Bayesian epistasis association mapping via SNP imputation

Genetic mutations may interact to increase the risk of human complex diseases. Mapping of multiple interacting disease loci in the human genome has recently shown promise in detecting genes with little main effects. The power of interaction association mapping, however, can be greatly influenced by the set of single nucleotide polymorphism (SNP) genotyped in a case-control study. Previous imputation methods only focus on imputation of individual SNPs without considering their joint distribution of possible interactions. We present a new method that simultaneously detects multilocus interaction associations and imputes missing SNPs from a full Bayesian model. Our method treats both the case-control sample and the reference data as random observations. The output of our method is the posterior probabilities of SNPs for their marginal and interacting associations with the disease. Using simulations, we show that the method produces accurate and robust imputation with little overfitting problems. We further show that, with the type I error rate maintained at a common level, SNP imputation can consistently and sometimes substantially improve the power of detecting disease interaction associations. We use a data set of inflammatory bowel disease to demonstrate the application of our method.     

Bayesian methods for quantitative trait loci mapping based on model selection: approximate analysis using the Bayesian information criterion.

We describe an approximate method for the analysis of quantitative trait loci (QTL) based on model selection from multiple regression models with trait values regressed on marker genotypes, using a modification of the easily calculated Bayesian information criterion to estimate the posterior probability of models with various subsets of markers as variables. The BIC-delta criterion, with the parameter delta increasing the penalty for additional variables in a model, is further modified to incorporate prior information, and missing values are handled by multiple imputation. Marginal probabilities for model sizes are calculated, and the posterior probability of nonzero model size is interpreted as the posterior probability of existence of a QTL linked to one or more markers. The method is demonstrated on analysis of associations between wood density and markers on two linkage groups in Pinus radiata. Selection bias, which is the bias that results from using the same data to both select the variables in a model and estimate the coefficients, is shown to be a problem for commonly used non-Bayesian methods for QTL mapping, which do not average over alternative possible models that are consistent with the data.     

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.     

Association Studies with Imputed Variants Using Expectation-Maximization Likelihood-Ratio Tests

Genotype imputation has become standard practice in modern genetic studies. As sequencing-based reference panels continue to grow, increasingly more markers are being well or better imputed but at the same time, even more markers with relatively low minor allele frequency are being imputed with low imputation quality. Here, we propose new methods that incorporate imputation uncertainty for downstream association analysis, with improved power and/or computational efficiency. We consider two scenarios: I) when posterior probabilities of all potential genotypes are estimated; and II) when only the one-dimensional summary statistic, imputed dosage, is available. For scenario I, we have developed an expectation-maximization likelihood-ratio test for association based on posterior probabilities. When only imputed dosages are available (scenario II), we first sample the genotype probabilities from its posterior distribution given the dosages, and then apply the EM-LRT on the sampled probabilities. Our simulations show that type I error of the proposed EM-LRT methods under both scenarios are protected. Compared with existing methods, EM-LRT-Prob (for scenario I) offers optimal statistical power across a wide spectrum of MAF and imputation quality. EM-LRT-Dose (for scenario II) achieves a similar level of statistical power as EM-LRT-Prob and, outperforms the standard Dosage method, especially for markers with relatively low MAF or imputation quality. Applications to two real data sets, the Cebu Longitudinal Health and Nutrition Survey study and the Women’s Health Initiative Study, provide further support to the validity and efficiency of our proposed methods.     

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.     

Multiple Imputation Approaches for the Analysis of Dichotomized Responses in Longitudinal Studies with Missing Data

Summary Often a binary variable is generated by dichotomizing an underlying continuous variable measured at a specific time point according to a prespecified threshold value. In the event that the underlying continuous measurements are from a longitudinal study, one can use the repeated‐measures model to impute missing data on responder status as a result of subject dropout and apply the logistic regression model on the observed or otherwise imputed responder status. Standard Bayesian multiple imputation techniques ( Rubin, 1987 , in Multiple Imputation for Nonresponse in Surveys) that draw the parameters for the imputation model from the posterior distribution and construct the variance of parameter estimates for the analysis model as a combination of within‐ and between‐imputation variances are found to be conservative. The frequentist multiple imputation approach that fixes the parameters for the imputation model at the maximum likelihood estimates and construct the variance of parameter estimates for the analysis model using the results of Robins and Wang (2000, Biometrika 87, 113–124) is shown to be more efficient. We propose to apply ( Kenward and Roger, 1997 , Biometrics 53, 983–997) degrees of freedom to account for the uncertainty associated with variance–covariance parameter estimates for the repeated measures model.     

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.     

A Unified Approach to Genotype Imputation and Haplotype-Phase Inference for Large Data Sets of Trios and Unrelated Individuals

We present methods for imputing data for ungenotyped markers and for inferring haplotype phase in large data sets of unrelated individuals and parent-offspring trios. Our methods make use of known haplotype phase when it is available, and our methods are computationally efficient so that the full information in large reference panels with thousands of individuals is utilized. We demonstrate that substantial gains in imputation accuracy accrue with increasingly large reference panel sizes, particularly when imputing low-frequency variants, and that unphased reference panels can provide highly accurate genotype imputation. We place our methodology in a unified framework that enables the simultaneous use of unphased and phased data from trios and unrelated individuals in a single analysis. For unrelated individuals, our imputation methods produce well-calibrated posterior genotype probabilities and highly accurate allele-frequency estimates. For trios, our haplotype-inference method is four orders of magnitude faster than the gold-standard PHASE program and has excellent accuracy. Our methods enable genotype imputation to be performed with unphased trio or unrelated reference panels, thus accounting for haplotype-phase uncertainty in the reference panel. We present a useful measure of imputation accuracy, allelic R², and show that this measure can be estimated accurately from posterior genotype probabilities. Our methods are implemented in version 3.0 of the BEAGLE software package.     

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.     

Multiple Imputation of Missing Composite Outcomes in Longitudinal Data

In longitudinal randomised trials and observational studies within a medical context, a composite outcome—which is a function of several individual patient-specific outcomes—may be felt to best represent the outcome of interest. As in other contexts, missing data on patient outcome, due to patient drop-out or for other reasons, may pose a problem. Multiple imputation is a widely used method for handling missing data, but its use for composite outcomes has been seldom discussed. Whilst standard multiple imputation methodology can be used directly for the composite outcome, the distribution of a composite outcome may be of a complicated form and perhaps not amenable to statistical modelling. We compare direct multiple imputation of a composite outcome with separate imputation of the components of a composite outcome. We consider two imputation approaches. One approach involves modelling each component of a composite outcome using standard likelihood-based models. The other approach is to use linear increments methods. A linear increments approach can provide an appealing alternative as assumptions concerning both the missingness structure within the data and the imputation models are different from the standard likelihood-based approach. We compare both approaches using simulation studies and data from a randomised trial on early rheumatoid arthritis patients. Results suggest that both approaches are comparable and that for each, separate imputation offers some improvement on the direct imputation of a composite outcome.     

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.     

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.     

A Model for Adjusting for Nonignorable Verification Bias in Estimation of the ROC Curve and Its Area with Likelihood‐Based Approach

Summary In estimation of the ROC curve, when the true disease status is subject to nonignorable missingness, the observed likelihood involves the missing mechanism given by a selection model. In this article, we proposed a likelihood‐based approach to estimate the ROC curve and the area under the ROC curve when the verification bias is nonignorable. We specified a parametric disease model in order to make the nonignorable selection model identifiable. With the estimated verification and disease probabilities, we constructed four types of empirical estimates of the ROC curve and its area based on imputation and reweighting methods. In practice, a reasonably large sample size is required to estimate the nonignorable selection model in our settings. Simulation studies showed that all four estimators of ROC area performed well, and imputation estimators were generally more efficient than the other estimators proposed. We applied the proposed method to a data set from research in Alzheimer's disease.     

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.     

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.     

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.     

Effects of Different Missing Data Imputation Techniques on the Performance of Undiagnosed Diabetes Risk Prediction Models in a Mixed-Ancestry Population of South Africa

Background

Imputation techniques used to handle missing data are based on the principle of replacement. It is widely advocated that multiple imputation is superior to other imputation methods, however studies have suggested that simple methods for filling missing data can be just as accurate as complex methods. The objective of this study was to implement a number of simple and more complex imputation methods, and assess the effect of these techniques on the performance of undiagnosed diabetes risk prediction models during external validation.

Methods

Data from the Cape Town Bellville-South cohort served as the basis for this study. Imputation methods and models were identified via recent systematic reviews. Models’ discrimination was assessed and compared using C-statistic and non-parametric methods, before and after recalibration through simple intercept adjustment.

Results

The study sample consisted of 1256 individuals, of whom 173 were excluded due to previously diagnosed diabetes. Of the final 1083 individuals, 329 (30.4%) had missing data. Family history had the highest proportion of missing data (25%). Imputation of the outcome, undiagnosed diabetes, was highest in stochastic regression imputation (163 individuals). Overall, deletion resulted in the lowest model performances while simple imputation yielded the highest C-statistic for the Cambridge Diabetes Risk model, Kuwaiti Risk model, Omani Diabetes Risk model and Rotterdam Predictive model. Multiple imputation only yielded the highest C-statistic for the Rotterdam Predictive model, which were matched by simpler imputation methods.

Conclusions

Deletion was confirmed as a poor technique for handling missing data. However, despite the emphasized disadvantages of simpler imputation methods, this study showed that implementing these methods results in similar predictive utility for undiagnosed diabetes when compared to multiple imputation.     

Multiple imputation for model checking: completed-data plots with missing and latent data

In problems with missing or latent data, a standard approach is to first impute the unobserved data, then perform all statistical analyses on the completed dataset--corresponding to the observed data and imputed unobserved data--using standard procedures for complete-data inference. Here, we extend this approach to model checking by demonstrating the advantages of the use of completed-data model diagnostics on imputed completed datasets. The approach is set in the theoretical framework of Bayesian posterior predictive checks (but, as with missing-data imputation, our methods of missing-data model checking can also be interpreted as "predictive inference" in a non-Bayesian context). We consider the graphical diagnostics within this framework. Advantages of the completed-data approach include: (1) One can often check model fit in terms of quantities that are of key substantive interest in a natural way, which is not always possible using observed data alone. (2) In problems with missing data, checks may be devised that do not require to model the missingness or inclusion mechanism; the latter is useful for the analysis of ignorable but unknown data collection mechanisms, such as are often assumed in the analysis of sample surveys and observational studies. (3) In many problems with latent data, it is possible to check qualitative features of the model (for example, independence of two variables) that can be naturally formalized with the help of the latent data. We illustrate with several applied examples.     

Measuring the fit of sequence data to phylogenetic model: allowing for missing data

It is fundamentally important to assess the fit of data to model in phylogenetic and evolutionary studies. Phylogenetic methods using molecular sequences typically start with a multiple alignment. It is possible to measure the fit of data to model expectations of data, for example, via the likelihood-ratio (G) test or the X(2) test, if all sites in all sequences have an unambiguous residue. However, nearly all alignments of interest contain sites (columns of the alignment) with missing data, that is, ambiguous nucleotides, gaps, or unsequenced regions, which must presently be removed before using the above tests. Unfortunately, this is often either undesirable or impractical, as it will discard much of the data. Here, we show how iterative ML estimators may directly estimate the site-pattern probabilities for columns with missing data, given only standard i.i.d. assumptions. The optimization may use an EM or Newton algorithm, or any other hill-climbing approach. The resulting optimal likelihood under the unconstrained or multinomial model may be compared directly with the likelihood of the data coming from the model (a G statistic). Alternatively the modified observed and the expected frequencies of site patterns may be compared using a X(2) test. The distribution of such statistics is best assessed using appropriate simulations. The new method is applicable to models using codons or paired sites. The methods are also useful with Hadamard conjugations (spectral analysis) and are illustrated with these and with ML evolutionary models that allow site-rate variability.