- METHOD
- Open Access

# Controlling bias and inflation in epigenome- and transcriptome-wide association studies using the empirical null distribution

- Maarten van Iterson
^{1}Email authorView ORCID ID profile, - Erik W. van Zwet
^{2}, - the BIOS Consortium and
- Bastiaan T. Heijmans
^{1}

**Received:**23 June 2016**Accepted:**12 December 2016**Published:**27 January 2017

## Abstract

We show that epigenome- and transcriptome-wide association studies (EWAS and TWAS) are prone to significant inflation and bias of test statistics, an unrecognized phenomenon introducing spurious findings if left unaddressed. Neither GWAS-based methodology nor state-of-the-art confounder adjustment methods completely remove bias and inflation. We propose a Bayesian method to control bias and inflation in EWAS and TWAS based on estimation of the empirical null distribution. Using simulations and real data, we demonstrate that our method maximizes power while properly controlling the false positive rate. We illustrate the utility of our method in large-scale EWAS and TWAS meta-analyses of age and smoking.

## Keywords

- Epigenome- and transcriptome-wide association studies
- Bias
- Inflation
- Empirical null distribution
- Gibbs sampler
- Meta-analysis

## Background

The large-scale analysis of epigenome and transcriptome data in population studies is thought to answer fundamental questions about genome biology and will be instrumental in linking genetic and environmental influences to disease etiology [1, 2]. Worldwide, research groups are now joining forces to generate and analyze such data [3–7] complementary to the vast resources of genetic data that are already available and have been used successfully in genome-wide association studies (GWAS). While the analysis tool box for GWAS has matured, the development of effective methodology for the analysis of epigenome- and transcriptome-wide association studies (EWAS and TWAS) is a nascent field of research. In an EWAS, DNA methylation levels of typically hundreds of thousands of CpG dinucleotides are individually tested for association with an outcome of interest, while in a TWAS this is done for expression levels of tens of thousands of genes. Currently, EWAS and TWAS analysis heavily relies on approaches specifically designed for GWAS. However, epigenome and transcriptome data are crucially different from genetic data. They are quantitative measures (and not discrete like genotypes) that are subject to major confounding effects of technical batches and biological influences, including cellular heterogeneity [2, 8]. Furthermore, molecular phenotypes such as DNA methylation and gene expression often show stronger associations with phenotypic traits or complex diseases than genotypic markers.

A key aspect of the analysis of ome-wide association studies is the control of test-statistic inflation. Inflation of test statistics leads to an overestimation of the level of statistical significance and dramatically increases the number of false positive findings [9]. This has always been a major concern in GWAS, but inflated test statistics are also observed in EWAS [10, 11]. Often the level of inflation exceeds that observed in GWAS, yet it is generally not corrected [12]. In GWAS, test-statistic inflation is commonly addressed using genomic control in which the inflated test statistics are divided by the genomic inflation factor. The genomic inflation factor estimates the amount of inflation by comparing observed test statistics across all genetic variants to those expected under the hypothesis of no effect [9]. Recent work pointed out crucial limitations of genomic control in GWAS [13, 14]. Notably, the genomic inflation factor was shown to provide an invalid estimate of test-statistic inflation when the outcome of interest is associated with many, small genetic effects [13]. In EWAS and TWAS, this is the rule rather than the exception. Moreover, test statistics may not only be subject to inflation but also to bias [15], which is not corrected for when using genomic control. Bias of test statistics leads to a shift in the distribution of effect sizes and is driven by confounding [16, 17], a prominent feature of EWAS and TWAS but much less of a concern in GWAS [18]. Thus, this calls for the development of new methods specifically designed to address test-statistic inflation and bias in EWAS and TWAS analyses.

Although generally ignored, genomic control will overestimate the actual inflation unless it is estimated on the basis of genetic variants not associated with the outcome of interest [9, 19]. A Bayesian outlier model [20] was proposed to solve this issue; it estimates inflation while assuming a fixed and small number of 10 associated genetic variants. Although this is an improvement for GWAS with few associations, it will not be sufficient to solve the overestimation of inflation in EWAS and TWAS, which typically yield substantially more associations. Nor does it address the occurrence of test-statistic bias. In the statistical literature, alternative methods have been proposed in the context of large-scale multiple hypothesis testing where an empirical null distribution is used for inference [16, 21–23]. The utility of these approaches in EWAS and TWAS, however, remains to be evaluated.

Here, we use simulation studies and large-scale methylome (*n*=2203) and transcriptome (*n*=1910) data [24, 25] to show that correcting inflated test statistics by applying genomic control is too conservative for EWAS and TWAS and that test-statistic bias cannot be ignored. Moreover, we demonstrate that test-statistic bias and inflation are represented by the mean and standard deviation of the empirical null distribution and propose a Bayesian method for its estimation. Application of state-of-the-art batch correction methods, including *SVA* [26], *RUV* [27], and *CATE* [17], were not able to remove all test-statistic bias and inflation. Hence, the resulting test statistics require empirical calibration to achieve optimal statistical power while controlling the number of false positives at the desired level. We develop a Bayesian method for estimation of the empirical null distribution and propose a bias and inflation correction implemented as an R/Bioconductor [28, 29] package *BACON*. Finally, we show the utility of our method by performing an EWAS and TWAS meta-analysis of two commonly studied outcomes: age and smoking status.

## Results

### The genomic inflation factor is not suitable to measure inflation in EWAS/TWAS

However, the genomic inflation factor appeared to be correlated with the expected number of true associations. For example, the genomic inflation factor was higher for age than smoking status, and previous studies showed that age is associated with many more differentially methylated sites and differentially expressed genes than smoking status [3–7]. For the analysis of age, the genomic inflation factor was higher for LL than LLS, which can be attributed to the higher statistical power for LL (age range 21 years) than LLS (age range 9 years).

### EWAS/TWAS not only suffer from inflation but also from test-statistic bias

*P*values, are frequently used to visualize inflation (Fig. 1), the alternative representation through a histogram of test statistics reveals a second artifact, namely a bias of the test statistics (Fig. 3 a and Additional file 1: Figure S1). This bias is visible as a deviation of the mode of the observed test statistics from zero, which is the mode of the standard normal distribution. Since the majority of features (being genetic variants, CpGs, or genes) are assumed not to be associated with the outcome of interest, test statistics obtained from a linear model should follow a standard normal distribution (i.e., centered at zero). We observed test-statistic bias in the EWAS and TWAS of age and smoking irrespective of cohort and outcome (Additional file 1: Figure S1). Genomic control does not address bias because it uses a normal distribution with the mode fixed at zero (Additional file 2). The misspecification of the observed distribution of test statistics by genomic control is illustrated in Fig. 3 c. Note that even permutation-based approaches, which are often assumed to rescue violations of assumptions regarding the theoretical null distribution, do not result in a proper null distribution, and both test-statistic bias and inflation persist [16, 30] (Fig. 3 d). We mathematically derived that unobserved confounding factors introduce bias in the analysis of high-dimensional data (Additional file 2), thus expanding on earlier work by Rao [15].

### Estimating test-statistic bias and inflation

### Correction for unobserved covariates reduces test-statistic bias and inflation

*P*values are calculated using the empirical null distribution instead of the standard normal or the inflated normal that is used by the genomic control method. A complication of the genomic inflation factor is that it estimates the variance of the null distribution (\(\lambda _{{\chi ^{2}_{1}}}\)), whereas the standard deviation (\(\sqrt {\lambda _{{\chi ^{2}_{1}}}}\)) is required for genomic control on normally distributed test statistics resulting from linear models with a continuous outcome (here, DNA methylation and gene expression data). Furthermore, it is important to note that bias not only results in incorrect test statistics and

*P*values but also results in biased estimates of effect sizes. To evaluate the performance of the two-stage method, we conducted a numerical simulation. To account for unmeasured confounding, we selected

*CATE*, a state-of-the-art method that was shown to have superior performance in estimating unobserved covariates as compared with alternative methods [17]. Our Bayesian method in combination with

*CATE*yielded the highest power with the fraction of false positives close to the nominal level (0.058±0.0052). In contrast, methods that ignore unobserved covariates led to high false positive rates and methods that use genomic control resulted in low power (Table 2). Also, the test-statistic calibration that has been proposed to use in combination with

*CATE*[17] was conservative, resulting in low power, which is in line with the fact that this method is closely related to genomic control.

Correction for unobserved covariates reduces test-statistic bias and inflation

Method | Genomic infl. factor | Bayesian infl. factor (bias) |
---|---|---|

\(\sqrt {\lambda _{{\chi _{1}^{2}}}}\) | ||

1. No | 1.322 | 1.229 (0.000) |

2. Known | 1.237 | 1.169 (0.080) |

3. PC (1) | 1.257 | 1.183 (0.048) |

4. PC (2) | 1.222 | 1.147 (-0.002) |

5. PC (3) | 1.160 | 1.090 (-0.139) |

6. SVA (3) | 1.181 | 1.116 (0.022) |

7. RUV-Res (3) | 1.332 | 1.166 (0.086) |

8. RUV-Emp (3) | 1.197 | 1.130 (0.021) |

9. CATE (2) | 1.161 | 1.077 (0.053) |

Bias and inflation correction after adjustment for confounding factors yields optimal power

Method | False positive rate | Power |
---|---|---|

mean (stdev) | mean (stdev) | |

No confounding adjustment | ||

No correction | 0.720 (0.0360) | 0.720 (0.049) |

Genomic control | 0.001 (0.0020) | 0.005 (0.007) |

Bayesian control | 0.029 (0.0076) | 0.050 (0.018) |

Confounding adjustment | ||

No correction | 0.060 (0.0056) | 0.860 (0.037) |

Calibration | 0.030 (0.0042) | 0.770 (0.053) |

Bayesian control | 0.058 (0.0052) | 0.860 (0.041) |

oracle | 0.052 (0.0052) | 0.850 (0.039) |

Empirical null estimates from correlated test statistics yield proper control of the false positives rate without any reduction in power

Method | False positive rate | Power |
---|---|---|

mean (stdev) | mean (stdev) | |

Uncorrelated | ||

No correction | 0.050 (0.003) | 0.770 (0.020) |

Genomic control | 0.028 (0.003) | 0.710 (0.020) |

Bayesian control | 0.052 (0.003) | 0.770 (0.020) |

Correlated | ||

No correction | 0.040 (0.030) | 0.770 (0.020) |

Genomic control | 0.023 (0.006) | 0.730 (0.090) |

Bayesian control | 0.054 (0.020) | 0.800 (0.060) |

### Fixed-effect meta-analysis with control for bias and inflation

*CATE*. Also estimates of inflation using genomic control were both higher and considerably more variable across analyses and cohorts than the estimates obtained using our Bayesian method (Table 4). The Bayesian method fully removed all bias and inflation. Critically, bias (< |0.03|) and inflation (< 1.14) remained minimal in the meta-analysis as compared with a meta-analysis using genomic control (Table 4). The latter contrasts to approaches in which inflation is not addressed at all or those using genomic control: both can result in high levels of inflation and bias in the meta-analysis that often are considerably higher than in the individual cohorts. The top hits identified for age and smoking included those consistently reported in previous studies [3–7]. Furthermore, the simultaneous performance of an EWAS and TWAS in a large-scale meta-analysis showed a remarkable overlap in results between the two study types of 410 and 57 genes for age and smoking, respectively (assigning the nearest gene to a CpG site) (Additional files 4–7: Tables S3a-d). For example, both DNA methylation near and expression of

*CD248*,

*DNMT3A*, and

*FBLN2*were associated with age (Fig. 5 a), while the same was true for

*GPR15*,

*AHRR*and

*CLDND1*for smoking (Fig. 5 b). In total 15,967 (3.5

*%*) CpG sites and 1020 (2.7

*%*) genes were significantly associated with age (Bonferroni-corrected

*P*values < 0.05). For smoking, the number of associated CpGs and genes were 1128 (0.25

*%*) and 301 (0.80

*%*), respectively.

Bias and inflation of test statistics for EWAS and TWAS across four cohorts on age and smoking status

EWAS | TWAS | ||||
---|---|---|---|---|---|

Age | Smoking | Age | Smoking | ||

infl. bias \(\left (\sqrt {\lambda _{{\chi ^{2}_{1}}}}\right)\) | infl. bias \(\left (\sqrt {\lambda _{{\chi ^{2}_{1}}}}\right)\) | infl. bias \(\left (\sqrt {\lambda _{{\chi ^{2}_{1}}}}\right)\) | infl. bias \(\left (\sqrt {\lambda _{{\chi ^{2}_{1}}}}\right)\) | ||

Uncorrected | CODAM | 1.17 0.100 (1.19) | 1.02 0.040 (1.03) | 1.13 -0.030 (1.20) | 1.05 0.100 (1.06) |

LL | 1.45 -0.500 (1.94) | 1.07 0.009 (1.08) | 1.17 0.040 (1.39) | 1.15 0.080 (1.22) | |

LLS | 1.30 0.100 (1.36) | 1.05 -0.200 (1.08) | 1.18 0.050 (1.26) | 1.15 -0.010 (1.17) | |

RS | 1.34 0.700 (1.57) | 0.99 -0.100 (1.01) | 1.11 -0.005 (1.12) | 1.10 -0.010 (1.12) | |

Corrected | CODAM | 1.01 -0.000 (1.01) | 1.00 0.000 (1.01) | 1.02 -0.010 (1.06) | 1.00 0.000 (1.00) |

LL | 1.00 -0.000 (1.27) | 1.00 0.000 (1.01) | 1.02 0.010 (1.19) | 1.02 0.010 (1.06) | |

LLS | 1.02 0.007 (1.05) | 1.00 -0.003 (1.01) | 1.03 0.001 (1.07) | 1.02 -0.010 (1.02) | |

RS | 1.00 0.000 (1.02) | 0.99 0.000 (1.01) | 1.02 -0.006 (1.01) | 1.01 0.001 (1.02) | |

1.19 -0.030 (1.47) | 1.05 0.020 (1.10) | 1.04 0.030 (1.28) | 1.06 -0.002 (1.14) | ||

meta-analysis |

We implemented our Bayesian method as an R/Bioconductor [28, 29] package *BACON*. *BACON* provides valid estimates of bias and inflation in large-scale analyses including EWAS and TWAS, yields corrected test statistics, and supports the streamlined application of the method to fixed-effect meta-analyses.

## Discussion and conclusion

We describe a novel Bayesian method to detect and correct for bias and inflation in epigenome- and transcriptome-wide association studies. Our method has the crucial characteristic that it is largely independent of the fraction of true associations in the data. We showed that the application of genomic control results in spurious associations because it does not address bias and, moreover, reduces power because it is sensitive to the number of true associations and thus commonly overestimates the levels of inflation. The performance of our method towards estimating the empirical null distribution of test statistics outperforms existing methods [16] by taking advantage of prior knowledge of the distribution and the composition of test statistics.

Methods that try to estimate unmeasured covariates [17, 26, 27] and those that try to recover the empirical null distribution [16] rely on the same principle. They extract information from features that are assumed not to be associated with the outcome of interest. Methods to estimate unknown covariates (e.g., *RUV*, *SVA*, and *CATE* as we used here) either use negative controls or assume the number of associated features to be sparse and, interestingly, they can be unified in a single mathematical framework [17]. Genomic control [9] yields a valid estimate of the inflation factor when calculated from features that are known not to be associated with the phenotype of interest. Similarly, the estimation of the empirical null distribution requires that the vast majority of features follow the null distribution [16]. Our Bayesian method is designed to be flexible in dealing with larger fractions of true associations, which turns out to be crucial in particular for EWAS and TWAS meta-analyses.

Our work extends the work of Devlin and Roeder [9], who originally propose to use genomic control to tackle test-statistic inflation for GWAS, and links their method to the pioneering work of Efron [16] on estimating an empirical null distribution for high-dimensional data inference. Hence, although specifically applied to EWAS and TWAS, our statistical method may have implications for any field focusing on statistical inference for high-dimensional data, whether it be omics types or imaging data.

Our method of estimating bias and inflation may resolve a common inconsistency in the current analysis of EWAS and TWAS. While it is becoming the norm to report inflation factors calculated using the traditional genomic control approach, inflation is rarely actually dealt with in the analysis, presumably because this is deemed to be too conservative. However, inflation may be substantial, in particular in a meta-analysis, and current practice is bound to introduce false positive findings. We show that estimating the inflation factor using the genomic inflation factor results both in an overestimation of the actual inflation (i.e., it is indeed conservative) and in imprecise estimates contributing to the previously unexplained, high variability across studies. Our method provides a realistic estimate of inflation that does not suffer from a high variability. Moreover, our method is the first to address the previously unrecognized issue of bias in test statistics. In conclusion, our method optimally reduces the number of false positive findings while preserving statistical power and can be seamlessly incorporated into existing work-flows for the analysis of EWAS, TWAS, and other omics data.

## Methods

### Data sets

DNA methylation data and RNA-seq data were generated within the Biobank-based Integrative Omics Studies Consortium (http://wiki.bbmri.nl/wiki/BIOS_start-). The data comprise four biobanks: Cohort on Diabetes and Atherosclerosis Maastricht (CODAM, n ≈180) [37], LifeLines (LL, n ≈700) [38], the Leiden Longevity Study (LLS, n ≈600) [39], and the Rotterdam Study (RS, n ≈600) [40]. Sample identity of DNA methylation and gene expression data was confirmed using genotype data. Both RNA-seq fastq files and DNA methylation idat files are available from the European Genome-phenome Archive (EGA) under accession number [EGA:EGAC00001000277] together with phenotypes and measured cell counts used in these analyses. Data were generated by the Human Genotyping facility (HugeF) of ErasmusMC, the Netherlands (www.glimDNA.org).

### RNA-seq data preprocessing

A detailed description of the RNA-seq data processing can be found in Zhernakova et al. [24]. Briefly, total RNA from whole blood, depleted of globin transcripts, was sequenced (2×50-bp) using the Illumina HiSeq 2000 platform, and read alignment was performed using STAR (v2.3.0). Subsequently, RNA-seq counts were normalized using *TMM* [41] and transformed to log2 counts per million. Genes that yielded zero counts for all samples across cohorts were removed, which resulted in 45,867 genes (ENSEMBLv73). For all analyses, genes with the lowest overall variance were excluded (5*%* lowest).

### 450K DNA methylation data preprocessing

The generation of genome-wide DNA methylation data is described by Bonder et al. [25]. Briefly, 500 ng of genomic DNA was bisulfite modified using the EZ DNA Methylation kit (Zymo Research, Irvine, CA, USA) and hybridized on Illumina 450K arrays according to the manufacturer’s protocols. The original idat files were generated by the Illumina iScan BeadChip scanner. Subsequently, sample quality control was performed using *MethylAid* [42]. Ambiguously mapped probes [43], probes with a high detection *P* value (> 0.01), probes with a low bead count (< 3 beads), and probes with a low success rate (missing in > 95*%* of the samples) were set to missing. Samples containing an excess of missing probes (> 5*%*) were excluded from the analysis. Subsequently, per cohort, imputation [44] was performed to impute the missing values. Functional normalization [45], as implemented in the *minfi* package [46], was used per cohort. All analyses were performed on M values. Detailed description of the 450K DNA methylation preprocessing steps are available from the git-repo Leiden450K [47].

### White blood cell count prediction

White blood cell counts (WBC), i.e., neutrophils, lymphocytes, monocytes, eosinophils, and basophils, were measured by the standard WBC differential as part of the complete blood count (CBC). A minority of samples were lacking CBC measurements. Since DNA methylation levels are informative of the white blood cell composition [48], we build a linear predictor to infer the white blood cell composition of those samples lacking WBC measurements (Additional file 2). Predicted cell counts were used in the meta-analysis. For the analyses of the cohort subsets, individuals with measured cell counts were selected.

### Association analyses

All association analyses were performed using *limma*’s lmFit function [49]. Since the sample sizes of our data were all above > 100, the empirical Bayes step was skipped. T test statistics were transformed to *P* values using a standard normal distribution. For the analysis of RNA-seq data, we first applied a *voom*-transformation [50] on the *TMM*-normalized counts while controlling for known covariates including age, gender, smoking status measured cell counts, and a technical covariate introducing a batch effect (the flow-cell identifier of the sequencing machine). For the analysis of DNA methylation data, the functional normalized beta-values [45] were transformed to M values, and again lmFit was used to obtain test statistics for the covariate of interest. Here we included age, gender, smoking status, measured cell counts, and array position as known covariates.

### Genomic control and the genomic inflation factor

*χ*

^{2}-distribution with one degree of freedom) divided by the theoretical median, . For example, let

*w*

_{1},

*w*

_{2},⋯,

*w*

_{ p }be a set of

*p*test statistics, following a \({\chi _{1}^{2}}\)-distribution with one degree of freedom; the following estimator was proposed to quantify the amount of inflation:

Furthermore, it was proposed to control the inflated test statistics by dividing the test statistics by the estimated amount of inflation; this approach is referred to as *genomic control* [9].

In EWAS/TWAS test statistics are usually obtained from inference on the coefficients of linear regression models (instead of a trend test), i.e., *t*-test statistics that can be assumed approximately to follow a standard normal distribution (instead of a *χ*
^{2}-distribution). Therefore, applying genomic control to these test statistics entails dividing by the square root of the genomic inflation factor, \(\sqrt {\lambda _{{\chi ^{2}_{1}}}}\) (instead of \(\lambda _{{\chi ^{2}_{1}}}\)).

### Estimation of the unobserved covariates

To investigate whether adding estimated unobserved covariates reduces bias and inflation, we performed EWAS/TWAS with (1) only the covariate of interest, (2) known covariates (e.g., white blood cell counts), and either (3) known covariates plus one, (4) plus two, or (5) plus three principal components estimated from the data, and (6) known covariates with estimated unobserved covariates using *CATE* [17]. For TWAS, we additionally used *RUV* [27, 32] and *SVA* [26]. For EWAS, *iSVA* [33] and *RUVm* [34] were used. All algorithms were used with default parameters except for *CATE*, which was run using calibrate=FALSE.

### Simulation studies

#### The impact of true association on the genomic inflation factor

One hundred sets of 2000 test statistics were generated from a normal mixture distribution with different mixture coefficients (0.8, 0.90, and 0.95). The majority of the null test statistics were drawn from a standard normal, *N*(0,1), while the alternative test statistics were drawn from a normal distribution, *N*(*μ*,1), with *μ*∼*N*(0,3). An equal number of positive and negative associations were simulated. For each set of test statistics, inflation factors were calculated to investigate the impact of the number of true associations (Additional file 3). Additional file 3 shows the performance and robustness of *BACON* in estimating the empirical null distribution when different data generating approaches are used.

#### Comparing different methods that estimate the empirical null distribution

Efron proposed two methods for estimation of the empirical null distribution from a set of test statistics [16]. In order to compare the performance of those methods with our Bayesian method, sets of test statistics were generated, similar to the approach described above, but under different scenarios: scenario “equal” with equal proportion of positive and negative associations (0.05, 0.05), scenario “skewed” with only positive associations (prop. 0.1), scenario “small” similar to scenario equal with only 0.01 proportion of true associations, and scenario “close” where the distribution for the means had expected value of 1 (instead of 3). For each scenario, 2000 test statistics were generated 100 times. To estimate the empirical null distributions as proposed by Efron, we used the *locfdr*
*R* package. For both methods, maximum likelihood and moment matching, default parameter settings were used (Additional file 3).

#### Simulation with unobserved confounding factors

We used the simulation setup of Wang et al. [17] to generate data with confounding factors. Briefly, data *Y*
_{
n×p
} for *n*=100 samples and *p*=2000 features were generated according to the following model: *Y*
_{
n×p
}=*X*
_{
n×1}
*β*
^{
T
}+*Z*
_{
n×r
}
*γ*
^{
T
}+*E*, where *Z*, represents the *r*=5 unobserved confounding factors model as *Z*|*X*=*X*
*α*
^{
T
}+*D*, with *α* representing the strength of confounding. Furthermore, a continuous covariate of interest, *X*, was sampled from the normal distribution. Effects were introduced by fixing 90% of the *β*s at zero while the remaining were different from zero. Both *E* and *D* represent Gaussian noise. A detailed description of the simulation setup is given by Wang et al. and is available as an *R* function *gen.sim.dat* from the package *CATE* (Additional file 3: section 5).

#### Simulation with correlated test statistics

Correlated test statistics were generated according to the approach of Efron [51] introducing a block-correlation structure among test statistics. The uncorrelated test statistics with effects generated from the normal mixture were added to the test statistics with block-correlation structure (Additional file 3: section 3.3 and section 5). The same number of repeated simulations, 100, number of test statistics, 2000, and proportion of null features, 0.9 were used.

#### The Gibbs sampler

with 9−1 parameters (the mixture proportions are constrained to sum to one, \(\sum _{j=1}^{3}\epsilon _{j} = 1\)), and *ϕ*(*x*;*μ*
_{
j
},*σ*
_{
j
}) being the density of \(\mathcal {N}(\mu _{j}, {\sigma _{j}^{2}})\). Furthermore, one component represents the empirical null distribution with its estimated mean (i.e., bias) and standard deviation (i.e., inflation). We propose to use a Gibbs sampling algorithm [31, 52, 53] to estimate the parameters of the mixture distribution.

Conjugate prior distributions are used for the means, *μ*
_{
j
}, variances, \({\sigma _{j}^{2}}\), and mixture proportions, *ε*
_{
j
}. Hence, we assume a normal distribution,
, for the means, an inverse gamma distribution, \({\sigma ^{2}_{j}} \sim \mathcal {IG}(\alpha _{j}, \beta _{j})\), for the variances, and a Dirichlet distribution, \((\epsilon _{1}, \epsilon _{2}, \epsilon _{3}) \sim \mathcal {D}(\gamma _{1}, \gamma _{2}, \gamma _{3})\), for the mixture proportions. Well chosen hyper-priors ensure that the occurrence of labeling switching is minimized; i.e., during sampling from the posterior, the null component is switched with one of the alternative components. That is, we take informative hyper-priors for means, the null component, *λ*
_{1}=0, and for the alternative components *λ*
_{2}=−3 and *λ*
_{3}=3 all *τ*’s are equal to 100. The hyper-priors for the variance parameters are equal for all components *α*=1.28 and *β*=0.36 and were taken from Raftery [54]. For the Dirichlet distribution, widely used uniform noninformative prior parameters were chosen: *γ*
_{1}=*γ*
_{2}=*γ*
_{3}=1. Furthermore, data-dependent starting values are used to start the algorithm at a good initial point. These are based on the median and median absolute deviation (MAD) of the test statistics. A burn-in period of 3000 iterations was used as well as 2000 subsequent samples to estimate the parameters of the mixture distribution using the mean.

Given test statistics *x*
_{
i
} (z-scores or transformed to z-scores) for *i*=1,⋯,*p*, prior distributions with hyper-parameters, and starting values for the posterior distributions, the Gibbs sampling algorithm is run in the following way:

*t*=1,⋯,5000,

- 1.
Generate the missing (unobserved) data: \(z_{ij} \sim \mathcal {M}(\tilde {p}_{ij})\) from a multinomial distribution, with parameter

*p*_{ ij }=*ε*_{ j }*ϕ*(*x*_{ i };*μ*_{ j },*σ*_{ j }), \(\tilde {p}_{ij}\) represents the normalized proportion \(\left (\sum _{j=1}^{3} = \tilde {p}_{ij} = 1\right)\). - 2.
Obtain and

- 3.Generate samples from the posteriors according to:$$ {}\begin{aligned} & \epsilon_{j} \sim \mathcal{D}(\gamma_{j}+n_{j}),\\ &\mu_{j}|{\sigma_{j}^{2}} \sim \mathcal{N}\left(\frac{\lambda_{j}\tau_{j} + s_{j}}{n_{j} + \tau_{j}}, \frac{{\sigma_{j}^{2}}+s_{j}}{n_{j} + \tau_{j}}\right),\\ &\sigma^{-2}_{j} \sim \Gamma \left(\alpha\,+\,\frac{1}{2}(n_{j}+1), \left(\beta + \frac{1}{2}\tau_{j}(\mu_{j}-\lambda_{j})^{2} \,+\, \frac{1}{2}{s_{j}^{2}}\right)^{-1}\right). \end{aligned} $$(3)

The latter mimics sampling from an inverse gamma distribution. For clarity, an iteration superscript is omitted. We assume that 3000 iterations (burn-in period) are sufficient for the Markov properties to hold and that the samples from the conditional distributions can be assumed to be samples from the joint parameter distribution. We implemented the Gibbs sampling algorithm in *C* and can either use weighted multinomial sampling method for binned test statistics or a fast sampling method [55] if all individual test statistics are used (user-defined). Optionally, test statistics following a distribution different from the normal distribution can be used by transforming them to z-scores. For example, test statistics *w*
_{1},⋯,*w*
_{
p
} that follow under the null hypothesis a *χ*
^{2}-distribution with *ν* degrees of freedom can be transformed to z-scores using \(\Phi ^{-1}(F_{\chi ^{2}_{\nu }}(w_{i}))\) [16] (Additional file 3).

## Declarations

### Acknowledgements

Samples were contributed by LifeLines (http://lifelines.nl/lifelines-research/general), the Leiden Longevity Study (http://www.leidenlangleven.nl), the Netherlands Twin Registry (http://www.tweelingenregister.org), the Rotterdam Study (http://www.erasmus-epidemiology.nl/research/ergo.htm), and the CODAM study (http://www.carimmaastricht.nl/). All analyses were carried out on the Dutch national e-infrastructure with the support of SURF Cooperative. The BIOS Consortium comprises Bastiaan T. Heimans, Peter A.C. ’t Hoen, Joyce van Meurs, Rick Jansen, Lude Franke, Dorret I. Boomsma, Rene Pool, Jenny van Dongen, Jouke J Hottenga, Marleen M.J. van Greevenbroek, Coen D.A. Stehouwer, Carla J.H. van der Kallen, Casper G. Schalkwijk, Cisca Wijmenga, Sasha Zhernakova, Ettje F. Tigchelaar, P. Eline Slagboom, Marian Beekman, Joris Deelen, Diana van Heernst, Jan H. Veldink, Leonard H. van den Berg, Cornelia M. van Duijn, Bert A. Hofman, Aaron Isaacs, Andre G. Uitterlinden P. Mila Jhamai, Michael Verbiest, H. Eka D. Suchiman, Marijn Verkerk, Ruud van der Breggen, Jeroen van Rooij, Nico Lakenberg, Hailiang Mei, Maarten van Iterson, Michiel van Galen, Jan Bot, Dasha V. Zhernakova, Peter van ’t Hof, Patrick Deelen, Irene Nooren, Matthijs Moed, Martijn Vermaat, Rene Luijk, Marc Jan Bonder, Freerk van Dijk, Wibowo Arindrarto, Szymon M. Kielbasa, Morris A. Swertz, Erik W. van Zwet, and Peter-Bram ’t Hoen.

**The BIOS consortium** The mission of the BIOS Consortium is to create a large-scale data infrastructure and to bring together BBMRI researchers focusing on integrative omics studies in Dutch Biobanks (www.bbmri.nl/?p=259).

**Participants:**
**Management Team:** Bastiaan T. Heijmans (chair)^{1}, Peter A.C. ’t Hoen^{2}, Joyce van Meurs^{3}, Rick Jansen^{5}, Lude Franke^{6}.

**Cohort collection:** Dorret I. Boomsma^{7}, René Pool^{7}, Jenny van Dongen^{7}, Jouke J. Hottenga^{7} (Netherlands Twin Register); Marleen MJ van Greevenbroek^{8}, Coen D.A. Stehouwer^{8}, Carla J.H. van der Kallen^{8}, Casper G. Schalkwijk^{8} (Cohort study on Diabetes and Atherosclerosis Maastricht); Cisca Wijmenga^{6}, Lude Franke^{6}, Sasha Zhernakova^{6}, Ettje F. Tigchelaar^{6} (LifeLines Deep); P. Eline Slagboom^{1}, Marian Beekman^{1}, Joris Deelen^{1}, Diana van Heemst^{9} (Leiden Longevity Study); Jan H. Veldink10, Leonard H. van den Berg10 (Prospective ALS Study Netherlands); Cornelia M. van Duijn^{4}, Bert A. Hofman^{11}, Aaron Isaacs^{4}, André G. Uitterlinden^{3} (Rotterdam Study).

**Data Generation:** Joyce van Meurs (Chair)^{3}, P. Mila Jhamai^{3}, Michael Verbiest^{3}, H. Eka D. Suchiman^{1}, Marijn Verkerk^{3}, Ruud van der Breggen^{1}, Jeroen van Rooij^{3}, Nico Lakenberg^{1}.

**Data management and computational infrastructure:** Hailiang Mei (Chair)^{12}, Maarten van Iterson^{1}, Michiel van Galen^{2}, Jan Bot^{13}, Dasha V. Zhernakova^{5}, Rick Jansen^{4}, Peter van ’t Hof^{12}, Patrick Deelen^{5}, Irene Nooren^{13}, Peter A.C. ’t Hoen^{2}, Bastiaan T. Heijmans^{1}, Matthijs Moed^{1}.

**Data analysis group:** Lude Franke (Co-Chair)^{6}, Martijn Vermaat^{2}, Dasha V. Zhernakova^{6}, René Luijk^{1}, Marc Jan Bonder^{6}, Maarten van Iterson^{1}, Patrick Deelen^{6}, Freerk van Dijk^{14}, Michiel van Galen^{2}, Wibowo Arindrarto^{12}, Szymon M. Kielbasa^{15}, Morris A. Swertz^{14}, Erik. W van Zwet^{15}, Rick Jansen^{5}, Peter-Bram ’t Hoen (Co-Chair)^{2}, Bastiaan T. Heijmans (Co-Chair)^{1}.

**Institutes involved:** (1) Molecular Epidemiology Section, Department of Medical Statistics and Bioinformatics, Leiden University Medical Center, Leiden, The Netherlands

(2) Department of Human Genetics, Leiden University Medical Center, Leiden, The Netherlands

(3) Department of Internal Medicine, ErasmusMC, Rotterdam, The Netherlands

(4) Department of Genetic Epidemiology, ErasmusMC, Rotterdam, The Netherlands

(5) Department of Psychiatry, VU University Medical Center, Neuroscience Campus Amsterdam, Amsterdam, The Netherlands

(6) Department of Genetics, University of Groningen, University Medical Centre Groningen, Groningen, The Netherlands

(7) Department of Biological Psychology, VU University Amsterdam, Neuroscience Campus Amsterdam, Amsterdam, The Netherlands

(8) Department of Internal Medicine and School for Cardiovascular Diseases (CARIM), Maastricht University Medical Center, Maastricht, The Netherlands

(9) Department of Gerontology and Geriatrics, Leiden University Medical Center, Leiden, The Netherlands

(10) Department of Neurology, Brain Center Rudolf Magnus, University Medical Center Utrecht, Utrecht, The Netherlands

(11) Department of Epidemiology, ErasmusMC, Rotterdam, The Netherlands

(12) Sequence Analysis Support Core, Leiden University Medical Center, Leiden, The Netherlands

(13) SURFsara, Amsterdam, the Netherlands

(14) Genomics Coordination Center, University Medical Center Groningen, University of Groningen, Groningen, the Netherlands

(15) Medical Statistics Section, Department of Medical Statistics and Bioinformatics, Leiden University Medical Center, Leiden, The Netherlands

### Funding

This work was done within the framework of the Biobank-Based Integrative Omics Studies (BIOS) Consortium funded by BBMRI-NL, a research infrastructure financed by the Dutch government (NWO 184.021.007).

### Availability of data and material

RNA-seq fastq files, DNA methylation idat files, phenotypes, and measured cell counts are available from the European Genome-phenome Archive (EGA) under accession number [EGA:EGAC00001000277]. Our Bayesian method is available as an R/Bioconductor package *BACON*: http://bioconductor.org/packages/bacon/under the open source license GPL (≥2). Source code of all analyses and simulations are deposited at https://git.lumc.nl/molepi/biasandinflationwith DOI 10.5281/zenodo.162160.

### Authors’ contributions

MvI and BTH designed the study. MvI and EvZ developed the statistical method. MvI performed all statistical analyses and implemented the software. MvI and BTH wrote the paper. All authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

### Ethics approval and consent to participate

The study was approved by the institutional review boards of the participating centers (CODAM, Medical Ethical Committee of the Maastricht University; LL, Ethics committee of the University Medical Centre Groningen; LLS, Ethical committee of the Leiden University Medical Center; and RS, Institutional review board (Medical Ethics Committee) of the Erasmus Medical Center). All participants have given written informed consent, and the experimental methods comply with the Helsinki Declaration.

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.

## Authors’ Affiliations

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