Imputer's Model Versus Analyst's Model

The MI Procedure

Imputer's Model Versus Analyst's Model

Schafer (1997, pp. 139-143) provides comprehensive coverage of this topic, and the following discussion is largely based on his work.

Multiple imputation inference assumes that the model you used to analyze the multiply imputed data (the analyst's model) is the same as the model used to impute missing values in multiple imputation (the imputer's model). But in practice, the two models may not be the same.

For example, consider the same trivariate data set with variables Y₁ and Y₂ fully observed, and a variable Y₃ with missing values. An imputer creates multiple imputations with the model Y₃ = Y₁ Y₂. However, the analyst can later use the simpler model Y₃=Y₁. In this case, the analyst assumes more than the imputer. That is, the analyst assumes there is no relationship between variables Y₃ and Y₂.

The effect of the discrepancy between the models depends on whether the analyst's additional assumption is true. If the assumption is true, the imputer's model still applies. The inferences derived from multiple imputations will still be valid, although they may be somewhat conservative because they reflect the additional uncertainty of estimating the relationship between Y₃ and Y₂.

On the other hand, suppose that the analyst models Y₃=Y₁, and there is a relationship between variables Y₃ and Y₂. Then the model Y₃ = Y₁ will be biased and is inappropriate. Appropriate results can be generated only from appropriate analyst's models.

Another type of discrepancy occurs when the imputer assumes more than the analyst. For example, suppose that an imputer creates multiple imputations with the model Y₃ = Y₁, but the analyst later fits a model Y₃ = Y₁ Y₂. When the assumption is true, the imputer's model is a correct model and the inferences still hold.

On the other hand, suppose there is a relationship between Y₃ and Y₂. Imputations created under the incorrect assumption that there is no relationship between Y₃ and Y₂ will make the analyst's estimate of the relationship biased toward zero. Multiple imputations created under an incorrect model can lead to incorrect conclusions.

Thus, generally you should include as many variables as you can when doing multiple imputation. The precision you lose when you include unimportant predictors is usually a relatively small price to pay for the general validity of analyses of the resultant multiply imputed data set (Rubin 1996).

Note that it is good practice to include a description of the imputer's model with the multiply imputed data set. That way, the analysts will have information about the variables involved in the imputation and which relationships among the variables have been implicitly set to zero.

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