Statistical Haplotypes Based on Functional Sequence Data Analysis for Genome-Wide Association Studies ^†

Abstract

Functional data analysis has demonstrated significant success in time series analysis. In recent biomedical research, it has also been used to analyze sequence variations in genome-wide association studies (GWAS). The observations of genetic variants, called single-nucleotide polymorphisms (SNPs), of an individual are distributed over the loci of a DNA sequence. Thus, it can be regarded as a realization of a stochastic process, which is no different from a time series. However, SNPs are usually coded as the number of minor alleles, which are categorical. The usual least-square smoothing in FDA only works well when the data is continuous and normally distributed. The normality assumption will be violated for categorical SNP data. In this work, we propose a two-step method for smoothing categorical SNPs using a novel method and constructing haplotypes having strong associations with the disease using functional generalized linear models. We show its effectiveness through a real-world PennCATH dataset.

1. Introduction

Functional data analysis (FDA) is a common tool for analyzing complex datasets that are collected over a sequence, such as time and location. It is based on the assumption that there is an underlying stochastic process represented by a continuous function $x_i\left(t\right)$, but the data can only be observed at discrete time points or locations. To overcome this challenge, FDA represents the datasets as functions or curves by smoothing the observations. FDA has found extensive application in many fields related to time series analysis. Examples include predicting stock prices, discovering geophysical and meteorological patterns, and forecasting traffic volumes. Recently, there has been increasing interest in using FDA in biomedical research, such as genome-wide association studies (GWAS).

GWAS aim to find associations between single-nucleotide polymorphisms (SNPs) and diseases or traits by case–control studies, providing insights into the genetic risk factors for diseases and offering opportunities for preventive measures and treatments [1]. As a result, GWAS plays a critical role in personal genomics, and thousands of GWAS have been conducted over the past few decades. Fan et al. [2] have highlighted that genetic variant data can be viewed as a collection of random variables forming a stochastic process, and can be treated as functional data. One of the key advantages of FDA in GWAS is its ability to naturally incorporate the correlation, linkage, and linkage disequilibrium (LD) information of the genetic variants into the association tests. FDA can capture the complex dependency structure and higher order LD among the genetic variants, which is often missed by other methods such as the sequence kernel association test (SKAT) [3] and its optimal unified test (SKAT-O) [4]. Another advantage of the functional data representation is its suitability for large-scale genomic data, providing a computationally efficient way to test the association between multiple variants and the phenotype.

In their work, Fan et al. [2] proposed two methods for fitting the discrete SNP values: either using ordinary least-square smoothers, or approximating SNPs with K functional principal components. The effectiveness of FDA in GWAS, as demonstrated by Fan et al., has also been confirmed in the subsequent studies [5,6,7]. Nevertheless, SNPs are typically coded as the number of minor alleles, resulting in categorical data that violates the normality assumption required for the least-square smoothing. The objective of this paper is to propose a more robust approach to represent genetic variants using functional data analysis, which addresses the problem of normality assumption violation in the least-square smoothing method. In addition, we introduce a two-step method that integrates our novel smoothing method with functional generalized linear models to identify haplotypes, i.e., blocks of SNPs, that are strongly associated with the phenotype.

Our proposed approach is described in detail in the following sections. In Section 2, we describe the PennCATH dataset used in our experiments and explain our two-step method, which includes curve smoothing and variable selection by regression analysis, for identifying haplotypes associated with the phenotype. Section 3 presents the performance of our method, compared with single SNP test results and haplotypes constructed based on genetic information. In Section 4, we discuss the strengths and limitations of our approach, as well as possible future directions. We conclude the work in Section 5 by summarizing our findings and emphasizing the potential impact of our approach on genetics and genomics research.

2. Materials and Methods

2.1. GWAS Data

In GWAS, SNPs are often coded as the number of minor alleles observed at a given locus. For a particular locus, there are two alleles on a homologous chromosome pair, and the SNPs can be coded as 0, 1, or 2. As an example, if 90% of the genomes have nucleotide A at the locus, and 10% have T, this locus would be an SNP with two alleles. The nucleotide A, the more common allele, is referred to as the major allele or reference allele, while T, the less common allele, will be the minor allele or non-reference allele. Consequently, a genotype of $AA$ is coded as 2, $AT$ or $TA$ as 1, and $TT$ as 0. Section 3 of this paper will present a GWAS example using the PennCATH dataset. PennCath [8] is a GWAS of coronary artery disease (CAD) conducted by the University of Pennsylvania Medical Centre. The study has recruited 3850 patients undergoing cardiac catheterization and coronary angiography between 1 July 1998 and 31 March 2003. All of the patients have provided written informed consent. Their age, gender, ethnicity, medical history, physical exams, and other clinical data have been extracted from their medical records. The samples in PennCath are genotyped by the calling algorithm “Birdseed”, offered by the Affymetrix Genome-Wide Human SNP Array 6.0 platform. The dataset comprises 656,890 SNPs from 1401 patients after a quality control procedure [9]: 933 patients with coronary artery disease (CAD) and 468 with no or minimal CAD. In this work, 3758 SNPs on Chromosome 9p21 are extracted and divided into 4 chunks of equal window size. It is expected to find strong associations between some of the SNPs at around 9p21.3 (position 19.9Mb to 25.6 Mb) and CAD according to previous studies [10,11,12].

2.2. Curve Smoothing

The characteristics of FDA make it natural to apply FDA to numerical data, but it may not seem to be applicable to categorical data such as SNPs. To address this issue, we make a different assumption about the underlying stochastic curves. Instead of assuming that all the categorical values $k=0,\cdots ,K-1$ came from a single underlying curve $x_i\left(t\right)$, we assume there are K probability curves $p_{ik}\left(t\right)$ associated with the continuum t. Hence, an observation ${\mathrm{y}}_{ij}$ can be seen as a sample drawn from a categorical distribution:

$${\mathrm{y}}_{ij}\sim \mathrm{Categorical}(p_{i0}\left(t_{ij}\right),...,p_{iK-1}\left(t_{ij}\right))$$

(1)

The probability curves for any sample i at any position t need to follow the law of total probability and lie between 0 and 1:

$$\begin{array}{c}\hfill \sum _{k=0}^{K-1}{p}_{ik}\left(t\right)=1\\ \hfill 0<p_{i0}\left(t\right),...,p_{iK-1}\left(t\right)<1\end{array}$$

In the case where the categorical observations $k=0,...K-1$ can be regarded as the number of “success” events in $K-1$ trials, $y_{ij}$ can be treated as a sample from the binomial distribution:

$${\mathrm{y}}_{ij}\sim \mathrm{Binomial}(n=K-1,p=p_i\left(t_{ij}\right))$$

(2)

in which the probability of “success” depends on the value of a single probability curve $p_i\left(t_{ij}\right)$. The bounds of 0 and 1 for probabilities still apply in this setting. In both (1) and (2), the rationale behind smooth probability curves assumption naturally accommodated the correlations among adjacent SNPs: for any two SNPs j and $j^{\prime }$ of an individual i, if the SNPs have close positions $t_{ij}$ and $t_{i{j}^{\prime }}$, they should have similar probabilities for minor alleles. Therefore, the probabilities are assumed to change smoothly as the position varies.

In minor-allele SNP coding, SNPs are coded as categorical variables 0, 1, and 2. There are at least two ways to represent SNPs with functional data. The first method is to consider them as 3 unordered categories with probabilities

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill P({\mathrm{SNP}}_{ij}=0)& =p_{ij0}\hfill \\ \hfill P({\mathrm{SNP}}_{ij}=1)& =p_{ij1}\hfill \\ \hfill P({\mathrm{SNP}}_{ij}=2)& =p_{ij2}\hfill \end{array}\end{array}$$

which has weak assumptions on interdependence among the categories. The observations of SNPs can then be adapted into the FDA framework based on Assumption (1) with $K=3$. However, each sample will be determined by two probability curves, which may give too much flexibility in fitting the data. In Assumption (2), SNPs are regarded as the outcome of 2 binomial trials with probability p, which is also a reasonable interpretation since each SNP is equivalent to the number of occurrences of the less common allele in two alleles at a position. With the Hardy–Weinberg Equilibrium (HWE) principle, which states that “genotype frequencies in a population remain constant between generations in the absence of disturbance by outside factors” [13], the occurrences of minor alleles in the two alleles are independent of one another, and the probability of each outcome will be as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill P({\mathrm{SNP}}_{ij}=0)& ={(1-p_{ij})}^2\hfill \\ \hfill P({\mathrm{SNP}}_{ij}=1)& =2p_{ij}(1-p_{ij})\hfill \\ \hfill P({\mathrm{SNP}}_{ij}=2)& =p_{ij}^2\hfill \end{array}\end{array}$$

In GWAS, the check of violations of HWE is usually part of the quality control. The departures from HWE can indicate potential genotyping errors [14,15], and it is a common practice to remove those SNPs from the studies. Therefore, it is reasonable to make HWE assumptions about the probabilities for minor alleles, and the functional representation can also be built on (2). In either setting, the probabilities vary between individuals and over SNP positions. Therefore, we can find the associations between the differences in their probability curves and the target disease. In this paper, we will be converting the observed SNP into functional data based on Assumption (2).

Smoothing for probability curves of SNP data is an example of curve smoothing with constraints. One solution to this is to transform them into unconstrained curves. For SNP data, fitting the probability curves can be achieved by fitting the following unconstrained log-odds curves:

$$g_i\left(t\right)=log\frac{p_i\left(t\right)+\alpha }{(1-p_i\left(t\right))+\alpha }$$

(3)

where $\alpha$ is a Laplace smoothing parameter set to 0.01 to avoid zero probabilities. Once the log-odds curves have been smoothed, they can easily be transformed back to probability curves $p_i\left(t\right)$ by the logistic function. To smooth the log-odds curves, we can use maximum likelihood estimation (MLE) to solve for the coefficients iteratively with gradient descent. Alternatively, we can smooth the empirical log-odds values (−4.62, 0, 4.62) that are transformed from the empirical probabilities (0, 0.5, 1) for categorical SNP observations of (0, 1, 2), respectively. This transformation brings the values closer to a normal distribution, enabling us to obtain a better approximation via least-square smoothing.

2.3. Variable Selection through Regression Analysis

Finite-dimensional regression models such as GLMs and MLMs are commonly used in GWAS. These models treat the phenotype or disease as the response variable, and the SNPs as explanatory variables. By examining the coefficients and p-values of the SNPs, we can determine which SNPs are strongly associated with the phenotype/disease. Similar to the finite-dimensional regression models, coefficients and p-values in functional regression models can also indicate the relationship between phenotype and covariates through the “scalar-on-function” model. Depending on the response data type, the model can be a functional linear regression or a functional logistic regression. When the phenotype is continuous $V_i$, such as height or blood pressure, a functional linear model can be employed, where ${\beta }_0$ is a scalar intercept term and $\beta \left(t\right)$ is the functional coefficient of the probability curve:

$$\begin{array}{c}\hfill V_i={\beta }_0+\int \beta \left(t\right)p_i\left(t\right)dt+{ϵ}_i\end{array}$$

(4)

On the other hand, when the phenotype $V_i$ is binary, such as a disease indicator, the probability of the individual having the phenotype, ${\pi }_i=Pr(V_i=1)$, can be modeled by:

$$log(\frac{{\pi }_i}{1-{\pi }_i})={\beta }_0+\int \beta \left(t\right)p_i\left(t\right)dt$$

(5)

Due to the complication of GWAS data, there may be some confounding factors making it difficult to find the true relationships between probability curves and the target disease. These factors can be accounted for by including them in the regression model. Therefore, the final disease-SNP model that accounts for age, sex, and population substructure can be formulated as:

$$log(\frac{{\pi }_i}{1-{\pi }_i})={\beta }_0+{\beta }_1{\mathrm{age}}_i+{\beta }_2{\mathrm{sex}}_i+{\beta }_3{\mathrm{pc}}_{i1}+\dots +{\beta }_{12}{\mathrm{pc}}_{i,10}+\int \beta \left(t\right)p_i\left(t\right)dt$$

(6)

The number of basis functions $q^{\left(\beta \right)}$ for $\beta \left(t\right)$ needs to be carefully selected. If $q^{\left(\beta \right)}$ is too large, $\beta \left(t\right)$ will have more curvature and may be capturing noise instead of signals in $p_i\left(t\right)$. A common method to determine $q^{\left(\beta \right)}$ is to select the value that minimizes $AIC=-2l+2q^{\left(\beta \right)}$ such that each additional basis function in $\beta \left(t\right)$ should significantly increase the log-likelihood in order to justify a lower AIC value.

In regression analysis, determining if an explanatory variable has a statistically significant association with the response variable requires taking into account not only the coefficient but also the standard errors of the estimated coefficients. The same principle applies to functional regression analysis. The statistically significant SNPs can be found by examining the confidence band of $\beta \left(t\right)$, which connects the point-wise confidence intervals of $\beta \left(t\right)$ at each position t:

$${\widehat{\mathbf{b}}}^T\mathit{\varphi }\left(t\right)\pm z_{\alpha /2}\sqrt{{\displaystyle \sum _{k=1}^{q^{\left(\beta \right)}}}{\displaystyle \sum _{l=1}^{q^{\left(\beta \right)}}}{\varphi }_k\left(t\right){\varphi }_l\left(t\right)Cov(b_k,b_l)}$$

where $\mathit{\varphi }\left(t\right)$ is a set of basis functions for $\beta \left(t\right)$, and $\mathbf{b}$ is a vector of coefficients corresponding to the basis functions.

3. Results

In our experiments, we aim to evaluate the performance of our two-step method in identifying important SNPs by comparing it with single SNP tests and genetic-based haplotypes using real-world data from the PennCATH study of coronary artery disease (CAD). We will use two different approaches to create genetic-based haplotypes: a fixed number of SNPs and a fixed window size. For these haplotypes, we will conduct GWAS by fitting logistic models. In contrast, for our FDA-based haplotypes, we will use functional logistic models to analyze all SNPs at once and identify haplotypes with strong associations with the target disease.

3.1. Single SNP Tests

The single SNP test is an association analysis that regresses the phenotype on each SNP separately. While it can identify SNPs with large marginal effects on the phenotype, it does not consider epistatic effects among the SNPs. The test results shown in Figure 1 (left) are based on logistic models fitted on each SNP using CAD as the response, adjusted for confounding factors. By setting the threshold for p-value to ${10}^{-3}$, we identify 9 significant SNPs between position 22064465 and 22125503. Additional information about the significant SNPs and their test results is provided in

Table 1. Single SNP tests found 9 SNPs strongly associated with CAD in chromosome 9p21 with p-value $<{10}^{-3}$.

SNP	Position	Estimate	Std. Error	t-Value	p-Value
rs6475606	22081850	−0.445673	0.092841	−4.800395	1.583531 $\times {10}^{-6}$
rs1333049	22125503	−0.442759	0.094169	−4.701767	2.579192 $\times {10}^{-6}$
rs4977574	22098574	−0.436480	0.094321	−4.627610	3.699095 $\times {10}^{-6}$
rs2891168	22098619	−0.432042	0.094221	−4.585422	4.530707 $\times {10}^{-6}$
rs1333042	22103813	−0.431915	0.094976	−4.547645	5.424963 $\times {10}^{-6}$
rs10757272	22088260	−0.421598	0.094232	−4.474050	7.675192 $\times {10}^{-6}$
rs1333048	22125347	−0.404207	0.094842	−4.261892	2.027037 $\times {10}^{-5}$
rs10757270	22072719	0.389317	0.094121	4.136343	3.528847 $\times {10}^{-5}$
rs8181047	22064465	−0.425314	0.109852	−3.871706	1.080763 $\times {10}^{-4}$

.

3.2. Genetic Haplotypes

We used two common approaches to construct the genetic-based haplotypes: (1) blocks containing a fixed number of SNPs and (2) fixed genomics window of size. In these two approaches, we have constructed blocks containing 5, 10, 20, and 50 SNPs and blocks with window sizes 10 kb, 20 kb, 50 kb, and 100 kb, respectively. It ends up with 1392 blocks generated for (1) and 2394 blocks generated for (2).

To compute the p-value of each haplotype, we fit a GLM for each haplotype while adjusting for confounding factors and compare it to a base model with only an intercept and confounding factors. However, due to missing values in our genotype data, samples containing missing values for any SNP in a haplotype were excluded when fitting the models. To ensure sufficient data for fitting GLMs, we imputed missing values using the most frequent value (mode) for each SNP before computing the p-values.

Applying the same p-value threshold of ${10}^{-3}$ as we did for single SNP tests, we found 9 significant blocks associated with CAD using Approach (1) and 11 significant blocks using Approach (2). Figure 1 (middle) displays the positions of the significant blocks, which are primarily located around 22 Mb with a few at around 24 Mb. These findings are consistent with our results from single SNP tests, which identified significant single SNPs near 22 Mb. The haplotypes at around 24 Mb have larger p-values (>${10}^{-4}$) as compared with the haplotypes at 22 Mb. Therefore, they could be considered as noise instead of true associations.

Table 2. Significant genetic haplotypes found by fixed SNP and fixed window size methods.

Method	Haplotype No.	Position	Number of SNPs	p-Value
SNP-5	1	22073334–22088260	5	1.522134 $\times {10}^{-5}$
	2	22098574–22125503	5	1.941005 $\times {10}^{-4}$
	3	22128709–22133251	5	9.663982 $\times {10}^{-5}$
	4	24095607–24112974	5	1.712341 $\times {10}^{-4}$
SNP-10	5	22073334–22125503	10	2.645748 $\times {10}^{-4}$
	6	22128709–22136489	10	6.936598 $\times {10}^{-5}$
	7	24075082–24112974	10	9.016075 $\times {10}^{-4}$
SNP-20	8	22073334–22136489	20	1.599449 $\times {10}^{-4}$
SNP-50	9	24018470–24233746	50	8.881007 $\times {10}^{-4}$
LEN-10k	10	22070008–22080008	2	1.605103 $\times {10}^{-4}$
	11	22080008–22090008	4	5.748821 $\times {10}^{-6}$
	12	22090008–22100008	2	1.189989 $\times {10}^{-5}$
	13	22100008–22110008	1	3.743386 $\times {10}^{-6}$
	14	22120008–22130008	3	1.541748 $\times {10}^{-5}$
LEN-20k	15	22060008–22080008	5	1.955793 $\times {10}^{-4}$
	16	22080008–22100008	6	1.896116 $\times {10}^{-5}$
	17	22100008–22120008	1	3.743386 $\times {10}^{-6}$
	18	22120008–22140008	15	4.555725 $\times {10}^{-4}$
LEN-50k	19	22100008–22150008	17	2.301992 $\times {10}^{-4}$
LEN-100k	20	22100008–22200008	30	9.809913 $\times {10}^{-4}$

provides further details on the significant blocks, including their p-values.

3.3. Two-Step FDA Approach

In the two-step approach, we construct the FDA-based haplotypes based on the functional GLM coefficients, as explained in Section 2. This involved smoothing the log-odds curves $g\left(t\right)$ using 64 cubic spline basis functions to approximate the genotype, with the positions of the SNPs in the genome treated as the continuum t for the functions. Then, we transform the log-odds curves back to probability curves and fit the functional logistic model as described in Equation (6). In regression analysis, we used AIC to select the appropriate number of basis functions for $\beta \left(t\right)$, resulting in $q^{\left(\beta \right)}=4$ cubic spline basis functions with $AIC=1440.4$. The confidence bands of the estimated coefficient curve are displayed in Figure 1 (right). However, in a large number of SNPs, small signals in the SNPs may be overlooked. In fact, the confidence bands in Figure 1 barely found any signals in SNPs. As a result, we divided the SNPs into four chunks with an equal window size of approximately 3.3 Mb per chunk and fit the curves independently. For each chunk, the number of basis functions for SNPs $q^{\left(\mathrm{SNP}\right)}$ was set to 64. The number of bases for $\beta \left(t\right)$ in the functional logistic models of the four chunks were chosen as $q^{\left(\beta \right)}=18$ ($AIC=1442.4$), $q^{\left(\beta \right)}=5$ ($AIC=1444.3$), $q^{\left(\beta \right)}=4$ ($AIC=1442.2$), and $q^{\left(\beta \right)}=4$ ($AIC=1442.0$) in the model selection. The final estimated coefficient curves and their confidence bands for all four chunks are presented in Figure 2.

The plots suggest that SNPs located around 22 Mb exhibit strong associations with the phenotype, CAD. The confidence bands indicate strong positive associations of 68 SNPs in the position range of 21986218 to 22219365 at the $5\%$ significance level. When the significance level is $1\%$, the number of SNPs with significant associations is reduced to 62 in the position range of 21988896 to 22176961. It is worth noting that there are some weak associations between CAD and a small number of SNPs located at 19914792 to 19915135, 20028452 to 20057787, and 21747672 to 21869079. These associations are significant at $5\%$ significance level but not at $1\%$, and therefore, are considered noises rather than signals.

4. Discussion

As stated in Section 2.1, we anticipated finding associations between SNPs in 9p21.3 and CAD. In our experiments, all three methods identified significant SNPs and haplotypes within this region (19.9 Mb to 25.6 Mb). Specifically, the SNPs with the smallest p-values in single SNP tests were located between 22.06 Mb and 22.13 Mb, while most of the strongly associated haplotypes in genetic haplotypes were found at 22.06 Mb to 22.20 Mb. Similarly, in FDA haplotypes, the most significant haplotype was found around 21.98 Mb to 22.22 Mb. Notably, in the region where these associations were detected, two genes, cyclin-dependent kinase 2A and 2B: CDKN2A (21.97 Mb to 21.99 Mb) and CDKN2B (22.00 Mb to 22.01 Mb), have been implicated in conferring risk for CAD in multiple studies [16,17,18], thus validating our findings.

5. Conclusions

In this paper, we presented a novel method for smoothing the categorical SNP data in GWAS and demonstrated its effectiveness in identifying haplotypes strongly associated with disease using the PennCATH dataset. The FDA haplotypes successfully retrieved all significant regions in the SNPs while maintaining low false-positive rates at low significance levels. Both functional and non-functional methods were successful, but the main advantage of the functional method is its ability to efficiently identify important SNPs without an exhaustive search while naturally incorporating spatial information and correlation among SNPs into the regression models. Additionally, in the presence of missing values, it implicitly imputes the missing SNP values when smoothing the curves, eliminating the need to exclude or impute missing values in SNPs. This prevents it from getting biased results when missing values are not handled correctly. There is not a large discrepancy between the curves smoothed directly by the least-square method and the curves smoothed by our novel approach. However, our approach, which considers the data as binomial probability curves, has better interpretability and resolves the issue of violating the normality assumption.