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.     

Application of Multiple Imputation for Missing Values in Three-Way Three-Mode Multi-Environment Trial Data

It is a common occurrence in plant breeding programs to observe missing values in three-way three-mode multi-environment trial (MET) data. We proposed modifications of models for estimating missing observations for these data arrays, and developed a novel approach in terms of hierarchical clustering. Multiple imputation (MI) was used in four ways, multiple agglomerative hierarchical clustering, normal distribution model, normal regression model, and predictive mean match. The later three models used both Bayesian analysis and non-Bayesian analysis, while the first approach used a clustering procedure with randomly selected attributes and assigned real values from the nearest neighbour to the one with missing observations. Different proportions of data entries in six complete datasets were randomly selected to be missing and the MI methods were compared based on the efficiency and accuracy of estimating those values. The results indicated that the models using Bayesian analysis had slightly higher accuracy of estimation performance than those using non-Bayesian analysis but they were more time-consuming. However, the novel approach of multiple agglomerative hierarchical clustering demonstrated the overall best performances.     

Estimating treatment effects with partially observed covariates using outcome regression with missing indicators

Missing data is a common issue in research using observational studies to investigate the effect of treatments on health outcomes. When missingness occurs only in the covariates, a simple approach is to use missing indicators to handle the partially observed covariates. The missing indicator approach has been criticized for giving biased results in outcome regression. However, recent papers have suggested that the missing indicator approach can provide unbiased results in propensity score analysis under certain assumptions. We consider assumptions under which the missing indicator approach can provide valid inferences, namely, (1) no unmeasured confounding within missingness patterns; either (2a) covariate values of patients with missing data were conditionally independent of treatment or (2b) these values were conditionally independent of outcome; and (3) the outcome model is correctly specified: specifically, the true outcome model does not include interactions between missing indicators and fully observed covariates. We prove that, under the assumptions above, the missing indicator approach with outcome regression can provide unbiased estimates of the average treatment effect. We use a simulation study to investigate the extent of bias in estimates of the treatment effect when the assumptions are violated and we illustrate our findings using data from electronic health records. In conclusion, the missing indicator approach can provide valid inferences for outcome regression, but the plausibility of its assumptions must first be considered carefully.     

Impact of missing genotype data on Monte-Carlo simulation based haplotype analysis

In the context of haplotype association analysis of unphased genotype data, methods based on Monte-Carlo simulations are often used to compensate for missing or inappropriate asymptotic theory. Moreover, such methods are an indispensable means to deal with multiple testing problems. We want to call attention to a potential trap in this usually useful approach: The simulation approach may lead to strongly inflated type I errors in the presence of different missing rates between cases and controls, depending on the chosen test statistic. Here, we consider four different testing strategies for haplotype analysis of case-control data. We recommend to interpret results for data sets with non-comparable distributions of missing genotypes with special caution, in case the test statistic is based on inferred haplotypes per individual. Moreover, our results are important for the conduction and interpretation of genome-wide association studies.     

Correcting bias due to missing stage data in the non-parametric estimation of stage-specific net survival for colorectal cancer using multiple imputation

BackgroundPopulation-based net survival by tumour stage at diagnosis is a key measure in cancer surveillance. Unfortunately, data on tumour stage are often missing for a non-negligible proportion of patients and the mechanism giving rise to the missingness is usually anything but completely at random. In this setting, restricting analysis to the subset of complete records gives typically biased results. Multiple imputation is a promising practical approach to the issues raised by the missing data, but its use in conjunction with the Pohar-Perme method for estimating net survival has not been formally evaluated.MethodsWe performed a resampling study using colorectal cancer population-based registry data to evaluate the ability of multiple imputation, used along with the Pohar-Perme method, to deliver unbiased estimates of stage-specific net survival and recover missing stage information. We created 1000 independent data sets, each containing 5000 patients. Stage data were then made missing at random under two scenarios (30% and 50% missingness).ResultsComplete records analysis showed substantial bias and poor confidence interval coverage. Across both scenarios our multiple imputation strategy virtually eliminated the bias and greatly improved confidence interval coverage.ConclusionsIn the presence of missing stage data complete records analysis often gives severely biased results. We showed that combining multiple imputation with the Pohar-Perme estimator provides a valid practical approach for the estimation of stage-specific colorectal cancer net survival. As usual, when the percentage of missing data is high the results should be interpreted cautiously and sensitivity analyses are recommended.     

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.     

Conditional and unconditional categorical regression models with missing covariates

We consider methods for analyzing categorical regression models when some covariates (Z) are completely observed but other covariates (X) are missing for some subjects. When data on X are missing at random (i.e., when the probability that X is observed does not depend on the value of X itself), we present a likelihood approach for the observed data that allows the same nuisance parameters to be eliminated in a conditional analysis as when data are complete. An example of a matched case-control study is used to demonstrate our approach.     

On Analysis of Proportions in Tables with Spoiled Data

This paper discusses analysis of dispersion tests in tables with spoiled data, that is, where some cells are missing. BENNETT (1987) gave a method for estimating the missing cells in a two-way table and illustrated it with a simple example. This paper uses GLIM to estimate the models and indicates that the General Linear Model performs the computations without need to estimate the missing values. The binomial error model also may be used for this problem, and is the more natural approach to the problem.     

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.     

Likelihood-based association analysis for nuclear families and unrelated subjects with missing genotype data

Missing data occur in genetic association studies for several reasons including missing family members and uncertain haplotype phase. Maximum likelihood is a commonly used approach to accommodate missing data, but it can be difficult to apply to family-based association studies, because of possible loss of robustness to confounding by population stratification. Here a novel likelihood for nuclear families is proposed, in which distinct sets of association parameters are used to model the parental genotypes and the offspring genotypes. This approach is robust to population structure when the data are complete, and has only minor loss of robustness when there are missing data. It also allows a novel conditioning step that gives valid analysis for multiple offspring in the presence of linkage. Unrelated subjects are included by regarding them as the children of two missing parents. Simulations and theory indicate similar operating characteristics to TRANSMIT, but with no bias with missing data in the presence of linkage. In comparison with FBAT and PCPH, the proposed model is slightly less robust to population structure but has greater power to detect strong effects. In comparison to APL and MITDT, the model is more robust to stratification and can accommodate sibships of any size. The methods are implemented for binary and continuous traits in software, UNPHASED, available from the author.     

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.     

Retrieving a mitogenomic mammal tree using composite taxa

Supermatrices are often characterized by a large amount of missing data. One possible approach to minimize such missing data is to create composite taxa. These taxa are formed by sampling sequences from different species in order to obtain a composite sequence that includes a maximum number of genes. Although this approach is increasingly used, its accuracy has rarely been tested and some authors prefer to analyze incomplete supermatrices by coding unavailable sequences as missing. To further validate the composite taxon approach, it was applied to complete mitochondrial matrices of 102 mammal species representing 93 families with varying amount of missing data. On average, missing data and composite matrices showed similar congruence to model trees obtained from the complete sequence matrix. As expected, the level of congruence to model trees decreased as missing data increased, with both approaches. We conclude that the composite taxon approach is worth considering in a phylogenomic context since it performs well and reduces computing time when compared to missing data matrices.     

Does adding characters with missing data increase or decrease phylogenetic accuracy?

Missing data are a widely recognized nuisance factor in phylogenetic analyses, and the fear of missing data may deter systematists from including characters that are highly incomplete. In this paper, I used simulations to explore the consequences of including sets of characters that contain missing data. More specifically, I tested whether the benefits of increasing the number of characters outweigh the costs of adding missing data cells to a matrix. The results show that the addition of a set of characters with missing data is generally more likely to increase phylogenetic accuracy than decrease it, but the potential benefits of adding these characters quickly disappear as the proportion of missing data increases. Furthermore, despite the overall trend, adding characters with missing data does decrease accuracy in some cases. In these situations, the missing data entries are not themselves misleading, but their presence may mimic the effects of limited taxon sampling, which can positively mislead. Criteria are discussed for predicting whether adding characters with missing data may increase or decrease accuracy. The results of this study also suggest that accuracy can be increased to a surprising degree by (1) "filling the holes" in a data matrix as much as possible (even when relatively few taxa are missing data), and (2) adding fewer characters scored for all taxa rather than adding a larger number of characters known for fewer taxa. Missing data can also be eliminated from an analysis through the exclusion of incomplete taxa rather than incomplete characters, but this approach may reduce the usefulness of the analysis and (in some cases) the accuracy of the estimated trees.     

A Semiparametric Missing‐Data‐Induced Intensity Method for Missing Covariate Data in Individually Matched Case–Control Studies

Summary In individually matched case–control studies, when some covariates are incomplete, an analysis based on the complete data may result in a large loss of information both in the missing and completely observed variables. This usually results in a bias and loss of efficiency. In this article, we propose a new method for handling the problem of missing covariate data based on a missing‐data‐induced intensity approach when the missingness mechanism does not depend on case–control status and show that this leads to a generalization of the missing indicator method. We derive the asymptotic properties of the estimates from the proposed method and, using an extensive simulation study, assess the finite sample performance in terms of bias, efficiency, and 95% confidence coverage under several missing data scenarios. We also make comparisons with complete‐case analysis (CCA) and some missing data methods that have been proposed previously. Our results indicate that, under the assumption of predictable missingness, the suggested method provides valid estimation of parameters, is more efficient than CCA, and is competitive with other, more complex methods of analysis. A case–control study of multiple myeloma risk and a polymorphism in the receptor Inter‐Leukin‐6 (IL‐6‐α) is used to illustrate our findings.     

Sensitivity analysis of longitudinal binary data with non-monotone missing values

This paper highlights the consequences of incomplete observations in the analysis of longitudinal binary data, in particular non-monotone missing data patterns. Sensitivity analysis is advocated and a method is proposed based on a log-linear model. A sensitivity parameter that represents the relationship between the response mechanism and the missing data mechanism is introduced. It is shown that although this parameter is identifiable, its estimation is highly questionable. A far better approach is to consider a range of plausible values and to estimate the parameters of interest conditionally upon each value of the sensitivity parameter. This allows us to assess the sensitivity of study's conclusion to assumptions regarding the missing data mechanism. The method is applied to a randomized clinical trial comparing the efficacy of two treatment regimens in patients with persistent asthma.     

Accounting for Missing Data in Noncancer Risk Assessment

Noncancer risk assessments are generally forced to rely on animal bioassay data to estimate a Tolerable Daily Intake or Reference Dose, as a proxy for the threshold of human response. In cases where animal bioassays are missing from a complete data base, the critical NOAEL (no-observed-adverse-effect level) needs to be adjusted to account for the impact of the missing bioassay(s). This paper presents two approaches for making such adjustments. One is based on regression analysis and seeks to provide a point estimate of the adjustment needed. The other relies on non-parametric analysis and is intended to provide a distributional estimate of the needed adjustment. The adjustment needed is dependent on the definition of a complete data base, the number of bioassays missing, the specific bioassays which are missing, and the method used for interspecies scaling. The results from either approach can be used in conjunction with current practices for computing the TDI or RfD, or as an element of distributional approaches for estimating the human population threshold.     

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.     

Improving missing value imputation of microarray data by using spot quality weights

Background  

Microarray technology has become popular for gene expression profiling, and many analysis tools have been developed for data interpretation. Most of these tools require complete data, but measurement values are often missing A way to overcome the problem of incomplete data is to impute the missing data before analysis. Many imputation methods have been suggested, some na?ve and other more sophisticated taking into account correlation in data. However, these methods are binary in the sense that each spot is considered either missing or present. Hence, they are depending on a cutoff separating poor spots from good spots. We suggest a different approach in which a continuous spot quality weight is built into the imputation methods, allowing for smooth imputations of all spots to larger or lesser degree.     

Classification trees as an alternative to linear discriminant analysis

Linear discriminant analysis (LDA) is frequently used for classification/prediction problems in physical anthropology, but it is unusual to find examples where researchers consider the statistical limitations and assumptions required for this technique. In these instances, it is difficult to know whether the predictions are reliable. This paper considers a nonparametric alternative to predictive LDA: binary, recursive (or classification) trees. This approach has the advantage that data transformation is unnecessary, cases with missing predictor variables do not require special treatment, prediction success is not dependent on data meeting normality conditions or covariance homogeneity, and variable selection is intrinsic to the methodology. Here I compare the efficacy of classification trees with LDA, using typical morphometric data. With data from modern hominoids, the results show that both techniques perform nearly equally. With complete data sets, LDA may be a better choice, as is shown in this example, but with missing observations, classification trees perform outstandingly well, whereas commercial discriminant analysis programs do not predict classifications for cases with incompletely measured predictor variables and generally are not designed to address the problem of missing data. Testing of data prior to analysis is necessary, and classification trees are recommended either as a replacement for LDA or as a supplement whenever data do not meet relevant assumptions. It is highly recommended as an alternative to LDA whenever the data set contains important cases with missing predictor variables.