Multiple Imputation describes a strategy for analyzing incomplete data that accounts for uncertainty in the missing data by replacing (imputing) each missing value by several ‘candidates’. The actual implementation of any Multiple Imputation method is typically computationally expensive which is why the concept has only really caught on around the verge of the new millennium, when the first algorithms for Multiple Imputation had become accessible.
In this article, we are going to give a rough overview of the shortcomings of methods for handling missing data prior to Rubin’s work in the late 1970s, and we explore the conceptual innovations that might have lead to Multiple Imputation based on an example, where mean imputation is the steppingstone for more advanced methods. The general concept of Multiple Imputation is explained using a simulated trivariate data set, and the imputation model is based on the standard Bayesian linear model, in order to explain the method as illustrative as possible.
Missing data are a ubiquitous problem in statistical analyses that has become an important research field in applied statistics. A highly useful technique to handle missing values in many settings is multiple imputation, that was first proposed by Rubin (1977, 1978) and extended in Rubin (1987). Due to the ongoing improvement in computer power in the last 10 years, multiple imputation has become a well known and often used tool in statistical analyses. However, obtaining significance levels from multiply-imputed data is still a problem, because the application of multiple imputation requires normally distributed or t-distributed complete-data estimators. Today there are basically three methods that extend the suggestions given in Rubin (1987). First, Li, Raghunathan, and Rubin (1991) proposed a procedure, in which significance levels are created by computing a modified Wald-test statistic that is then referred to an F-distribution. This procedure is essentially calibrated, but it requires access to the completed-data estimates and their variance-covariance matrices that may not be available with standard software. Second, Meng and Rubin (1992) proposed a complete-data two-stage-likelihood-ratio-test-based procedure that requires access to the code for the calculation of the log-likelihood-ratio statistics. Common statistical software does not provide access to the code in their standard analyses routines. Third, Li, Meng, Raghunathan, and Rubin (1991) developed an improved version of a method in Rubin (1987) that only requires the chi-square-statistics from a usual complete-data Wald-test. This method is only approximately calibrated and has a substantial loss of power compared to the previous two. There thus exist several procedures to generate significance levels in general from multiply-imputed data, but none of them has satisfactory applicability. Since many statistical analyses are based on hypothesis tests, especially on the Wald-test in regression analyses, it is very important to find a method that retains the advantages and overcomes the disadvantages of the existing procedures. Developing such a method was the aim of the present thesis. After providing a short introduction to multiple imputation theory and the existing methods to generate significance levels from multiply-imputed data, we present a new procedure that is based on a z-transformation. We analytically show that the z-transformation works for a one-sided z-test. By simulation we show that our new z-transformation is working well for all one-dimensional tests, because they can be linked to an F-test or to a z-test, respectively. Despite the success of this new z-transformation procedure in several practical settings, problems arise when multi-dimensional tests are performed. We develop and discuss a possible procedure to fix these problems. Based on a comprehensive simulation study we discover an interesting general statistical problem: Using a chi-square-distribution rather than an F-distribution, can lead to a not negligible error for small sample sizes, especially with larger dimensions of the estimator. This problem seems to be unnoticed until now. In addition, we show the influence of the sample size for generating accurate significance levels from multiply imputed data. We present an adjusted procedure, the componentwise-moment-based method, to easily calculate correct significance levels from multiply-imputed data under some assumptions. This procedure is related to one of the three existing methods and uses the small-sample degrees of freedom given in Barnard and Rubin (1999) componentwise. We examine this new method and the already existing procedures in detail and compare them with each other by an extensive simulation study with 55296 different situations given by the factorial design of our simulation study. We also compare the results with former simulation studies of Li, Raghunathan, Meng, and Rubin (1991, 1992), where they simulated draws from the theoretically calculated distributions of the test statistics. To analyze the behaviour of the four methods we perform an ANOVA to identify the most important factors for the distribution of the multiple imputation p-values. Afterwards we examine the rejection rates of the four methods with their originally proposed degrees of freedom and some further "method and degrees-of-freedom"-combinations and compare them with the particular nominal levels. We summarize the results of the simulation study in detail. Based on these results we give some practical advices for the data collector, for the imputer and for the data analyst, about how to calculate correct significance levels from multiply-imputed data. Finally, we give an overview of challenging tasks left for future research.
Censoring of variables is a common problem with microdata. This situation often arises with wage and income variables due to manifold reasons. The data may not be available due to difficulties during the data collection process, it may be artificially censored to ensure confidentiality, or it may just not be reliable because of high wage earners tending above average not to answer income questions. An important example for this problem is the German IAB Employment Sample (IABS), which is based on administrative data from the social security systems. Here, right-censoring of wages occurs due to the contribution limit in the German social security system. If earnings are to be analyzed from right-censored or top-coded data, standard models cannot be applied. We treat this problem as a missing data problem and use multiple imputation approaches to impute the censored wages by draws of a random variable from a truncated distribution, based on Markov chain Monte Carlo techniques. In this dissertation thesis new single and multiple imputation methods allowing for heteroscedasticity are suggested. Whereas one goal of the thesis is to present new imputation approaches that are applicable for right-censored wages, a main objective is also to confirm the validity of multiple imputation approaches for right-censored wages in general and to show the superiority of the new multiple imputation approach considering heteroscedasticity in a wide range of situations. To assess the validity of this approach, we also develop alternative approaches using uncensored wage information from a survey (German Structure of Earnings Survey, GSES). Simulation studies are performed to compare the different imputation approaches under different situations and to show the superiority of the new approach working without external information. Additionally, analyses that were done with the IABS are replicated to demonstrate the validity of imputed wage data.
Multiple Imputation (MI) is a general purpose approach to impute partially incomplete data. The proposed method - Bayesian Bootstrap Predictive Mean Matching - is a variant that incorporates the robustifying properties of a nearest neighbour technique (Predictive Mean Matching) into MI.