Full information maximum likelihood estimation in factor analysis with a large number of missing values

We consider the problem of full information maximum likelihood (FIML) estimation in factor analysis when a majority of the data values are missing. The expectation–maximization (EM) algorithm is often used to find the FIML estimates, in which the missing values on manifest variables are included in complete data. However, the ordinary EM algorithm has an extremely high computational cost. In this paper, we propose a new algorithm that is based on the EM algorithm but that efficiently computes the FIML estimates. A significant improvement in the computational speed is realized by not treating the missing values on manifest variables as a part of complete data. When there are many missing data values, it is not clear if the FIML procedure can achieve good estimation accuracy. In order to investigate this, we conduct Monte Carlo simulations under a wide variety of sample sizes.     

A non-iterative least squares estimation of missing values in experimental designs

In this paper an attempt has been made to obtain a systematic method of estimating the missing values in experimental designs. When the observations are missing in a particular pattern (in RBD and LSD) explicit expressions are given for the estimators of the missing values. This procedure is compared with Yate's iterative procedure by numerical examples.     

Semiparametric predictive mean matching

Predictive mean matching is an imputation method that combines parametric and nonparametric techniques. It imputes missing values by means of the Nearest Neighbor Donor with distance based on the expected values of the missing variables conditional on the observed covariates, instead of computing the distance directly on the values of the covariates. In ordinary predictive mean matching the expected values are computed through a linear regression model. In this paper a generalization of the original predictive mean matching is studied. Here the expected values used for computing the distance are estimated through an approach based on Gaussian mixture models. This approach includes as a special case the original predictive mean matching but allows one to deal also with nonlinear relationships among the variables. In order to assess its performance, an empirical evaluation based on simulations is carried out.     

On the distribution of summary statistics for missing data

Under an assumption that missing values occur randomly in a matrix, formulae are developed for the expected value and variance of six statistics that summarize the number and location of the missing values. For a seventh statistic, a regression model based on simulated data yields an estimate of the expected value. The results can be used in the development of methods to control the Type I error and approximate power and sample size for multilevel and longitudinal studies with missing data.     

A likelihood-based approach for multivariate one-sided tests with missing data

Inequality-restricted hypotheses testing methods containing multivariate one-sided testing methods are useful in practice, especially in multiple comparison problems. In practice, multivariate and longitudinal data often contain missing values since it may be difficult to observe all values for each variable. However, although missing values are common for multivariate data, statistical methods for multivariate one-sided tests with missing values are quite limited. In this article, motivated by a dataset in a recent collaborative project, we develop two likelihood-based methods for multivariate one-sided tests with missing values, where the missing data patterns can be arbitrary and the missing data mechanisms may be non-ignorable. Although non-ignorable missing data are not testable based on observed data, statistical methods addressing this issue can be used for sensitivity analysis and might lead to more reliable results, since ignoring informative missingness may lead to biased analysis. We analyse the real dataset in details under various possible missing data mechanisms and report interesting findings which are previously unavailable. We also derive some asymptotic results and evaluate our new tests using simulations.     

Detecting covariates with non-random missing values in a survey of primary education in Madagascar

A major survey of the determinants of access to primary education in Madagascar was carried out in 1994. The probability of enrolment, probability of admission, delay before beginning school, probability of repeating a year and probability of dropping out were studied. The results of the survey are briefly described. In the analysis, one major problem was non-random missing values in the covariates. Some simple methods were developed for detecting whether a response variable depends on the missingness of a given covariate and whether eliminating the missing values would distort the resulting model. A way of incorporating covariates with randomly missing values was used such that the individuals having the missing values did not need to be eliminated. These methods are described and examples are given on how they were applied for one of the key covariates that had a large number of non-random missing values and for one for which the values appear to be randomly missing.     

A cautionary note on the analysis of randomized block designs with a few missing values

The randomized block design is routinely employed in the social and biopharmaceutical sciences. With no missing values, analysis of variance (AOV) can be used to analyze such experiments. However, if some data are missing, the AOV formulae are no longer applicable, and iterative methods such as restricted maximum likelihood (REML) are recommended, assuming block effects are treated as random. Despite the well-known advantages of REML, methods like AOV based on complete cases (blocks) only (CC-AOV) continue to be used by researchers, particularly in situations where routinely only a few missing values are encountered. Reasons for this appear to include a natural proclivity for non-iterative, summary-statistic-based methods, and a presumption that CC-AOV is only trivially less efficient than REML with only a few missing values (say≤10%). The purpose of this note is two-fold. First, to caution that CC-AOV can be considerably less powerful than REML even with only a few missing values. Second, to offer a summary-statistic-based, pairwise-available-case-estimation (PACE) alternative to CC-AOV. PACE, which is identical to AOV (and REML) with no missing values, outperforms CC-AOV in terms of statistical power. However, it is recommended in lieu of REMLonly if software to implement the latter is unavailable, or the use of a “transparent” formula-based approach is deemed necessary. An example using real data is provided for illustration.     

A Bayesian model for longitudinal count data with non-ignorable dropout

Asthma is an important chronic disease of childhood. An intervention programme for managing asthma was designed on principles of self-regulation and was evaluated by a randomized longitudinal study.The study focused on several outcomes, and, typically, missing data remained a pervasive problem. We develop a pattern-mixture model to evaluate the outcome of intervention on the number of hospitalizations with non-ignorable dropouts. Pattern-mixture models are not generally identifiable as no data may be available to estimate a number of model parameters. Sensitivity analyses are performed by imposing structures on the unidentified parameters.We propose a parameterization which permits sensitivity analyses on clustered longitudinal count data that have missing values due to non-ignorable missing data mechanisms. This parameterization is expressed as ratios between event rates across missing data patterns and the observed data pattern and thus measures departures from an ignorable missing data mechanism. Sensitivity analyses are performed within a Bayesian framework by averaging over different prior distributions on the event ratios. This model has the advantage of providing an intuitive and flexible framework for incorporating the uncertainty of the missing data mechanism in the final analysis.     

Weighted modified first order regression procedures for estimation in linear models with missingX-observations

This paper considers the estimation of coefficients in a linear regression model with missing observations in the independent variables and introduces a modification of the standard first order regression method for imputation of missing values. The modification provides stochastic values for imputation and, as an extension, makes use of the principle of weighted mixed regression. The proposed procedures are compared with two popular procedures—one which utilizes only the complete observations and the other which employs the standard first order regression imputation method for missing values. A simulation experiment to evaluate the gain in efficiency and to examine interesting issues like the impact of varying degree of multicollinearity in explanatory variables is proceeded. Some work on the case of discrete regressor variables is in progress and will be reported in a future article to follow.     

Analysis of ranked data in randomized blocks when there are missing values

Data consisting of ranks within blocks are considered for randomized block designs when there are missing values. Tied ranks are possible. Such data can be analysed using the Skillings–Mack test. Here we suggest a new approach based on carrying out an ANOVA on the ranks using the general linear model platform available in many statistical packages. Such a platform allows an ANOVA to be calculated when there are missing values. Indicative sizes and powers show the ANOVA approach performs better than the Skillings–Mack test.     

Missing values: sparse inverse covariance estimation and?an?extension to sparse regression

We propose an ℓ ₁-regularized likelihood method for estimating the inverse covariance matrix in the high-dimensional multivariate normal model in presence of missing data. Our method is based on the assumption that the data are missing at random (MAR) which entails also the completely missing at random case. The implementation of the method is non-trivial as the observed negative log-likelihood generally is a complicated and non-convex function. We propose an efficient EM algorithm for optimization with provable numerical convergence properties. Furthermore, we extend the methodology to handle missing values in a sparse regression context. We demonstrate both methods on simulated and real data.     

Calibrated imputation in surveys under a quasi-model-assisted approach

Summary.  We propose to use calibrated imputation to compensate for missing values. This technique consists of finding final imputed values that are as close as possible to preliminary imputed values and are calibrated to satisfy constraints. Preliminary imputed values, potentially justified by an imputation model, are obtained through deterministic single imputation. Using appropriate constraints, the resulting imputed estimator is asymptotically unbiased for estimation of linear population parameters such as domain totals. A quasi-model-assisted approach is considered in the sense that inferences do not depend on the validity of an imputation model and are made with respect to the sampling design and a non-response model. An imputation model may still be used to generate imputed values and thus to improve the efficiency of the imputed estimator. This approach has the characteristic of handling naturally the situation where more than one imputation method is used owing to missing values in the variables that are used to obtain imputed values. We use the Taylor linearization technique to obtain a variance estimator under a general non-response model. For the logistic non-response model, we show that ignoring the effect of estimating the non-response model parameters leads to overestimating the variance of the imputed estimator. In practice, the overestimation is expected to be moderate or even negligible, as shown in a simulation study.     

Multiple imputation for ordinal longitudinal data with monotone missing data patterns

Missing data often complicate the analysis of scientific data. Multiple imputation is a general purpose technique for analysis of datasets with missing values. The approach is applicable to a variety of missing data patterns but often complicated by some restrictions like the type of variables to be imputed and the mechanism underlying the missing data. In this paper, the authors compare the performance of two multiple imputation methods, namely fully conditional specification and multivariate normal imputation in the presence of ordinal outcomes with monotone missing data patterns. Through a simulation study and an empirical example, the authors show that the two methods are indeed comparable meaning any of the two may be used when faced with scenarios, at least, as the ones presented here.     

Longitudinal data analysis in the presence of informative sampling: weighted distribution or joint modelling

ABSTRACT

Weighted distributions, as an example of informative sampling, work appropriately under the missing at random mechanism since they neglect missing values and only completely observed subjects are used in the study plan. However, length-biased distributions, as a special case of weighted distributions, remove the subjects with short length deliberately, which surely meet the missing not at random mechanism. Accordingly, applying length-biased distributions jeopardizes the results by producing biased estimates. Hence, an alternate method has to be used such that the results are improved by means of valid inferences. We propose methods that are based on weighted distributions and joint modelling procedure and compare them in analysing longitudinal data. After introducing three methods in use, a set of simulation studies and analysis of two real longitudinal datasets affirm our claim.     

Interpolation,outliers mid inverse autocorrelations

The paper addresses the problem of estimating missing observations in an infinite realization of a linear, possibly nonstationary, stochastic processes when the model is known. The general case of any possible distribution of missing observations in the time series is considered, and analytical expressions for the optimal estimators and their associated mean squared errors are obtained. These expressions involve solely the elements of the inverse or dual autocorrelation function of the series.

This optimal estimator -the conditional expectation of the missing observations given the available ones- is equal to the estimator that results from filling the missing values in the series with arbitrary numbers, treating these numbers as additive outliers, and removing with intervention analysis the outlier effects from the invented numbers.     

A comparison of various software tools for dealing with missing data via imputation

In real-life situations, we often encounter data sets containing missing observations. Statistical methods that address missingness have been extensively studied in recent years. One of the more popular approaches involves imputation of the missing values prior to the analysis, thereby rendering the data complete. Imputation broadly encompasses an entire scope of techniques that have been developed to make inferences about incomplete data, ranging from very simple strategies (e.g. mean imputation) to more advanced approaches that require estimation, for instance, of posterior distributions using Markov chain Monte Carlo methods. Additional complexity arises when the number of missingness patterns increases and/or when both categorical and continuous random variables are involved. Implementation of routines, procedures, or packages capable of generating imputations for incomplete data are now widely available. We review some of these in the context of a motivating example, as well as in a simulation study, under two missingness mechanisms (missing at random and missing not at random). Thus far, evaluation of existing implementations have frequently centred on the resulting parameter estimates of the prescribed model of interest after imputing the missing data. In some situations, however, interest may very well be on the quality of the imputed values at the level of the individual – an issue that has received relatively little attention. In this paper, we focus on the latter to provide further insight about the performance of the different routines, procedures, and packages in this respect.     

Consistency of Normal Distribution Based Pseudo Maximum Likelihood Estimates When Data Are Missing at Random

This paper shows that, when variables with missing values are linearly related to observed variables, the normal-distribution-based pseudo MLEs are still consistent. The population distribution may be unknown while the missing data process can follow an arbitrary missing at random mechanism. Enough details are provided for the bivariate case so that readers having taken a course in statistics/probability can fully understand the development. Sufficient conditions for the consistency of the MLEs in higher dimensions are also stated, while the details are omitted.     

On the Use of Adaptive Nearest Neighbors for Missing Value Imputation

A popular nonparametric treatment of missing value imputation uses methods based on k-nearest neighbors, where the number k of nearest neighbors is fixed without any consideration of the local features of missing values. This article proposes an alternative imputation method based on adaptive nearest neighbors, which takes into account the local features of the data. The proposed method adapts the number of neighbors in imputing the missing values according to the location of the missing values. Efficiency evaluation is then gauged through simulation studies using both simulated and real data. It is shown that the proposed method has distinct advantages over the imputation method based on k-nearest neighbors.     

Quasi-maximum likelihood estimation of GARCH models in the presence of missing values

This work presents a new method to deal with missing values in financial time series. Previous works are generally based in state-space models and Kalman filter and few consider ARCH family models. The traditional approach is to bound the data together and perform the estimation without considering the presence of missing values. The existing methods generally consider missing values in the returns. The proposed method considers the presence of missing values in the price of the assets instead of in the returns. The performance of the method in estimating the parameters and the volatilities is evaluated through a Monte Carlo simulation. Value at risk is also considered in the simulation. An empirical application to NASDAQ 100 Index series is presented.     

Robust multivariate mixture regression models with incomplete data

Multivariate mixture regression models can be used to investigate the relationships between two or more response variables and a set of predictor variables by taking into consideration unobserved population heterogeneity. It is common to take multivariate normal distributions as mixing components, but this mixing model is sensitive to heavy-tailed errors and outliers. Although normal mixture models can approximate any distribution in principle, the number of components needed to account for heavy-tailed distributions can be very large. Mixture regression models based on the multivariate t distributions can be considered as a robust alternative approach. Missing data are inevitable in many situations and parameter estimates could be biased if the missing values are not handled properly. In this paper, we propose a multivariate t mixture regression model with missing information to model heterogeneity in regression function in the presence of outliers and missing values. Along with the robust parameter estimation, our proposed method can be used for (i) visualization of the partial correlation between response variables across latent classes and heterogeneous regressions, and (ii) outlier detection and robust clustering even under the presence of missing values. We also propose a multivariate t mixture regression model using MM-estimation with missing information that is robust to high-leverage outliers. The proposed methodologies are illustrated through simulation studies and real data analysis.