Intercontinental distribution of Plagiochila corrugata (Plagiochilaceae, Hepaticae) inferred

The use of the Gibbs sampler with fully conditionally specified models, where the distribution of each variable given the other variables is the starting point, has become a popular method to create imputations in incomplete multivariate data. The theoretical weakness of this approach is that the specified conditional densities can be incompatible, and therefore the stationary distribution to which the Gibbs sampler attempts to converge may not exist. This study investigates practical consequences of this problem by means of simulation. Missing data are created under four different missing data mechanisms. Attention is given to the statistical behavior under compatible and incompatible models. The results indicate that multiple imputation produces essentially unbiased estimates with appropriate coverage in the simple cases investigated, even for the incompatible models. Of particular interest is that these results were produced using only five Gibbs iterations starting from a simple draw from observed marginal distributions. It thus appears that, despite the theoretical weaknesses, the actual performance of conditional model specification for multivariate imputation can be quite good, and therefore deserves further study. Key words: multivariate missing data, multiple imputation, distributional compatibility, Gibbs sampling, simulation, proper imputation 2 1

Data were collected prospectively on parameters re-lated to first calving on 18 farms located in Northeast-ern Pennsylvania. This project was designed to study possible residual effects of calf management practices and events occurring during the first 16 wk of life on age, BW, skeletal growth, and body condition score at first calving. Multiple imputation method for handling missing data was incorporated in these analyses. This method has the advantage over ad hoc single imputa-tions because the appropriate error structure is main-tained. Much similarity was found between the multi-ple imputation method and a traditional mixed model analysis, except that some estimates from the multiple imputation method seemed more logical in their effects

The use of the Gibbs sampler with fully conditionally specified models, where the distribution of each variable given the other variables is the starting point, has become a popular method to create imputations in incomplete multivariate data. The theoretical weakness of this approach is that the specified conditional densities can be incompatible, and therefore the stationary distribution to which the Gibbs sampler attempts to converge may not exist. This study investigates practical consequences of this problem by means of simulation. Missing data are created under four different missing data mechanisms. Attention is given to the statistical behavior under compatible and incompatible models. The results indicate that multiple imputation produces essentially unbiased estimates with appropriate coverage in the simple cases investigated, even for the incompatible models. Of particular interest is that these results were produced using only five Gibbs iterations starting from a simple draw from observed marginal distributions. It thus appears that, despite the theoretical weaknesses, the actual performance of conditional model specification for multivariate imputation can be quite good, and therefore deserves further study.