Microarray missing data imputation based on a set theoretic framework and biological knowledge

Gene expressions measured using microarrays usually suffer from the missing value problem. However, in many data analysis methods, a complete data matrix is required. Although existing missing value imputation algorithms have shown good performance to deal with missing values, they also have their limitations. For example, some algorithms have good performance only when strong local correlation exists in data while some provide the best estimate when data is dominated by global structure. In addition, these algorithms do not take into account any biological constraint in their imputation. In this paper, we propose a set theoretic framework based on projection onto convex sets (POCS) for missing data imputation. POCS allows us to incorporate different types of a priori knowledge about missing values into the estimation process. The main idea of POCS is to formulate every piece of prior knowledge into a corresponding convex set and then use a convergence-guaranteed iterative procedure to obtain a solution in the intersection of all these sets. In this work, we design several convex sets, taking into consideration the biological characteristic of the data: the first set mainly exploit the local correlation structure among genes in microarray data, while the second set captures the global correlation structure among arrays. The third set (actually a series of sets) exploits the biological phenomenon of synchronization loss in microarray experiments. In cyclic systems, synchronization loss is a common phenomenon and we construct a series of sets based on this phenomenon for our POCS imputation algorithm. Experiments show that our algorithm can achieve a significant reduction of error compared to the KNNimpute, SVDimpute and LSimpute methods.     

Missing data imputation and haplotype phase inference for genome-wide association studies

Imputation of missing data and the use of haplotype-based association tests can improve the power of genome-wide association studies (GWAS). In this article, I review methods for haplotype inference and missing data imputation, and discuss their application to GWAS. I discuss common features of the best algorithms for haplotype phase inference and missing data imputation in large-scale data sets, as well as some important differences between classes of methods, and highlight the methods that provide the highest accuracy and fastest computational performance.     

DNA microarray data imputation and significance analysis of differential expression

MOTIVATION: Significance analysis of differential expression in DNA microarray data is an important task. Much of the current research is focused on developing improved tests and software tools. The task is difficult not only owing to the high dimensionality of the data (number of genes), but also because of the often non-negligible presence of missing values. There is thus a great need to reliably impute these missing values prior to the statistical analyses. Many imputation methods have been developed for DNA microarray data, but their impact on statistical analyses has not been well studied. In this work we examine how missing values and their imputation affect significance analysis of differential expression. RESULTS: We develop a new imputation method (LinCmb) that is superior to the widely used methods in terms of normalized root mean squared error. Its estimates are the convex combinations of the estimates of existing methods. We find that LinCmb adapts to the structure of the data: If the data are heterogeneous or if there are few missing values, LinCmb puts more weight on local imputation methods; if the data are homogeneous or if there are many missing values, LinCmb puts more weight on global imputation methods. Thus, LinCmb is a useful tool to understand the merits of different imputation methods. We also demonstrate that missing values affect significance analysis. Two datasets, different amounts of missing values, different imputation methods, the standard t-test and the regularized t-test and ANOVA are employed in the simulations. We conclude that good imputation alleviates the impact of missing values and should be an integral part of microarray data analysis. The most competitive methods are LinCmb, GMC and BPCA. Popular imputation schemes such as SVD, row mean, and KNN all exhibit high variance and poor performance. The regularized t-test is less affected by missing values than the standard t-test. AVAILABILITY: Matlab code is available on request from the authors.     

Collateral missing value imputation: a new robust missing value estimation algorithm for microarray data

MOTIVATION: Microarray data are used in a range of application areas in biology, although often it contains considerable numbers of missing values. These missing values can significantly affect subsequent statistical analysis and machine learning algorithms so there is a strong motivation to estimate these values as accurately as possible before using these algorithms. While many imputation algorithms have been proposed, more robust techniques need to be developed so that further analysis of biological data can be accurately undertaken. In this paper, an innovative missing value imputation algorithm called collateral missing value estimation (CMVE) is presented which uses multiple covariance-based imputation matrices for the final prediction of missing values. The matrices are computed and optimized using least square regression and linear programming methods. RESULTS: The new CMVE algorithm has been compared with existing estimation techniques including Bayesian principal component analysis imputation (BPCA), least square impute (LSImpute) and K-nearest neighbour (KNN). All these methods were rigorously tested to estimate missing values in three separate non-time series (ovarian cancer based) and one time series (yeast sporulation) dataset. Each method was quantitatively analyzed using the normalized root mean square (NRMS) error measure, covering a wide range of randomly introduced missing value probabilities from 0.01 to 0.2. Experiments were also undertaken on the yeast dataset, which comprised 1.7% actual missing values, to test the hypothesis that CMVE performed better not only for randomly occurring but also for a real distribution of missing values. The results confirmed that CMVE consistently demonstrated superior and robust estimation capability of missing values compared with other methods for both series types of data, for the same order of computational complexity. A concise theoretical framework has also been formulated to validate the improved performance of the CMVE algorithm. AVAILABILITY: The CMVE software is available upon request from the authors.     

Missing value estimation for DNA microarray gene expression data: local least squares imputation

MOTIVATION: Gene expression data often contain missing expression values. Effective missing value estimation methods are needed since many algorithms for gene expression data analysis require a complete matrix of gene array values. In this paper, imputation methods based on the least squares formulation are proposed to estimate missing values in the gene expression data, which exploit local similarity structures in the data as well as least squares optimization process. RESULTS: The proposed local least squares imputation method (LLSimpute) represents a target gene that has missing values as a linear combination of similar genes. The similar genes are chosen by k-nearest neighbors or k coherent genes that have large absolute values of Pearson correlation coefficients. Non-parametric missing values estimation method of LLSimpute are designed by introducing an automatic k-value estimator. In our experiments, the proposed LLSimpute method shows competitive results when compared with other imputation methods for missing value estimation on various datasets and percentages of missing values in the data. AVAILABILITY: The software is available at http://www.cs.umn.edu/~hskim/tools.html CONTACT: hpark@cs.umn.edu     

Two-pass imputation algorithm for missing value estimation in gene expression time series

Gene expression microarray experiments frequently generate datasets with multiple values missing. However, most of the analysis, mining, and classification methods for gene expression data require a complete matrix of gene array values. Therefore, the accurate estimation of missing values in such datasets has been recognized as an important issue, and several imputation algorithms have already been proposed to the biological community. Most of these approaches, however, are not particularly suitable for time series expression profiles. In view of this, we propose a novel imputation algorithm, which is specially suited for the estimation of missing values in gene expression time series data. The algorithm utilizes Dynamic Time Warping (DTW) distance in order to measure the similarity between time expression profiles, and subsequently selects for each gene expression profile with missing values a dedicated set of candidate profiles for estimation. Three different DTW-based imputation (DTWimpute) algorithms have been considered: position-wise, neighborhood-wise, and two-pass imputation. These have initially been prototyped in Perl, and their accuracy has been evaluated on yeast expression time series data using several different parameter settings. The experiments have shown that the two-pass algorithm consistently outperforms, in particular for datasets with a higher level of missing entries, the neighborhood-wise and the position-wise algorithms. The performance of the two-pass DTWimpute algorithm has further been benchmarked against the weighted K-Nearest Neighbors algorithm, which is widely used in the biological community; the former algorithm has appeared superior to the latter one. Motivated by these findings, indicating clearly the added value of the DTW techniques for missing value estimation in time series data, we have built an optimized C++ implementation of the two-pass DTWimpute algorithm. The software also provides for a choice between three different initial rough imputation methods.     

Missing values in mass spectrometry based metabolomics: an undervalued step in the data processing pipeline

Missing values in mass spectrometry metabolomic datasets occur widely and can originate from a number of sources, including for both technical and biological reasons. Currently, little is known about these data, i.e. about their distributions across datasets, the need (or not) to consider them in the data processing pipeline, and most importantly, the optimal way of assigning them values prior to univariate or multivariate data analysis. Here, we address all of these issues using direct infusion Fourier transform ion cyclotron resonance mass spectrometry data. We have shown that missing data are widespread, accounting for ca. 20% of data and affecting up to 80% of all variables, and that they do not occur randomly but rather as a function of signal intensity and mass-to-charge ratio. We have demonstrated that missing data estimation algorithms have a major effect on the outcome of data analysis when comparing the differences between biological sample groups, including by t test, ANOVA and principal component analysis. Furthermore, results varied significantly across the eight algorithms that we assessed for their ability to impute known, but labelled as missing, entries. Based on all of our findings we identified the k-nearest neighbour imputation method (KNN) as the optimal missing value estimation approach for our direct infusion mass spectrometry datasets. However, we believe the wider significance of this study is that it highlights the importance of missing metabolite levels in the data processing pipeline and offers an approach to identify optimal ways of treating missing data in metabolomics experiments.

     

Missing values in mass spectrometry based metabolomics: an undervalued step in the data processing pipeline

Missing values in mass spectrometry metabolomic datasets occur widely and can originate from a number of sources, including for both technical and biological reasons. Currently, little is known about these data, i.e. about their distributions across datasets, the need (or not) to consider them in the data processing pipeline, and most importantly, the optimal way of assigning them values prior to univariate or multivariate data analysis. Here, we address all of these issues using direct infusion Fourier transform ion cyclotron resonance mass spectrometry data. We have shown that missing data are widespread, accounting for ca. 20% of data and affecting up to 80% of all variables, and that they do not occur randomly but rather as a function of signal intensity and mass-to-charge ratio. We have demonstrated that missing data estimation algorithms have a major effect on the outcome of data analysis when comparing the differences between biological sample groups, including by t test, ANOVA and principal component analysis. Furthermore, results varied significantly across the eight algorithms that we assessed for their ability to impute known, but labelled as missing, entries. Based on all of our findings we identified the k-nearest neighbour imputation method (KNN) as the optimal missing value estimation approach for our direct infusion mass spectrometry datasets. However, we believe the wider significance of this study is that it highlights the importance of missing metabolite levels in the data processing pipeline and offers an approach to identify optimal ways of treating missing data in metabolomics experiments.     

Model selection of generalized estimating equations with multiply imputed longitudinal data

Longitudinal data often encounter missingness with monotone and/or intermittent missing patterns. Multiple imputation (MI) has been popularly employed for analysis of missing longitudinal data. In particular, the MI‐GEE method has been proposed for inference of generalized estimating equations (GEE) when missing data are imputed via MI. However, little is known about how to perform model selection with multiply imputed longitudinal data. In this work, we extend the existing GEE model selection criteria, including the “quasi‐likelihood under the independence model criterion” (QIC) and the “missing longitudinal information criterion” (MLIC), to accommodate multiple imputed datasets for selection of the MI‐GEE mean model. According to real data analyses from a schizophrenia study and an AIDS study, as well as simulations under nonmonotone missingness with moderate proportion of missing observations, we conclude that: (i) more than a few imputed datasets are required for stable and reliable model selection in MI‐GEE analysis; (ii) the MI‐based GEE model selection methods with a suitable number of imputations generally perform well, while the naive application of existing model selection methods by simply ignoring missing observations may lead to very poor performance; (iii) the model selection criteria based on improper (frequentist) multiple imputation generally performs better than their analogies based on proper (Bayesian) multiple imputation.     

Imputation methods for missing data for polygenic models

Methods to handle missing data have been an area of statistical research for many years. Little has been done within the context of pedigree analysis. In this paper we present two methods for imputing missing data for polygenic models using family data. The imputation schemes take into account familial relationships and use the observed familial information for the imputation. A traditional multiple imputation approach and multiple imputation or data augmentation approach within a Gibbs sampler for the handling of missing data for a polygenic model are presented.We used both the Genetic Analysis Workshop 13 simulated missing phenotype and the complete phenotype data sets as the means to illustrate the two methods. We looked at the phenotypic trait systolic blood pressure and the covariate gender at time point 11 (1970) for Cohort 1 and time point 1 (1971) for Cohort 2. Comparing the results for three replicates of complete and missing data incorporating multiple imputation, we find that multiple imputation via a Gibbs sampler produces more accurate results. Thus, we recommend the Gibbs sampler for imputation purposes because of the ease with which it can be extended to more complicated models, the consistency of the results, and the accountability of the variation due to imputation.     

How to Improve Postgenomic Knowledge Discovery Using Imputation

While microarrays make it feasible to rapidly investigate many complex biological problems, their multistep fabrication has the proclivity for error at every stage. The standard tactic has been to either ignore or regard erroneous gene readings as missing values, though this assumption can exert a major influence upon postgenomic knowledge discovery methods like gene selection and gene regulatory network (GRN) reconstruction. This has been the catalyst for a raft of new flexible imputation algorithms including local least square impute and the recent heuristic collateral missing value imputation, which exploit the biological transactional behaviour of functionally correlated genes to afford accurate missing value estimation. This paper examines the influence of missing value imputation techniques upon postgenomic knowledge inference methods with results for various algorithms consistently corroborating that instead of ignoring missing values, recycling microarray data by flexible and robust imputation can provide substantial performance benefits for subsequent downstream procedures.     

Non-targeted UHPLC-MS metabolomic data processing methods: a comparative investigation of normalisation,missing value imputation,transformation and scaling

Introduction

The generic metabolomics data processing workflow is constructed with a serial set of processes including peak picking, quality assurance, normalisation, missing value imputation, transformation and scaling. The combination of these processes should present the experimental data in an appropriate structure so to identify the biological changes in a valid and robust manner.

Objectives

Currently, different researchers apply different data processing methods and no assessment of the permutations applied to UHPLC-MS datasets has been published. Here we wish to define the most appropriate data processing workflow.

Methods

We assess the influence of normalisation, missing value imputation, transformation and scaling methods on univariate and multivariate analysis of UHPLC-MS datasets acquired for different mammalian samples.

Results

Our studies have shown that once data are filtered, missing values are not correlated with m/z, retention time or response. Following an exhaustive evaluation, we recommend PQN normalisation with no missing value imputation and no transformation or scaling for univariate analysis. For PCA we recommend applying PQN normalisation with Random Forest missing value imputation, glog transformation and no scaling method. For PLS-DA we recommend PQN normalisation, KNN as the missing value imputation method, generalised logarithm transformation and no scaling. These recommendations are based on searching for the biologically important metabolite features independent of their measured abundance.

Conclusion

The appropriate choice of normalisation, missing value imputation, transformation and scaling methods differs depending on the data analysis method and the choice of method is essential to maximise the biological derivations from UHPLC-MS datasets.

     

Infinite hidden Markov models for multiple multivariate time series with missing data

Exposure to air pollution is associated with increased morbidity and mortality. Recent technological advancements permit the collection of time-resolved personal exposure data. Such data are often incomplete with missing observations and exposures below the limit of detection, which limit their use in health effects studies. In this paper, we develop an infinite hidden Markov model for multiple asynchronous multivariate time series with missing data. Our model is designed to include covariates that can inform transitions among hidden states. We implement beam sampling, a combination of slice sampling and dynamic programming, to sample the hidden states, and a Bayesian multiple imputation algorithm to impute missing data. In simulation studies, our model excels in estimating hidden states and state-specific means and imputing observations that are missing at random or below the limit of detection. We validate our imputation approach on data from the Fort Collins Commuter Study. We show that the estimated hidden states improve imputations for data that are missing at random compared to existing approaches. In a case study of the Fort Collins Commuter Study, we describe the inferential gains obtained from our model including improved imputation of missing data and the ability to identify shared patterns in activity and exposure among repeated sampling days for individuals and among distinct individuals.     

Missing value imputation for microRNA expression data by using a GO-based similarity measure

Missing values are commonly present in microarray data profiles. Instead of discarding genes or samples with incomplete expression level, missing values need to be properly imputed for accurate data analysis. The imputation methods can be roughly categorized as expression level-based and domain knowledge-based. The first type of methods only rely on expression data without the help of external data sources, while the second type incorporates available domain knowledge into expression data to improve imputation accuracy. In recent years, microRNA (miRNA) microarray has been largely developed and used for identifying miRNA biomarkers in complex human disease studies. Similar to mRNA profiles, miRNA expression profiles with missing values can be treated with the existing imputation methods. However, the domain knowledge-based methods are hard to be applied due to the lack of direct functional annotation for miRNAs. With the rapid accumulation of miRNA microarray data, it is increasingly needed to develop domain knowledge-based imputation algorithms specific to miRNA expression profiles to improve the quality of miRNA data analysis. We connect miRNAs with domain knowledge of Gene Ontology (GO) via their target genes, and define miRNA functional similarity based on the semantic similarity of GO terms in GO graphs. A new measure combining miRNA functional similarity and expression similarity is used in the imputation of missing values. The new measure is tested on two miRNA microarray datasets from breast cancer research and achieves improved performance compared with the expression-based method on both datasets. The experimental results demonstrate that the biological domain knowledge can benefit the estimation of missing values in miRNA profiles as well as mRNA profiles. Especially, functional similarity defined by GO terms annotated for the target genes of miRNAs can be useful complementary information for the expression-based method to improve the imputation accuracy of miRNA array data. Our method and data are available to the public upon request.     

Multiply imputing missing values arising by design in transplant survival data

In this article, we address a missing data problem that occurs in transplant survival studies. Recipients of organ transplants are followed up from transplantation and their survival times recorded, together with various explanatory variables. Due to differences in data collection procedures in different centers or over time, a particular explanatory variable (or set of variables) may only be recorded for certain recipients, which results in this variable being missing for a substantial number of records in the data. The variable may also turn out to be an important predictor of survival and so it is important to handle this missing-by-design problem appropriately. Consensus in the literature is to handle this problem with complete case analysis, as the missing data are assumed to arise under an appropriate missing at random mechanism that gives consistent estimates here. Specifically, the missing values can reasonably be assumed not to be related to the survival time. In this article, we investigate the potential for multiple imputation to handle this problem in a relevant study on survival after kidney transplantation, and show that it comprehensively outperforms complete case analysis on a range of measures. This is a particularly important finding in the medical context as imputing large amounts of missing data is often viewed with scepticism.     

A Bayesian network approach incorporating imputation of missing data enables exploratory analysis of complex causal biological relationships

Bayesian networks can be used to identify possible causal relationships between variables based on their conditional dependencies and independencies, which can be particularly useful in complex biological scenarios with many measured variables. Here we propose two improvements to an existing method for Bayesian network analysis, designed to increase the power to detect potential causal relationships between variables (including potentially a mixture of both discrete and continuous variables). Our first improvement relates to the treatment of missing data. When there is missing data, the standard approach is to remove every individual with any missing data before performing analysis. This can be wasteful and undesirable when there are many individuals with missing data, perhaps with only one or a few variables missing. This motivates the use of imputation. We present a new imputation method that uses a version of nearest neighbour imputation, whereby missing data from one individual is replaced with data from another individual, their nearest neighbour. For each individual with missing data, the subsets of variables to be used to select the nearest neighbour are chosen by sampling without replacement the complete data and estimating a best fit Bayesian network. We show that this approach leads to marked improvements in the recall and precision of directed edges in the final network identified, and we illustrate the approach through application to data from a recent study investigating the causal relationship between methylation and gene expression in early inflammatory arthritis patients. We also describe a second improvement in the form of a pseudo-Bayesian approach for upweighting certain network edges, which can be useful when there is prior evidence concerning their directions.     

Dealing with missing data in family-based association studies: a multiple imputation approach

To test for association between a disease and a set of linked markers, or to estimate relative risks of disease, several different methods have been developed. Many methods for family data require that individuals be genotyped at the full set of markers and that phase can be reconstructed. Individuals with missing data are excluded from the analysis. This can result in an important decrease in sample size and a loss of information. A possible solution to this problem is to use missing-data likelihood methods. We propose an alternative approach, namely the use of multiple imputation. Briefly, this method consists in estimating from the available data all possible phased genotypes and their respective posterior probabilities. These posterior probabilities are then used to generate replicate imputed data sets via a data augmentation algorithm. We performed simulations to test the efficiency of this approach for case/parent trio data and we found that the multiple imputation procedure generally gave unbiased parameter estimates with correct type 1 error and confidence interval coverage. Multiple imputation had some advantages over missing data likelihood methods with regards to ease of use and model flexibility. Multiple imputation methods represent promising tools in the search for disease susceptibility variants.     

Gaussian mixture clustering and imputation of microarray data

MOTIVATION: In microarray experiments, missing entries arise from blemishes on the chips. In large-scale studies, virtually every chip contains some missing entries and more than 90% of the genes are affected. Many analysis methods require a full set of data. Either those genes with missing entries are excluded, or the missing entries are filled with estimates prior to the analyses. This study compares methods of missing value estimation. RESULTS: Two evaluation metrics of imputation accuracy are employed. First, the root mean squared error measures the difference between the true values and the imputed values. Second, the number of mis-clustered genes measures the difference between clustering with true values and that with imputed values; it examines the bias introduced by imputation to clustering. The Gaussian mixture clustering with model averaging imputation is superior to all other imputation methods, according to both evaluation metrics, on both time-series (correlated) and non-time series (uncorrelated) data sets.