 By J.L. Schafer

ISBN-10: 0412040611

ISBN-13: 9780412040610

Provides a unified, Bayesian method of the research of incomplete multivariate facts, protecting datasets within which the variables are non-stop, express or either. contains genuine information examples and useful suggestion.

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Additional resources for Analysis of Incomplete Multivariate Data

Sample text

2) over the predictive distribution P(Ymis|Yobs,θ , where θ is a preliminary estimate of the unknown parameter. 3) where ( ) ∫ ( ) Q θ | θ (t ) = l(θ | Y )P Ymis | Yobs , θ (t ) dYmis and ( ) ∫ ( ) H θ | θ (t ) = log P(Ymis | Yobs , θ )P Ymis | Yobs , θ (t ) dYmis. A central result of Dempster, Laird, and Rubin (1977) is that if (t+1) (t) be the value of θ that maximizes Q(θ|θ ), then we let θ θ (t+1) is a better estimate than θ in the sense that its observed(t) data loglikelihood is at least as high as that of θ , (t) ( ) ( ) l θ (t +1) | Yobs ≥ l θ (t ) | Yobs .

5 General ignorable procedures Virtually all of the missing-data procedures used in statistical practice, both ad hoc approaches and principled ones, rely at least implicitly on an assumption of ignorability. Often the assumptions made are even stronger. For example, the casedeletion method used by many statistical packages (omitting all incomplete cases from the analysis) may introduce bias into inferences about θ unless the missing data are MCAR. Even if MCAR holds, case deletion may still be grossly inefficient.

3 that the rate of convergence in this example is determined by n0|(n0+n1), the proportion of observations that are missing. More generally, the convergence rate of EM is governed by the fractions of information about components of θ missing due to nonresponse. (t+1) (t) Example 2: Two binary variables with missing data on both. Suppose that Y1 and Y 2 are two potentially related dichotomous variables, each taking values 1 or 2. If the n units in a sample are iid, the complete data may, without loss of information, be reduced to an array of counts x=(x 11, x12, x21, x22) having a multinomial distribution, where xij is the number of sample units having Y1 = i and Y2 = j.