1 Introduction

We present a package for estimation of cis-eQTL effect sizes, using a new model called ACME which respects biological understanding of cis-eQTL action. The model, fully presented and validated in (Palowitch et al. 2017), involves an additive effect of allele count and multiplicative component random noise (hence “ACME”: Additive-Contribution, Multiplicative-Error), and is defined as

\[y_i = \log(\beta_0 + \beta_1 s_i) + Z_i^T \gamma + \epsilon_i\]

where

\(y_i\) – log-transformed normalized gene read count from sample \(i\)
\(s_i\) – minor allele count for the SNP of sample \(i\)
\(Z_i\) – the \(p \times 1\) vector of covarites for the sample \(i\)
\(\beta_0\) – unknown “baseline” mean of raw expression
\(\beta_1\) – unknown linear contribution of \(s_i\) on the raw expression
\(\gamma\) – unknown \(p \times 1\) covariate coefficient
\(\epsilon_i\) – noise

We estimate the model using a fast iterative algorithm.
The algorithm estimates the model which is nonlinear only with respect to \(\eta = \beta_1 / \beta_0\)

\[y_i = \log(1 + s_i \eta) + \log(\beta_0) + Z_i^T \gamma + \epsilon_i\]

2 Installation

ACMEeqtl can be installed with the following command.

install.packages("ACMEeqtl")

3 Using the Package

ACMEeqtl package provides functions for analysis of a single gene-SNP pair as well as fast parallel testing of all local gene-SNP pairs.

library(ACMEeqtl)

3.1 Testing a Single Gene-SNP Pair

First we generate sample gene expression, SNP allele counts, and a set of covariates.

# Model parameters
beta0 = 10000
beta1 = 50000

# Data dimensions
nsample = 1000
ncvrt = 19

### Data generation
### Zero average covariates
cvrt = matrix(rnorm(nsample * ncvrt), nsample, ncvrt)
cvrt = t(t(cvrt) - colMeans(cvrt))

# Generate SNPs
s = rbinom(n = nsample, size = 2, prob = 0.2)

# Generate log-normalized expression
y = log(beta0 + beta1 * s) + 
    cvrt %*% rnorm(ncvrt) + 
    rnorm(nsample)

We provide two equivalent functions for model estimation.

effectSizeEstimationR – fully coded in R
effectSizeEstimationC – faster version with core coded in C.

z1 = effectSizeEstimationR(s, y, cvrt)
z2 = effectSizeEstimationC(s, y, cvrt)

pander(rbind(z1,z2))

	beta0	beta1	nits	SSE	SST	Ftest	eta	SE_eta
z1	9956	47308	6	963	1760	810	4.75	0.366
z2	9956	47308	6	963	1760	810	4.75	0.366

Variables z1, z2 show that the estimation was done in 6 iterations, with estimated parameters

\(\hat\beta_0\) = 9955.8 (true parameter is 10000)
\(\hat\beta_1\) = 47307.8 (true parameter is 50000)

3.2 Testing All Local Gene-SNP Pairs

First we generate a eQTL dataset in filematrix format (see filematrix package).

tempdirectory = tempdir()
z = create_artificial_data(
    nsample = 100,
    ngene = 100,
    nsnp = 501,
    ncvrt = 1,
    minMAF = 0.2,
    saveDir = tempdirectory,
    returnData = FALSE,
    savefmat = TRUE,
    savetxt = FALSE,
    verbose = FALSE)

## Loading required namespace: RSQLite

In this example, we use 2 CPU cores (threads) for testing of all gene-SNP pairs within 100,000 bp.

multithreadACME(
    genefm = "gene",
    snpsfm = "snps",
    glocfm = "gene_loc",
    slocfm = "snps_loc",
    cvrtfm = "cvrt",
    acmefm = "ACME",
    cisdist = 1.5e+06,
    threads = 2,
    workdir = file.path(tempdirectory, "filematrices"),
    verbose = FALSE)

Now the filematrix ACME holds estimations for all local gene-SNP pairs.

fm = fm.open(file.path(tempdirectory, "filematrices", "ACME"))
TenResults = fm[,1:10]
rownames(TenResults) = rownames(fm)
close(fm)

pander(t(TenResults))

geneid	snp_id	beta0	beta1	nits	SSE	SST	F	eta	SE
1	1	125	-43.5	6	88.5	104	16.8	-0.347	0.0453
1	2	84.1	9.72	6	103	104	0.508	0.115	0.176
1	3	89.4	2.26	5	104	104	0.0264	0.0253	0.159
1	4	78.7	20.6	5	102	104	2.15	0.262	0.211
2	4	57.1	-8.94	4	154	156	0.957	-0.157	0.135
2	5	52.9	-3.1	4	156	156	0.115	-0.0586	0.163
2	6	81.9	-36.4	10	112	156	37.5	-0.444	0.0207
2	7	46.8	8.59	4	155	156	0.722	0.184	0.244
2	8	49.2	2.82	3	156	156	0.091	0.0573	0.199
2	9	54.2	-4.28	5	155	156	0.235	-0.079	0.151

3.3 Testing Multi-SNP Model for All Local Gene-SNP Pairs

Now we can estimate multi-SNP ACME models for each gene:

multisnpACME(
    genefm = "gene",
    snpsfm = "snps",
    glocfm = "gene_loc",
    slocfm = "snps_loc",
    cvrtfm = "cvrt",
    acmefm = "ACME",
    workdir = file.path(tempdirectory, "filematrices"),
    genecap = Inf,
    verbose = FALSE)

Now the filematrix ACME_multiSNP holds estimations for all multi-SNP models.

fm = fm.open(file.path(tempdirectory, "filematrices", "ACME_multiSNP"))
TenResults = fm[,1:10]
rownames(TenResults) = rownames(fm)
close(fm)

pander(t(TenResults))

geneid	snp_id	beta0	betas	forward_adjR2
1	1	111	-45.7	0.139
1	4	111	14.5	0.149
1	2	111	14	0.151
2	6	91.4	-36.7	0.272
2	8	91.4	5.14	0.284
2	9	91.4	-8.36	0.285
2	4	91.4	-7.75	0.291
3	11	121	162	0.218
3	12	121	-24.3	0.22
4	17	57	41.4	0.0669

Note that each multi-SNP model will contain at least one SNP, even if that initial SNP was not significant under the single-SNP models. This initial SNP will be the one with the highest adjusted-R\(^2\) value among the single-SNP models. However, after the initial SNP, further SNPs are added only if the combined model’s adjusted-R\(^2\) is greater than that from the previous combined model.

References

Palowitch, John, Andrey Shabalin, Yi-Hui Zhou, Andrew B Nobel, and Fred A Wright. 2017. “Estimation of Cis-eQTL Effect Sizes Using a Log of Linear Model.” Biometrics. Wiley Online Library.

Interpretable eQTL Effect Sizes Using a Log of Linear Model

2018-01-13

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