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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.

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I was going through my morning rituals coffee, e-mail, voice mail when the lieutenant brought me a new case le and said, I believe you re going to like this one. As a police investigator I usually received relatively standard cookie-cutter, even fraud cases, but on the humid spring day when I read the report led by Eric Mathis, I was forced to switch up my routine and take heed of the peculiar circumstances. I reviewed the initial report. This was an opportunity to wear my Dr. Holmes hat and get into some real investigative work, especially for an Internet fraud; these crimes were still relatively new at the time and I looked forward to expanding my skills. Eric Mathis stated in his report, A former employee is trying to put

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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).

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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.

no or very limited translation and the amount of rotation will vary with the type and extent of the ankylosis.6,16 As a patient with TMJ ankylosis attempts to open beyond the restriction, the ankylosed TMJ will have very limited translation, the contralateral TMJ will continue to translate, and the practitioner should observe the mandible de ecting to the affected side. During the range-of-motion evaluation, the practitioner should also observe a marked limitation to the contralateral side. Digital palpation of the affected TMJ during maximal movements will also re ect no or very limited translation of the condyle.6

EXAMPLE B.1 Let V and W be vector spaces over the same eld F. The Cartesian product V W consists of the set of ordered pairs fv; wg with v 2 V and w 2 W. V W is a vector space. Vector addition and scalar multiplication on V W are de ned as fv; wg fp; qg fv p; w qg afv; wg fav; awg h fhv ; hw g where a 2 F, v and p 2 V, w and q 2 W, hv and hw are the null elements of V and W, respectively. The Cartesian product vector space can be extended to include any number of vector spaces.

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