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- New methods for generating significance levels from multiply-imputed data (2010)
- 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.