Gene-based Association Analysis of Survival Traits via Functional Regression based Mixed Effect Cox Models for Related Samples

Summary

The importance to integrate survival analysis into genetics and genomics is widely recognized, but only a small number of statisticians have produced relevant work toward this research direction. For unrelated population data, functional regression models have been developed to test for association between a quantitative/dichotomous/survival trait and genetic variants in a gene region. In major gene association analysis, these models have higher power than sequence kernel association tests (SKAT). In this paper, we extend this approach to analyze censored traits for family data or related samples using functional regression based mixed effect Cox models (FamCoxME). The FamCoxME model the effect of major gene as fixed mean via functional data analysis techniques, the local gene or polygene variations or both as random, and the correlation of pedigree members by kinship coefficients or genetic relationship matrix (GRM) or both. The association between the censored trait and the major gene is tested by likelihood ratio tests (FamCoxME FR LRT). Simulation results indicate that the LRT control the type I error rates accurately/conservatively and have good power levels when both local gene or polygene variations are modeled. The proposed methods were applied to analyze a breast cancer data set from the Consortium of Investigators of Modifiers of BRCA1 and BRCA2 (CIMBA). The FamCoxME provides a new tool for gene-based analysis of family-based studies or related samples.

1. Introduction

Understanding the determinants of time to events (e.g., age at onset) is of great importance in dissecting complex disorders, but statistical methods for identifying genetic variants that affect disease progression are not well developed. Some statisticians have recognized the importance of using time to event data into genetic risk assessment and some research in such analyses has been done. For gene-based association studies using sequencing data, the available methods can analyze survival traits of unrelated samples (Chen et al., 2014; Fan et al., 2016a; Tzeng et al., 2014). There are limited gene-based approaches to analyze censored data from studies of either familial or cryptically related individuals. To our knowledge, Leclerc et al. (2015) can analyze related samples but it targets on regions containing a small number of variants while Chien et al. (2017) does not release related software.

Familial or cryptically related data are useful in association analysis since they may oversample affected individuals who are likely to harbor risk variants (Amin et al., 2007; Chiu et al., 2018; Lange, 2002). Moreover, these data are available in the studies of the Consortium of Investigators of Modifiers of BRCA1 and BRCA2 (CIMBA), lung cancer, mental health, and eye diseases (Cough et al., 2013; dataset of breast cancer; Leutenegger et al., 2003). Analyzing the existing data by powerful methods can identify important variants and elucidate genetic architecture. However, investigators can not adequately analyze them due to the lack of powerful methods and software to handle cryptic relatedness, large samples, and high dimension sequencing data. In practice, genetic studies of survival outcomes have been proposed and conducted for cancers in cryptically related samples, but only common variants are mainly used so far.

Compared with genetic studies based only on unrelated individuals, family-based association studies have unique advantages and strengths: controlling population stratification, studying parent-of-origin effects, identifying rare causal variants, and detecting de novo mutations. Cryptically related samples can be viewed as a large pedigree with hidden relatedness and should not be analyzed as unrelated individuals. To model correlation of familial or cryptic relatedness, mixed models are useful in analyzing common variants, but the existing mixed models can not be applied to analyze sequencing data directly due to a large number of rare variants.

Many genetic analyses use individual single nucleotide polymorphisms (SNPs) in mapping genes such as genome wide association studies (GWAS). In the presence of a large number of rare variants, gene-based analysis is more powerful than testing of individual variants (Lee et al., 2012, Li and Leal, 2008; Madsen and Browning, 2009; Morris and Zeggini, 2010). For unrelated samples, functional regression (FR) test statistics perform markedly better than sequence kernel association test (SKAT) procedure in major gene association analysis (Fan et al., 2013, 2014, 2016a,b; Luo, Boerwinkle, and Xiong, 2011; Luo, Zhu, and Xiong, 2012, 2013; Svishcheva, Belonogova, and Axenovich, 2015; Vsevolozhskaya et al., 2014, 2016). However, functional models and related tests are not well-developed to analyze familial or cryptic related data for survival traits.

Motivated by real data analysis needs and the elegant performance of functional and mixed models, we develop FR based mixed effect Cox models in this paper to analyze familial and cryptically related censored data at the gene level (FamCoxME). The FamCoxME model the effect of major gene as fixed mean via functional data analysis techniques, the local gene or polygene variations or both as random, and the correlation of pedigree members by kinship coefficients or genetic relationship matrix (GRM) or both. The models cope with high dimensionality and relatedness and can be utilized to elucidate the genetic architecture of cancers and complex disorders. The association between the censored trait and the major gene is tested by likelihood ratio tests (FamCoxME FR LRT). Extensive simulations are performed to evaluate the type I error rates and power performance of FamCoxME FR LRT. The models are applied to analyze an ovarian cancer dataset in CIMBA.

2. Methods

Here we describe FR based mixed effect Cox models for association analysis of censored traits with sequencing data from pedigrees or a combination of population and pedigrees. Consider a sample of n individuals from multiple extended pedigrees and individual singletons. The n individuals can be cryptically related. The pedigree may include members who are sequenced and phenotyped, and members who are not sequenced or phenotyped. In the sample, assume that n individuals are phenotyped and sequenced in a genomic region that has m variants. We assume that the m variants are located in a region with ordered physical positions 0 ≤ u₁ < … < u_m, and that the physical position of each variant u_j is known. To make the notation simple, we normalize the region [u₁, u_m] to be [0, 1]. For the i-th individual, let T_i denote the survival time, and C_i denote the respective right-censoring time. Let y_i = min(T_i, C_i) be the observed time-to-event and censoring indicator δ_i = 1_(yi=Ti). In addition, let X_i = (x_i(u₁), … , x_i(u_m))′ denote a genotype vector of the m variants and Z_i = (z_i1, … , z_ic)′ denote a c × 1 vector of fixed effect covariates. For the genotypes, we assume that x_i(u_j) (= 0, 1, 2) is the number of minor alleles of the individual at the j-th variant located at the position u_j.

To model random variations and correlation among the n individuals, we consider the contributions from both local gene and polygenes. To model correlation among the n individuals due to polygenic effect, we consider two types of correlation matrices: a kinship matrix and an empirical GRM. For pedigree data, the pedigree members who are not sequenced or phenotyped are used to calculate relations among the pedigree members, i.e., kinship coefficients. For the n individuals who are phenotyped and sequenced, let Ω be a n × n matrix containing diagonal elements Ω_ii = 1 and off-diagonal elements Ω_ij = 2ϕ_ij. The parameter ϕ_ij is the kinship coefficient between individuals i and j, the probability that a randomly chosen allele at a given locus from individual i is identical by descent (IBD) to a randomly chosen allele from individual j conditional on their ancestral relationship.

For population data with structure or a combination of population and pedigree data, the GRM can be calculated based on marker data other than those of the local gene to account for population structure and cryptic relatedness. For individuals i and j from different pedigrees, the kinship coefficient is ϕ_ij = 0. However, the genetic relationship coefficient of individuals i and j can be non-zero since they may be cryptically related. In the following, the kinship matrix or the empirical GRM is denoted by Ω.

To model correlation among the n individuals due to the local gene under the consideration, we estimate regional proportion of alleles shared identical by state (IBS) of the i-th and the j-th individuals by marker information as follows (Yang et al., 2010; Zhu and Xiong, 2012)

$$\begin{gathered} {\pi }_{ij}=\frac{1}{m}\sum _{\mathcal{l}=1}^m{\pi }_{ij\mathcal{l}} \\ =\{\begin{array}{ll}\frac{1}{m}{\sum }_{\mathcal{l}=1}^m\frac{\left[x_i\left(t_{\mathcal{l}}\right)-2p_{\mathcal{l}}\right]\left[x_j\left(t_{\mathcal{l}}\right)-2p_{\mathcal{l}}\right]}{2p_{\mathcal{l}}\left(1-p_{\mathcal{l}}\right)},\hfill & i\ne j\hfill \\ 1+\frac{1}{m}\sum _{\mathcal{l}=1}^m\frac{{\left[x_i\left(t_{\mathcal{l}}\right)\right]}^2-\left(1+2p_{\mathcal{l}}\right)x_i\left(t_{\mathcal{l}}\right)+2p_{\mathcal{l}}^2}{2p_{\mathcal{l}}\left(1-p_{\mathcal{l}}\right)},\hfill & i=j\hfill \end{array}, \end{gathered}$$ (1)

where

$${\pi }_{ij\mathcal{l}}=\{\begin{array}{ll}\frac{\left[x_i\left(t_{\mathcal{l}}\right)-2p_{\mathcal{l}}\right]\left[x_j\left(t_{\mathcal{l}}\right)-2p_{\mathcal{l}}\right]}{2p_{\mathcal{l}}\left(1-p_{\mathcal{l}}\right)},\hfill & i\ne j\hfill \\ 1+\frac{{\left[x_i\left(t_{\mathcal{l}}\right)\right]}^2-\left(1+2p_{\mathcal{l}}\right)x_i\left(t_{\mathcal{l}}\right)+2p_{\mathcal{l}}^2}{2p_{\mathcal{l}}\left(1-p_{\mathcal{l}}\right)},\hfill & i=j\hfill \end{array}$$

is the proportion of alleles shared IBS at ℓ-th variant by the i-th and j-th individuals. Let Π be an n × n matrix containing diagonal elements π_ij for the pedigree.

2.1. Functional Regression Based Mixed Effect Cox Models

In addition to the time-to-event observation y_i and covariates of the i-th individual, we denote the i-th individual’s genetic variant function (GVF) as X_i(u), u ∈ [0, 1]. Note that the data set includes n discrete realizations or observations X_i of the genotypes. Using the genetic variant information X_i, we can estimate the related genetic variant function X_i(u), which will be discussed below. To relate the genetic variant function to the time-to-event observation adjusting for covariates, we consider the following FR based mixed effect Cox proportional hazard model

$${\lambda }_i\left(s|Z_i,X_i,g_i,G_i\right)={\lambda }_0(s)\exp \left(Z_i^{\prime }\alpha +{\int }_0^1{X}_i(u)\beta (u)du+g_i+G_i\right),$$ (2)

where λ₀(s) is the baseline hazard function, α is a c × 1 vector of fixed regression coefficients of covariates, β(u) is the genetic effect of genetic variant function X_i(u) at the position u, (g₁, … , g_n)′ is a normal random vector with mean 0 and covariance matrix ${\sigma }_g^2\Pi$, and (G₁, … , G_n)′ is random vector with mean 0 and covariance matrix ${\sigma }_G^2\mathrm{\Omega }$. Here ${\sigma }_g^2$ and ${\sigma }_G^2$ are local and polygenic variances, respectively.

In the Cox model (2), the genetic variant functions X_i(u) are assumed to be smooth. This assumption can be relaxed by considering the following mixed effect beta-smooth only Cox model

$${\lambda }_i\left(s|Z_i,X_i,g_i,G_i\right)={\lambda }_0(s)\exp \left(Z_i^{\prime }\alpha +\sum _{j=1}^m{x}_i\left(u_j\right)\beta \left(u_j\right)+g_i+G_i\right),$$ (3)

where the genetic effect function β(u) is assumed to be continuous/smooth and so it is called beta-smooth only Cox model. The integration term ${\int }_0^1{X}_i(u)\beta (u)du$ in Cox model (2) is replaced by a summation term ${\sum }_{j=1}^m{x}_i\left(u_j\right)\beta \left(u_j\right)$ in the above model (3), and we make no assumption about smoothness of the genetic variant functions X_i(u). We use the raw genotype data X_i = (x_i(u₁), … , x_i(u_m))′ directly in the beta-smooth only Cox model (3).

Fan et al. (2016a) proposed FR models to analyze survival traits for unrelated population data. Therefore, there were no random terms g_i and G_i in the models of Fan et al. (2016a). In this paper, the models (2) and (3) are designed for pedigree data or related samples. In addition to the fixed effect terms, the random terms g_i and G_i are utilized to model the correlation among the pedigree members. To omit the random terms of (g₁, … , g_n)′ from the models, the models (2) and (3) can be revised as

$${\lambda }_i\left(s|Z_i,X_i,G_i\right)={\lambda }_0(s)\exp \left(Z_i^{\prime }\alpha +{\int }_0^1{X}_i(u)\beta (u)du+G_i\right),$$ (4)

$${\lambda }_i\left(s|Z_i,X_i,G_i\right)={\lambda }_0(s)\exp \left(Z_i^{\prime }\alpha +\sum _{j=1}^m{x}_i\left(u_j\right)\beta \left(u_j\right)+G_i\right).$$ (5)

To omit the random terms of (G₁, … , G_n)′ from the models, the models (2) and (3) can be revised as

$${\lambda }_i\left(s|Z_i,X_i,g_i\right)={\lambda }_0(s)\exp \left(Z_i^{\prime }\alpha +{\int }_0^1{X}_i(u)\beta (u)du+g_i\right),$$ (6)

$${\lambda }_i\left(s|Z_i,X_i,g_i\right)={\lambda }_0(s)\exp \left(Z_i^{\prime }\alpha +\sum _{j=1}^m{x}_i\left(u_j\right)\beta \left(u_j\right)+g_i\right).$$ (7)

2.2. Revised Mixed Effect Cox Models

The genetic effect function β(u) in Cox models (2) and (3) is assumed to be smooth, i.e., β(u) is a continuous function of physical position u. One may expand it using B-spline or Fourier basis functions. Formally, let us expand the genetic effect function β(u) by a series of K_β basis functions ${\psi }_1(u),\dots ,{\psi }_{K_{\beta }}(u)$ as $\beta (u)=\left({\psi }_1(u),\dots ,{\psi }_{K_{\beta }}(u)\right){\left({\beta }_1,\dots ,{\beta }_{K_{\beta }}\right)}^{\prime }=\psi {(u)}^{\prime }\beta$, where $\beta ={\left(|{\beta }_1,\dots ,{\beta }_{K_{\beta }}\right)}^{\prime }$ is a K_β × 1 vector of coefficients and $\psi (u)={\left({\psi }_1(u),\dots ,{\psi }_{K_{\beta }}(u)\right)}^{\prime }$. We consider two types of basis functions: (1) the B-spline basis: ψ_k(u) = B_k(u), k = 1, … , K_β; and (2) the Fourier basis: ψ₁(u) = 1, ψ_2r+1(u) = sin(2πru), and ψ_2r(u) = cos(2πru), r = 1, … , (K_β − 1)/2. Here for Fourier basis, K_β is taken as a positive odd integer [Ramsay and Silverman, 2005].

To estimate the genetic variant functions X_i(u) from the genotypes X_i, we use an ordinary linear square smoother [de Boor, 2001; Ferraty and Romain, 2010; Horváth and Kokoszka, 2012]. Let ϕ_k(u), k = 1, … , K, be a series of K basis functions, such as the B-spline basis and Fourier basis functions. Let Φ denote the m × K matrix containing the values ϕ_k(u_j), and we let ϕ(u) = (ϕ₁(u), … , ϕ_K(u))′. Using the discrete realizations X_i = (x_i(u₁), … , x_i(u_m))′, we estimate the genetic variant function X_i(u) using an ordinary linear square smoother as follows

$${\widehat{X}}_i(u)=\left(x_i\left(u_1\right),\dots ,x_i\left(u_m\right)\right)\mathrm{\Phi }{\left[{\mathrm{\Phi }}^{\prime }\mathrm{\Phi }\right]}^{-1}\varphi (u).$$ (8)

Assume that the genetic effect β(u) is expanded by a series of basis functions ψ_k(u), k = 1, … , K_β, as β(u) = ψ(u)′β. Replacing X_i(u) in the FR based mixed effect Cox model (2) by ${\widehat{X}}_i(u)$ in (8) and β(u) by the expansion, we have a revised mixed effect Cox model

$$\begin{gathered} {\lambda }_i\left(s|Z_i,X_i,g_i,G_i\right) \\ ={\lambda }_0(s)\exp \left(Z_i^{\prime }\alpha +\left(x_i\left(u_1\right),\dots ,x_i\left(u_m\right)\right)\mathrm{\Phi }{\left[{\mathrm{\Phi }}^{\prime }\mathrm{\Phi }\right]}^{-1}{\int }_0^1\varphi (u){\psi }^{\prime }(u)du\beta +g_i+G_i\right) \\ ={\lambda }_0(s)\exp \left(Z_i^{\prime }\alpha +W_i^{\prime }\beta +g_i+G_i\right), \end{gathered}$$ (9)

where $W_i^{\prime }=\left(x_i\left(u_1\right),\dots ,x_i\left(u_m\right)\right)\mathrm{\Phi }{\left[{\mathrm{\Phi }}^{\prime }\mathrm{\Phi }\right]}^{-1}{\int }_0^1\varphi (u){\psi }^{\prime }(u)du$. In the statistical packages R, codes to calculate Φ[Φ′Φ]⁻¹ and ${\int }_0^1\varphi (u){\psi }^{\prime }(u)du$ are readily available [Ramsay et al., 2009].

For the beta-smooth only mixed effect Cox model (3), β(u_j) is introduced as the genetic effect at the position u_j. In this article, we assume that the genetic effect function β(u) is a continuous function of the physical position u. Therefore, β(u_j), j = 1, 2, … , m, are the values of function β(u) at the m physical positions. Expanding β(u_j) by B-spline or Fourier basis functions as above, the mixed effect Cox model (3) can be revised as

$${\lambda }_i\left(s|Z_i,X_i,g_i,G_i\right)\text{}={\lambda }_0(s)\exp \left(Z_i^{\prime }\alpha +\left[\sum _{j=1}^m{x}_i\left(u_j\right)\left({\psi }_1\left(u_j\right),\dots ,{\psi }_{K_{\beta }}\left(u_j\right)\right)\right]{\left({\beta }_1,\dots ,{\beta }_{K_{\beta }}\right)}^{\prime }+g_i+G_i\right)\text{}={\lambda }_0(s)\exp \left(Z_i^{\prime }\alpha +W_i^{\prime }\beta +g_i+G_i\right),$$ (10)

where $W_i^{\prime }={\sum }_{j=1}^m{x}_i\left(u_j\right)\left({\psi }_1\left(u_j\right),\dots ,{\psi }_{K_{\beta }}\left(u_j\right)\right)$. In the same manner, the models (4) and (5) can be revised to obtain

$${\lambda }_i\left(s|Z_i,X_i,g_i\right)={\lambda }_0(s)\exp \left(Z_i^{\prime }\alpha +W_i^{\prime }\beta +G_i\right).$$ (11)

In the same manner, the models (6) and (7) can be revised to

$${\lambda }_i\left(s|Z_i,X_i,g_i\right)={\lambda }_0(s)\exp \left(Z_i^{\prime }\alpha +W_i^{\prime }\beta +g_i\right).$$ (12)

2.3. Test Statistics and Parameters

To test for association between the m genetic variants and the survival trait, the null hypothesis is $H_0:\beta ={\left({\beta }_1,\dots ,{\beta }_{K_{\beta }}\right)}^{\prime }=0$. By fitting the mixed effect Cox models (9), (10), (11), and (12), we may test the null H₀ : β = 0 by a χ²-distributed LRT (FamCoxME FR LRT) statistic with K_β degrees of freedom [Cox, 1972; Cox and Oakes, 1984; Therneau and Grambsch, 2000].

In the data analysis and simulations, we use functions in the fda R package to create the basis functions. The order of the B-spline basis was 4, the number of B-spline basis functions was K = K_β = 10, and the number of Fourier basis functions was K = K_β = 11. To make sure that the results are valid and stable, we examined a wide range of parameters: 6 ≤ K = K_β ≤ 13 for B-spline and Fourier basis functions.

2.4. Simulation Studies

Extensive simulations were performed to evaluate the performance of the proposed FamCoxME FR LRT statistics. In our simulations, we de ne rare variants to be the genetic variants whose minor allele frequencies (MAFs) are less than or equal to 0.03. Two scenarios were considered: (1) some causal variants are rare and some are common; (2) all causal variants are rare. The pedigree structures are described below.

Pedigree Template of 25 Families.

We simulated 25 families by randomly choosing progeny sizes from a negative binomial distribution. Each child within the second generation has a 25% chance of having offspring. The structure of the pedigrees included 228 individuals (119 males and 109 females; 70 founders and 158 nonfounders). The pedigree size ranged from 4 to 24 with an average value of 9.12.

Pedigree Template of 50 Families.

By doubling the 25 families, the pedigree structures included 456 individuals (238 males and 218 females; 140 founders and 316 nonfounders) within the 50 families.

Genetic Variants.

The sequence data are of European ancestry from 10,000 chromosomes covering 1 Mb regions using the calibrated coalescent model as programmed in COSI. The sequence data were generated using COSI’s calibrated best-fit models, and the generated European haplotypes mimic CEPH Utah individuals with ancestry from northern and western Europe in terms of site frequency spectrum and LD patterns [Figure 4 in Schaffner et al., 2005]. Genetic regions of 6, 9, and 12 kb length were randomly selected for empirical type I error and power calculations. To evaluate empirical type I error and power levels, we randomly sampled two haplotypes for each founder. For each nonfounder, we chose one haplotype at random from his or her parents. Genotypes were constructed by summing up two haplotypes for each individual to determine the number of minor alleles.

Type I Error Simulations.

For a constant a > 0, let U_i ~ U(0, a) denote a uniform random variable on (0, a). To evaluate the type I error rates of the proposed LRT statistics, we generated baseline survival time from a Weibull (2, 2) by [Bender et al., 2005]

$$T_i\left(z_{i1},z_{i2},G_i\right)=\sqrt{-\frac{4\log U_i}{\exp \left(0.005\left(z_{i1}-50\right)+0.05z_{i2}+G_i\right)}},$$ (13)

where U_i was uniformly distributed random variable U(0, 1), z_i1 is a continuous covariate from a normal distribution N(50, 5²), z_i2 is a dichotomous covariate taking values 0 and 1 with a probability of 0.5, and (G₁, … , G_n)′ is generated as a normal vector with mean 0 and a covariance matrix ${\sigma }_G^2\mathrm{\Omega },{\sigma }_G=0.2$. Four censoring schemes were considered: (1) C_i = ∞, no censoring, (2) C_i ~ U(0, 10), (3) C_i ~ U(0, 5), and (4) C_i ~ U(0, 3). The time-to-event time is calculated by y_i = min(T_i, C_i) and the censoring indicator is calculated by δ_i = 1_(Ti≤Ci) for a random sample T_i, C_i, i = 1, 2 … , n. The proportions of censored observations in 4 censoring schemes are 0, 17.5%, 35.0%, and 56.5%, respectively.

Genotypes were selected from variants in 6, 9, and 12 kb subregions which were randomly selected from the 1 Mb region. Note that the trait values are not related to the genotypes, and so the null hypothesis holds. For each combination of a pedigree template and a censoring scheme, 1,010 independent seeds were used to calculate type I error rates and 1,000 datasets were generated for a seed. The simulations were carried on National Institutes of Health (NIH) high-performance computational capabilities of the Biowulf cluster which killed a simulation if it took more than 10 days. Thus, we used 1,010 independent seeds to make sure enough datasets were generated to calculate a valid type I error rate. For a combination of a pedigree template and a censoring scheme, 10⁶ phenotype-genotype datasets or slightly more were generated. For each data set, we fit the proposed Cox models (2), (3), (4), (5), (6), and (7). The related FamCoxME FR LRT statistics and related p-values were calculated. After the simulations were complete, an empirical type I error rate was calculated as the proportion of the total p-values which were smaller than a given significant level α.

Empirical Power Simulations.

To evaluate the power of the proposed FamCoxME FR LRT statistics, we simulated data sets under the alternative hypothesis by randomly selecting 6, 9, and 12 kb subregions to obtain causal genetic variants. For each sample dataset, a subset of q causal variants located in the selected subregion was then randomly selected, yielding genotypes X_i = (x_i(u₁), … , x₍u_q))′. Then, we generated the survival time by

$$T_i\left(z_{i1},z_{i2},X_i,G_i\right)=\sqrt{-\frac{4\log U_i}{\exp \left(0.005\left(z_{i1}-50\right)+0.05z_{i2}+\cdots +{\beta }_q{x}_i\left(u_q\right)\right)+G_i}},$$ (14)

where z_i1 and z_i2 were the same as in the type I error model (13), X_i = (x_i(u₁), … , x_i(u_q))′ were genotypes of the i-th individual at the causal variants, and the βs are additive effects for the causal variants defined as follows. We used |β_j| = c| log₁₀(MAF_j)|, where MAF_j was the MAF of the jth variant. Three different settings were considered: 5%, 10%, and 15% of variants in the 6 kb subregion are chosen as causal variants. When 5%, 10%, and 15% of the variants were causal and all causal variants are rare, c = log(90)/k, log(70)/k, and log(50)/k, respectively. When 5%, 10%, and 15% of the variants were causal and some causal variants are common and the rest are rare, c = log(90)/(2k), log(70)/(2k), and log(50)/(2k), respectively. For the template of 50 two- or three- generation families with a total of 456 related individuals, the constants k and genetic effect sizes decrease as region sizes increase

$$k=\{\begin{array}{ll}1.25\hfill & \text{ if region size }=6\text{kb,}\hfill \\ 1.5\hfill & \text{ if region size }=9\text{kb,}\hfill \\ 1.75\hfill & \text{ if region size }=12\text{kb.}\hfill \end{array}$$ (15)

In addition to varying the percentage of causal variants in the subregion, we also varied the direction of effect. We considered situations where (i) all causal variants have positive effects; (ii) 20%/80% causal variants have negative/positive effects; and (iii) 50%/50% causal variants have negative/positive effects. For each setting, 1,000 datasets were simulated to calculate the empirical power as the proportion of p-values which are smaller than a given α level. For each dataset, the causal variants are the same for all the individuals in the dataset, but we allow the causal variants to be different from dataset to dataset.

2.5. Real Data Analysis: Application to CIMBA Ovarian Cancer Data

To evaluate the proposed FamCoxMe FR LRT, we analyzed an ovarian cancer data set from the CIMBA data, which aims to identify genetic factors associated with breast cancer risk in BRCA1 and BRCA2 mutation carriers. These mutation carriers were recruited through cancer genetics clinics or research studies of high risk families in 25 countries. In total, 7,912 women of European ancestry were available for analysis. The sample consists of 5,381 clusters, in which 1,401 have a size greater than one and 3,980 are singletons. For clusters which have a size greater than one, the individuals within a cluster are cryptically related and GRM was used to model their correlations. Among the clusters which have more than one individuals, the cluster size varies between 2 and 38, for a total of 3,932 subjects.

The SNP set analyzed comprises 186 variants across the KCNAB1 locus on chromosome 3 (positions 156158932–156647297). SNPs were genotyped on the collaborative oncological gene environment study (iCOGS) custom array. iCOGS methodology and quality control procedures are detailed elsewhere [Couch et al., 2013]. The MAF of the SNPs ranges from0.0019 to 0.4986. The entries of the IBD matrix were estimated using the genotype data of the iCOGS array other that the tested KCNAB1 region [Amin et al., 2007]. The phenotype of each individual is defined by age at ovarian cancer diagnosis or age at last follow-up. The observations are right censored if any of the following three events occurs before ovarian cancer diagnosis: breast cancer diagnosis, bilateral prophylacticmastectomy, or lost to follow-up. The censoring rate is 7.3%.

3. Results

3.1. Empirical Type I Error Rates

The empirical type I error rates for the proposed FamCoxMe FR LRT statistics are reported in Tables 1 and 2 at four nominal significance levels α = 0.05, 0.01, 0.001, and 0.0001. In Table 1, all variants were used to generate genotype data but none of them relates to the trait. In Table 2, only rare variants were used to generate genotype data.

Table 1: Empirical type I error rates of the FamCoxME FR LRT Statistics at nominal levels α = 0:05; 0:01; 0:001; and 0:0001 using the 50 two- or three- generation families with a total of 456 related individuals as a template, when region sizes are 6 kb, 9 kb, and 12 kb, and some variants are common and the rest are rare.

The order of B-spline basis was 4, and the number of basis functions of B-spline was K = K_β = 10; the number of Fourier basis functions was K = K_β = 11.

Region Size (# Variants)	The censoring scheme	Nominal Level α	Model both polygenic ${\sigma }_G^2$ and ${\sigma }_g^2$ local				Model local variance ${\sigma }_G^2$ only				Model local variance ${\sigma }_g^2$ only
			Cox model (2)		Cox model (3)		Cox model (4)		Cox model (5)		Cox model (6)		Cox model (7)
			B-spline	Fourier	B-spline	Fourier	B-spline	Fourier	B-spline	Fourier	B-spline	Fourier	B-spline	Fourier
6 kb (117)	∞	0.05	0.049773	0.050839	0.049823	0.050855	0.062691	0.064135	0.062752	0.064150	0.059405	0.060981	0.059483	0.061007
		0.01	0.009028	0.009217	0.009034	0.009214	0.013798	0.014226	0.013812	0.014231	0.011506	0.011840	0.011511	0.011844
		0.001	0.000719	0.000675	0.000719	0.000675	0.001588	0.001552	0.001583	0.001551	0.000994	0.000961	0.000996	0.000963
		0.0001	5.87E-05	5.21E-05	5.84E-05	5.20E-05	0.000192	0.000170	0.000190	0.000171	9.81E-05	7.72E-05	0.000101	7.90E-05
	U(0, 10)	0.05	0.050198	0.050935	0.050207	0.050948	0.063058	0.064201	0.063102	0.064224	0.058486	0.059601	0.058519	0.059635
		0.01	0.008868	0.008974	0.008861	0.008977	0.013690	0.014031	0.013692	0.014048	0.010907	0.011213	0.010905	0.011217
		0.001	0.000686	0.000724	0.000687	0.000725	0.001507	0.001596	0.001513	0.001600	0.000928	0.000963	0.000925	0.000963
		0.0001	6.05E-05	4.39E-05	6.02E-05	4.39E-05	0.000183	0.000173	0.000184	0.000174	8.67E-05	6.59E-05	8.63 E-05	6.58E-05
	U(0, 5)	0.05	0.051437	0.052462	0.051482	0.052494	0.064406	0.065868	0.064482	0.065909	0.058569	0.060164	0.058649	0.060205
		0.01	0.009304	0.009524	0.009315	0.009542	0.014240	0.014557	0.014258	0.014571	0.011104	0.011395	0.011119	0.011415
		0.001	0.000766	0.000762	0.000766	0.000764	0.001622	0.001660	0.001622	0.001666	0.000948	0.000961	0.000947	0.000961
		0.0001	6.46E-05	5.59E-05	6.73E-05	5.68E-05	0.000187	0.000182	0.000189	0.000183	8.89E-05	7.19E-05	9.13E-05	7.18E-05
	U(0, 3)	0.05	0.054675	0.056454	0.054765	0.056486	0.068074	0.070144	0.068159	0.070197	0.061059	0.062962	0.061161	0.063020
		0.01	0.010040	0.010452	0.010054	0.010450	0.015201	0.015879	0.015223	0.015895	0.011536	0.012110	0.011553	0.012119
		0.001	0.000844	0.000862	0.000841	0.000860	0.001725	0.001902	0.001728	0.001909	0.001009	0.001058	0.001013	0.001055
		0.0001	6.48E-05	7.70E-05	6.63E-05	7.48E-05	0.000194	0.000225	0.000195	0.000229	7.59E-05	9.19E-05	7.93E-05	8.88E-05
9 kb (176)	∞	0.05	0.050766	0.051208	0.050764	0.051207	0.061611	0.062415	0.061608	0.062413	0.060787	0.061822	0.060784	0.061815
		0.01	0.009397	0.009466	0.009393	0.009466	0.013251	0.013445	0.013248	0.013445	0.012023	0.012202	0.012022	0.012202
		0.001	0.000785	0.000767	0.000784	0.000767	0.001432	0.001452	0.001432	0.001452	0.001131	0.001133	0.001131	0.001134
		0.0001	5.16E-05	4.56E-05	5.16E-05	4.56E-05	0.000138	0.000138	0.000138	0.000138	8.73E-05	7.63 E-05	8.73E-05	7.63E-05
	U(0, 10)	0.05	0.050881	0.051275	0.050879	0.051277	0.061778	0.062301	0.061776	0.062304	0.059597	0.060468	0.059597	0.060468
		0.01	0.009451	0.009620	0.009446	0.009622	0.013304	0.013596	0.013305	0.013598	0.011677	0.012015	0.011680	0.012018
		0.001	0.000811	0.000814	0.000808	0.000814	0.001464	0.001496	0.001462	0.001497	0.001090	0.001122	0.001093	0.001123
		0.0001	6.39E-05	5.88E-05	6.29E-05	5.88E-05	0.000152	0.000163	0.000152	0.000163	9.88E-05	8.97E-05	0.000102	8.97E-05
	U(0, 5)	0.05	0.051383	0.052038	0.051402	0.052039	0.062361	0.063130	0.062382	0.063129	0.058792	0.059901	0.058811	0.059899
		0.01	0.009669	0.009786	0.009670	0.009785	0.013624	0.013728	0.013628	0.013729	0.011664	0.011923	0.011663	0.011921
		0.001	0.000801	0.000846	0.000803	0.000846	0.001484	0.001521	0.001487	0.001521	0.001047	0.001091	0.001048	0.001090
		0.0001	6.87E-05	5.77E-05	6.96E-05	5.77E-05	0.000157	0.000174	0.000159	0.000174	9.36E-05	8.75E-05	9.35E-05	8.65E-05
	U(0, 3)	0.05	0.053260	0.053318	0.053291	0.053322	0.064147	0.064552	0.064183	0.064556	0.059207	0.059642	0.059256	0.059645
		0.01	0.010009	0.009987	0.010025	0.009987	0.014020	0.014081	0.014036	0.014080	0.011628	0.011694	0.011650	0.011695
		0.001	0.000872	0.000867	0.000873	0.000867	0.001559	0.001573	0.001559	0.001572	0.001067	0.001074	0.001066	0.001074
		0.0001	7.65E-05	8.03E-05	7.74E-05	8.03E-05	0.000176	0.000190	0.000176	0.000189	0.000105	0.000108	0.000104	0.000108
12 kb (235)	∞	0.05	0.051113	0.051861	0.051113	0.051861	0.060704	0.061716	0.060704	0.061716	0.061268	0.062323	0.061258	0.062328
		0.01	0.009727	0.009978	0.009728	0.009978	0.013096	0.013444	0.013096	0.013444	0.012484	0.012935	0.012483	0.012933
		0.001	0.000892	0.000867	0.000892	0.000867	0.001433	0.001462	0.001433	0.001462	0.001235	0.001240	0.001237	0.001240
		0.0001	9.55E-05	9.34E-05	9.54E-05	9.34E-05	0.000173	0.000171	0.000173	0.000171	0.000131	0.000113	0.000132	0.000113
	U(0, 10)	0.05	0.051300	0.052430	0.051306	0.052430	0.060971	0.062214	0.060977	0.062214	0.060243	0.061634	0.060245	0.061634
		0.01	0.009689	0.009954	0.009691	0.009954	0.013027	0.013300	0.013027	0.013300	0.012047	0.012522	0.012048	0.012522
		0.001	0.000786	0.000879	0.000787	0.000879	0.001377	0.001497	0.001377	0.001497	0.001081	0.001191	0.001081	0.001190
		0.0001	5.46E-05	7.24E-05	5.56E-05	7.24E-05	0.000138	0.000155	0.000138	0.000155	8.34E-05	0.000114	8.43 E-05	0.000114
	U(0, 5)	0.05	0.051835	0.052237	0.051838	0.052237	0.061499	0.062182	0.061501	0.062182	0.059208	0.060040	0.059212	0.060039
		0.01	0.009680	0.009914	0.009684	0.009914	0.013069	0.013345	0.013074	0.013345	0.011765	0.012078	0.011770	0.012080
		0.001	0.000854	0.000853	0.000856	0.000853	0.001398	0.001430	0.001401	0.001430	0.001106	0.001131	0.001109	0.001131
		0.0001	5.74E-05	7.52E-05	5.94E-05	7.52E-05	0.000132	0.000155	0.000136	0.000155	7.53E-05	0.000101	7.62E-05	0.000101
	U(0, 3)	0.05	0.051937	0.052373	0.051942	0.052373	0.061822	0.062142	0.061831	0.062142	0.058012	0.058653	0.058024	0.058653
		0.01	0.009959	0.009927	0.009965	0.009927	0.013255	0.013432	0.013263	0.013432	0.011573	0.011666	0.011578	0.011665
		0.001	0.000914	0.000885	0.000914	0.000885	0.001445	0.001464	0.001445	0.001464	0.001104	0.001091	0.001105	0.001091
		0.0001	5.95E-05	8.13E-05	5.85E-05	8.13E-05	0.000127	0.000154	0.000127	0.000154	9.32E-05	0.000103	9.32E-05	0.000103

Table 2: Empirical type I error rates of the FamCoxME FR LRT Statistics at nominal levels α = 0:05; 0:01; 0:001; and 0:0001 using the 50 two- or three- generation families with a total of 456 related individuals as a template, when region sizes are 6 kb, 9 kb, and 12 kb, and all variants are rare.

The order of B-spline basis was 4, and the number of basis functions of B-spline was K = K_β = 10; the number of Fourier basis functions was K = K_β = 11.

Region Size (# Variants)	The censoring scheme	Nominal Level α	Model both polygenic ${\sigma }_G^2$ and ${\sigma }_g^2$ local				Model local variance ${\sigma }_G^2$ only				Model local variance ${\sigma }_g^2$ only
			Cox model (2)		Cox model (3)		Cox model (4)		Cox model (5)		Cox model (6)		Cox model (7)
			B-spline	Fourier	B-spline	Fourier	B-spline	Fourier	B-spline	Fourier	B-spline	Fourier	B-spline	Fourier
6 kb (106)	∞	0.05	0.051735	0.053520	0.052028	0.053686	0.067600	0.069811	0.067988	0.070068	0.060031	0.062197	0.060360	0.062414
		0.01	0.009166	0.009573	0.009242	0.009615	0.015477	0.015929	0.015610	0.015993	0.011117	0.011709	0.011192	0.011747
		0.001	0.000762	0.000797	0.000774	0.000797	0.001919	0.002013	0.001963	0.002012	0.001003	0.001002	0.001021	0.001001
		0.0001	6.95E-05	6.16E-05	8.29E-05	6.66E-05	0.000279	0.000313	0.000313	0.000302	9.62E-05	0.000102	0.000108	9.65E-05
	U(0, 10)	0.05	0.052385	0.054618	0.052768	0.054722	0.068422	0.071200	0.068921	0.071450	0.059649	0.062366	0.060071	0.062523
		0.01	0.009356	0.009873	0.009448	0.009886	0.015574	0.016469	0.015706	0.016557	0.011137	0.011714	0.011212	0.011742
		0.001	0.000761	0.000815	0.000778	0.000810	0.002041	0.002124	0.002061	0.002158	0.000949	0.001001	0.000961	0.000994
		0.0001	7.39E-05	8.49E-05	8.68E-05	8.03E-05	0.000329	0.000355	0.000334	0.000375	9.21E-05	0.000107	9.83 E-05	9.83 E-05
	U(0, 5)	0.05	0.056842	0.060443	0.057291	0.060666	0.073463	0.077578	0.073963	0.077884	0.063636	0.067878	0.064096	0.068158
		0.01	0.010506	0.011187	0.010605	0.011234	0.017103	0.018338	0.017273	0.018435	0.012129	0.012975	0.012212	0.013026
		0.001	0.000863	0.000945	0.000892	0.000963	0.002206	0.002356	0.002235	0.002362	0.001058	0.001118	0.001069	0.001131
		0.0001	7.41E-05	0.000102	9.93E-05	0.000117	0.000373	0.000383	0.000380	0.000391	0.000101	0.000121	0.000116	0.000126
	U(0, 3)	0.05	0.070466	0.077778	0.070894	0.077912	0.088008	0.095945	0.088529	0.096091	0.077786	0.085371	0.078200	0.085655
		0.01	0.014043	0.016189	0.014134	0.016239	0.021916	0.024733	0.022065	0.024737	0.015960	0.018263	0.016038	0.018271
		0.001	0.001223	0.001627	0.001268	0.001711	0.002941	0.003536	0.002984	0.003554	0.001433	0.001819	0.001473	0.001802
		0.0001	0.000137	0.000276	0.000168	0.000367	0.000475	0.000681	0.000496	0.000717	0.000157	0.000273	0.000184	0.000252
9 kb (159)	∞	0.05	0.052254	0.052590	0.052277	0.052597	0.065083	0.065993	0.065118	0.066008	0.061004	0.062038	0.061035	0.062047
		0.01	0.009584	0.009601	0.009597	0.009605	0.014277	0.014534	0.014295	0.014538	0.011782	0.011971	0.011791	0.011973
		0.001	0.000800	0.000791	0.000801	0.000793	0.001606	0.001650	0.001611	0.001653	0.001063	0.001045	0.001063	0.001045
		0.0001	6.13E-05	6.73E-05	6.11E-05	6.80E-05	0.000163	0.000195	0.000164	0.000196	8.88E-05	8.73E-05	9.08 E-05	8.72E-05
	U(0, 10)	0.05	0.053015	0.053791	0.053045	0.053809	0.065889	0.067120	0.065931	0.067136	0.060806	0.062020	0.060873	0.062037
		0.01	0.009980	0.010016	0.009975	0.010018	0.014706	0.015077	0.014725	0.015075	0.011970	0.012228	0.011992	0.012224
		0.001	0.000862	0.000905	0.000850	0.000905	0.001797	0.001836	0.001790	0.001831	0.001080	0.001177	0.001089	0.001165
		0.0001	8.94E-05	8.87E-05	8.40E-05	8.53E-05	0.000287	0.000282	0.000279	0.000273	0.000122	0.000148	0.000132	0.000128
	U(0, 5)	0.05	0.055511	0.056743	0.055623	0.056774	0.068744	0.070519	0.068866	0.070539	0.062614	0.064166	0.062715	0.064199
		0.01	0.010678	0.010850	0.010701	0.010866	0.015710	0.016151	0.015733	0.016151	0.012645	0.012804	0.012669	0.012819
		0.001	0.000991	0.000969	0.000993	0.000982	0.001974	0.002035	0.001967	0.002029	0.001239	0.001240	0.001249	0.001248
		0.0001	9.71E-05	9.04E-05	9.66E-05	0.000100	0.000295	0.000302	0.000282	0.000293	0.000122	0.000136	0.000126	0.000134
	U(0, 3)	0.05	0.064170	0.066714	0.064291	0.066742	0.077961	0.081075	0.078132	0.081119	0.070795	0.073738	0.070927	0.073825
		0.01	0.013087	0.013765	0.013117	0.013778	0.018578	0.019793	0.018656	0.019805	0.015005	0.015747	0.015042	0.015768
		0.001	0.001227	0.001359	0.001228	0.001363	0.002401	0.002664	0.002435	0.002668	0.001499	0.001640	0.001486	0.001636
		0.0001	0.000102	0.000136	0.000103	0.000148	0.000311	0.000380	0.000340	0.000382	0.000155	0.000160	0.000143	0.000156
12 kb (212)	∞	0.05	0.052414	0.053135	0.052420	0.053135	0.063401	0.064484	0.063411	0.064486	0.061682	0.062721	0.061693	0.062719
		0.01	0.010015	0.010020	0.010014	0.010021	0.013867	0.014087	0.013865	0.014088	0.012519	0.012686	0.012516	0.012688
		0.001	0.000928	0.000900	0.000928	0.000900	0.001569	0.001572	0.001569	0.001573	0.001242	0.001215	0.001242	0.001215
		0.0001	7.22E-05	7.82E-05	7.22E-05	7.82E-05	0.000166	0.000176	0.000166	0.000177	0.000108	0.000112	0.000106	0.000112
	U(0, 10)	0.05	0.053154	0.053681	0.053163	0.053685	0.064272	0.065219	0.064303	0.065232	0.061222	0.062298	0.061268	0.062293
		0.01	0.010259	0.010313	0.010260	0.010314	0.014135	0.014524	0.014156	0.014529	0.012456	0.012605	0.012485	0.012601
		0.001	0.000906	0.000942	0.000906	0.000940	0.001660	0.001676	0.001679	0.001678	0.001178	0.001221	0.001189	0.001218
		0.0001	7.07E-05	9.07E-05	7.16E-05	8.80E-05	0.000186	0.000226	0.000206	0.000227	0.000108	0.000140	0.000118	0.000139
	U(0, 5)	0.05	0.054780	0.055327	0.054796	0.055327	0.066037	0.066878	0.066062	0.066882	0.062056	0.063002	0.062052	0.062998
		0.01	0.010636	0.010661	0.010641	0.010660	0.014827	0.014933	0.014852	0.014937	0.012649	0.012716	0.012655	0.012724
		0.001	0.000929	0.000964	0.000925	0.000960	0.001711	0.001781	0.001720	0.001786	0.001228	0.001262	0.001231	0.001266
		0.0001	8.77E-05	8.75E-05	8.68E-05	8.67E-05	0.000244	0.000258	0.000250	0.000257	0.000133	0.000137	0.000136	0.000140
	U(0, 3)	0.05	0.060170	0.060531	0.060218	0.060537	0.071801	0.072469	0.071888	0.072486	0.066545	0.067295	0.066598	0.067284
		0.01	0.012197	0.012374	0.012213	0.012386	0.016533	0.016888	0.016587	0.016908	0.014033	0.014274	0.014053	0.014281
		0.001	0.001154	0.001132	0.001156	0.001142	0.001994	0.002022	0.002031	0.002037	0.001449	0.001452	0.001451	0.001457
		0.0001	0.000115	9.26E-05	0.000116	0.000101	0.000269	0.000283	0.000299	0.000293	0.000167	0.000143	0.000173	0.000152

When some variants are common and the rest are rare, Tables 1 shows that the FamCoxMe FR LRT statistics of the Cox models (2), (3), (6), and (7) control the type I error rates correctly, whether the genotype data are smoothed or not and regardless of which basis functions are used to smooth the GVF and β(u). When all variants are rare, Table 2 shows that the type I error rates are well controlled for the Cox models (2) and (3) except for the heaviest censoring level scheme U(0, 3) and region sizes of 6 kb and 9 kb, and the type I error rates of (6) and (7) are slightly higher. The type I error rates of models (4) and (5) are inflated in both Tables 1 and 2. Therefore, only modeling the random term G_i may lead to a high false positive rate.

The results of the Cox models (2), (4), and (6) are very similar to those of beta-smooth only models (3), (5), and (7), respectively. Therefore, the FamCoxME FR LRT statistics do not strongly depend on whether the genotype data are smoothed or not, or which basis functions are used.

3.2. Statistical Power Evaluation

The power of the proposed FamCoxME FR LRT statistics was evaluated by using the simulated sequence data. Since the type I error rates of LRT statistics of Cox models (2) and (3) are well-controlled when the region size is 12 kb, we reported in Figures 1 and 2 the power levels for the region. In Figure 1, some causal variants are rare and some are common. In Figure 2, all causal variants are rare.

Figure 1: The empirical power of the statistics at = 0.01 using the 50 two- or three- generation families with a total of 456 related individuals as a template, when some variants are common and the rest are rare, genetic effect sizes are given by (15), and the region size is 12 kb.

The order of B-spline basis was 4, and the number of basis functions of B-spline was K = K_β = 10; the number of Fourier basis functions was K = K_β = 11. Abbreviation: “pct” = percent, “Neg pct” means percentage of causal variants which have negative effects.

Figure 2: The empirical power of the statistics at α = 0.01 using the 50 two- or three- generation families with a total of 456 related individuals as a template, when all variants are rare, genetic effect sizes are given by (15), and the region size is 12 kb.

The order of B-spline basis was 4, and the number of basis functions of B-spline was K = K_β = 10; the number of Fourier basis functions was K = K_β = 11. Abbreviation: “pct” = percent, “Neg pct” means percentage of causal variants which have negative effects.

We compared the power of FamCoxME FR LRT statistics of the models (2) and (3) by B-spline and Fourier basis functions. In the two FamCoxME FR LRT statistics to use B-spline (or Fourier) basis functions, one is to smooth both genetic variant functions and genetic effect function β(u) in model (2), and the other is only to smooth the genetic effect function β(u) (i.e., beta-smooth only model (3)). The four LRT statistics have similar power.

When some causal variants are rare and some are common, the power levels are pretty high in Figure 1. Relatively, the power levels in Figure 2 are lower when all causal variants are rare.

3.3. Real Data Analysis: Application to CIMBA Ovarian Cancer Data

Table 3 shows the results of association analysis of CIMBA ovarian cancer data for the gene KCNAB1 using the proposed FamCoxME FR LRT of models (2) and (3). We analyzed the data three times in the gene region for each case of “All Subjects” and “Clusters Only”: (1) all 186 genetic variants, (2) 110 common variants only, and (3) 76 rare variants only. Here the rare variants are defined as those that the MAF ≤ 0.05, and common variants are defined as those that the MAF > 0.05.

Table 3: Application to CIMBA Ovarian Cancer Data.

In all subjects, 7,912 women of European ancestry were available for analysis. The sample of all subjects consists of 5,381 clusters, in which 1,401 have a size greater than one and 3,980 are singletons. In “Cluster Only”, the 1,401 clusters which have more than one individuals are analyzed. The order of B-spline basis was 4, and the number of basis functions of B-spline was K = K_β = 10; the number of Fourier basis functions was K = K_β = 11.

Type of Variants	Number of Variants	Type of Data	The	Model both polygenic ${\sigma }_G^2$ and local ${\sigma }_g^2$
			Sample Size	Cox model (2)		Cox model (3)
				B-spline	Fourier	B-spline	Fourier
All	186	All Subjects	7,912	1.51E-05	0.000354	1.51E-05	0.000354
		Cluster Only	3,932	0.002830	0.002508	0.002830	0.002508
Rare	76	All Subjects	7,912	6.42E-07	3.61E-06	6.42E-07	3.61E-06
		Cluster Only	3,932	0.000254	0.000136	0.000254	0.000136
Common	110	All Subjects	7.912	0.002373	0.008107	0.002373	0.008107
		Cluster Only	3,932	0.010498	0.005116	0.010498	0.005116

The most significant results were found in the analysis of 76 rare variants using all subjects, with a p-value 6.42 × 10⁻⁷ by B-spline basis functions for both models (2) and (3) and a p-value 3.61 × 10⁻⁶ by Fourier basis functions. From the analysis of CIMBA data, we may see that rare variants rather than common variants in the KCNAB1 gene region may play an important role for ovarian cancer. An analysis by combining rare and common variants together may dilute or reduce the signal of the rare variants.

In the Table 3, the results of the FamCoxME FR LRT statistics of beta-smooth only model (3) are identical to those of model (2) by smoothing both genetic variant functions X_i(u) and genetic effect function β(u). Thus, whether the genetic variant functions are smoothed or not does not have much impact on the results. This shows that the functional regression models perform very stable as shown in the simulations.

4. Discussion

In this article, we developed mixed effect Cox proportional hazard models and related FamCoxME FR LRT statistics for gene-based association analysis of survival traits to analyze familial and cryptically related samples. Extensive simulations are performed to evaluate empirical type I error rates and power of the LRT statistics. We show that the FamCoxME FR LRT statistics control the type I error well when variations and correlations of both local gene and polygenes are modelled. The FamCoxME FR LRT statistics have good power to analyze related samples. The proposed methods were applied to analyze a CIMBA ovarian cancer dataset and it was found that rare variants play an important role in ovarian cancer, and this helps to elucidate cancer risk and progression.

In the proposed functional regression based Cox models, the random variations and correlations of the local gene or polygene contributions or both are modelled to account for familial relatedness. To handle high dimension genetic data, the genetic effects are treated as a function of the physical position and the genetic variant data are viewed as stochastic functions of the physical position [Ross, 1996]. It was found that only modeling the polygenic variation G_i may in ate the type I errors while simultaneously modelling variations and correlations of both local gene and polygene contributions can stabilize the type one errors.

To fit the proposed Cox models, one needs to estimate parameters and the procedure can be slower than kernel-based tests. In our simulation studies on our linux system, it takes about 120 hours or 5 days to analyze 10³ phenotype-genotype datasets to calculate the four FamCoxME FR LRT statistics in Tables 1 and 2 for 50 pedigree template. For the CIMBA ovarian cancer dataset, it takes about a week to nish the analysis. Hence, the computational burden is heavy. The models can be used to analyze candidate genes for large samples. For the whole genome and whole exome association studies and moderate samples, the models and related test statistics can be utilized by dividing large number of gene regions to be small number ones to be analyzed in a parallel way to speed up the analysis.

Functional regression models are utilized to perform association analysis for quantitative and dichotomous traits for both population and related samples. This paper fills the gaps by using functional and mixed models to analyze related samples of survival traits and high dimension genetic data. The models can be used to dissect architecture of complex disorders by analyzing survival traits.