Functional Linear Models for Association Analysis of Quantitative Traits

Functional Linear Model

Consider n individuals who are sequenced in a genomic region that has m variants. We assume that the m variants are located in a region with ordered physical locations . Here, we assume that each variant's physical location is known, e.g., in terms of the number of base pairs. To make the notation simpler, we normalized the region to be [0, 1]. For the i-th individual, let denote a quantitative trait, denote the genotype of the m variants, and denote the covariates. Hereafter in this article, ′ denotes the transpose of a vector or matrix. For the genotypes, we assume that ( = 0, 1, 2) is the number of minor alleles of the individual at j-th variant located at the location .

In addition to the quantitative traits and covariates, we denote the i-individual's genetic variant function (GVF) as . Notice that the sample includes n discrete realizations or observations of the human genome. By using the genetic variant information , we may estimate the related genetic variant function , which will be discussed below. To relate the genetic variant function to the phenotype adjusting for covariates, we consider the following functional linear model

(1)

where α₀ is the overall mean, α is a vector of regression coefficients of covariates, is the genetic effect of genetic variant function at the location t, and is an error term that is normally distributed with a mean of zero and a variance of Note that the genetic effect function is a function of the physical location t.

Estimation of Genetic Variant Function

To estimate the genetic variant functions , we consider the following three discrete realizations: (1) to model the additive effect of the minor alleles, define , (2) to model the dominant effect of the minor alleles, define when , and when , (3) to model the recessive effect of the minor alleles, define when , and when . Using the discrete realizations , we may estimate the genetic variant function , by ordinary linear square smoother [Ramsay and Silverman, 1996, Chapter 4]. Specifically, let be a series of basis functions. Denote the m by K matrix Φ as containing the values . Then, is estimated by

(2)

where is a column vector of basis functions. Corresponding to the above three discrete realizations, the genetic variant functions are called additive, dominant, and recessive, respectively. In this article, we consider two types of basis functions: (1) the B-spline basis: and (2) the Fourier basis: and . Here for Fourier basis, K is taken as a positive odd integer [de Boor, 2001; Ferraty and Romain, 2010; Horváth and Kokoszka, 2012; Ramsay and Silverman, 1996; Ramsay et al., 2009].

The second approach to estimate the genetic variant functions is to utilize functional principal component analysis (FPCA) techniques [Goldsmith et al., 2011; Horváth and Kokoszka, 2012; Ramsay and Silverman, 1996; Ramsay et al., 2009]. To briefly introduce the main idea of FPCA, let be the covariance function of the genetic variant function. One may want to notice that the covariance function can be estimated by the observed genotype data [Horváth and Kokoszka, 2012; Ramsay and Silverman, 1996]. Let be the spectral decomposition of , where are the nonincreasing eigenvalues and are the corresponding orthonormal eigenfunctions. An approximation for , based on a truncated Karhunen–Loève expansion, is where K is the truncation lag, that can be estimated by the observed genotype data, and .

The third approach to deal with the genetic variant functions is to use the genotype data directly. Then, the focus is to estimate the genetic effect function , and we call this approach as beta-smooth only approach. The related revised models are presented below.

Revised Functional Linear Model

First, let us talk about the case of expanding by B-spline basis or Fourier basis functions. In a similar way to estimate the genetic variant function , we may expand the genetic effect by a series of basis functions as , where is a vector of coefficients . Note that the basis functions to expand can be different from those to expand the genetic variant functions. Thus, and can be different. Let . Replacing in 1 by in 2 and by the expansion, we have a revised function linear model

(3)

where . In the above revised regression model, one needs to calculate and in order to get . In the statistical packages R or Matlab, there are readily available codes to calculate them [Ramsay et al., 2009]. Thus, the revised regression model 3 transforms the initial model 1 to be useful for practical data analysis.

In the case of FPCA, we expand the genetic effect by linear spline basis as where are knots in the interval [0, 1]; and indicates if t is larger than , i.e., if and 1 if . Thus, is a column vector of linear spline basis. In addition, we denote . Then, the revised model in the case of FPCA is

(4)

When we use B-spline (or Fourier) basis functions to estimate the genetic variant function in the model 3, the genetic effect function is expanded by B-spline (or Fourier) basis functions. In the model 4, is expanded by the linear spline basis. This provides a wide range of choices to make comparison.

In the beta-smooth only case, the model 1 is revised as

(5)

In the above model, the integration term in model 1 is replaced by a summation term , and we make no assumption about smoothness of the genetic variant functions . The genetic effect function is assumed to be smooth and one may estimate it by B-spline or Fourier or linear spline basis functions. Replacing by the expansion , the model 5 can be revised as

(6)

where . The revised model 6 is straightforward and less technical. It turns out that model 6 performs very similar to the corresponding model 3 in our real data analysis and simulation studies.

F-distributed Tests of Fixed Effect Functional Linear Models

We first consider the fixed effect models 3, 4, and 6, i.e., we treat the regression coefficients β as unknown constant parameters. Therefore, the revised regression models 3, 4, and 6 are treated as usual multiple linear regressions that model the genetic effect of genetic variant functions adjusted for covariates. To test the association between the m genetic variants and the quantitative trait, the null hypothesis is . By using the standard statistical approach, we may test the null by a -distributed statistic with degrees of freedom (Weisberg, 2005). An alternative approach is to use likelihood ratio tests (LRT) to test the association, which is χ²-distributed with degrees of freedom. In Luo et al. (2012a, b), χ²-distributed score statistics were used to test the association without adjusting for covariates.

Functional Kernel Score Tests of Mixed Effect Functional Linear Models

In the second analysis, we treat the regression coefficients β as a random vector. We assume that each follows a normal distribution with a mean of zero and a variance τ, and are identically independent. Therefore, models 3, 4, and 6 are treated as linear-mixed effect models with α₀ and α as fixed effect components, and β as a random component. Denote the model matrix of the regression coefficients β. Then, models 3, 4, and 6 can be viewed as linear-square kernel machine regression with a kernel proposed in Liu et al. [2007].

To test the association between the m genetic variants and the quantitative trait, one may test a null hypothesis A variance-component functional kernel score test as follows can be used to test the association

(7)

where is a vector of trait values, is the prediction mean of Y under the null H₀, and is the estimation of under the null. That is, , where is the covariate matrix, and and are estimated under the null model by regressing Y on the covariate matrix Z. As pointed out by the authors of SKAT, an important advantage of the score test is that it only fits the null model and computationally it is simple and routine [Kwee et al., 2008; Lee et al., 2012a, 2012b; Liu et al., 2008; Wu et al., 2010, 2011]. The test statistic follows a mixture of distributions. To facilitate the inference, one can approximate the distribution of S by a scaled χ² distribution , where δ is scale parameter and ν is the degree of freedom [Davies, 1980; Duchesne and Lafaye De Micheaux, 2010; Lin, 1997; Liu et al., 2009]. It can be shown that the mean and variance of are given by

Notice that μ and are unknown in practice, and they are estimated/replaced by and to get . To account for this, we replace the mean by by the same argument as Kwee et al. [2008], where and is the identity matrix. In addition, the variance is replaced by , where and . Solving the equations and gives the approximations of the scale parameter and the degree of freedom by

Real Data Analysis of the Trinity Students Study

We analyzed the effect of 36 SNP variants in one enzyme gene region on three biochemical traits in a sample of 2,232 individuals from the Trinity Students Study (see below for a brief description of the study). Since the raw traits were not normally distributed, we transformed the three traits by inverse normal rank transformation. We adjusted for three factors: gender, a continuous covariate of another chemical compound known to affect these biochemical traits, and a dichotomous covariate to indicate if supplements containing these biochemical factors was used. We tested the association between the transformed individual traits and the 36 SNPs by F-test statistics of fixed effect models and variance-component functional kernel score tests of mixed effect models using both B-spline basis and Fourier basis. To make comparisons with the existing methods in the literature, we applied SKAT in R package to test the association by both SKAT and SKAT-O.

Details concerning the Trinity Student Study data collection, SNP genotyping, and quality control procedures are given in Stone et al. [2011] and Mills et al. [2011]. Briefly, this study enrolled a cohort of 2,524 healthy, ethnically Irish individuals, attending the University of Dublin, Trinity College during the academic year in 2003–2004. These students ranged in age from 18 to 28 years. Dietary and supplement questionnaires and blood samples were collected from participants. Ethical approval was obtained from the Dublin Federated Hospitals Research Ethics Committee (affiliated with the Trinity College), and reviewed by the Office of Human Subjects Research at the National Institutes of Health. Written informed consent was obtained from participants before recruitment. SNP genotyping was conducted at the Center for Inherited Disease Research (CIDR) (Baltimore, Maryland), using Illumina 1M HumanOmni1-Quadv1-0B chips on DNA from 2,438 study samples, 14 blind duplicates, and 105 HapMap controls. The HapMap samples had a 99.71% concordance rate with their known genotypes and the blind duplicate sample pairs had a concordance rate of 99.997%. Samples were excluded based on: (1) incomplete phenotype information , (2) gender discrepancy between self-report and genotypes , (3) aberrant ploidy of sex chromosomes (, one XYY male and two XX/XO mosaic females), and (4) less than 95% call rate using all SNPs with at least 95% call rate. Further quality assessment was performed on 1,008,829 SNPs. SNPs were dropped that (1) had less than 98% call rate, (2) had any Mendelian errors using HapMap trios (), (3) had discordant genotypes using HapMap controls (), (4) had discordant genotypes from two or more pairs among the study duplicates () allowing for one error, (5) were monomorphic or (6) had low minor allele frequency (MAF < 0.01). SNPs with deviation from Hardy–Weinberg equilibrium (P-value < 10E-4) were flagged for future reference but kept in the analysis. Thirty-six high-quality SNPs within one enzyme gene were chosen for the demonstration of the methods described in this paper.

Numerical Simulations

Two sets of simulations were performed to evaluate the performance of the proposed methods when sample sizes range from 250 to 2,000. The first one uses the sequence data used in Wu et al. [2011] and Lee et al. [2012a, 2012b] for two scenarios in empirical power calculation: (1) the causal variants are all rare; (2) the causal variants are both rare and common. The sequence data are with European ancestry from 10,000 chromosomes covering 1 Mb regions using the calibrated coalescent model programed in COSI [Schaffner et al., 2005]. Using the same strategy of the simulations in Wu et al. [2011] and Lee et al. [2012a, 2012b], genetic regions of 3 kb length were randomly selected in the simulations for type I error calculation and power calculations.

The second set of simulations is based on the GWAS data of the Trinity Students Study. We consider the enzyme gene used in the real data analysis. In the GWAS data of the Trinity Students Study, 36 SNPs are located in the region. The MAF of the 36 SNPs ranges from 0.05428 to 0.4108. Thus, the 36 SNPs can be treated as common variants if the same cutoff for rare variants is taken to be 0.03 as in literature of Wu et al. [2011] and Lee et al. [2012a, 2012b].

Simulations Based on COSI Sequence Data

In this part of simulations, we used the same strategy as Wu et al. [2011] and Lee et al. [2012a, 2012b] to generate phenotype data and the same COSI sequence data were used. This guarantees the comparison with SKAT and SKAT-O to be valid.

Type I error Simulations. To evaluate the robustness of the proposed models and tests, we generated phenotype datasets by using the model

(8)

where Z₁ is a dichotomous covariate taking values 0 and 1 with a probability of 0.5, Z₂ is a continuous covariate from a standard normal distribution N(0, 1), and ε follows a standard normal distribution N(0, 1). To obtain genotype data, 3 kb subregions were randomly selected in the 1 Mb region and the ordered genotypes were these SNPs in the 3 kb subregions. Notice that the trait values are not related to the genotypes, and so the null hypothesis holds. The sample size of the datasets were taken as 250, 500, 1,000, 1,500, 2,000, respectively. For each sample size case, 10⁶ phenotype-genotype datasets were generated to fit the proposed models and to calculate the test statistics and related P-values. Then, an empirical type I error rate was calculated as the proportion of 10⁶ P-values that were smaller than a given α level (i.e., 0.05, 0.01, and 0.001, 0.0001, respectively).

Empirical Power Simulations. To evaluate the power performance of the proposed models and tests, we simulated datasets under the alternative hypothesis by randomly selecting 3 kb subregions to obtain causal variants for the phenotype values as follows. Once a 3 kb subregion was selected from the 1 Mb region, a subset of P causal variants located in the 3 kb subregion was then randomly selected to obtain ordered genotypes . Then, we generated the quantitative traits by

(9)

where , and ε are the same as in the type I error model 8, and the βs are additive effect for the causal variants defined as follows. We used , where was the MAF of the j-th variant. As in Wu et al. [2011] and Lee et al. [2012a], three different settings were considered: 10%, 20%, and 50% of variants in the 3 kb subregion are chosen as causal variants. When 10%, 20%, and 50% of the variants were causal, , and log (2), respectively. For each setting, 2,000 datasets were simulated to calculate the empirical power as the proportion of P-values that are smaller than a given α level (i.e., 0.05, 0.01, and 0.001, respectively).

One may want to notice that only rare variants with MAF <0.03 were used as causal variants in the simulations of Wu et al. [2011] and Lee et al. [2012a]. In our simulations, we considered two scenarios: (1) the causal variants are all rare, and (2) the causal variants are both rare and common. For each scenario, we performed power calculations by using two strategies: (1) only rare variants were used to calculate the empirical power levels, and (2) both rare and common variants were used.

Simulations Based on the SNP Data of the Trinity Students Study

Type I error Simulations. The 36 SNP genotype data in the enzyme gene region of the Trinity Students Study were used to evaluate the robustness of the proposed models and tests for the common variant case. We generated 10⁶ phenotype datasets by using the model

(10)

where Z₁ is the gender of the individual in the dataset taking values 0 and 1, Z₂ is a continuous biochemical covariate from the dataset, Z₃ is a dichotomous covariate to indicate if supplement was used, and ε follows a standard normal distribution N(0, 1). The coefficients (–0.5, 0.002, 0.5) were chosen based on an empirical analysis of a trait of the Trinity Students Study data. Then, a sample of individuals was selected and the genotypes of 36 SNPs of these individuals were taken from the Trinity Students Study database. Notice that the simulated trait values are not related to the genotypes, and so it can be used to test the null. Using the genotypes and the simulated phenotype traits, we fitted the proposed models and calculated the tests to get related P-values. In addition, empirical type I error rates were calculated as the proportions of P-values which were smaller than a given α level (i.e., 0.05, 0.01, and 0.001, 0.0001, respectively).

Empirical Power Simulations. To evaluate the power performance of the proposed models and related tests for a single common variant as causal variant, we chose one SNP of the 36 SNPs as the causal variant. Based on the genotype G (=0, 1, 2, respectively) of the SNP, we defined X as follows: (1) for a mode of additive inheritance, ; (2) for a mode of dominant inheritance, when and when ; (3) for a mode of recessive inheritance, when and when . We generated phenotype traits by using the model

(11)

where , and ε are the same as in the type I error model 10, and is the genetic effect of the minor allele of the causal SNP (i.e., additive effect, dominant effect, and recessive effect for the three modes of inheritance, respectively).

For each value of genetic effect , we considered five sample sizes of 250, 500, 1,000, 1,500, 2,000, respectively. For each combination of a genetic effect and a sample size, 2,000 datasets were simulated to calculate the empirical power as the proportion of P-values that were smaller than a given α level (i.e., 0.05, 0.01, and 0.001, respectively).

Application to the SNP Data of the Trinity Students Study

Table 1 presents the results of SNP data of the enzyme gene of the Trinity Students Study. We analyzed the data by three types of genetic variant functions: additive, dominant, and recessive. For all three traits, the results of fixed effect F-distributed tests are far better than those of SKAT and SKAT-O since the P-values of the fixed effect F-distributed test statistics in columns 3–7 of Table 1 were much smaller than those of SKAT and SKAT-O in columns 8 and 9 of Table 1 for most entries. The results of beta-smooth only are identical to those of smoothing both the genetic variant functions and the genetic effect function except for trait C using Fourier basis functions. Therefore, whether smoothing the genetic variant functions or not does not have much impact on the results.

Table 1. Results of association analysis of three traits of the Trinity Students Study in the region of an enzyme gene

		P-values of the proposed F-tests
		Basis of both GVF and			Basis of beta-smooth only		P-values
Type of GVF	Trait	B-spline basis	Fourier basis	FPCA	B-spline basis	Fourier basis	SKAT	SKAT-O
Add	A	1.73E-13	7.89E-13	1.54E-15	1.73E-13	7.89E-13	7.13E-08	2.16E-10
	B	3.44E-13	1.80E-11	1.58E-13	3.44E-13	1.80E-11	4.34E-05	2.69E-05
	C	1.11E-11	9.91E-10	8.67E-11	1.11E-11	9.91E-10	5.80E-05	1.26E-05
Dom	A	1.83E-10	1.71E-10	3.68E-12	1.83E-10	1.71E-10	3.10E-07	4.23E-10
	B	4.66E-09	3.35E-07	3.62E-09	4.66E-09	3.35E-07	0.000152	0.000154
	C	1.54E-08	5.93E-07	9.52E-08	1.54E-08	5.93E-07	8.99E-05	1.84E-05
Rec	A	5.26E-11	9.92E-11	4.94E-13	5.26E-11	3.72E-10	7.28E-06	6.32E-06
	B	2.75E-06	4.38E-06	4.04E-06	2.75E-06	5.70E-06	0.063910	0.095384
	C	2.26E-07	5.19E-08	1.07E-07	2.26E-07	3.84E-07	0.075480	0.108617

• The P-values of the proposed F-tests were based on the fixed effect models, and the P-values of SKAT and SKAT-O were based of R package SKAT. When type of GVF = Add, Dom, and Rec, the genetic variant function was taken as additive, dominant, and recessive, respectively. Abbreviation: GVF, genetic variant function.

The results of functional kernel score tests of mixed effect models are presented in the Table S1 of Supplementary Materials. The results of functional kernel score tests are somewhat mixed: for additive genetic variant function and Fourier basis, the functional kernel score tests perform better than SKAT. For dominant and recessive genetic variant functions and Fourier basis, the functional kernel score tests may perform poorer than SKAT. For B-spline basis, the functional kernel score tests may sometimes perform poorer, or sometimes better than SKAT.

Type I Error Simulation Results

The empirical type I error rates are reported in Tables 2 and 3 in the main text, and Tables S2, S3, and S4 in the Supplementary Materials. For each entry of empirical type I error rates, we generated 10⁶ datasets. Results of four different , and 0.0001 levels were reported. When additive genetic variant function was used, Tables 2 and 3 and supplementary Table S2 reported the results based on COSI sequence data and the enzyme gene SNP data of the Trinity Students Study, respectively. In Tables S3 and S4 of the Supplementary Materials, we used dominant and recessive genetic variant functions, respectively, based on the enzyme gene SNP data.

Table 2. Simulation results of type I error rates of the proposed F-tests of the fixed effect models based on sequence data generated by COSI, when the genetic variant functions were taken as additive. The results of “Basis of both GVF and ” were based on the model 3, the results of “FPCA” were based on the model 4, and the results of “Basis of beta-smooth only” were based on the model 6, respectively

		Basis of both GVF and			Basis of beta-smooth only
Nominal level α	Sample size	B-sp basis	Fourier basis	FPCA	B-sp basis	Fourier basis
0.05	250	0.049816	0.049839	0.049850	0.048148	0.046349
	500	0.049955	0.050035	0.049817	0.049779	0.046926
	1000	0.050112	0.050188	0.050266	0.050141	0.049293
	1500	0.050000	0.049933	0.050065	0.049967	0.049800
	2000	0.050211	0.050143	0.050370	0.050191	0.050163
0.01	250	0.009757	0.009925	0.009832	0.009458	0.009205
	500	0.009930	0.010094	0.010031	0.009900	0.009414
	1000	0.010164	0.010005	0.010058	0.010139	0.009794
	1500	0.010012	0.010058	0.010053	0.010003	0.009988
	2000	0.009926	0.010061	0.010129	0.009928	0.010064
0.001	250	0.000987	0.000995	0.001028	0.000968	0.000916
	500	0.001018	0.001007	0.000986	0.001012	0.000946
	1000	0.001037	0.000988	0.001018	0.001044	0.000965
	1500	0.000949	0.000998	0.000997	0.000943	0.000996
	2000	0.000931	0.000967	0.001054	0.000931	0.000961
0.0001	250	0.000102	0.000109	9.70E-05	9.20E-05	1.00E-04
	500	0.000106	9.80E-05	9.60E-05	0.000107	9.90E-05
	1000	0.000108	8.80E-05	9.30E-05	0.000108	8.60E-05
	1500	0.000110	8.60E-05	0.000101	0.000112	8.90E-05
	2000	9.40E-05	0.000106	0.000120	9.40E-05	0.000104

Table 3. Simulation results of type I error rates of four tests based on 36 SNPs of the Trinity Students Study when the additive genetic variant functions were used

		Basis of both GVF and		Basis of FKST
Nominal level α	Sample size	B-spline basis	Fourier basis	B-spline basis	Fourier basis
0.05	250	0.049813	0.049989	0.049077	0.049759
	500	0.049970	0.049750	0.048870	0.048709
	1000	0.050406	0.049664	0.049154	0.049170
	1500	0.050334	0.050181	0.049428	0.049935
	2000	0.050218	0.049991	0.049077	0.050262
0.01	250	0.009906	0.010043	0.011951	0.010759
	500	0.010025	0.009932	0.011644	0.011102
	1000	0.010093	0.010030	0.011803	0.010734
	1500	0.009936	0.010024	0.012013	0.010522
	2000	0.010119	0.010067	0.011886	0.011785
0.001	250	0.001036	0.001017	0.001808	0.001216
	500	0.001030	0.001021	0.001632	0.001416
	1000	0.000996	0.001052	0.001634	0.001311
	1500	0.000916	0.001009	0.001674	0.001125
	2000	0.001016	0.000983	0.001672	0.001604
0.0001	250	0.000113	0.000117	0.000293	0.000143
	500	8.10E-05	0.000113	0.000234	0.000190
	1000	0.000102	0.000106	0.000241	0.000176
	1500	9.80E-05	9.50E-05	0.000245	0.000109
	2000	9.60E-05	7.20E-05	0.000260	0.000252

• The results of “Basis of both GVF and ” were based on the F-distributed test statistics of the fixed effect model 3 and the results of “Basis of FKST” were based on the variance-component functional kernel score test 7 of the mixed effect models, for B-spline basis and Fourier basis, respectively. FKST, functional Kernel score tests.

For the proposed F-distributed test statistics of the fixed effect functional linear models, all empirical type I error rates are around the nominal α levels (columns 3–7 of Table 2, columns 3 and 4 of Table 3 and supplementary Tables S2, S3, and S4). Therefore, the F-distributed test statistics of fixed effect functional linear models controlled type I error rates correctly over all sample sizes and all significance levels. For the variance-component functional kernel score tests of mixed effect functional linear models, the empirical type I error rates are around the nominal and levels, but are generally higher than the nominal and levels (columns 5 and 6 of Table 3 supplementary Tables S2, S3, and S4). In addition to the results reported, we also calculated the type I error rates of LRT statistics of the fixed effect functional linear models. For large sample size cases (1, 500 and 2, 000), the type I error rates of LRT statistics are around all the nominal , and 0.0001 levels; when the sample sizes are 250, 500, and 1, 000, the type I error rates are around the nominal levels, but higher than the smaller nominal 0.001 and 0.0001 levels (data not shown).

Therefore, the fixed effect functional linear models and related F-tests are very robust and can be useful in the whole genome-wide and whole exome association studies, while the mixed effect functional linear models can be useful in candidate gene analysis. One may want to notice that the proposed methods are reasonably robust for small sample size cases. Interestingly, the variance-component functional kernel score tests of mixed effect models are counter-conservative at the nominal and levels in Table 3 and supplementary Tables S2, S3, and S4, based on both the enzyme gene SNP and COSI sequence data, since large sample sizes can lead to higher type I error rates.

Statistical Power of the Proposed Tests and SKAT

We compared the power performance of the proposed tests with SKAT and SKAT-O based on both the simulated COSI sequence data and the SNP data of the Trinity Students Study. The proposed tests are those considered in the type I error simulations, i.e., the F-distributed test statistics of fixed effect functional linear models and two variance-component functional kernel score tests of mixed effect functional linear models, respectively. The empirical power levels are reported in the figures both in the main text and in the Supplementary Materials. In Figures 1–6, and supplementary Figures S1– S6, the results of the proposed F-tests of the fixed effect models are compared with those of SKAT and SKAT-O based on the COSI sequence data. Supplementary Figures S7– S11 compared the power performance of the proposed tests with single causal SNP regression, SKAT and SKAT-O based on the enzyme gene SNP data.

In the legend of all the Figures, “GVF&Beta, B-sp” (or “GVF&Beta, F-sp”) means that both genetic variant function and genetic effect function were smoothed by B-spline (or Fourier) basis functions, “Beta, B-sp” (or “Beta, F-sp”) means that only the genetic effect function was smoothed by B-spline (or Fourier) basis functions (i.e., beta-smooth only), “FKST” means that functional kernel score tests were used to calculate the power level based on the mixed effect models, “B-sp” means B-spline basis was used, and “F-sp” means Fourier basis was used. In addition, the F-test statistics are used except for “FKST.”

Simulation Results Based on COSI Sequence Data When the Causal Variants Can Be Both Rare and Common. In Figures 1, 2, 3, and supplementary Figures S1, S2, and S3, the causal variants can be both rare and common. In Figures 1, 2, and 3, both rare and common variants were used in the analysis while only rare variants were used in the analysis in supplementary Figures S1, S2, and S3. In Figure 1 and supplementary Figure S1, all causal variants have positive effects; when 20%/80% causal variants have negative/positive effects, we present the results in the Figure 2 and supplementary Figure S2; when 50%/50% causal variants have negative/positive effects, the results are presented in the Figure 3 and supplementary Figure S3.

The proposed F-distributed test statistics of the fixed effect functional linear models have higher power than that of SKAT and SKAT-O, except that the SKAT-O has slightly higher power for small and moderate sample size cases of in plots (a3), (b3), and (c3) of a single supplementary Figure S1. When both rare and common variants were used in the analysis (Fig. 1, 2, and 3), the power of the proposed F-distributed test statistics of fixed effect functional linear models are usually much higher than that of SKAT and SKAT-O for small and moderate sample size cases of . If only rare variants were used in the analysis, the power will be dramatically reduced. To see this, let us compare the corresponding power levels of Figure 1 and supplementary Figure S1 (or Fig. 2 and supplementary Fig. S2 or Fig. 3 and supplementary Fig. S3). Figure 1 (or Fig. 2 or Fig. 3) provides higher power levels than those of supplementary Figure S1 (or supplementary Fig. S2 or supplementary Fig. S3).

Simulation Results Based on COSI Sequence Data When the Causal Variants Are Only Rare Variants. In Figures 4, 5, 6 and supplementary Figures S4, S5, and S6, the causal variants are only rare variants. In Figures 4, 5, and 6, only rare variants were used in the analysis while both rare and common variants were used in supplementary Figures S4, S5, and S6. In Figure 4 and supplementary Figure S4, all causal variants have positive effects; when 20%/80% causal variants have negative/positive effects, we present the results in Figure 5 and supplementary Figure S5; when 50%/50% causal variants have negative/positive effects, the results are presented in Figure 6 and supplementary Figure S6.

The proposed F-distributed test statistics of fixed effect functional linear models have higher power than that of SKAT and SKAT-O, except that the SKAT-O has slightly higher power for small and moderate sample size cases of in some plots (a3), (b3), and (c3) of Figure 4 and supplementary Figure S4. When only rare variants are causal, if only rare variants were used in analysis, the tests have higher power levels in Figure 4 (or 5 or 6) than those in supplementary Figure S4 (or S5 or S6) if both rare and common variants in the analysis.

Statistical Power Based on the Enzyme Gene SNP Data of the Trinity Students Study. Supplementary Figures S7, S8, S9, S10, and S11 report the results of power simulations based on the enzyme gene SNP data of the Trinity Students Study and model 11, when one SNP is the only causal variant for the modes of additive, dominant, and recessive inheritance.

As expected, single causal SNP regression model provides the highest power compared with the proposed tests, and SKAT and SKAT-O. For the mode of additive inheritance, the proposed test statistics have substantially higher power than that of SKAT and SKAT-O except for small sample sizes . Interestingly, the functional kernel score test of B-spline offers the second highest power for small and moderate sample sizes of and 500 as shown in graphs (a1), (a2), and (a3) of supplementary Figures S7 and S8. For the mode of dominant inheritance, the two F-distributed tests of the fixed effect models and the functional kernel score test of B-spline have higher power than that of SKAT and SKAT-O. For the mode of recessive inheritance, only the two F-distributed tests of fixed effect models offer good power when the sample size . Both the two functional kernel score tests of mixed effect models and SKAT/SKAT-O have minimal power.

General Observation. The results of the proposed F-tests are very similar in the Figures 1, 2, and 3 (or in the supplementary Figs. S1, S2, and S3, or in the Figs. 4, 5, and 6, or in the supplementary Figure S4, S5, and S6), when all causal variants have positive effect, 20%/80%, and 50%/50% causal variants have negative/positive effects, respectively. Therefore, the proposed tests are very robust to the proportion of causal variants that were positively/negatively related to the trait. Although SKAT-O has higher power than the proposed F-tests in some plots (a3), (b3), and (c3) of supplementary Figures S1 and S4 and Figure 4, the power levels of SKAT-O are not higher anymore in the other figures. This shows that the presence of both negative and positive effects of causal variants has little impact on the proposed F-tests, but can affect SKAT-O much more severely.

In total, we compared five F-test statistics of the fixed effect models: two are based on B-spline basis functions, two are based on Fourier basis functions, and one is based on FPCA. In the two F-tests to use B-spline (or Fourier) basis functions, one is to smooth both the genetic variant functions and the genetic effect function , and the other is only to smooth the genetic effect function (i.e., beta-smooth only). Generally, the five F-test statistics of the fixed effect functional linear models have similar power, although the two tests of Fourier basis have slightly higher power and the test of FPCA has slightly lower power and the two tests of B-spline basis have power levels right in the middle. Such as in the real data analysis, the power levels of beta-smooth only are almost identical to those of smoothing both the genetic variant functions and genetic effect function by B-spline basis (or Fourier basis). Therefore, the proposed F-tests of the fixed effect models have superior performance, and the tests do not strongly depend on whether the genotype data are smoothed or not. In addition, the F-tests do not strongly depend on which basis functions are used.

For the fixed effect functional linear models, we calculated the empirical power levels of the LRT statistics, which provide very similar power levels as the F-tests (data not shown). We also explored the power performance of the variance-component functional kernel score tests of mixed effect functional linear models for COSI sequence data, and it was found that they have less power than the proposed F-tests and SKAT (data not shown).