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.     

Generalizing treatment effects with incomplete covariates: Identifying assumptions and multiple imputation algorithms

We focus on the problem of generalizing a causal effect estimated on a randomized controlled trial (RCT) to a target population described by a set of covariates from observational data. Available methods such as inverse propensity sampling weighting are not designed to handle missing values, which are however common in both data sources. In addition to coupling the assumptions for causal effect identifiability and for the missing values mechanism and to defining appropriate estimation strategies, one difficulty is to consider the specific structure of the data with two sources and treatment and outcome only available in the RCT. We propose three multiple imputation strategies to handle missing values when generalizing treatment effects, each handling the multisource structure of the problem differently (separate imputation, joint imputation with fixed effect, joint imputation ignoring source information). As an alternative to multiple imputation, we also propose a direct estimation approach that treats incomplete covariates as semidiscrete variables. The multiple imputation strategies and the latter alternative rely on different sets of assumptions concerning the impact of missing values on identifiability. We discuss these assumptions and assess the methods through an extensive simulation study. This work is motivated by the analysis of a large registry of over 20,000 major trauma patients and an RCT studying the effect of tranexamic acid administration on mortality in major trauma patients admitted to intensive care units. The analysis illustrates how the missing values handling can impact the conclusion about the effect generalized from the RCT to the target population.     

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.     

Smoothed empirical likelihood inference for ROC curve in the presence of missing biomarker values

This paper considers statistical inference for the receiver operating characteristic (ROC) curve in the presence of missing biomarker values by utilizing estimating equations (EEs) together with smoothed empirical likelihood (SEL). Three approaches are developed to estimate ROC curve and construct its SEL-based confidence intervals based on the kernel-assisted EE imputation, multiple imputation, and hybrid imputation combining the inverse probability weighted imputation and multiple imputation. Under some regularity conditions, we show asymptotic properties of the proposed maximum SEL estimators for ROC curve. Simulation studies are conducted to investigate the performance of the proposed SEL approaches. An example is illustrated by the proposed methodologies. Empirical results show that the hybrid imputation method behaves better than the kernel-assisted and multiple imputation methods, and the proposed three SEL methods outperform existing nonparametric method.     

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.     

Filling the gap in functional trait databases: use of ecological hypotheses to replace missing data

Functional trait databases are powerful tools in ecology, though most of them contain large amounts of missing values. The goal of this study was to test the effect of imputation methods on the evaluation of trait values at species level and on the subsequent calculation of functional diversity indices at community level using functional trait databases. Two simple imputation methods (average and median), two methods based on ecological hypotheses, and one multiple imputation method were tested using a large plant trait database, together with the influence of the percentage of missing data and differences between functional traits. At community level, the complete‐case approach and three functional diversity indices calculated from grassland plant communities were included. At the species level, one of the methods based on ecological hypothesis was for all traits more accurate than imputation with average or median values, but the multiple imputation method was superior for most of the traits. The method based on functional proximity between species was the best method for traits with an unbalanced distribution, while the method based on the existence of relationships between traits was the best for traits with a balanced distribution. The ranking of the grassland communities for their functional diversity indices was not robust with the complete‐case approach, even for low percentages of missing data. With the imputation methods based on ecological hypotheses, functional diversity indices could be computed with a maximum of 30% of missing data, without affecting the ranking between grassland communities. The multiple imputation method performed well, but not better than single imputation based on ecological hypothesis and adapted to the distribution of the trait values for the functional identity and range of the communities. Ecological studies using functional trait databases have to deal with missing data using imputation methods corresponding to their specific needs and making the most out of the information available in the databases. Within this framework, this study indicates the possibilities and limits of single imputation methods based on ecological hypothesis and concludes that they could be useful when studying the ranking of communities for their functional diversity indices.     

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.     

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.     

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.

     

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.     

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.     

Multiple imputation for missing values through conditional Semiparametric odds ratio models

Multiple imputation is a practically useful approach to handling incompletely observed data in statistical analysis. Parameter estimation and inference based on imputed full data have been made easy by Rubin's rule for result combination. However, creating proper imputation that accommodates flexible models for statistical analysis in practice can be very challenging. We propose an imputation framework that uses conditional semiparametric odds ratio models to impute the missing values. The proposed imputation framework is more flexible and robust than the imputation approach based on the normal model. It is a compatible framework in comparison to the approach based on fully conditionally specified models. The proposed algorithms for multiple imputation through the Markov chain Monte Carlo sampling approach can be straightforwardly carried out. Simulation studies demonstrate that the proposed approach performs better than existing, commonly used imputation approaches. The proposed approach is applied to imputing missing values in bone fracture data.     

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     

Towards clustering of incomplete microarray data without the use of imputation

MOTIVATION: Clustering technique is used to find groups of genes that show similar expression patterns under multiple experimental conditions. Nonetheless, the results obtained by cluster analysis are influenced by the existence of missing values that commonly arise in microarray experiments. Because a clustering method requires a complete data matrix as an input, previous studies have estimated the missing values using an imputation method in the preprocessing step of clustering. However, a common limitation of these conventional approaches is that once the estimates of missing values are fixed in the preprocessing step, they are not changed during subsequent processes of clustering; badly estimated missing values obtained in data preprocessing are likely to deteriorate the quality and reliability of clustering results. Thus, a new clustering method is required for improving missing values during iterative clustering process. RESULTS: We present a method for Clustering Incomplete data using Alternating Optimization (CIAO) in which a prior imputation method is not required. To reduce the influence of imputation in preprocessing, we take an alternative optimization approach to find better estimates during iterative clustering process. This method improves the estimates of missing values by exploiting the cluster information such as cluster centroids and all available non-missing values in each iteration. To test the performance of the CIAO, we applied the CIAO and conventional imputation-based clustering methods, e.g. k-means based on KNNimpute, for clustering two yeast incomplete data sets, and compared the clustering result of each method using the Saccharomyces Genome Database annotations. The clustering results of the CIAO method are more significantly relevant to the biological gene annotations than those of other methods, indicating its effectiveness and potential for clustering incomplete gene expression data. AVAILABILITY: The software was developed using Java language, and can be executed on the platforms that JVM (Java Virtual Machine) is running. It is available from the authors upon request.     

Imputation strategies for missing continuous outcomes in cluster randomized trials

In cluster randomized trials, intact social units such as schools, worksites or medical practices - rather than individuals themselves - are randomly allocated to intervention and control conditions, while the outcomes of interest are then observed on individuals within each cluster. Such trials are becoming increasingly common in the fields of health promotion and health services research. Attrition is a common occurrence in randomized trials, and a standard approach for dealing with the resulting missing values is imputation. We consider imputation strategies for missing continuous outcomes, focusing on trials with a completely randomized design in which fixed cohorts from each cluster are enrolled prior to random assignment. We compare five different imputation strategies with respect to Type I and Type II error rates of the adjusted two-sample t -test for the intervention effect. Cluster mean imputation is compared with multiple imputation, using either within-cluster data or data pooled across clusters in each intervention group. In the case of pooling across clusters, we distinguish between standard multiple imputation procedures which do not account for intracluster correlation and a specialized procedure which does account for intracluster correlation but is not yet available in standard statistical software packages. A simulation study is used to evaluate the influence of cluster size, number of clusters, degree of intracluster correlation, and variability among cluster follow-up rates. We show that cluster mean imputation yields valid inferences and given its simplicity, may be an attractive option in some large community intervention trials which are subject to individual-level attrition only; however, it may yield less powerful inferences than alternative procedures which pool across clusters especially when the cluster sizes are small and cluster follow-up rates are highly variable. When pooling across clusters, the imputation procedure should generally take intracluster correlation into account to obtain valid inferences; however, as long as the intracluster correlation coefficient is small, we show that standard multiple imputation procedures may yield acceptable type I error rates; moreover, these procedures may yield more powerful inferences than a specialized procedure, especially when the number of available clusters is small. Within-cluster multiple imputation is shown to be the least powerful among the procedures considered.     

Multiple imputation to minimise bias from missing stage information in estimates of early cancer diagnosis in England: a population-based study

IntroductionMonitoring early diagnosis is a priority of cancer policy in England. Information on stage has not always been available for a large proportion of patients, however, which may bias temporal comparisons. We previously estimated that early-stage diagnosis of colorectal cancer rose from 32% to 44% during 2008–2013, using multiple imputation. Here we examine the underlying assumptions of multiple imputation for missing stage using the same dataset.MethodsIndividually-linked cancer registration, Hospital Episode Statistics (HES), and audit data were examined. Six imputation models including different interaction terms, post-diagnosis treatment, and survival information were assessed, and comparisons drawn with the a priori optimal model. Models were further tested by setting stage values to missing for some patients under one plausible mechanism, then comparing actual and imputed stage distributions for these patients. Finally, a pattern-mixture sensitivity analysis was conducted.ResultsData from 196,511 colorectal patients were analysed, with 39.2% missing stage. Inclusion of survival time increased the accuracy of imputation: the odds ratio for change in early-stage diagnosis during 2008–2013 was 1.7 (95% CI: 1.6, 1.7) with survival to 1 year included, compared to 1.9 (95% CI 1.9–2.0) with no survival information. Imputation estimates of stage were accurate in one plausible simulation. Pattern-mixture analyses indicated our previous analysis conclusions would only change materially if stage were misclassified for 20% of the patients who had it categorised as late.InterpretationMultiple imputation models can substantially reduce bias from missing stage, but data on patient’s one-year survival should be included for highest accuracy.     

Research and implementation of animations evaluation system

A scientific and systematical evaluation of animations will be able to reflect the integrated strength of animations. Based on the popularity of animations in the market and the quality of animations, and following the principles of index selection, we propose an animations evaluation system (AES) by using multiple social research methods. We establish and analyze a mathematical model based on analytic hierarchy process and fuzzy comprehensive evaluation. We then treat the missing data of the market repercussion index and the professional evaluation index by using hot deck imputation and mean imputation. Finally, with sample test to the data extracted from Chinese market, experimental results show that AES and the mathematical model is feasible and the missing data treatment is reasonable. Furthermore, we prove that social values and artistic values of animations, which are both part and parcel of the integrated strength of animations, can be objectively reflected by AES.     

Multiple imputation when records used for imputation are not used or disseminated for analysis

When some of the records used to estimate the imputation modelsin multiple imputation are not used or available for analysis,the usual multiple imputation variance estimator has positivebias. We present an alternative approach that enables unbiasedestimation of variances and, hence, calibrated inferences insuch contexts. First, using all records, the imputer samplesm values of the parameters of the imputation model. Second,for each parameter draw, the imputer simulates the missing valuesfor all records n times. From these mn completed datasets, theimputer can analyse or disseminate the appropriate subset ofrecords. We develop methods for interval estimation and significancetesting for this approach. Methods are presented in the contextof multiple imputation for measurement error.     

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.