Summary
Genetic pathway analysis has become an important tool for investigating the association between a group of genetic variants and traits. With dense genotyping and extensive imputation, the number of genetic variants in biological pathways has increased considerably and sometimes exceeds the sample size 
. Conducting genetic pathway analysis and statistical inference in such settings is challenging. We introduce an approach that can handle pathways whose dimension 
could be greater than 
. Our method can be used to detect pathways that have nonsparse weak signals, as well as pathways that have sparse but stronger signals. We establish the asymptotic distribution for the proposed statistic and conduct theoretical analysis on its power. Simulation studies show that our test has correct Type I error control and is more powerful than existing approaches. An application to a genome-wide association study of high-density lipoproteins demonstrates the proposed approach.
Keywords: Genetic pathway analysis, Genetic variant, High-dimensional inference, Nonsparse signal, Power analysis, Sparse signal
1. Introduction
Genetic association analysis plays an important role in identifying genetic variants that are associated with traits. Genetic variants are often analysed by single-variant-based methods, using approaches such as Armitage’s trend test. Pathway-based analysis has become a popular tool for analysing genetic variant data (Chen et al., 2011b), whereby multiple genetic variants in the genes in a prespecified pathway are examined. There are several reasons to consider pathway analysis for association studies. First, pathways are generally defined using biological knowledge and thus are more likely to be functionally relevant (Zhong et al., 2010). Second, by analysing multiple variants simultaneously, pathway analysis has the potential to accumulate weak signals into stronger ones, while single-variant-based methods lack power in such a situation. Third, because the number of pathways is much smaller than the number of variants, the multiple-testing burden can be dramatically reduced.
One of the main challenges in pathway analysis is to deal with the high dimensionality. With increasingly dense genotyping and extensive imputation, the number of variants 
in genetic pathways has grown so rapidly that it can be larger than the sample size 
. This is seen in our real-data example, where the sample size is around 4000 while the number of single nucleotide polymorphisms in a pathway can be as large as 25 000. In such high dimensions, statistical testing methods that were designed for moderate 
, such as the likelihood ratio test, tend to have low power or may be inapplicable. To deal with the high dimensionality in pathway analysis, one potential approach is the burden test (Morgenthaler & Thilly, 2007), in which one simply sums the genotypes into a single predictor and then subjects this predictor to regression analysis. The burden test works well if all the variants have similar effect sizes, but this assumption rarely holds in real situations. Another common approach to dealing with high dimensions is to use principal component analysis in the regression modelling. One first derives the principal components from the genetic pathway under consideration, and then uses the leading components for association analysis (Buas et al., 2017). The disadvantages of this approach are that principal components with large variations need not be associated with the traits; it is rarely clear how many principal components to include; the interpretation of the regression coefficients can be difficult; and when 
, the estimated principal components may not be consistent (Shen et al., 2016). Complementary to the aforementioned approaches, kernel machine methods such as the sequence kernel association test (Wu et al., 2011) can also be applied to genetic pathway analysis. However, the latter test has been used primarily to analyse moderate-sized variant sets, and its performance in cases where 
is substantially larger is unclear. Other methods that have been developed for testing a group of genetic features in high-dimensional settings (Chen & Qin, 2010; Chen et al., 2011a; Gregory et al., 2015) focus on testing the mean difference between two groups rather than conducting association analysis.
In addition to the high-dimensional challenge, another difficulty in pathway analysis is power maximization under multiple plausible alternative hypotheses. For pathway analysis, the alternative hypothesis concerns both the number and the magnitudes of the nonzero genetic signals, which are generally unknown (Zhang, 2015). A situation often considered for genetic signals is that a pathway harbours potentially many variants with weak effects, called the nonsparse-signal situation. The sequence kernel association test can aggregate multiple signals and is potentially applicable to such a setting. Another possibility is that a genetic pathway contains only a few strong signals, called the sparse-signal situation. Several methods have been proposed to deal with this case, such as the 
test (Conneely & Boehnke, 2007), which first examines each variant individually and then seeks to obtain the 
-value for the maximum of the observed statistics. However, the 
test has little power in the nonsparse situation, while the sequence kernel association test loses power in the sparse situation.
In this paper, we propose a method for conducting high-dimensional genetic pathway analysis, where the dimension 
of the pathway can go to infinity and could exceed the sample size 
. Our approach can be used to identify pathways that harbour a large number of weak signals, i.e., nonsparse signals, as well as genetic pathways that contain only a few strong signals, i.e., sparse signals, or a mixture of weak and strong signals. We establish the asymptotic properties of the proposed statistics in high dimensions and conduct theoretical analysis of their power.
2. Methods
2.1. Model and statistics
Suppose that the data consist of a continuous trait vector 
, an adjusting covariates matrix 
and a genotype matrix 
for a genetic pathway; that is, the pathway being considered contains 
genetic variants. Suppose that the true regression model is
 |
where 
is the coefficient vector for 
, with 
being the intercept, 
is the coefficient vector for 
, and 
is a vector of independent Gaussian errors with mean zero and variance 
. The design matrices 
and 
are considered fixed. The dimension 
of the adjusting covariates is assumed to be finite, while the dimension 
of the genotype matrix can go to infinity.
We are interested in testing the global null hypothesis 
against the alternative 
. Tests such as the likelihood ratio test and Wald test consider all the 
variants jointly and tend to perform poorly when 
is large; the statistics may not exist when 
. Marginal statistics are easy to calculate and have been widely used to evaluate the significance of each individual variant. Recall that in a marginal analysis, one first fits a regression model for a given variant, say the 
th, by 
(
) and then obtains the marginal score statistic as
 |
with 
, where 
is the identity matrix. To conduct a pathway analysis, it is natural to consider the sum of all the squared marginal statistics, 
In fact, it can be shown that 
is equivalent to the sequence kernel association test statistic, if the estimator 
of 
is ignored in the latter. However, our proposed approach is not focused on 
per se, but rather uses 
to develop a suite of statistics for high-dimensional settings, particularly for the case of 
for a constant 
.
Under the null hypothesis 
, it can be shown that 
and var
, where 
, with 
a diagonal matrix whose elements are 
(
), and 
is the Frobenius norm. For the moment we assume that 
is known, but later on we will address the practical situation where 
needs to be estimated. We propose to standardize 
, which yields
 |
(1) |
where the superscript 
emphasizes that both 
and 
can go to infinity; it will be suppressed below for ease of notation. Expression (1) suggests that 
may converge to normality as 
gets large. However, the central limit theorem does not directly apply here because the 
are correlated. In fact, the correlation matrix for the 
, 
, can be shown to have the form
 |
and it can further be shown that 
. In Lemma 1 we show that under proper conditions, 
is standard normal as both 
and 
go to infinity.
Before presenting Lemma 1, we define some notation. For a vector 
, let 
be the 
-norm of the vector for 
. For any 
matrix 
, denote the induced 
-norm by 
. When 
is an 
matrix, we denote its maximum and minimum eigenvalues by 
and 
, respectively.
Lemma 1.
Let
. If
(2) then under
, the statistic
in distribution.
Remark 1.
Here we have no constraint on the order of
with respect to
, providing they both go to infinity. Condition (2) is mild for genetic studies. By Hölder’s inequality,
, where
is the maximum absolute column sum of the matrix. When the correlation structure in
is not overly strong, as is the case for the power-decay structure, i.e.,
for some
, then one can show that
. Here
, the correlation of
and
, can be interpreted as the linkage disequilibrium of genetic variants after adjusting for covariates
; when there are no adjusting covariates,
reduces to the linkage disequilibrium matrix of
. The power decay structure indicates that two distant genetic variants have virtually no linkage disequilibrium, which is indeed what is observed in genetic studies, particularly in the human genome data (International HapMap Consortium, 2005). Similar structures have also been used in other articles on genetic studies, such as Dai et al. (2012). Our proposed statistic naturally takes linkage disequilibrium into account, because
. The linkage disequilibrium can influence both the denominator and the numerator of
, so the impact of the linkage disequilibrium on the power of the proposed test is influenced by the size and density of the genetic signals. However, the linkage disequilibrium will not affect the validity of the test or its asymptotic properties, because the calculation of
does not involve inversion of the linkage disequilibrium matrix, and the normality of the proposed statistics requires only that distant variants tend to have linkage disequilibrium approaching zero. In practice, variants in a gene tend to be in linkage disequilibrium, while those for different genes are generally not in linkage disequilibrium; this type of structure is covered in Lemma 1.
So far we have assumed that the noise level 
is known. To make our proposal practical, it is tempting to replace 
with a consistent estimator 
. It turns out that the validity of doing so depends on the order of 
relative to 
. In the following, we elaborate on this and propose different statistics to accommodate different ratios 
.
We first consider the situation where 
for some 
, i.e., 
is of smaller order than 
. The following lemma shows that if we replace 
with a consistent estimator 
, normality still holds.
Lemma 2.
Suppose that (2) holds. Let
be a root-
-consistent estimator of
such that
. Then under
, as
such that
for
,
in distribution.
Next, we consider the situation in which 
for some constant 
. The normality of 
no longer holds because 
becomes excessively large; see the proof of Lemma 2 for more details. In light of this, we propose a new statistic
 |
(3) |
where 
, with 
being the number of adjusting covariates as mentioned earlier. The 
in the numerator of 
is replaced by 
in 
. The motivation behind this is that 
estimates 
under 
. We discovered that this replacement of 
enables one to overcome the limitation of 
in high dimensions. The following theorem shows that 
follows a normal distribution for 
.
Theorem 1.
Suppose that (2) holds. For any consistent estimator
, as
such that
, if
for some constant
, then under
we have
in distribution.
Theorem 1 allows one to conduct statistical inference for pathway analysis when 
, although 
should not be excessively larger than 
. The condition 
is necessary to prevent the 
in the denominator equalling zero, as it can be shown that 
. To obtain a consistent estimator for 
under 
in a high-dimensional setting, Fan et al. (2012) proposed a refitted crossvalidation method based on procedures that satisfy the sure screening property. When the sparsity of the model is completely unknown, we can also estimate 
by the moment-based estimators of Dicker (2014), which are root-
consistent when 
.
2.2. Power loss in the presence of sparse signals
The proposed statistic 
can handle situations where the association signals are spread out over a large number of genetic variants. However, the power of 
will be relatively low for the sparse-signal situation, in which a few genetic variants carry strong signals while all the others have zero coefficients. Fan et al. (2015) proposed the power-enhancement principle, the fundamental idea of which is to include a screening statistic that goes to zero under 
, but is nonzero under the sparse alternatives 
. Motivated by this principle, we propose a statistic that strengthens 
and is able to guard against potential power loss in the sparse-signal situation.
We define a screening set 
where 
is a threshold chosen to be slightly larger than the maximum estimation error of the marginal estimator, i.e., 
. Then, a power-enhancement component 
is
 |
where 
denotes the sign of 
. Our statistic that is able to detect both nonsparse and sparse signals is 
.
Since 
has the same sign as 
, 
always has power at least that of 
. The threshold 
needs to ensure that the screening set 
is empty with probability approaching 1 under 
, so that the size of 
will be asymptotically equivalent to that of 
. Then, under 
, if an estimator is large enough that 
is nonempty, one can gain power. For Gaussian and sub-Gaussian errors, 
can be chosen to be 
, as suggested by Fan et al. (2015). The power-enhancement procedure in Fan et al. (2015) deals with a consistent estimator under 
, which is not available in our procedure, while our approach builds upon marginal estimators which are inconsistent under 
. Nevertheless, the size of our proposed statistic is asymptotically equivalent to that of 
under 
; in the next subsection, we will show that under the sparse alternatives 
can be powerful even when 
is not.
Lemma 3.
Under the same conditions as in Theorem 1, if
where
as
, then under the null hypothesis
we have
in distribution. Thus, the sizes of
and
are asymptotically equivalent.
To select 
in practice, we propose an adaptive procedure to accommodate different correlation structures. We first generate a vector of 
random errors 
from the standard normal distribution. Then we compute the maximum of the marginal estimators as 
. Finally, we repeat these two steps many times and set 
based on all the replicates. McKeague & Qian (2015) also used an adaptive approach to determine threshold parameters for high-dimensional testing.
2.3. Power analysis
In this subsection we investigate the asymptotic power of the proposed tests 
and 
for nonsparse and sparse alternatives. Under 
, let 
be the set of nonzero coefficients, and let 
. Define the subvector 
and the submatrix 
. Let 
be the diagonal matrix with nonzero elements 
for 
. Similarly, let 
, 
and 
denote the corresponding quantities for 
.
The following theorem states that the sum-of-squares type of statistic 
has high power for the nonsparse-signal situation when the accumulated signals are sufficiently large.
Theorem 2.
Suppose that all the conditions in Theorem 1 hold. Consider a nonsparse alternative
in which
for a sufficiently large constant
. If
and
for some constants
, then as
,
, where
is the
-quantile of the standard normal distribution.
While 
can have high power under nonsparse alternatives, it may lose power under sparse alternatives. In the following theorem we show that 
, which adds a power-enhancement term 
to 
, can be powerful under both nonsparse and sparse alternatives.
Theorem 3.
Assume that the conditions in Theorem 2 hold. Consider a sparse alternative
in which
for a sufficiently large constant
. If
for some constant
, then under either the nonsparse alternative
or the sparse alternative
, as
,
.
In practice, we recommend use of 
for detecting both weak and strong signals. However, if one wishes to distinguish between the sparse signals and the nonsparse signals, one can examine the values of 
and 
. If 
is larger than 
, then the power-enhancement component 
is nonzero and there exist strong signals in the pathway. If 
, then there are no strong signals in the pathway and the significance is driven by weak signals.
2.4. Incorporating biological information into 
and 
The statistics 
and 
give equal weight to all the variants. In some applications, one may wish to assign different weights based on prior information. For example, if the effect of a genetic variant is related to its minor allele frequency, one may assign a weight 
to this variant, where 
is the minor allele frequency for the 
th variant. In other cases, one may assign functional scores to different variants to reflect their biological functions. In lieu of these considerations, we propose incorporating prior biological information into our proposed statistics as follows.
Let 
(
) be prespecified positive weights, and let 
be the diagonal matrix with elements 
. Next, define 
and 
. Let 
and 
Similar to 
, we define a statistic 
. The following result shows the asymptotic normality of 
.
Corollary 1.
Suppose that (2) holds and
as
. Assume that as
,
. For any consistent estimator
, if
for some constant
, then under
in distribution.
As was done for 
, we can add 
to 
to guard against potential power loss in the presence of strong signals. Thus, our proposed statistics can readily accommodate prior biological information and still preserve their theoretical properties.
2.5. Edgeworth expansion for extreme significance levels
Genetic studies sometimes involve a large number of pathways, so the significance level can be much lower than 0.05. For example, in our real-data analysis, the significance level is 0.0003. At such levels, the normal distribution in Lemma 3 may be a poor approximation. We therefore propose a two-term Edgeworth expansion to characterize the tail probability of 
with higher accuracy. Recall that under 
, 
. It is known that 
follows a mixed chi-squared distribution with weights 
, where 
are the eigenvalues of 
. Using the Edgeworth expansion for independent random variables with varying distributions (Feller, 1971, p. 546), we can derive the following two-term expansion for 
:
 |
(4) |
where 
for 
. Further, 
. Then, under the conditions in Theorem 1, the last remainder term in (4) can be shown to be 
. This expansion tends to be more accurate than the normal approximation, as the remainder term of the normal approximation is typically 
. Directly calculating (4) involves computing the 
, which can be onerous when 
and 
are large. Instead, we can use the identity 
for 
. We call the test that uses (4) to approximate the 
-value for 
the 
test. Similarly, we can apply an Edgeworth expansion to 
, and we call the resulting test 
.
3. Simulation studies
Monte Carlo simulations were conducted to evaluate the performance of the proposed tests, 
and 
, in high-dimensional settings and to compare them with the Bonferroni test, the burden test, principal component analysis, and the sequence kernel association test.
We generated the genotype matrix 
similarly to He et al. (2016). For each person, we first generated a block-diagonal covariance matrix with each block being a 
matrix 
. We considered compound symmetric 
with diagonal elements 
and off-diagonal elements 0.5 and autoregressive 
with 
th off-diagonal element 0.6
. Then we trichotomized the simulated vector into genotype values of 
according to the Hardy–Weinberg equilibrium.
We generated the trait by setting 
(
), where 
is an adjusting covariate and 
with 
. For the sample size 
and the dimension 
, we considered both the 
and the 
cases by setting 
with 
and setting 
with 
.
The null model is 
(
). To simulate the data under various alternatives, we assume that 
has 
nonzero signals with support set 
. The magnitude of the signal 
was set to be 
, and half of the 
had positive signs while the other half had negative signs. The magnitude of the signals varied from 0.03 to 0.05 under these set-ups, and the proportion of nonnull variants among all the variants, 
, was set to 5%, 10%, 15% or 20%.
The tests 
and 
were conducted as described in § 2. For variance estimation, we applied the refitted crossvalidation method (Fan et al., 2012) to obtain 
. The threshold parameter 
in the power-enhancement component 
was estimated over 1000 replicates. We used two versions of principal component analysis. In the first, we used the five leading principal components and performed a likelihood ratio test. In the second version, we included the principal components that explain 50% of the total variance for the likelihood ratio test. The reason for considering the second version is that, in practice, a few principal components may not always capture the majority of the variance, as seen in Avery et al. (2011). We also included the sequence kernel association test without any weights.
The Type I errors of the tests were calculated over 10 000 replications, and the power was based on 1000 replications. Table 1 displays the Type I errors of the tests for the models considered. It can be seen that the Bonferroni test, principal component analysis using the five leading components, the sequence kernel association test, and our tests 
and 
all have their Type I errors controlled. Principal component analysis using components that explain 50% of the variance appears to have an inflated Type I error; this is likely due to the fact that many principal components are needed to account for the 50% of variance, and hence the likelihood ratio test has a large degree of freedom. Because of its inflated Type I error, this method was excluded from the subsequent experiments.
Table 1.
Type I error 
of the tests at level 
|
|
Corr. | Bonf. | Burden | PCA | PCA50 | SKAT |
|
|
|---|---|---|---|---|---|---|---|---|---|
| 500 | 300 | CS | 4.66 | 4.90 | 5.00 | 5.97 | 2.79 | 4.84 | 5.10 |
| AR | 4.47 | 5.30 | 4.91 | 6.40 | 2.77 | 5.03 | 5.13 | ||
| 500 | CS | 4.80 | 4.92 | 5.06 | 6.59 | 2.04 | 4.89 | 4.97 | |
| AR | 4.64 | 4.80 | 5.45 | 6.59 | 1.64 | 4.76 | 4.84 | ||
| 1000 | CS | 4.71 | 5.02 | 5.00 | 7.22 | 0.75 | 4.75 | 4.93 | |
| AR | 4.96 | 4.98 | 5.10 | 7.31 | 0.68 | 4.83 | 4.93 | ||
| 1000 | 800 | CS | 5.07 | 4.86 | 5.20 | 6.49 | 2.47 | 5.03 | 5.12 |
| AR | 4.78 | 5.26 | 5.25 | 6.49 | 2.18 | 4.92 | 5.01 | ||
| 1000 | CS | 4.92 | 5.08 | 4.71 | 6.13 | 2.02 | 4.85 | 5.04 | |
| AR | 5.24 | 5.00 | 4.90 | 6.60 | 1.81 | 4.88 | 4.99 | ||
| 1500 | CS | 4.92 | 5.08 | 5.27 | 7.07 | 1.38 | 4.95 | 5.14 | |
| AR | 5.22 | 5.08 | 5.26 | 6.85 | 1.04 | 4.82 | 4.98 |
Corr., correlation structure; CS, compound symmetric; AR, autoregressive; Bonf., Bonferroni test; PCA, principal component analysis using the five leading components; PCA50, principal component analysis using components that explain 50% of the variance; SKAT, the sequence kernel association test.
For the compound symmetric dependence structure, plots of the power for different sample sizes and dimensions against the proportion 
of nonzero signals are shown in Fig. 1. When the ratio 
is small, all the methods have low power. This indicates that when signals are sparse and weak, it is highly difficult to detect the association for the pathway considered. As 
increases, the power improves for all methods, because more variants carry association signals in the studied pathway. However, the tests 
and 
always have higher power than the other approaches in these settings. The results for the autoregressive structure show a similar pattern.
Fig. 1.
Power of the Bonferroni test (
), the burden test (
), principal component analysis (
), the sequence kernel association test (
), 
(
) and 
(
) at level 0.05 for different sample sizes and dimensions plotted against the proportion of nonzero signals, 
. The compound symmetric dependence structure is considered.
We then considered the situation in which a genetic pathway contains both weak and strong signals. We simulated weak signals as described earlier, and then simulated a strong signal with 
. Table 2 shows that both 
and 
compete favourably with the other statistics, and 
has higher power than 
.
Table 2.
Power 
of the tests under mixed signals at level 
|
|
Corr. | Bonf. | Burden | PCA | SKAT |
|
|
|---|---|---|---|---|---|---|---|---|
| 500 | 300 | CS | 36.0 | 5.7 | 15.6 | 30.8 | 35.1 | 39.8 |
| AR | 35.7 | 5.4 | 12.0 | 24.0 | 28.6 | 34.1 | ||
| 500 | CS | 34.8 | 5.3 | 15.7 | 33.6 | 43.5 | 46.7 | |
| AR | 35.2 | 5.5 | 12.6 | 25.0 | 35.6 | 39.4 | ||
| 1000 | CS | 32.4 | 5.2 | 18.5 | 39.2 | 61.8 | 63.6 | |
| AR | 31.1 | 5.0 | 14.8 | 26.8 | 51.0 | 53.5 | ||
| 1000 | 800 | CS | 38.4 | 5.2 | 16.0 | 54.0 | 60.1 | 62.4 |
| AR | 36.1 | 5.0 | 12.3 | 42.4 | 50.1 | 53.3 | ||
| 1000 | CS | 36.7 | 5.0 | 17.0 | 59.2 | 67.2 | 69.0 | |
| AR | 36.0 | 4.8 | 12.7 | 45.8 | 56.2 | 59.2 | ||
| 1500 | CS | 35.4 | 5.1 | 18.1 | 67.3 | 79.4 | 80.9 | |
| AR | 34.5 | 4.6 | 13.1 | 52.6 | 69.1 | 71.5 |
Corr., correlation structure; CS, compound symmetric; AR, autoregressive; Bonf., Bonferroni test; PCA, principal component analysis using the five leading components; SKAT, the sequence kernel association test.
We conducted simulation studies to examine the performance of the 
test, which involves a two-term Edgeworth expansion and is expected to be more accurate than 
in controlling Type I error at extreme significance levels. We set the significance level to 0.0001 and evaluated the Type I error with 1 000 000 simulations. The threshold parameter 
in the power enhancement was estimated over 50 000 replicates. Table 3 shows that at level 0.0001, the 
statistic tends to have inflated Type I error due to the less accurate characterization of the tail probability. In contrast, 
can control the Type I error well at 0.0001 when the sample size and dimension are sufficiently large.
Table 3.
Type I error 
of the tests at level 
|
|
Corr. | Bonf. | Burden | PCA | SKAT |
|
|
|---|---|---|---|---|---|---|---|---|
| 500 | 500 | CS | 1.01 | 1.08 | 1.31 | 0.01 | 5.06 | 1.04 |
| AR | 0.96 | 1.16 | 1.16 | 0.00 | 4.08 | 0.97 | ||
| 1000 | 1000 | CS | 1.34 | 1.01 | 1.20 | 0.00 | 3.19 | 1.08 |
| AR | 1.30 | 1.08 | 1.10 | 0.00 | 2.86 | 1.05 | ||
| 1500 | 1500 | CS | 1.20 | 1.11 | 1.00 | 0.00 | 2.29 | 0.95 |
| AR | 1.11 | 1.04 | 1.34 | 0.00 | 2.23 | 0.95 |
Corr., correlation structure; CS, compound symmetric; AR, autoregressive; Bonf., Bonferroni test; PCA, principal component analysis using the five leading components; SKAT, the sequence kernel association test.
4. Real-data analysis
We analysed the high-density lipoprotein cholesterol data from the Genomics and Randomized Trials Network in the Women’s Health Initiative (Coviello et al., 2012). The overall goal of the study is to identify novel genetic factors that contribute to the incidence of myocardial infarction, stroke and diabetes. DNA samples were genotyped on the HumanOmni-Quad platform, and genotypes were imputed with reference panels. Genetic variants that have imputations 
and minor allele frequency greater than 5% were included. We focused on the 3990 samples of Caucasian ancestry.
We first tested whether our approach can capture existing genetic pathways that are known to be involved in high-density lipoprotein metabolism. Assmann & Gotto (2004) listed a pathway involved in the generation and conversion of high-density lipoprotein. The pathway includes 11 genes: APOA1, APOE, LCAT, LIPC, CETP, PLTP, SCARB, LRP1, LDLR, ABCA1 and ABCF1. We mapped the genetic variants to these genes and obtained 629 variants for this pathway. We adjusted for the following covariates: age, hormone replacement therapy arm, smoking status, body mass index, and the first two principal components for ancestry (Asselbergs et al., 2012). The 
-values for the pathway analysis are displayed in Table 4. Several methods yielded low 
-values, including the Bonferroni test, the sequence kernel association test and the proposed tests 
and 
. The test 
yielded the lowest 
-value. The 
-value of the test 
is lower than that of 
because a number of variants in the CETP and LIPC genes were observed to carry strong association signals that exceed the power-enhancement threshold.
Table 4.
Real-data analysis: 
-values of the tests for the known lipid pathway
| Bonf. | Burden | PCA | SKAT |
|
|
|
|---|---|---|---|---|---|---|
 -value |
|
0.252 | 0.034 |
|
|
|
Bonf., Bonferroni test; PCA, principal component analysis using the five leading components; SKAT, the sequence kernel association test.
Next, we investigated the associations between the KEGG pathways and high-density lipoprotein. The KEGG database contains 186 pathways, which represent a wide variety of cellular processes and molecular functions; for more details see http://www.genome.jp/kegg/pathway.html. We excluded one pathway from our analysis due to overlapping, so our real-data analysis includes 185 pathways. Figure 2 provides an overview of the number of variants in each of the 185 pathways. The median number of variants in these pathways is around 3000. A number of pathways have more than 10 000 variants, with some containing nearly 25 000.
Fig. 2.
The number of single-nucleotide polymorphisms, SNPs, in each of the 185 KEGG pathways.
To control for the familywise Type I error, the threshold of significance was set to 
, i.e., a Bonferroni correction. Table 5 shows the pathways that pass the significance threshold in any of the tests. The 
approach identified three pathways: arachidonic acid metabolism, metabolism of xenobiotics by cytochrome P450, and drug metabolism by cytochrome P450. The 
statistic yielded the same values as 
, indicating that no signal exceeds the power-enhancement threshold in the studied pathways. The sequence kernel association test detected only the arachidonic acid metabolism pathway, while the other methods identified no significant pathway.
Table 5.
The 
-values 
of the tests for the three significant KEGG pathways; 
-values lower than 
are indicated by 
 -value |
|||||||
|---|---|---|---|---|---|---|---|
| #SNPs | Bonf. | Burden | PCA | SKAT |
|
|
|
| Arach. acid metab. | 2590 | 5.11 | 0.57 | 0.07 | 0.02
|
|
|
| Metab. xenobio. | 2254 | 6.30 | 0.46 | 0.18 | 0.07 |
|
|
| Drug metab. | 2385 | 7.86 | 0.39 | 0.16 | 0.04 |
|
|
#SNPs, number of single-nucleotide polymorphisms; Bonf., Bonferroni test; PCA, principal component analysis using the five leading components; SKAT, the sequence kernel association test; Arach. acid metab., arachidonic acid metabolism pathway; Metab. xenobio., metabolism of xenobiotics by cytochrome P450; Drug metab., drug metabolism by cytochrome P450.
The arachidonic acid metabolism pathway contains 2590 variants in 55 genes. A recent biological study suggested that this pathway is an important regulator of cholesterol metabolism (Demetz et al., 2014). The linkage disequilibrium plot of the genetic variants of this pathway in Fig. 3(a) shows that variants in proximity to each other tend to have strong correlations, while those far apart have barely detectable correlations. To gain more insight, we plot the marginal 
-values for all 2590 variants in Fig. 3(b). There are a number of variants with 
-values between 
and 
, but none of them reaches genome-wide significance. Instead, the proposed 
statistic was able to aggregate these relatively mild signals into a stronger one, which leads to the detection of the arachidonic acid metabolism pathway. The variants that contribute to the significance of this pathway, the linkage disequilibrium plots and marginal 
-values of variants in the other two pathways, metabolism of xenobiotics by cytochrome P450 and drug metabolism by cytochrome P450, are given in the Supplementary Material.
Fig. 3.
Analysis of the arachidonic acid metabolism pathway: (a) linkage disequilibrium plot; (b) marginal 
-values of the 2590 single-nucleotide polymorphisms, SNPs, in the pathway, where the dashed line represents the Bonferroni threshold.
5. Discussion
Our approach can be extended to deal with non-Gaussian errors as long as the errors satisfy the moment condition 
for some constant 
and 
. In such a situation, we can adjust the denominator of 
in (3) from 
to 
, where 
is the kurtosis of the errors and 
is the 
th diagonal entry of 
. Then, using the results in Bhansali et al. (2007), we can show the asymptotic normality of the adjusted test statistic accordingly. Our approach can be also extended to accommodate genetic interactions.
Screening techniques have been used in genetic association studies to filter out irrelevant variants; see, for example, Li et al. (2014) and Cui et al. (2015). However, these screening procedures are typically used as a variable-selection step to reduce dimensions, not for statistical testing. In contrast, our screening statistic is directly integrated into the test statistic and is designed for statistical testing. Our approach has focused on the fixed design, which is commonly considered in genetic studies. It will be interesting to develop similar methods under the random design, although it remains challenging to establish the asymptotic properties of the proposed statistics in high dimensions.
Supplementary Material
Acknowledgement
This research was supported by the U.S. National Institutes of Health. We thank the Women’s Health Initiative investigators for sharing the data. The Women’s Health Initiative programme is funded by the National Heart, Lung and Blood Institute. Correspondence should be addressed to QH. We thank the editor, associate editor and reviewers for helpful comments.
Supplementary material
Supplementary material available at Biometrika online includes technical proofs, together with additional simulation results and real-data analysis.
References
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