Statistical Assumptions for Multiple Imputation

The MI Procedure

Statistical Assumptions for Multiple Imputation

The MI procedure assumes that the data are from a continuous multivariate distribution and contain missing values that can occur on any of the variables. It also assumes that the data are from a multivariate normal distribution when either the regression method or the MCMC method is used.

Suppose Y is the n×p matrix of complete data, which is not fully observed, and denote the observed part of Y by Y_obs and the missing part by Y_mis. The SAS MI and MIANALYZE procedures assume that the missing data are missing at random (MAR), that is, the probability that an observation is missing can depend on Y_obs, but not on Y_mis (Rubin 1976; 1987, p. 53).

To be more precise, suppose that R is the n×p matrix of response indicators whose elements are zero or one depending on whether the corresponding elements of Y are missing or observed. Then the MAR assumption is that the distribution of R can depend on Y_obs but not on Y_mis.

$p( R | Y_{obs}, Y_{mis} )=p( R | Y_{obs} )$

For example, consider a trivariate data set with variables Y₁ and Y₂ fully observed, and a variable Y₃ that has missing values. MAR assumes that the probability that Y₃ is missing for an individual can be related to the individual's values of variables Y₁ and Y₂, but not to its value of Y₃. On the other hand, if a complete case and an incomplete case for Y₃ with exactly the same values for variables Y₁ and Y₂ have systematically different values, then there exists a response bias for Y₃, and MAR is violated.

The MAR assumption is not the same as missing completely at random (MCAR), which is a special case of MAR. Under the MCAR assumption, the missing data values are a simple random sample of all data values; the missingness does not depend on the values of any variables in the data set.

Furthermore, the MI and MIANALYZE procedures assume that the parameters ${{\theta}}$ of the data model and the parameters ${\phi}$ of the model for the missing data indicators are distinct. That is, knowing the values of ${{\theta}}$ does not provide any additional information about ${\phi}$ , and vice versa. If both the MAR and distinctness assumptions are satisfied, the missing-data mechanism is said to be ignorable (Rubin 1987, pp. 50 -54; Schafer 1997, pp. 10 -11) .

Chapter Contents
Previous
Next
Top