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     

Multiple imputation for multivariate data with missing and below-threshold measurements: time-series concentrations of pollutants in the Arctic

Many chemical and environmental data sets are complicated by the existence of fully missing values or censored values known to lie below detection thresholds. For example, week-long samples of airborne particulate matter were obtained at Alert, NWT, Canada, between 1980 and 1991, where some of the concentrations of 24 particulate constituents were coarsened in the sense of being either fully missing or below detection limits. To facilitate scientific analysis, it is appealing to create complete data by filling in missing values so that standard complete-data methods can be applied. We briefly review commonly used strategies for handling missing values and focus on the multiple-imputation approach, which generally leads to valid inferences when faced with missing data. Three statistical models are developed for multiply imputing the missing values of airborne particulate matter. We expect that these models are useful for creating multiple imputations in a variety of incomplete multivariate time series data sets.     

Missing value estimation methods for DNA microarrays

MOTIVATION: Gene expression microarray experiments can generate data sets with multiple missing expression values. Unfortunately, many algorithms for gene expression analysis require a complete matrix of gene array values as input. For example, methods such as hierarchical clustering and K-means clustering are not robust to missing data, and may lose effectiveness even with a few missing values. Methods for imputing missing data are needed, therefore, to minimize the effect of incomplete data sets on analyses, and to increase the range of data sets to which these algorithms can be applied. In this report, we investigate automated methods for estimating missing data. RESULTS: We present a comparative study of several methods for the estimation of missing values in gene microarray data. We implemented and evaluated three methods: a Singular Value Decomposition (SVD) based method (SVDimpute), weighted K-nearest neighbors (KNNimpute), and row average. We evaluated the methods using a variety of parameter settings and over different real data sets, and assessed the robustness of the imputation methods to the amount of missing data over the range of 1--20% missing values. We show that KNNimpute appears to provide a more robust and sensitive method for missing value estimation than SVDimpute, and both SVDimpute and KNNimpute surpass the commonly used row average method (as well as filling missing values with zeros). We report results of the comparative experiments and provide recommendations and tools for accurate estimation of missing microarray data under a variety of conditions.     

LSimpute: accurate estimation of missing values in microarray data with least squares methods

Microarray experiments generate data sets with information on the expression levels of thousands of genes in a set of biological samples. Unfortunately, such experiments often produce multiple missing expression values, normally due to various experimental problems. As many algorithms for gene expression analysis require a complete data matrix as input, the missing values have to be estimated in order to analyze the available data. Alternatively, genes and arrays can be removed until no missing values remain. However, for genes or arrays with only a small number of missing values, it is desirable to impute those values. For the subsequent analysis to be as informative as possible, it is essential that the estimates for the missing gene expression values are accurate. A small amount of badly estimated missing values in the data might be enough for clustering methods, such as hierachical clustering or K-means clustering, to produce misleading results. Thus, accurate methods for missing value estimation are needed. We present novel methods for estimation of missing values in microarray data sets that are based on the least squares principle, and that utilize correlations between both genes and arrays. For this set of methods, we use the common reference name LSimpute. We compare the estimation accuracy of our methods with the widely used KNNimpute on three complete data matrices from public data sets by randomly knocking out data (labeling as missing). From these tests, we conclude that our LSimpute methods produce estimates that consistently are more accurate than those obtained using KNNimpute. Additionally, we examine a more classic approach to missing value estimation based on expectation maximization (EM). We refer to our EM implementations as EMimpute, and the estimate errors using the EMimpute methods are compared with those our novel methods produce. The results indicate that on average, the estimates from our best performing LSimpute method are at least as accurate as those from the best EMimpute algorithm.     

Non-Iterative Least Squares Estimation of Missing Values in Cross-Over Designs Without Residual Effect

In the present study an attempt has been made to esttimate several missing values in cross-over designs. When the observations are missing in a particular pattern, explicit expressions are given for the estimators of the missing values. This procedure is illustrated with the help of a numerical example.     

Analysis of multilocus fingerprinting data sets containing missing data

Missing data are commonly encountered using multilocus, fragment‐based (dominant) fingerprinting methods, such as random amplified polymorphic DNA (RAPD) or amplified fragment length polymorphism (AFLP). Data sets containing missing data have been analysed by eliminating those bands or samples with missing data, assigning values to missing data or ignoring the problem. Here, we present a method that uses random assignments of band presence–absence to the missing data, implemented by the computer program famd (available from http://homepage.univie.ac.at/philipp.maria.schlueter/famd.html ), for analyses based on pairwise similarity and Shannon's index. When missing values group in a data set, sample or band elimination is likely to be the most appropriate action. However, when missing values are scattered across the data set, minimum, maximum and average similarity coefficients are a simple means of visualizing the effects of missing data on tree structure. Our approach indicates the range of values that a data set containing missing data points might generate, and forces the investigator to consider the effects of missing values on data interpretation.     

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.     

How many spots with missing values can be tolerated in quantitative two-dimensional gel electrophoresis when applying univariate statistics?

Quantitative proteomic comparisons require a sufficient number of samples to reach an acceptable level of significance. But 2D gel electrophoresis commonly results in incomplete data sets due to spots with missing values reducing thereby the number of parallel measurements for individual proteins. Here we investigated how many missing values per spot can be tolerated. The number of spots in common between all gels was found to decrease with the number of parallel gels in a non-linear fashion. Increasing numbers of missing values were associated with a moderate increase in the quantitative variation of spot volumes. Based on the missing value pattern in 20 gels we performed an analysis of the multiple testing power for the hypothetical scenario of a comparative 2DE study with six or twelve parallel gels. The calculation considered the statistical power of the individual spot as well as the number of spots included in the analysis. The power increased with inclusion of spots with higher number of missing values and showed an optimum at a specific minimum number of spot replicates. The results suggest that proteins with missing values can be included in a univariate analysis as long as a sufficient number of parallel gels are made.     

NS-kNN: a modified <Emphasis Type="Italic">k</Emphasis>-nearest neighbors approach for imputing metabolomics data

Introduction

A common problem in metabolomics data analysis is the existence of a substantial number of missing values, which can complicate, bias, or even prevent certain downstream analyses. One of the most widely-used solutions to this problem is imputation of missing values using a k-nearest neighbors (kNN) algorithm to estimate missing metabolite abundances. kNN implicitly assumes that missing values are uniformly distributed at random in the dataset, but this is typically not true in metabolomics, where many values are missing because they are below the limit of detection of the analytical instrumentation.

Objectives

Here, we explore the impact of nonuniformly distributed missing values (missing not at random, or MNAR) on imputation performance. We present a new model for generating synthetic missing data and a new algorithm, No-Skip kNN (NS-kNN), that accounts for MNAR values to provide more accurate imputations.

Methods

We compare the imputation errors of the original kNN algorithm using two distance metrics, NS-kNN, and a recently developed algorithm KNN-TN, when applied to multiple experimental datasets with different types and levels of missing data.

Results

Our results show that NS-kNN typically outperforms kNN when at least 20–30% of missing values in a dataset are MNAR. NS-kNN also has lower imputation errors than KNN-TN on realistic datasets when at least 50% of missing values are MNAR.

Conclusion

Accounting for the nonuniform distribution of missing values in metabolomics data can significantly improve the results of imputation algorithms. The NS-kNN method imputes missing metabolomics data more accurately than existing kNN-based approaches when used on realistic datasets.

     

A Bayesian missing value estimation method for gene expression profile data

MOTIVATION: Gene expression profile analyses have been used in numerous studies covering a broad range of areas in biology. When unreliable measurements are excluded, missing values are introduced in gene expression profiles. Although existing multivariate analysis methods have difficulty with the treatment of missing values, this problem has received little attention. There are many options for dealing with missing values, each of which reaches drastically different results. Ignoring missing values is the simplest method and is frequently applied. This approach, however, has its flaws. In this article, we propose an estimation method for missing values, which is based on Bayesian principal component analysis (BPCA). Although the methodology that a probabilistic model and latent variables are estimated simultaneously within the framework of Bayes inference is not new in principle, actual BPCA implementation that makes it possible to estimate arbitrary missing variables is new in terms of statistical methodology. RESULTS: When applied to DNA microarray data from various experimental conditions, the BPCA method exhibited markedly better estimation ability than other recently proposed methods, such as singular value decomposition and K-nearest neighbors. While the estimation performance of existing methods depends on model parameters whose determination is difficult, our BPCA method is free from this difficulty. Accordingly, the BPCA method provides accurate and convenient estimation for missing values. AVAILABILITY: The software is available at http://hawaii.aist-nara.ac.jp/~shige-o/tools/.     

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.     

A Note on Estimation of Several Missing Values in Two Way Classified Data with Many Observations Per Cell

In this paper an attempt has been made to reduce the computational complexities involved in estimation of several missing values. As a result it has been shown that one can estimate m missing values by developing only k (≤m) linear equations, where m and k are respectively the number of missing values and missing cells. The procedure is also illustrated with the help of a numerical example.     

Non-Iterative Least Squares Estimation of Missing Values in Hyper-Graeco-Latin Square Designs

In this paper an attempt has been made to obtain explicit expressions for the estimators of the several missing values in hyper-graeco-latin square designs. Further it has been shown that the estimates of the missing values in latin square designs and graeco-latin square designs are obtained as a particular case of the estimates of the missing values in hyper-graeco-latin square designs.     

Non-Iterative Least Squares Estimation of Missing Values in Replicated Latin Square Designs

In this paper an attempt has been made to estimate several missing values in replicated latin square designs. The explicit computable expressions for the non-iterative least squares estimates of the missing values are presented for particular patterns of missing values.     

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.     

Treatment of missing values for multivariate statistical analysis of gel-based proteomics data

The presence of missing values in gel-based proteomics data represents a real challenge if an objective statistical analysis is pursued. Different methods to handle missing values were evaluated and their influence is discussed on the selection of important proteins through multivariate techniques. The evaluated methods consisted of directly dealing with them during the multivariate analysis with the nonlinear estimation by iterative partial least squares (NIPALS) algorithm or imputing them by using either k-nearest neighbor or Bayesian principal component analysis (BPCA) before carrying out the multivariate analysis. These techniques were applied to data obtained from gels stained with classical postrunning dyes and from DIGE gels. Before applying the multivariate techniques, the normality and homoscedasticity assumptions on which parametric tests are based on were tested in order to perform a sound statistical analysis. From the three tested methods to handle missing values in our datasets, BPCA imputation of missing values showed to be the most consistent method.     

Treatment of nonignorable missing data when modeling unobserved heterogeneity with finite mixture models

Multiple imputation has become a widely accepted technique to deal with the problem of incomplete data. Typically, imputation of missing values and the statistical analysis are performed separately. Therefore, the imputation model has to be consistent with the analysis model. If the data are analyzed with a mixture model, the parameter estimates are usually obtained iteratively. Thus, if the data are missing not at random, parameter estimation and treatment of missingness should be combined. We solve both problems by simultaneously imputing values using the data augmentation method and estimating parameters using the EM algorithm. This iterative procedure ensures that the missing values are properly imputed given the current parameter estimates. Properties of the parameter estimates were investigated in a simulation study. The results are illustrated using data from the National Health and Nutrition Examination Survey.     

Robust Estimation of Area Under ROC Curve Using Auxiliary Variables in the Presence of Missing Biomarker Values

Summary In medical research, the receiver operating characteristic (ROC) curves can be used to evaluate the performance of biomarkers for diagnosing diseases or predicting the risk of developing a disease in the future. The area under the ROC curve (ROC AUC), as a summary measure of ROC curves, is widely utilized, especially when comparing multiple ROC curves. In observational studies, the estimation of the AUC is often complicated by the presence of missing biomarker values, which means that the existing estimators of the AUC are potentially biased. In this article, we develop robust statistical methods for estimating the ROC AUC and the proposed methods use information from auxiliary variables that are potentially predictive of the missingness of the biomarkers or the missing biomarker values. We are particularly interested in auxiliary variables that are predictive of the missing biomarker values. In the case of missing at random (MAR), that is, missingness of biomarker values only depends on the observed data, our estimators have the attractive feature of being consistent if one correctly specifies, conditional on auxiliary variables and disease status, either the model for the probabilities of being missing or the model for the biomarker values. In the case of missing not at random (MNAR), that is, missingness may depend on the unobserved biomarker values, we propose a sensitivity analysis to assess the impact of MNAR on the estimation of the ROC AUC. The asymptotic properties of the proposed estimators are studied and their finite‐sample behaviors are evaluated in simulation studies. The methods are further illustrated using data from a study of maternal depression during pregnancy.     

Doubly robust estimates for binary longitudinal data analysis with missing response and missing covariates

Longitudinal studies often feature incomplete response and covariate data. Likelihood-based methods such as the expectation-maximization algorithm give consistent estimators for model parameters when data are missing at random (MAR) provided that the response model and the missing covariate model are correctly specified; however, we do not need to specify the missing data mechanism. An alternative method is the weighted estimating equation, which gives consistent estimators if the missing data and response models are correctly specified; however, we do not need to specify the distribution of the covariates that have missing values. In this article, we develop a doubly robust estimation method for longitudinal data with missing response and missing covariate when data are MAR. This method is appealing in that it can provide consistent estimators if either the missing data model or the missing covariate model is correctly specified. Simulation studies demonstrate that this method performs well in a variety of situations.     

A probabilistic treatment of the missing spot problem in 2D gel electrophoresis experiments

Two-dimensional SDS-PAGE gel electrophoresis using post-run staining is widely used to measure the abundances of thousands of protein spots simultaneously. Usually, the protein abundances of two or more biological groups are compared using biological and technical replicates. After gel separation and staining, the spots are detected, spot volumes are quantified, and spots are matched across gels. There are almost always many missing values in the resulting data set. The missing values arise either because the corresponding proteins have very low abundances (or are absent) or because of experimental errors such as incomplete/over focusing in the first dimension or varying run times in the second dimension as well as faulty spot detection and matching. In this study, we show that the probability for a spot to be missing can be modeled by a logistic regression function of the logarithm of the volume. Furthermore, we present an algorithm that takes a set of gels with technical and biological replicates as input and estimates the average protein abundances in the biological groups from the number of missing spots and measured volumes of the present spots using a maximum likelihood approach. Confidence intervals for abundances and p-values for differential expression between two groups are calculated using bootstrap sampling. The algorithm is compared to two standard approaches, one that discards missing values and one that sets all missing values to zero. We have evaluated this approach in two different gel data sets of different biological origin. An R-program, implementing the algorithm, is freely available at http://bioinfo.thep .lu.se/MissingValues2Dgels.html.