# An integrative imputation method based on multi-omics datasets

**Part of the following topical collections:**

## Abstract

### Background

Integrative analysis of multi-omics data is becoming increasingly important to unravel functional mechanisms of complex diseases. However, the currently available multi-omics datasets inevitably suffer from missing values due to technical limitations and various constrains in experiments. These missing values severely hinder integrative analysis of multi-omics data. Current imputation methods mainly focus on using single omics data while ignoring biological interconnections and information imbedded in multi-omics data sets.

### Results

In this study, a novel multi-omics imputation method was proposed to integrate multiple correlated omics datasets for improving the imputation accuracy. Our method was designed to: 1) combine the estimates of missing value from individual omics data itself as well as from other omics, and 2) simultaneously impute multiple missing omics datasets by an iterative algorithm. We compared our method with five imputation methods using single omics data at different noise levels, sample sizes and data missing rates. The results demonstrated the advantage and efficiency of our method, consistently in terms of the imputation error and the recovery of mRNA-miRNA network structure.

### Conclusions

We concluded that our proposed imputation method can utilize more biological information to minimize the imputation error and thus can improve the performance of downstream analysis such as genetic regulatory network construction.

### Keywords

Multi-omics data Imputation Integrative analysis Ensemble learning## Background

Recent advances in high-throughput technologies prompt the production of a variety of ‘omics’ data such as transcriptomics, proteomics and metabolomics from the same set of subject tissues/cells, facilitating the discovery of various levels of risk genetic factors for the analysis of human complex diseases. However, due to technical limitations of these high throughput technologies and experimental designs, the presence of missing values remains an inevitable and prevalent problem in large-scale profiling experiments [1]. For example, proteomics data suffers significantly from missing values due to the imperfect identification of coding sequences within a genome and the limited sensitivity of current peptide detection technologies [2, 3, 4]. Current technologies allow the detection of only one-third to one-half of all coded proteins and thus leave a significant number of proteins experimentally undetected [5, 6, 7]. In miRNA array, it is often observed that a large portion of miRNAs are expressed below the detection limit, resulting in missing data in the output [8, 9]. In general, there are three types of missing mechanisms [10, 11]: the first one is data missing completely at random (MCAR), where data missing is due to some factors unrelated to the experimental questions. The causes of missing are usually unobserved in the experiment. The second mechanism is data missing at random (MAR), where missing depends on some variables, which can be measured in the experiment such as different slides, media or experimental conditions for assaying expression data. The last one is missing not at random (MNAR) where data missing is caused by some unobserved variables; they may be highly related to the experiment, for instance, low-abundance expression may remain undetected due to the detection bias of instruments.

A number of studies have indicated that missing values in large-scale omics data can drastically hinder downstream analyses, such as unsupervised clustering of genes [12], detection of differentially expressed genes [13], supervised classification of clinical samples [14], construction of gene regulatory networks [15], genome wide association studies [16] and detection of differentially methylated regions [17]. Missing values in multi-omics data can also obstruct integrative analysis of multi-omics data, leading to difficulty in the interpretation of complex diseases. Therefore, it is highly demanded to impute the missing values before performing integrative analysis of multi-omics data.

However, most current imputation methods mainly focus on single omics data as reviewed in [18], including global methods such as Bayesian principle component analysis (BPCA) [19], and singular value decomposition imputation (SVDimpute) [20], local methods such as k nearest neighbor imputation (KNNimpute) [20], local least square imputation (LLS) [21] and iterative local least square imputation (iLLS) [22], and hybrid methods which combine both global and local methods such Lincmb [23]. The main limitation of these imputation methods is that they only focused on utilizing the information from single omics data. Thus there is an increasing interest in incorporating additional information for the imputation, e.g., biological databases or other omics data [24]. For transcriptomics datasets, a priori information about the functional similarities in term of GO (Gene Ontology) was used for missing value imputation, based on the idea that functionally related genes tend to express in a modular fashion [25]. Experimental results indicated that the imputation accuracy can be enhanced by incorporating GO information, even when the missing rate was large [26, 27]. Other knowledge based impute methods such as integrative missing value estimation method (iMISS) employed the information from multiple external references data to find consistent and reliable neighboring genes of a target missing gene for better estimation [28]. Meta-data imputation method combined multiple available microarray datasets of a species to select top closest columns to impute missing column in target missing matrix [29]. These knowledge-based imputation methods usually require the features (e.g., genes) correspond with each other across diverse datasets and they still focus on one type of omics data. There are some endeavors to incorporate the relationships between diverse omics data into the imputation. Nie et al. proposed a Zero-inflated Poisson regression model to use the correlation between transcriptomics and proteomics datasets for imputing the missing proteomics data [30]. Torres-Garcia et al. published a stochastic Gradient Boosted Trees (GBT) approach to uncover possible nonlinear relationships between transcriptomics and proteomics data. GBT was used to predict those protein abundance not experimentally detected based on the predictors such as mRNA abundance, cellular role, molecular weight, sequence length, protein length, GC content and triplet codon counts [4, 31]. Histone acetylation information was combined into KNNimpute and LLS to improve the missing value estimation of gene expression data [32]. Artificial neural network approach was also applied to impute the missing values of the proteins using the relations between transcriptomics and proteomics data in the study [33].

By extending these methods to incorporate informative features from other types of omics data for the imputation, we developed an ensemble learning based algorithm to deal with missing values. Our multi-omics imputation method takes advantage of the correlation across different omics data with the assumption that the missing feature from one type of omics data can be explained by its neighboring features from the same omics data as well as the features from other omics data. Some prior biological knowledge about interactions among different levels of omics data (e.g., GO, protein-protein interaction database) can also be incorporated. In addition, to account for the situation that there are multiple omics data having missing values, we extended our multi-omics imputation method to simultaneously impute multiple missing omics data. We compared our method with five single omics data imputation methods with respect to different noise levels, sample sizes and missing rates. Moreover, we further evaluated the impact of different imputation methods on downstream analysis, e.g., mRNA-miRNA network reconstruction. The results consistently confirmed the advantage and efficiency of our multi-omics imputation method in terms of the imputation error and the recovery of mRNA-miRNA network.

## Methods

In this study, we take three kinds of omics data (e.g., mRNA, microRNA and DNA methylation) as an example to elaborate our method. For each omics data, it is represented by a matrix \( {G}_i\in {R}^{p_i\times n},i=1,2,..,m \), where *i* indicates the type of omics data, *p* _{ i } is the number of rows of each matrix corresponding to different types of features (e.g., gene expression) and *n* is the number of columns corresponding to different subjects. The missing point at the *m*-th feature on the *l*-th subject is denoted by *G* _{ i } ^{ m,l } , m = *1*, *2*, …, *p* _{ i }, *l* = *1*, *2*, …, *n*. In the following, we first introduce diverse single omics imputation methods, and then propose multi-omics data imputation and its extension to more general case.

### Single omics imputation

*g*

_{ t }∈

*R*

^{ n }in \( {G}_1\in {R}^{p_1\times n} \) contains missing values located in the first

*s*subjects. Hence,

*g*

_{ t }

^{ miss }∈

*R*

^{1 × s }is the missing vector in the target gene and

*g*

_{ t }

^{ c }∈

*R*

^{1 × (n − s)}is complete vector containing non-missing values. To estimate the missing vector

*g*

_{ t }

^{ miss }, firstly, we compute the distance (Euclidean distance)

*d*

_{ t, j }between the target gene

*t*and other gene

*j*(or eigengene

*j*[20]); secondly, top

*k*close genes (or eigengenes), denoted by

*Gk*= [

*Gk*

^{ miss },

*Gk*

^{ c }] ∈

*R*

^{ k × n }are used for imputation. Specifically, KNNimpute estimates g

_{t}

^{miss}by averaging the weighted values of neighboring genes or eigengenes while the other methods tend to use linear regression as in (2)

*Gk*

^{ miss }∈

*R*

^{ k × s }is the submatrix of

*Gk*corresponding to the missing location in the target gene; and

*β*is the coefficient vector to weight the contribution of neighboring genes/eigengenes, which can be estimated by the following least square minimization:

*β*= (

*Gk*

^{ c })

^{†}

*g*

_{ t }

^{ c }, where (

*Gk*

^{ c })

^{†}is the pseudo inverse of

*Gk*

^{ c }.

### Multi-omics data imputation

*G*

_{1}), mRNA (

*G*

_{2}) and DNA methylation (

*G*

_{3}), which have been identified to be correlated with each other in their elements or components [34, 35]. As shown in Fig. 1, we built an integrative model based on ensemble learning [36, 37], which generally consisted of three steps: the first step was ensemble learning which generated a set of basic models; the second was the ensemble pruning, where models were pruned to remove some models with little contributions; the final step was the integration of multiple models into a new prediction model. In this study, we built a predictive model on a set of basic models for missing value estimation, as shown in Fig. 1. Multiple constraints were imposed on each basic model (e.g., non-negativity constraint) to reduce the overfitting as well as the influence of those basic models with little contribution.

The basic models were generated based on three types of imputations, i.e., self-imputation and cross imputation by *G* _{2} and *G* _{3} respectively. The self-imputation was to impute *G* _{1} by itself using single-omics imputation method as mentioned in the section of “Single omics imputation”. The cross-imputation was to impute *G* _{1} by other omics data, i.e., *G* _{2}. Because of the scale difference among different types of omics data, we intended to impute each missing feature in G_{1} individually by exploiting the correlated information from *G* _{2}. For each target gene *g* _{ t } = [*g* _{ t } ^{ miss } , *g* _{ t } ^{ c } ] in *G* _{1}, it was combined with correlated features in *G* _{2} to be a new missing matrix *H*. Matrix *H* was then imputed by self-imputation methods to estimate *g* _{ t } ^{ miss } . Eventually, we obtained three imputation outputs for all missing values in *G* _{1} by different omics data, denoted by *G* _{1 ← 1}, *G* _{1 ← 2} and *G* _{1 ← 3} respectively. Moreover, prior knowledge from accessible databases can also provide extra information (e.g. protein-protein interactions (PPI), co-expressed genes) to improve the imputation accuracy. In this study, we took advantage of information from PPI to partially avoid the overfitting for LLS and iLLS in *G* _{2}. For example, to impute the expression data of target gene *g* _{ t } by other genes in *G* _{2}, we collected those genes that had STRING scores > 0.9 [38, 39] with *g* _{ t } in STRING database and had significant correlation (*p* < 0.05) in *G* _{2}. We keep these genes in the prediction model for LLS and iLLS during the feature selection procedure. This partially decreased the selection of genes which had random correlations with target gene in *G* _{2}. To further improve the accuracy of prediction, we considered the generation of heterogeneous learning algorithms that can ensure a level of diversity among the basic models. Diverse models can provide different predictions, which can be combined for better performance. We subsampled features in omics data and re-run the imputation *B* times to get multiple imputed matrices, {*G* _{1 ← 1,b }, *G* _{1 ← 2,b }, *G* _{1 ← 3,b }}, *b* = 1, 2, …, *B*.

*β*

_{1 ← i,b },

*i*= 1, 2, …,

*m*,

*b*= 1, 2, …,

*B*are the weights for different basic imputation models, and

*j*indicates missing location in target gene. Since all these models aim to impute the same missing values, their outputs are highly correlated. Instead of using ridge regression, we imposed non-negative regularization on the coefficients to handle the high multi-colinearity among variables in the model, which has been found to be more reliable and consistent [37]. To avoid the over-fitting issue, we adopted bootstrapping to randomly generate faking missing values at the locations which were not overlapped with true missing locations. The weights were estimated by (4) based on the imputed and true values on the faking missing points (Additional file 1 B). The averaged value of each weight on

*T*times bootstrapping was used for prediction. We set

*T*to be 30 in the following experiments.

### Extension of multi-omics data imputation

Algorithm for iterative multi-omics imputation

A: Initialize with replacing all missing values in all matrices |

B: for each iteration h, |

(1). |

a. Self-impute |

b. Self-impute |

c. Self-impute |

(2). Determine the sum of square of difference on the missing locations j between { |

\( {\delta}^h={\displaystyle \sum_j{\displaystyle \sum_i{\left({G}_i^{j,\left(h-1\right)}-{G}_i^{j,(h)}\right)}^2}} \) |

C. If |

In the simple case that only one omics data contains missing values, there is only one step in the iteration. In the case of missing values in multiple omics data, compared with performing single omics imputation separately, the advantage of our extended multi-omics imputation is its ability of reusing data in both self-imputation and cross-imputation processes. In current single-omics imputation methods, most of them use row average to impute missing value initially for deriving the neighboring gene/eigengene only once, which can cause biases in the final imputation. Instead of using one-time row average initialization, in each iteration, the self-imputation (e.g., *G* _{1 ← 2} ^{(h)} ) is implemented based on the completed matrix from the previous iteration *G* _{1 ← 2} ^{(h − 1)} , which is updated iteratively to reduce the bias. In addition, information from other omics data will be incorporated by the cross-imputation, which can further improve the imputation accuracy, especially when large missing data exist in *G* _{1 ← 2} ^{(h)} .

### Evaluation metric

*G*

_{ i }

^{ j }and

*Ĝ*

_{ i }

^{ j }are the true and imputed value of the

*j*-th missing point in the

*i*-th omics data respectively; and N is the number of missing points in all datasets. In addition, we performed a paired

*t*-test for each factor exploration, measuring the significance of NRMSE difference between the methods along with each simulation.

## Results

### Simulation scheme

We performed simulation analysis to evaluate the performance of our proposed methods based on the MCAR missing mechanism. Simulation data were derived from the cancer genomic atlas (TCGA; http://cancergenome.nih.gov/) database on Glioma cancer study containing 50 subjects with 5939 mRNAs, 104 microRNAs and 5013 DNA methylation sites. We have removed missing values in all of these data, yielding complete data matrices. Based on these observation matrices, a certain percent of entries (e.g., 1, 5 %) was randomly set to have missing values. To evaluate the effect of sample size (e.g., 10, 20 subjects), a specific number of samples was first selected randomly and then missing matrices were generated. Similarly, for different noise levels, a random noise from normal distribution *N*(0, *σ* _{ e } ^{2} ) with different standard deviations was added to the observed matrices for missing matrices generation. Each type of missing matrix generation was repeated 50 times, and then several imputation methods were applied for comparison.

First, we compared the single-omics based method with multi-omics based imputation method when only one type of omics data (e.g., miRNA) contained missing values and the other datasets (e.g., mRNA and DNA methylation) were complete. Then, to consider more realistic situation when more than one type of omics data contained missing values, we simulated both mRNA and miRNA datasets with missing values. For simplicity, we set the same missing rate on both datasets. Single-omics-based method was applied to each type of missing matrix, while our iterative multi-omics method was directly used for imputing both datasets simultaneously. Finally, the imputation accuracy was evaluated on both simulations.

### Parameters setting

For KNNimpute method, we set the neighboring size to be 15 as suggested in [20, 28]. Both BPCA and SVDimpute depend on the number of principal axes (eigenvectors). The number of eigenvectors selected in SVD imputation was set to be 20 % of the number of samples as tested in [20]; but the number of PCs used in BPCA could be set more loosely as suggested in the earlier study [19] which suggested a safety number of PCs to be *k* = D-1,where D was the number of samples. The automatic relevance determination prior applied in Bayesian estimation can reduce the redundant dimension automatically. For local regression based methods, LLS impute [21] and iLLS impute [22], a procedure of estimating the optimal number k was applied. Prior to imputation, missing values were initially estimated by row average and some faking missing values were generated with true values known. The methods searched optimal *k* (LLS) or ratio (iLLS) value from a given range (e.g., *k* between 2 and D-1 or ratio between 0 and 1) with the lowest estimation error.

### Comparison on single missing omics data

We simulated one omics data (e.g., miRNA) with missing values while keeping the other omics datasets (e.g., mRNA and DNA methylation) to be complete. Five popular imputation methods were applied to single missing omics data, which were then compared with our proposed multi-omics imputation method. Three simulations were performed to study the effects of three factors on imputation accuracy: missing rate, sample size and noise level.

### Effect of missing rate

#### Effect of sample size

#### Effect of noise level

Due to technical limitations, there are a variety of noises introduced in the collected data, which may cause difficulty in imputation. To test the robustness of diverse imputation methods to noise, we simulated five datasets by adding different levels of Gaussian noise with varying standard deviations (std) from 0.1 to 1. The sample size was set to be 50 and missing rate was 5 % in all datasets.

### Comparisons on multiple missing omics data

To further evaluate the imputation methods when there are missing values in multiple omics data, we simulated mRNA and miRNA datasets with different missing rates of 1, 5 and 10 % respectively. Five single-omics methods were used to impute each missing data separately and then compared with our iterative multi-omics imputation method, which can impute both missing matrices simultaneously.

### Comparisons for network analysis

Besides the comparison of imputation accuracy between different methods, we further evaluated the influence of different imputation methods on downstream analysis, e.g., reconstruction of regulatory network between mRNA and miRNA, which is significant for exploring the interactions between different omics. There are a variety of methods proposed for reconstructing genetic regulatory networks. We applied a web tool, miRNA and genes integrated analysis (MAGIA) [44], to reconstruct fundamental post-transcriptional regulatory networks between miRNAs and mRNAs for Glioma cancer. The Pearson correlation was used as the measure of interactions between each pair of miRNA and mRNA expressions on the matched design. Then the combination of two target prediction algorithms (PicTar and PITA) was applied to predict the target of miRNA and thus the mRNA-miRNA regulatory network was built based on significant test on each interaction.

We evaluated network reconstruction as a binary classification task (prediction of absence or presence of mRNA-miRNA interaction). The original network was constructed by using completed matrices. Then for each network built on the imputed matrices, a receiver operating characteristics (ROC) curve was derived by varying correlation threshold. The area under curve (AUC) was calculated to evaluate the influence of different imputation methods on regulatory network. Higher AUC indicates better capability of preserving significant interactions while lower AUC means worse effect of imputation methods on network structure reconstruction.

*p*< 1 × 10

^{−4}) were selected to be significant. There are 655 mRNA-miRNA interactions selected corresponding to the correlation value less than −0.55. For each pair of imputed mRNA and miRNA matrices by different methods, their interactions were reconstructed and compared with significant interaction set to obtain false positive rate (FPR) and true positive rate (TPR). Figure 6 shows the ROC plots, which compared the performance of iterative multi-omics imputation and single-omics imputation (i.e., KNNimpute) algorithms on recovering true mRNA-miRNA interactions. The curves were plotted by choosing different correlation thresholds. When missing rate is small, e.g., 1 %, both imputation methods give similar network structures with relatively better performance of multi-omics method for the two higher missing rates. The difference between two ROCs increases as the missing rate increases, showing the advantage of our iterative method for preserving significant mRNA-miRNA interactions

## Discussion

In this work, our multi-omics imputation method is able to combine the estimations from various basic models linearly to estimate missing values, e.g., self-imputation and cross-imputation. Multi-omics imputation method aims to employ information from different sources (e.g., diverse omics data). Each source is expected to contribute partially to the estimation. Similar to the contribution measure in [23], source contribution can be reflected in the coefficient weight in the final linear combination model as shown in (4). We evaluated the coefficient value of each source in the simulation of varying missing rate (data not shown). The weights of sources in most methods (except KNNimpute) are stable with the change of missing rate. miRNA shows higher weights than the other omics, indicating that self-information is still most important for imputation in this simulation. The other omics data also contain useful information and contribute to the improvement of imputation with appropriate weights. In addition to building basic model by different omics data, there are some other basic models such as multiple single-omics imputation algorithms that can also be incorporated as basic models and combined with cross-imputation estimation. Similar to the results discussed in [23], other types of single-omics imputation algorithms may also provide complementary information to improve imputation accuracy.

The cross-imputation part of the proposed multi-omics imputation method is based on regression model which requires the subjects cross omics data to be matched, while this case may not always hold in real data analysis. For the case that some subjects are shared across multiple omics data while some are omics data specific, we suggest the following imputation strategy: if those missing points are located on the subjects having only one type of omics data, only self-imputation part can be used in the proposed method, which degrades to single-omics imputation; for other missing points happened on subjects assayed with more than one types of omics data, both self-imputation and cross-imputation can be applied and combined in our method.

Since most of single-omics imputation methods were evaluated by simulations based on MCAR missing mechanism, we applied the same simulation strategy in this study for evaluation of multi-omics imputation performance. However, this assumption may not always hold in reality. Two other missing mechanisms (MAR and MNAR) may also be possible in expression data, as mentioned in Introduction. We further evaluated the performance of imputation methods with MAR missing mechanism by assuming data missing in some specific genes or proteins (e.g., existing on neighboring genes or probes). This missing may be due to the contamination of slides in the experiment, which can be documented for users. Additional file 1: Figure S1 shows the results of comparisons. There is no significant difference between two simulation schemes when missing rate is low (<0.2), indicating the insensitivity of the imputation methods to the way of how missing values are generated. When missing rate is high (i.e., 0.2), NRMSE in MAR imputation is much higher than that with MCAR missing, because of more information loss of these neighboring genes in MAR simulation (they all contain missing values). More importantly, the multi-omic method still shows significant improvement over single-omics methods since extra information can be provided by other omics data. For more complicated MNAR missing mechanisms, the loss of valuable information causing the missingness may make the most of imputation methods to be suboptimal and the estimation of missing patterns to be infeasible [1, 11]. Although this case is not discussed in this study, it is still expected that imputation method incorporating informative features from different types of omics data will outperform single-omics imputation methods, given the advantage of integrating more information from multiple omics data.

## Conclusions

We proposed a novel multi-omics imputation framework, which can take advantage of information from multiple omics data for improving the imputation accuracy. With the production of vast multi-omics data, there is increasing knowledge about complex biological relationship among multiple levels of omics (e.g., co-expression or co-regulation among gene expression, miRNA expression and transcriptional factors). We proposed multi-omics imputation method to exploit the underlying cross-omics relationship for missing value imputation. Experimental results confirmed the advantage of our multi-omics based method over five single-omics imputation methods (KNNimpute, BPCA, SVDimpute, LLS and iLLS) consistently in all three different scenarios in terms of lower value of NRMSE. To handle multiple omics data with missing values, we extended the imputation method, so it can utilize the relationship among multi-omics data iteratively to impute multiple missing omics data simultaneously. Compared with conventional single-omics methods for imputing each omics data separately, our iterative method is able to improve the imputation accuracy significantly in each missing omics data, especially for lower dimensional omics datasets, e.g., miRNA. In addition, the evaluation of mRNA-miRNA regulatory network demonstrated that our iterative method outperforms all five single-omics methods in uncovering the relationship across omics data, which is therefore significant for the study of biological regulatory mechanisms.

## Abbreviations

AUC, the area under curve; BPCA, Bayesian principle component analysis; FPR, false positive rate; GBT, Gradient Boosted Trees; GO, Gene Ontology; iLLS, iterative local least square; iMISS, integrative missing value estimation method; KNNimpute, k nearest neighboring impute; LLS, local least square; MAGIA, miRNA and genes integrated analysis; MAR, missing at random; MCAR, missing completely at random; MNAR, missing not at random; NRMSE, normalized root mean squared error; PPI, protein-protein interactions; ROC, the receiver operating characteristics curve; SVDimpute, singular value decomposition impute; TCGA, the cancer genomic atlas; TPR, true positive rate

## Notes

### Acknowledgements

This work was supported by NIH [R01 GM109068, R01 MH 107354 and R01 MH104680] and NSF (**#**1539067).

### Funding

This work was supported by NIH [R01 GM109068, R01 MH 107354 and R01 MH104680] and NSF (**#**1539067). Non-financial competing interests are included.

### Availability of data and materials

The Glioma cancer data is available at the cancer genomic atlas (TCGA; http://cancergenome.nih.gov/).

### Authors’ contributions

DL and JZ designed the experiments and methods. JY and CX implemented the comparisons with other algorithms. HD and YW participated in the design and coordination of study and drafted the manuscript. All authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

### Consent for publication

Not applicable.

### Ethics approval and consent to participate

Not applicable.

## Supplementary material

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