let the SNPs under testing have different MAFs and LD structures across the four subpopulations. Specifically, when generating the SNPs following the steps described at the beginning of the Simulations section, we used four different sets of MAFs and LD structures (represented by Σ_p×p) as follows,

• MAFs = {0.0118, 0.0304, 0.0476, 0.0450, 0.0393, 0.0076, 0.0459, 0.0054} and Σ_p×p = {0.2^|k−l|} in subpopulation 1;

• MAFs = {0.0378, 0.0421, 0.0312, 0.0467, 0.0156, 0.0085, 0.0362, 0.0336} and Σ_p×p = {0.5^|k−l|} in subpopulation 2;

• MAFs = {0.0157, 0.0299, 0.0220, 0.0119, 0.0272, 0.0093, 0.0063, 0.0354} and Σ_p×p = {0.7^|k−l|} in subpopulation 3;

• MAFs = {0.0257, 0.0323, 0.0289, 0.0252, 0.0078, 0.0319, 0.0439, 0.0193} and Σ_p×p = {0.6^|k−l|} in subpopulation 4,

where every MAF was generated from Unif(0.005, 0.05). The first four and eight values of each set of MAFs were used when p was 4 and 8 respectively. The survival time of subject i (i = 1, …, n) was generated from an exponential distribution with the following hazard function,

$$\text{exp}\left\{\sum _{k=1}^p\left({\beta }_0+{\mathit{\beta }}_1^T{Z}_i\right)G_{ik}+{\mathit{\beta }}_2^T{Z}_i\right\},$$ (5)

where Z_i = (Z_i1, Z_i2, Z_i3)^T is a three-dimensional vector of dummy variables that codes the subpopulation subject i belongs to. For the size and the power assessments, we used respectively (β₀, ${\beta }_1^T$) = (0, 0, 0, 0) and (β₀, ${\beta }_1^T$) = (0.0005, 0.4, 0.002, 0.0005). We set β₂ = (0.2, 0.2, 0.1)^T, L = 0.1 and R = 0.5. The IBS kernel was used to measure the genetic similarity, and the identity kernel was used to measure the subpopulation similarity in HWV-IC. Table 5 shows that the empirical sizes of HWV-IC and WV-IC are both around the nominal level, but HWV-IC had higher power by accounting for the genetic heterogeneity across the subpopulations.

Table 5.

Comparison of performance of WV-IC and HWV-IC (r = 0) in testing genetic association under genetic heterogeneity across four observable subpopulations and no left truncation.

	Empirical Size (Power)
	p=4, n=500	p=4, n=1000
HWV-IC	0.043 (0.219)	0.056 (0.496)
WV-IC	0.044 (0.173)	0.056 (0.331)
	p=8, n=500	p=8, n=1000
HWV-IC	0.053 (0.233)	0.048 (0.627)
WV-IC	0.048 (0.180)	0.053 (0.412)

Empirical size and power of HWV-IC in the presence of genetic heterogeneity across two latent subpopulations.

In this simulation, we investigated the empirical size and power of HWV-IC under genetic heterogeneity across two latent subpopulations and no left truncation. The survival time of j-th subject in i-th subpopulation was generate from an exponential distribution with the following hazard rate,

$$\text{exp}\left\{\sum _{k=1}^p{G}_{ijk}{\beta }_{ik}+0.05Z_{ij1}+0.05Z_{ij2}\right\},$$ (6)

where β_i1 = … = β_ip(i = 1, 2) represent the effects of G_k’s (k = 1, …, p) in subpopulation i and vary depending on the simulation scenario. One covariate, $X_1={\left\{X_{i1}\right\}}_{i=1}^n$, was simulated to infer the subpopulation. Specifically, X_i1 = a_i + 1 + e, where e ~ N(0, 0.5) and a_i ~ Bernoulli(0.5) is an indicator variable such that a_i = 1 indicates that the i-th subject is from the first subpopulation. β_ik = 0 (i = 1, 2; k = 1, …, p) in size assessment, while different values were given to β_ik’s according to the different heterogeneity scenarios considered in power evaluation, as shown in Table 7. The IBS kernel was used to measure genetic similarity and the Gaussian kernel for subpopulation similarity. Table 6 shows that the empirical size of HWV-IC is close to the nominal level. Table 7 shows that the power of HWV-IC increases with the sample size and the heterogeneity size, measured by |β_1k − β_2k|.

Table 7.

Power of HWV-IC with r = 0 in testing genetic effects under genetic heterogeneity across two latent subpopulations and no left truncation. Various heterogeneity scenarios were considered, determined by the values of β_1k and β_2k, including the same effect size and the same effect direction (T1), identical sizes but opposite directions (T2), no effect in one subpopulation while positive effect in the other (T3), different sizes and opposite directions (T4), and different sizes but the same direction (T5).

	Heterogeneity Scenario
	T1		T2		T3		T4		T5
β 1k	0.05	0.1	−0.05	−0.1	0	0	−0.025	−0.025	0.02	0.02
β 2k	0.05	0.1	0.05	0.1	0.05	0.1	0.05	0.1	0.05	0.1
p=4, n=500	0.153	0.508	0.079	0.213	0.063	0.185	0.083	0.201	0.079	0.202
p=4, n=1000	0.257	0.862	0.136	0.402	0.110	0.369	0.108	0.305	0.162	0.440
p=8, n=500	0.210	0.790	0.232	0.735	0.139	0.441	0.167	0.437	0.142	0.450
p=8, n=1000	0.423	0.989	0.477	0.976	0.230	0.801	0.280	0.788	0.258	0.812

Table 6.

Empirical size of HWV-IC with r = 0 in testing genetic effects under genetic heterogeneity across two latent subpopulations and no left truncation.

	Empirical Size
	p=4, n=500	p=4, n=1000
HWV-IC	0.042	0.049
	p=8, n=500	p=8, n=1000
HWV-IC	0.047	0.045

Empirical size and power of HWV in the presence of genetic heterogeneity across twenty latent subpopulations.

In this simulation, we investigated the performance of HWV-IC under similar settings to the previous section but increased the number of latent subpopulations to 20. The genetic effects of G_k’s (k = 1, …, p) in each of the 20 subpopulations, were set to satisfy β_i1 = … = β_ip(i = 1, …, 20). The genetic effects were zero in empirical size assessment and were sampled from a uniform distribution with mean μ_β and variance ${\sigma }_{\beta }^2$ in power assessment. We simulated 25 covariates, X_d (d = 1, …, 25), to infer the subpopulation. The X_d of the j-th subject in i-th subpopulation was generated by X_ijd = a_id + δ_ij, where a_id (d = 1, …, 25) were randomly sampled from {1, …, 20} without replacement and δ_ij ~ N(0, 0.5). We used the IBS kernel to measure the genetic similarity and the Gaussian kernel to measure the subpopulation similarity in HWV-IC. Table 8 shows that the empirical size of HWV-IC is close to the nominal level. Table 9 shows that as the average effect size (μ_β) increases, the power of HWV-IC increases. For a fixed average effect size, as the genetic heterogeneity size (σ_β) increases, HWV-IC gains power.

Table 8.

Empirical size of HWV-IC with r = 0 in testing genetic effects under genetic heterogeneity across twenty latent subpopulations and no left truncation.

	Empirical Size
	p=4, n=500	p=4, n=1000
HWV-IC	0.051	0.050
	p=8, n=500	p=8, n=1000
HWV-IC	0.045	0.052

Table 9.

Power of HWV-IC with r = 0 under genetic heterogeneity across twenty latent subpopulations and no left truncation.

	Power
μ β	0	0	0	0.025	0.025	0.025	0.05	0.05	0.05
σ β	0.05	0.1	0.15	0.05	0.1	0.15	0.05	0.1	0.15
p=4, n=500	0.063	0.148	0.369	0.105	0.240	0.483	0.224	0.422	0.640
p=4, n=1000	0.108	0.376	0.804	0.252	0.562	0.906	0.532	0.796	0.969
p=8, n=500	0.179	0.766	0.993	0.283	0.862	0.996	0.549	0.933	0.999
p=8, n=1000	0.434	0.989	1.000	0.687	0.999	1.000	0.908	1.000	1.000

Empirical size and power of HWV-IC in the presence of genetic heterogeneity across individual genome profiles.

In this simulation, we investigated the performance of HWV-IC when the subpopulation structure is “continuous”. Specifically, we let the genetic effect vary across individual genome profiles instead of a small number of subpopulations, e.g., males and females. We generated the survival time of the i-th (i = 1, …, n) subject from an exponential distribution with the following hazard rate,

$$\text{exp}\left(\sum _{k=1}^p{G}_{ik}{\beta }_{ik}+0.05Z_{i1}+0.05Z_{i2}\right),$$ (7)

where β_i1 = β_i2 = … = β_ip, and they were set to be zero in empirical size assessment and randomly sampled from a uniform distribution with mean μ_β and variance ${\sigma }_{\beta }^2$ in power assessment. The values of μ_β and ${\sigma }_{\beta }^2$ vary to result in different simulation scenarios. We simulated a set of 1000 SNPs for each subject, ${\left\{X_{id}\right\}}_{d=1}^{1000}(i=1,\dots ,n)$, to represent the genome profile. The genome profile was generated in two steps: 1) for each 1 ≤ d ≤ 1000, sample ${\tilde{X}}_d=\left({\tilde{X}}_{1d},\dots ,{\tilde{X}}_{nd}\right)$ from a multivariate normal distribution, MVN(0, Σ), where Σ is a n × n covariance matrix with the (i, j)-th element being Σ_ij = I(i = j) under the null hypothesis (no genetic association) and Σ_ij = exp(−|β_i1 − β_j1|/σ_β) under the alternative; 2) X_id is then obtained by categorizing ${\tilde{X}}_{id}$ into three levels, 0, 1 and 2, using rank-based cut-off values (i = 1, …, n). The cut-off values were selected to achieve the Hardy-Weinberg equilibrium and the pre-specified minor allele frequency that was randomly sampled from Unif(0.05, 0.2). (L, R) were set to (0.2, 0.6) and (0.1, 0.5) for empirical size assessment and power assessment respectively. We used the IBS kernel to measure both the genetic and the subpopulation similarity in HWV-IC. Table 10 shows that the empirical size of HWV-IC is around the nominal level. Table 11 shows that the power of HWV-IC increases with the mean effect size (μ_β) and, for a fixed mean effect size, HWV-IC becomes more powerful as the genetic heterogeneity (σ_β) increases.

Table 10.

Empirical size of HWV-IC with r = 0 in testing genetic effects under genetic heterogeneity across individual genome profiles and no left truncation.

	Empirical Size
	p=4, n=500	p=4, n=1000
HWV-IC	0.043	0.052
	p=8, n=500	p=8, n=1000
HWV-IC	0.047	0.049

Table 11.

Power of HWV-IC with r = 0 under genetic heterogeneity across individual genome profiles and no left truncation.

	Power
μ β	0	0	0	0.02	0.02	0.02	0.04	0.04	0.04
σ β	0.04	0.08	0.12	0.04	0.08	0.12	0.04	0.08	0.12
p=4, n=500	0.062	0.076	0.103	0.089	0.116	0.149	0.170	0.190	0.252
p=4, n=1000	0.069	0.087	0.223	0.125	0.204	0.369	0.330	0.442	0.590
p=8, n=500	0.094	0.280	0.693	0.170	0.388	0.765	0.347	0.589	0.850
p=8, n=1000	0.117	0.750	0.996	0.326	0.856	0.999	0.699	0.957	1.000

Testing G–G/G–E interaction

In this simulation, we compared the performance of WVI-IC and the regular likelihood ratio test (LRT) in testing interactions between G, a SNP set, and H, a SNP set or a binary variable (Bernoulli(0.5), acting as an environment variable). The survival times were generated from an exponential distribution with the following hazard rate,

$$\text{exp}\left(\sum _{k=1}^{p}0.02G_k+\sum _{l=1}^{q}0.02H_l+\sum _{m=1}^{pq}{(GH)}_m{\beta }_m\right),$$ (8)

where G_k, H_l, and (GH)_m are respectively the k-, l-, and m-th element in G, H and their interaction term GH. β_m = 0 (m = 1, …, pq) under the null hypothesis of no G–H interaction, while β_m = 0.08 under the alternative hypothesis that the interaction exists. The cross-product kernel was used to measure the genetic similarity and the identity kernel was used to measure the environment similarity. Tables 12 and 13 show that the empirical size of WVI-IC is close to the nominal level while that of LRT is larger than the nominal level when the number of elements in the interaction term is large. In all the simulation scenarios, WVI-IC is more powerful than LRT.

Table 12.

Empirical sizes and powers of WVI-IC and LRT in testing G–G interaction under no left truncation.

	Empirical Size (power)
	p=4, q=2, n=500	p=4, q=2, n=1000
WVI-IC	0.046 (0.458)	0.052 (0.793)
LRT	0.069 (0.305)	0.073 (0.519)
	p=6, q=3, n=500	p=6, q=3, n=1000
WVI-IC	0.060 (0.864)	0.057 (0.989)
LRT	0.115 (0.590)	0.076 (0.863)

Table 13.

Empirical sizes and powers of WVI-IC and LRT in detecting G–E interaction under no left truncation.

	Empirical Size (power)
	p=4, q=1, n=500	p=4, q=1, n=1000
WVI-IC	0.051 (0.236)	0.056 (0.435)
LRT	0.055 (0.204)	0.054 (0.350)
	p=8, q=1, n=500	p=8, q=1, n=1000
WVI-IC	0.047 (0.450)	0.043 (0.768)
LRT	0.069 (0.341)	0.068 (0.575)

Empirical sizes of WV-IC and HWV-IC under stringent p-value thresholds

Genome-wide association studies with genotyping or sequencing data usually test the associations between hundreds of thousands of genetic variants and a phenotype, causing the multiple testing problem. Common approaches to address the multiple testing issue, such as Bonferroni correction and Benjamini-Hochberg Procedure (Benjamini & Hochberg, 1995), lead to stringent p-value thresholds when applied to those association analyses. In this simulation, we investigated the sizes of WV-IC and HWV-IC under stringent p-value thresholds (much smaller than 0.05). The simulation setting for WV-IC was the same as that for assessing its size and power under 0.05 level. The simulation setting for HWV-IC was the same as that for assessing its size and power in the presence of genetic heterogeneity across observable subpopulations under 0.05 level. In both scenarios, 150K Monte Carlo samples with n = 1000 and p = 4 were generated to calculate the empirical sizes. Table 14 shows that the empirical sizes of WV-IC and HWV-IC are close to the stringent p-value thresholds. These results indicate that WV-IC and HWV-IC are suitable for large-scale genetic association analyses.

Table 14.

Empirical sizes of WV-IC and HWV-IC under stringent p-value thresholds and no left truncation.

	Empirical Size
Threshold	WV-IC	HWV-IC
0.005	0.0046	0.0047
0.0005	0.00039	0.00053
0.00005	0.000044	0.000047

Application to a dental caries dataset

The etiology of dental caries has a substantial genetic component. Estimates of dental caries heritability range from 30% to 70%, with higher heritability found for primary versus permanent dentition caries (Bretz et al., 2005; Shaffer et al., 2012; Vieira, Modesto, & Marazita, 2014; Wang et al., 2010). However, genetic studies of early childhood caries (ECC) before age 6 have not found any association variant with a genome-wide significance. All the existing genome-wide association studies of ECC (Ballantine et al., 2017; Orlova et al., 2019; Shaffer et al., 2011) tested the genetic association in a single-locus manner. In this paper, we performed a genome-wide association analysis of an ECC dataset using the proposed multi-marker association tests. The dataset analyzed was extracted from a master dataset, Dental Caries: Whole Genome Association and Gene × Environment Studies, which was obtained from dbGaP (accession number: phs000095.v3.p1). The master dataset contains caries assessment data and whole genome genotyping data of 5418 subjects from four study sites: PITT, IOWA, DRDR and GEIRS. The number of typed SNPs is 601,273. For each subject in the master dataset, caries data were from only one dental exam. The dataset analyzed is the dataset of participants who were younger than age 6 at the dental exam. None of them were from DRDR since there was no such participant from that site in the master database. The phenotype is the age to ECC, which is subject to case 1 interval censoring.

SNP-level and subject-level filterings were conducted using PLINK 1.9. Specifically, a SNP was removed if one of the three criteria was met: 1) MAF < 0.01, 2) HWE test’s p-value < 10⁻⁶, and 3) missing rate > 2%. A subject was removed if his/her genotype missing rate is greater than 2%. After the quality control, there are 1125 subjects each with 553,194 SNPs in the final analysis dataset. 385 of the 1125 subjects had developed ECC by the time of the dental exam. The missing genotypes of a SNP were imputed by sampling from a binomial distribution, $Bin(2,\widehat{\text{MAF}})$, where $\widehat{\text{MAF}}$ was the sample minor allele frequency estimated using the non-missing genotypes.

We performed gene-based genome-wide association analyses. The 553,194 SNPs were grouped into 23008 genes based on the reference genome hg18. Specifically, the SNPs located within a gene or its upstream/downstream of 5000 base pairs were grouped to the gene. We then tested the association of each of the 23008 genes with age to ECC. To improve power and/or reduce confounding for the genetic association analyses, we adjusted for sex, race, study cohort (PITT, IOWA or GEIRS), and the top 10 principal components of the SNP data. We used WV-IC and HWV-IC for the association analyses. In the analyses using HWV-IC, we considered four heterogeneity sources: race, sex, genetic background, and home water fluoride level. Race has six categories: White, Asian, Black, American Indian, Bi- or multi-Racial, and Other. The genetic background was represented by a random sample of 200K SNPs from the whole genome. Home water fluoride level was dichotomized as ‘sufficient’ (> 0.7 mg/L) and ‘insufficient’ (≤ 0.7 mg/L). When considering genetic heterogeneity between the two home water fluoride levels, the analysis dataset was a subset consisting of 652 subjects from two study sites, PITT and IOWA, due to missing home water fluoride data in DRDR. In all the analyses, we considered two transformation models: the proportional odds (r = 1) and proportional hazards (r = 0) models. The false discover rate (FDR) was controlled using Benjamini-Hochberg procedure (Benjamini & Hochberg, 1995) without the assumption that the tests were independent of each other. The corresponding p-value threshold is $l_i=(i\alpha ){\left\{m{\sum }_{k=1}^m(1/k)\right\}}^{-1}$ for i = 1, …, m, where m = 23008 and α, the target FDR, is 10%. The Cross-product and the IBS kernels were both used to measure the genetic similarity. The genetic background similarity was measured by the IBS kernel. The identity kernel was used to measure the subpopulation similarity when the subpopulations were defined by race, sex or the dichotomized home water fluoride level.

The data analysis was preformed in R 4.0.2 on the computers (28-core CPU at 2.4 GHz and 115 GB RAM) of MSU High Performance Computing Center (HPCC). The average computation speed of the association tests was less than one second per gene. In our analysis, the total 23008 genes were divided into 231 tasks, each testing 100 genes (the last task tested only 8 genes). Therefore, the computation time of each task was less than 2 minutes on average. We carried out the 231 tasks simultaneously by using 231 computers of MSU HPCC. As a result, the real analysis was completed in a few minutes. Had all the tasks been carried out sequentially, the real analysis would take 6.4 hours at most.

The analysis results are summarized in Table 15. None of the genes reached the significance level after the multiple-testing adjustment. Nevertheless, MPPED2 and TSPAN2 are the two genes that were most frequently found to have the strongest association with age to ECC. MPPED2 was implicated in the first GWAS of childhood caries (Shaffer et al., 2011), and subsequently showed evidence of association in a meta-analysis of five independent replication samples (Stanley et al., 2014), although its role in caries etiology remains unknown. TSPAN2 has not been reported in the literature of ECC genetics before. The functions of these two genes are summarized in Table 16.

Table 15.

Top five genes discovered by WV-IC and HWV-IC from a gene-based genome-wide association analysis of the DC-WGAGE dataset. PH and PO stand for the proportional hazard model and the proportional odds model respectively. CP and IBS stand for cross-product kernel and IBS kernel respectively. Various types of heterogeneity were considered, including no genetic heterogeneity (S1), heterogeneity across races (S2), heterogeneity between the two home water fluoride levels (S3), heterogeneity between sexes (S4), and heterogeneity across genetic backgrounds (S5).

	Scenario	Genes and p-values
PO+CP	S1	MPPED2	OR2B11	NECAB3	E2F1	CYB5R2
		5.01E-05	7.61E-05	8.62E-05	8.62E-05	1.69E-04
	S2	MPPED2	HOXC13-AS	CYB5R2	NECAB3	E2F1
		3.21E-05	1.29E-04	1.37E-04	1.79E-04	1.79E-04
	S3	TSPAN2	CRCT1	HOXC13-AS	LCE5A	TSHB
		1.17E-05	2.23E-05	2.55E-05	2.73E-05	4.78E-05
	S4	MPPED2	OR2B11	NECAB3	E2F1	CDK5RAP1
		5.88E-05	9.91E-05	1.10E-04	1.11E-04	1.65E-04
	S5	MPPED2	OR2B11	NECAB3	E2F1	CYB5R2
		4.94E-05	7.84E-05	8.70E-05	8.70E-05	1.69E-04
PH+CP	S1	MPPED2	OR2B11	NECAB3	E2F1	CYB5R2
		8.23E-05	8.48E-05	8.90E-05	8.90E-05	9.26E-05
	S2	MPPED2	CYB5R2	LINC00511	HOXC13-AS	NCR3LG1
		5.75E-05	9.12E-05	1.69E-04	1.90E-04	2.08E-04
	S3	TSPAN2	HOXC13-AS	CRCT1	ECRG4	LCE5A
		1.27E-05	3.05E-05	4.53E-05	5.07E-05	5.50E-05
	S4	CYB5R2	MPPED2	TSPAN2	NCR3LG1	OR2B11
		2.16E-04	2.21E-04	2.49E-04	2.64E-04	2.73E-04
	S5	TSPAN2	MPPED2	CYB5R2	TPM3P9, ZNF761, ZNF765	NCR3LG1
		1.77E-04	1.93E-04	2.02E-04	2.07E-04	2.09E-04
PO+IBS	S1	LOC101927989	MPPED2	PAX9	NECAB3	E2F1
		4.68E-05	4.89E-05	7.04E-05	7.61E-05	7.61E-05
	S2	MPPED2	STARD5	LOC101927989	CCDC185	CYB5R2
		2.78E-05	3.56E-05	1.19E-04	1.31E-04	1.50E-04
	S3	TSPAN2	CRCT1	TSHB	LCE5A	HOXC13-AS
		5.84E-06	5.63E-05	6.04E-05	6.23E-05	6.48E-04
	S4	LOC101927989	CCDC185	MPPED2	PAX9	NECAB3
		5.03E-05	6.12E-05	6.46E-05	8.93E-05	9.72E-05
	S5	LOC101927989	MPPED2	PAX9	NECAB3	E2F1
		4.79E-05	4.85E-05	7.13E-05	7.79E-05	7.79E-05
PH+IBS	S1	CCDC185	MPPED2	NECAB3	E2F1	PAX9
		5.76E-05	6.89E-05	7.58E-05	7.58E-05	8.95E-05
	S2	STARD5	MPPED2	CCDC185	CYB5R2	PPM1E
		4.28E-05	4.36E-05	7.93E-05	1.07E-04	1.65E-04
	S3	TSPAN2	HOXC13-AS	TSHB	CRCT1	ECRG4
		6.21E-06	7.55E-05	7.83E-05	1.14E-04	1.25E-04
	S4	CCDC185	STARD5	TSPAN2	MPPED2	CYB5R2
		2.04E-05	1.02E-04	1.63E-04	2.01E-04	2.51E-04
	S5	CCDC185	STARD5	TSPAN2	MPPED2	CYB5R2
		2.79E-05	7.71E-05	1.12E-04	1.60E-04	2.14E-04
p-value threshold		4.09E-07	8.18E-07	1.23E-06	1.64E-06	2.05E-06

Table 16.

Functions of MPPED2 and TSPAN2 according to Entrez.

Gene	Function
MPPED2	This gene likely encodes a metallophosphoesterase. The encoded protein may play a role in brain development.
TSPAN2	The protein encoded by this gene is a member of the transmembrane 4 superfamily. The proteins mediate signal transduction events that play a role in the regulation of cell development, activation, growth and motility.

Discussion

In this paper, we developed the first set of multi-marker tests for genetic associations and G–G/G–E interactions with interval-censored and possibly left-truncated survival outcomes. The new tests can adjust for covariates based on a semiparametric transformation model. The proposed HWV-IC can also account for genetic heterogeneity to increase the power of association testing.

Our methods are directly applicable to interval-censored competing risks data if the observation process is independent of the process of competing risks given the covariates. Specifically, by applying our methods to the reduced interval-censored competing risks data pertaining to the failure cause of interest (Hudgens, Li, & Fine, 2014), one can test the effect of a marker set or a G–G/G–E interaction on the cumulative incidence function of that cause.

It is worthwhile to extend the proposed tests to multivariate survival phenotypes. This extension has applications to genetic studies of chronic diseases that can occur to multiple sites in a human body, such as dental caries, diabetic retinopathy, and age-related macular degeneration. Joint analysis of the failure times at the different sites could increase the power to detect the association of such a disease with SNPs, genes or biological pathways. The extension hinges on finding an appropriate phenotype similarity for multivariate interval-censored survival endpoints.

Data Availability Statement

The GWAS data in the real application are available from https://www.ncbi.nlm.nih.gov/projects/gap/cgi-bin/study.cgi?study_id=phs000095.v3.p1 with the permission of dbGaP.