# Detecting differentially methylated loci for multiple treatments based on high-throughput methylation data

- Zhongxue Chen
^{1}Email author, - Hanwen Huang
^{2}and - Qingzhong Liu
^{3}

**15**:142

https://doi.org/10.1186/1471-2105-15-142

© Chen et al.; licensee BioMed Central Ltd. 2014

**Received: **10 December 2013

**Accepted: **6 May 2014

**Published: **15 May 2014

## Abstract

### Background

Because of its important effects, as an epigenetic factor, on gene expression and disease development, DNA methylation has drawn much attention from researchers. Detecting differentially methylated loci is an important but challenging step in studying the regulatory roles of DNA methylation in a broad range of biological processes and diseases. Several statistical approaches have been proposed to detect significant methylated loci; however, most of them were designed specifically for case-control studies.

### Results

Noticing that the age is associated with methylation level and the methylation data are not normally distributed, in this paper, we propose a nonparametric method to detect differentially methylated loci under multiple conditions with trend for Illumina Array Methylation data. The nonparametric method, Cuzick test is used to detect the differences among treatment groups with trend for each age group; then an overall p-value is calculated based on the method of combining those independent p-values each from one age group.

### Conclusions

We compare the new approach with other methods using simulated and real data. Our study shows that the proposed method outperforms other methods considered in this paper in term of power: it detected more biological meaningful differentially methylated loci than others.

### Keywords

Cuzick test Nonparametric test Trend test## Background

DNA methylation is an epigenetic mark that has important effects on transcriptional regulation, chromosomal stability, genomic imprinting, and X-inactivation, [1, 2]. In addition, it is associated with many human diseases, including various types of cancer [3–10].

Due to the recent advances of BeadArray technology, high-throughput genome-wide methylation data can be routinely generated by Infinium Methylation Assays. This provides good opportunities for researchers to simultaneously study hundreds of thousands of DNA methylation loci. However, it also requires sophisticated and advanced statistical methods to analyze this kind of data.

The raw data generated from BeadArray are fluorescent intensities for each locus; they need appropriate preprocesses, such as background correction and normalization. Then a summarized β-value is generated from about 30 replicates in the same array as follows: $\mathit{\beta}=\frac{max\left(\mathit{M},0\right)}{max\left(\mathit{U},0\right)+100}$, where M is the average signal from a methylated allele while U is that from an unmethylated allele. Obviously, the β-values are continuous numbers between 0 and 1, with 0 stands for totally unmethylated and 1 for completely methylated.

Due to the non-normality of the β-value [11–13], those commonly used statistical methods, such as t-test for case control designs, ANOVA for multiple conditions, or linear regression with age as a predictor, usually have low power to detect differentially methylated loci [13, 14]. Some statistical approaches with or without adjusting the age-effect, which has been found highly associated with DNA methylation [15, 16], have been proposed to detect differentially methylated loci for case-control designs [11–13]. However, very little work has been done for the situation where there are three or more groups (e.g., conditions, or treatments). In a previous study, we compared some statistical tests with age-effect adjustment for DNA methylation data with three treatments, and found that the method based on the nonparametric Kruskal-Wallis (KW) test is usually more powerful than other methods, such as ANOVA and regression method [14]. However, if there is a trend associated with treatments or conditions, KW based test is no longer the optimal method since it ignores this information. In this case, a more powerful statistical approach is desirable.

In this paper, we propose a new statistical approach to detecting differentially methylated loci for methylation data with multiple conditions with trend. In this method, we also adjust the age-effect in a similar way that we used before. More specifically, we first group subjects into several categories based on their age; we then apply a nonparametric trend test and get a one-sided p-value for each age category. An overall p-value is then obtained through combining those individual p-values. The performance of the new approach is assessed through comparing it with other methods using simulated data and a real methylation data set with three treatments. The R code for the new method is provided (please see the Additional file 1: R code).

## Methods

### Existing methods

In a recent paper, we have proposed several methods based on combining independent p-values to adjust the effect of age for genome-wide methylation data with multiple conditions [14]. Since those commonly used methods, such as regression models with age as continuous or categorical covariate, have poor performances [12], we compare the proposed approach with the following ones, which have the best performances among current methods based on our previous study [14].

### Combined ANOVA test

*K*conditions (treatments) and

*G*age groups. For each age group

*g (g = 1,2,…,G)*, we apply an ANOVA test and obtain a p-value ${\mathit{p}}_{\mathit{g}}^{\mathit{A}}\mathit{NOVA}.$ We know that under the null hypothesis that this locus is not differentially methylated among K conditions in any age group, $-2log\left({\displaystyle \prod _{\mathit{g}=1}^{\mathit{G}}}{\mathit{p}}_{\mathit{g}}^{\mathit{ANOVA}}\right)$ has a chi-square distribution with 2G degrees of freedom (df) since these G p-values are independent. Therefore, the overall p-value can be obtained through combining p-values by Fisher test [17]:

### Combined KW test

### Methods for combining p-values

Besides the Fisher method mentioned above, we can also use Z-test to combine p-values from independent tests [18–20]. We calculated the weighted Z statistic using individual p-values from each age group: $\mathit{Z}={\displaystyle \sum _{\mathit{g}=1}^{\mathit{G}}{\mathit{n}}_{\mathit{g}}{\mathrm{\Phi}}^{-1}\left(1-{\mathit{p}}_{\mathit{g}}\right)}/{\displaystyle \sum _{\mathit{g}=1}^{\mathit{G}}{\mathit{n}}_{\mathit{g}}^{2}}$, where *n*_{
g
} is the total sample size in age group *g* and Φ is the cumulative distribution function (CDF) of the standard normal distribution. It can be shown that under the null hypothesis this statistic has the standard normal distribution. The overall p-value is then calculated as 1- Φ(Z) by the one-sided z-test.

### The proposed method

where N_{g} is the total number of subjects in age group g, *r*_{
gi
} is the rank of the i^{th} of the N_{g} subjects, *s*_{
gk
} is the score of the k^{th} (*k* = 1, 2, …, *K*) treatment, K is the number of treatments, ${\mathit{p}}_{\mathit{gk}}=\frac{{\mathit{n}}_{\mathit{gk}}}{{\mathit{N}}_{\mathit{g}}}$, and *n*_{
gk
} is the number of subjects in the k^{th} treatment within the g^{th} age group. For the k^{th} treatment, we assign a score *s*_{
gk
} to each of the *n*_{
gk
} subjects. In this paper, we set *s*_{
gk
} = *k* (*k* = 1, 2, …, *K*), that is, we use scores 1,2,…,K.

It can be shown that under the null hypothesis, the statistic C_{g} (g = 1,2,…,G ) in (3) has an asymptotic standard normal distribution [21]. If there is an increasing trend over the K treatments, we should use the one-sided p-value, *p*_{r,g} = *Prob*(*Z* > *c*_{
g
}) = 1 - Φ(*c*_{
g
}). On the other hand if there is a decreasing trend over the K treatments, we should use the other one-sided p-value, *p*_{l,g} = *Prob*(*Z* < *c*_{
g
}) = Φ(*c*_{
g
}). The statistic $\phantom{\rule{0.25em}{0ex}}{\mathit{W}}_{1}=-2log\left({\displaystyle \prod _{\mathit{g}=1}^{\mathit{G}}}{\mathit{p}}_{\mathit{l},\mathit{g}}\right)$ has an asymptotic chi-square distribution with 2G df under the null hypothesis according to Fisher [17]. Similarly, under the null hypothesis the statistic ${\mathit{W}}_{2}=-2log\left({\displaystyle \prod _{\mathit{g}=1}^{\mathit{G}}}\left(1-{\mathit{p}}_{\mathit{l},\mathit{g}}\right)\right)$ also has an asymptotic chi-square distribution with 2G df.

*W*

_{1}or

*W*

_{2}to calculate the overall p-value. However, if the trend direction is unknown, which is usually the case in practice, we will use the maximum of the two statistics:

Since *W*_{1} and *W*_{2} are correlated, the null distribution of *W* is not easy to obtain. However, we have the following nice result [22–27].

**Theorem 1**Under the null hypothesis, the survival function of

*W*in (4) is asymptotically bounded by

where $\mathit{\gamma}=1-{\mathit{\chi}}_{2\mathit{G}}^{2}\left(\mathit{w}\right),$ and ${\mathit{\chi}}_{2\mathit{G}}^{2}\left(.\right)$ is the cumulative distribution function of the chi-square distribution with 2G df.

The above theorem can be proved using the concept of associated variables due to Esary, Proschan and Walkup [28] and Theorem 2 of Owen [29]. From theorem 1, the overall p-value can be estimated by the upper bound 2*γ*. It is easily seen that when the true p-value of W is small, the difference between the true and the estimated p-values is negligible.

where the weight ${\mathit{w}}_{\mathit{g}}=\frac{{\mathit{n}}_{\mathit{g}}}{\mathit{N}}$, and *n*_{
g
} (g = 1,2,…,G) is the number of total subjects of the K treatments within the g^{th} age group. The validity of (6) is easily seen: under the null hypothesis *c*_{
g
} and therefore *w*_{
g
}*c*_{
g
} has an asymptotic standard normal distribution; a two-sided p-value then can be obtained through (6).

### Simulation settings

To assess the performance of the proposed method, we use simulated data to compare the proposed test with current methods in terms of controlling type I error rate and power. We assume there are three different treatments (i..e, K = 3) and six age groups (i.e., G = 6). For each treatment we assume the β-value has the same following distributions over the six age groups: (i) uniform U(a,b) where 0 ≤ a < b ≤ 1, (ii) truncated normal TN (*μ*, *σ*^{2}, 0, 1) (or simply TN (*μ*, *σ*^{2}), and (iii) Beta distribution Beta (c,d) with various parameters. We consider relatively small sample sizes in our simulation study. To reflect practical situations, we either choose 20 samples for each of the three treatments (sample size = (20, 20, 20)), or set the sample sizes as 15, 20, and 25 (sample size = (15, 20, 25)), respectively, for the three treatments. Since the proposed test is designed to detect differentially methylated loci when there is a monotonic trend over the treatments, we simulate β-values with increasing or decreasing mean values over the three treatments for the alternative hypotheses. For example, in simulation, we first generate 20 β-values (sample size = (20, 20, 20)) from three uniform distributions (denoted by a = (0,0,0.25), b = (1,1,1)), U(0,1), U(0,1), and U(0.25, 1) for each of the three treatment groups. The significance level is set to be 0.05 in simulation study. The type I error rate and power are estimated by the proportions of rejection with 10^{4} replicates.

### A real data set

The real methylation data set of the United Kingdom Ovarian Cancer Population Study (UKOPS) [16], which is one of the largest available Illumina methylation data sets, will be used for real data application. This data set originally includes 274 healthy controls, 131 pre-treatment cases, and 135 post treatment cases. If the DNA methylation of a locus is positively associated with the disease, we would expect that the methylation rates are increasing from control to post-treatment then to pre-treatment. On the other hand, if the association is negative, there would be a decreasing trend over the three conditions: control, post-treatment, and pre-treatment. In either of the two situations, we can use the proposed test.

**Number of samples in age by treatment group used in the paper after data quality control step**

Age group | Control | Pre-treat | Post-treat | Total |
---|---|---|---|---|

50_55 | 14 | 15 | 16 | 45 |

55_60 | 61 | 17 | 25 | 103 |

60_65 | 64 | 17 | 22 | 103 |

65_70 | 35 | 17 | 21 | 73 |

70_75 | 63 | 24 | 22 | 109 |

75_over | 20 | 18 | 9 | 47 |

Total | 257 | 108 | 115 | 480 |

## Results

### Simulation results

**Empirical size for each method at significance level 0.05 with 10**
^{
4
}
**replicates from the simulation study**

Simulation setting (3 treatments) | Combined ANOVA | Combined K-W | New | ||
---|---|---|---|---|---|

Distribution | Sample size | Parameters | |||

Uniform U(a,b) | (20,20,20) | a = (0,0,0), b = (1,1,1) | 0.051 | 0.045 | 0.047 |

a = (0,0,0), b = (0.5,0.5,0.5) | 0.051 | 0.045 | 0.045 | ||

a = (0.5,0.5,0.5), b = (1,1,1) | 0.055 | 0.046 | 0.047 | ||

(15,20,25) | a = (0,0,0), b = (1,1,1) | 0.052 | 0.043 | 0.046 | |

a = (0,0,0), b = (0.5,0.5,0.5) | 0.052 | 0.044 | 0.050 | ||

a = (0.5,0.5,0.5), b = (1,1,1) | 0.049 | 0.040 | 0.043 | ||

Truncated Normal TN (μ ,σ | (20,20,20) | $\mathit{\mu}$= (0.5,0.5,0.5), $\phantom{\rule{0.12em}{0ex}}\mathit{\sigma}$ = (1,1,1)/5 | 0.050 | 0.043 | 0.048 |

$\mathit{\mu}$= (0.5,0.5,0.5), $\phantom{\rule{0.12em}{0ex}}\mathit{\sigma}$ = (1,2,3)/5 | 0.058 | 0.050 | 0.045 | ||

$\mathit{\mu}$= (0.2,0.2,0.2), $\phantom{\rule{0.12em}{0ex}}\mathit{\sigma}$ = (1,1,1)/5 | 0.049 | 0.050 | 0.043 | ||

$\mathit{\mu}$= (0.8, 0.8, 0.8), $\phantom{\rule{0.12em}{0ex}}\mathit{\sigma}$ = (1,1,1)/5 | 0.046 | 0.043 | 0.048 | ||

(15,20,25) | $\mathit{\mu}$= (0.5,0.5,0.5), $\phantom{\rule{0.12em}{0ex}}\mathit{\sigma}$ = (1,1,1)/5 | 0.050 | 0.046 | 0.045 | |

$\mathit{\mu}$= (0.5,0.5,0.5), $\phantom{\rule{0.12em}{0ex}}\mathit{\sigma}$ = (1,1.2,1.3)/5 | 0.053 | 0.041 | 0.033 | ||

$\mathit{\mu}$= (0.2,0.2,0.2), $\phantom{\rule{0.12em}{0ex}}\mathit{\sigma}$ = (1,1,1)/5 | 0.050 | 0.046 | 0.051 | ||

$\mathit{\mu}$= (0.8, 0.8, 0.8), $\phantom{\rule{0.12em}{0ex}}\mathit{\sigma}$ = (1,1,1)/5 | 0.049 | 0.044 | 0.048 | ||

Beta (c,d) | (20,20,20) | c = (1,1,1), d = (1,1,1) | 0.050 | 0.044 | 0.049 |

c = (1,1,1), d = (5,5,5) | 0.046 | 0.045 | 0.043 | ||

c = (5,5,5), d = (1,1,1) | 0.048 | 0.044 | 0.045 | ||

c = (5,5,5), d = (5,5,5) | 0.049 | 0.041 | 0.044 | ||

(15,20,25) | c = (1,1,1), d = (1,1,1) | 0.049 | 0.044 | 0.046 | |

c = (1,1,1), d = (5,5,5) | 0.045 | 0.042 | 0.047 | ||

c = (5,5,5), d = (1,1,1) | 0.049 | 0.049 | 0.048 | ||

c = (5,5,5), d = (5,5,5) | 0.052 | 0.044 | 0.052 |

**Empirical power for each method at significance level 0.05 with 10**
^{
4
}
**replicates from the simulation study**

Simulation setting (3 treatments) | Combined ANOVA | Combined K-W | New | New | New | ||
---|---|---|---|---|---|---|---|

Distribution | Sample size | Parameters | |||||

Uniform U(a,b) | (20,20,20) | a = (0,0,0.25), b = (1,1,1) | 0.699 | 0.607 | 0.877 | 0.962 | 0.069 |

a = (0,0.1,0.1), b = (0.5,0.5,0.5) | 0.450 | 0.339 | 0.724 | 0.830 | 0.726 | ||

a = (0.6,0.6,0.5), b = (1,1,1) | 0.460 | 0.338 | 0.695 | 0.821 | 0.027 | ||

(15,20,25) | a = (0,0,0.25), b = (1,1,1) | 0.809 | 0.692 | 0.926 | 0.980 | 0.957 | |

a = (0,0.1,0.1), b = (0.5,0.5,0.5) | 0.433 | 0.319 | 0.618 | 0.758 | 0.218 | ||

a = (0.6,0.6,0.5), b = (1,1,1) | 0.482 | 0.380 | 0.754 | 0.854 | 0.860 | ||

Truncated Normal TN ( | (20,20,20) |
| 0.451 | 0.394 | 0.743 | 0.862 | 0.052 |

| 0.773 | 0.642 | 0.962 | 0.954 | 0.200 | ||

| 0.691 | 0.656 | 0.918 | 0.976 | 0.054 | ||

| 0.402 | 0.374 | 0.696 | 0.820 | 0.032 | ||

(15,20,25) |
| 0.464 | 0.428 | 0.786 | 0.886 | 0.948 | |

| 0.735 | 0.643 | 0.959 | 0.952 | 0.713 | ||

| 0.775 | 0.738 | 0.949 | 0.981 | 0.827 | ||

| 0.382 | 0.364 | 0.756 | 0.838 | 0.852 | ||

Beta (c,d) | (20,20,20) | c = (1,1,1), d = (30,40,50) | 0.596 | 0.442 | 0.889 | 0.723 | 0.432 |

c = (1,1.2,1.5), d = (40,40,40) | 0.490 | 0.609 | 0.962 | 0.920 | 0.329 | ||

c = (30,40,50), d = (1,1,1) | 0.578 | 0.450 | 0.899 | 0.745 | 0.420 | ||

c = (40,40,40), d = (1,1.2,1.5) | 0.488 | 0.620 | 0.972 | 0.924 | 0.369 | ||

(15,20,25) | c = (1,1,1), d = (30,40,50) | 0.608 | 0.405 | 0.861 | 0.727 | 0.998 | |

c = (1,1.2,1.5), d = (40,40,40) | 0.426 | 0.602 | 0.952 | 0.912 | 0.559 | ||

c = (30,40,50), d = (1,1,1) | 0.618 | 0.409 | 0.888 | 0.752 | 0.458 | ||

c = (40,40,40), d = (1,1.2,1.5) | 0.450 | 0.606 | 0.958 | 0.919 | 0.995 |

### Results from the real data application

^{-3}, 10

^{-4}, 10

^{-5}, 10

^{-6}, 10

^{-7}, and 10

^{-8}. The results are reported in Table 4. For each of the given cutoff p-values, the proposed test always detects more loci than the other methods. In addition, most of the loci detected by the combined ANOVA and KW tests were also detected by the proposed test. For example, when the cutoff p-value is 10

^{-5}, the combined ANOVA test , the combined KW test, and the proposed test detected 479, 551, and 1283 loci, respectively, when Fisher method was used to combine p-values. Out of the 479 loci detected by the combined ANOVA test, 471 were also detected by the new test; out of the 551 loci detected by the combined KW test, only 7 were not detected by the proposed test.

**Number of significant differentially methylated loci detected by each method for each given cutoff p-value**

Method | 1e-3 | 1e-4 | 1e-5 | 1e-6 | 1e-7 | 1e-8 | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

F | Z | F | Z | F | Z | F | Z | F | Z | F | Z | |

T1 (Combined ANOVA) | 981 | 1079 | 655 | 690 | 479 | 499 | 350 | 375 | 257 | 275 | 189 | 208 |

T2 (Combined KW) | 1359 | 1340 | 823 | 859 | 551 | 590 | 381 | 401 | 261 | 277 | 172 | 185 |

T3 (New) | 2915 | 3117 | 1855 | 1951 | 1283 | 1310 | 905 | 929 | 674 | 686 | 513 | 521 |

T1 and T2 | 926 | 980 | 615 | 656 | 442 | 474 | 306 | 338 | 221 | 235 | 152 | 167 |

T1 and T3 | 931 | 1018 | 639 | 670 | 471 | 491 | 346 | 367 | 252 | 269 | 187 | 206 |

T2 and T3 | 1294 | 1279 | 806 | 832 | 544 | 577 | 377 | 396 | 259 | 276 | 170 | 184 |

T1, T2, and T3 | 895 | 954 | 605 | 642 | 437 | 468 | 303 | 336 | 220 | 234 | 151 | 166 |

This indicates that the proposed test is more powerful than other methods that are compared in this study. It is noticeable that the methods of combining independent p-values (i.e., Fisher test and Z-test) have similar performance here, although the Z-test usually gives a few more significant loci.

## Discussion

^{-3}from the proposed test. From Figure 1, we can see there is a decreasing trend among the three treatments (i.e., for the β-value, pre-treatment < post-treatment < control) for all of the six age groups; while from for those loci with large p-values, such trend does not exist for any of the six age groups (see Additional file 2: Figure S1).

Although many methods can detect those loci which are strongly differentially methylated among different treatments, it is important to detect loci having small effects as they are biological meaningful and provide useful information for set-based analyses, such as gene, gene-set, and pathway analyses which use those detected differentially methylated loci as input [30].

To use the Cuzick test, we need to assign a score for each of the treatment. Here we assign 1, 2, and 3 to the control, post-treatment, and pre-treatment, respectively. In practice, if we have the information of the effects for each treatment, we can use this information to assign scores. For example, for the K-1 treatments 2, 3, …, K, if the effect sizes are m_{2}, …, m_{K} compared to treatment 1, we can assign scores 0, m_{2}, …, m_{K} to those treatments for the proposed test. However, if we only know that there is a monotonic trend, we can choose 1, 2, …, K (equivalent to 0, 1, …, K-1) as the scores. Although, the performance of the proposed test can be improved by assigning optimal scores, which are determined by the true effects, to the treatments; in general, it is impractical to obtain the optimal scores. In addition, the optimal scores for each locus may not be the same across age groups (see Figure 1).

Like other large scale data, such as microarray data and genome-wide association study data, the multiple comparison is an important but challenging issue. Although some procedures have been proposed to control either family-wise error rate or false discovery rate, it remains an open topic in this area. One possible direction is to use the so-called “effective number” estimated from correlations among the loci [31].

## Conclusions

We propose a new statistical approach to detecting methylated loci for high-throughput methylation data with multiple groups. This approach is based on the nonparametric Cuzick test, which is robust and powerful if there exists a trend over groups. Through simulated and real data, we show that the proposed test outperforms existing methods.

## Declarations

### Acknowledgements

The authors would like to thank the Editor and five reviewers for their constructive comments, which resulted in an improved presentation of this paper. The first author also acknowledges the support from the faculty research funds awarded by the School of Public Health, Indiana University Bloomington.

## Authors’ Affiliations

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