Dear Fay: you could take the estimate of the mean of the missing values in pattern from EM and then add multivariate normal error with covariance matrix given by the residual covariance of the missing values given the observed values. The vector of errors could be drawn as a sequence of univariate normal draws, conditioning on the draws of the previously imputed errors.
This is what Rubin calls improper MI, since it does not propagate errors in the parameters. It works reasonably well if there is not much missing data, but less well when there is a lot of missing data, yielding anti-conservative confidence coverage. A relatively simple way to make the method proper without doing a full Gibbs algorithm is to compute the mean and covariance matrix for each set of imputes from a different bootstrapped sample (ie a simple random sample with replacement from the complete and incomplete cases). In effect the parameters are then drawn from a bootstrap distribution, which is asymptotically equivalent to drawing from the posterior distribution, and hence proper. For more details of these methods see the MI chapter of the revision of Little and Rubin (2002), which has more material on MI than the first edition. Best, Rod Little On Thu, 14 Nov 2002, Fay Hughes wrote: > Hi, > > I was wondering whether anyone could help me, I am currently > investigating the EM Algorithm as a way of dealing with missing data, > but it appears as if in order to do broad based/exploratory inference on > the data I need to run the Data Augmentation algorithm after EM and then > perform Multiple Imputation by combining my results with Rubins (1987) > rules. > > I was wondering if there is any other way of preforming imputations > directly from the results of EM, ie. Assuming a multivariate normal > model and obtain the MLE's via EM, then impute in some way a certain > number of different data sets (directly from the multivarate normal with > MLE's) and then after having done standard complete tests on the > complete datasets to combine them using the same rules as MI. So in > other words, what imputation techniques would allow me to use Rubins > rules to yield valid inferences. > > Any help, anyone could give would be appreciated. > Fay Hughes (Miss) > Statistics Masters Student > University of Natal > Durban > South Africa > > > ___________________________________________________________________________________ Roderick Little Richard D. Remington Collegiate Professor (734) 936-1003 Department of Biostatistics Fax: (734) 763-2215 U-M School of Public Health M4045 SPH II [email protected] 1420 Washington Hgts http://www.sph.umich.edu/~rlittle/ Ann Arbor, MI 48109-2029
