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and, by constructing likelihoods based on the observed data, they avoid the need to impute the missing values Weaknesses are that the inferences are asymptotic and potentially vulnerable to model misspecification Also, current widely available software does not handle missing data in covariates or nonignorable models (although the ML approach can be applied to handle such problems) An alternative to these methods is to multiply-impute the missing values with draws from their predictive distribution under a model Multiple imputation (MI) allows the imputation uncertainty to be propagated in inferences by using MI combining rules A principled version of this approach, assuming MAR, is to impute using the posterior predictive distribution p(Ymis |Yobs) based on the model for Y and a prior distribution p( ) for the parameters of this model This approach uses the Bayesian paradigm to propagate uncertainty in the parameters.

Sequence Q0 and its variants are reproduced in Fig 729 The receiving events in Q0 are numbered and shown with their race sets Table 73 is the race table for sequence Q0 The three columns represent the three receiving events whose race sets are nonempty Each row represents a race variant of Q0 Consider the second row, which is (0, 1, 1) Each value indicates how the sending partner of the corresponding receiving event in Q0 is changed to create variant V2 The value 0 indicates that the sending partner of receiving event r2 will be left unchanged The value 1 indicates that the sending partner of receiving event r3 will be changed to s3 , which is the rst (and only) send event in race(r3 ) The value 1 indicates that receiving event r4 will be removed from V2.

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The imputations can be used for MI inferences, or can be viewed as a stepping stone to simulating directly the posterior distribution p( | Yobs) of the parameters, as in data augmentation (Tanner and Wong, 1987; Schafer, 1997; Little and Rubin, 2002, chapter 10) When the imputations and complete-data inferences are based on the same model, say f(Y | ), and the prior distribution p( ) has support over the full parameter space of , this approach is asymptotically equivalent to the ML or REML approach mentioned above This fact follows from the well-known large-sample equivalence of Bayes and ML, discussed for example in Little and Rubin (2002, chapter 6).

In general, let r be the receiving event corresponding to column j , V the race variant derived from row i, and v the value in row i column j Value v indicates how receiving event r is changed to derive variant V : 1 v = 1 indicates that r is removed from V..

The Bayes/MI approach can be a useful advance over ML for small samples; for example, in simple settings it yields inferences based on the Student t distribution that allow for uncertainty in estimating the variance, which are not a feature of the ML approach A useful feature of MI is that there is no requirement that the model for creating the imputations be the same as the analysis model applied to the filled-in data This idea featured prominently in Rubin s initial proposal of MI, which was in the context of imputation of public-use datasets where the imputer could not necessarily be expected to have the same analysis in mind as the analyst.

The Catalog session bean is responsible for accessing the catalog and returning a list of items that ACME sells. Listing 23 12 shows the source for the Catalog interface. Listing 23 12: The Catalog interface

A number of thorny theoretical issues arise when the imputation and analysis models differ and may not be mutually congenial (Rubin, 1987, 1996; Meng, 1994; Fay, 1996; Rao, 1996; Robins and Wang, 2000), but the added flexibility opens up some attractive missing-data solutions for the practitioner The resulting methods are not in all cases theoretically pristine , but a well-thought-out imputation model is likely to yield inferences with good properties Here are some examples: (1) Missing data in covariates As noted above, the ML approach as currently implemented in standard software does not allow for missing covariates One possibility for dealing with missing data in the covariates is to multiply-impute them conditional on the value of Yobs and observed values of the covariates, using software such as IVEware (Raghunathan et al, 2001).

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