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.     

Improving cluster-based missing value estimation of DNA microarray data

We present a modification of the weighted K-nearest neighbours imputation method (KNNimpute) for missing values (MVs) estimation in microarray data based on the reuse of estimated data. The method was called iterative KNN imputation (IKNNimpute) as the estimation is performed iteratively using the recently estimated values. The estimation efficiency of IKNNimpute was assessed under different conditions (data type, fraction and structure of missing data) by the normalized root mean squared error (NRMSE) and the correlation coefficients between estimated and true values, and compared with that of other cluster-based estimation methods (KNNimpute and sequential KNN). We further investigated the influence of imputation on the detection of differentially expressed genes using SAM by examining the differentially expressed genes that are lost after MV estimation. The performance measures give consistent results, indicating that the iterative procedure of IKNNimpute can enhance the prediction ability of cluster-based methods in the presence of high missing rates, in non-time series experiments and in data sets comprising both time series and non-time series data, because the information of the genes having MVs is used more efficiently and the iterative procedure allows refining the MV estimates. More importantly, IKNN has a smaller detrimental effect on the detection of differentially expressed genes.     

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.     

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.     

Evaluation of missing data imputation methods for human osteometric measurements

It is not uncommon for biological anthropologists to analyze incomplete bioarcheological or forensic skeleton specimens. As many quantitative multivariate analyses cannot handle incomplete data, missing data imputation or estimation is a common preprocessing practice for such data. Using William W. Howells' Craniometric Data Set and the Goldman Osteometric Data Set, we evaluated the performance of multiple popular statistical methods for imputing missing metric measurements. Results indicated that multiple imputation methods outperformed single imputation methods, such as Bayesian principal component analysis (BPCA). Multiple imputation with Bayesian linear regression implemented in the R package norm2, the Expectation–Maximization (EM) with Bootstrapping algorithm implemented in Amelia, and the Predictive Mean Matching (PMM) method and several of the derivative linear regression models implemented in mice, perform well regarding accuracy, robustness, and speed. Based on the findings of this study, we suggest a practical procedure for choosing appropriate imputation methods.     

Multiple imputation methods for handling incomplete longitudinal and clustered data where the target analysis is a linear mixed effects model

Multiple imputation (MI) is increasingly popular for handling multivariate missing data. Two general approaches are available in standard computer packages: MI based on the posterior distribution of incomplete variables under a multivariate (joint) model, and fully conditional specification (FCS), which imputes missing values using univariate conditional distributions for each incomplete variable given all the others, cycling iteratively through the univariate imputation models. In the context of longitudinal or clustered data, it is not clear whether these approaches result in consistent estimates of regression coefficient and variance component parameters when the analysis model of interest is a linear mixed effects model (LMM) that includes both random intercepts and slopes with either covariates or both covariates and outcome contain missing information. In the current paper, we compared the performance of seven different MI methods for handling missing values in longitudinal and clustered data in the context of fitting LMMs with both random intercepts and slopes. We study the theoretical compatibility between specific imputation models fitted under each of these approaches and the LMM, and also conduct simulation studies in both the longitudinal and clustered data settings. Simulations were motivated by analyses of the association between body mass index (BMI) and quality of life (QoL) in the Longitudinal Study of Australian Children (LSAC). Our findings showed that the relative performance of MI methods vary according to whether the incomplete covariate has fixed or random effects and whether there is missingnesss in the outcome variable. We showed that compatible imputation and analysis models resulted in consistent estimation of both regression parameters and variance components via simulation. We illustrate our findings with the analysis of LSAC data.     

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.     

A multiple kernel density clustering algorithm for incomplete datasets in bioinformatics

Background

While there are a large number of bioinformatics datasets for clustering, many of them are incomplete, i.e., missing attribute values in some data samples needed by clustering algorithms. A variety of clustering algorithms have been proposed in the past years, but they usually are limited to cluster on the complete dataset. Besides, conventional clustering algorithms cannot obtain a trade-off between accuracy and efficiency of the clustering process since many essential parameters are determined by the human user’s experience.

Results

The paper proposes a Multiple Kernel Density Clustering algorithm for Incomplete datasets called MKDCI. The MKDCI algorithm consists of recovering missing attribute values of input data samples, learning an optimally combined kernel for clustering the input dataset, reducing dimensionality with the optimal kernel based on multiple basis kernels, detecting cluster centroids with the Isolation Forests method, assigning clusters with arbitrary shape and visualizing the results.

Conclusions

Extensive experiments on several well-known clustering datasets in bioinformatics field demonstrate the effectiveness of the proposed MKDCI algorithm. Compared with existing density clustering algorithms and parameter-free clustering algorithms, the proposed MKDCI algorithm tends to automatically produce clusters of better quality on the incomplete dataset in bioinformatics.

     

Quantifying the impact of fixed effects modeling of clusters in multiple imputation for cluster randomized trials

In cluster randomized trials (CRTs), identifiable clusters rather than individuals are randomized to study groups. Resulting data often consist of a small number of clusters with correlated observations within a treatment group. Missing data often present a problem in the analysis of such trials, and multiple imputation (MI) has been used to create complete data sets, enabling subsequent analysis with well-established analysis methods for CRTs. We discuss strategies for accounting for clustering when multiply imputing a missing continuous outcome, focusing on estimation of the variance of group means as used in an adjusted t-test or ANOVA. These analysis procedures are congenial to (can be derived from) a mixed effects imputation model; however, this imputation procedure is not yet available in commercial statistical software. An alternative approach that is readily available and has been used in recent studies is to include fixed effects for cluster, but the impact of using this convenient method has not been studied. We show that under this imputation model the MI variance estimator is positively biased and that smaller intraclass correlations (ICCs) lead to larger overestimation of the MI variance. Analytical expressions for the bias of the variance estimator are derived in the case of data missing completely at random, and cases in which data are missing at random are illustrated through simulation. Finally, various imputation methods are applied to data from the Detroit Middle School Asthma Project, a recent school-based CRT, and differences in inference are compared.     

Clustering microarray gene expression data using weighted Chinese restaurant process

MOTIVATION: Clustering microarray gene expression data is a powerful tool for elucidating co-regulatory relationships among genes. Many different clustering techniques have been successfully applied and the results are promising. However, substantial fluctuation contained in microarray data, lack of knowledge on the number of clusters and complex regulatory mechanisms underlying biological systems make the clustering problems tremendously challenging. RESULTS: We devised an improved model-based Bayesian approach to cluster microarray gene expression data. Cluster assignment is carried out by an iterative weighted Chinese restaurant seating scheme such that the optimal number of clusters can be determined simultaneously with cluster assignment. The predictive updating technique was applied to improve the efficiency of the Gibbs sampler. An additional step is added during reassignment to allow genes that display complex correlation relationships such as time-shifted and/or inverted to be clustered together. Analysis done on a real dataset showed that as much as 30% of significant genes clustered in the same group display complex relationships with the consensus pattern of the cluster. Other notable features including automatic handling of missing data, quantitative measures of cluster strength and assignment confidence. Synthetic and real microarray gene expression datasets were analyzed to demonstrate its performance. AVAILABILITY: A computer program named Chinese restaurant cluster (CRC) has been developed based on this algorithm. The program can be downloaded at http://www.sph.umich.edu/csg/qin/CRC/.     

A multi-stage approach to clustering and imputation of gene expression profiles

MOTIVATION: Microarray experiments have revolutionized the study of gene expression with their ability to generate large amounts of data. This article describes an alternative to existing approaches to clustering of gene expression profiles; the key idea is to cluster in stages using a hierarchy of distance measures. This method is motivated by the way in which the human mind sorts and so groups many items. The distance measures arise from the orthogonal breakup of Euclidean distance, giving us a set of independent measures of different attributes of the gene expression profile. Interpretation of these distances is closely related to the statistical design of the microarray experiment. This clustering method not only accommodates missing data but also leads to an associated imputation method. RESULTS: The performance of the clustering and imputation methods was tested on a simulated dataset, a yeast cell cycle dataset and a central nervous system development dataset. Based on the Rand and adjusted Rand indices, the clustering method is more consistent with the biological classification of the data than commonly used clustering methods. The imputation method, at varying levels of missingness, outperforms most imputation methods, based on root mean squared error (RMSE). AVAILABILITY: Code in R is available on request from the authors.     

A generalized multivariate approach to pattern discovery from replicated and incomplete genome-wide measurements

Estimation of pairwise correlation from incomplete and replicated molecular profiling data is an ubiquitous problem in pattern discovery analysis, such as clustering and networking. However, existing methods solve this problem by ad hoc data imputation, followed by aveGation coefficient type approaches, which might annihilate important patterns present in the molecular profiling data. Moreover, these approaches do not consider and exploit the underlying experimental design information that specifies the replication mechanisms. We develop an Expectation-Maximization (EM) type algorithm to estimate the correlation structure using incomplete and replicated molecular profiling data with a priori known replication mechanism. The approach is sufficiently generalized to be applicable to any known replication mechanism. In case of unknown replication mechanism, it is reduced to the parsimonious model introduced previously. The efficacy of our approach was first evaluated by comprehensively comparing various bivariate and multivariate imputation approaches using simulation studies. Results from real-world data analysis further confirmed the superior performance of the proposed approach to the commonly used approaches, where we assessed the robustness of the method using data sets with up to 30 percent missing values.     

A robust approach based on Weibull distribution for clustering gene expression data

Background

Clustering is a widely used technique for analysis of gene expression data. Most clustering methods group genes based on the distances, while few methods group genes according to the similarities of the distributions of the gene expression levels. Furthermore, as the biological annotation resources accumulated, an increasing number of genes have been annotated into functional categories. As a result, evaluating the performance of clustering methods in terms of the functional consistency of the resulting clusters is of great interest.

Results

In this paper, we proposed the WDCM (Weibull Distribution-based Clustering Method), a robust approach for clustering gene expression data, in which the gene expressions of individual genes are considered as the random variables following unique Weibull distributions. Our WDCM is based on the concept that the genes with similar expression profiles have similar distribution parameters, and thus the genes are clustered via the Weibull distribution parameters. We used the WDCM to cluster three cancer gene expression data sets from the lung cancer, B-cell follicular lymphoma and bladder carcinoma and obtained well-clustered results. We compared the performance of WDCM with k-means and Self Organizing Map (SOM) using functional annotation information given by the Gene Ontology (GO). The results showed that the functional annotation ratios of WDCM are higher than those of the other methods. We also utilized the external measure Adjusted Rand Index to validate the performance of the WDCM. The comparative results demonstrate that the WDCM provides the better clustering performance compared to k-means and SOM algorithms. The merit of the proposed WDCM is that it can be applied to cluster incomplete gene expression data without imputing the missing values. Moreover, the robustness of WDCM is also evaluated on the incomplete data sets.

Conclusions

The results demonstrate that our WDCM produces clusters with more consistent functional annotations than the other methods. The WDCM is also verified to be robust and is capable of clustering gene expression data containing a small quantity of missing values.     

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.     

Comprehensive cluster analysis with Transitivity Clustering

Transitivity Clustering is a method for the partitioning of biological data into groups of similar objects, such as genes, for instance. It provides integrated access to various functions addressing each step of a typical cluster analysis. To facilitate this, Transitivity Clustering is accessible online and offers three user-friendly interfaces: a powerful stand-alone version, a web interface, and a collection of Cytoscape plug-ins. In this paper, we describe three major workflows: (i) protein (super)family detection with Cytoscape, (ii) protein homology detection with incomplete gold standards and (iii) clustering of gene expression data. This protocol guides the user through the most important features of Transitivity Clustering and takes ～1 h to complete.     

Statistical Inference for Hardy-Weinberg Proportions in the Presence of Missing Genotype Information

In genetic association studies, tests for Hardy-Weinberg proportions are often employed as a quality control checking procedure. Missing genotypes are typically discarded prior to testing. In this paper we show that inference for Hardy-Weinberg proportions can be biased when missing values are discarded. We propose to use multiple imputation of missing values in order to improve inference for Hardy-Weinberg proportions. For imputation we employ a multinomial logit model that uses information from allele intensities and/or neighbouring markers. Analysis of an empirical data set of single nucleotide polymorphisms possibly related to colon cancer reveals that missing genotypes are not missing completely at random. Deviation from Hardy-Weinberg proportions is mostly due to a lack of heterozygotes. Inbreeding coefficients estimated by multiple imputation of the missings are typically lowered with respect to inbreeding coefficients estimated by discarding the missings. Accounting for missings by multiple imputation qualitatively changed the results of 10 to 17% of the statistical tests performed. Estimates of inbreeding coefficients obtained by multiple imputation showed high correlation with estimates obtained by single imputation using an external reference panel. Our conclusion is that imputation of missing data leads to improved statistical inference for Hardy-Weinberg proportions.     

Latent class regression: Inference and estimation with two‐stage multiple imputation

Latent class regression (LCR) is a popular method for analyzing multiple categorical outcomes. While nonresponse to the manifest items is a common complication, inferences of LCR can be evaluated using maximum likelihood, multiple imputation, and two‐stage multiple imputation. Under similar missing data assumptions, the estimates and variances from all three procedures are quite close. However, multiple imputation and two‐stage multiple imputation can provide additional information: estimates for the rates of missing information. The methodology is illustrated using an example from a study on racial and ethnic disparities in breast cancer severity.     

On the statistical analysis of the GS-NS0 cell proteome: imputation, clustering and variability testing

We have undertaken two-dimensional gel electrophoresis proteomic profiling on a series of cell lines with different recombinant antibody production rates. Due to the nature of gel-based experiments not all protein spots are detected across all samples in an experiment, and hence datasets are invariably incomplete. New approaches are therefore required for the analysis of such graduated datasets. We approached this problem in two ways. Firstly, we applied a missing value imputation technique to calculate missing data points. Secondly, we combined a singular value decomposition based hierarchical clustering with the expression variability test to identify protein spots whose expression correlates with increased antibody production. The results have shown that while imputation of missing data was a useful method to improve the statistical analysis of such data sets, this was of limited use in differentiating between the samples investigated, and highlighted a small number of candidate proteins for further investigation.     

A nonparametric procedure for the two-factor mixed model with missing data

We develop a nonparametric imputation technique to test for the treatment effects in a nonparametric two-factor mixed model with incomplete data. Within each block, an arbitrary covariance structure of the repeated measurements is assumed without the explicit parametrization of the joint multivariate distribution. The number of repeated measurements is uniformly bounded whereas the number of blocks tends to infinity. The essential idea of the nonparametric imputation is to replace the unknown indicator functions of pairwise comparisons by the corresponding empirical distribution functions. The proposed nonparametric imputation method holds valid under the missing completely at random (MCAR) mechanism. We apply the nonparametric imputation on Brunner and Dette's method for the nonparametric two-factor mixed model and this extension results in a weighted partial rank transform statistic. Asymptotic relative efficiency of the nonparametric imputation method with the complete data versus the incomplete data is derived to quantify the efficiency loss due to the missing data. Monte Carlo simulation studies are conducted to demonstrate the validity and power of the proposed method in comparison with other existing methods. A migraine severity score data set is analyzed to demonstrate the application of the proposed method in the analysis of missing data.